Theoretical Studies on Extended Higgs Sectors Towards Future Precision Measurements (Springer Theses) 9789819913237, 9819913233

Citation preview

Springer Theses Recognizing Outstanding Ph.D. Research

Masashi Aiko

Theoretical Studies on Extended Higgs Sectors Towards Future Precision Measurements

Springer Theses Recognizing Outstanding Ph.D. Research

Aims and Scope The series “Springer Theses” brings together a selection of the very best Ph.D. theses from around the world and across the physical sciences. Nominated and endorsed by two recognized specialists, each published volume has been selected for its scientific excellence and the high impact of its contents for the pertinent field of research. For greater accessibility to non-specialists, the published versions include an extended introduction, as well as a foreword by the student’s supervisor explaining the special relevance of the work for the field. As a whole, the series will provide a valuable resource both for newcomers to the research fields described, and for other scientists seeking detailed background information on special questions. Finally, it provides an accredited documentation of the valuable contributions made by today’s younger generation of scientists.

Theses may be nominated for publication in this series by heads of department at internationally leading universities or institutes and should fulfill all of the following criteria • They must be written in good English. • The topic should fall within the confines of Chemistry, Physics, Earth Sciences, Engineering and related interdisciplinary fields such as Materials, Nanoscience, Chemical Engineering, Complex Systems and Biophysics. • The work reported in the thesis must represent a significant scientific advance. • If the thesis includes previously published material, permission to reproduce this must be gained from the respective copyright holder (a maximum 30% of the thesis should be a verbatim reproduction from the author’s previous publications). • They must have been examined and passed during the 12 months prior to nomination. • Each thesis should include a foreword by the supervisor outlining the significance of its content. • The theses should have a clearly defined structure including an introduction accessible to new PhD students and scientists not expert in the relevant field. Indexed by zbMATH.

Masashi Aiko

Theoretical Studies on Extended Higgs Sectors Towards Future Precision Measurements Doctoral Thesis accepted by Osaka University, Osaka, Japan

Author Dr. Masashi Aiko High Energy Accelerator Research Organization (KEK) Tsukuba, Ibaraki, Japan

Supervisor Prof. Shinya Kanemura Department of Physics Osaka University Osaka, Japan

ISSN 2190-5053 ISSN 2190-5061 (electronic) Springer Theses ISBN 978-981-99-1323-7 ISBN 978-981-99-1324-4 (eBook) https://doi.org/10.1007/978-981-99-1324-4 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Supervisor’s Foreword

It is my greatest pleasure to introduce the work of Dr. Masashi Aiko for publication in the Springer Thesis. His thesis is culmination of what he has done during his Ph.D. course at Osaka University, which is on theoretical studies for extended Higgs sectors. Since the Higgs boson discovery at LHC in 2012, enormous knowledge about this particle has been accumulated by a lot of experimental and theoretical efforts. However, the nature of electroweak symmetry breaking and the structure of the Higgs sector remain in mystery. The property of the Higgs boson which was clarified up to now is not contradict with the prediction in the standard model under the experimental and theoretical uncertainties. This, however, does not imply that the SM Higgs sector is truly correct. In fact, various Higgs models which have non-minimal structures can explain current experimental data, and such extended Higgs sectors can closely relate to new physics paradigms which may solve the hierarchy problem and also can explain phenomena which the standard model cannot explain such as neutrino oscillation, dark matter and baryon asymmetry of the Universe. In other words, from the shape of the non-minimal Higgs sector, we can expect that the direction of physics beyond the standard model is determined. Therefore, it is quite important to explore the structure of the Higgs sector by current and future experiments in order to understand the nature of electroweak symmetry breaking and further new physics beyond the standard model. An obvious and important way to determine the structure of a non-minimal Higgs sector is to directly detect additional Higgs bosons, which are generally predicted in extended Higgs sectors, at high energy collider experiments like LHC and its high luminosity operation HL-LHC. On the other hand, another important approach is to precisely determine the property of the already-discovered Higgs boson with the mass of 125 GeV. Today, Higgs factories such as the ILC, FCCee, CEPC and CLIC are being proposed for future precision measurements of the Higgs sectors. With the precision measurements with the order of 1% or better, deviations from the standard model predictions are expected to be detected in the Higgs observables such as decay branching ratios or production cross sections. By the pattern of the direction and magnitudes of these deviations, we may be able to obtain important v

vi

Supervisor’s Foreword

information on the structure of the non-minimal Higgs sector and also the direction of new physics beyond the standard model. Dr. Masashi Aiko performed important works on radiative corrections in various extended Higgs sectors, which are extremely important to explore the Higgs sector along with the above-mentioned approach. His thesis is an excellent summary of his achievement, in which starting with the introduction of the standard model he systematically explains the renormalization theory in models with extended Higgs sectors, and then evaluate NLO radiative corrections to various decay rates and evaluate a full set of the decay branching ratios of the SM-like Higgs boson in these extended Higgs models including NLO QCD and QED effects. Production process at future electron–positron colliders (the Higgs-strahlung process) was also evaluated at one loop for polarized electron and positron beams in the two Higgs doublet models. In extended Higgs sectors, there are additional Higgs bosons like a pair of charged Higgs states and CP-odd scalar states. He also performed a comprehensive study for the NLO calculation of decay rates of these additional scalar states. These calculations become important when additional Higgs bosons will be discovered at current and future hadron and lepton colliders. The synergy between HL-LHC and future Higgs factories is also discussed to explore the Higgs sector. His achievement for these systematic and comprehensive studies is full of originality and really valuable for exploring the Higgs physics and new physics beyond the standard model. The work presented here will certainly have a significant impact in forthcoming several decades. Osaka, Japan December 2022

Prof. Shinya Kanemura

Parts of this thesis have been published in the following journal articles: This thesis is based on the research during my Ph.D. course at Osaka University. These works have been published in the following journal articles: [1] M. Aiko, S. Kanemura, M. Kikuchi, K. Mawatari, K. Sakurai and K. Yagyu, “Probing extended Higgs sectors by the synergy between direct searches at the LHC and precision tests at future lepton colliders”, Nucl. Phys. B, Vol. 966, p. 115375, 2021. [2] M. Aiko, S. Kanemura and K. Mawatari. “Next-to-leading-order corrections to the Higgs strahlung process from electron-positron collisions in extended Higgs models”, Eur. Phys. J. C, Vol. 81, No. 11, p. 1000, 2021. [3] M. Aiko, S. Kanemura and K. Sakurai, “Radiative corrections to decays of charged Higgs bosons in two Higgs doublet models”, Nucl. Phys. B, Vol. 973, p. 115581, 2021. [4] M. Aiko, S. Kanemura and K. Sakurai, “Radiative corrections to decay branching ratios of the CP-odd Higgs boson in two Higgs doublet models”, Nucl. Phys. B, Vol. 986, p. 116047, 2023.

vii

Acknowledgements

I would like to express my sincere gratitude to my supervisor, Prof. Shinya Kanemura for providing me with a lot of instruction in the research of particle physics. I am much obliged to Prof. Kentarou Mawatari, Prof. Kei Yagyu, Prof. Mariko Kikuchi and Dr. Kodai Sakurai for the fruitful collaborations. I would like to appreciate Prof. Tetsuya Onogi, Prof. Masaharu Aoki and Prof. Satoshi Yamaguchi for their careful reading of my thesis. I am also grateful to faculty members of the Particle Physics Theory group of Osaka University, Prof. Ryosuke Sato, Prof. Minoru Tanaka, Prof. Norihiro Iizuka, Prof. Hidenori Fukaya and Prof. Yutaka Hosotani. I also thank colleagues in the laboratory for providing me with a good environment during my Ph.D. program. Finally, I would like to express my gratitude to my family, particularly my parents, for their kind support and encouragement.

ix

Contents

1

Introduction and Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Summary of the Author’s Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Part I 2

1 1 4 5 5

Higgs Physics at the Leading Order

Review of the Standard Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Particle Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Quarks and Leptons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Gauge Bosons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.3 Higgs Boson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Classical Lagrangian of the SM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Higgs Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Weak and Electromagnetic Currents . . . . . . . . . . . . . . . . . . . 2.3 Quantization of the Standard Model . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Gauge Fixing in the Rξ Gauge . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Fadeev-Popov Ghost . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 Becchi-Rouet-Stora-Tyutin (BRST) Symmetry . . . . . . . . . . 2.4 Higgs Sector in the SM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Higgs Boson Couplings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Global Custodial Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.3 Perturbative Unitarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.4 Vacuum Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.5 Flavor Changing Neutral Current and GIM Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.6 Electroweak Precision Test . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.7 Decay of Higgs Boson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11 11 11 13 14 14 15 17 18 18 19 21 23 23 24 27 29 30 30 32 33

xi

xii

3

4

Contents

Review of the Extended Higgs Models . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Two Higgs Doublet Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Two-Higgs Doublet Model with the Softly-Broken Z 2 Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 Higgs Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.3 Mass Eigenbasis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.4 Theoretical Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Higgs Couplings and the Alignment Limit . . . . . . . . . . . . . . . . . . . . 3.3 Custodial Symmetry in the 2HDMs . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Constraints from Experimental Data . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Electroweak Precision Tests . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 Signal Strengths of the SM-Like Higgs Boson . . . . . . . . . . . 3.4.3 Flavor Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Synergy Between Direct Searches at the LHC and Precision Tests at Future Lepton Colliders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Decays of the SM-Like Higgs Boson . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Decay Rates for h → f f¯ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2 Decay Rates for h → V V ∗ → V f f¯ . . . . . . . . . . . . . . . . . . . 4.1.3 Decay Rates for h → γ γ , Z γ , gg . . . . . . . . . . . . . . . . . . . . 4.2 Decays of the Additional CP-Even Higgs Boson . . . . . . . . . . . . . . . 4.2.1 Decay Rates for H → f f¯ . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Decay Rates for H → W + W − , Z Z . . . . . . . . . . . . . . . . . . . 4.2.3 Decay Rates for H → Z A, W ± H ∓ . . . . . . . . . . . . . . . . . . . 4.2.4 Decay Rates for H → hh, A A, H + H − . . . . . . . . . . . . . . . . 4.2.5 Decay Rates for H → γ γ , Z γ , gg . . . . . . . . . . . . . . . . . . . . 4.3 Decays of the CP-Odd Higgs Boson . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Decay Rates for A → f f¯ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Decay Rates for A → Z h, Z H, W ± H ∓ . . . . . . . . . . . . . . . 4.3.3 Decay Rates for A → W W, Z Z , Z γ , γ γ , gg . . . . . . . . . . 4.4 Decays of the Charged Higgs Bosons . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Decay Rates for H ± → f f¯  . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2 Decay Rates for H ± → W ± φ (φ = h, H, A) . . . . . . . . . . . 4.4.3 Decay Rates for H ± → W ± Z , W γ . . . . . . . . . . . . . . . . . . . 4.5 Total Decay Widths and Decay Branching Ratios . . . . . . . . . . . . . . 4.6 Direct Searches at the LHC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.1 Production Cross Sections for the Additional Higgs Bosons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.2 Constraints from the Direct Searches at the LHC Run-II with 36 fb−1 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7 Combined Results of Direct Searches at the HL-LHC and Precision Tests at the ILC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

35 36 37 38 41 43 43 46 49 49 49 50 50 53 53 54 55 56 60 60 62 63 65 65 69 69 70 72 75 76 78 79 80 84 86 89 93 97

Contents

Part II

xiii

Next-to-Leading-Order Electroweak Corrections in Higgs Physics

5

Renormalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Renormalization of the Standard Model . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 Tadpole Renormalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.2 Higgs Sector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.3 Gauge Sector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.4 Fermion Sector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Renormalization of the Two-Higgs Doublet Model . . . . . . . . . . . . . 5.2.1 Tadpole Renormalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Higgs Sector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

103 103 104 105 106 109 110 110 112 118

6

One-Loop Calculations for Decays of the SM-Like Higgs Boson . . . 6.1 Decay Rates with Higher Order Corrections . . . . . . . . . . . . . . . . . . . 6.1.1 Form Factors for Vertex Functions . . . . . . . . . . . . . . . . . . . . . 6.1.2 Decay Rates for h → f f¯ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.3 Decay Rates for h → Z Z ∗ → Z f f¯ . . . . . . . . . . . . . . . . . . . 6.1.4 Decay Rates for h → W W ∗ → W f f¯ . . . . . . . . . . . . . . . . . 6.2 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Deviation in Partial Decay Widths . . . . . . . . . . . . . . . . . . . . . 6.2.2 Deviation in the Total Decay Width . . . . . . . . . . . . . . . . . . . . 6.2.3 Deviation in Branching Ratios . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

119 119 120 123 124 126 127 127 129 130 131

7

Higgs Strahlung Process in Electron–Positron Colliders . . . . . . . . . . . 7.1 Electroweak Corrections to the Process e+ e− → h Z . . . . . . . . . . . 7.1.1 Kinematics of the Higgs Strahlung Process . . . . . . . . . . . . . 7.1.2 Convention of Spinors and Polarization Vectors . . . . . . . . . 7.1.3 Helicity Amplitudes and Helicity-Dependent Cross-Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.4 Tree-Level Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.5 One-Loop Contributions to the Form Factors . . . . . . . . . . . . 7.2 Numerical Results for the Cross-section . . . . . . . . . . . . . . . . . . . . . . 7.2.1 Standard Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.2 Two Higgs Doublet Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Numerical Results for the Cross Section Times Decay Branching Ratios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

133 133 134 135

8

One-Loop Calculations for Decays of the Charged Higgs Bosons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Decay Rates with Higher Order Corrections . . . . . . . . . . . . . . . . . . . 8.1.1 Form Factors for Vertex Functions of the Charged Higgs Bosons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.2 Decay Rates for H ± → f f  . . . . . . . . . . . . . . . . . . . . . . . . . .

137 140 142 145 145 146 150 154 155 155 156 159

xiv

Contents

8.1.3 Decay Rates for H ± → W ± φ . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Next-to-Leading Order Electroweak Corrections to the Decay Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Next-to-Leading Order Electroweak Corrections to the Decay Branching Ratios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.1 Scenario A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.2 Scenario B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.3 Size of the Squared One-Loop Amplitude for H + → W + h . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

One-Loop Calculations for Decays of the CP-Odd Higgs Boson . . . . 9.1 Decay Rates with Higher-Order Corrections . . . . . . . . . . . . . . . . . . . 9.1.1 Form Factors for Vertex Functions of CP-Odd Higgs Boson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.2 Decay Rates for A → f f¯ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.3 Decay Rates for A → V φ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Next-to-Leading-Order Electroweak Corrections to the Decay Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Next-to-Leading-Order Electroweak Corrections to the Decay Branching Ratios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.1 Scenario A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.2 Scenario B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.3 Discrimination of Types of the Yukawa Interaction in the 2HDMs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

161 162 165 167 169 172 173 175 175 175 178 179 180 184 186 189 191 192

10 Conclusion and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 Appendix A: Input Parameters and Basics of Quantum Chromodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 Appendix B: Feynman Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 Appendix C: Loop Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 Appendix D: One Particle Irreducible Diagrams in the Two-Higgs Doublet Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221

Chapter 1

Introduction and Summary

1.1 Introduction The standard model (SM) of particle physics successfully describes the nature of fundamental particles. The way of three known interactions, strong, weak, and electromagnetic interactions, are determined by gauge principle [1, 2], which imposes the invariance of the theory under the local gauge transformations. The strong interaction is described by Quantum Chromodynamics (QCD) [3–6], while the GlashowWeinberg-Salam (GWS) model [7–9] of the electroweak interactions provides a unified description for the weak and electromagnetic interactions. Gauge symmetry prohibits the masses of fundamental particles, especially gauge bosons, while the observed particles are massive. The Higgs mechanism [10–14] successfully provides the masses of particles based on the spontaneous breaking down of the gauge symmetry triggered by a fundamental scalar field. The SM has been tested in detail, especially in the collider experiments. The large electron-positron (LEP) collider [15] revealed the non-abelian structure of the weak interaction. It also tested the quantum corrections of the top quark and the Higgs boson even before their discoveries through the precision study of the weak gauge bosons. The hadron collider such as the Tevatron and the Large Hadron Collider (LHC) discovered the top quark [16, 17] and the Higgs boson [18, 19] with the masses consistent with the results of the electroweak precision tests at the LEP. Since the discovery of the Higgs boson, it has turned out that its properties are in agreement with those in the SM within theoretical and experimental uncertainties [20, 21]. Despite the success of the SM, we are convinced that the SM is not a fundamental theory because of the following reasons. First of all, the SM does not contain gravity. Second, there are phenomena that cannot be explained within the SM, such as the existence of dark matter [22], baryon asymmetry of the universe [22, 23], and tiny but non-zero neutrino masses [24–26]. Third, there are conceptual problems in the SM such as the hierarchy problem [27–29], the strong CP problem [30], no unified description for the gauge group [31] and the flavor structure [32], and so on. Therefore, the SM must be replaced by a more fundamental theory. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 M. Aiko, Theoretical Studies on Extended Higgs Sectors Towards Future Precision Measurements, Springer Theses, https://doi.org/10.1007/978-981-99-1324-4_1

1

2

1 Introduction and Summary

In seeking a more fundamental theory, one may realize that the structure of the Higgs sector, which determines the concrete realization of the Higgs mechanism, is still a mystery while the Higgs boson was found. There is no theoretical principle to insist on the minimal structure of the Higgs sector as introduced in the SM, and we can construct models with extended Higgs sectors that are consistent with current experimental data. Since the extended Higgs sector often appears in various models such as the minimal supersymmetric standard model (MSSM) and the composite Higgs models, we can narrow down the scenario of new physics by unraveling the nature of the Higgs sector. In addition, new physics models with non-minimal Higgs sectors can solve the above-mentioned problems in the SM through electroweak baryogenesis, the introduction of weakly interacting scalar dark matter, radiatively generalization of the neutrino masses, and so on. Therefore, we can test these scenarios by studying the Higgs sector and reveal mechanisms that solve the problems in the SM. Thus, the nature of the Higgs sector is related to various aspects of particle physics, and the determination of its structure is one of the central interests of current and future high-energy physics. The clear evidence of extended Higgs sectors is a discovery of additional Higgs bosons. Its properties such as the mass, the electric charge, and the parity give important information about the Higgs sector. In a wide variety of search channels, various efforts have been devoted to discovering such new particles although no observation of such new particles has been reported, leading to constraints on the parameters of extended Higgs models. However, there are various scenarios that have not been explored yet, and the direct search for new particles is one of the key programs especially at the LHC as well as at its luminosity-upgraded operation (HL-LHC). In addition to the direct search, we can indirectly test the extended Higgs sector. Even before the discoveries of the top quark and the Higgs boson, we had roughly estimated their masses by studying the quantum effects on the electroweak observables. In addition, even before the discovery of the SM-like Higgs boson, the extended Higgs sector is also constrained by electroweak precision measurements and various flavor measurements. The electroweak precision measurements, especially the measurement of the electroweak rho parameter, constrain the multiplet structure of the Higgs fields. In addition, the radiative correction to the electroweak rho parameter indicates that new physics models should respect the global custodial symmetry at least approximately. On the other hand, flavor measurements, especially the suppression of the flavor-changing neutral current (FCNC), give information about the structure of the Yukawa interactions. Thus, the indirect test of new particles is a powerful way to explore new physics models. The discovery of the SM-like Higgs boson has opened a new window to test the Higgs sector. The evidence of extended Higgs sectors indirectly appears in the observables of the SM-like Higgs boson such as product cross-sections, decay branching ratios, and total decay widths. Thus, we can investigate the Higgs sector by measuring the various Higgs observables and studying deviations from the SM predictions. If deviations are observed, we can discriminate the extended Higgs models by studying the correlations among deviations in the Higgs observables. In addition, from a magnitude of the deviation, we can theoretically deduce an upper limit on the typical

1.1 Introduction

3

mass scale of additional Higgs bosons, and this would set the next target of the energy scale at next-generation collider experiments. Therefore, the discovered Higgs boson is a probe of new physics. Detailed studies of the SM-like Higgs boson are also one of the key programs for the current and future collider experiments. At the LHC, the Higgs boson couplings have been measured with typically order ten percent accuracy [20, 21], and they have been in agreement with those in the SM up to now. This Higgs signal measurement leads us to study the so-called Higgs alignment scenario [33] in extended Higgs models, where SM-like Higgs boson couplings take their SM values. In most of the extended Higgs models, the Higgs alignment scenario can be realized by following two ways. The first one is the alignment due to the decoupling of additional Higgs bosons. In the decoupling scenario, a new scale introduced in extended Higgs models is much larger than the electroweak scale, and their effects decouple from electroweak observables [34]. The second one is the alignment of the SM-like Higgs boson couplings without decoupling. In the non-decoupling scenario, the inner parameters of extended Higgs models, especially mixing among the SM-like Higgs boson and additional Higgs bosons, take the values so that the SM-like Higgs boson couplings take their SM values. Since the light degrees of freedom are favorable to solving the problems in the SM, the non-decoupling scenario is an interesting possibility, and it is the main target in the current and future collider experiments. In this thesis, we show that direct searches of additional Higgs bosons and precision tests of SM-like Higgs boson’s properties play complemental roles. The direct search of additional Higgs bosons gives a lower bound for the mass scale of the additional Higgs bosons. In the approximate Higgs alignment scenario, decays of additional Higgs bosons into the SM-like Higgs boson can be a dominant decay process, and such Higgs-to-Higgs decays are quite useful channels for the direct search of additional Higgs bosons. On the other hand, the deviations in the SM-like Higgs boson couplings give an upper bound for the mass scale of the additional Higgs bosons. Thus, we can explore the extended Higgs sector, especially in the approximately SM-like scenario by utilizing the synergy between the direct search of Higgs-to-Higgs decay at future hadron colliders and the indirect search at future lepton colliders. The accuracy of the Higgs measurements will be improved at the HL-LHC and further significantly at future lepton colliders such as the International Linear Collider (ILC) [35–38], the Future Circular Collider (FCC-ee) [39], and the Circular Electron Positron Collider (CEPC) [40]. Since a few percent accuracies are typically expected in these future experiments, theoretical predictions at the lowest order of perturbation are not enough, and we need to include the higher-order quantum corrections. In addition, the current SM-like situation makes the study of higher-order corrections further important. This is because higher-order corrections would sizably change the theoretical predictions from the lowest-order analysis. Therefore, theoretical studies of the extended Higgs models including the higher-order corrections are necessary for the investigation of the Higgs sector in future Higgs precision measurements. In this thesis, we show that higher-order corrections play an important role in the physics of the discovered Higgs boson and the additional Higgs bosons. For the

4

1 Introduction and Summary

SM-like Higgs boson, we study the Higgs strahlung process from electron-positron collisions. At future lepton colliders, the production cross-section, especially that for the Higgs strahlung process, will be measured with a few percent accuracies in addition to the decay branching ratios and the total decay widths of the SM-like Higgs boson. We study the higher-order corrections for the production cross-section in the approximate Higgs alignment scenario, and we show that these effects are sizable and quantitatively change the theoretical predictions. For the additional Higgs bosons, we study the Higgs-to-Higgs decays. In the Higgs alignment scenario, the Higgs-toHiggs decays are suppressed at the lowest order, and higher-order corrections can be comparable to lowest-order contributions. We study the higher-order corrections for decays of the charged and CP-odd Higgs bosons in the approximate Higgs alignment scenario, and we discuss the impact of higher-order corrections, especially focusing on the Higgs-to-Higgs decay.

1.2 Summary of the Author’s Work We present the following results in this thesis. (I) Synergy between direct searches for additional Higgs bosons at the hadron colliders and precision measurements of the SM-like Higgs boson properties at future electron-positron colliders. [in Chap. 4] In the two Higgs doublet models (2HDM), we concretely show that most of the parameter space can be explored by the direct searches of the additional Higgs bosons and also by the theoretical constraints if the future collider experiments measure the deviations in the coupling constants of the SM-like Higgs boson. (II) Cross-section for the Higgs strahlung process from electron-positron collisions including the full next-to-leading order electroweak corrections. [in Chap. 7] We evaluate the production cross section for e+ e− → h Z process with arbitrary sets of electron and Z boson polarization including the full next-to-leading order electroweak corrections in the 2HDM. We discuss the discrimination of the 2HDMs through the precision measurement of the cross-section times branching ratio at the future colliders. (III) Decay branching ratios of the charged Higgs bosons including NLO EW corrections, and higher-order QCD corrections. [in Chap. 8] We evaluate the decay branching ratios of charged Higgs bosons including NLO EW corrections and higher-order QCD corrections in the 2HDMs. We discuss the impact of higher-order corrections on the partial decay widths and the decay branching ratios in the approximate Higgs alignment scenario. (IV) Decay branching ratios of the CP-odd Higgs boson including NLO EW corrections, and higher-order QCD corrections. [in Chap. 9]

1.3 Organization

5

We evaluate the decay branching ratios of the CP Higgs boson including NLO EW corrections and higher-order QCD corrections in the 2HDMs. We discuss the impact of higher-order corrections on the partial decay widths and the decay branching ratios in the approximate Higgs alignment scenario. These results are based on the following author’s work in collaboration with Prof. Shinya Kanemura, Prof. Kentarou Mawatari, Prof. Kei Yagyu, Prof. Mariko Kikuchi, and Dr. Kodai Sakurai. [41] M. Aiko, S. Kanemura, M. Kikuchi, K. Mawatari, K. Sakurai and K. Yagyu, “Probing extended Higgs sectors by the synergy between direct searches at the LHC and precision tests at future lepton colliders”, Nucl. Phys. B, Vol. 966, p. 115375, 2021. [42] M. Aiko, S. Kanemura and K. Mawatari. “Next-to-leading-order corrections to the Higgs strahlung process from electron-positron collisions in extended Higgs models”, Eur. Phys. J. C, Vol. 81, No. 11, p. 1000, 2021. [43] M. Aiko, S. Kanemura and K. Sakurai, “Radiative corrections to decays of charged Higgs bosons in two Higgs doublet models”, Nucl. Phys. B, Vol. 973, p. 115581, 2021. [44] M. Aiko, S. Kanemura and K. Sakurai, “Radiative corrections to decay branching ratios of the CP-odd Higgs boson in two Higgs doublet models”, Nucl. Phys. B, Vol. 986, p. 116047, 2023.

1.3 Organization This thesis is organized as follows. In Chaps. 2 and 3, we review the SM and the 2HDMs. In Chap. 4, we study the synergy between direct searches for additional Higgs bosons at the hadron colliders and precision measurements of the SM-like Higgs boson properties at future electron-positron colliders. In Chap. 5, we review the gauge-independent renormalization of the SM and the extended Higgs models. In Chap. 6, we review the decays of the discovered Higgs boson including the higherorder corrections. In Chap. 7, we study the Higgs strahlung production process at electron-proton colliders including the quantum corrections. In Chaps. 8 and 9, we study the radiative corrections for the decays of charged and CP-odd Higgs bosons and discuss their impacts. We conclude this thesis in Chap. 10.

References 1. Yang CN, Mills RL (1954) Conservation of isotopic spin and isotopic gauge invariance. Phys Rev 96:191–195 2. Utiyama R (1956) Invariant theoretical interpretation of interaction. Phys Rev 101:1597–1607

6

1 Introduction and Summary

3. Fritzsch H, Gell-Mann M, Leutwyler H (1973) Advantages of the color octet gluon picture. Phys Lett B 47:365–368 4. Gross DJ, Wilczek F (1973) Asymptotically free gauge theories - I. Phys Rev D 8:3633–3652 5. Gross DJ, Wilczek F (1974) Asymptotically free gauge theories - II. Phys Rev D 9:980–993 6. Politzer HD (1973) Reliable perturbative results for strong interactions? Phys Rev Lett 30:1346–1349 7. Glashow SL (1961) Partial symmetries of weak interactions. Nucl Phys 22:579–588 8. Weinberg S (1967) A model of leptons. Phys Rev Lett 19:1264–1266 9. Salam A (1968) Weak and electromagnetic interactions. Conf Proc C 680519:367–377 10. Englert F, Brout R (1964) Broken symmetry and the mass of gauge vector mesons. Phys Rev Lett 13:321–323 11. Higgs PW (1964) Broken symmetries and the masses of gauge bosons. Phys Rev Lett 13:508– 509 12. Higgs PW (1964) Broken symmetries, massless particles and gauge fields. Phys Lett 12:132– 133 13. Guralnik GS, Hagen CR, Kibble TWB (1964) Global conservation laws and massless particles. Phys Rev Lett 13:585–587 14. Higgs PW (1966) Spontaneous symmetry breakdown without massless bosons. Phys Rev 145:1156–1163 15. Schael S et al (2006) Precision electroweak measurements on the Z resonance. Phys Rept 427:257–454 16. Abe F et al (1995) Observation of top quark production in p¯ p collisions. Phys Rev Lett 74:2626–2631 17. Abachi S et al (1995) Observation of the top quark. Phys Rev Lett 74:2632–2637 18. Aad G et al (2012) Observation of a new particle in the search for the Standard Model Higgs boson with the ATLAS detector at the LHC. Phys Lett B 716:1–29 19. Chatrchyan S et al (2012) Observation of a new boson at a mass of 125 GeV with the CMS experiment at the LHC. Phys Lett B 716:30–61 20. Sirunyan AM et al √ (2019) Combined measurements of Higgs boson couplings in proton– proton collisions at s = 13 TeV. Eur Phys J C 79(5):421 21. Aad G et al (2020) Combined measurements of √ Higgs boson production and decay using up to 80 fb−1 of proton-proton collision data at s = 13 TeV collected with the ATLAS experiment. Phys Rev D 101(1):012002 22. Aghanim N et al (2020) Planck 2018 results. VI. Cosmological parameters. Astron Astrophys 641:A6. [Erratum: Astron.Astrophys. 652, C4 (2021)] 23. Fields BD, Olive KA, Yeh TH, Young C (2020) Big-Bang nucleosynthesis after Planck. JCAP 03:010. [Erratum: JCAP 11, E02 (2020)] 24. Fukuda Y et al (1998) Evidence for oscillation of atmospheric neutrinos. Phys Rev Lett 81:1562–1567 25. Ahmad QR et al (2001) Measurement of the rate of νe + d → p + p + e− interactions produced by 8 B solar neutrinos at the Sudbury Neutrino Observatory. Phys Rev Lett 87:071301 26. Ahmad QR et al (2002) Direct evidence for neutrino flavor transformation from neutral current interactions in the Sudbury Neutrino Observatory. Phys Rev Lett 89:011301 27. Weinberg S (1979) Implications of dynamical symmetry breaking. Phys Rev D 13:974–996. [Addendum: Phys.Rev.D 19, 1277–1280 (1979)] 28. Gildener E (1976) Gauge symmetry hierarchies. Phys Rev D 14:1667 29. Susskind L (1979) Dynamics of spontaneous symmetry breaking in the Weinberg-Salam Theory. Phys Rev D 20:2619–2625 30. Peccei RD, Quinn HR (1977) CP conservation in the presence of instantons. Phys Rev Lett 38:1440–1443 31. Georgi H, Glashow SL (1974) Unity of all elementary particle forces. Phys Rev Lett 32:438– 441 32. Froggatt CD, Nielsen HB (1979) Hierarchy of quark masses, Cabibbo angles and CP violation. Nucl Phys B 147:277–298

References

7

33. Gunion JF, Haber HE (2003) The CP conserving two Higgs doublet model: the approach to the decoupling limit. Phys Rev D 67:075019 34. Appelquist T, Carazzone J (1975) Infrared singularities and massive fields. Phys Rev D 11:2856 35. The international linear collider technical design report-Volume 2: Physics 6 36. Fujii K et al (2017) Physics case for the 250 GeV stage of the international linear collider 10 37. Asai S, Tanaka J, Ushiroda Y, Nakao M, Tian J, Kanemura S, Matsumoto S, Shirai S, Endo M, Kakizaki M (2017) Report by the committee on the scientific case of the ILC operating at 250 GeV as a Higgs factory 10 38. Fujii K et al (2019) Tests of the standard model at the international linear collider 8 39. Bicer M et al (2014) First look at the physics case of TLEP. JHEP 01:164 40. Ahmad M et al (2015) CEPC-SPPC preliminary conceptual design report. 1. Physics and detector 3 41. Aiko M, Kanemura S, Kikuchi M, Mawatari K, Sakurai K, Yagyu K (2021) Probing extended Higgs sectors by the synergy between direct searches at the LHC and precision tests at future lepton colliders. Nucl Phys B 966:115375 42. Aiko M, Kanemura S, Mawatari K (2021) Next-to-leading-order corrections to the Higgs strahlung process from electron–positron collisions in extended Higgs models. Eur Phys J C 81(11):1000 43. Aiko M, Kanemura S, Sakurai K (2021) Radiative corrections to decays of charged Higgs bosons in two Higgs doublet models. Nucl Phys B 973:115581 44. Aiko M, Kanemura S, Sakurai K (2023) Radiative corrections to decay branching ratios of the CP-odd Higgs boson in two Higgs doublet models. Nucl Phys B 986:116047

Part I

Higgs Physics at the Leading Order

Chapter 2

Review of the Standard Model

In this chapter, we review the standard model (SM) of particle physics. The SM is based on two pillars; the gauge principle [1, 2] and the Higgs mechanism [3–7]. The gauge principle determines the structure of the gauge interactions among the fundamental particles. The Higgs mechanism provides the masses of the particles through the spontaneous breaking down of the gauge symmetry. We also review the phenomenological aspects of the Higgs boson in the SM. We follow the sign notation in H-COUP program [8, 9] which corresponds to η = η = ηe = −1, η Z = ηθ = ηY = +1 and ηG = +1 in Ref. [10].

2.1 Particle Contents In this section, we list the particle contents of the SM and their representation under the gauge group G = SU (3)C × SU (2) L × U (1)Y .

2.1.1 Quarks and Leptons There are six flavors of quarks and leptons in the SM. Based on the SU (2) L multiplet structure, we classify the quarks and leptons into three generations. The representations of the fermion multiplets are summarized in Table 2.1. In this thesis, we do not introduce the right-handed neutrinos ν iR . The left-handed quarks, Q i = (u i , d i )TL , are the fundamental representation of SU (3)C and SU (2) L gauge symmetry.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 M. Aiko, Theoretical Studies on Extended Higgs Sectors Towards Future Precision Measurements, Springer Theses, https://doi.org/10.1007/978-981-99-1324-4_2

11

12

2 Review of the Standard Model

 Q1 =

u d

 ,

Q2 =

L

  c , s L

 Q3 =

t b

 .

(2.1)

L

The right-handed quarks, u iR and d Ri , are also the fundamental representation of SU (3)C but singlet of SU (2) L . Under the SU (3)C gauge transformation, the quarks, q (= u, d, c, s, t, b) L/R , are transformed as q → UC q = e−igs T

a

βa

q, (a = 1, 2, · · · , 8),

(2.2)

where T a = λa /2 are SU (3)C generators with the Gell-Mann matrices λa . The infinitesimal transformation, q → q + δq is given by δC q = −igs T a β a q.

(2.3)

Similarly, under the SU (2) L gauge transformation, the left-handed quarks, Q i , are transformed as Q i → U L Q i = e−igτ

α

a a

Q i , (a = 1, 2, 3),

(2.4)

where τ i = σ i /2 are SU (2) L generators with the Pauli matrices σ i . The infinitesimal transformation is given by δ L Q i = −igτ a α a q.

(2.5)

Under the U (1)Y gauge transformation, the quarks are transformed as 

q → UY q = e−ig Y α q, 4

(2.6)

where the values of Y are given in Table 2.1. The infinitesimal transformations are given by δY q = −ig  Y α 4 q.

(2.7)

In summary, the infinitesimal transformations of left and right-handed quarks are given by   δ Q i = −i gs T a β a + gτ a α a + g  Y α 4 Q i ,   δq R = −i gs T a β a + g  Y α 4 q R .

(2.8) (2.9)

All of the leptons are singlet under the SU (3)c . The left-handed leptons L i = (ν i , ei )TL are the fundamental representation of SU (2) L  L = 1

νe e



 , L

L = 2

νμ μ



 , L

L = 3

ντ τ

 , L

(2.10)

2.1 Particle Contents

13

Table 2.1 Representation of fermion fields under the SM gauge symmetry. Index i represents the generation of quarks and leptons SU (3)C SU (2) L U (1)Y Q i = (u i , d i )TL u iR d Ri L i = (ν i , i )TL iR

3 3 3 1 1

2 1 1 2 1

16 23 −13 −12 −1

3 and 2 are fundamental representation of SU (3)C and SU (2) L , respectively. Index R represents the right-handed fermions, while Q and L correspond to left-handed quarks and leptons, respectively Table 2.2 Representation of gauge fields under the SM gauge symmetry SU (3)C SU (2) L U (1)Y G aμ Wμi Bμ

8 1 1

1 3 1

0 0 0

SU (3)c index a = 1, 2, · · · 8, and SU (2) L index i = 1, 2, 3. 8 and 3 are adjoint representation of SU (3)c and SU (2) L respectively

while the right-handed leptons, iR , are singlet. Similar to the quarks, the infinitesimal transformations of the left and right-handed leptons are given by   δL i = −i gτ a α a + g  Y α 4 L i , δiR

= −ig



Y α 4 iR .

(2.11) (2.12)

2.1.2 Gauge Bosons There are three kinds of gauge fields in the SM. The transformation properties of gauge fields are summarized in Table 2.2. Under the gauge transformation, the gauge fields are transformed as i (∂μ UC )UC−1 , gs i Wμi τ i → U L Wμi τ i U L−1 − (∂μ U L )U L−1 , g Bμ → Bμ − ∂μ α 4 .

G aμ T a → UC G aμ T a UC−1 −

(2.13) (2.14) (2.15)

14

2 Review of the Standard Model

Table 2.3 Representation of Higgs fields under the SM gauge symmetry SU (3)C SU (2) L U (1)Y

c

1 1

2 2

12 −12

c is charge conjugation of

The infinitesimal transformations are given by δC G aμ = −∂μ β a + gs f abc β b G cμ , δ L Wμi

= −∂μ α + g i

i jk

α

j

(2.16)

Wμk ,

(2.17)

δY Bμ = −∂μ α , 4

(2.18)

where f abc and  abc are the structure constants of SU (3)c and SU (2) L , respectively.

2.1.3 Higgs Boson There is one Higgs doublet field in the SM. The transformation property of the Higgs field is summarized in Table 2.3. The infinitesimal transformation of the Higgs field is given by   δ = −i gτ i α i + g  Y α 4 .

(2.19)

The charge conjugation of Higgs field c is defined as

c = iσ2 ∗ .

(2.20)

c behaves as a doublet under the SU (2) L transformation, while it has opposite hypercharge as . The infinitesimal transformation of c is given by   δ c = −i gτ i α i − g  Y α 4 .

(2.21)

2.2 Classical Lagrangian of the SM The classical Lagrangian of the standard model is given by 1  a aμν 1  a aμν 1 G G − W W − Bμν B μν 4 a=1 μν 4 a=1 μν 4 8

LSM = −

i

3

i

i

i

+ Q (i D)Q i + u iR (i D)u iR + d R (i D)d Ri + L (i D)L i +  R (i D)iR

2.2 Classical Lagrangian of the SM

15

λ + (Dμ )† D μ + μ2 † − ( † )2 2   i j ij i j ij i j c − Yu Q u R + Yd Q d R + Yei j L  R + h.c. .

(2.22)

a The field strength tensors G aμν , Wμν , Bμν are defined as

G aμν = ∂μ G aν − ∂ν G aμ + gs f abc G bμ G cν , a Wμν a Bμν

= =

∂μ Wνa − ∂ν Wμa ∂μ Bνa − ∂ν Bμa .

+ g

abc

Wμb Wνc ,

(2.23) (2.24) (2.25)

The covariant derivative is defined as Dμ = ∂μ − igs T a G aμ − igτ a Wμa − ig  Y Bμ .

(2.26)

2.2.1 Higgs Mechanism In the SM, the Higgs potential is given by V ( ) = −μ2 † +

λ † 2 ( ) , 2

(2.27)

where μ2 and λ are real parameters, and λ is positive due to the stability of the Higgs potential. If μ2 is positive, the Higgs potential has a non-trivial minimum, and SU (2) L × U (1)Y gauge symmetry spontaneously breaks down into U (1)em . We parametrize as   G+   , (2.28)

= √1 v + h + i G0 2 where v is vacuum expectation value (VEV), and G ± and G 0 are the NambuGoldstone (NG) bosons [11, 12]. The Higgs potential can be written by 1 1 1 V ( ) = −Th h + m 2h h 2 + m 2G 0 (G 0 )2 + m 2G ± G + G − 2 2 2 + cubic and quartic terms.

(2.29)

16

2 Review of the Standard Model

The tree-level tadpole is given by   ∂ V ( ) 1 2 2 = μ − λv v. Th = − ∂h 0 2

(2.30)

By imposing the stationary condition, Th = 0, we have μ2 = λv 2 /2.

(2.31)

The mass of the Higgs boson h is given by m 2h =

∂ 2 V ( ) Th = λv 2 − , ∂h 0 v

(2.32)

while the masses of the NG bosons are given by m 2G ±

∂ 2 V ( ) Th ∂ 2 V ( ) Th 2 = = − , m G0 = =− . 2 − 0 ∂G + ∂G 0 v v ∂G 0

(2.33)

We obtain the mass terms of the gauge bosons from the kinetic term of the Higgs field   3μ  

gv 2  g 2 gg  v2  3 W Wμ Bμ LHiggs ⊃ Wμ+ W −μ + , (2.34) gg  g 2 Bμ 2 8 where we have defined charged weak bosons W ± as  1  Wμ± ≡ √ Wμ1 ∓ i Wμ2 . 2

(2.35)

We define mass eigenstates Z μ and Aμ as 

Zμ Aμ



 ≡

cW −sW sW cW



Wμ3 Bμ

 ,

(2.36)

with the shorthand notation for the trigonometric functions as sW = sin θW and cW = cos θW . The weak mixing angle θW is defined by tan θW ≡ g  /g. The masses of the gauge bosons are given by mW =

1 1 gv, m Z = g Z v, m γ = 0, 2 2

(2.37)

with g Z ≡ g/cW . We obtain the mass terms of the quarks and leptons from the Yukawa interaction terms

2.2 Classical Lagrangian of the SM

17



LYukawa ⊃ −

i



fL

f =u,d,e

1 ij √ Yf v 2

 j

f R + h.c.

(2.38)

Since any n-dimensional square complex matrix can be diagonalized by a biunitary transformation, we can define the mass eigenstate of the quarks and leptons as, i j (m) j

u iL = U L u L d Li

=

eiL

=

i j (m) j

, u iR = U R u R

i j (m) j DL dL , i j (m) j E L eL ,

,

i j (m) j = DR dR , i j (m) j eiR = E R e R ,

d Ri

(2.39) (2.40) (2.41)

such that v j diag(m u , m c , m t ) = √ (U L† )ik Yuk U R , 2 v j diag(m d , m s , m b ) = √ (D L† )ik Ydk D R , 2 v j diag(m e , m μ , m τ ) = √ (E L† )ik Yek E R . 2

(2.42) (2.43) (2.44)

2.2.2 Weak and Electromagnetic Currents We obtain the fermion’s electromagnetic, weak-charged and weak-neutral currents from the kinetic terms of the fermions. In the mass eigenstate, they are given by

g μ μ μ Aμ + √ J+ Wμ+ + h.c. + g Z J Z Z μ , L ⊃ e Jem 2

(2.45)

with e ≡ gsW = g  cW . The electromagnetic current is given by μ Jem = δi j



Qf f

(m)i

γ μ f (m) j ,

(2.46)

f =u,d,e

where the electromagnetic charge Q f is determined by the Nishijima-Gell-Mann’s formula [13–15], Q f ≡ τ 3f + Y f .

(2.47)

The weak-charged current is defined by μ

J+ = VCKM u (m)i γ μ PL d (m) j + δi j ν i γ μ PL e(m) j , ij

(2.48)

18

2 Review of the Standard Model

with Cabibbo-Kobayashi-Maskawa (CKM) matrix [16, 17] ij

kj

VCKM = (U L† )ik D L .

(2.49)

Since there is no mass term for the neutrinos, we have rotated the left-handed neutrinos so that the leptonic charged current becomes flavor diagonal. The weakneutral current is given by 

μ

J Z = δi j

f

(m)i

γ μ (v f − a f γ 5 ) f (m) j ,

(2.50)

f =u,d,e

with vf =

1 3 1 2 , a f = τ 3f . τ f − Q f sW 2 2

(2.51)

Since the weak-neutral current is flavor diagonal, it conserves all quark flavors in the SM [18]. In low-energy processes, we can approximate the propagators of the weak gauge bosons as igμν −igμν → 2 . 2 2 p − mV mV

(2.52)

From Eqs. (2.48) and (2.50), we obtain the effective four-fermi operators, g2 μ g2 μ J J − Z2 J Z J Z μ 2 + −μ 2m W 2m Z √ √ μ μ ≡ −2 2G F J+ J−μ − ρ2 2G F J Z J Z μ ,

Le f f = −

(2.53)

where we have defined the Fermi constant G F and the electroweak rho parameter ρ as GF = √

1 2v 2

, ρ=

m 2W . 2 m 2Z cW

(2.54)

In the SM, ρ = 1 at the tree level.

2.3 Quantization of the Standard Model 2.3.1 Gauge Fixing in the Rξ Gauge We need to make the gauge fixing following the Faddeev-Popov prescription [19] in order to quantize the SM. We adopt the Rξ gauge [20], where the mixing terms

2.3 Quantization of the Standard Model

19

between the gauge bosons and the NG bosons cancel out.   L ⊃ im W Wμ− ∂ μ G + − Wμ+ ∂ μ G − + m Z Z μ ∂ μ G 0 .

(2.55)

The gauge fixing terms are given by 1 1 1 LGF = − FG2 − F+ F− − FZ2 − FA2 , 2 2 2

(2.56)

where FGa = (ξG )−1/2 ∂ μ G aμ , F± = (ξW ) FZ = (ξ Z ) FA = (ξ A )

−1/2

−1/2

−1/2

∂

μ

Wμ±

(2.57) ∓ i (ξW )

μ

∂ Z μ − (ξ Z )

1/2

1/2

±

mW G ,

mZG , 0

μ

∂ Aμ .

(2.58) (2.59) (2.60)

We choose the gauge parameters as ξG = ξW = ξ Z = ξ A = ξ.

(2.61)

In this thesis, we adopt the ’t Hooft-Feynman gauge [21], where ξ = 1.

2.3.2 Fadeev-Popov Ghost In order to restore the unitarity, the Fadeev-Popov ghosts [19] must be introduced as  4  ∂(δ FGa ) b  ∂(δ F+ ) ∂(δ F− ) ∂(δ FZ ) ∂(δ FA ) i ω + + c ¯ + c ¯ + c ¯ c ¯ c + − Z A ∂β b ∂α i ∂α i ∂α i ∂α i a,b=1 i=1   ≡ ω¯ a Mab ωb + c¯+ M+i ci + c¯− M−i ci + c¯ Z M Zi ci + c¯ A M Ai ci , (2.62)

Lghost =

8 

ω¯ a

where ωa denote the SU (3)c ghosts, and c± , c Z and c A are electroweak ghosts. We have defined the gauge parameters α ± , α Z , α A as 1 α ± = √ (α 1 ∓ iα 2 ), α Z = α 3 cW − α 4 sW , α A = α 3 sW + α 4 cW . 2

(2.63)

We define the variation of the gauge fields as δG aμ b ω = Dμab ωb , δβb

δViμ j c ≡ Dμi j c j . δα j

(2.64)

20

2 Review of the Standard Model

We have ∂(δ FGa ) b ∂ FGa δG cμ b ∂ FGa ω = ω = Dμcb ωb , ∂β b ∂G cμ δβb ∂G cμ ∂(δ Fi ) j ∂ Fi δVkμ j ∂ Fi δ k j c = c + c Mi j c j = j j ∂α ∂ Vkμ δα ∂ k δα j ∂ Fi δVkμ j = c Dμi j c j , (i = ±, Z , A), ∂ Vkμ δα j

Mab ωb =

(2.65)

(2.66)

where Vi = W ± , Z , A and i = G ± , G, h. The infinitesimal gauge transformations of gauge fixing terms are given by δ FGa = (ξG )−1/2 ∂ μ δG aμ , δ F± = (ξW ) δ FZ = (ξ Z ) δ FA = (ξ A )

−1/2

−1/2

−1/2

∂

μ

δWμ±

μ

(2.67) ∓ i (ξW )

∂ δ Z μ − (ξ Z )

1/2

1/2

±

m W δG ,

m Z δG,

(2.68) (2.69)

μ

∂ δ Aμ ,

(2.70)

where the infinitesimal gauge transformations of the mass eigenstate gauge bosons are given by δG aμ = −∂μ β a + g f abc β b G cμ , δ Aμ = δ Zμ = δWμ±

=

−∂μ α A + ie(Wμ+ α − − Wμ− α + ), −∂μ α Z + igcW (Wμ+ α − − Wμ− α + ), −∂μ α + ± ig[α ± (Z μ cW + Aμ sW ) −

(2.71) (2.72) (2.73) (α Z cW +

α A sW )Wμ± ],

(2.74)

and those of the NG bosons are given by 1 1 δG = − g(α − G + + α + G − ) + g Z α Z (v + h), 2 2  i i ± ± δG = ∓ g(v + h ± i G)α + g Z c2W G ± α Z + ieG ± α A . 2 2

(2.75) (2.76)

For the latter convenience, we also give the infinitesimal gauge transformations of mass-eigenstate fields. That of the Higgs boson is given by δh =

i 1 g(α + ϕ − − α − ϕ + ) − g Z α Z ϕ Z . 2 2

(2.77)

2.3 Quantization of the Standard Model

21

Those of the left and right-handed fermions, ψ L and ψ R , are given by   g 2 δψ L = −i gs T a β a + √ (τ + α + + τ − α − ) + eQ ψ L α A + g Z (τ3 − Q ψ L sW )α Z ψ L , (2.78) 2   a a 2 δψ R = −i gs T β + eQ ψ R α A − g Z Q ψ R sW αZ ψR . (2.79)

2.3.3 Becchi-Rouet-Stora-Tyutin (BRST) Symmetry Since we have performed the gauge fixing, Lagrangian is no longer invariant under the gauge transformation. However, Lagrangian is invariant under the BRST transformation. The BRST transformation is a special kind of gauge transformation with gauge parameter δα i = ci δλ, where δλ is a global c-number variable which anticommutes with the ghost fields, δBRST Vμi =

δVμi

c j δλ ≡ s(Vμi )δλ, δα j δφ i j δBRST φ i = c δλ ≡ s(φ i )δλ, δα j

(2.80) (2.81)

where Vμi are gauge fields, and φi are matter fields (fermions and bosons). We have introduced the Slavnov operator s, which has the following properties: 1. For a product of two fields, we have s(X Y ) = s(X )Y + (−1)GN(X ) X s(Y ),

(2.82)

where the ghost number, GN(X ) , is zero for gauge and matter fields, +1 for ci (ghosts), and −1 for c¯i (anti-ghosts). 2. The s operator raises the mass dimension by one unit. 3. The s operator does not change the charge. 4. The s operator is nilpotent, s 2 (X ) = 0.

(2.83)

By replacing gauge parameter αi to corresponding ghost fields ci , we obtain the BRST transformation of mass eigenstate gauge bosons as, s(G aμ ) = −∂μ ca + g f abc cb G cμ ,   s(Aμ ) = −∂μ c A + ie Wμ+ c− − Wμ− c+ ,   s(Z μ ) = −∂μ c Z + igcW Wμ+ c− − Wμ− c+ ,

s(Wμ± ) = −∂μ c± ± ig c± (Z μ cW + Aμ sW ) − (c Z cW + c A sW )Wμ± .

(2.84) (2.85) (2.86) (2.87)

22

2 Review of the Standard Model

For the NG bosons and the Higgs boson, we have 1 1 s(G) = − g(c− G + + c+ G − ) + g Z c Z (v + h), 2 2  i i ± ± ± ± s(G ) = ∓ g(v + h + i G)c + g Z cW G c Z + ieG c A , 2 2 i 1 + − − + s(h) = g(c G − c G ) − g Z c Z G Z . 2 2

(2.88) (2.89) (2.90)

For the fermions, we have   g 2 )c s(ψ L ) = −i gs T a ca + √ (τ + c+ + τ − c− ) + eQ ψ L c A + g Z (τ3 − Q ψ L sW Z ψL , 2   2 c s(ψ R ) = −i gs T a ca + eQ ψ R c A − g Z Q ψ R sW Z ψR .

(2.91) (2.92)

From the nilpotency of the gauge fields, s 2 (Vμi ) = 0, the BRST transformations of the SU (3)C and SU (2) L ghost fields are defined by gs abc b c f ω ω δλ, 2 g δBRST ci = s(ci )δλ = f i jk c j ck δλ. 2

δBRST ωa = s(ωa )δλ =

(2.93) (2.94)

We have gs abc b c f ω ω, 2 s(c A ) = −iec+ c− ,

s(ωa ) =

+ −

s(c Z ) = −gcW c c ,

s(c± ) = ∓i gcW c Z c± + ec A c± .

(2.95) (2.96) (2.97) (2.98)

We define the BRST transformations of the anti-ghost fields so that LGF+Ghost is invariant under the BRST transformations, δBRST c¯i = s(c¯i )δλ = Fi δλ.

(2.99)

We have s(ω¯ a ) = FGa , s(c¯ A ) = FA , s(c¯ Z ) = FZ , s(c¯± ) = F± .

(2.100)

2.4 Higgs Sector in the SM

23

2.4 Higgs Sector in the SM 2.4.1 Higgs Boson Couplings The interaction terms of the Higgs boson and the weak gauge bosons are given by L⊃

2m 2W μν m2 g hWμ+ Wν− + Z g μν h Z μ Z ν . v v

(2.101)

Since the Higgs boson is neutral under the SU (3)c color and the U (1)em charge, there is no tree-level interaction with the gluon and the photon. The interaction terms of the Higgs boson and the fermions are given by LYukawa ⊃ −

 mf f f h. v f =u,d,e

(2.102)

Since the mass matrices and the Yukawa interaction terms are diagonalized simultaneously, the interactions of the Higgs boson and fermions conserve the quark flavors in the SM. The Higgs boson couplings to gauge bosons and fermions are proportional to their mass. Thus, its couplings to the weak gauge bosons, top and bottom quark, and tau lepton are relevant for Higgs physics. The Higgs boson couplings are tested based on the kappa framework [26], where the Higgs boson couplings are modified from the SM values through the scaling factors. For a given production process or decay mode j, the corresponding scaling factor κ j is defined by κ 2j =

σ jobs σ jSM

or κ 2j =

 obs j  SM j

,

(2.103)

where σ j is production cross section and  j is partial decay width. The observed results at the LHC are given in Table 2.4. We assume that the effective couplings with photon and gluon are also free parameters, and they are parametrized by κγ and κg . In addition, we assume that the branching ratio of the decay into BSM particles is zero. In Fig. 2.1, we show the observed results for the scaling factors by the ATLAS experiments with 80 fb−1 [22]. The dotted line corresponds to the SM prediction, where κV = κ f = 1. For the bottom quark, we have used its MS running mass evaluated at the scale of the Higgs bosons mass, while we have used pole-mass for the top quark. The black error bars represent 68% CL intervals for the measured parameters. The observed Higgs couplings with weak gauge bosons and third-generation fermions are consistent with the SM predictions. The current uncertainties of the measured κ values are 10% and 10–20% level for κV and κ f , respectively. These uncertainties are significantly reduced in future collider experiments. For example,

24

2 Review of the Standard Model

Table 2.4 Summary for the current measurements and expected 1σ accuracies of the κ values at future colliders Current (ATLAS, CMS) HL-LHC (ATLAS, ILC250 ILC500 (1σ [%]) CMS) κZ

κb κt κc κτ κμ κg κγ κZ γ

(1.11 ± 0.08, 1.00 ± 0.11) (1.05 ± 0.09, −1.13+0.16 −0.13 ) +0.27 (1.03+0.19 , 1.17 −0.17 −0.31 ) +0.15 (1.09−0.14 , 0.98 ± 0.14) (–, –) (1.05+0.16 −0.15 , 1.02 ± 0.17) +0.59 (–, 0.80−0.80 ) +0.11 (0.99−0.10 , 1.18+0.16 −0.14 ) (1.05 ± 0.09, 1.07+0.14 −0.15 ) (–, –)

κh

(–, –)

κW

(2.6, 2.4)

0.38

0.30

(3.1, 2.6)

1.8

0.40

(6.2, 6.0) (6.3, 5.5) (–, –) (3.7, 2.8) (7.7, 6.7) (4.2, 4.0) (3.7, 2.9) (12.7, –)

1.8 – 2.4 1.9 5.6 2.2 1.1 16

0.60 6 1.2 0.80 5.1 0.97 1.0 16

(–, –)

–

27

For the current measurements, we refer to the values, assuming that the branching ratio of the decay into BSM particles is zero, which are given by the ATLAS experiments with 80 fb−1 [22] and the CMS experiments with 35.9 fb−1 [23]. For the HL-LHC, we refer to the expected accuracies given in Ref. [24] using systematic uncertainties at the Run-II experiment. For the ILC250, we refer to the expected accuracies given by the ILC with 250 GeV and 2000 fb−1 [25]. For the ILC500, the √ expected accuracies are based on√ the results of the ILC250 combining the simulations at s = 350 GeV with 200 fb−1 and those at s = 500 GeV with 4000 fb−1 [25]

κ Z is expected to be measured with a few percent at the HL-LHC, and it is less than 1% at the ILC, as shown in Table 2.4.

2.4.2 Global Custodial Symmetry The Higgs potential in the SM given in Eq. (2.27) respects not only the local SU (2) L × U (1)Y gauge symmetry but also global O(4) symmetry. In the symmetric phase, we parametrize as 1

= √ 2



Since the Higgs potential consists of | |2 = global S O(4) transformation of φi φi → Oi j φ j ,



φ1 + iφ2 φ3 + iφ4

.

 i

(2.104)

φi2 /2, this is invariant under the

Oi j Oik = δik .

(2.105)

2.4 Higgs Sector in the SM

25

Fig. 2.1 Observed results for the scaling factors by the ATLAS experiments with 80 fb−1 . The dotted line corresponds to the SM prediction. The black error bars represent 68% CL intervals for the measured parameters. We made this figure based on Fig. 15 in Ref. [22]

In the broken phase, the VEV of φ3 breaks the global S O(4) symmetry into S O(3) symmetry leading to three NG bosons. The S O(4) and S O(3) groups are homomorphic to SU (2) × SU (2) and SU (2) groups, respectively. Therefore, we can regard the breaking pattern of the global symmetry as SU (2) × SU (2) symmetry into SU (2) symmetry. We call the unbroken S O(4) ∼ SU (2) L × SU (2) R symmetry as custodial symmetry, while this terminology is originally introduced for the residual SU (2)V symmetry [27]. One can study this global symmetry structure by introducing a bi-doublet field, M = ( c , ).

(2.106)

The bi-doublet field transforms under the SU (2) L × U (1)Y transformation as, SU (2) L : M → eigαa (x)τa M, U (1)Y : M → Me

−ig  Y α4 (x)σ3

(2.107) ,

(2.108)

26

2 Review of the Standard Model

and Tr(M † M) = 2| |2 is gauge invariant. The Higgs potential is written by V ( ) = −

λ μ2 Tr(M † M) + Tr2 (M † M), 2 8

(2.109)

and it is invariant under the global SU (2) L × SU (2) R transformation of the bidoublet field, M → U L MU R† .

(2.110)

From Eq. (2.108), we can see that global U (1)Y is a subgroup of global SU (2) R . In the broken phase, the bi-doublet field becomes, v

M = √ 2



 10 , 01

(2.111)

and this breaks both SU (2) L and SU (2) R , U L M = M, MU R† = M.

(2.112)

However, M leaves the unbroken subgroup SU (2)V , which is corresponding to simultaneous transformation of SU (2) L and SU (2) R with UV UV MUV† = M.

(2.113)

Thus the VEV of the bi-doublet filed breaks global SU (2) L × SU (2) R symmetry into SU (2)V . The custodial symmetry is not the global symmetry of the whole SM Lagrangian. The covariant derivative of the bi-doublet field is written by Dμ M = ∂μ M − igWμ τ a M + ig  Y Bμ Mσ3 .

(2.114)

Since U R† does not commute with σ3 , the hypercharge interaction breaks the custodial symmetry. In order to study the Yukawa interaction terms, we introduce the left-handed doublet Q iL = Q i and the right-handed doublet Q iR = (u i , d i )TR . The Yukawa interaction terms for quarks are written by 1 1 i i ij j ij j LYuk ⊃ − (Yui j + Yd )Q L M Q R − (Yui j − Yd )Q L Mσ3 Q R . 2 2

(2.115)

Thus, the custodial symmetry is broken by the second term, which is proportional to the mass difference between up and down-type quarks after the symmetry breaking. Since there is no right-handed neutrino, the Yukawa interaction terms for leptons maximally break the custodial symmetry.

2.4 Higgs Sector in the SM

27

2.4.3 Perturbative Unitarity The perturbative unitarity bound [28, 29] gives the upper limit on the mass of the Higgs boson in the SM. From Eq. (2.32), we can see that a heavier Higgs boson leads to a larger Higgs quartic coupling. This indicates that there is an upper bound for the mass of the Higgs boson above which the perturbative expansion for the Higgs quartic coupling is not verified.

Unitarity Condition The breakdown of the perturbative expansion violates the unitarity of the S matrix, S † S = 1. In order to isolate the interaction, we parametrize the S matrix as S = 1 + i T and obtain T † T = −i(T − T † ).

(2.116)

For the initial state |i = |k1 , k2 , · · · and the final state | f  = | p1 , p2 , · · ·, the scattering amplitude satisfies the unitarity condition,    −i M(i → f ) − M∗ ( f → i) =



d X M∗ ( f → X )M(i → X ),

X

(2.117) where we used the completeness relationship 1=

 X    X

i=1

d 3 qi 1 (2π )3 2E i

 |X  X |

(2.118)

with an intermediate state |X  = |q1 , q2 , · · ·. The scattering amplitude is defined by

f | i T |i = (2π )4 δ 4



ki −



 pi iM(i → f ),

(2.119)

and the phase space integral is defined by 

 d X =

X   i=1

d 3 qi 1 (2π )3 2E i

 (2π )4 δ 4



ki −

  qi .

(2.120)

For the two-body to two-body scattering, we can define the partial wave amplitude a J as

28

2 Review of the Standard Model

M(i → f ) = 16π

∞  (2J + 1)a J PJ (cos θ ),

(2.121)

j=0

with the Legendre polynomials PJ (cos θ ). This gives the unitarity bound for the partial wave amplitude  1 1 − |2 + |a 2→n |2 = , |a 2→2 J J 2 4 n>2

(2.122)

in a representation where the elastic scattering amplitudes are diagonal. Thus, the two-body elastic scattering amplitudes lie on a circle of radius η j /2 centered at (0, 1/2) in the complex plane, with  ηj = 1 − 4



1/2 |a 2→n |2 J

.

(2.123)

n>2

Lee-Quigg-Thacker Bound √ The high-energy region where s  m V , the longitudinal polarization vector of massive gauge boson V behaves as μ L

  pμ mW . = +O √ mV s

(2.124)

Therefore, it is expected that the amplitudes of four longitudinal gauge bosons scattering behave as  M= A

E mV

4

 +B

E mV

2 + C.

(2.125)

Therefore, they potentially break the unitarity bound at a high-energy limit. However, one can show that the coefficients A and B become zero after summing over the relevant Feynman diagrams. Thus, the unitarity bound constrains the magnitude of C, and it gives the upper bound for the mass of the Higgs boson, which is the so-called Lee-Quigg-Thacker bound [28, 29]. Instead of evaluating the scattering amplitudes of longitudinal gauge bosons, we use the equivalence theorem [30]. This theorem states that the longitudinal gauge bosons can be replaced by the corresponding NG bosons in the high-energy limit, M(W L± ,



mW Z L )  M(G , G ) + O √ s ±

0

 .

(2.126)

2.4 Higgs Sector in the SM

29

√ In the high-energy limit with m h  s, contact interaction diagrams are only relevant. The partial wave amplitudes are given by ⎛

3 G F m 2h ⎜ 0 a0 = − √ ⎜ 8π 2 ⎝ 0 0

0 1 0 0

0 0 1 0

⎞ 0 0⎟ ⎟ 0⎠ 1

!⎫ ⎧ 1 + − 1 √ G G + (hh + G 0 G 0 ) ⎪ 2 2 ⎪ ⎪ 1 + − 1 !⎪ ⎬ ⎨ √ G G − (hh + G 0 G 0 ) 2 2 . ! 1 ⎪ ⎪ hh − G!0 G 0 ⎪ ⎪ 2 ⎭ ⎩ hG 0

(2.127)

From the largest eigenvalue, we obtain the upper bound for the mass of the Higgs boson as % √ 4π 2 mh ≤ ∼ 500 GeV. (2.128) 3G F The observed Higgs boson mass m h  125 GeV satisfies the Lee-Quigg-Thacker bound, and we can verify the perturbative calculation within the SM.

2.4.4 Vacuum Stability At LO, the parameters of the Higgs potential in the SM should satisfy the following conditions, μ2 > 0, λ > 0.

(2.129)

The first condition is necessary to make the origin of the potential to be unstable and to occur the spontaneous breaking down of the gauge symmetry. The second condition guarantees that the Higgs potential is bounded from below with large field values. However, the value of the coupling constant is scale-dependent. Therefore, one needs to study the scale dependence of the parameters of the Higgs potential by using the renormalization equation in order to investigate the potential structure at a large energy scale. Figure 2.2 shows the scale dependence of λ(μ). We have reproduced Fig. 1 in Ref. [31] by using the one-loop beta functions and obtained consistent results. We vary the value of the input parameters m t and αs between the ±3σ . The black line in both Fig. 2.2 shows the evolution of the λ for the central value of m t and αs . The blue and red band show 3σ width of m t and αs respectively. The value of λ is close to zero and breaks vacuum stability around 1010 GeV when we use the central value of m t and αs . However, the value of the vacuum stability breaking scale is highly dependent on the value of the input parameter m t , and it can be varied between 108 and 1016 GeV.

30

2 Review of the Standard Model

Fig. 2.2 RG evolution of λ coupling varying m t and αs . We made this figure based on Fig. 1 in Ref. [31]

2.4.5 Flavor Changing Neutral Current and GIM Mechanism As we have discussed in Sects. 2.2.2 and 2.4.1, both of the weak-neutral current and the Yukawa interaction conserve the quark flavors in the SM. Therefore, there is no flavor-changing neutral current (FCNC) in the SM. FCNC processes such as K 0 − K¯ 0 mixing occur through the charged-current loop diagram. However, one-loop amplitudes are suppressed not only by the loop factor 1/4π but also by the mass-squared differences among the virtual quarks such as (m 2c − m 2u )/v 2 . This is because the mass differences of quarks are the only sources for the flavor violation in the SM since the weak-charged current also conserves the quark flavor if the masses of the quarks are degenerate. Thus, FCNC processes in the SM are highly suppressed as a consequence of the quark flavor structure, and this is the so-called GIM mechanism [18].

2.4.6 Electroweak Precision Test The electroweak S, T and U parameters [32, 33] parametrize the radiative corrections appearing in the two-point functions of weak gauge bosons. Since these parameters

2.4 Higgs Sector in the SM

31

are independent of external particles in the process, they are so-called oblique parameters. These parameters are useful to study new particles because they can capture the non-decoupling effects of new particles. The two-point functions of gauge bosons are composed of the transverse and longitudinal parts, AB (q) μν

  qμ qν qμ qν TAB (q 2 ) + 2  LAB (q 2 ), = −gμν + 2 q q

(2.130)

where AB = γ γ , Z γ , Z Z , W W . The transverse part can be parametrized as γγ

T (q 2 ) = e2 TQ Q (q 2 ),   Zγ QQ 2 2 2 T (q 2 ) = eg Z 3Q T (q ) − sW T (q ) ,   3Q 2 QQ 2 2 2 4 TZ Z (q 2 ) = g 2Z 33 (q ) − 2s  (q ) + s  (q ) , T W T W T 2 TW W (q 2 ) = g 2 11 T (q ).

(2.131) (2.132) (2.133) (2.134)

The S, T and U parameters are defined by   2 33 S = 16π Re 3Q T,γ (m Z ) − T,Z (0) , √

4 2G F 11 T = Re 33 T (0) − T (0) , αem

11 U = 16π Re 33 T,Z (0) − T,W (0) ,

(2.135) (2.136) (2.137)

with AB (q 2 ) = T,V

TAB (q 2 ) − TAB (m 2V ) . q 2 − m 2V

(2.138)

In terms of the two-point functions of γ , Z and W bosons, S, T and U parameters are given by & ' Zγ γγ 2 2 − sW TZ Z (m 2Z ) − TZ Z (0) cW α T (m 2Z ) T (m 2Z ) S = −Re − − , 2 2 cW sW 4sW cW m 2Z m 2Z m 2Z (2.139)   WW ZZ T (0) T (0) , (2.140) αT = −Re − m 2W m 2Z  WW 2 ZZ 2 ZZ T (m W ) − TW W (0) α 2 T (m Z ) − T (0) U = −Re − c W 2 4sW m 2W m 2Z

32

2 Review of the Standard Model

Fig. 2.3 Constraints in the oblique parameters S and T fixing U = 0 (blue contour). The masses of the Higgs boson and top quark are m h = 125 GeV and m t = 172.5 GeV, respectively. Individual  measurements (yellow), the Z constraints are also shown from the asymmetry and direct sin2 θeff partial and total widths (green) and W mass and width (red). This figure is reprinted from Ref. [34] under the Creative Commons Attribution 4.0 International License. © The Author(s) 2018 Zγ γγ 2  T (m 2Z ) 2 T (m Z ) . − 2sW cW − sW m 2Z m 2Z

(2.141)

These parameters have been studied by the electroweak precision tests at LEP/SLC experiments, and they provided the information of masses of the top quark and Higgs boson even before their discovery. Figure 2.3 taken from Ref. [34] shows that the global fit for S and T the parameter. We can see that the SM predictions for S and T parameter with observed Higgs boson and top quark masses, m h = 125 GeV and m t = 172.5 GeV, are consistent with the current experimental data at 95% CL.

2.4.7 Decay of Higgs Boson The Higgs couplings to gauge bosons and fermions are proportional to their masses. Therefore, the Higgs boson tends to decay into heavy particles if such decay modes are kinematically allowed. We here qualitatively discuss the decay property of the Higgs boson, while the theoretical calculations including the higher-order corrections are discussed in Chap. 6. The theoretical predictions for the Higgs branching ratio including the electroweak and QCD higher-order corrections have been performed [36]. We also list the pre-

2.4 Higgs Sector in the SM

33

Table 2.5 The theoretical predictions for the branching ratios of the Higgs boson with m h = 125 GeV bb¯ τ τ¯ μμ¯ cc¯ gg γγ Zγ WW ZZ 58.2%

6.27%

0.0218% 2.89%

8.19%

0.227%

0.153%

21.4%

2.62%

The predicted total width is 4.09 MeV. These values are taken from Ref. [35]. We omit the estimated errors for simplicity

dicted values for the Higgs branching ratio with the discovered mass m h = 125 GeV in Table 2.5. In Table 2.5, we list the value for the Higgs branching ratio with m h = 125 GeV [35]. These results have been evaluated by using the programs HDECAY [37, 38] and PROPHECY4F [39]. HDECAY evaluates the partial decay width for the Higgs two-body decays into a pair of fermions h → f f¯ and on-shell gauge boson pair h → V V , as well as the loop-induced processes h → γ γ , Z γ , gg. On the other hand, PROPHECY4F evaluates those for the Higgs off-shell four-body decays h → V ∗ V ∗ → 4 f . The detail of the higher-order calculations and estimation of uncertainties are explained in Ref. [36].

References 1. Yang C-N, Mills RL (1954) Conservation of isotopic spin and isotopic gauge invariance. Phys Rev 96:191–195 2. Utiyama R (1956) Invariant theoretical interpretation of interaction. Phys Rev 101:1597–1607 3. Englert F, Brout R (1964) Broken symmetry and the mass of gauge vector mesons. Phys Rev Lett 13:321–323 4. Higgs PW (1964) Broken Symmetries and the Masses of Gauge Bosons. Phys Rev Lett 13:508–509 5. Higgs PW (1964) Broken symmetries, massless particles and gauge fields. Phys Lett 12:132– 133 6. Guralnik GS, Hagen CR, Kibble TWB (1964) Global conservation laws and massless particles. Phys Rev Lett 13:585–587 7. Higgs PW (1966) Spontaneous symmetry breakdown without massless bosons. Phys Rev 145:1156–1163 8. Kanemura S, Kikuchi M, Sakurai K, Yagyu K (2018) H-COUP: a program for one-loop corrected Higgs boson couplings in non-minimal Higgs sectors. Comput Phys Commun 233:134– 144 9. Kanemura S, Kikuchi M, Mawatari K, Sakurai K, Yagyu K (2020) H-COUP version 2: a program for one-loop corrected Higgs boson decays in non-minimal Higgs sectors. Comput Phys Commun 257:107512 10. Romao JC, Silva JP (2012) A resource for signs and Feynman diagrams of the standard model. Int J Mod Phys A 27:1230025 11. Nambu Yoichiro (1960) Quasiparticles and Gauge invariance in the theory of superconductivity. Phys Rev 117:648–663 12. Goldstone J (1961) Field theories with superconductor solutions. Nuovo Cim 19:154–164 13. Nakano T, Nishijima K (1953) Charge independence for V-particles. Prog Theor Phys 10:581– 582 14. Nishijima K (1955) Charge independence theory of V particles. Prog Theor Phys 13(3):285– 304

34

2 Review of the Standard Model

15. Gell-Mann M (1953) Isotopic spin and new unstable particles. Phys Rev 92:833–834 16. Cabibbo N (1963) Unitary symmetry and leptonic decays. Phys Rev Lett 10:531–533 17. Kobayashi M, Maskawa T (1973) CP violation in the renormalizable theory of weak interaction. Prog Theor Phys 49:652–657 18. Glashow SL, Iliopoulos J, Maiani L (1970) Weak interactions with Lepton-Hadron symmetry. Phys Rev D 2:1285–1292 19. Faddeev LD, Popov VN (1967) Feynman diagrams for the Yang-Mills field. Phys Lett B 25:29–30 20. Fujikawa K, Lee BW, Sanda AI (1972) Generalized renormalizable Gauge formulation of spontaneously broken Gauge theories. Phys Rev D 6:2923–2943 21. Hooft G (1971) Renormalizable Lagrangians for massive Yang-Mills fields. Nucl Phys B 35:167–188 22. Aad G et al (2020) Combined measurements of √ Higgs boson production and decay using up to 80 fb−1 of proton-proton collision data at s = 13 TeV collected with the ATLAS experiment. Phys Rev D 101(1):012002 23. Sirunyan AM et al √ (2019) Combined measurements of Higgs boson couplings in proton– proton collisions at s = 13 TeV. Eur Phys J C 79(5):421 24. Cepeda M et al (2019) Report from working group 2: Higgs physics at the HL-LHC and HE-LHC. CERN Yellow Rep Monogr 7:221–584 25. Fujii K et al (2017) Physics case for the 250 GeV stage of the international linear collider 26. Andersen JR et al (2013) Handbook of LHC Higgs cross sections: 3. Higgs properties 27. Sikivie P, Susskind L, Voloshin MB, Zakharov VI (1980) Isospin breaking in technicolor models. Nucl Phys B 173:189–207 28. Lee BW, Quigg C, Thacker HB (1977) The strength of weak interactions at very high-energies and the Higgs boson mass. Phys Rev Lett 38:883–885 29. Lee BW, Quigg C, Thacker HB (1977) Weak interactions at very high-energies: the role of the Higgs boson mass. Phys Rev D 16:1519 30. Cornwall JM, Levin DN, Tiktopoulos G (1974) Derivation of Gauge invariance from highenergy unitarity bounds on the s matrix. Phys Rev D 10:1145 [Erratum: Phys Rev D 11:972 (1975)] 31. Degrassi G, Di Vita S, Elias-Miro J, Espinosa JR, Giudice GF, Isidori G, Strumia A (2012) Higgs mass and vacuum stability in the Standard Model at NNLO. JHEP 08:098 32. Peskin ME, Takeuchi T (1990) A new constraint on a strongly interacting Higgs sector. Phys Rev Lett 65:964–967 33. Peskin ME, Takeuchi T (1992) Estimation of oblique electroweak corrections. Phys Rev D 46:381–409 34. Haller J, Hoecker A, Kogler R, Mönig K, Peiffer T, Stelzer Jörg (2018) Update of the global electroweak fit and constraints on two-Higgs-doublet models. Eur Phys J C 78(8):675 35. de Florian D et al (2016) Handbook of LHC Higgs cross sections: 4. Deciphering the nature of the Higgs sector, vol 2 36. Denner A, Heinemeyer S, Puljak I, Rebuzzi D, Spira M (2011) Standard model Higgs-Boson branching ratios with uncertainties. Eur Phys J C 71:1753 37. Djouadi A, Kalinowski J, Spira M (1998) HDECAY: a program for Higgs boson decays in the standard model and its supersymmetric extension. Comput Phys Commun 108:56–74 38. Djouadi A, Kalinowski J, Muehlleitner M, Spira M (2019) HDECAY: twenty++ years after. Comput Phys Commun 238:214–231 39. Denner A, Dittmaier S, Mück A (2020) PROPHECY4F 3.0: a Monte Carlo program for Higgsboson decays into four-fermion final states in and beyond the Standard Model. Comput Phys Commun 254:107336

Chapter 3

Review of the Extended Higgs Models

In this chapter, we review the two-Higgs doublet model (2HDM). Before moving on to the descriptions of 2HDM, we give the motivations for studying the extended Higgs sectors. Unraveling the structure of the Higgs sector is one of the promising ways to approach a more fundamental theory. While the SM-like Higgs boson was found, the structure of the Higgs sector is still a mystery. There is no theoretical principle to insist on the minimal structure of the Higgs sector as introduced in the SM. In addition, non-minimal Higgs sectors are often introduced in various new physics models, which can solve problems beyond the SM. Therefore, the determination of the Higgs sector is one of the important tasks to narrow down a scenario of new physics. Although there are various ways to extend the Higgs sector in general, the electroweak rho parameter constrains the possible multiplet structure of the extended Higgs models. The rho parameter is defined by ρ=

m 2Z

m 2W . cos2 θW

(3.1)

It is unity at the tree level in the SM. On the other hand, it deviates from unity in extended Higgs models in general. The rho parameter at the tree level is given by   ρtree =

i

 Ti (Ti + 1) − Yi2 vi2  , 2 i Yi2 vi2

(3.2)

in the model with an arbitrary number of scalar fields φi with a hypercharge Yi , an isospin Ti and a VEV vi . Since the observed value of the rho parameter is close to unity, the extended Higgs models in which the rho parameter becomes unity at the tree level are promising. There are two ways to make the rho parameter unity at the tree level. The first way is the introduction of the Higgs multiplets which satisfy © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 M. Aiko, Theoretical Studies on Extended Higgs Sectors Towards Future Precision Measurements, Springer Theses, https://doi.org/10.1007/978-981-99-1324-4_3

35

36

3 Review of the Extended Higgs Models

Ti (Ti + 1) − 3Yi2 = 0.

(3.3)

In such models, the rho parameter becomes unity independently of the value of Higgs VEVs. The HSM and 2HDM belong to this class. The second way is assuming an alignment among the Higgs VEVs. The Georgi-Machacek (GM) model [1], in which we introduce the additional real and complex triplet fields with Yi = 0 and Yi = 1, belongs to this class. We note that the rho parameter does not change if additional Higgs multiplets do not develop the VEVs, such as the IDM. In this thesis, we study the 2HDM in detail as a representative of the extended Higgs models.

3.1 Two Higgs Doublet Model In the 2HDM, we have two SU (2) L doublet scalar fields 1 and 2 with the hypercharge Y = 1/2. The Lagrangian of the 2HDM is given by Y L2HDM = LSM + Lkin 2HDM (1 , 2 ) + L2HDM (1 , 2 ) − V2HDM (1 , 2 ),

(3.4)

where LSM is the Lagrangian of the SM without the Higgs fields. The kinetic terms of the Higgs doublets are given by 2  2      =  +  Lkin   . D D μ 1 μ 2 2HDM

(3.5)

The Yukawa Lagrangian is given by =− L2HDM Y



 ˜ i u R + Yd Q¯ L i d R + Ye L¯ L i e R + h.c. , Yu Q¯ L 

(3.6)

i

The Higgs potential is given by  2  2

    V = m 21 1  + m 22 2  − m 23 †1 2 + h.c.  2  2  2 λ1  4 λ2  4       + 1  + 2  + λ3 1  2  + λ4 †1 2  2 2 

2   λ5 † † † † 1 2 + 1 1 + λ7 1 1 λ6 1 2 + h.c. . + 2

(3.7)

In the most general 2HDM, flavor-changing neutral currents (FCNCs) appear at tree level, and it is severely constrained by experiments. In order to avoid such FCNCs, we introduce a discrete Z 2 symmetry, where two doublets transform as [2, 3] 1 → 1 , 2 → −2 .

(3.8)

3.1 Two Higgs Doublet Model

37

One can introduce the soft breaking term of the Z 2 symmetry in the Higgs potential without spoiling the desirable property of the flavor sector.

3.1.1 Two-Higgs Doublet Model with the Softly-Broken Z2 Symmetry The Higgs potential under the softly-broken Z 2 symmetry is given by  2  2

    V = m 21 1  + m 22 2  − m 23 †1 2 + h.c.

  2  2  

2 λ5 λ1  4 λ2  4      † 2 † 1 2 + h.c. , + 1  + 2  + λ3 1  2  + λ4 1 2  + 2 2 2 (3.9) where m 23 is the softly breaking parameter of the Z 2 symmetry. While m 23 and λ5 are generally complex, we take them to be real and consider the CP-conserving scenario in this thesis. We parametrize the doublet fields as  i =

 ωi+ , (i = 1, 2), √1 (vi + h i + i z i ) 2

(3.10)

 where v1 and v2 are the VEVs of two doublets with v = v12 + v22 . We define the ration of VEVs as tan β = v2 /v1 with the domain of β to be 0 < β < π/2. In terms of the component fields, the Higgs potential can be written as V2HDM = −Th 1 h 1 − Th 2 h 2      +     1 1 h1 z ω1 h 1 h 2 Mh2 z 1 z 2 Mz2 1 + ω1− ω2− Mω2 + + h2 z2 ω2+ 2 2 + cubic and quartic terms. (3.11) Since we are considering the charge-conserving vacuum, there are no linear terms for the charged fields ωi+ . In addition, there are no linear terms for the CP-odd fields z i due to the CP conservation in the Higgs potential. The tadpole terms for h 1 and h 2 are given by 

 ∂ V2HDM  v2 2 2 2 2 2 (λ −m = vc + M s − c + λ s ) , β 1 β 345 β 1 β h 1 0 2 

 ∂ V2HDM  v2 2 2 2 2 2 = vsβ −m 2 + M cβ − (λ2 sβ + λ345 cβ ) , Th 2 = − h 2 0 2 Th 1 = −

(3.12) (3.13)

38

3 Review of the Extended Higgs Models

where λ345 = λ3 + λ4 + λ5 . The dimensionful parameter M 2 = m 23 /(sβ cβ ) describes the softly-breaking scale of the Z 2 symmetry. By imposing the stationary conditions Th 1 = Th 2 = 0, m 21 and m 22 are written as, v2 (λ1 cβ2 + λ345 sβ2 ), 2 v2 m 22 = M 2 cβ2 − (λ2 sβ2 + λ345 cβ2 ). 2

m 21 = M 2 sβ2 −

(3.14) (3.15)

The mass matrices of the Higgs fields in the Z 2 basis are given by  Mh2 =

M 2 sβ2 + λ1 cβ2 v 2 −M 2 sβ cβ + λ345 sβ cβ v 2 −M 2 sβ cβ + λ345 sβ cβ v 2 M 2 cβ2 + λ2 sβ2 v 2









sβ2



 −

0 Th 1 /vcβ 0 Th 2 /vsβ

−sβ cβ 0 Th 1 /vcβ , − 0 Th 2 /vsβ −sβ cβ cβ2       sβ2 −sβ cβ λ4 + λ5 2 0 Th 1 /vcβ v . Mω2 = M 2 − − 0 Th 2 /vsβ −sβ cβ cβ2 2 Mz2 = M 2 − λ5 v 2

,

(3.16)







(3.17) (3.18)

Since Mz2 and Mω2 have the same structure, we can diagonalize them by using the same angle β, while we need to introduce another angle α to diagonalize CP even states. We define the mass eigenstates of Higgs fields as 

h1 h2



 = R(α)

H h



 ,

z1 z2



 = R(β)

G0 A



 ,

ω1± ω2±



 = R(β)

G± H±

 , (3.19)

with the rotational matrix  R(θ ) =

cθ −sθ sθ cθ

 .

(3.20)

3.1.2 Higgs Basis We introduce the Higgs basis, where only one of the Higgs doublets acquires its VEV,     H1 1 = R(β) . (3.21) 2 H2

3.1 Two Higgs Doublet Model

39

From Eqs. (3.17) and (3.18), the mass matrices of the CP-odd and charged Higgs bosons are diagonalized in the Higgs basis. The Higgs potential can be expressed as V (H1 , H2 ) = Y12 H1† H1 + Y22 H2† H2 − Y32 (H1† H2 + H2† H1 ) 1 1 + Z 1 (H1† H1 )2 + Z 2 (H2† H2 )2 + Z 3 (H1† H1 )(H2† H2 ) 2 2 + Z 4 (H1† H2 )(H2† H1 )    1 Z 5 (H1† H2 )2 + Z 6 H1† H1 + Z 7 H2† H2 H1† H2 + h.c. , + 2

(3.22)

where each coefficient is written by the parameters in Z 2 basis by Y12 = m 21 cos2 β + m 22 sin2 β − m 23 sin 2β, Y22

=

m 21

sin β + 2

m 22

cos β + 2

m 23

sin 2β,

1 2 (m − m 22 ) sin 2β + m 23 cos 2β, 2 1 1 = λ1 cos4 β + λ2 sin4 β + λ345 sin2 2β, 2 1 = λ1 sin4 β + λ2 cos4 β + λ345 sin2 2β, 2 1 = sin2 2β[λ1 + λ2 − 2λ345 ] + λ3 , 4 1 = sin2 2β[λ1 + λ2 − 2λ345 ] + λ4 , 4 1 = sin2 2β[λ1 + λ2 − 2λ345 ] + λ5 , 4 1 = − sin 2β[λ1 cos2 β − λ2 sin2 β − λ345 cos 2β], 2 1 = − sin 2β[λ1 sin2 β − λ2 cos2 β + λ345 cos 2β]. 2

(3.23) (3.24)

Y32 =

(3.25)

Z1

(3.26)

Z2 Z3 Z4 Z5 Z6 Z7

(3.27) (3.28) (3.29) (3.30) (3.31) (3.32)

Two parameters among Yi and Z i are not free parameters, and we can write them in terms of the other parameters. We have the following two relations, 1 1 (3.33) Z 6 + Z 7 = − sin 2β(λ1 − λ2 ) = − (Z 1 − Z 2 ) tan 2β, 2 2 1 1 Z 6 − Z 7 = − (λ1 + λ2 − 2λ345 ) sin 2β cos 2β = − (Z 1 + Z 2 − 2Z 345 ) tan 4β. 2 4 (3.34)

40

3 Review of the Extended Higgs Models

We parametrize the Higgs doublets as  H1 =

 G+   , √1 v + φ1 + i G 0 2

 H2 =

 H+ , √1 (φ2 + i A) 2

(3.35)

where G ± and G 0 are the Nambu-Goldstone bosons while H ± and A are the physical charged and CP-odd Higgs bosons. The Higgs potential can be written as V2HDM = −Tφ1 φ1 − Tφ2 φ2    0  1 0  2 1 φ1 G φ1 φ2 Mφ2 G A Modd + + φ2 A 2 2    − −  2 G+ + G H M± H+ + cubic and quartic terms. The tadpole terms for φ1 and φ2 are given by     Th 1 Tφ1 = R(β) . Th 2 Tφ2 with



∂ V2HDM  1  = v −Y12 − Z 1 v 2 , 0 φ1 2 

∂ V2HDM  1 Tφ2 ≡ −  = v Y32 − Z 6 v 2 , 0 φ2 2 Tφ1 ≡ −

(3.36)

(3.37)

(3.38) (3.39)

By imposing the stationary conditions, we obtain 1 1 Y12 = − Z 1 v 2 , Y32 = Z 6 v 2 . 2 2 The mass matrices of the Higgs fields in the Higgs basis are given by     Z 6v2 Z 1v2 Tφ1 φ1 Tφ1 φ2 2 t 2 + , Mφ = R(β) Mh R(β) = Tφ2 φ1 Tφ2 φ2 Z 6 v 2 Y22 + 21 Z 345 v 2     TG 0 G 0 TG 0 A 0 0 2 Modd + , = R(β)t Mz2 R(β) = T AG 0 T A A 0 m 2A     TG ± G ± TG ± H ± 0 0 2 t 2 + , M± = R(β) Mω R(β = TH ± G ± TH ± H ± 0 m 2H ±

(3.40)

(3.41) (3.42) (3.43)

where the masses of H ± and A are given by m 2H ± = Y22 +

1 1 Z 3 v 2 = M 2 − (λ4 + λ5 )v 2 , 2 2

(3.44)

3.1 Two Higgs Doublet Model

41

1 m 2A = Y22 + (Z 3 + Z 4 − Z 5 )v 2 = M 2 − λ5 v 2 . 2

(3.45)

The tadpole contributions are given by 

Tφ1 φ1 Tφ1 φ2 Tφ2 φ1 Tφ2 φ2





   TG 0 G 0 TG 0 A TG ± G ± TG ± H ± = T AG 0 T A A TH ± G ± TH ± H ±   0 Th 1 /vcβ R(β) = R(β)t 0 Th 2 /vsβ   (−sβ Tφ1 + cβ Tφ2 )/v (cβ Tφ1 + sβ Tφ2 )/v = . (−sβ Tφ1 + cβ Tφ2 )/v (Th 1 sβ2 /cβ + Th 2 cβ2 /sβ )/v =

(3.46)

3.1.3 Mass Eigenbasis In general, the mass matrix of the CP-even states is not diagonalized in the Higgs basis, and we need further rotation to obtain CP-even mass eigenstates h and H , 

φ1 φ2



 = R(α − β)

H h

 .

(3.47)

The Higgs potential can be written as V2HDM = −Th h − TH H +

 2 1 H h Meven 2



H h

 +

1 0  2 G A Modd 2

 0     2 G+ G + G − H − M± + H A

+ cubic and quartic terms.

(3.48)

The tadpole terms for h and H are given by 

TH Th



 = R(−α + β)

Tφ1 Tφ2



 = R(−α)

Th 1 Th 2

 .

(3.49)

The mass matrix of the CP-even Higgs fields in the mass eigenbasis is given by 2 Meven = R(α − β)t Mφ2 R(α − β) = R(α)t Mh2 R(α)     2 TH H TH h mH 0 + , = Th H Thh 0 m 2h

(3.50)

42

3 Review of the Extended Higgs Models

Table 3.1 Charge assignment of the softly-broken Z 2 symmetry and the mixing factors in Yukawa interactions Z 2 charge

Mixing factor

1

2

QL

LL

uR

dR

eR

ζu

ζd

ζe

Type-I

+

−

+

+

−

−

−

cot β

cot β

cot β

Type-II

+

−

+

+

−

+

+

cot β

− tan β

− tan β

Type-X + (lepton specific)

−

+

+

−

−

+

cot β

cot β

− tan β

Type-Y + (flipped)

−

+

+

−

+

−

cot β

− tan β

cot β

The masses of the CP-even Higgs bosons and the mixing angle β − α are given by     2  2  + M2φ 22 sβ−α − M2φ 12 s2(β−α) , (3.51) m 2H = M2φ 11 cβ−α  2 2  2 2  2 2 m h = Mφ 11 sβ−α + Mφ 22 cβ−α + Mφ 12 s2(β−α) , (3.52)  2 −2 Mφ 12 tan2(β − α) = . (3.53) (M2φ )11 − (M2φ )22 We define the domain of β − α to be 0 ≤ β − α ≤ π so that sβ−α is always positive and cβ−α has the opposite sign from (M2φ )12 [4]. The tadpole contributions are given by 

TH H TH h Th H Thh





0 Th 1 /vcβ 0 Th 2 /vsβ



= R(α) R(α)   (Th 1 cα2 /cβ + Th 2 sα2 /sβ )/v −sα cα (Th 1 /cβ − Th 2 /sβ )/v = . −sα cα (Th 1 /cβ − Th 2 /sβ )/v (Th 1 sα2 /cβ + Th 2 cα2 /sβ )/v (3.54) t

We identify h and H as the observed Higgs boson with the mass 125 GeV and an additional CP-even Higgs boson, respectively. The eight parameters in the Higgs potential are expressed by the following six input parameters m H , m A, m H± ,

M 2 , tan β, sβ−α ,

(3.55)

and the two parameters m h and v are fixed by experiments. In addition, we have a degree of freedom of the sign of cβ−α . Under the Z 2 symmetry, 2HDM can be classified into four independent types of Yukawa interactions so-called Type-I, Type-II, Type-X, and Type-Y 2HDMs as given in Table 3.1 [5, 6]. The Yukawa Lagrangian is generally written by ˜ u u R − Yd Q¯ L d d R − Ye L¯ L e e R + h.c., LY = − Yu Q¯ L 

(3.56)

˜ u = iσ2 ∗u , and u,d,e are 1 or 2 depending on the types of 2HDMs. where 

3.2 Higgs Couplings and the Alignment Limit

43

3.1.4 Theoretical Constraints The parameters in the Higgs potential are constrained by perturbative unitarity, vacuum stability, and the condition to avoid wrong vacua. For the perturbative unitarity bound, there are twelve independent eigenvalues of the s-wave amplitude matrix [7–10].   1  0 3(λ1 + λ2 ) ± 9(λ1 − λ2 )2 + 4(2λ3 + λ4 )2 , (3.57) = a1,± 32π   1 0 a2,± (3.58) (λ1 + λ2 ) ± (λ1 − λ2 )2 + 4λ24 , = 32π 

 1 0 (3.59) (λ1 + λ2 ) ± (λ1 − λ2 )2 + 4λ25 , a3,± = 32π 1 0 a4,± (λ3 + 2λ4 ± 3λ5 ), = (3.60) 16π 1 0 (λ3 ± λ4 ), a5,± = (3.61) 16π 1 0 (λ3 ± λ5 ). a6,± = (3.62) 16π The vacuum stability bound is sufficiently and necessarily satisfied by imposing the following conditions [11–15] λ1 > 0, λ2 > 0,

 λ1 λ2 + λ3 + MIN(0, λ4 + λ5 , λ4 − λ5 ) > 0.

(3.63)

In addition, the wrong vacua can be avoided by taking M 2 ≥ 0 [16]. We thus only take the positive value of M 2 in the following discussion.

3.2 Higgs Couplings and the Alignment Limit In the mass eigenstate, the interaction terms among the gauge bosons and the scalar bosons are given by,  2  2m W + − m 2Z Wμ Wν + Zμ Zν Lint = [sβ−α h + cβ−α H ]g μν v v   μν ± ∓ 2 + g G Wμ em W Aν − gm Z sW Z ν   μ + gφ1 φ2 V (∂ μ φ1 ) φ2 − φ1 ∂ μ φ2 Vμ + gφ1 φ2 V1 V2 gμν φ1 φ2 V1 V2ν ,

(3.64)

where gφ1 φ2 V and gφ1 φ2 V1 V2 are given in Table B.4. The Yukawa interaction terms among the fermions and the scalar bosons are given by

44

3 Review of the Extended Higgs Models

  mf  f f f f (ζh h + ζ H H ) − 2i I f f γ5 f (G 0 + ζ f A) v f =u,d,e   √    md  mu u PL d G + + ζu H + − u PR d G + + ζd H + + h.c. + 2Vud v v √ m   + + (3.65) − 2 ν PR  G + ζ H + h.c. v

Lint = −

where f

ζh = sβ−α + ζ f cβ−α ,

(3.66)

f ζH

(3.67)

= cβ−α − ζ f sβ−α ,

and ζ f is the Type-dependent mixing factor given in Table 3.1. For the neutral Higgs φ bosons, we define the scaling factor κ X as φ

κX =

gφ X X gh SM X X

, φ = h, H, A,

(3.68)

where gh SM X X and gφ X X are tree-level couplings in the SM and the 2HDMs. The scaling factors are given by κVh = sβ−α , κVH = cβ−α , κVA = 0, κ hf

=

f ζh ,

κ Hf

=

f ζH ,

κVA

= −2i I f ζ f .

(3.69) (3.70)

When sβ−α = 1, the couplings of h with various SM particles become SM-like. We call this SM-like limit, sβ−α = 1, as the alignment limit in this thesis. In the kappa framework, the scaling factors in the 2HDM are given by f

κV = κVh = sβ−α , κ f = κ hf = ζh .

(3.71)

In Fig. 3.1, we show the tree-level predictions for the correlation between the scaling factors for the leptons and down-type quarks in the four types of 2HDMs. The sign of cβ−α is negative in the left panel while positive in the right panel. For illustration purposes, we slightly shift the lines in Type-I and Type-II along with κe = κd . From the definition of the scaling factors, the point with κe = κd = 1 corresponds to the SM-like limit. In Type-I, both κd and κe are close to unity when tan β is large due to the cot β suppression in the Yukawa couplings. When cβ−α is negative, both κd and κe are smaller than the SM values, while they are larger than unity when cβ−α is positive. In Type- II, both κd and κe deviate from SM values with large tan β due to the tan β enhancement in the Yukawa couplings. When cβ−α is negative, both κd and κe are larger than unity, while they are smaller than unity when cβ−α is positive. In Type-X (Y) with large tan β, κe (κd ) becomes large due to the tan β enhancement, while κd (κe ) is close to unity due to the cot β suppression. When cβ−α is negative, κd (κe ) is smaller than unity, while κe (κd ) is larger than unity in

3.2 Higgs Couplings and the Alignment Limit

45

Fig. 3.1 Tree-level predictions for the correlation between the scaling factors for the leptons and down-type quarks in the four types of 2HDMs. The sign of cβ−α is negative in the left panel while positive in the right panel. We made these figures based on Figs. 10 and 11 in Ref. [17]

Fig. 3.2 Tree-level predictions for the correlation between the scaling factors for the leptons and up-type quarks in the four types of 2HDMs. The sign of cβ−α is negative in the left panel while positive in the right panel. We made these figures based on Figs. 10 and 11 in Ref. [17]

Type-X (Type-Y). On the other hand, when cβ−α is positive, κe (κd ) is smaller than unity, while κd (κe ) is larger than unity in Type-X (Type-Y). In Fig. 3.2, we show the tree-level predictions for the correlation between the scaling factors for the leptons and up-type quarks in the four types of 2HDMs. For illustration purposes, we slightly shift the lines in Type-I and Type-Y along with κe = κd . In Type-I and Type-Y, we have the same correlation for κu and κe . When tan β is large, both κu and κe are close to unity due to the cot β suppression. When cβ−α is negative, both κd and κe are smaller than the SM values, while they are larger than unity when cβ−α is positive. Similarly, in Type-II and Type-X, we have the same

46

3 Review of the Extended Higgs Models

correlation for κu and κe . When tan β is large, κe deviate from unity due to the tan β enhancement, while κu is close to unity due to the cot β suppression. When cβ−α is negative (positive), κu (κe ) is smaller than unity, while κe (κu ) is larger than unity. Thus, each type of 2HDMs shows different correlations for the scaling factor, and we can discriminate the types of Yukawa interaction by measuring the Higgs bosons couplings. As we have discussed in Sect. 7.2, the current observed data are consistent with the SM predictions. The current constraints on the scaling factors are summarized in Table. 2.4. The values of sβ−α and tan β are constrained by the Higgs signal measurements, and the near alignment region with sβ−α ≈ 1 is favored in the 2HDMs. The alignment limit can be achieved in different two ways [18–20]; (i) decoupling of additional Higgs bosons, and (ii) alignment without decoupling. In scenario (i), we take the decoupling limit: M 2 f (λi )v 2 . Then, we have tan 2(β − α) 

−2Z 6 v 2  0, −M 2

(3.72)

where we have used that Z 6 does not depend on the M. Eq. (3.72) implies sin (β − α) = 1, and the couplings of h become SM like. In this scenario, masses of the additional Higgs bosons are close to M, and they are decoupled from the electroweak physics. In scenario (ii), the off-diagonal component of the mass matrix for the CP-even state is equal to zero;   1 Z 6 = − sin 2β λ1 cos2 β − λ2 sin2 β − λ345 cos 2β = 0, 2

(3.73)

where we have used abbreviation λ345 = λ3 + λ4 + λ5 . In this scenario, the additional Higgs bosons need not be decoupled, and masses of these particles can be taken around the electroweak scale. Therefore, this scenario is testable in current and future experiments [17, 21]. The simple realization of the condition in Eq. (3.73) is taking the natural alignment conditions [22], λ1 = λ2 = λ345 ,

(3.74)

where the alignment is realized independently of the value of tan β.

3.3 Custodial Symmetry in the 2HDMs In this section, we discuss the custodial symmetry in the 2HDM. We introduce the bi-doublet fields [23] in the Higgs basis, Mi = (iσ2 Hi∗ , Hi ), (i = 1, 2).

(3.75)

3.3 Custodial Symmetry in the 2HDMs

47

They transform under the SU (2) L × U (1)Y gauge transformations as follows, SU (2) L : Mi → exp [igαa (x)τa ]Mi , U (1)Y : Mi → Mi exp [−ig Y α4 (x)σ3 ]. (3.76) We note that the following bi-doublets can also be used, Mi P ≡ Mi exp [−iχ σ3 ] = Mi diag(e−iχ , eiχ ), with 0 ≤ χ < 2π.

(3.77)

to construct the gauge-invariant Higgs potential. Since we define the VEV of H1 as real and positive, we consider M1 and M2 = M2 P as building blocks of the Higgs potential. The SU (2) L × SU (2) R transformations of M1 and M2 are given by M1 → L M1 R † ,

M2 → L M2 R † ,

(3.78)

where L ∈ SU (2) L , R ∈ SU (2) R . Four SU (2) L × U (1)Y invariants are given by Tr(M1† M1 ) = 2|H1 |2 , † Tr(M 2 M2 ) Tr(M1† M2 ) Tr(M1† M2 σ3 )

(3.79)

= 2|H2 | , 2

= =

e H1† H2 eiχ H1† H2 iχ

(3.80) + −

−iχ

e H2† H1 , e−iχ H2† H1 .

(3.81) (3.82)

Tr(M1† M1 ), Tr(M †2 M2 ) and Tr(M1† M2 ) are hermitian and SU (2) L × SU (2) R invariant. On the other hand, Tr(M1† M2 σ3 ) is anti-hermitian and does not respect SU (2) L × SU (2) R symmetry. In terms of these invariants, the Higgs potential is written by 1 2 1 † Y Tr(M1† M1 ) + Y22 Tr(M 2 M2 ) − Re(Y32 e−iχ )Tr(M1† M2 ) 2 1 2 1 1 1 † † + Z 1 Tr 2 (M1† M1 ) + Z 2 Tr 2 (M 2 M2 ) + Z 3 Tr(M1† M1 )Tr(M 2 M2 ) 8 8 4 1 + [Z 4 + Re(Z 5 e−2iχ )]Tr 2 (M1† M2 ) 4 1 † + [Re(Z 6 e−iχ ) Tr(M1† M1 ) + Re(Z 7 e−iχ ) Tr(M 2 M2 )]Tr(M1† M2 ) 2 1 − i Im(Y32 e−iχ ) Tr(M1† M2 σ3 ) − [Z 4 − Re(Z 5 e−2iχ )]Tr 2 (M1† M2 σ3 ) 4 i + Im(Z 5 e−2iχ ) Tr(M1† M2 )Tr(M1† M2 σ3 ) 2 i † + [Im(Z 6 e−iχ ) Tr(M1† M1 ) + Im(Z 7 e−iχ ) Tr(M 2 M2 )]Tr(M1† M2 σ3 ). 2

V (M1 , M2 ) =

(3.83)

48

3 Review of the Extended Higgs Models

If the Higgs potential respects the SU (2) L × SU (2) R symmetry, we have Im(Y32 e−iχ ) = Im(Z 5 e−2iχ ) = Im(Z 6 e−iχ ) = Im(Z 7 e−iχ ) = 0, Z 4 = Re(Z 5 e

−2iχ

).

(3.84) (3.85)

When the Higgs potential is CP invariant, Yi2 and Z i are real. Therefore, we have Z4 = Z5 or

for χ = 0, π,

(3.86)

Z 4 = −Z 5 and Y32 = Z 6 = Z 7 = 0

for χ = π/2, 3π/2.

(3.87)

In terms of the parameters in Eq. (3.9), we have λ4 = λ 5 ,

for χ = 0, π,

(3.88)

or λ4 = −λ5 , λ1 = λ2 = λ3

for χ = π/2, 3π/2.

(3.89)

The case in Eq. (3.88) (m A = m H ± ) is introduced in Ref. [24]. On the other hand, the case in Eq. (3.89) is introduced in Ref. [25], and it is so-called twisted custodial symmetry. In the twisted-custodial symmetric scenario, the Higgs potential is given by V (H1 , H2 ) = Y12 H1† H1 + Y22 H2† H2 +

1 Z S (H1† H1 + H2† H2 )2 − Z AS (H1† H2 − H2† H1 )2 , 2

(3.90)

where Z S = Z 1 = Z 2 = Z 3 and Z AS = Z 4 = −Z 5 . The masses of the Higgs bosons are given by m 2h = Z S v 2 . m 2H ± = m 2H = M 2 = Y2 + m 2A = M 2 + Z AS v 2 .

(3.91) 1 Z S v2 . 2

(3.92) (3.93)

In this scenario, the Higgs alignment, sβ−α = 1, is realized by Z 6 = 0. In addition, T becomes small by m H ± = m H and sβ−α = 1. The CP quantum numbers of H and A cannot be determined only from the Higgs potential when Z 6 = Z 7 = 0 [26]. However, if neutral Higgs-fermion interactions are CP conserving, as in the case we are considering, we can determine such that H is the CP-even and A is the CP-odd. In the twisted custodial symmetric scenario, H ± is degenerate with the CP-even scalar H . Therefore, this scenario is different from Case II in Ref. [24] where H should be regarded as the CP-odd state. Therefore, we can treat tan β as a free parameter differently from Case II where tan β = 1 is required.

3.4 Constraints from Experimental Data

49

The Yukawa couplings and the U (1)Y gauge coupling violate the custodial symmetry. Therefore, the custodial symmetry is not the symmetry of the whole 2HDM Lagrangian. The relations among the Higgs quartic couplings given in Eq. (3.89) are broken by quantum corrections. Such violating effects indicate the peculiarity of the scenario, where the Higgs potential exactly respects the twisted custodial symmetry at the electroweak scale. In Ref. [27], the possibility of the approximate realization of the twisted-custodial symmetry at the electroweak scale, starting from a twisted-custodial symmetric theory at high scale  is discussed. The scenario with the softly-broken O(8) symmetry at a high scale is discussed in Ref. [22].

3.4 Constraints from Experimental Data We here discuss relevant experimental constraints on the 2HDMs. In Chap. 4, we study current and future constraints from direct searches of the additional Higgs bosons.

3.4.1 Electroweak Precision Tests The constraint from the electroweak precision tests is imposed by the S and T parameters [28, 29]. The new physics contributions in the 2HDMs are defined by S = S2HDM − SSM and T = T2HDM − TSM . The analytical formulae for S and T are given in Refs. [30–33]. The experimental data are given in Ref. [34], S = 0.04 ± 0.08, T = 0.08 ± 0.07,

(3.94)

where the U parameter is fixed to zero. The reference values of the masses of the SM Higgs boson and the top quark are m h,ref = 125 GeV and m t,ref = 172.5 GeV, respectively. The correlation coefficient in χ 2 analysis is +0.92. We require S and T to be within 95% CL.

3.4.2 Signal Strengths of the SM-Like Higgs Boson Measurements of signal strengths for the SM-like Higgs boson constrain the parameter space of the 2HDMs. We evaluate the decay rates of the SM-like Higgs boson, (h → X Y ), including the NLO EW and higher-order QCD corrections by using H-COUP v2 [35]. The analytic expressions for (h → X Y ) are given in Ref. [36]. We define the scaling factors at the one-loop level,

50

3 Review of the Extended Higgs Models

  2HDM  LO+EW+QCD (h → X Y ) . κX =  SM LO+QCD (h → X Y )

(3.95)

We require that the scaling factors for X Y = bb, τ τ, γ γ , gg and Z Z ∗ to be consistent with the values presented in Table 11a of Ref. [37] at 95 % CL.

3.4.3 Flavor Constraints The mass of the charged Higgs bosons is constrained by the B meson flavor violating decay, B → X s γ [38, 39]. For Type-II and Type-Y, m H ±  800GeV is excluded for tan β > 1 [38, 39]. For Type-I and Type-X, the constraint is weaker than that for Type-II and Type-Y. The excluded regions are given in lower tan β regions, e.g., m H ±  400 (180) GeV for tan β = 1 (1.5) [38]. For Type-II, high tan β regions are also constrained by the B meson rare leptonic decay, Bs → μ+ μ− [40]. For m H ± = 800 GeV, the region with tan β  10 is excluded [34]. Comprehensive studies for constraints from various flavor observables such as B meson decays, D meson decays and B0 − B¯ 0 mixing, are performed in Refs. [34, 41].

References 1. Georgi H, Machacek M (1985) Doubly charged Higgs bosons. Nucl Phys B 262:463–477 2. Glashow SL, Weinberg S (1977) Natural conservation laws for neutral currents. Phys Rev D 15:1958 3. Paschos EA (1977) Diagonal neutral currents. Phys Rev D 15:1966 4. Bernon J, Gunion JF, Haber HE, Jiang Y, Kraml S (2015) Scrutinizing the alignment limit in two-Higgs-doublet models: mh =125 GeV. Phys Rev D 92(7):075004 5. Barger VD, Hewett JL, Phillips RJN (1990) New constraints on the charged Higgs sector in two Higgs doublet models. Phys Rev D 41:3421–3441 6. Aoki M, Kanemura S, Tsumura K, Yagyu K (2009) Models of Yukawa interaction in the two Higgs doublet model, and their collider phenomenology. Phys Rev D 80:015017 7. Kanemura S, Kubota T, Takasugi E (1993) Lee-Quigg-Thacker bounds for Higgs boson masses in a two doublet model. Phys Lett B 313:155–160 8. Akeroyd AG, Arhrib A, Naimi E-M (2000) Note on tree level unitarity in the general two Higgs doublet model. Phys Lett B 490:119–124 9. Ginzburg IF, Ivanov IP (2005) Tree-level unitarity constraints in the most general 2HDM. Phys Rev D 72:115010 10. Kanemura S, Yagyu K (2015) Unitarity bound in the most general two Higgs doublet model. Phys Lett B 751:289–296 11. Deshpande NG, Ma E (1978) Pattern of symmetry breaking with two Higgs doublets. Phys Rev D 18:2574 12. Klimenko KG (1985) On necessary and sufficient conditions for some Higgs potentials to be bounded from below. Theor Math Phys 62:58–65 13. Sher M (1989) Electroweak Higgs potentials and vacuum stability. Phys Rep. 179:273–418 14. Nie S, Sher M (1999) Vacuum stability bounds in the two Higgs doublet model. Phys Lett B 449:89–92

References

51

15. Kanemura S, Kasai T, Okada Y (1999) Mass bounds of the lightest CP even Higgs boson in the two Higgs doublet model. Phys Lett B 471:182–190 16. Barroso A, Ferreira PM, Ivanov IP, Santos R (2013) Metastability bounds on the two Higgs doublet model. JHEP 06:045 17. Kanemura S, Tsumura K, Yagyu K, Yokoya H (2014) Fingerprinting nonminimal Higgs sectors. Phys Rev D 90:075001 18. Kanemura S, Tohyama H-A (1998) Nondecoupling effects of Higgs bosons on e+ e− → W(L)+ W(L)− in the two doublet model. Phys Rev D 57:2949–2956 19. Gunion JF, Haber HE (2003) The CP conserving two Higgs doublet model: the Approach to the decoupling limit. Phys Rev D 67:075019 20. Kanemura S, Okada Y, Senaha E, Yuan CP (2004) Higgs coupling constants as a probe of new physics. Phys Rev D 70:115002 21. Kanemura S, Yokoya H, Zheng Y-J (2014) Complementarity in direct searches for additional Higgs bosons at the LHC and the International Linear Collider. Nucl Phys B 886:524–553 22. Bhupal Dev and PS, Pilaftsis A (2014) Maximally symmetric two Higgs doublet model with natural standard model alignment. JHEP 12:024 [Erratum: JHEP 11:147 (2015)] 23. Sikivie P, Susskind L, Voloshin MB, Zakharov VI (1980) Isospin breaking in technicolor models. Nucl Phys B 173:189–207 24. Pomarol A, Vega R (1994) Constraints on CP violation in the Higgs sector from the rho parameter. Nucl Phys B 413:3–15 25. Gerard JM, Herquet M (2007) A Twisted custodial symmetry in the two-Higgs-doublet model. Phys Rev Lett 98:251802 26. Haber HE, O’Neil D (2011) Basis-independent methods for the two-Higgs-doublet model III: the CP-conserving limit, custodial symmetry, and the oblique parameters S, T, U. Phys Rev D 83:055017 27. Aiko M, Kanemura S (2021) New scenario for aligned Higgs couplings originated from the twisted custodial symmetry at high energies. JHEP 02:046 28. Peskin ME, Takeuchi T (1990) A new constraint on a strongly interacting Higgs sector. Phys Rev Lett 65:964–967 29. Peskin ME, Takeuchi T (1992) Estimation of oblique electroweak corrections. Phys Rev D 46:381–409 30. Bertolini S (1986) Quantum effects in a two Higgs doublet model of the electroweak interactions. Nucl Phys B 272:77–98 31. Grimus W, Lavoura L, Ogreid OM, Osland P (2008) The Oblique parameters in multi-Higgsdoublet models. Nucl Phys B 801:81–96 32. Kanemura S, Okada Y, Taniguchi H, Tsumura K (2011) Indirect bounds on heavy scalar masses of the two-Higgs-doublet model in light of recent Higgs boson searches. Phys Lett B 704:303–307 33. Kanemura S, Kikuchi M, Yagyu K (2015) Fingerprinting the extended Higgs sector using one-loop corrected Higgs boson couplings and future precision measurements. Nucl Phys B 896:80–137 34. Haller J, Hoecker A, Kogler R, Mönig K, Peiffer T, Stelzer J (2018) Update of the global electroweak fit and constraints on two-Higgs-doublet models. Eur Phys J C 78(8):675 35. Kanemura S, Kikuchi M, Mawatari K, Sakurai K, Yagyu K (2020) H-COUP version 2: a program for one-loop corrected Higgs boson decays in non-minimal Higgs sectors. Comput Phys Commun 257:107512 36. Kanemura S, Kikuchi M, Mawatari K, Sakurai K, Yagyu K (2019) Full next-to-leading-order calculations of Higgs boson decay rates in models with non-minimal scalar sectors. Nucl Phys B 949:114791 37. Aad G et al (2020) Combined measurements of √ Higgs boson production and decay using up to 80 fb−1 of proton-proton collision data at s = 13 TeV collected with the ATLAS experiment. Phys Rev D 101(1):012002 38. Misiak M, Steinhauser M (2017) Weak radiative decays of the B meson and bounds on M H ± in the two-Higgs-doublet model. Eur Phys J C 77(3):201

52

3 Review of the Extended Higgs Models

39. Misiak M, Rehman A, Steinhauser M (2020) Towards B → X s γ at the NNLO in QCD without interpolation in mc . JHEP 06:175 40. Cheng X-D, Yang Y-D, Yuan X-B (2016) Revisiting Bs → μ+ μ− in the two-Higgs doublet models with Z 2 symmetry. Eur Phys J C 76(3):151 41. Enomoto T, Watanabe R (2016) Flavor constraints on the Two Higgs doublet models of Z2 symmetric and aligned types. JHEP 05:002

Chapter 4

Synergy Between Direct Searches at the LHC and Precision Tests at Future Lepton Colliders

In this chapter, we discuss the complementarity between direct searches for additional Higgs bosons at the (HL-)LHC and precision measurements of the Higgs boson properties at future electron-positron colliders such as the ILC. The direct search of additional Higgs bosons gives a lower bound for the mass scale of the additional Higgs bosons. On the other hand, if the couplings of the SM-like Higgs boson slightly deviate from those in the SM, the indirect search gives an upper bound for the mass scale of the additional Higgs bosons through the theoretical constraints. The decays of additional Higgs bosons into the SM-like Higgs boson can be a dominant decay process in the approximate alignment scenario, and such Higgs-to-Higgs decays are quite useful for the direct search of additional Higgs bosons. In the 2HDM, we concretely show that most of the parameter space can be explored by utilizing the synergy between the direct search of Higgs-to-Higgs decay at future hadron colliders and the indirect search at future lepton colliders. This chapter is mainly based on Ref. [1].

4.1 Decays of the SM-Like Higgs Boson In this section, we give the analytic expressions for the decay rates of the SM-like Higgs boson including higher-order QCD corrections.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 M. Aiko, Theoretical Studies on Extended Higgs Sectors Towards Future Precision Measurements, Springer Theses, https://doi.org/10.1007/978-981-99-1324-4_4

53

54

4 Synergy Between Direct Searches at the LHC and Precision Tests …

4.1.1 Decay Rates for h → f f¯ The SM-like Higgs boson decays into a pair of leptons and quarks. At the LO, the h f f¯ vertex is given by hS,tree ff = −

mf h κ . v f

(4.1)

The decay rate for h → f f¯ ( f = t) is given by   f (h → f f¯) = LO (h → f f¯) 1 + QCD,h ,

(4.2)

f where LO (h → f f¯) and QCD,h denote the LO decay rate and QCD corrections, respectively. The LO decay rate is given by

LO (h → f f¯) =

Ncf

m h  S,tree 2 3/2   λ 8π h f f



m 2f m 2f , m 2h m 2h

 ,

(4.3)

f

with Nc = 3 (1) for f being quarks (leptons). The kinematical factor is defined by λ(x, y) = (1 − x − y)2 − 4x y.

(4.4)

For decays into leptons, we use the pole masses of leptons in hS,tree f f . On the other hand, for decays into quarks, f = q (q = t), we replace a quark mass in the Yukawa coupling to a running mass m q (μ) evaluated at μ = m h , h f f¯ → −

m q (m h )κqh v

.

(4.5)

Thereby, leading logarithmic corrections are resumed [2]. We use the pole masses of leptons and quarks in λ(x, y). For decays into quarks, we include QCD corrections up to NNLO in the MS scheme. The QCD corrections are given by q,MS

QCD,h = qq + h .

(4.6)

The first term qq is evaluated in the chiral limit, m q  m h . The NLO correction is given in Refs. [2], while the NNLO correction is given in Refs. [3–7],

4.1 Decays of the SM-Like Higgs Boson

qq

55

  17 3 μ2 αs (μ) CF + ln 2 = π 4 2 mh    2  

Nf αs (μ) 10801 39 65 2 19 + − ζ(3) − N f − ζ(3) − π 2 − , π 144 2 24 3 12 18 (4.7)

with C F = 4/3 and the Riemann zeta function ζ(n). We take the number of active flavors N f as N f = 5 and the renormzalization scale μ as μ = m h . The second term h includes top-quark loop contributions. In the heavy top-mass limit, m h  m t , it is given by  φ =

αs (μ) π

2 

 m 2φ m q2 (μ) 1 2 2 1.57 − log 2 + log , 3 9 mt m 2φ

(4.8)

and we take μ = m h .

4.1.2 Decay Rates for h → V V ∗ → V f f¯ The SM-like Higgs boson decays into a pair of on-shell and off-shell weak gauge bosons. At the LO, the h Z Z and hW W vertices are given by h1,tree ZZ =

2m 2W h 2m 2Z h 1,tree κ Z , hW κW . W = v v

(4.9)

The decay rate for h → V V ∗ is given by (h → V V ∗ ) =



(h → V f f¯).

(4.10)

f

The decay rate for h → V f f¯ is given by

(h → Z f f¯) = LO (h → Z f f¯) 1 + QCD ,

(h → W f f¯  ) = LO (h → W f f¯  ) 1 + QCD ,

(4.11) (4.12)

where LO (h → V f f¯) and QCD denote the LO decay rate and QCD corrections, respectively. In the following, we neglect the masses of the final-state fermions. Then, LO (h → V f f¯) is given by [8]

56

4 Synergy Between Direct Searches at the LHC and Precision Tests …

√  2  m  4 2G F m h 2 Z 2  1,tree  ¯ , LO (h → Z f f ) = (v f + a f )h Z Z  F 3 768π mh √   2G F m h  1,tree 2 mW LO (h → W f f¯  ) = , F   hW W 768π 3 mh

(4.13) (4.14)

where F(x) is given by   2 3x − 1 3(1 − 8x 2 + 20x 4 ) arccos √ 2x 3 4x 2 − 1   47 2 13 1 − (1 − x 2 ) x − + 2 − 3(1 − 6x 2 + x 4 ) ln x. 2 2 x

F(x) =

(4.15)

For decays into light quarks, the QCD correction in the MS scheme is given by [9] QCD = C F

3αs (μ) , 4π

(4.16)

and we take μ = m h . The QCD correction is the same for h → Z f f¯ and h → W f f¯  This is because the QCD corrections only appear in the Z f f¯ and W f¯ f  vertices, and gluon-loop contributions in the Z f f¯ vertex are the same as those in W f¯ f  vertex in the massless limit of fermions [10].

4.1.3 Decay Rates for h → γγ, Zγ, gg The SM-like Higgs boson decays into a pair of γγ, Z γ and gg at the one-loop level. At the LO, the loop-induced hVV  vertices are given by μν 1  hVV hVV  ( p1 , p2 , q) = g μν   +

Fig. 4.1 Momentum assignment for the renormalized h VV  vertex

μ

p1ρ p2σ 3 p1ν p2 2   hVV  + i μνρσ hVV  . 2 q q2

(4.17)

4.1 Decays of the SM-Like Higgs Boson

57

where p1 and p2 denote the incoming four-momentum of the gauge bosons V and V  , and q is the outgoing four-momentum of the SM-like Higgs boson (see Fig. 4.1). We take 0123 = +1 and assume that the external gauge bosons are on-shell. For the μ on-shell photon and gluon with a four-momentum pi , the Ward-Takahashi identity holds μν

hVV  ( p1 , p2 , q) = 0, piμ

(4.18)

and we obtain 2 2 2 2  hVV  ( p1 , p2 , q ) = −

2q 2   1  ( p 2 , p 2 , q 2 ). q 2 − p12 − p22 hVV 1 2

(4.19)

When we neglect the effects of the CP violation, we have 3 2 2 2  hVV  ( p1 , p2 , q ) = 0.

(4.20)

At the LO, the decay rate for h → γγ is given by 2 1  1   (0, 0, m 2h )  16πm h hγγ √ 2 m3  2  f 2G F αem λ + −  h  h I h (τ ) + = Nc Q 2f κhf I Fh (τ f ) − H H h I Sh (τ H ± ) , κV W W 3 v 256π

LO (h → γγ) =

f

(4.21) with τi = m 2h /(4m i2 ), (i = W, f, H ± ). The loop functions are given by φ

IW (τW ) =

2m 2W

 6+

m 2φ

 + (12m 2W − 6m 2φ )C0 (0, 0, φ; W, W, W )

m 2φ m 2W   −2 2 = − 2τW + 3τW + 3(2τW − 1) f (τW ) τW ,     m 2φ 8m 2f φ 2 C0 (0, 0, φ; f, f, f ) I F (τ f ) = − 2 1 + 2m f − 2 mφ  −2  = 2 τ f + (τ f − 1) f (τ f ) τ f ,  2v 2  φ I S (τ H ± ) = 2 1 + 2m 2H ± C0 (0, 0, φ; H ± , H ± , H ± ) mφ = − [τ H ± − f (τ H ± )] τ H−2± ,

(4.22)

(4.23)

(4.24)

with f (τ ) defined as ⎧ √ ⎨arcsin2 ( τ ) (τ ≤ 1),  2 √ f (τ ) = −1 1 1+ 1−τ ⎩− ln √ −1 − iπ 4 1− 1−τ

(τ > 1).

(4.25)

58

4 Synergy Between Direct Searches at the LHC and Precision Tests …

The quark-loop contributions receive QCD corrections. In the large top-mass limit, the QCD corrections up to NNLO are included by replacing I Fh as [11, 12]  I Fh

→

I Fh

αs (μ) − 1− π



αs (μ) π

2

 2   mh 31 7 + ln , 24 4 4m 2t

(4.26)

and we take μ = m h . At the LO, the decay rate for h → Z γ is given by  2  m2 1  1  h Z γ (m 2Z , 0, m 2h ) 1 − Z2  8πm h mh √ 3  2 2 3 2G F αem m h m = 1 − Z2 3 128π mh  2  λH + H −h v h   ± , λ H ± ) , × κhV JWh (τW , λW ) + Ncf Q f v f κhf JFh (τ f , λ f ) + J (τ H S 2 m ± H f

LO (h → Z γ) =

(4.27) with τi = 4m i2 /m 2h and λi = 4m i2 /m 2Z . The loop functions are given by φ

JW (τW , λW ) =

2m 2W



 c2W

5+

m 2φ

sW cW (m 2φ − m 2Z ) 2m 2W × 1 + 2m 2W C0 (0, Z , φ; W, W, W ) +



s2 − W c2W

 1+

m 2φ



2m 2W

(φ; W, W ) − B (Z ; W, W )] [B 0 0 m 2φ − m 2Z    s2  2 m φ − m 2Z )C0 (0, Z , φ; W, W, W − 2c2W 3 − W 2 cW      2   2 sW sW 2 2 1+ − 5 + (τ , λ ) + 4 3 − (τ , λ ) , I = cW I 1 W W 2 W W τW c2 τW c2W W m 2Z

(4.28)

8m 2f 1 φ 1 + (4m 2f − m 2φ + m 2Z )C0 (0, Z , φ; f, f, f ) J F (τ f , λ f ) = − 2 2 2 sW cW (m φ − m Z )

m2 + 2 Z 2 [B0 (φ; f, f ) − B0 (Z ; f, f )] mφ − m Z  4  I1 (τ f , λ f ) − I2 (τ f , λ f ) , cW  m 2H ± c2W φ JS (τ H ± , λ H ± ) = − 1 + 2m 2H ± C0 (0, Z , φ; H ± , H ± , H ± ) sW cW (m φ − m 2Z )    m2 + 2 Z 2 B0 (φ; H ± , H ± ) − B0 (Z ; H ± , H ± ) mφ − m Z =

=

c2W I1 (τ H ± , λ H ± ). 2cW

(4.29)

(4.30)

4.1 Decays of the SM-Like Higgs Boson

59

The function I1 (τ , λ) and I2 (τ , λ) are given by I1 (τ , λ) =

  τλ τ 2 λ  −1 τ 2 λ2  f (τ −1 ) − f (λ−1 ) + g(τ ) − g(λ−1 ) , + 2 2 2(τ − λ) 2(τ − λ) (τ − λ)

(4.31)

  τλ f (τ −1 ) − f (λ−1 ) , I2 (τ , λ) = − 2(τ − λ)

(4.32)

where f (τ ) is defined in Eq. (4.25), while g(τ ) is defined as g(τ ) =

√ √ τ −1 − 1 arcsin( τ ) √ 1−τ −1 2

ln

√ 1+√1−τ −1 1− 1−τ −1

(τ ≥ 1), − iπ (τ < 1).

(4.33)

We do not include QCD corrections for h → Z γ. The decay rate for h → gg is given by 

αs (μ) (1) Eh + (h → gg) = LO (h → gg) 1 + π



αs (μ) π

2

 E h(2)

,

(4.34)

where LO (h → gg) is given by √ 2 2 2G F αs2 (μ)m 3h   h h 1  1  2   κ I (τ ) LO (h → gg) =  , hgg (0, 0, m h ) =  q q q 16πm h 128π 3 q (4.35) and I Fh is given in Eq. (4.23). We include QCD corrections up to NNLO in the large top-mass limit. The NLO QCD corrections, E h(1) , can be decomposed as [12], E h(1) = E hvirt (m t → ∞) + E hreal (m t → ∞),

(4.36)

where the first and second terms denote the contribution from virtual-gluon loops and that from real-gluon emissions, respectively. These are expressed as E φvirt (m t → ∞) =

μ2 11 33 − 2N f + ln 2 , 2 6 mφ

(4.37)

E φreal (m t → ∞) =

73 7 − Nf, 4 6

(4.38)

and we take μ = m h . The NNLO QCD correction, E φ(2) , is given by [13]

60

4 Synergy Between Direct Searches at the LHC and Precision Tests …

495 19 m 2t 149533 363 − ζ(2) − ζ(3) − ln 2 288 8 8 8 mφ     127 1 5 2 m 2t 4157 11 2 + ζ(2) + ζ(3) − ln 2 + N f − ζ(2) . + Nf − 72 2 4 3 mφ 108 6 (4.39)

E φ(2) =

4.2 Decays of the Additional CP-Even Higgs Boson In this section, we give the analytic expressions for the decay rates of the additional CP-even Higgs boson including higher-order QCD corrections.

4.2.1 Decay Rates for H → f f¯ When 2m f ≤ m H , the additional CP-even Higgs boson decays into a pair of fermions. At the LO, the H f f¯ vertex is given by  HS,tree ff = −

mf H κ . v f

(4.40)

The decay rates for H → f f¯ are given by   f (H → f f¯) = LO (H → f f¯) 1 + QCD,H ,

(4.41)

where the LO decay rate is given by m H  S,tree 2 3/2 LO (H → f f¯) = Ncf  λ  8π H f f



m 2f

m 2f

, m 2H m 2H

 ,

(4.42)

Similarly to the h → f f¯ decay, we use the pole masses of leptons in  HS,tree f f , while to a running mass m (μ) evaluated at μ = mH. we replace a quark mass in  HS,tree q ff We use the pole masses of leptons and quarks in λ(x, y). For decays into quarks, we include QCD corrections. In the MS scheme, the NNLO QCD corrections are given by q,MS

QCD,H = qq +  H ,

(4.43)

where qq and  H are given in Eqs. (4.7) and (4.8) with μ = m H , respectively. We take N f = 5 for m H ≤ m t , while N f = 6 for m t < m H .

4.2 Decays of the Additional CP-Even Higgs Boson

61

The top-quark mass effects become significant for the decay into a pair of top quarks. In the on-shell scheme, the NLO QCD correction is given by [14, 15] t,OS QCD,H

αs (μ) = CF π



 L(β H ) 1 3 2 4 2 − (3 + 34β H − 13β H ) ln ρ H + 2 (7β H − 1) , βH 16β 3H 8β H

(4.44) with βφ = λ1/2 (m 2t /m 2H , m 2t /m 2H ), ρφ = (1 − βφ )/(1 + βφ ).

(4.45) (4.46)

The function L(βφ ) is given by L(βφ ) = (1 + βφ2 ) 4Li2 (ρφ ) + 2Li2 (−ρφ ) + 3 ln ρφ ln − 3βφ ln

2 + 2 ln ρφ ln βφ 1 + βφ

4 − 4βφ ln βφ , 1 − βφ2



(4.47)

where Li2 (x) is the dilog function. When m t  m H , the top quark can be regarded as massless, and we can apply the QCD corrections in the MS scheme. In order to treat the transition between the two regions, we use the linear interpretation function [16],     pole t,MS 2 MS ¯ (H → t t¯) = R 2 LO (H → t t¯) 1 + t,OS QCD,H + (1 − R )LO (H → t t ) 1 + QCD,H ,

(4.48) pole with R = 2m t /m H . In the evaluation of LO (H → t t¯), we use the pole mass for the MS top Yukawa coupling, while the running mass is used for LO (H → t t¯). The QCD t,MS correction in the MS scheme QCD,H is evaluated as similar to the decays into the light quarks. When m t + m W ≤ m H < 2m t , the additional CP-even Higgs boson decays into a pair of on-shell and off-shell top quarks, H → tt ∗ → tbW . The decay rate is given by [17, 18] +

+

(H → tt ∗ ) = 2

yt

yb dyt

yt−

dyb yb−

d ¯ − ), (H → t t¯∗ → t bW dyt dyb

(4.49)

with the scaling variables y f = 1 − 2E f /m H , ( f = t, b). The charge conjugate final state, t¯bW + , doubles the partial decay width. The kinematic boundaries are given by

62

4 Synergy Between Direct Searches at the LHC and Precision Tests …

yt+ = (1 − xt )2 − xt2 , yt−

= (xb + x W ) − 2

(4.50)

xt2 ,

(4.51)

− yt )(yt + xt2 − xb2 + (1 − 2 + yb± = xt2 − xb2 + x W 2(yt + xt2 )   2 xb2 xW 1 ± λ1/2 (yt + xt2 , xt2 )λ1/2 , , 2 yt + xt2 yt + xt2 2xt2

2 xW )

(4.52)

where xi = m i /m H (i = t, b, W ). The Dalitz density is given by 2 2 2 3 0 d ¯ − ) = 3G F m t ζ H m H (H → t t¯∗ → t bW , 2 3 dyt dyb 64π yt + κt γt

(4.53)

with 0 = −yt3 + (1 − 5κt + κW − yb )yt2 + [2κt (1 − 6κW ) − yb (κt − 2κW )] yt + (κt − κW )(κt + 2κW )(1 − 4κt ), (4.54) and κi = m i2 /m 2H (i = t, b, W ). The reduced decay width for the virtual top quark is defined by γt = t2 /m 2H with the total decay width of the top quark t . In the numerical evaluation, we use the on-shell mass of the top quark in the Yukawa interaction.

4.2.2 Decay Rates for H → W + W − , Z Z When 2m V ≤ m H (V = W ± , Z ), the additional CP-even Higgs boson decays into a pair of gauge bosons. At the LO, the H V V vertices are given by  1,tree HZZ =

2m 2W H 2m 2Z H κ Z ,  1,tree κW . HWW = v v

(4.55)

The decay rates for H → V V are given by    2  m 2V m V m 2V m 4V m 3H  1,tree 2 1/2 , LO (H → V V ) = δV ,   1 − 4 2 + 12 4 λ 128πm 4V H V V mH mH m 2H m 2H (4.56) where δV = 2 (1) for V = W (Z ). When m V ≤ m H < 2m V , the additional CP-even Higgs boson decays into a pair of on-shell and off-shell gauge bosons. The decay rate for H → V V ∗ is given by

4.2 Decays of the Additional CP-Even Higgs Boson

(H → V V ∗ ) =



63

(H → V f f¯)

(4.57)

f

The decay rate for H → V V ∗ → V f f¯ is given by (H → Z f f¯) = LO (H → Z f f¯)(1 + QCD ), (H → W f f¯  ) = LO (H → W f f¯  )(1 + QCD ),

(4.58) (4.59)

where the LO decay rates are given by √ 2  m   2G F m H 2 4 Z 2  1,tree   , (v + a ) LO (H → Z f f¯) =  f f HZZ F 768π 3 mH √   2G F m H  1,tree 2 mW  ¯ LO (H → W f f ) = ,  H W W  F 768π 3 mH

(4.60) (4.61)

with F(x) given in Eq. (4.15). The QCD corrections are given in Eq. (4.16).

4.2.3 Decay Rates for H → Z A, W ± H ∓ When m Z + m A ≤ m H , the additional CP-even Higgs boson decays into a pair of CP-odd Higgs boson and Z boson. The decay rate for H → Z A is given by

LO (H → Z A) =

   tree 2  H Z A  m 3 16π

H m 2Z

 λ

3/2

m 2A m 2Z , m 2H m 2H

 ,

(4.62)

where  tree H Z A are given by  tree H Z A = ig H AZ = i

mZ sβ−α . v

(4.63)

When m A ≤ m H < m Z + m A , the additional CP-even Higgs boson decays into a pair of on-shell CP-odd Higgs boson and off-shell Z boson. The decay rate for H → AZ ∗ → A f f¯ is given by [17] x+

+

(H → AZ ∗ ) =

x f

 f¯ dx f

x −f

dx f¯ x −f¯

d (H → AZ ∗ → A f f¯), dx f dx f¯

(4.64)

with the scaling variables x f ( f¯) = 2E f ( f¯) /m H . We neglect the masses of the finalstate fermions. The kinematic boundaries are given by 0 ≤ x f ≤ 1 − κA,

(4.65)

64

4 Synergy Between Direct Searches at the LHC and Precision Tests …

1 − x f − κ A ≤ x f¯ ≤ 1 −

κA , 1− xf

(4.66)

with κ A = m 2A /m 2H . The Dalitz density is given by   9G F  tree 2 2 40 4 d 7 10 2 (H → AZ ∗ → A f f¯) = √ m m + − s s  H Z A  Z H dx f dx f¯ 12 9 W 27 W 16 2π 3 (1 − x f )(1 − x f¯ ) − κ A × , (4.67) (1 − x f − x f¯ − κ A + κ Z )2 + κ Z γ Z

where κ Z = m 2Z /m 2H . The reduced decay width for the virtual Z boson is defined by γ Z =  2Z /m 2H with the total decay width of the Z boson  Z . When m W + m H ± ≤ m H , the additional CP-even Higgs boson decays into a pair of charged Higgs bosons H ± and W ± bosons. The decay rate for H → W ± H ∓ is given by  2  tree   2 2   H W ∓ H ±  m 3 H 3/2 m H ± m W ± ∓ , λ , LO (H → W H ) = 8π m 2W m 2H m 2H

(4.68)

where  tree H W ∓ H ± = ig H H ± W ∓ = ∓

mW sβ−α . v

(4.69)

When m H ± ≤ m H < m W + m H ± , the additional CP-even Higgs boson decays into a pair of on-shell charged Higgs bosons H ± and off-shell W ± boson. The decay rate for H → H ± W ∓∗ → H ± f f¯ is given by x+

+

(H → H ± W ∓∗ ) = 2

x f

 f¯ dx f

x −f

dx f¯ x −f¯

d (H → H + W −∗ → H + f f¯ ), dx f dx f¯ (4.70)

with the scaling variables x f ( f¯ ) = 2E f ( f¯ ) /m H . The charge conjugate final state, H − f¯ f  , doubles the partial decay width. We neglect the masses of the final state fermions. The kinematic boundaries are given by 0 ≤ x f ≤ 1 − κH ± , κH ± , 1 − x f − κ H ± ≤ x f¯ ≤ 1 − 1− xf

(4.71) (4.72)

4.2 Decays of the Additional CP-Even Higgs Boson

65

with κ H ± = m 2H ± /m 2H . The Dalitz density is given by 2 9G F  tree d  (H → H + W −∗ → H + f f¯ ) = √  H W ∓ H ±  m 2W m H 3 dx f dx f¯ 16 2π (1 − x f )(1 − x f¯ ) − κ H ± × , (1 − x f − x f¯ − κ H ± + κW )2 + κW γW

(4.73)

where κW = m 2W /m 2H . The reduced decay width for the virtual W boson is defined 2 /m 2H with the decay width of the W ± boson W . by γW = W

4.2.4 Decay Rates for H → hh, A A, H + H − When 2m S ≤ m H (S = h, A, H ± ), the additional CP-even Higgs boson decays into a pair of Higgs bosons. At the LO, the h SS vertices are given by tree tree  tree H hh = 2λ H hh ,  H A A = 2λ H A A ,  H H + H − = λ H H + H − .

(4.74)

The decay rates for H → SS are given by

LO (H → SS) = δ S

   tree 2  H SS  32πm H

 λ1/2

m 2S

m 2S

, m 2H m 2H

 ,

(4.75)

where δ S = 2 (1) for V = H ± (h, A). We do not consider three-body decays, H → SS ∗ .

4.2.5 Decay Rates for H → γγ, Zγ, gg The additional CP-even Higgs boson decays into γγ, Z γ, and gg at the one-loop level. Similar to the SM-like Higgs boson, the loop-induced H VV  vertices are given by μν   1H VV  +  H VV  ( p1 , p2 , q) = g μν 

μ

p1ρ p2σ 3 p1ν p2 2    H VV  + i μνρσ  H VV  , 2 q q2

(4.76)

where the momentum assignments are the same as those in Fig. 4.1. From the WardTakahashi identity, we have   2H VV  ( p12 , p22 , q 2 ) = −

2q 2   1  ( p 2 , p 2 , q 2 ), q 2 − p12 − p22 H VV 1 2

(4.77)

66

4 Synergy Between Direct Searches at the LHC and Precision Tests …

for the on-shell photon and gluon. When we neglect the effects of the CP violation, we have   3H VV  ( p12 , p22 , q 2 ) = 0.

(4.78)

At the LO, the decay rate for H → γγ is given by 2 1  1   H γγ (0, 0, m 2H )  16πm H √ 2 2 m3   f 2G F αem λH + H − H H  H H H = Nc Q 2f κ Hf I FH (τ f ) − I S (τ H ± ) , κV IW (τW ) + 3 256π v

LO (H → γγ) =

f

(4.79) with τi = m 2H /(4m i2 ) (i = W, f, H ± ). The loop functions, IiH , are given in Eqs. (4.22), (4.23) and (4.24). The QCD corrections for quark-loop contributions up to NLO are included by replacing I FH as [11, 12, 19] 

 4τq μ2 αs (μ) , I FH (τq ) → I FH (τq ) 1 + C1H (τq ) + C2H (τq ) ln π m 2H

(4.80)

The NLO QCD correction in the MS scheme is given by [12, 19], 3 H θ(1 + θ + θ2 + θ3 ) 108Li4 (θ) + 144Li4 (−θ) − 64 [Li3 (θ) + Li3 (−θ)] ln θ I F (τ )C1H (τ ) = − 4 (1 − θ)3

1 4 + 14Li2 (θ) ln2 θ + 8Li2 (−θ) ln2 θ + ln θ + 4ζ(2) ln2 θ + 16ζ(3) ln θ + 18ζ(4) 12

2 θ(1 + θ) − 32Li + (−θ) + 16Li (−θ) ln θ − 4ζ(2) ln θ 3 2 (1 − θ)2 −

4θ(7 − 2θ + 7θ)2 8θ(3 − 2θ + 3θ)2 Li3 (θ) + Li2 (θ) ln θ 4 (1 − θ) (1 − θ)4

+

θ(3 + 25θ − 5θ2 + 3θ3 ) 3 2θ(5 − 6θ + 5θ2 ) ln (1 − θ) ln2 θ + ln θ 4 (1 − θ) 3(1 − θ)5

+

4θ(1 − 14θ + θ2 ) 12θ2 12θ(1 + θ) 20θ ζ(3) + ln2 θ − ln θ − , 4 (1 − θ) (1 − θ)4 (1 − θ)3 (1 − θ)2

(4.81)

with √ 1 − τ −1 − 1 . θ ≡ θ(τ ) = √ 1 − τ −1 + 1

(4.82)

For analytic continuation, we take τ → τ + i0. The analytic expression for I FH C2H (τ ) is given by [19]

4.2 Decays of the Additional CP-Even Higgs Boson

67

 3 H 3  I F (τ )C2H (τ ) = 2 τ + (τ − 2) f (τ ) − (τ − 1)τ f  (τ ) , 4 τ

(4.83)

where f (τ ) is given in Eq. (4.25), while f  (τ ) is given by  arcsin(√τ ) 

f (τ ) =

τ (1−τ ) √ 1 2 1−τ −1

 ln

(τ ≤ 1),

√ 1+√1−τ −1 1− 1−τ −1

− iπ



(4.84)

(τ > 1).

At the LO, the decay rate for H → Z γ is given by   2 m2 1  1   H Z γ (m 2Z , 0, m 2H ) 1 − 2Z  8πm H mH  3 √ 3 2 2 m 2G F αem m H = 1 − 2Z 128π 3 mH  2  f λ H + H − H g Z c2W H   H (τ , λ ) + H × κVH JW Nc Q f v f κ H JS (τ H ± , λ H ± ) , W W f J F (τ f , λ f ) − v 2

LO (H → Z γ) =

f

(4.85) with τi = 4m i2 /m 2H (i = W, f, H ± ) and λi = 4m i2 /m 2Z . The loop functions, JiH , are given in Eqs. (4.28), (4.29) and (4.30), respectively. We do not include QCD corrections for H → Z γ. The decay rate for H → gg including the NNLO QCD corrections is given by 

αs (μ) (1) Eφ + (H → gg) = LO (φ → gg) 1 + π



αs (μ) π

2

 E φ(2)

,

(4.86)

where LO (H → gg) is given by LO (H → gg) =

√ 2 2 2G F αs2 (μ)m 3H   H H 1  1   κq Iq (τq ) ,  H gg (0, 0, m 2H ) =   3 16πm H 128π q

(4.87)

and I FH is given in Eq. (4.23). The NLO QCD correction E (1) H can be decomposed as [12]   virt  real  E (1) H = E H m t →∞ + E H m t →∞ + E H ,

(4.88)

where the first and second terms are given in Eqs. (4.37) and (4.38), respectively. The last term E H vanishes in the large top-quark mass limit. It can be decomposed into the following three parts, ggg

gq q¯

E H = E virt H + E H + N f E H .

(4.89)

68

4 Synergy Between Direct Searches at the LHC and Precision Tests …

The analytic expression for E virt H is given by [12, 19]  H H κ I (τ ) B1H (τq ) + B2H (τq ) ln q q f F  H H q κ f I F (τq )

 E virt H = Re

m 2H m q2

 −

11 , 2

(4.90)

with 3 H θ(1 + θ)2 72H (1, 0, −1, −; θ) + 6 ln (1 − θ) ln3 θ − 36ζ(2)Li2 (θ) I F (τ )B1H (τ ) = 4 (1 − θ)4 − 36ζ(2) ln (1 − θ) ln θ − 108ζ(3) ln (1 − θ) − 64Li3 (−θ)

+ 32Li2 (−θ) ln θ − 8ζ(2) ln θ 36θ(5 + 5θ + 11θ2 + 11θ3 ) 36θ(5 + 5θ + 7θ2 + 7θ3 ) Li4 (θ) − Li4 (−θ) 5 (1 − θ) (1 − θ)5  4(1 + θ)θ(23 + 41θ2 )  + Li3 (θ) + Li3 (−θ) ln θ 5 (1 − θ)

−

−

2 3 2θ(5 + 5θ + 23θ2 + 23θ3 ) 2 θ − 16θ(1 + θ + θ + θ ) Li (−θ) ln2 θ Li (θ) ln 2 2 (1 − θ)5 (1 − θ)5

+

θ(5 + 5θ − 13θ2 − 13θ3 ) 4 θ(1 + θ − 17θ2 − 17θ3 ) ln θ + ζ(2) ln2 θ 24(1 − θ)5 (1 − θ)5

+

2θ(11 + 11θ − 43θ2 − 43θ3 ) 36θ(1 + θ − 3θ2 − 3θ3 ) ζ(3) ln θ + ζ(4) (1 − θ)5 (1 − θ)5

−

2θ(55 + 82θ + 55θ2 ) 2θ(51 + 74θ + 51θ2 ) Li3 (θ) + Li2 (θ) ln θ 4 (1 − θ) (1 − θ)4

+

θ(6 + 59θ + 58θ2 + 33θ3 ) 3 θ(47 + 66θ + 47θ2 ) ln (1 − θ) ln2 θ + ln θ 4 (1 − θ) 3(1 − θ)5

2θ(31 + 34θ + 31θ2 ) 3θ(3 + 22θ + 3θ2 ) 2 ζ(3) + ln θ 4 (1 − θ) 2(1 − θ)4 94θ 24θ(1 + θ) ln θ − , − (1θ)3 (1 − θ)2  3 H 6  I (τ )B2H (τ ) = 2 τ + (τ − 2) f (τ ) − (τ − 1)τ f  (τ ) , 4 F τ +

(4.91) (4.92)

where H (1, 0, −1, −; θ) is the Harmonic Polylogarithm function, and θ and f (τ ) ggg are given by Eqs. (4.82) and (4.25), respectively. Those for the real emissions E H gq q¯ and E H are given in Ref. [12], which are expressed in the form with a double integral with respect to phase space variables. According to Ref. [12], the factor E H is dominantly determined by the contribution from the virtual gluon loop gq q¯ ggg E virt H . Therefore, we neglect the contributions from E H and E H . For NNLO QCD corrections, we include the expression in the heavy top-mass limit, which is given in Eq. (4.39).

4.3 Decays of the CP-Odd Higgs Boson

69

4.3 Decays of the CP-Odd Higgs Boson In this section, we give the analytic expressions for the decay rates of the additional CP-odd Higgs boson including higher-order QCD corrections.

4.3.1 Decay Rates for A → f f¯ When 2m f ≤ m A , the additional CP-odd Higgs boson decays into a pair of fermions. At the LO, the A f f¯ vertex is given by  AP,tree f f = i2I f

mf ζf. v

(4.93)

The decay rates for A → f f¯ are given by   f (A → f f¯) = LO (A → f f¯) 1 + QCD,A ,

(4.94)

where the decay rate at LO is given by LO (A → f f¯) =

Ncf

m A  P,tree 2 1/2   λ 8π A f f



m 2f m 2f , m 2A m 2A

 ,

(4.95)

Similarly to the h → f f¯ decay, we use the pole masses of leptons in  AP,tree f f , while P,tree we replace a quark mass in  A f f to a running mass m q (μ) evaluated at μ = m A . We use the pole masses of leptons and quarks in λ(x, y). For decays into quarks, we include QCD corrections. In the MS scheme, the NNLO QCD corrections are given by q,MS

QCD,A = qq +  A ,

(4.96)

where qq is given in Eqs. (4.7) with μ = m A . The second term,  A , is given by κA  A = tA κq



αs (m A ) π

 2  2 1 2 m q (m A ) m 2A 3.83 − ln 2 + ln . 6 mt m 2A

(4.97)

We take N f = 5 for m A ≤ m t , while N f = 6 for m t < m A . The top-quark mass effects become significant for the decay into a pair of top quarks. In the on-shell scheme, the NLO QCD corrections are given by [14, 15]

70

4 Synergy Between Direct Searches at the LHC and Precision Tests …

t,OS QCD,A =

αs (μ) CF π



L(β A ) 1 3 2 4 2 (7 − β − (19 + 2β + 3β ) ln ρ + ) , A A A A βA 8 16β 3A (4.98)

with β A , ρ A and L(β A ) are given in Eqs. (4.45)–(4.47), respectively. Similarly to the H → t t¯ decay, we use the linear interpretation function,     pole t,MS 2 MS ¯ (A → t t¯) = R 2 LO (A → t t¯) 1 + t,OS QCD,A + (1 − R )LO (A → t t ) 1 + QCD,A ,

(4.99) with R = 2m t /m A . When m t + m W ≤ m A < 2m t , the additional CP-odd Higgs boson decays into a pair of on-shell and off-shell top quarks, A → tt ∗ → tbW . The decay rate is given by [17, 18] +

+

(A → tt ∗ ) = 2

yt

yb dyt

yt−

dyb yb−

d ¯ − ), (A → t t¯∗ → t bW dyt dyb

(4.100)

with the scaling variables yt (b) = 1 − 2E t (b) /m A . The kinematic boundaries are given in Eqs. (4.50)–(4.52) with xi = m i /m A (i = t, b, W ). The Dalitz density is given by 3G 2F m 2t ζ 2f m 3A 0 d ¯ −) = (A → t t¯∗ → t bW , dyt dyb 64π 3 yt2 + κt γt

(4.101)

where 0 is given by 0 = −yt3 + (1 − κt + κW − yb )yt2 + [2κt − yb (κt − 2κW )]yt + (κt − κW )(κt + 2κW ),

(4.102) with κi = m i2 /m 2A (i = t, b, W ).

4.3.2 Decay Rates for A → Zh, Z H, W ± H ∓ When m Z + m φ ≤ m A , the CP-odd Higgs boson decays into a pair of CP-even Higgs bosons φ (= h, H ) and Z boson. The decay rate for A → Z φ (φ = h, H ) is given by

4.3 Decays of the CP-Odd Higgs Boson

LO (A → Z φ) =

71

   tree 2  AZ φ  m 3

A

16π

m 2Z

 λ3/2

m 2φ m 2Z , m 2A m 2A

 ,

(4.103)

where  tree AZ φ are given by mZ mZ cβ−α ,  tree sβ−α . AZ H = −ig H AZ = −i v v

 tree AZ h = −igh AZ = i

(4.104)

When m φ ≤ m A < m Z + m φ , the CP-odd Higgs boson decays into a pair of onshell CP-even Higgs boson φ (= h, H ) and off-shell Z boson. The decay rate for A → φZ ∗ → φ f f¯ is given by [17] x+

+

(A → φZ ∗ ) =

x f

 f¯ dx f

x −f

dx f¯ x −f¯

d (A → φZ ∗ → φ f f¯), dx f dx f¯

(4.105)

with the scaling variables x f ( f¯) = 2E f ( f¯) /m A . We neglect the masses of the finalstate fermions. The kinematic boundaries are given by 0 ≤ x f ≤ 1 − κφ , κφ , 1 − x f − κφ ≤ x f¯ ≤ 1 − 1− xf

(4.106) (4.107)

with κφ = m 2φ /m 2A . The Dalitz density is given by   7 10 2 d 9G F  tree 2 2 40 4 ∗ ¯ − sW + sW (A → φZ → φ f f ) = √  AZ φ  m Z m A dx f dx f¯ 12 9 27 16 2π 3 (1 − x f )(1 − x f¯ ) − κφ × , (4.108) (1 − x f − x f¯ − κφ + κ Z )2 + κ Z γ Z where κ Z = m 2Z /m 2A and γ Z =  2Z /m 2A . When m W + m H ± ≤ m A , the CP-odd Higgs boson decays into a pair of charged Higgs bosons H ± and W ± bosons. The decay rate for A → W ± H ∓ is given by  2  tree   2 2   AW ∓ H ±  m 3 A 3/2 m H ± m W ± ∓ , λ , LO (A → W H ) = 8π m 2W m 2A m 2A

(4.109)

where  tree AW ∓ H ± = ig AH ± W ∓ = −i

mW . v

(4.110)

72

4 Synergy Between Direct Searches at the LHC and Precision Tests …

Fig. 4.2 Momentum assignment for the renormalized AV1 V2 vertex

p1

V1

p2

V2

q A

When m H ± ≤ m A < m W + m H ± , the CP-odd Higgs boson decays into a pair of on-shell charged Higgs bosons H ± and off-shell W ± boson. The decay rate for A → H ± W ∓∗ → H ± f f¯ is given by x+

+

(A → H ± W ∓∗ ) = 2

x f

 f¯ dx f

x −f

dx f¯ x −f¯

d (A → H + W −∗ → H + f f¯ ), dx f dx f¯ (4.111)

with the scaling variables x f ( f¯ ) = 2E f ( f¯ ) /m A . The charge conjugate final state, H − f¯ f  , doubles the partial decay width. We neglect the masses of the final state fermions. The kinematic boundaries are given by 0 ≤ x f ≤ 1 − κH ± , κH ± , 1 − x f − κ H ± ≤ x f¯ ≤ 1 − 1− xf

(4.112) (4.113)

with κ H ± = m 2H ± /m 2A . The Dalitz density is given by 2 d 9G F  tree  (A → H + W −∗ → H + f f¯ ) = √  AW ∓ H ±  m 2W m A dx f dx f¯ 16 2π 3 (1 − x f )(1 − x f¯ ) − κ H ± × , (1 − x f − x f¯ − κ H ± + κW )2 + κW γW (4.114) 2 where κW = m 2W /m 2A and γW = W /m 2A .

4.3.3 Decay Rates for A → W W, Z Z, Zγ, γγ, gg The AV1 V2 vertices (V1 , V2 ) = (W ± , W ∓ ), (Z , Z ), (Z , γ) and (γ, γ) are loopinduced. They can be expressed as

4.3 Decays of the CP-Odd Higgs Boson

73

μν ν μ 2,1PI μνρσ   AV1 V2 ( p1 , p2 , q) = g μν  1,1PI p1ρ p2σ  3,1PI AV1 V2 + p1 p2  AV1 V2 + i

AV1 V2 ,

(4.115)

where p1 and p2 denote the incoming four-momentum of the gauge bosons Vi (i = 1, 2), and q is the outgoing four-momentum of the CP-odd Higgs boson (see Fig. 4.2). When we neglect the effects of the CP violation, the 1PI vertices satisfy 2,1PI  1,1PI AV1 V2 =  AV1 V2 = 0.

(4.116)

In the CP-conserving 2HDMs, radiative corrections consist of fermion loops only [20]. This is because the bosonic sector separately conserves P and C symmetries [21], and AV1 V2 couplings are prohibited. On the other hand, the fermion sector breaks them, and  3,1PI AV1 V2 is induced. The AV1 V2 vertices are given in Appendix D.3.10. The LO decay rates for A → Vi V j are given by [20] m 3A λ3/2 (A → Vi V j ) = 32π(1 + δi j )



m 2Vi m 2V j , m 2A m 2A



   3,1PI 2  AVi V j  .

(4.117)

For A → γγ, we include the QCD corrections. The decay rate for A → γγ is given by 2 m 3   G F αem Q 2 κA I FA (τ ) √ A  128 2π 3    2   q  4τq μ2 αs (m A ) 2 A A A A  , Nc Q q κq I F (τq ) 1 + + C1 (τq ) + C2 (τq ) ln  2 π m q A

(A → γγ) =

(4.118) with τ f = m 2A /(4m 2f ). The loop function I FA (τ f ) is given by I FA (τ f ) =

2 f (τ f ), τf

(4.119)

where f (τ ) is given in Eq. (4.25). The NLO QCD correction in the MS scheme is given by [12, 19], I FA (τ )C1A (τ ) = −

 θ(1 + θ2 ) 128  Li3 (θ) + Li3 (−θ) ln θ 72Li4 (θ) + 96Li4 (−θ) − 3 (1 − θ)3 (1 + θ)

16 1 4 8 32 28 Li (θ) ln2 θ + Li (−θ) ln2 θ + ln θ + ζ(2) ln2 θ + ζ(3) ln θ + 12ζ(4) 3 2 3 2 18 3 3 56 64 32 θ − Li (θ) − Li (−θ) + 16Li2 (θ) ln θ + Li (−θ) ln θ + 3 3 3 3 3 2 (1 − θ)2

20 8 8 + ln (1 − θ) ln2 θ − ζ(2) ln θ + ζ(3) 3 3 3



+

+

2θ(1 + θ) 3 ln θ, 3(1 − θ)3

(4.120)

74

4 Synergy Between Direct Searches at the LHC and Precision Tests …

where θ is given by Eq. (4.82). The analytic expression for I FA C2A (τ ) is given by [19] I FA (τ )C2A (τ ) =

 2 f (τ ) − τ f  (τ ) . τ

(4.121)

The decay rate A → gg is given by 

αs (m A ) (1) (A → gg) = LO (A → gg) 1 + EA + π



αs (m A ) π

2

 E (2) A

, (4.122)

where the decay rate at LO is given by √ 2 2G F αs2 m 3A   A A  κ I (τ ) LO (A → gg) =  q F q  , 128π 3 q

(4.123)

with the loop function I FA (τ ) defined in Eq. (4.119). The NLO QCD correction E (1) A in (4.122) can be decomposed as [12]   virt  real  E (1) A = E A m t →∞ + E A m t →∞ + E A .

(4.124)

The first and second terms on the right-hand side denote the contributions from virtual gluon loops and those from real gluon emissions in the large top-mass limit, respectively. At μ = m A , these are given by [12]  

 E virt A 

m t →∞

= 6,

 

 E real A 

m t →∞

=

73 7 − Nf. 4 6

(4.125)

The third term, E A , vanishes in the large top-mass limit. It can be decomposed as ggg

gq q¯

E A = E virt A + E A + N f E A .

(4.126)

The analytic expression for E virt A is given by [19]  A 2I ζ I (τ ) B1A (τq ) + B2A (τq ) ln f f q q F  A q 2I f ζ f I F (τq )

 E virt A = Re

m 2A m q2

 − 6,

(4.127)

4.4 Decays of the Charged Higgs Bosons

75

with θ 48H (1, 0, −1, −; θ) + 4 ln (1 − θ) ln3 θ − 24ζ(2)Li2 (θ) (1 − θ)2 128 220 Li3 (θ) − Li3 (−θ) − 24ζ(2) ln (1 − θ) ln θ − 72ζ(3) ln (1 − θ) − 3 3 64 94 ln (1 − θ) ln2 θ2 + 68Li2 (θ) ln θ + Li2 (−θ) ln θ + 3

3 124 16 ζ(3) + 3 ln2 θ − ζ(2) ln θ + 3 3

I FA (τ )B1A (τ ) =

24θ(5 + 11θ2 ) 24θ(5 + 7θ2 ) Li4 (θ) − Li4 (−θ) 3 (1 − θ) (1 + θ) (1 − θ)3 (1 + θ)  8θ(23 + 41θ2 )  Li + (θ) + Li (−θ) ln θ 3 3 3(1 − θ)3 (1 + θ) −

−

32θ(1 + θ2 ) 4θ(5 + 23θ2 ) 2θ− (θ) ln Li Li2 (−θ) ln2 θ 2 3(1 − θ)3 (1 + θ) 3(1 − θ)3 (1 + θ)

+

θ(5 − 13θ2 ) 2θ(1 − 17θ2 ) ln4 θ + ζ(2) ln2 θ 3 36(1 − θ) (1 + θ) 3(1 − θ)3 (1 + θ)

+

4θ(11 − 43θ2 ) 24θ(1 − 3θ2 ) 2θ(2 + 11θ) 3 ln θ, ζ(3) ln θ + ζ(4) + 3 3 3(1 − θ) (1 + θ) (1 − θ) (1 + θ) 3(1 − θ)3

(4.128)  4 f (τ ) − τ f  (τ ) , I FA (τ )B2A (τ ) = τ

(4.129)

where θ, f (τ ) and f  (τ ) are given by Eqs. (4.82), (4.25) and (4.84), respectively. As similar to the H → gg decay, We neglect the contributions from the real emissions gq q¯ ggg E A and E A because they are subdominant [12]. The NNLO QCD correction E (2) A in (4.122) is evaluated in the large top-mass limit. It is given by [22] 495 51959 363 (2) − ζ(2) − ζ(3) + N f EA = 96 8 8 

+ N 2f

 251 1 − ζ(2) . 216 6



m 2 (m A ) 5 473 11 + ζ(2) + ζ(3) − ln t 2 − 8 2 4 mA



(4.130)

4.4 Decays of the Charged Higgs Bosons In this section, we give the analytic expressions for the decay rates of the charged Higgs boson including higher-order QCD corrections.

76

4 Synergy Between Direct Searches at the LHC and Precision Tests …

4.4.1 Decay Rates for H ± → f f¯  When 2m f ≤ m H ± , the charged Higgs bosons decay into a pair of fermions. The decay rates for H ± → f f¯ are given by  2  2  2 √2G m ± m m  f f F H   λ1/2 (H ± → f f¯ ) = Ncf V f f   , 8π m 2H ± m 2H ±      m 2f + m 2f   2 2  RR 2 2 LL m 1 +  + m 1 +  × 1− ζ ζ   f f QCD f f QCD m 2H ±  

m f m f RL . (4.131) + 4 2 m f m f  ζ f ζ f  1 + QCD m H± Similarly to the h → f f¯ decay, we use the pole masses of leptons in the Yukawa coupling, while we replace a quark mass to a running mass m q (μ) evaluated at μ = m H ± . We use the pole masses of leptons and quarks in λ(x, y). For decays into quarks, we include QCD corrections. In the MS scheme, the NNLO QCD QCD H± QCD corrections are expressed by the same factor, QCD R R =  L L =  R L = q , and it is given by [3–7], qq  ,MS QCD,H ±

αs (m H ± ) 23 CF + = π 4



αs (m H ± ) π

2 (35.94 − 1.36N f ).

(4.132)

The top-quark mass effects become significant for the decay into quarks including the top quark. In the on-shell scheme, the NLO QCD corrections are given by [15, 23] RR QCD = CF

αs (m H ± ) + αs (m H ± ) + αs (m H ± ) − LL RL = CF = CF qq  , QCD q  q , QCD qq  , π π π

(4.133)

where ( 23 − μ f − μ f  )λ(μ f , μ f  ) + 5μ f μ f  3 − 2μ f + 2μ f  μf 9 +  = + + ln (x f x f  ) + B f f  , ln ff 4 4 μf 2λ1/2 (μ f , μ f  )(1 − μ f − μ f  ) −  = 3 + ff

μf −μf 2

ln

μf μf

+

λ(μ f , μ f  ) + 2(1 − μ f − μ f  ) 2λ1/2 (μ f , μ f  )

ln (x f x f  ) + B f f  ,

(4.134) (4.135)

with x f = 2μ f /[1 − μ f − μ f  + λ1/2 (μ f , μ f  )]. The B f f  function is symmetric under i ↔ j, and it is given by

4.4 Decays of the Charged Higgs Bosons

77

1−μf −μf  B f f  = 1/2 4Li2 (x f x f  ) − 2Li2 (−x f ) − 2Li2 (−x f  ) + 2 ln x f x f  ln (1 − x f x f  ) λ (μ f , μ f  )   xf xf  ln x f x f  − ln x f ln (1 + x f ) − ln x f  ln (1 + x f  ) − 4 ln (1 − x f x f  ) + 1− xf xf     xf λ(μ f , μ f  ) + μ f − μ f  ln x f  + (i ↔ j) . (4.136) + ln (1 + x f ) − λ(μ f , μ f  ) 1+xf

When a down-type quark is massless, m f  → 0, we have  B f f  = −2Li2

−μ f 1−μf



ln (1 − μ f ) + ln (1 − μ f ) ln − 1−μf



μf 1−μf

 +

μf μf ln , 1−μf 1−μf

(4.137) 2μ2f − 7μ f + 3

μf ln 4(1 − μ f ) 1−μf     μf −μ f ln (1 − μ f ) − 2Li2 + ln (1 − μ f ) ln − . 1−μf 1−μf 1−μf

+f f  =

9 + 4

(4.138)

Similarly to the H → t t¯ decay, we use the linear interpretation function, ¯ = R 2  pole (H ± → t D) ¯ + (1 − R 2 ) MS (H ± → t D), ¯ (H ± → t D)

(4.139)

with R = (m t + m D )/m H ± (D = d, s, b). We regard the d and s quarks as massless ¯ In the evaluation of  OS (H ± → t D), ¯ we and use Eq (4.138) for  pole (H ± → t D). use the pole masses in the Yukawa coupling, while the running masses are used in ¯  MS (H ± → t D). When m W + m b + m D ≤ m H ± < m t + m D , the charged Higgs bosons decay into a pair of on-shell down-type quark and an off-shell top quark. the decay rate for H + → t ∗ D → W + bD is given by [17] +

(H + → t ∗ D) =

+

x D¯

xb dx D¯

− xD ¯

dxb xb−

d (H + → t ∗ D → W + bD), dx D¯ dxb

(4.140)

with the scaling variables x D¯ (b) = 2E D¯ (b) /m H ± . We neglect the masses of the downtype quarks. The kinematic boundaries are given by 0 ≤ x D¯ ≤ 1 − κW , κW 1 − x D¯ − κW ≤ xb ≤ 1 − 1 − x D¯

(4.141) (4.142)

78

4 Synergy Between Direct Searches at the LHC and Precision Tests …

with κW = m 2W /m 2H ± . The Dalitz density is given by 3G 2F m 4t ζt2 m 2W d (H + → t ∗ D → W + bD) = d x D¯ d xb 32π 3 m H ± (1 − xb )(1 − x D¯ ) + κW (2xb + 2x D¯ − 3 + 2κt ) × , κW [(1 − x D¯ − κt )2 + κt γt ]

(4.143) where κt = m 2t /m 2H ± and γt = t2 /m 2H ± .

4.4.2 Decay Rates for H ± → W ± φ (φ = h, H, A) When m φ + m W ≤ m H ± , the charged Higgs boson decays into a pair of neutral Higgs bosons φ (= h, H, A) and W ± bosons. The decay rates for H ± → W ± φ are given by

±

±

LO (H → W φ) =

 2  tree   H ± W ∓ φ  m 3 ± 16π

H m 2W

 λ

3/2

m 2φ

m2 , 2W 2 m H± m H±

 ,

(4.144)

where  tree H ± W ∓ φ = −igφH ± W ∓ .

(4.145)

When m φ ≤ m H ± < m φ + m W , the charged Higgs bosons decay into a pair of onshell neutral Higgs bosons and off-shell W boson. The decay rates for H ± → W ±∗ φ are given by x+

+

(H + → W +∗ φ) =

x f

 f¯ dx f

x −f

dx f¯ x −f¯

d (H + → W +∗ φ → f f¯ φ), (4.146) dx f dx f¯

with the scaling variables x f ( f¯ ) = 2E f ( f¯ ) /m H ± . We neglect the masses of the final state fermions. The kinematic boundaries are given by 0 ≤ x f ≤ 1 − κW , κW . 1 − x f − κW ≤ x f¯ ≤ 1 − 1− xf

(4.147) (4.148)

4.4 Decays of the Charged Higgs Bosons

79

with κW = m 2W /m 2H ± . The Dalitz density is given by 2 d 9G F  tree  (H ± → W ±∗ h → f f  h) = √  H ± W ∓ φ  m 2W m H ± 3 d x1 d x2 16 2π (1 − x f )(1 − x f¯ ) − κφ × , (1 − x f − x f¯ − κφ + κW )2 + κW γW (4.149) 2 where κφ = m 2φ /m 2H ± and γW = W /m 2H ± .

4.4.3 Decay Rates for H ± → W ± Z, W γ The H ± V W ∓ vertices (V = Z , γ) are loop-induced. They can be expressed as [24, 25] μν μ 2   H ± V W ∓ ( p1 , p2 , q) = g μν   1H ± V W ∓ + p1ν p2   H ± V W ∓ + i μνρσ p1ρ p2σ   3H ± V W ∓ , (4.150)

where p1 and p2 denote the incoming four-momentum of W ± and V , and q is the outgoing four-momentum of the charged Higgs bosons (see Fig. 4.3). We have assumed that the external gauge bosons are on-shell. From the Ward-Takahashi identity, we have   1H ± γW ∓ = − p1 · p2  2H ± γW ∓ ,

(4.151)

for the on-shell photon. The form factors are written by the loop contributions, i,loop i   iH ± V W ∓ =  H ± V W ∓ =  i,1PI H ± V W ∓ + δ H ± V W ∓ .

(4.152)

The 1PI diagram contributions are given in Appendix D.3.7. The counterterms originated from G ± V W ∓ vertices are given by

Fig. 4.3 Momentum assignment for the renormalized H + V μ W −ν vertex

p1

Wμ+

q H

+

p2

Vν

80

4 Synergy Between Direct Searches at the LHC and Precision Tests …

1 gG ± V W ∓ δ Z G + H − , 2 = δ 3H ± V W ∓ = 0,

δ 1H ± V W ∓ =

(4.153)

δ 2H ± V W ∓

(4.154)

where the wave function renormalization counterterm δ Z G + H − is given in Eq. (5.77). The decay rates for H ± → W ± Z are given by [24, 25] 1 (H → W Z ) = λ1/2 16πm H ± ±

±



m 2W m 2Z , m 2H ± m 2H ±

  2  2      MT T  + M L L  , (4.155)

where     2 2 m 4 ± 2  m 2W m 2Z  3      , (4.156)  H ± Z W ∓  , MT T  = 2 1H ± Z W ∓  + H λ 2 2 2 m H± m H±     2  2 m 4H ±  m 2H ±  m 2W + m 2Z m 2W m 2Z   1 2   . 1 − + ,   λ M L L  = ± ∓ ± ∓  H ZW H ZW  2 4m 2W m 2Z m 2H ± m 2H ± m 2H ±

(4.157)

The decay rates for H ± → W ± γ are given by (H ± → W ± γ) =

m 3H ± 32π

 1−

m 2W m 2H ±

3  2  2   2     H ± γW ∓  +  3H ± γW ∓  .

(4.158)

4.5 Total Decay Widths and Decay Branching Ratios In this section, we discuss the total widths and the decay branching ratios of the Higgs bosons in the 2HDMs. We describe the behavior of the total widths and the branching ratios in cases with the alignment limit, sβ−α = 1 and without taking the alignment limit, sβ−α = 0.995. We assume that the masses of the additional Higgs bosons m  ≡ m H = m A = m H ± are degenerate with M. We take m  to be m  = 200 GeV or 800 GeV with scanning tan β as 0.5 < tan β < 50. We note that m  = 200 GeV for Type-II and Type-Y are already excluded by the constraint from the flavor physics without depending on tan β [26, 27]. Parameter regions with tan β  2 are also excluded in Type-I and Type-X. However, we dare to omit these constraints to compare results among the four types of 2HDMs. In Fig. 4.4, we show the total decay widths of Higgs bosons as a function of tan β. The solid and dashed lines show the results of m  = M = 200 GeV and m  = M = 800 GeV, respectively. We take sβ−α = 1 in the top panels, sβ−α = 0.995 with cβ−α < 0 in the middle panels, and sβ−α = 0.995 with cβ−α > 0 in the bottom panels.

4.5 Total Decay Widths and Decay Branching Ratios

81

Fig. 4.4 Total widths of h, H , A and H ± as a function of tan β in Type-I, Type-II, Type-X and Type-Y 2HDMs. Solid and dashed lines show results of m  = M = 200 GeV and m  = M = 800 GeV, respectively. We take sβ−α = 1 in the top panels, sβ−α = 0.995 with cβ−α < 0 in the middle panels, and sβ−α = 0.995 with cβ−α > 0 in the bottom panels

For the h decays with sβ−α = 1, the couplings with fermions and weak gauge bosons coincide with those in the SM at tree level, and the total decay width does not depend on tan β. When sβ−α = 0.995 with cβ−α < 0, the total width increases as tan β increases in Type-II, Type-X and Type-Y. This is because h → bb¯ is enhanced by tan β in Type-II and Type-Y, while h → τ τ¯ is enhanced in Type-II and Type-X. For the heavy Higgs bosons H, A and H ± , the total widths vary in the both cases of sβ−α = 1 and sβ−α = 1. While those in Type-I monotonically decrease except for H with a mass of 800 GeV, there appears a dip at a certain value of tan β for each additional Higgs boson in Type-II, Type-X, and Type-Y. In Fig. 4.5, we show the tan β dependence of the decay branching ratios of the Higgs bosons in the alignment limit, sβ−α = 1, with m  = M = 200 GeV. For the h decays, there is no tan β dependence of all the decay modes, since all the scaling factors κhX are unity when sβ−α = 1. For the H and A decays, we note that the squared scaling factors of the fermion couplings for H and A are common, and these are simply expressed by ζ f parameters; i.e., |κ Hf |2 = |κ Af |2 = ζ 2f when sβ−α = 1. In the case with m  = M = 200 GeV, the decay mode into a pair of the top quarks does not open for the H (A) decays. Therefore, the main decay mode of H and A is H (A) → bb¯ for tan β > 1 except for Type-X with tan β  1. For Type-X, the main decay mode is H (A) → τ τ¯ for tan β  1 due to the tan β enhancement for the leptonic decays, which also causes H (A) → μμ¯ with about 0.3% for tan β 

82

4 Synergy Between Direct Searches at the LHC and Precision Tests …

Fig. 4.5 Decay branching ratios for h, H , A and H ± as a function of tan β in the case of m  = M = 200 GeV and sβ−α = 1

4. The difference from H decays appears in the decay into gg due to the QCD corrections. BR(A → gg) is relatively larger than BR(H → gg) because the NLO QCD correction is more significant than H → gg. This mainly comes from the fact that the NLO QCD correction is more significant than H → gg, although the expressions at the LO are also different between H and A. Apart from the neutral Higgs bosons, H ± decays into a pair of the top quark and a down-type quark. While the decay into tb is the main decay mode for Type-I and Type-Y, the decay into τ ν can also be a main decay mode with tan β  5 for Type-II and Type-X due to the tan β enhancement. In Fig. 4.6, the branching ratios in the case of sβ−α = 1 and m  = M = 800 GeV are shown as a function of tan β. For h decays, the behavior does not change much from the case with m  = M = 200 GeV. This is because most of the decay rates do not depend on the mass of the additional Higgs bosons at the LO, while H ± contributes to h → γγ and Z γ. Main difference from Fig. 4.5 is appearance of H (A) → t t¯. It dominates the branching ratios of H and A for Type-I with any value

4.5 Total Decay Widths and Decay Branching Ratios

83

Fig. 4.6 Decay branching ratios for h, H , A and H ± as a function of tan β in the case of m  = M = 800 GeV and sβ−α = 1

of tan β. On the other hand, for Type-II, Type-X, and Type-Y, H (A) → t t¯ can be dominant only for tan β  10 due to the cot β suppression. Next, we discuss the cases with sβ−α = 1. In these cases, the branching ratios of h(H ) → f f¯ vary with a sign of cβ−α . In addition, additional decay modes, such as H → V V (V = W, Z ) H → hh, A → Z h and H ± → W ± h, shall appear for decays of the additional Higgs bosons. In Fig. 4.7, we show the tan β dependence of the decay branching ratios of Higgs bosons with sβ−α = 0.995 and m  = M = 200 GeV. For decays of h and H , predictions with cβ−α < 0 and cβ−α > 0 are separately plotted by solid and dotted lines, respectively. For h, one can see clear tan β dependence for all the decay modes. The branching ratio for h → bb¯ increases by tan β. For H , H → Z Z and H → W + W − , which are proportional to m 3H as seen in Eq. (4.56), can dominate. Whereas, H → bb¯ (τ τ¯ ) overcomes them for large tan β in Type-II and Type-Y (Type-II and Type-X). Similarly, A → h Z ∗ and H ± → hW ±∗ can be sizable in Type-I with large tan β.

84

4 Synergy Between Direct Searches at the LHC and Precision Tests …

Fig. 4.7 Decay branching ratios for h, H , A and H ± as a function of tan β in the case of m  = M = 200 GeV and sβ−α = 0.995. Solid lines show results of cβ−α < 0 and dotted lines are those of cβ−α > 0

In Fig. 4.8, we show the decay branching ratios with sβ−α = 0.995 and m  = M = 800 GeV. While H → t t¯ is the main decay mode with low tan β, H → hh, which is proportional to m 3H , can dominate. The branching ratio for H → t t¯ and H → hh are close to 0 at tan β ∼ 10 and tan β ∼ 16, respectively, when cβ−α > 0. This is because the scaling factor κtH and the scalar coupling λ H hh vanish at those values of tan β. We note that the value of λ H hh depends on the value of M. Therefore, the decay width for H → hh can change if we consider the non-degenerate case; i.e., m  = M.

4.6 Direct Searches at the LHC In this section, we present constraints on the parameter space in the 2HDMs from direct searches for heavy Higgs bosons with the LHC Run-II data.

4.6 Direct Searches at the LHC

85

Fig. 4.8 Decay branching ratios for h, H , A and H ± as a function of tan β in the case of m  = M = 800 GeV and sβ−α = 0.995. Solid lines show results of cβ−α < 0 and dotted lines are those of cβ−α > 0

We here briefly summarize the procedure of how we obtain the constraints on the parameters in the 2HDMs from model-independent analyses for heavy Higgs boson searches at the LHC. First, we compute production cross sections of heavy neutral Higgs bosons, φ = H and A, for the gluon-fusion process ( pp → φ) and the ¯ at the bottom-quark associated (or bottom-quark annihilate) process ( pp → φ(bb)) NNLO in QCD by using Sushi-1.7.0 [37, 38]. For the charged Higgs boson production pp → t H ± , we use the values given at the NLO QCD by the Higgs cross section working group [39], based on Refs. [40–43]. Second, we calculate the decay branching ratios of the Higgs bosons including higher-order QCD corrections. Finally, we compute the production cross sections times the branching ratios for each parameter point for each search channel at the LHC listed in Table 4.1, and we compare with the upper limits at 95% CL with 36 fb−1 data to obtain the constraints. We assume that the masses of the heavy Higgs bosons are degenerate with M, i.e. m  = M. We consider the value of sβ−α as 1 and 0.995 both for cβ−α < 0 and cβ−α > 0. We note that we use the expected upper limits, not the observed ones,

86

4 Synergy Between Direct Searches at the LHC and Precision Tests …

Table 4.1 List of constraints used in this study from direct searches for heavy Higgs bosons at the 13 TeV LHC Constrained quantity Applicable mass region (GeV) References σ(φ) × BR(φ → τ τ ) σ(φ(bb)) × BR(φ → τ τ ) σ(φ(bb)) × BR(φ → bb) σ(φ) × BR(φ → tt) σ(H ) × BR(H → hh) × BR(h → bb)2 σ(H ) × BR(H → W W ) σ(H ) × BR(H → Z Z ) σ(A) × BR(A → Z h) × BR(h → bb) σ(A(bb)) × BR(A → Z h) × BR(h → bb) σ(t H ± ) × BR(H ± → tb) σ(t H ± ) × BR(H ± → τ ν)

200 < m  200 < m  450 < m  400 < m  260 < m 

< 2000 < 2000 < 1400 < 5000 < 2000

Figure 7a in [28] Figure 7b in [28] Figure 8 in [29] Figure 14 in [30] Figure 9a in [31]

200 < m  < 2000 200 < m  < 2000 200 < m  < 2000

Figure 5 in [32] Figure 6 in [33] Figure 6a in [34]

200 < m  < 2000

Figure 6b in [34]

200 < m  < 2000 200 < m  < 2000

Figure 8 in [35] Figure 8a in [36]

from the LHC analyses for the HL-LHC projection, which is discussed in Sect. 4.7. Although we use the ATLAS data listed in Table 4.1, similar limits have been reported by the CMS experiment [44–49]. We also note that, although new analyses with full Run-II data (139 fb−1 ) are available for some channels; e.g., φ → τ τ¯ [50], we use the upper limit with 36 fb−1 data for a fair comparison with the other channels.

4.6.1 Production Cross Sections for the Additional Higgs Bosons Before discussing the current constraints on the parameter space from direct searches, we present production rates for the heavy Higgs bosons at the 13 TeV LHC. In Fig. 4.9, we show the cross sections for H via the gluon fusion process (left two columns) and via the bottom-quark associated process (right two columns) on the m  –tan β plane. We only show the cases in the Type-I and Type-II 2HDMs since the leptonic sector is irrelevant to these production processes, and the productions in Type-X and Type-Y are the same as in Type-I and Type-II, respectively. The value of sβ−α is set to be 1 and 0.995 with cβ−α < 0 in the top and bottom panels. For the gluon-fusion process, shown in the left two columns in Fig. 4.9, the Higgs bosons are produced via quark loops. Therefore, the difference in the down-type quark Yukawa couplings between Type-I and Type-II leads to different dependence on the model parameters. In Type-I, where the top-quark loop is entirely dominant, the larger tan β leads the smaller the cross-section for a fixed mass. The threshold

Fig. 4.9 Production cross sections for H at the 13 TeV LHC on the m  –tan β plane. Panels in two columns from the left (right) show the production via the gluon fusion (the bottom-quark associated) in the Type-I and Type-II 2HDMs, where the value of sβ−α is set to be 1 and 0.995 with cβ−α < 0 in the top and bottom panels. The cross sections are shown with different colors from blue to red, corresponding to from 10−5 to 102 pb

4.6 Direct Searches at the LHC 87

88

4 Synergy Between Direct Searches at the LHC and Precision Tests …

Fig. 4.10 Production cross sections for H at the 13 TeV LHC on the m  –tan β plane, where the value of sβ−α is set to be 1 and 0.995 with cβ−α > 0 in the top and bottom panels

Fig. 4.11 Production cross sections for A at the 13 TeV LHC on the m  –tan β plane. Panels from the left to the right show the production via the gluon fusion in the Type-I and Type-II, and via the bottom-quark associated process in the Type-I and Type-II, respectively

enhancement of the top-quark loop appears at m  ∼ 2m t . In Type-II, the top-quark loop is dominant for small tan β, while the production via the bottom-quark loop becomes dominant for large tan β due to the tan β enhancement on bottom-Yukawa coupling. The sβ−α dependence of the cross sections is very small for small tan β. In the large tan β region, on the other hand, the cross sections for a fixed mass tend to be larger than those in sβ−α = 1. The production via the bottom-quark associated process, shown in the right two columns in Fig. 4.9, is entirely subdominant in Type-I, while that becomes dominant for large tan β in Type-II. In Fig. 4.10, similar to Fig. 4.9, but for cβ−α > 0, we show the production rates. In this case, the cross sections show a peculiar tan β dependence except for the bassociate process in Type-II. This is because both the top and the bottom Yukawa in Type-I and the top Yukawa in Type-II vanish for a certain sβ−α and tan β; e.g., tan β ∼ 10 for the sβ−α = 0.995 case. In Fig. 4.11, we show the production rates for A. The production processes are the same as those for H , shown in Figs. 4.9 and 4.10, namely the gluon fusion process (left two columns) and the bottom-quark associated process (right two columns). Different from H , the production rates only depend on tan β because of the Yukawa structure. The global parameter dependence of the cross sections via the gluon fusion is similar to that for H with sβ−α = 1, but the production rate for A is slightly larger

4.6 Direct Searches at the LHC

89

Fig. 4.12 Production cross sections for the charged Higgs boson H ± at the 13 TeV LHC on the m  –tan β plane in the Type-I (left) and Type-II (right)

than that for H at each point on the m  –tan β plane. The parameter dependence of the cross sections via the bottom-quark annihilation is as same as for H with sβ−α = 1. In Fig. 4.12, we show the production rates for H ± . At the LHC, H ± are mainly produced in association with a top quark via gb → t H ± for m H ± > m t . Similar to the productions for A, the cross-section only depends on tan β. For a fixed mass, in Type-I, the larger tan β leads to a smaller production rate. In Type-II, the larger tan β leads to a smaller production rate up to tan β ∼ 7 similar to the Type-I case. However, for tan β  7, the production rate becomes larger for larger tan β due to the tan β enhancement of the bottom-Yukawa coupling. We here briefly mention other production processes of the heavy Higgs bosons. Although we assume m H ± = m H in this study, if the H/A → H ± W ∓ decay is kinematically allowed, the production via gg → H → H ± W ∓ can be comparable with that via gb → t H ± [51–54]. Heavy Higgs bosons are also produced in electroweak processes such as H A, H ± h/H/A, and H + H − [55, 56], as well as in loop induced processes such as H ± W ∓ [57] and H + H − [58].

4.6.2 Constraints from the Direct Searches at the LHC Run-II with 36 fb−1 Data We here discuss constraints on the parameter space in each 2HDM from direct searches for heavy Higgs bosons with the 36 fb−1 LHC Run-II data listed in Table 4.1. In Fig. 4.13, we show the exclusion regions at 95 % CL on the m  –tan β plane in the Type-I, Type-II, Type-X, and Type-Y 2HDMs. The value of sβ−α is set to be 1 and 0.995 with cβ−α < 0 in the top and bottom panels. The shaded regions with dotted, solid, and dashed border lines denote the exclusion regions for H , A, and H ± , respectively.

Fig. 4.13 Regions on the m  –tan β plane excluded at 95% CL in the Type-I, Type-II, Type-X, and Type-Y 2HDMs (from the left to the right panels) via direct searches for heavy Higgs bosons with the 36 fb−1 LHC Run-II data. The value of sβ−α is set to be 1 and 0.995 with cβ−α < 0 in the top and bottom panels

90 4 Synergy Between Direct Searches at the LHC and Precision Tests …

Fig. 4.14 Regions on the m  –tan β plane excluded at 95% CL in the Type-I, Type-II, Type-X, and Type-Y 2HDMs (from the left to the right panels) via direct searches for heavy Higgs bosons with the 36 fb−1 LHC Run-II data. The value of sβ−α is set to be 1 and 0.995 with cβ−α > 0 in the top and bottom panels

4.6 Direct Searches at the LHC 91

92

4 Synergy Between Direct Searches at the LHC and Precision Tests …

The shape of each exclusion region can be understood by the production rates, shown in Figs. 4.9, 4.10, 4.11 and 4.12, times branching ratios, depicted in Figs. 4.5, 4.6, 4.8. We explain the characteristics of the exclusion regions for A, H , and H ± in order. Regarding the CP-odd Higgs boson A; • For large tan β, exclusion regions appear in Type-II and Type-Y, where the production through the bottom-quark loop, as well as the bottom-quark associated production, becomes dominant due to the tan β enhancement. • A → τ τ¯ is significant for m A < 2m t . In addition, it becomes significant in Type-II for large tan β. Although A → τ τ¯ becomes the main decay mode for large tan β in Type-X as similar to Type-II, the production rate is too small to constrain parameter regions.1 For 2m t < m A , A → t t¯ becomes relevant for small tan β in all types of 2HDMs.2 • Since A → Z h only appears for the non-alignment case, the exclusion regions are remarkably different for the alignment and non-alignment cases. Regarding the additional CP-even Higgs boson H ; • The production rate via the gluon fusion for H is smaller than that for A, as we have mentioned. Moreover, in the non-alignment case, the branching ratios of H → f f¯ for low tan β are smaller than those for A due to H → W W, Z Z , which do not occur for A at LO. Therefore, the constraints are slightly weaker than those for ¯ → τ τ¯ , A. We do not present the exclusions explicitly for the H → τ τ¯ , H (bb) ¯ → bb¯ and H → t t¯ channels. H (bb) • For 2m W,Z < m H and/or 2m h < m H in the non-alignment case, the peculiar decay modes are H → W W , H → Z Z , and H → hh. The excluded region from H → W W is similar to H → Z Z but smaller, so we do not show it explicitly. • The decay rate of H → hh depends on M 2 as we have mentioned in Sect. 4.5. For a non-degenerate case M = m  , the exclusion region from the H → hh channel can be different from that for the degenerate case. Regarding to the charged Higgs boson H ± ; • In the low tan β region (tan β  5), H ± → tb is dominant for all types of 2HDMs. Therefore, the exclusions of the low-mass and low-tan β region from the H ± → tb channel are almost the same for all the panels. • In the large tan β region, the constraint from H ± → τ ν becomes significant in Type-II. Although H ± → τ ν becomes the main decay mode for large tan β in Type-X as similar to Type-II, the production rate is too small to constrain parameter regions. • We note that, in Type-II and Type-Y, there are independent constraints from flavor observables on the mass of charged Higgs bosons, m H ±  800 GeV, as mentioned in Sect. 3.1. Four-τ final states from the pp → H A process in Type-X can be relevant [59]. Because there is no specific analysis for the spin-0 resonance in the t t¯ final state in the LHC Run-II, we use the limit for Z  [30], which is valid from the Run-I 8 TeV analysis [60].

1 2

4.7 Combined Results of Direct Searches at the HL-LHC …

93

Table 4.2 95% CL allowed range of tan β for the case with cβ−α < 0 (cβ−α > 0) from the signal strength of the discovered Higgs boson at the LHC [61] sβ−α Type-I Type-II Type-X Type-Y 0.995

tβ ≥ 0.54 (tβ ≥ 0.54)

– 0.43 ≤ tβ ≤ 4.1 – (–) (0.57 ≤ tβ ≤ 1.6) (0.42 ≤ tβ ≤ 4.2)

The hyphen denotes no allowed region

In Fig. 4.14, we show the exclusion regions for the cβ−α > 0 case. A remarkable difference from the cβ−α < 0 case is that the constraints for H in the non-alignment case are much weaker for around tan β ∼ 7 − 10 due to the strong suppression of the production rates. Although BR(A → Z h) does not depend on the sign of cβ−α , the exclusion regions for cβ−α > 0 in Type-II and Y are smaller than those for cβ−α < 0. ¯ depends on the sign of cβ−α . It has a singular behavior This is because BR(h → bb) for cβ−α > 0, as shown in Figs. 4.6 and 4.8. Before closing this section, we briefly discuss the constraints from the signal strength for the discovered Higgs boson measured at the LHC Run-II experiment. It provides independent constraints on the parameter space from those given by the direct searches discussed in this section. Measurements of the signal strength set constraints on the scale factors, κ, defined in Sect. 3.1, which can be translated into those on sβ−α and tan β. In Table 4.2, we summarize the 95% CL allowed range of tan β in the 2HDMs with fixed values of sβ−α . The κ values are extracted from Ref. [61], which are presented in Table 2.4. We see that except for the Type-I 2HDM, it gives severe constraints on the value of stan β. This is because κb and/or κτ can sizably differ from unity in the Type-II, Type-X, and Type-Y 2HDMs even for the approximate alignment case.

4.7 Combined Results of Direct Searches at the HL-LHC and Precision Tests at the ILC In this section, we discuss combined studies on direct searches for heavy Higgs bosons at the HL-LHC and precision measurements of the SM-like Higgs boson couplings at the ILC. The current measurements of the κ values at the LHC Run-II and the expected 1σ accuracies of their measurements at the HL-LHC and the ILC are summarized in Table 2.4 in Sect. 2.4.1. For direct searches of heavy Higgs bosons, we perform sensitivity projections to −1 the HL-LHC with √ 3000 fb of integrated luminosity. We rescale currently expected sensitivities√ by 3000/36 ∼ 9.1. We also perform a further rescaling of the sensitivity from s = 13 TeV to 14 TeV by taking into account the ratio of the signal cross sections, σ(m  )14 TeV /σ(m  )13 TeV . In the latter rescaling, we assume that signal and background increase by the same amount from 13 TeV to 14 TeV. We note that this assumption might be conservative, particularly for the high-mass region.

94

4 Synergy Between Direct Searches at the LHC and Precision Tests …

The detailed projection with systematic uncertainties for the φ → τ τ¯ channel was performed in the report for the HL-LHC [62]. It has shown that we have a higher sensitivity for m   1200 GeV compared with our sensitivity projections. In Fig. 4.15, we show the expected exclusion regions at 95% CL in the TypeI, Type-II, Type-X, and Type-Y 2HDMs (from the left to the right panels) on the m  –tan β plane. We analyze the same search channels for the current constraints in Figs. 4.13 and 4.14. The value of sβ−α is set to be 1 and 0.995 with cβ−α < 0 in the top and bottom panels. The shaded regions with solid, dotted, and dashed border lines denote the exclusion regions for A, H , and H ± , respectively. The global picture of the exclusion regions from the direct searches is similar to the current exclusions, but much wider parameter regions are excluded. Especially for the non-alignment case, the large portion in the parameter space is excluded via the H → hh and A → Z h channels, which set the lower-mass limit with a given tan β. We note that we here assume the masses of the additional Higgs bosons are degenerate with M. The exclusion region from the H → hh channel can be changed depending on M 2 . On the other hand, the exclusion region from A → Z h is not changed even in the case with m 2 = M 2 . The black-shaded regions in the non-alignment case are excluded from the constraints of perturbative unitarity and vacuum stability. Here, we assume that deviation on the SM-like Higgs boson couplings with weak bosons is measured as κhV = sβ−α = 0.995 ± 0.0038,

(4.159)

where the precision at the ILC250 is adopted. Within the above 1σ range for the κhV , we scan the value of M 2 with M 2 > 0. We note that, for sβ−α = 0.995, the alignment limit is included within the 2σ error. For cβ−α < 0, the third condition of the vacuum stability bound given in Eq. (3.63) sets an upper limit on M. A bound on m  is slightly larger than that on M almost independent of the value of tan β. For example, for sβ−α = 0.995, M  680 and 780 GeV is excluded for m  = 800 and 1000 GeV, respectively. We note that the value of m 2 − M 2 becomes larger for a larger value of m  to satisfy the vacuum stability bound. On the other hand, the unitarity bound excludes a larger difference between M 2 and m 2 since it leads to strong Higgs self-couplings. Therefore, for a fixed value of sβ−α and tan β, there is a critical value of m 2 , above which we have no value of M 2 satisfying both of the unitarity and the vacuum stability bound. The constraint on m  becomes stronger when the value of tan β differs from unity. This is because the λ1 or λ2 coupling becomes large, and the unitarity bound sets more   severe constraints on M 2 − m 2 . We find that, for the non-alignment case, the entire parameter space we consider can be explored by combining the constraints from the direct searches for the additional Higgs bosons at the HL-LHC, especially for H → hh and A → Z h, and that from the precision measurements of the SM-like Higgs boson couplings at the ILC. In Fig. 4.16, we show the expected exclusion regions for the cβ−α > 0 case. Different from the cβ−α < 0 case, a narrow parameter region remains even for low

Fig. 4.15 Regions on the m  –tan β plane expected to be excluded at 95% CL in the Type-I, Type-II, Type-X, and Type-Y 2HDMs (from the left to the right panels) via direct searches for heavy Higgs bosons at the HL-LHC and precision measurements of the Higgs boson couplings at the ILC. The value of sβ−α is set to be 1 and 0.995 with cβ−α < 0 in the top and bottom panels

4.7 Combined Results of Direct Searches at the HL-LHC … 95

Fig. 4.16 Regions on the m  –tan β plane expected to be excluded at 95% CL in the Type-I, Type-II, Type-X, and Type-Y 2HDMs (from the left to the right panels) via direct searches for heavy Higgs bosons at the HL-LHC and precision measurements of the Higgs boson couplings at the ILC. The value of sβ−α is set to be 1 and 0.995 with cβ−α > 0 in the top and bottom panels

96 4 Synergy Between Direct Searches at the LHC and Precision Tests …

References

97

m  under the constraints from the direct searches. A remarkable difference from the cβ−α > 0 case is the excluded region via H → hh for tan β > 4. This is because of the singular behaviors of the branching ratios for H , shown in Fig. 4.8, and BR(→ hh) is suppressed around tan β ∼ 7 − 10 in all types of 2HDMs. In addition, there are the singular behaviors of the production cross section for H and the branching ratios for h around tan β ∼ 7 − 10, shown in Figs. 4.10 and 4.8. There is also a remarkable difference in the constraints from the measurement of the SM-like Higgs boson couplings. In a low tan β region, the condition of the vacuum stability bound, λ2 > 0, sets an upper limit on M 2 for a fixed value of m 2 as M 2  m 2 . As tan β gets large, a constraint by λ2 > 0 becomes milder. When tan β exceeds a certain value, an upper limit on M 2 is almost fixed to be m 2 due to the condition λ1 > 0 instead of λ2 > 0. This tan β dependence on the vacuum stability bound provides the two peaks of the upper limit on m  as shown in Fig. 4.16. As a result, some parameter regions remain uncovered by both the HL-LHC and the ILC250, and more data and/or precision are needed to be explored. We here comment on the case, where the common mass of the additional Higgs bosons m  and M are not degenerate. The partial decay width for H → hh depends on the value of M, and the exclusion region for H might be changed if M = m  . However, most of the parameter regions excluded by H → hh are also excluded by A → Z h, which does not depend on the value of M. Therefore, our main conclusion does not change even if m  = M. In summary, we find that a wide range of parameter space in the 2HDMs can be explored by combining the direct searches for the additional Higgs bosons at the HL-LHC and the precision measurements of the SM-like Higgs boson couplings at the ILC. If we observed any deviations for the Higgs boson couplings at the ILC, we can deduce an upper bound on the mass scale of the additional Higgs bosons. Therefore, we would be able to find additional Higgs bosons at the HL-LHC or reject a certain type of new physics model.

References 1. Aiko M, Kanemura S, Kikuchi M, Mawatari K, Sakurai K, Yagyu K (2021) Probing extended Higgs sectors by the synergy between direct searches at the LHC and precision tests at future lepton colliders. Nucl Phys B 966:115375 2. Braaten E, Leveille JP (1980) Higgs boson decay and the running mass. Phys Rev D 22:715 ¯ to order ααs . Phys Lett B 751:442– 3. Mihaila L, Schmidt B, Steinhauser M (2015) (H → bb) 447 4. Gorishnii SG, Kataev AL, Larin SA, Surguladze LR (1990) Corrected three loop QCD correction to the correlator of the quark scalar currents and tot (H 0 → Hadrons). Mod Phys Lett A 5:2703–2712 5. Gorishnii SG, Kataev AL, Larin SA, Surguladze LR (1991) Scheme dependence of the next to next-to-leading QCD corrections to Gamma(tot) (H0 –> hadrons) and the spurious QCD infrared fixed point. Phys Rev D 43:1633–1640 6. Chetyrkin KG, Kwiatkowski A (1996) Second order QCD corrections to scalar and pseudoscalar Higgs decays into massive bottom quarks. Nucl Phys B 461:3–18

98

4 Synergy Between Direct Searches at the LHC and Precision Tests …

7. Larin SA, van Ritbergen T, Vermaseren JAM (1995) The Large top quark mass expansion for Higgs boson decays into bottom quarks and into gluons. Phys Lett B 362:134–140 8. Keung W-Y, Marciano WJ (1984) Higgs scalar decays: H –> W+- X. Phys Rev D 30:248 9. Kniehl BA (1994) Higgs phenomenology at one loop in the standard model. Phys Rep 240:211– 300 10. Denner A (1993) Techniques for calculation of electroweak radiative corrections at the one loop level and results for W physics at LEP-200. Fortsch Phys 41:307–420 11. Dawson S, Kauffman R (1994) QCD corrections to Higgs boson production: nonleading terms in the heavy quark limit. Phys Rev D 49:2298–2309 12. Spira M, Djouadi A, Graudenz D, Zerwas PM (1995) Higgs boson production at the LHC. Nucl Phys B 453:17–82 13. Chetyrkin KG, Kniehl BA, Steinhauser M (1997) Hadronic Higgs decay to order αs4 . Phys Rev Lett 79:353–356 14. Drees M, Hikasa K (1990) Heavy quark thresholds in Higgs physics. Phys Rev D 41:1547 15. Djouadi A (2008) The Anatomy of electro-weak symmetry breaking. II. The Higgs bosons in the minimal supersymmetric model. Phys Rep 459:1–241 16. Djouadi A, Kalinowski J, Spira M (1998) HDECAY: a program for Higgs boson decays in the standard model and its supersymmetric extension. Comput Phys Commun 108:56–74 17. Djouadi A, Kalinowski J, Zerwas PM (1996) Two and three-body decay modes of SUSY Higgs particles. Z Phys C 70:435–448 18. Barradas E, Diaz-Cruz JL, Gutierrez A, Rosado A (1996) Three body decays of Higgs bosons in the MSSM. Phys Rev D 53:1678–1683 19. Harlander R, Kant P (2005) Higgs production and decay: analytic results at next-to-leading order QCD. JHEP 12:015 20. Gunion John F, Haber Howard E, Kao Chung (1992) Searching for the CP odd Higgs boson of the minimal supersymmetric model at hadron supercolliders. Phys Rev D 46:2907–2917 21. Gunion JF, Haber HE (1986) Higgs Bosons in supersymmetric models. 2. Implications for phenomenology. Nucl Phys B 278:449 [Erratum: Nucl Phys B 402:569–569 (1993)] 22. Chetyrkin KG, Kniehl BA, Steinhauser M, Bardeen WA (1998) Effective QCD interactions of CP odd Higgs Bosons at three loops. Nucl Phys B 535:3–18 23. Djouadi A, Gambino P (1996) QCD corrections to Higgs boson self energies and fermionic decay widths. Phys Rev D 51:218–228 [Erratum: Phys Rev D 53:4111 (1996)] 24. Peyranere MC, Haber HE, Irulegui P (1991) H+− → W+− gamma and H+− → W+− Z in two Higgs doublet models. 1. The Large fermion mass limit. Phys Rev D 44:191–201 25. Kanemura S (2000) Enhancement of loop induced H ± W ∓ Z 0 vertex in two Higgs doublet model. Phys Rev D 61:095001 26. Misiak M, Steinhauser M (2017) Weak radiative decays of the B meson and bounds on M H ± in the two-Higgs-doublet model. Eur Phys J C 77(3):201 27. Misiak M, Rehman A, Steinhauser M (2020) Towards B → X s γ at the NNLO in QCD without interpolation in mc . JHEP 06:175 28. Aaboud M et al (2018) Search for additional heavy neutral Higgs and Gauge bosons in the √ Ditau final state produced in 36 fb−1 of pp collisions at s = 13 TeV with the ATLAS detector. JHEP 01:055 29. Aad G et al (2020) Search for heavy neutral Higgs bosons produced in association with b√ quarks and decaying into b-quarks at s = 13 TeV with the ATLAS detector. Phys Rev D 102(3):032004 30. Aaboud Morad et al (2018) Search for heavy particles √ decaying into top-quark pairs using lepton-plus-jets events in proton–proton collisions at s = 13 TeV with the ATLAS detector. Eur Phys J C 78(7):565 ¯ ¯ 31. Aaboud M et al (2019) Search √ for pair production of Higgs bosons in the bbbb final state using proton-proton collisions at s = 13 TeV with the ATLAS detector. JHEP 01:030 32. Aaboud M et al (2018) √ Search for heavy resonances decaying into W W in the eνμν final state in pp collisions at s = 13 TeV with the ATLAS detector. Eur Phys J C 78(1):24

References

99

33. Aaboud M et al (2018) Search for heavy√ZZ resonances in the + − + − and + − ν ν¯ final states using proton–proton collisions at s = 13 TeV with the ATLAS detector. Eur. Phys. J. C 78(4):293 34. Aaboud M et al (2018) Search for heavy resonances decaying √ into a W or Z boson and a Higgs boson in final states with leptons and b-jets in 36 fb−1 of s = 13 TeV pp collisions with the ATLAS detector. JHEP 03:174 [Erratum: JHEP 11:051 (2018)] 35. Aaboud M et al (2018) Search for charged Higgs bosons decaying into top and bottom quarks √ at s = 13 TeV with the ATLAS detector. JHEP 11:085 ± 36. Aaboud M et al (2018) Search for charged Higgs bosons decaying via H ± → √τ ντ in the −1 τ +jets and τ +lepton final states with 36 fb of pp collision data recorded at s = 13 TeV with the ATLAS experiment. JHEP 09:139 37. Harlander RV, Liebler S, Mantler H (2013) SusHi: a program for the calculation of Higgs production in gluon fusion and bottom-quark annihilation in the Standard Model and the MSSM. Comput Phys Commun 184:1605–1617 38. Harlander R, Liebler S, Mantler H (2017) SusHi Bento: beyond NNLO and the heavy-top limit. Comput Phys Commun 212:239–257 39. de Florian D (2016) c sections: 4. Deciphering the nature of the Higgs sector, vol 2 40. Berger E, Han T, Jiang J, Plehn T (2005) Associated production of a top quark and a charged Higgs boson. Phys Rev D 71:115012 41. Dittmaier S, Kramer M, Spira M, Walser M (2011) Charged-Higgs-boson production at the LHC: NLO supersymmetric QCD corrections. Phys Rev D 83:055005 42. Flechl M, Klees R, Kramer M, Spira M, Ubiali M (2015) Improved cross-section predictions for heavy charged Higgs boson production at the LHC. Phys Rev D 91(7):075015 43. Degrande C, Ubiali M, Wiesemann M, Zaro Marco (2015) Heavy charged Higgs boson production at the LHC. JHEP 10:145 44. Khachatryan V et al √ (2017) Search for heavy resonances decaying to tau lepton pairs in protonproton collisions at s = 13 TeV. JHEP 02:048 45. Sirunyan AM et al (2018) Search for beyond the standard model Higgs bosons decaying into √ a bb pair in pp collisions at s = 13 TeV. JHEP 08,:113 46. Sirunyan AM et al (2017) Search for tt resonances √ in highly boosted lepton+jets and fully hadronic final states in proton-proton collisions at s = 13 TeV. JHEP 07,:001 47. Sirunyan AM et al (2018) Search for a massive resonance √ decaying to a pair of Higgs bosons in the four b quark final state in proton-proton collisions at s = 13 TeV. Phys Lett B 781:244–269 48. Khachatryan Vardan et al (2017) Search for heavy resonances decaying into a vector boson and a Higgs boson in final states with charged leptons, neutrinos, and b quarks. Phys Lett B 768:137–162 49. Sirunyan√ AM et al (2019) Search for a heavy pseudoscalar boson decaying to a Z and a Higgs boson at s = 13 TeV. Eur Phys J C 79(7):564 50. Aad G et al (2020) Search for heavy Higgs bosons decaying into two tau leptons with the √ ATLAS detector using pp collisions at s = 13 TeV. Phys Rev Lett 125(5):051801 51. Dicus DA, Hewett JL, Kao C, Rizzo TG (1989) W+− H−+ production at hadron colliders. Phys Rev D 40:787 52. Barrientos Bendezu AA, Kniehl BA (1999) W+− H−+ associated production at the large hadron collider. Phys Rev D 59:015009 53. Moretti S, Odagiri K (1999) The Phenomenology of W+− H−+ production at the large hadron collider. Phys Rev D 59:055008 54. Akeroyd AG et al (2017) Prospects for charged Higgs searches at the LHC. Eur Phys J C 77(5):276 55. Moretti S (2002) Pair production of charged Higgs scalars from electroweak gauge boson fusion. J Phys G 28:2567–2582 56. Alves A, Plehn T (2005) Charged Higgs boson pairs at the CERN LHC. Phys Rev D 71:115014 57. Brein O, Hollik W, Kanemura S (2001) The MSSM prediction for W+− H−+ production by gluon fusion. Phys Rev D 63:095001

100

4 Synergy Between Direct Searches at the LHC and Precision Tests …

58. Brein O, Hollik W (2000) Pair production of charged MSSM Higgs bosons by gluon fusion. Eur Phys J C 13:175–184 59. Kanemura S, Tsumura K, Yokoya H (2012) Multi-tau-lepton signatures at the LHC in the two Higgs doublet model. Phys Rev D 85:095001 60. Aad G et al (2015) A search for tt resonances using lepton-plus-jets events in proton-proton √ collisions at s = 8 TeV with the ATLAS detector. JHEP 08:148 61. Aad G et al (2020) Combined measurements √ of Higgs boson production and decay using up to 80 fb−1 of proton-proton collision data at s = 13 TeV collected with the ATLAS experiment. Phys Rev D 101(1):012002 62. Cepeda M et al (2019) Report from working Group 2: Higgs physics at the HL-LHC and HE-LHC. CERN Yellow Rep Monogr 7:221–584

Part II

Next-to-Leading-Order Electroweak Corrections in Higgs Physics

Chapter 5

Renormalization

In this chapter, we review the renormalization of the ultraviolet (UV) divergences in higher-order calculations. Before moving to the detail of the renormalization scheme, we give a qualitative discussion here. We separate the parameters in the bare Lagrangian into the renormalized parameters and counterterms g B = Z g g = g + δg.

(5.1)

The finite renormalized parameters are fixed by a set of renormalization conditions, and counterterms absorb the UV divergences in higher-order corrections. In addition, we redefine the bare fields in terms of the renormalized fields. 1/2

φ0 = Z φ φ.

(5.2)

Expanding the renormalization constant as Zi = 1 + δ Zi ,

(5.3)

the Lagrangian is split into a renormalized and counterterm part L(φ0 , g0 ) = L(φ, g) + δL(φ, g, δ Z φ , δg),

(5.4)

which results in all Green functions being finite at a given order.

5.1 Renormalization of the Standard Model In this section, we review the renormalization scheme for the EWSM which has been developed in Refs. [1, 2]. We choose the particle masses and the fine-structure constant as the physical parameters. We separate the bare parameters in the symmetric Lagrangian into the physical parameters and the counterterms. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 M. Aiko, Theoretical Studies on Extended Higgs Sectors Towards Future Precision Measurements, Springer Theses, https://doi.org/10.1007/978-981-99-1324-4_5

103

104

5 Renormalization

5.1.1 Tadpole Renormalization In this section, we discuss the renormalization of the tadpole in the SM based on [3, 4]. We shit the bare VEV as v B → v B + v.

(5.5)

We note that v corresponds to the redefinition of v B , and it is not the counterterm. Then, the bare tadpole term is shifted as Th,B → Th,B + Th , where Th,B and Th are given as   λB 2 vB vB , Th,B = μ2B − 2    v v v 1+ v + Th,B . Th = −λ B v 2B v 1 + vB 2v B vB

(5.6) (5.7)

We choose v B so that the bare tadpole term vanishes, Th,B = 0. Then, the bare parameters satisfy the tree-level relations such as m 2h,B = λ B v 2B , and Th becomes  Th =

−m 2h,B v

v 1+ vB

  v 1+ v. 2v B

(5.8)

We shift the bare Higgs mass and the bare Higgs field as, 1/2

m 2h,B = m 2h + δm 2h , h B = Z hh h = (1 + δ Z hh )1/2 h.

(5.9)

Then, the linear term of h in the Lagrangian becomes 1/2

L ⊃ Th h B = Th Z hh h,

(5.10)

and the renormalized one-point function of h is given by 1/2  h = h1PI + Th Z hh ,

(5.11)

where h1PI is the one-particle irreducible contribution. We determine v so that h does not develop a VEV at each order of perturbation, and it is equivalent to  h = 0. At the one-loop level, we have

v =

h1PI i = i 1PI = m 2h −m 2h h

h

(5.12)

5.1 Renormalization of the Standard Model

105

From this expression, we can interpret terms with v as contributions of tadpoleinserted diagrams, which contain the tadpole connected by the Higgs propagator with zero momentum transfer. We note that v appears for all terms in the Lagrangian that depend on the v B . This results in terms with v to two-point and three-point functions involving the scalars and vector bosons, and fermionic two-point functions. For example, the shift of h Z Z coupling is given by i igh Z Z ,B → igh Z Z ,B + g 2Z v = igh Z Z ,B + i4ghh Z Z v 2 = igh Z Z ,B + ihTad ZZ.

(5.13)

The last term corresponds to the h Z Z diagram with tadpole contribution,

ihTad ZZ =

h

.

(5.14)

5.1.2 Higgs Sector The inverse propagator of the Higgs boson is given by     hh ( p 2 ) , hh ( p 2 ) = i p 2 − m 2h + 

(5.15)

where the renormalized self-energy of the Higgs boson is given by  2  2 2 2  hh ( p 2 ) = 1PI  hh ( p ) + 6λhhh v + δ Z hh p − m h − δm h  2  2 Tad 2 2 = 1PI hh ( p ) + hh + δ Z hh p − m h − δm h ,

(5.16)

where Tad hh is the tadpole inserted self-energy of the Higgs boson. We impose the following on-shell conditions. Re  hh (m 2h ) = 0,

d 2  Re hh ( p ) = 1. 2 dp p2 =m 2h

(5.17)

d  hh ( p 2 ) Re  2 2 = 0. 2 dp p =m h

(5.18)

These are equivalent to  hh (m 2h ) = 0, Re 

106

5 Renormalization

Then, the counterterms δm 2h and δ Z hh are determined as   2 Tad δm 2h = Re 1PI hh (m h ) + hh , d 1PI 2 . δ Z hh = − 2 Re hh ( p ) dp 2 2

(5.19) (5.20)

p =m h

5.1.3 Gauge Sector In the mass-eigenbasis, we shift the bare parameters as m 2W,B = m 2W + δm 2W , m 2Z ,B = m 2Z + δm 2Z , e B = e + δe, v B = v + δv (5.21) The bare gauge fields are shifted as   1 = 1 + δ Z W Wμ± , 2 ⎛

1 1 1 + δZZ δ Z Zγ   ⎜ ⎜ 2 2 Z Bμ 

=⎜ ⎜1 2 A Bμ δsW ⎝ 1+ δ Zγ Z + 2 cW sW ± W Bμ

(5.22) ⎞

2 δsW ⎟  cW sW ⎟ Z μ ⎟ ⎟ Aμ . 1 ⎠ δ Zγ 2

−

(5.23)

These counterterms are related to the counterterms in the weak eigenbasis, a W Bμ = (Z W )1/2 Wμa ,

g B = Z g (Z W )

−3/2

B Bμ = (Z B )1/2 Bμ ,

g,

g B

= Z g (Z B )

(5.24)

−3/2 

g,

(5.25)

and not all of the counterterms in the mass-eigenbasis are independent. When we take δ Z γ and δ Z γ Z as independent parameters, δ Z W and δ Z Z are determined by cW δ Zγ Z , sW 2 c2 − sW δ Z Z = δ Zγ + W δ Zγ Z . cW sW

δ ZW = δ Zγ +

(5.26) (5.27)

2 In addition, from the tree-level relation, sW,B = 1 − m 2W,B /m 2Z ,B , we obtain 2 δsW cW = cW sW sW



δm 2 δm 2Z − 2W 2 mZ mW

 .

(5.28)

5.1 Renormalization of the Standard Model

107

Since wave-function renormalization constants are gauge-dependent, mixing angles defined by wave-function renormalization constants are also gauge-dependent. 2 is gauge-independent since it is defined in terms of the mass counHowever, δsW 2 terterms of the weak gauge bosons. From the tree-level relation, v 2B = m 2W,B sW,B / (π αem,B ), the counterterm for the VEV is given by  2  2 2 − cW δm 2W cW δv δm 2Z δe 1 sW . + 2 −2 = 2 v 2 e sW m 2W sW m 2Z

(5.29)

Therefore, there are only five independent counterterms, δm 2W , δm 2Z , δ Z γ , δ Z γ Z and δe, in the electroweak gauge sector. The inverse propagators of the gauge bosons are given by μν 2   μν V V  = −ig μν ( p 2 − m 2V )δV V  + i  V V  ( p ),

(5.30)

where V, V  = W, Z , γ . The self-energies of gauge bosons are expressed by the sum of the longitudinal polarization part and transverse polarization part,   pμ pν pμ pν L 2 μν T 2  μν   V V  ( p 2 ).    ( p ) = −g + ( p ) +   VV VV p2 p2

(5.31)

The longitudinal parts are irrelevant to the renormalization of physical particles. Thus, we omit the T and L subscript in the following. The renormalized self-energies  V V , are written as of gauge bosons,  2 Tad 2 2 2 2  W W ( p 2 ) = 1PI  W W ( p ) + W W ( p ) − δm W + δ Z W ( p − m W ), 2 Tad 2 2 2 2  Z Z ( p 2 ) = 1PI  Z Z ( p ) +  Z Z ( p ) − δm Z + δ Z Z ( p − m Z ), 2 2  γ γ ( p 2 ) = 1PI  γ γ ( p ) + δ Zγ p ,    2  m 2W δm 2W δm Z 1 2 2 2  Z γ ( p 2 ) = 1PI  p − . m ( p ) − δ Z − − Zγ Zγ 2 Z 2cW sW m 2Z m 2W

We impose the following on-shell conditions. Re  W W (m 2W ) = 0, d 2  Re  ( p ) γγ 2 = 1, 2 dp p =0

Re   Z Z (m 2Z ) = 0,

(5.32)

Re   Z γ (0) = 0.

(5.33)

These are equivalent to  W W (m 2W ) = 0, Re   Z Z (m 2Z ) = 0, Re  d  γ γ ( p 2 )   2 = 0, γ Z (0) = 0. 2 dp p =0

(5.34) (5.35)

108

5 Renormalization

From these renormalization condition, δm 2W , δm 2Z , δ Z γ and δ Z γ Z , are fixed as     1PI 2 2 Tad 2 Tad δm 2W = Re 1PI W W (m W ) + W W , δm Z = Re  Z Z (m Z ) +  Z Z , (5.36) d 1PI 2   ( p ) , (5.37) δ Zγ = − dp 2 γ γ p2 =0  1PI 2 2 Tad  1PI 2 cW  Z Z (m Z ) + Tad W W (m W ) + W W ZZ . (0) + Re − δ Z γ Z = − 2 1PI sW mZ γ Z m 2Z m 2W (5.38) μ

We determine δe so that the renormalized photon-electron-positron vertex  eeγ takes tree-level form at the Tomson limit, μ  eeγ (q 2 = 0, /p 1 = /p 2 = m e ) = ieγ μ .

(5.39)

1PI 1 d δe sW γ Z (0) 1PI 2 =  ( p ) − . 2 e 2 dp 2 γ γ cW m 2Z p =0

(5.40)

We obtain,

In this renormalization scheme, the residue of the gauge boson two-point function  V V (m 2V ), is not unity. Therefore, we need to add the derivative of the self-energy,   2 ±  to each external W and Z boson. Similarly, we need to add  Z γ (m Z ) if there are γ − Z mixing contributions.  In the evaluation of fermionic-loop conributions for 1PI γ γ (0) , we have logarithmic terms which depend on fermion masses. In order to avoid using the light quark masses, we use the following relation,  1PI γ γ (0) =

 1  1PI 2 1 1PI 2  γ γ (m 2Z ) =  (m ) + αem , γ γ (m Z ) −  2 mZ m 2Z γ γ Z

(5.41)

where αem is the shift of the structure constant which can be determined by experiments. In this expression, we can neglect terms depending on the light fermion 2 2 2 2 masses in 1PI γ γ (m Z )/m Z because these are of order m f /m Z . In our calculation, we input the value of the Fermi constant G F . For tree-level contribution, we replace v to G F by using the one-loop level relation, v2 = √

1 2G F (1 − r )

,

(5.42)

where r is obtained by the one-loop corrected muon decay rate as  W W (0) Re  αem r = + 2 m 2W 4π sW



 2 7 − 4sW 2 ln cW . 6+ 2 2sW

(5.43)

5.1 Renormalization of the Standard Model

109

√ For one-loop parts, we use v 2 = 1/ 2G F since the shift by r corresponds to a two-loop contribution.

5.1.4 Fermion Sector In the mass-eigenbasis, we shift the bare masses and bare fermion fields as 1 f 1 f m f,B = m f + δm f , ψ B,L = ψ L + δ Z L ψ L , ψ B,R = ψ R + δ Z R ψ R , (5.44) 2 2 and for ψ B = ψ B,L + ψ B,R , ψB = ψ +

 1 f f δ Z V − δ Z A γ5 ψ, 2

(5.45)

with f

δ ZV =

1 1 f f f f f (δ Z L + δ Z R ), δ Z A = (δ Z L − δ Z R ). 2 2

(5.46)

In this thesis, we do not take into account the mixing of fermions in one-loop calculations. The inverse propagators of the fermions are given by   f f ( p).  f f ( p) = i( /p − m f ) + i 

(5.47)

The self-energies of fermions are composed of vector, axial and scalar parts,  f f ( p) = /p   f f,A + m f   f f,S .  f f,V − /p γ5  

(5.48)

The renormalized self-energies of fermions are given by f 2  f f,V ( p 2 ) = 1PI  f f,V ( p ) + δ Z V ,

(5.49)

 f f,A ( p ) = 

(5.50)

2

2 1PI f f,A ( p )

+

f δ Z A,

f 2 Tad  f f,S ( p 2 ) = 1PI  f f,S ( p ) +  f f,S − δ Z V −

δm f . mf

(5.51)

We impose the following renormalization conditions,  f f,V (m 2f ) 

+

 f f,S (m 2f ) 

= 0,

/p + m f  Re  ( p)u( p) f f 2 p2 − m f

= iu( p) p2 =m 2f

(5.52)

110

5 Renormalization

These are equivalent to  f f,V (m 2 ) +   f f,S (m 2 ) = 0,  f f

d  f f,V ( p 2 )  2 2 = 0, d /p p =m f

d  f f,A ( p 2 )  2 2 = 0. d /p p =m f

(5.53) We renormalize the self-energies of fermions so that the residue of vector and axial parts become unity, but the coefficient of the axial part does not vanish at p 2 = m 2f . f

By these renormalization conditions, three independent counterterms, δm f , δ Z V and f δ Z A are determined as   2 1PI 2 Tad (5.54) δm f = m f 1PI f f,V (m f ) +  f f,S (m f ) +  f f,S ,   d d f 1PI 2 2 1PI 2 1PI 2  ( p ) +  ( p ) δ Z V = − f f,V (m f ) − 2m f , dp 2 f f,V dp 2 f f,S p2 =m 2f p2 =m 2f f

2 2 δ Z A = −1PI f f,A (m f ) + 2m f

d 1PI 2  ( p ) f f,A dp 2

(5.55) p2 =m 2f

.

(5.56)

5.2 Renormalization of the Two-Higgs Doublet Model 5.2.1 Tadpole Renormalization We shift the bare VEVs as v1,B → v1,B + v1,B , v2,B → v2,B + v2,B .

(5.57)

At the one-loop level, the bare tadpole terms are shifted as Th 1 ,B → Th 1 ,B + Th 1 ,B   v2 Th 1 = Th 1 ,B + − m 212 − λ1 v12 v1 + (m 212 − λ345 v1 v2 )v2 , v1 v1 Th 2 ,B → Th 2 ,B + Th 2 ,B   v1 Th 2 = Th 2 ,B + (m 212 − λ345 v1 v2 )v1 + − m 212 − λ2 v22 v2 . v2 v2

(5.58)

(5.59)

We choose v1,B and v2,B so that the bare tadpole terms vanish, Th 1 ,B = Th 2 ,B = 0. We obtain

5.2 Renormalization of the Two-Higgs Doublet Model



Th 1 Th 2



111

⎞

⎛

v 2 2 ⎜ −m 12 v 1 ⎜

− λ345 v1 v2 ⎟  v  1 ⎟ ⎠ v2 v 1 m 212 − λ345 v1 v2 −m 212 − λ2 v22 v   2 v 1 , = − Mh2 Th =Th =0 v2 1 2 −

=⎝

λ1 v12

m 212

(5.60)

where Mh2 is given by Eq. (3.16). Since Mh2 is diagonalized by the angle α, we obtain 

v H vh



 2 −1 = − Meven



TH Th



⎛

TH

⎞

⎜ − m2 ⎟ H ⎟ =⎜ ⎝ Th ⎠ , − 2 mh

(5.61)

2 where Meven is given by Eq. (3.50), and we have defined,



v H vh



 = R(α)

t

v1 v2



 ,

TH Th



 = R(α)

t



Th 1 Th 2

.

(5.62)

Then, the shift of VEVs in the Z 2 basis are written by 

v1 v2



⎛

TH

Th

⎞

⎜ m 2 cα − m 2 sα ⎟ H h ⎟. = −⎜ ⎝ TH Th ⎠ sα + 2 cα m 2H mh

(5.63)

The renormalized one-point functions of h and H are given by  h = h1PI + Th ,   H =  1PI H + TH ,

(5.64)

where h1PI and  1PI H are given in Appendix D.1. We determine vh and v H from  h = 0 and   H = 0. We have

vh =

i i 1PI = −m 2h h

h

, v H =

i i 1PI = −m 2H H

H

.

(5.65)

Similarly to the SM, we can interpret terms with vh and v H as contributions of tadpole-inserted diagrams.

112

5 Renormalization

5.2.2 Higgs Sector In the mass-eigenbasis, we shift the bare parameters as m 2φ,B = m 2φ + δm 2φ , α B = α + δα, β B = β + δβ,

M B2 = M 2 + δ M 2 , (5.66)

where φ = h, H, A, H ± . The bare Higgs fields H, h, G 0 , A, G ± and H ± are shifted as ⎞ 1 1   HB H ⎜ 1 + 2δ Z H H 2δ Z H h ⎟ H 1/2 = Z even =⎝ , ⎠ 1 hB 1 h h δ Z h H 1 + δ Z hh 2 2 ⎞ ⎛ 1 1  0    0 0 0 0 δ Z δ Z 1 + G G G A ⎟ G0 G GB ⎜ 1/2 2 2 = Z odd =⎝ ⎠ A , 1 AB A 1 δ Z AG 0 1 + δ Z A A 2 2 ⎛ ⎞ 1 1  ±   ±  G GB ⎜ 1 + 2δ Z G + G − 2δ Z G + H − ⎟ G ± 1/2 = Z = . ⎝ ⎠ ± 1 HB± H± 1 H± + − + − 1 + δZH H δZH G 2 2 



5.2.2.1





⎛

(5.67)

(5.68)

(5.69)

Relations between the Wave-Functions and Mixing Angles

For the renormalization of the mixing angles, we discuss the relations between the wave functions and mixing angles [5]. The bare CP-even states in the Z 2 basis and those in the mass-eigenbasis are related as 

HB hB



    1/2 1/2 Z h1 h1 Z h1 h2 h 1,B H = R(−αB ) = R(−δα)R(−α) R(α) 1/2 1/2 h 2,B h Z h2 h1 Z h2 h2   H 1/2 , (5.70) = R(−δα) Z˜ even h 

where we have defined

1/2 = R(−α) Z˜ even

1/2

1/2

Z h1 h1 Z h1 h2 1/2 1/2 Z h2 h1 Z h2 h2



R(α) =

1/2 1/2 Z˜ H H Z˜ H h 1/2 1/2 Z˜ h H Z˜ hh

 .

(5.71)

Then, the wave-function renormalization constants for the CP-even states are obtained as

5.2 Renormalization of the Two-Higgs Doublet Model

1/2 Z even

= R(−δα)

1/2 Z˜ H H 1/2 Z˜ h H

113 1/2 Z˜ H h 1/2 Z˜ hh

 .

(5.72)

1/2 We assume that Z˜ even is symmetric and expand Z˜ even as

1 1/2 Z˜ H H = 1 + δ Z H H , 2

1 1/2 Z˜ hh = 1 + δ Z hh , 2

1/2 1/2 Z˜ H h = Z˜ h H = δC h .

(5.73)

We obtain  1/2 Z even

=

1 + 21 δ Z H H δC h + δα δC h − δα 1 + 21 δ Z hh

 ,

(5.74)

Similarly, the wave-function renormalization constants for the CP-odd and charged scalar bosons are obtained as   1 + 21 δ Z G 0 G 0 δC A + δβ , (5.75) Z odd = δC A − δβ 1 + 21 δ Z A A   1 + 21 δ Z G + G − δC H ± + δβ . (5.76) Z± = δC H ± − δβ 1 + 21 δ Z H + H − We note that not all of δ Z G 0 A , δ Z AG 0 , δ Z G + H − and δ Z H + G − are independent. There are only three degrees of freedom, δC A , δC H ± and δβ, since the CP-odd and charged scalar bosons are simultaneously diagonalized by the β angle. We determine δ Z G + H − from the other counterterms as 1 δ Z G + H − = δC H + + δβ 2 1 = (δ Z H + G − + δ Z G 0 A − δ Z AG 0 ) . 2 5.2.2.2

(5.77)

CP-Even Higgs Bosons

The inverse propagators of the CP-even Higgs bosons are given by    H H ( p2 ) ,  H H ( p 2 ) = i p 2 − m 2H +   H h ( p 2 ),  H h ( p2 ) = i   h H ( p ), h H ( p ) = i    2  hh ( p 2 ) , hh ( p ) = i p 2 − m 2h +  2

2

(5.78) (5.79) (5.80) (5.81)

114

5 Renormalization

where the renormalized self-energies of the CP-even Higgs bosons are given by  2  2 Tad 2 2  H H ( p 2 ) = 1PI  (5.82) H H ( p ) +  H H + δ Z H H p − m H − δm H ,     1 1 2 Tad 2 2 2 2  H h ( p 2 ) = 1PI  H h ( p ) + H h + δ Z H h p − m H + δ Zh H p − mh , 2 2 (5.83)     1 1 2 Tad 2 2 2 2  h H ( p 2 ) = 1PI  h H ( p ) + h H + δ Z H h p − m H + δ Z h H p − m h , 2 2 (5.84)  2  2 1PI 2 Tad 2 2  hh ( p ) = hh ( p ) + hh + δ Z hh p − m h − δm h . (5.85)  The tadpole-inserted self-energies are given by Tad H H = 6λ H H H v H + 2λh H H vh ,

(5.86)

Tad Hh Tad hh

(5.87)

=

Tad hH

= 2λh H H v H + 2λ H hh vh ,

= 2λ H hh v H + 6λhhh vh .

(5.88)

We impose the following on-shell conditions. Re hh (m 2h ) = 0, Re  H H (m 2H ) = 0, d d 2 2 Re  ( p ) = 1, Re  ( p ) hh H H 2 2 = 1, d p2 d p2 p2 =m 2h p =m H Re  H h (m 2h ) = 0,

(5.89) (5.90)

Re h H (m 2H ) = 0.

(5.91)

These are equivalent to  H H (m 2H ) = 0,  hh (m 2h ) = 0, Re  Re  d d  hh ( p 2 )  H H ( p 2 ) Re  = 0, Re  2 2 = 0, 2 2 dp dp p2 =m 2h p =m H  H h (m 2h ) = 0, Re 

(5.92) (5.93)

 h H (m 2H ) = 0. Re 

(5.94)

Then, we obtain     2 Tad 2 1PI 2 Tad δm 2h = Re 1PI hh (m h ) + hh , δm H = Re  H H (m H ) +  H H , d d 2 ) 1PI ( p 2 ) δ Z hh = − 2 Re 1PI ( p , δ Z = − Re  HH hh HH 2 2 2 dp d p2

p =m 2H

p =m h

δZ Hh =

2 m 2H − m 2h

2 Tad Re [1PI H h (m h ) +  H h ], δ Z h H =

2 m 2h − m 2H

(5.95) ,

(5.96)

2 Tad Re [1PI h H (m H ) + h H ].

(5.97)

5.2 Renormalization of the Two-Higgs Doublet Model

5.2.2.3

115

CP-Odd Scalar Bosons

The inverse propagators of the CP-odd scalar bosons are given by    A A ( p2 ) ,  A A (q 2 ) = i p 2 − m 2A ) +   AG 0 ( p 2 ),  AG 0 ( p 2 ) = i   G 0 A ( p ), G 0 A ( p ) = i  2

2

(5.98) (5.99) (5.100)

where the renormalized self-energies of the CP-odd Higgs bosons are given by  2  2 Tad 2 2  A A ( p 2 ) = 1PI  A A ( p ) +  A A + δ Z A A p − m A − δm A ,   1 1  AG 0 ( p 2 ) = 1PI 0 ( p 2 ) + Tad 0 + δ Z AG 0 p 2 − m 2A + δ Z G 0 A p 2 ,  AG AG 2 2   1 1  G 0 A ( p 2 ) = 1PI0 ( p 2 ) + Tad0 + δ Z AG 0 p 2 − m 2A + δ Z G 0 A p 2 .  G A G A 2 2

(5.101) (5.102) (5.103)

The tadpole-inserted self-energies are given by Tad A A = 2λ H A A v H + 2λh A A vh , Tad AG 0

=

Tad G0 A

= 2λ AG 0 H v H + 2λ AG 0 h vh .

(5.104) (5.105)

We impose the following on-shell condition. Re

 A A (m 2A )

d 2 = 0, Re  A A ( p ) = 1, 2 dp p2 =m 2A

Re G 0 A (m 2A ) = 0,

(5.106)

Re  AG 0 (0) = 0.

(5.107)

d  A A ( p 2 ) Re  2 2 = 0, 2 dp p =m A

(5.108)

 AG 0 (0) = 0. Re 

(5.109)

These are equivalent to  A A (m 2A ) = 0, Re   G 0 A (m 2A ) = 0, Re  Then, we obtain   1PI 2 d Tad 1PI 2 = Re  A A (m A ) +  A A , δ Z A A = − 2 Re  A A ( p ) , dp p2 =m 2A (5.110)   1PI −2 2 Tad = 2 Re G 0 A (m 2A ) + Tad Re [1PI G 0 A , δ Z AG 0 = AG 0 (0) +  AG 0 ]. mA m 2A (5.111) δm 2A

δ Z G0 A

116

5.2.2.4

5 Renormalization

Charged Scalar Bosons

The inverse propagators of the charged scalar bosons are given by    H + H − ( p2 ) ,  H + H − ( p 2 ) = i p 2 − m 2H ± ) +   H + G − ( p 2 ),  H + G − ( p2 ) = i 

(5.112)

 G + H − ( p ), G + H − ( p ) = i 

(5.114)

2

2

(5.113)

where the renormalized self-energies of the charged scalar bosons are given by  2  2 Tad 2 2  H + H − ( p 2 ) = 1PI  H + H − ( p ) +  H + H − + δ Z H + H − p − m H ± − δm H ± , (5.115)  2  1 1 2 Tad 2 2  H + G − ( p 2 ) = 1PI  H + G− ( p ) + H + G− + δ Z H + G− p − m H ± + δ Z G+ H − p , 2 2 (5.116)   1 1 2 Tad 2 2 2  G + H − ( p 2 ) = 1PI  G + H − ( p ) + G + H − + δ Z H + G − p − m H ± + δ Z G + H − p . 2 2 (5.117) The tadpole-inserted self-energies are given by Tad H + H − = λ H H + H − v H + λh H + H − vh ,

(5.118)

Tad H + G−

(5.119)

=

Tad G+ H −

= λ H + G − H v H + λ H + G − h vh .

We impose the following on-shell condition. Re  H + H − (m 2H ± ) = 0,

d 2 + − Re  ( p ) H H 2 2 = 1, 2 dp p =m ±

(5.120)

H

Re  H + G − (0) = 0.

(5.121)

These are equivalent to  H + H − (m 2H ± ) = 0, Re 

d 2  Re  H + H − ( p ) = 0, 2 dp p2 =m 2 ±

(5.122)

H

 H + G − (0) = 0. Re 

(5.123)

Then, we obtain   d 2 δm 2H ± = Re 1PI (m 2H ± ) + Tad , δ Z H + H − = − 2 Re 1PI + H − ( p ) H+ H− H+ H− H dp p 2 =m 2

,

H±

δ Z H + G− =

2 m 2H ±

  . Re 1PI (0) + Tad H + G− H + G−

(5.124) (5.125)

5.2 Renormalization of the Two-Higgs Doublet Model

117

Since we determine δ Z G + H − from Eq. (5.77), the mixing self-energy of the charged scalar bosons does not vanish at p 2 = m 2H ± ,  G + H − (m 2 ± ) = 1PI+ − (m 2 ± ) + Tad Re  H G H H G+ H − +

m 2H ±  2

δ Z H + G − + δ Z G 0 A − δ Z AG 0

= 0



(5.126)

Therefore, we need to take into account the mixing contributions for processes with the charged Higgs bosons.

5.2.2.5

Mixing Angles

From relations between the wave-function renormalization constants and the mixing angles, we can determine the counterterms for the mixing angles as   1 2 1PI 2 Tad Tad Re 1PI H h (m h ) + h H (m H ) +  H h + h H , 2(m 2H − m 2h )   1 2 1PI Tad Tad δβ = − 2 Re 1PI G 0 A (m A ) +  AG 0 (0) + G 0 A +  AG 0 . 2m A

δα =

(5.127) (5.128)

Although we can renormalize UV divergences by using these counterterms, δα and δβ depend on the gauge choice. It is known that the gauge dependences in the mixing angles do not cancel in higher-order amplitudes, and they lead to gauge dependences for physical quantities such as partial decay widths [6]. Since physical quantities have to be gauge independent, we need to make the counterterms of mixing angles gauge-independent. This can be achieved by using the pinch technique (PT). In this scheme, we introduce gauge-independent pinched self-energies, in which gauge dependences are canceled by adding pinch terms. 1PI Tad PT ( p 2 ) + ab + ab ( p 2 ). ab ( p 2 ) = ab

(5.129)

By using the pinched mixing self-energies, we define the counterterms of mixing angles as,   1 Re  H h (m 2h ) + h H (m 2H ) , 2 − mh )   1 δβ = − 2 Re G 0 A (m 2A ) +  AG 0 (0) . 2m A

δα =

2(m 2H

(5.130) (5.131)

118

5.2.2.6

5 Renormalization

Softly-Braking Term of Z2 Symmetry

Different from the other counterterms, we cannot determine the δ M 2 in terms of the one-point and two-point functions. Therefore, we determine δ M 2 in the MS scheme. From the renormalized hhh vertex, δ M 2 is obtained as   2N f m 2 ζ 2  4M 2 − 2m 2H ± − m 2A 2m 2 + m 2 c δM2 1 s2α m 2H − m 2h f f = + + − 3 W 2 Z div 2 2 2 2 2 s2β M 16π v v v v f    1PI 2c2β h1PI + cβ−α − H2 sβ−α . (5.132) 2 vs2β mh mH div

References 1. Bohm M, Spiesberger H, Hollik W (1986) On the one loop renormalization of the electroweak standard model and Its application to leptonic processes. Fortsch Phys 34:687–751 2. Hollik WFL (1990) Radiative corrections in the standard model and their role for precision tests of the electroweak theory. Fortsch Phys 38:165–260 3. Fleischer J, Jegerlehner F (1983) Radiative corrections to Higgs production by e+ e− → Z H in the Weinberg-Salam model. Nucl Phys B 216:469–492 4. Denner A, Jenniches L, Lang J-N, Sturm C (2016) Gauge-independent M S renormalization in the 2HDM. JHEP 09:115 5. Kanemura S, Okada Y, Senaha E, Yuan CP (2004) Higgs coupling constants as a probe of new physics. Phys Rev D 70:115002 6. Krause M, Lorenz R, Muhlleitner M, Santos R, Ziesche H (2016) Gauge-independent renormalization of the 2-Higgs-doublet model. JHEP 09:143

Chapter 6

One-Loop Calculations for Decays of the SM-Like Higgs Boson

In this chapter, we review the EW corrections to the decay rates of the SM-like Higgs boson for various decay modes. In Sect. 6.1, we list the analytic expressions of the renormalized form factors for vertex functions of the SM-like Higgs boson. One-loop calculations for EW corrections are performed based on the on-shell renormalization scheme discussed in Chap. 5. The decay rates with NLO EW and QCD corrections are given in terms of the form factors. In Sect. 6.2, the numerical evaluations are shown for the decay rates and branching ratios of the SM-like Higgs boson obtained by using the H-COUP program [1, 2]. This chapter is the review of Refs. [3–6].

6.1 Decay Rates with Higher Order Corrections In this section, we discuss the decay rates with NLO EW corrections for the decays of the SM-like Higgs boson. The loop-induced decay processes, h → γ γ , Z γ , gg, and the QCD corrections for the decays of the SM-like Higgs boson are discussed in Sect. 4.1.

p1

f

p1

q

Vμ

q

h

h p2

f¯

p2

Vν

Fig. 6.1 Momentum assignment for the renormalized h f f¯ and hV V vertices

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 M. Aiko, Theoretical Studies on Extended Higgs Sectors Towards Future Precision Measurements, Springer Theses, https://doi.org/10.1007/978-981-99-1324-4_6

119

120

6 One-Loop Calculations for Decays …

6.1.1 Form Factors for Vertex Functions 6.1.1.1

h f f¯ Vertex

The renormalized h f f¯ vertex can be decomposed as [7]  hPf f¯ + /p 1 hVf1 f¯ + /p 2 hVf2 f¯ h f f¯ ( p1 , p2 , q) =  hS f f¯ + γ5 hAf1 f¯ + /p 2 γ5 hAf2 f¯ + /p 1 /p 2 hTf f¯ + /p 1 /p 2 γ5 hPfTf¯ , + /p 1 γ5

(6.1)

where p1 ( p2 ) denotes the incoming four-momentum of the fermion (anti-fermion), and q is the outgoing four-momentum of the SM-like Higgs boson (see the left panel of Fig. 6.1). The following relations hold for the on-shell fermions; i.e., p12 = p22 = m 2f ,  hPf f =  hPfTf = 0,  hVf1 f = − hVf2 f ,  hAf1 f = − hAf2 f .

(6.2)

The renormalized form factors are composed of the tree-level and the one-loop part as X,loop  hXf f¯ = hX,tree + h f f¯ , (X = S, P, V1 , V2 , A1 , A2 , T, P T ). f f¯

(6.3)

The tree-level coupling for h f f¯ vertices are given by, =− hS,tree f f¯

mf f ζ , hX,tree = 0 (X = S). f f¯ v h

(6.4)

The one-loop-level parts are decomposed into the contributions from 1PI diagrams and the counterterms, X,loop

h f f¯

= hX,1PI + δhXf f¯ . f f¯

(6.5)

The analytic expressions for the 1PI contributions are presented in Ref. [3], and we list them in Appendix D.3. The counterterms are given by 

δhS f f¯

f

δζ δm f δv δ Z hh f + hf + δ Z V + − mf v 2 ζh



δ Z Hh 2     δm f δv δ Z hh f S,tree δ Z H h = hS,tree − ζ +  − δα , − δβ + δ Z + f V f f¯ H f f¯ mf v 2 2 (6.6)

=

hS,tree f f¯

δhXf f¯ = 0

(X = S),

+  HS,tree f f¯

(6.7)

6.1 Decay Rates with Higher Order Corrections

121

where  HS,tree = −m f κ Hf /v, and we have used f f¯ f

δζh f

ζh 6.1.1.2

f

=−

ζH f

ζh

δα − ζ f δβ.

(6.8)

hV V Vertex

The renormalized hV V vertex is defined by   μ p1ν p2 2 μν 1 μνρσ p1ρ p2σ 3 2 2 2   hV   + + i

hV V ( p1 , p2 , q) = g μν  V hV V ( p1 , p2 , q ), m 2V hV V m 2V (6.9) μ

μ

where p1 and p2 are the incoming four-momenta of the weak bosons, and q μ is the outgoing four-momentum of the Higgs boson (see the right panel of Fig. 6.1). When we neglect the effects of the CP violation, we have 3 2 2 2  hV V ( p1 , p2 , q ) = 0.

(6.10)

The renormalized form factors are composed of the tree-level and one-loop-level parts as i,loop i,tree X  hV V = hV V + hV V , (i = 1, 2, 3).

(6.11)

The tree-level couplings are given by, 1,tree hV V =

2m 2V h 2,tree 3,tree κ , hV V = hV V = 0. v V

(6.12)

The one-loop-level parts are decomposed into the contributions from 1PI diagrams, tadpole diagrams, and the counterterms, i,loop

i,1PI i,Tad i hV V = hV V + hV V + δhV V

(6.13)

The analytic expressions for the 1PI diagrams are presented in Ref. [3], and we list them in Appendix D.3.3. The tadpole contributions are given by, 1,Tad hV V

  h1PI  1PI H = cV 2ghhV V 2 + gh H V V 2 , mh mH

2,Tad 3,Tad hV V = hV V = 0,

(6.14) (6.15)

122

6 One-Loop Calculations for Decays …

where c Z (W ) = 2 (1). The counterterms are given by  1 δhV V

=

1,tree hV V

δm 2V δv δ Z hh + δ ZV + − 2 v 2 mV



 +

 1,tree HV V

 δ Z Hh + δβ − δα , 2 (6.16)

2 3 δhV V = δhV V = 0.

6.1.1.3

(6.17)

V f f¯ Vertex

In the massless limit of external fermions, the renormalized V f f¯ vertex can be decomposed as

μ  V f f¯ ( p1 , p2 , q) = gV γ μ  VA f f¯ ( p12 , p22 , q 2 ), VV f f¯ − γ5

(6.18)

where p1 ( p2 ) is the incoming four-momentum of the fermion (anti-fermion), and q is the√outgoing four-momentum of the weak gauge boson. The coupling gV is g Z and g/ 2 for the Z and W bosons, respectively. The renormalized form factors are composed of the tree-level and one-loop-level parts as i,loop  Vi f f¯ = Vi,tree + V f f¯ , (i = V, A). f f¯

(6.19)

The tree-level contribution is given by If If 2 − Q f sW ,  ZA,tree = , f f¯ 2 2 1 A,tree = W f f¯  = . 2

=  ZV,tree f f¯

(6.20)

V,tree W f f¯ 

(6.21)

The one-loop-level parts are decomposed into the contributions from 1PI diagrams and the counterterms, i,loop

V f f¯ = Vi,1PI + δ iZ f f¯ . f f¯

(6.22)

The analytic expressions for the 1PI diagrams are presented in Ref. [6], and we list them in Appendix D.3.4. The counterterms are given by 

δ ZV f f¯

2 − s 2 δs 2 cW 1 δe f W W = vf + δ Z Z + δ ZV − 2 2 e 2 2cW sW   2 δsW 1 f 2 + a f δ Z A − Q f δsW , + Q f sW cW δ Z Z γ + 2 sW cW

(6.23)

6.1 Decay Rates with Higher Order Corrections 

δ ZA f

f¯

V δW f f¯

c2 − s 2 δe − W 2 W e 2cW  1 δe A = δW = − f f¯ 2 e = af

2 δsW 2 sW 2 δsW 2 2sW

+

1 f δ Z Z + δ ZV 2

123 f

+ v f δZ A,

(6.24)

 1  1 1 f f f f + δ ZW + δ ZV + δ Z A + δ ZV + δ Z A . 2 2 2

(6.25)

6.1.2 Decay Rates for h → f f¯ The decay rate for h → f f¯ ( f = t) is given by

f f (h → f f¯) = LO (h → f f¯) 1 + EW + QCD,h ,

(6.26)

where the LO decay rate is given by Eq. (4.3), while the QCD correction is given by f Eq. (4.6). The EW correction , EW , is decomposed into weak and QED corrections as f

f

f

EW = weak + QED .

(6.27)

f

The expression for weak is given by    ∗   m 2f 2 S,loop V ,loop T,loop f S,tree h f f + 2m f h f1 f − r.

weak =  + m 2h 1 − 2 h f f 2 Re h f f  S,tree  mh h f f 

(6.28) The QED correction is composed of the contributions of virtual-photon loops and those from the real-photon emissions. For the leptonic decays, the QED correction in the on-shell scheme is given by [8–10]

QED =

  αem 2 1 1 1+β 3 2 4 2 (3 + 34β − 13β ) log (7β − 1) , Q AQED (β) + + π β 16β 3 1−β 8β 2

(6.29) with      1+β 2 1−β 1−β + 2Li2 − − 3 log log AQED (β) = (1 + β ) 4Li2 1+β 1+β 1−β 1+β  4 1+β log β − 3β log − 2 log − 4β log β. (6.30) 1−β 1 − β2 2

124

6 One-Loop Calculations for Decays …

Fig. 6.2 Diagrams contributing to the h → Z Z ∗ → Z f f¯ (h → W W ∗ → W f f¯ ) mode at NLO. Each diagram denotes the contributions from hV V vertex corrections (a), oblique corrections (b), V f f¯ vertex corrections (c), h f f¯ vertex corrections (d), and box corrections (e). This figure is reprinted from Ref. [6] under the Creative Commons Attribution 4.0 International License. © 2019 The Authors

For the hadronic decays, the QED correction in the MS scheme [11] with the renormalization scale μ is given by q

QED =

αem 2 Q π q



17 3 μ2 + log 2 4 2 mh

 .

(6.31)

6.1.3 Decay Rates for h → Z Z ∗ → Z f f¯ Similarly to the tree-level calculation, we neglect the masses of final-state fermions. The decay rate for the h → Z Z ∗ → Z f f¯ process is given by   Z Z + QCD , (h → Z f f¯) = LO (h → Z f f¯) 1 + EW

(6.32)

where the LO decay rate is given by Eq. (4.13). The EW correction is composed of the weak and QED corrections, Z Z Z = weak + QED .

EW

(6.33)

6.1 Decay Rates with Higher Order Corrections

125

Z The weak correction weak is given by

Z

weak

  1,loop 2,loop h Z Z λ¯ (x Z , xs ) h Z Z (m 2Z , s, m 2h ) ds |M0Z |2 Re + 1,tree xZ h1,tree  ZZ hZ Z 0  1 2 2 ˆ ˆ h Z γ h Z γ v f Q f cW sW s − m Z ¯ + + λ(x Z , xs ) 1,tree (m 2Z , s, m 2h ) Re s v 2f + a 2f h1,tree h Z Z ZZ

2 = LO

(m h−m Z )2

V,loop

+

Re[v f  Z f f

1 + LO

A,loop

+ a f Z f f v 2f + a 2f

(m h−m Z )2

−

ˆ Z γ (s)  ˆ Z Z (s) v f Q f sW cW Re  Re  − s s − m 2Z v 2f + a 2f

umax

 du Re ThZf f + B Z

ds 0

](0, 0, s)

u min

ˆ Z Z (m 2Z ), − 2 r − Re

(6.34)

with the Mandelstam variable s = ( p f + p f¯ )2 . The squared tree-level amplitude |M0Z |2 is given by  2   g 2Z h1,tree v 2f + a 2f λ(x Z , xs ) + 12x Z xs 1/2 ZZ  Z 2 |M0 | = λ (x Z , xs ), 3x Z 256π 3 m 3h (xs − x Z )2

(6.35)

with x Z = m 2Z /m 2h , xs = s/m 2h . The kinematic factor λ¯ (x, y) is defined by ¯ λ(x, y) =

1−x −y λ(x, y) . 2 λ(x, y) + 12x y

(6.36)

The first and second lines denote the contributions from the diagram (a) in Fig. 6.2. The third line denotes the contributions from diagrams (b) and (c). In the fourth line, the ThZf f and B Z terms represent the contribution from the h f f¯ vertex corrections and the box diagrams shown as the diagrams (d) and (e) in Fig. 6.2, respectively. Both ThZf f and B Z depend on the Mandelstam variable u = ( p Z + p f¯ )2 in loop functions. Analytic expressions for the ThZf f and B Z terms are given in Ref. [6]. The integration range of u is given by u max,min =

m 2h [1 + x Z − xs ± λ1/2 (x Z , xs )]. 2

(6.37)

The fifth line denotes the finite contributions originated from the replacement of VEV and the wave function renormalization of the external Z boson. The QCD and QED corrections only enter the Z f f¯ vertex correction depicted in diagram (c) in Fig. 6.2. The QCD correction is given by Eq. (4.16), while the QED correction is given by Refs. [12]

126

6 One-Loop Calculations for Decays … Z

QED = Q 2f

3αem . 4π

(6.38)

Although diagrams (d) and (e) also receive both QCD and QED corrections, they vanish in the massless limit for the external fermions.

6.1.4 Decay Rates for h → W W ∗ → W f f¯ Similarly to the tree-level calculation, we neglect the masses of final-state fermions. The decay rate for the h → W W ∗ → W f f¯ process is given by   W W . + QCD (h → W f f¯  ) = LO (h → W f f¯  ) 1 + EW

(6.39)

W where the LO decay rate is given by Eq. (4.14). The EW correction EW is given by

  1,loop 2,loop ¯ 2  λ (x , x ) W s hW W hW W W ds |M0W |2 Re +

EW (m 2W , s, m 2h ) 1,tree 1,tree xW hW hW W W 0  ˆ W W (s) 2Re  V,loop A,loop + 2Re[W f f + W f f ](0, 0, s) − s − m 2W ⎡ ⎤ (m h−m W )2 u max   1 ⎢ ⎥ + ds du ThWf f + BW + (h → W f f¯ γ )⎦ ⎣ LO 1 = LO

(m h−m W )2

0

− 2 r −

u min

ˆ W W (m 2W ), Re

(6.40)

with the Mandelstam variable s = ( p f + p f¯  )2 . The squared tree-level amplitude |M0W |2 is given by    1,tree 2 g 2 hW W  λ(x W , x s ) + 12x W x s |M0W |2 = λ1/2 (x W , xs ), 3x W (xs − x W )2 512π 3 m 3h

(6.41)

with x W = m 2W /m 2h , xs = s/m 2h . Analytic expressions for the ThWf f and BW terms are given in Ref. [6]. The QCD correction only enters the W f f¯  vertex correction, and it is given by Eq. (4.16). Different from h → Z f f¯, we cannot separate the weak and QED corrections. This is because virtual photons appear together with virtual W bosons in vertex corrections. The IR divergences are canceled by adding the real photon emissions (h → W f f¯ γ ) [6].

6.2 Numerical Results

127

6.2 Numerical Results In this section, we discuss the predictions for the partial decay widths and branching ratios of the SM-like Higgs boson at NLO in the four types of 2HDM. The deviation in the branching ratios from those SM predictions can be parametrized by

μ X Y ≡

BRNP (h → X Y ) − 1. BRSM (h → X Y )

(6.42)

We can expand μ X Y up to NLO as

μ X Y ≈ R(h → X Y ) −  tot ,

(6.43)

where NP (h → X Y ) − 1, SM (h → X Y ) tot NP ≡ tot − 1. SM

R(h → X Y ) ≡

 tot

(6.44) (6.45)

Thus, the qualitative behavior of μ X Y can be understood by analyzing  X Y and

tot .

6.2.1 Deviation in Partial Decay Widths In Fig. 6.3, we show the correlation between between R(h → τ τ ) and R(h → bb) in the 2HDMs. The colored regions correspond to the predictions in these models; i.e., red, blue, green, and purple correspond to the Type-I, II, X, and Y 2HDMs, respectively. The contrast of the colors represents the mass scales of the additional Higgs bosons. The left (right) panel shows the results with cβ−α < 0 (cβ−α > 0) in the 2HDMs. The tree-level predictions with tan β = 1 (3) in the 2HDMs are shown by gray (black) lines with dots denoting sβ−α = 1, 0.995, 0.99, 0.98, 0.95 from the origin. The patterns of the deviations are mainly determined by the mixing on the treelevel couplings [13]. In the 2HDM, the decay widths show characteristic patterns due to the Yukawa structures. The size of the mixing parameters is constrained by theoretical constraints such as perturbative unitarity and vacuum stability. We can see that the darker-colored regions are included in the lighter-colored regions. Therefore, the upper bound of the additional Higgs boson masses can be deduced when a deviation is observed.

128

6 One-Loop Calculations for Decays …

Fig. 6.3 Correlation between R(h → τ τ ) and R(h → bb) in the Type-I (red), Type-II (blue), Type-X (green), Type-Y (purple) 2HDMs. The left (right) panel shows the case for cβ−α < 0 (> 0) in the 2HDMs. The tree-level predictions with tan β = 1 and 3 in the 2HDMs are also presented by gray and black lines, respectively. This figure is reprinted from Ref. [5] under the Creative Commons Attribution 4.0 International License. © 2018 The Author(s)

Fig. 6.4 Correlation between R(h → τ τ ) and R(h → cc) in the Type-I and Y (red) and Type-II and X (blue) 2HDMs. This figure is reprinted from Ref. [5] under the Creative Commons Attribution 4.0 International License. © 2018 The Author(s)

In Fig. 6.4, we show the correlation between R(h → τ τ ) and R(h → cc). At the tree level, the results in the Type-I (Type-II) 2HDM coincide with those in the Type-Y (Type-X) 2HDM, because the Yukawa structures for up-type quarks and leptons are the same. In contrast to the case in Fig. 6.3, the predictions in the Type-II and Type-Y 2HDM spread out since the Yukawa structures between up-type quarks and leptons are different. From the correlations among the three different fermionic

6.2 Numerical Results

129

Fig. 6.5 Correlation between R(h → τ τ ) and R(h → Z Z ∗ ) in the Type-I and Y (red) and Type-II and X (blue) 2HDMs. This figure is reprinted from Ref. [5] under the Creative Commons Attribution 4.0 International License. © 2018 The Author(s).

decay modes of the Higgs boson shown in Figs. 6.3 and 6.4, we can identify a type of the 2HDM independently of the model parameters when a deviation is observed in experiments. In Fig. 6.5, we show the correlation between R(h → τ τ ) and R(h → Z Z ∗ ). Similar to the discussion in Fig. 6.3, patterns of the deviations can be mainly determined by the mixing in the tree-level Higgs couplings. The results in the Type-I (TypeII) 2HDM coincide with those in the Type-Y (Type-X) 2HDM at the tree level, as the Yukawa structures for leptons are the same. An important remark on the 2HDMs is the difference of the magnitude of the deviations between h → f f and h → V V ∗ [3]. The 5% deviation for the coupling in the gauge sector (i.e. sβ−α = 0.95) gives rise to R(h → Z Z ∗ ) ∼ −10% to −15%, while R(h → f f ) ∼ ± 60% or more.

6.2.2 Deviation in the Total Decay Width In Fig. 6.6, we show the deviation in the total width as a function of tan β in four types of the 2HDMs with sβ−α = 0.99 and cβ−α < 0 (> 0) in the left (right) panel. The values of m  and M 2 are scanned within 300 ≤ m  ≤ 1000 GeV and 0 ≤ M 2 ≤ m 2 , respectively. In the case with cβ−α < 0 (the left panel), the width becomes larger as tan β increases except for the Type-I 2HDM. This is because some of the partial widths have a tan β enhancement, e.g., the h → bb¯ (h → τ τ¯ ) mode in the Type-II and TypeY (Type-II and Type-X) 2HDMs. In the Type-I 2HDM on the contrary, the total 2 SM , at the large tan β region. The region with tan β  11 is width approaches sβ−α eliminated by the theoretical constraints.

130

6 One-Loop Calculations for Decays …

Fig. 6.6 Deviation in the total width from the SM prediction in four types of the 2HDMs with sβ−α = 0.99 as a function of tan β. The left (right) panel shows the case of cβ−α < 0 (> 0). This figure is reprinted from Ref. [6] under the Creative Commons Attribution 4.0 International License. © 2019 The Authors

In the case with cβ−α > 0 (the right panel), the total width has the minimal value at tan β ∼ 7 in the Type-II, Type-X, and Type-Y 2HDMs, due to the cancellation between the sβ−α term and the cβ−α term in κ hf . This behavior is remarkably observed in the Type-II and Type-Y 2HDMs due to the h → bb¯ mode, which is the biggest partial width of h in the SM. We can also see that the deviation in the total width becomes zero at tan β  14, as we have (κ hf )2  1 for all the types of Yukawa interaction. In the Type-I 2HDM, the width approaches the SM value at a large value of tan β as in the case with cβ−α < 0.

6.2.3 Deviation in Branching Ratios We here discuss the case where the central value of μW W is consistent with zero at the 2σ level. The left (right) panel of Fig. 6.7 shows the correlation between

μτ τ and μbb ( μcc ) in the four types of the 2HDMs. The error for μW W is taken to consider about 2σ level region at ILC250 [14]. We scan 1.5 ≤ tan β ≤ 10, 0 ≤ M 2 ≤ m 2 and 300 (600) ≤ m  ≤ 1000 GeV for (darker) colored points. We see that the predictions in the 2HDMs are spread out in different directions depending on the type of Yukawa interactions. From the correlation between μτ τ and μbb , we can distinguish Type-II 2HDMs from Type-X and Type-Y 2HDMs. In order to distinguish Type-X and Type-Y, μcc is useful. Thus, the predictions in the four types of 2HDMs are separated from one another, and we can distinguish the four types of 2HDMs when the deviations are measured.

References

131

Fig. 6.7 Correlation between μτ τ and μbb in the Type-I (red), Type-II (blue), Type-X (green), Type-Y (magenta) 2HDMs. The left (right) panel shows the case with μW W = 0 ± 2% (0 ± 4%). This figure is reprinted from Ref. [6] under the Creative Commons Attribution 4.0 International License. © 2019 The Authors

References 1. Shinya K, Mariko K, Kodai S, Kei Y (2018) H-COUP: a program for one-loop corrected Higgs boson couplings in non-minimal Higgs sectors. Comput Phys Commun 233:134–144 2. Shinya K, Mariko K, Kentarou M, Kodai S, Kei Y (2020) H-COUP version 2: a program for one-loop corrected Higgs boson decays in non-minimal Higgs sectors. Comput Phys Commun 257:107512 3. Shinya K, Mariko K, Kei Y (2015) Fingerprinting the extended Higgs sector using one-loop corrected Higgs boson couplings and future precision measurements. Nucl Phys B 896:80–137 4. Shinya K, Mariko K, Kodai S, Kei Y (2017) Gauge invariant one-loop corrections to Higgs boson couplings in non-minimal Higgs models. Phys Rev D 96(3):035014 5. Shinya K, Mariko K, Kentarou M, Kodai S, Kei Y (2018) Loop effects on the Higgs decay widths in extended Higgs models. Phys Lett B 783:140–149 6. Shinya K, Mariko K, Kentarou M, Kodai S, Kei Y (2019) Full next-to-leading-order calculations of Higgs boson decay rates in models with non-minimal scalar sectors. Nucl Phys B 949:114791 7. Shinya K, Mariko K, Kei Y (2014) Radiative corrections to the Yukawa coupling constants in two Higgs doublet models. Phys Lett B 731:27–35 8. Kniehl BA (1992) Radiative corrections for H → f anti-f (γ ) in the standard model. Nucl Phys B 376:3–28 9. Dabelstein A, Hollik W (1992) Electroweak corrections to the fermionic decay width of the standard Higgs boson. Z Phys C 53:507–516 10. Bardin DYu, Vilensky BM, Khristova PKh (1991) Calculation of the Higgs boson decay width into fermion pairs. Sov J Nucl Phys 53:152–158 ¯ to order ααs . Phys Lett B 751:442–447 11. Luminita M, Barbara S, Matthias S (2015) (H → bb) 12. Kniehl BA (1994) Higgs phenomenology at one loop in the standard model. Phys Rep 240:211– 300 13. Shinya K, Koji T, Kei Y, Hiroshi Y (2014) Fingerprinting nonminimal Higgs sectors. Phys Rev D 90:075001 14. Fujii K et al (2017) Physics case for the 250 GeV stage of the international linear collider

Chapter 7

Higgs Strahlung Process in Electron–Positron Colliders

The e+ e− → h Z process is the dominant production process at the early stage of the future lepton colliders such as the ILC, and the production cross section will be measured with a few percent accuracies. Thus, it is important to perform the theoretical calculation compatible with future precision measurements by including the higher-order corrections. In this chapter, we discuss the cross section for the Higgs strahlung process e+ e− → h Z with arbitrary sets of electron and Z boson polarization at the full next-to-leading order. We systematically perform the complete one-loop calculations to the helicity amplitudes in the SM and 2HDMs based on the on-shell renormalization scheme discussed in Chap. 5. We present the full analytic results, as well as numerical evaluations. We find that the extended Higgs models can be classified by measuring the pattern of deviations from the SM prediction in the cross-section times decay branching ratio. This chapter is based on the author’s work in [1].

7.1 Electroweak Corrections to the Process e+ e− → hZ In this section, we define the notation for the e+ e− → h Z process and introduce helicity amplitudes based on the form-factor decomposition. The differential cross sections with arbitrary sets of electron and Z boson polarization are given in terms of the form factors. We discuss the size of the helicity-dependent total and differential cross-sections at the LO in the SM. Formulae of the form factors including the oneloop corrections are given in terms of renormalized quantities.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 M. Aiko, Theoretical Studies on Extended Higgs Sectors Towards Future Precision Measurements, Springer Theses, https://doi.org/10.1007/978-981-99-1324-4_7

133

134

7 Higgs Strahlung Process in Electron-Positron Colliders

Fig. 7.1 The process e+ e− → h Z with momentum and helicity assignments. The momenta pe and pe¯ are incoming, while kh and k Z are outgoing

e+ (pe¯, σe¯)

Z(kZ , λ)

e− (pe , σe )

h(kh )

7.1.1 Kinematics of the Higgs Strahlung Process The Higgs strahlung process e− ( pe , σe ) + e+ ( pe¯ , σe¯ ) → h(kh ) + Z (k Z , λ)

(7.1)

is depicted in Fig. 7.1. The momenta and helicities of the incoming electron and positron are denoted by ( pe , σe ) and ( pe¯ , σe¯ ), respectively. Correspondingly, (k Z , λ) is used for the outgoing Z boson, and kh is the momentum of the outgoing Higgs boson. The signs + and − of the variables σe and σe¯ refer to helicities +1/2 and −1/2, respectively. The helicity λ takes ± or 0. In the following discussion, we neglect the mass of the electron except for QED corrections, where mass singular logarithms appear. The Mandelstam variables are given by s = ( pe + pe¯ )2 = (k Z + kh )2 ,

(7.2)

t = ( pe − k Z ) = ( pe¯ − kh ) ,

(7.3)

u = ( pe − kh ) = ( pe¯ − k Z ) .

(7.4)

s + t + u = m 2Z + m 2h .

(7.5)

2

2

2 2

They satisfy

In the center-of-mass (CM) frame of the e+ e− collision, the momenta of each external particle are given by √

s (1, 0, 0, 1), 2 √ s μ pe¯ = (1, 0, 0, −1), 2 √   m 2Z − m 2h s μ kZ = 1+ , β sin θ, 0, β cos θ , 2 s √   m 2Z − m 2h s μ kh = 1− , −β sin θ, 0, −β cos θ , 2 s peμ

=

(7.6) (7.7) (7.8) (7.9)

7.1 Electroweak Corrections to the Process e+ e− → h Z

135

where β is defined by     k Z 

   1  s − (m Z + m h )2 s − (m Z − m h )2 , (7.10) E s √ with the beam energy E = s/2. We use the scattering angle θ between e− and Z boson β=

=

k Z = cos θ,  pe · 

(7.11)

where the hat indicates the unit vector. The θ angle is related to the t and u variables via 1 2 (m + m 2h − s) + 2 Z 1 u = (m 2Z + m 2h − s) − 2 t=

s β cos θ, 2 s β cos θ. 2

(7.12) (7.13)

7.1.2 Convention of Spinors and Polarization Vectors The incoming-electron spinor can be decomposed as

u( pe , σe )− , U ( pe , σe ) = (PL + PR )U ( pe , σe ) = u( pe , σe )+

(7.14)

where U ( pe , σe ) is a Dirac spinor, and it contains four elements. The chiralityeigenstate spinor u( pe , σe )α with a helicity σe and a chirality α is defined as

u( pe , σe )− , 0

0 . PR U ( pe , σe ) = u( pe , σe )+

PL U ( pe , σe ) =

(7.15) (7.16)

The u( pe , σe )α spinor contains two elements. In the massless limit, the chirality of the electron becomes equal to its helicity, σe = α. We have √ lim u( pe , σe )α = δσe ,α 2Eχσe (θ, φ),

m e →0

(7.17)

where χσe (θ, φ) is eigenvector of helicity operator (pe · σ)/|p|. For massless spinors, we omit helicity indices and introduce the short-hand notation u( pe , σe ) ≡ lim u( pe , σe )α = m e →0

√

2Eχσe (θ, φ).

(7.18)

136

7 Higgs Strahlung Process in Electron-Positron Colliders

We take the momentum of the incoming electron as parallel to the z-axis, and we set θ = φ = 0. Then, the u( pe , σe ) spinor becomes   0 , 1   √ √ 1 . u( pe , +) = 2Eχ+ (θ = 0, φ = 0) = 2E 0

√ √ u( pe , −) = 2Eχ− (θ = 0, φ = 0) = 2E

(7.19) (7.20)

The decomposition of the incoming-positron spinor is almost the same as the decomposition of the incoming-electron spinor, V ( pe¯ , σe¯ ) = (PL + PR )V ( pe¯ , σe¯ ) =

v( pe¯ , σe¯ )− . v( pe¯ , σe¯ )+

(7.21)

We note that the V ( pe¯ , σe¯ ) spinor is related to the U ( pe¯ , σe¯ ) spinor through the charge conjugation T

T

V ( pe¯ , σe¯ ) = CU ( pe¯ , σe¯ ) = iγ 2 γ 0 U ( pe¯ , σe¯ ) = iγ 2 U ∗ ( pe¯ , σe¯ ).

(7.22)

Therefore, we have V ( pe¯ , σe¯ ) =

v( pe¯ , σe¯ )− v( pe¯ , σe¯ )+



=

iσ 2 u ∗ ( pe¯ , σe¯ )+ . iσ 2 u ∗ ( pe¯ , σe¯ )−

(7.23)

In the massless limit, the chirality of the positron takes the opposite sign to its helicity,  √ v( pe¯ , σe¯ ) ≡ lim v( pe¯ , σe¯ )α = δ−σe¯ ,α − 2E χ−σe¯ (θ, φ) m e¯ →0 √ = − 2Eχ−σe¯ (θ, φ),

(7.24)

where we have introduced the short-hand notation as similar to Eq. (7.18). We take the momentum of the incoming positron as anti-parallel to the z-axis, and we set θ → π − θ = π and φ → φ + π = π. Then, the v( pe¯ , σe¯ ) spinor becomes   0 2E , 1   √ √ −1 v( pe¯ , +) = 2Eχ− (θ = π, φ = π) = 2E . 0 v( pe¯ , −) =

√

2Eχ+ (θ = π, φ = π) =

√

(7.25) (7.26)

In summary, we have U ( pe , −) =

u( pe , −)− 0



⎡ ⎤ 0 √ ⎢ 1 ⎥ ⎥ = 2E ⎢ ⎣0 ⎦, 0

(7.27)

7.1 Electroweak Corrections to the Process e+ e− → h Z

137

⎡ ⎤ 0 √ ⎢ 0 ⎥ 0 ⎥ = 2E ⎢ U ( pe , +) = ⎣1 ⎦, u( pe , +)+ 0 ⎡ ⎤ 0

√ ⎢ 0 ⎥ 0 ⎥ = 2E ⎢ V ( pe¯ , −) = ⎣0 ⎦, v( pe¯ , −)+ 1 ⎡ ⎤ −1

√ ⎢ 0 ⎥ v( pe¯ , +)− ⎥ V ( pe¯ , +) = = 2E ⎢ ⎣ 0 ⎦. 0 0



(7.28)

(7.29)

(7.30)

The polarization vectors of the outgoing Z boson with the helicity λ are given as 1 ε∗μ (k Z , ±) = √ [0, ∓ cos θ, i, ± sin θ] , (7.31) 2 √

s E Z |k Z | , sin θ, 0, cos θ = ε∗μ (k Z , 0) = [β, α sin θ, 0, α cos θ] , (7.32) mZ EZ 2m Z with α = 1 + (m 2Z − m 2h )/s.

7.1.3 Helicity Amplitudes and Helicity-Dependent Cross-Sections The helicity amplitudes for the e+ e− → h Z process vanish for σe = σe¯ in the massless limit of electron due to the chirality conservation. Therefore, we use σ = σe = −σe¯ for the non-vanishing amplitudes. The helicity amplitude Mσλ (s, t) can be decomposed by a set of basic matrix elements Mi,σλ (s, t) and corresponding form factors Fi,σ (s, t) as [2] Mσλ (s, t) =

3 

Fi,σ (s, t)Mi,σλ (s, t).

(7.33)

i=1

The basic matrix elements are given by μν

Mi,σλ = jσ,μ ( pe , pe¯ )Ti (s, t)ε∗ν (k Z , λ),

(7.34)

where the fermion current of the initial electron jσμ ( pe , pe¯ ) is defined as ¯ pe¯ )γ μ Pσ u( pe ) = jσμ ( pe , pe¯ ) = v(

√

s[0, 1, σi, 0],

(7.35)

138

7 Higgs Strahlung Process in Electron-Positron Colliders μν

with the chirality projection operator Pσ = (1 + σγ5 )/2. The three basis tensor Ti are defined by μν

μν

μ

μν

μ

T1 = g μν , T2 = k Z ( pe + pe¯ )ν , T3 = k Z ( pe − pe¯ )ν .

(7.36)

In the CM frame, the first element of the basic matrices M1,σλ becomes 

√ s 1 (1 ± σ cos θ) = σ 2s dσ,± (θ), 2 sα sα 1 =− sin θ = σ √ dσ,0 (θ), 2m Z 2m Z

M1, σ± = σ

(7.37)

M1, σ0

(7.38)

j

where dm  ,m (θ) is the Wigner’s d function. The second element M2,σλ is give by M2,σ± = 0, M2,σ0 = −

(7.39) s β s β 1 sin θ = σ √ dσ,0 (θ). 4m Z 2 2m Z 2 2

2 2

(7.40)

The third element M3,σλ is given by   s 2 sβ s sin2 θ = ±sβ d (θ), 2 2 3 2,0  s 2 αβ s 2 αβ  2 2 d2,1 (θ) − d2,−1 = cos θ sin θ = − (θ) . 2 4m Z 4m Z

M3, σ± = ±

(7.41)

M3, σ0

(7.42)

In terms of the form factors Fi,σ , the six helicity amplitudes are given by 

s sβ F1,σ (s, t) ± (σ ∓ cos θ)F3,σ (s, t) (σ ± cos θ), (7.43) 2 2

s sβ 2 sαβ Mσ0 (s, t) = − αF1,σ (s, t) + F2,σ (s, t) − cos θF3,σ (s, t) sin θ. 2m Z 2 2 (7.44)

Mσ± (s, t) =

We denote the tree and one-loop contributions to the helicity amplitude as M(0) σλ (s, t) =

3 

(0) Fi,σ (s, t)Mi,σλ ,

(7.45)

(1) Fi,σ (s, t)Mi,σλ .

(7.46)

i=1

M(1) σλ (s, t) =

3  i=1

7.1 Electroweak Corrections to the Process e+ e− → h Z

139

e+

Fig. 7.2 Tree-level diagram for the Higgs strahlung process e+ e− → h Z

Z Z∗

e−

h

The helicity-dependent differential cross-section at NLO in EW is given by  β  (0) dσ (0) (1)∗ 2 (σ, λ; s, t) = |M (s, t)| + 2 Re[M (s, t)M (s, t)] . (7.47) σλ σλ σλ d 64π 2 s The helicity-dependent cross-section σ(σ, λ; s) can be obtained by integrating Eq. (7.47) over the solid angle d = d cos θdφ. Since the electron and positron beams are not purely polarized in a realistic setup, we introduce the degree of polarization of initial electron Pe and positron Pe¯ . We follow the convention where a purely left-handed (right-handed) electron corresponds to Pe = −1 (+1). The polarized differential cross section is given by 1 dσ dσ (Pe , Pe¯ , λ; s, t) = (1 + σ Pe )(1 − σ Pe¯ ) (σ, λ; s, t). d 4 d σ=±

(7.48)

The unpolarized cross section σ(λ; s) corresponds to Pe = Pe¯ = 0. The planned polarization at the ILC [3] is (Pe , Pe¯ ) = (∓ 0.8, ± 0.3). The polarized cross-section can be rewritten in terms of the helicity-dependent cross-section as [4] 1 1 (1 − Pe )(1 + Pe¯ )σ(−, λ; s) + (1 + Pe )(1 − Pe¯ )σ(+, λ; s) 4 4 = σ(λ; s)(1 − Pe Pe¯ )(1 − Peff ALR ), (7.49)

σ(Pe , Pe¯ , λ; s) =

where the effective polarization Peff and the left-right asymmetry ALR are defined as Pe − Pe¯ , 1 − Pe Pe¯ σ(−, λ; s) − σ(+, λ; s) . = σ(−, λ; s) + σ(+, λ; s)

Peff =

(7.50)

ALR

(7.51)

By using Eq. (7.49), one can evaluate the effect of the beam polarization from the helicity-dependent cross sections. Therefore, in the following discussion, we focus on the unpolarized and helicity-dependent cross-sections to exhibit analytical behaviors.

140

7 Higgs Strahlung Process in Electron-Positron Colliders

7.1.4 Tree-Level Contributions At the LO, only one diagram of Fig. 7.2 is relevant in the massless limit of the electron. The contribution of the tree-level diagram to the form factors is expressed as (0) = Fi,σ

g Z h1,tree ZZ gσ f i(0) . s − m 2Z

(7.52)

The couplings g± are defined by 1 2 2 , g− = − + sW . g+ = sW 2

(7.53)

f 1(0) = 1,

(7.54)

The coefficients f i(0) are f 2(0) = f 3(0) = 0.

The lowest-order differential cross-section is given by 2 ⎧    1 β  (0) 2 ⎨2s dσ,± (θ) (λ = ±), dσLO   (σ, λ; s, t) = F d 64π 2 s 1,σ ⎩ s 2 α2 d 1 (θ)2 (λ = 0). 2 σ,0 2m

(7.55)

Z

In the left panel of Fig. 7.3, we show the helicity-dependent cross-sections at the LO as a function of√the CM energy. In the numerical evaluation, we use the treelevel relation v = ( 2G F )−1/2 and take G F as an input parameter. The solid and dashed lines correspond to the transversely (λ = ±) and the longitudinally (λ = 0) polarized Z bosons, respectively. The cross-sections peak just above the thresh√ old s = m Z + m h and monotonically decrease at higher energies. For energies well above the threshold, the longitudinally polarized Z boson dominates the crosssection. This is due to the factor s/m 2Z originating from the longitudinal polarization vector defined in Eq. (7.32). The cross-section for the left-handed electron is larger than that for the right-handed electron because the left-handed electron more strongly 2 couples √to the Z boson than the right-handed electron (g− /g+ )  1.8. At s = 250 GeV, the unpolarized cross-section is 242 fb. The polarized crosssection with (Pe , Pe¯ ) = (−0.8, 0.3) is 379 fb. Therefore, the beam polarization significantly changes the size of the cross-section. √ The angular distributions of the LO cross-section at s = 250 GeV are given 1 (θ). The cross-section in the right panel of Fig. 7.3. They are determined by dσ,λ for the transversely polarized Z boson is proportional to 1 + cos2 θ, and it takes maximal value in the forward-backward direction. On the other hand, that for the longitudinally polarized Z boson is proportional to 1 − cos2 θ, and it vanishes in the forward-backward direction and takes maximal value at cos θ = 0.

7.1 Electroweak Corrections to the Process e+ e− → h Z

141

Fig. 7.3 (Left) Helicity-dependent cross sections of e+ e− → h Z at LO in the SM as a function of the CM energy. The red lines show the results for the left-handed electron and the right-handed positron, while the blue lines show those for the right-handed electron and the left-handed positron. The solid (dashed) lines show the results for the transversely (longitudinally) polarized Z bosons. The black solid line corresponds to that for the unpolarized cross-section where the polarization of the Z boson is also summed. (Right) Helicity-dependent differential cross sections of e+ e− → h Z √ at LO in the SM as a function of cos θ at s = 250 GeV. The line colors and styles are the same as those of the left figure

e+

Z

e−

h (a)

(b)

(c)

(d)

(e)

(f)

Fig. 7.4 NLO corrections for e+ e− → h Z ; a Gauge boson self-energy, b Z ee¯ vertex, c h Z Z and h Z γ vertices, d–e hee¯ vertex, f Box diagrams

We note that values of predicted cross-sections at the LO depend on the input schemes. In our input scheme, the Fermi constant G F is an input parameter, and we obtain m W = 80.93 GeV as output by using the tree-level relation. Since this output value is larger than the observed value m W = 80.38 GeV [5], we have larger cross sections than those evaluated with the observed m W .

142

7 Higgs Strahlung Process in Electron-Positron Colliders

7.1.5 One-Loop Contributions to the Form Factors In Fig. 7.4, we show the NLO corrections for the e+ e− → h Z process. We here omit all Feynman diagrams which contain couplings of the Higgs and NG bosons to the external electron line because they are proportional to m e /m W . The one-loop (1) consist of (a) the Z boson self-energy and the contributions to the form factors Fi,σ Z γ mixing, (b) Z ee¯ vertex corrections, (c) h Z Z and h Z γ vertex corrections, (de) hee¯ vertex corrections, and (f) box diagrams contributions. In addition, we have the term − Z Z (m 2Z )/2 from the wave function renormalization of the on-shell Z boson [6, 7]. We also include the EW correction to the Fermi decay constant r due to the replacement of the EW VEV in the tree-level amplitude by G F , since the tree-level relation between these two parameters is no longer valid at the one-loop level. This replacement corresponds to the resummation of universal higher-order leading corrections such as large logarithms from light fermion masses [8]. (1) are given by The one-loop contributions to the form factors Fi,σ Zγ

hZγ

(1) ZZ Z ee¯ hZ Z hee¯ Box Fi,σ = Fi,σ + Fi,σ + Fi,σ + Fi,σ + Fi,σ + Fi,σ + Fi,σ 

r + Fi,σZ Z + Fi,σ ,

(7.56)

where the terms in the first line correspond to the contributions from the diagrams in Fig. 7.4, while the terms in the second line come from the renormalization procedure. The calculations of these EW corrections are performed based on the on-shell renormalization scheme discussed in Chap. 5. Apart from the UV divergences, there are infrared (IR) divergences in virtual photon loop contributions. In the calculation of individual photon loop contributions, we regularize them with a finite photon mass μ. The photon mass dependences in the one-loop calculation are exactly canceled by adding contributions of soft-photon emissions. The analytic expression of the real photon contribution with the softphoton approximation is given by Refs. [2, 8, 9], dσsoft

  

 α m 2e 4E 2 1 2 m 2e m 2e π2 1 + ln ln + ln + ln + , = dσLO − π μ2 s 2 s s 3 (7.57)

with the photon energy cutoff E. The dependence of E vanishes in the inclusive cross section where one also includes the contribution of hard photon emissions [8]. The inclusive cross section still depends on ln(m 2e /s), and this logarithmic term potentially takes a large value. This dependence can also be eliminated by introducing the electron structure functions as discussed in [10]. However, the treatment of hard photon emission highly depends on the experimental setup. The hard photon changes the kinematics of the process, and these effects would be eliminated by applying appropriate experimental cuts. In addition, if one considers the scenario with κhZ  1, these effects in extended Higgs models are almost the same as in the SM. Therefore,

7.1 Electroweak Corrections to the Process e+ e− → h Z

143

we do not consider the electromagnetic effects when we focus on the difference between the predictions in the extended models and those in the SM.

7.1.5.1

Expression of Form Factors Including One-Loop Corrections

We list the one-loop contributions to the form factors in terms of the renormalized quantities. The one-loop propagator corrections appear in the sum in Eq. (7.56) as the term   1,tree T  TZ γ (s)    g  (s) Z hZ Z SE Fi,σ = (7.58) −gσ Z Z 2 − Q e sW cW f i(0) , 2 s s − mZ s − mZ  TV V  (s) of the neutral vector bosons. The renorwith the renormalized self-energies  malized Z ee¯ corrections appear as Z ee¯ = Fi,σ

 V   g Z h1,tree ZZ   Z ee¯ − σ  ZAee¯ (m 2e , m 2e , s) f i(0) . 2 s − mZ

(7.59)

The renormalized h Z V (V = Z , γ) corrections appear as hZV = Fi,σ

gZ gZ γ(1) Q e sW cW f i (s), g f Z (1) (s) + 2 σ i s s − mZ

(7.60)

with h1 Z Z (m 2Z , s, m 2h ), f 1Z (1) (s) =  1 2 f 2Z (1) (s) = − 2   (m 2 , s, m 2h ), mZ hZ Z Z γ(1) f 1 (s) =  h1 Z γ (m 2Z , s, m 2h ),

(7.62)

1 2   (m 2 , s, m 2h ). m 2h h Z γ Z

(7.64)

γ(1)

f2

(s) = −

(7.61)

(7.63)

The renormalized hee¯ corrections appear as hee¯ Fi,σ = −g Z gσ



! !  (0) V1 2 2   A1 V2  A2 he e¯ + σ hee¯ (t, 0, m h ) − hee¯ + σ hee¯ (0, u, m h ) f i . (7.65)

The 1/t and 1/u terms originated from the fermion propagator are canceled by the vertex corrections [2]. However, the renormalized vertices depend on t and u, and they cause non-trivial cos θ dependence. There are five W boson-mediated and one Z boson-mediated box diagrams in the massless limit of the electron. The amplitudes of the W boson-mediated diagrams

144

7 Higgs Strahlung Process in Electron-Positron Colliders

can be written as  κV δσ− C k Fik (s, t)Mi,σλ , (k = 1, 2, · · · , 5), 2 16π i=1 3

Mkσλ =

(7.66)

with F1k (s, t) = F k (s, t),  1 k Fe (s, t) + Fe¯k (s, t) , F2k (s, t) = 2  1 k F3k (s, t) = F (s, t) − Fe¯k (s, t) . 2 e

(7.67) (7.68) (7.69)

The amplitude of the Z boson-mediated diagram has a different structure from others, and it can be written as M6σλ =

3 κV  3 6 6 g C Fi (s, t)Mi,σλ , 16π 2 i=1 σ

(7.70)

with F16 (s, t) = F 6 (s, t),  1 6 F26 (s, t) = F (s, t) + Fe¯6 (s, t) , 2 e  1 6 F36 (s, t) = F (s, t) − Fe¯6 (s, t) . 2 e

(7.71) (7.72) (7.73)

The expressions of C k and Fik (s, t) are given in Appendix D.4.1. In summary, the form factors at the one-loop level are given by 

 TZ Z (s)  V  2 2  A   h1,tree −g (m +  − σ  , m , s) σ Z ee¯ Z ee¯ e e ZZ s − m 2Z  f i(0) + gσ f i Z (1) (s)    TZ γ (s)  eQ e γ(1) (0) 1,tree f + f i (s) + −h Z Z s s − m 2Z i  ! !  V1 A1 V2 A2 2 2    − g Z gσ  he (t, 0, m (0, u, m + σ  ) −  + σ  ) f i(0) h h e¯ hee¯ hee¯ hee¯

g Z h1,tree 1 ZZ  2 − Re + Bi,σ (s, t) + g (m ) − r f i(0) , (7.74) σ ZZ Z 2 s − m 2Z

(1) Fi,σ =

gZ s − m 2Z

7.2 Numerical Results for the Cross-section

145

where Bi,σ (s, t) is given by   5  κV Bi,σ (s, t) = C k Fik (s, t) + gσ3 C 6 Fi6 (s, t) . δσ− 16π 2 k=1

(7.75)

7.2 Numerical Results for the Cross-section In this section, we begin with an analysis of the behavior of the NLO weak corrections to the cross-section of the e+ e− → h Z process in the SM. We then discuss the deviations from the SM predictions at NLO in the 2HDMs. (1) into the H-COUP program [11, We have newly implemented the formulae of Fi,σ 12], and enabled to evaluate the cross section of e+ e− → h Z including the higherorder corrections. In order to compare our results in the SM with the previous works [2, 13–15], we extend the H-COUP program and take m W as an input instead of G F . With this extension, we have confirmed that our results are in agreement with the previous results. In the following, we show the results obtained in the scheme where αem (0), G F and m Z are input parameters.

7.2.1 Standard Model We parametrize the one-loop corrections to the cross-section as σ(s) = σLO (1 + δweak + δem ),

(7.76)

where δweak and δem denote the relative weak and electromagnetic corrections, respectively. In the left panel of Fig. 7.5, we show δweak as a function of the CM energy. The weak corrections to the cross-section for the right-handed electron are positive, and they increase the cross-section by about 10%. On the other hand, those for the lefthanded electron are negative, and the size of these corrections strongly depends on the CM energy. The difference between the NLO corrections for right-handed and left-handed electrons comes from the negative contributions of the box diagrams. Among the six box diagrams, the five W boson-mediated diagrams only contribute to the helicity amplitudes for the left-handed electron, and they give negative contributions to the helicity-dependent cross sections. Their effects become √ relevant at higher energies and give large negative corrections. The peak around s  350 GeV corresponds to the threshold at 2m t in the top-loop contributions. In Refs. [14, 15], NNLO electroweak-QCD corrections have been estimated and the magnitude is about a percent.

146

7 Higgs Strahlung Process in Electron-Positron Colliders

Fig. 7.5 (Left) Weak corrections to the helicity-dependent cross sections in the SM. The red lines show the results for the left-handed electron and the right-handed positron, while the blue lines show those for the right-handed electron and the left-handed positron. The solid (dashed) lines show the results for the transversely (longitudinally) polarized Z bosons. The black solid line corresponds to that for the unpolarized cross-section where the polarization of the Z boson is also summed. (Right) √ Weak corrections to the helicity-dependent differential cross sections in the SM at s = 250 GeV. The line colors and styles are the same as those of the left figure

In the right panel of Fig. 7.5, we show the weak one-loop corrections to the differential cross sections as a function of the CM energy. From Eq. (7.75), we can see that only the hee¯ vertex √ and box corrections cause different cos θ dependence from those at the LO. At s = 250 GeV, this effect is not so large, and the angular 1 (θ) functions. At higher distribution of the Z boson is almost determined by the dσ,λ energies, the angular distribution of the Z boson is significantly modified through the box contributions [2]. However, the size of the cross sections decreases in such a higher energy region.

7.2.2 Two Higgs Doublet Model As we will discuss later, new physics contributions mainly come from  h1 Z Z vertex, and that appear independently of σ and λ. Therefore, we show the results of the unpolarized cross-section. In order to analyze new physics contributions in each EW renormalized quantity, we introduce  X defined as EW

X

EW = EW ¯ h Z Z , h Z γ, hee, ¯ Box, Z Z , r ), X,NP −  X,SM , (X = Z Z , Z γ, Z ee,

(7.77)

7.2 Numerical Results for the Cross-section

147

with EW X = σ X /σLO . We evaluate σ X by substituting M(1)X σλ (s, t) =

3 

X Fi,σ Mi,σλ

(7.78)

i=1 X into M(1) σλ (s, t) in Eq. (7.47), where Fi,σ is defined in Eq. (7.56). We also evaluate the ratios of the total cross-sections in the 2HDMs and that in the SM to exhibit the deviations from the SM prediction,

R h Z =

σNP − 1. σSM

(7.79)

First, we consider the alignment limit with sβ−α = 1, where R h Z = 0 at LO. We also assume that the additional Higgs bosons are degenerate in their mass, m  ≡ m H = m H ± = m A . We scan the remaining two input parameters, tan β and M 2 , under the perturbative unitarity bound, the vacuum stability bound, and the constraints on the S and T parameters. In order to analyze the theoretical behavior, we here dare to omit the constraints from the direct searches of the additional Higgs boson, the Higgs coupling measurements, and the flavor measurements. We show the predictions for R h Z in the Type-I 2HDM as a representative. This EW is because we have found that predictions for  X and R h Z are almost the same in all types of 2HDMs. Differences among the four types of 2HDMs appear through the down-type quark and lepton Yukawa interactions with the SM-like Higgs boson. In a EW dominant contribution, h Z Z , the top-quark loop dominates fermionic contributions, and there is no sizable difference among the four types of 2HDMs. hZ of In the left panel √ of Fig. 7.6, we show the predictions for R as a function m  in Type-I at s = 250 GeV. We here take tan β to 1.5, 3 and 5 and scan M 2 for 0 ≤ M 2 ≤ (3 TeV)2 . The sign of R h Z is negative except for m  ≤ 200 GeV. In the case with tan β = 1.5, the magnitude of R h Z becomes larger when m  is taken to be larger up to m  ≈ 400 GeV. The peak around m  ≈ 400 GeV corresponds to the point where the minimum value of M 2 changes from zero to non-zero due to the theoretical bounds. For 400 ≤ m  , the magnitude of R h Z monotonically decreases due to the theoretical constraints. In such a large mass region, m  is approximately equal to M, and the additional Higgs boson almost decouples following the decoupling theorem [16]. In the case with tan β = 3 and 5, we have similar behavior with tan β = 1.5. However, a possible magnitude of R h Z becomes smaller than that with tan β = 1.5 due to the stronger theoretical constraints. EW In the right panel of Fig. 7.6, we show  X as a function of m  . We here take tan β = 1.5 and scan M 2 for 0 ≤ M 2 ≤ (3 TeV)2 . We can see that the renormalized h Z Z vertex gives dominant contributions among the new physics contributions. The main new physics contributions come from the wave function renormalization factor of the SM-like Higgs boson δ Z h , which is determined by the self-energy of the 2 SM-like Higgs boson 1PI hh ( p ) as

148

7 Higgs Strahlung Process in Electron-Positron Colliders

Fig. 7.6 NP effects in the EW corrections as a function of masses of the additional Higgs bosons in √ the Type-I 2HDM with sβ−α = 1 at s = 250 GeV. We take m H = m A = m H ± . In the left panel, EW

we show R h Z with tan β = 1.5 (red), 3 (blue) and 5 (green). In the right panel, we show  X with tan β = 1.5. The solid and dashed curves denote the case with the maximal and minimal value of M 2 , respectively. The perturbative unitarity bound, the vacuum stability bound, and the constraints on the S and T parameters are imposed

 d 1PI 2  δ Z h = − 2 hh ( p ) . dp p2 =m 2h

(7.80)

2 In 1PI hh ( p ), there are two kinds of diagrams including Higgs self-interactions. One of them is proportional to λhh while the other is proportional to λ2h , where  = H, A, H ± . While the former does not contribute to δ Z h because the loop function A(m  ) does not depend on the external momentum, the latter contributes to δ Z h as,

δ Z h2HDM

−

δ Z hSM

=−

 =H,A,H ±

  λ2h d 2 2 2  B0 ( p ; m  , m  ) . 16π 2 d p 2 p2 =m 2h

(7.81)

These contributions give dominant contributions for R h Z if λh take large values. EW If m  ≤ 150 GeV,  X can be positive due to the 1PI diagram contributions. EW While δ Z h gives negative contributions to h Z Z , 1PI diagrams for the h Z Z vertex give positive contributions. In general, δ Z h gives larger contributions than 1PI diagram contributions. However 1PI diagram contributions can be larger than the EW contribution of δ Z h in the small m  regions, and h Z Z becomes positive. Next, we consider the non-alignment scenario with sβ−α = 1. In the top (bottom) left panel of Fig. 7.7, we show√R h Z as a function of m  in Type-I with sβ−α = 0.99 and cβ−α < 0 (cβ−α > 0) at s = 250 GeV. We here take tan β to 1.5, 3 and 5 and scan M 2 for 0 ≤ M 2 ≤ (3 TeV)2 . The LO cross-section decreases from its SM value 2  −0.02 at the LO due to the mixing of the CP-even scalars, and R h Z = −cβ−α when sβ−α = 0.99. We can see that the magnitude of one-loop effects is comparable with that of the LO contribution, and the NP effects sizably change the predictions for

7.2 Numerical Results for the Cross-section

149

Fig. 7.7 NP effects in the EW corrections as √ a function of masses of the additional Higgs bosons in the Type-I 2HDM with sβ−α = 0.99 at s = 250 GeV. We take m H = m A = m H ± . The top EW

left panel shows  X with tan β = 1.5 and cβ−α < 0. The top right panel shows R h Z with tan β = 1.5 (red), 3 (blue) and 5 (green) and cβ−α < 0. The bottom panels correspond to the case with cβ−α > 0. The solid and dashed curves denote the case with the maximal and minimal value of M 2 , respectively. The black dashed line shows the size of the LO deviation due to the mixing. Perturbative unitarity and vacuum stability bounds and the constraints on the S and T parameters are imposed

R h Z . In the non-alignment case, NP effects do not show the decoupling behavior in large m  regions. The magnitude of R h Z increases up to m  ≤ 900 GeV, above which we have no parameter regions satisfying the theoretical constraints. This is because non-zero mixing between the CP-even states leads to larger Higgs quartic couplings in the larger m  region. Since possible sizes of the Higgs quartic couplings are theoretically constrained, we have an upper bound on m  . The maximal value of m  is about 900 GeV for sβ−α = 0.99 with cβ−α < 0, while it is about 600 GeV with cβ−α > 0 for tan β = 1.5. In both signs of cβ−α , NLO corrections increase the magnitude of R h Z except for the region with a relatively lighter m  . The possible magnitude of R h Z decreases as tan β becomes large due to the theoretical constraints.

150

7 Higgs Strahlung Process in Electron-Positron Colliders

Table 7.1 Expected 1σ accuracy for the SM-like Higgs boson measurements at the ILC σ(e+ e− → h → bb¯ h → cc¯ h → τ τ¯ h → W W∗ h → Z Z∗ h → γγ Z h) 2% 1.3% 8.3% 3.2% 4.6% 18% 34% √ We quote the values at s = 250 GeV with 250 fb−1 for (Pe , Pe¯ ) = (−0.8, +0.3) in Table VI in [17]. Except for σ(e+ e− → Z h), the numbers correspond to the accuracy of σ(e+ e− → h Z ) × BR(h → X Y )

EW

In the top (bottom) right panel of Fig. 7.7, we show  X as a function of m  . We here take tan β = 1.5 and scan M 2 for 0 ≤ M 2 ≤ (3 TeV)2 . Similar to the alignEW ment case, the magnitude of h Z Z is larger than that of the other loop contributions independently of the sign of cβ−α . Finally, we mention the corrections to the angular distribution of the Z boson. At the NLO, the hee¯ vertex and the box contributions cause non-trivial cos θ dependence. In the limit of the massless electron, only the mixing of CP-even states modifies these contributions. However, as we can see from√ Figs. 7.7, the sizes of the hee¯ vertex and the box contributions are rather small at s = 250 GeV. Therefore, the angular distribution of Z bosons is almost the same as the predictions in the SM.

7.3 Numerical Results for the Cross Section Times Decay Branching Ratios In this section, we discuss predictions for the cross-section times decay branching ratios of the SM-like Higgs boson, σ(e+ e− → h Z ) × BR(h → X Y ), in the 2HDM including the NLO EW corrections. At future collider experiments such as the ILC, the cross-section of e+ e− → h Z can be measured without depending on the decay of the SM-like Higgs boson by utilizing the recoil mass technique [17, 18]. In addition, the cross-section times decay branching ratios of the SM-like Higgs boson can also be measured precisely. In Table 7.1, we summarize the expected accuracy of the cross√section times decay branching ratios of the SM-like Higgs boson at the ILC at s = 250 GeV with 250 fb−1 for (Pe , Pe¯ ) = (−0.8, +0.3). The values in Table 7.1 are taken from Table VI in [17]. In order to discuss deviations from the SM prediction, we evaluate the ratio of the total cross-section times the decay branching ratios, R hXZY =

σNP (e+ e− → h Z )BRNP (h → X Y ) − 1, σSM (e+ e− → h Z )BRSM (h → X Y )

(7.82)

where we assume the beam polarization (Pe , Pe¯ ) = (− 0.8, + 0.3). In the evaluation of the decay branching ratios of the SM-like Higgs boson with the one-loop EW and QCD corrections, we use the H-COUP program [11, 12]. We omit the QED

7.3 Numerical Results for the Cross Section Times Decay Branching Ratios

151

corrections in the evaluation for the total cross-section. The correlations of R hXZY are not changed even if we include the QED corrections following a realistic experimental setup. This is because they universally change the magnitude of σ(e+ e− → h Z ) × BR(h → X Y ) in the SM and the 2HDMs. We scan the six input parameters in the 2HDM following the constraints from the direct searches of the additional Higgs bosons and flavor constraints. For simplicity, we assume that the additional Higgs bosons are degenerate in their mass as in the previous subsection. The degenerate mass scale m  (= m H ± = m H = m A ) is scanned as 400 GeV ≤ m  < 2000 GeV for the Type-I and X 2HDMs, 800 GeV ≤ m  < 2000 GeV for the Type-II and Y 2HDMs.

(7.83) (7.84)

The lower bound of m  in the Type-I and Type-X 2HDMs comes from the direct search for A → τ τ¯ at the LHC [19]. While the parameter regions with tan β  2 are not excluded in Type-I, we take the above parameter regions for simplicity. In Type-II and Type-Y, the lower bound comes from the flavor experiments, especially from Bs → X s γ [20]. The other parameters are scanned as 0.98 ≤ sβ−α < 1, 2 ≤ tan β < 10, 0 ≤ M 2 < (m  + 250 GeV)2 .

(7.85)

The lower bound of tan β comes from the flavor experiments. We analyze both the positive and negative signs of cβ−α . In addition, over the above parameter spaces, we impose the constraints discussed in Chap. 3 such as perturbative unitarity, vacuum stability, and the constraints on the S and T parameters. We also take into account the current data of the signal strengths of the discovered Higgs boson at the LHC. We evaluate the decay rates of the SM-like Higgs boson with higher-order corrections by using the H-COUP program [11, 12]. We define the scaling factor at the one-loop level as " κX Y =

 2HDM (h → X Y ) .  SM (h → X Y )

(7.86)

We remove the parameter points where κ X Y deviates from the observed data at 95% CL. In Table 2.4, we summarize the current measurements of κ X Y factors at 1σ accuracy. In Type-II, Type-X, and Type-Y, we have parameter points where Yukawa coupling constants take the negative sign with a large value of tan β and cβ−α > 0, and we would have sizable deviations both in the Higgs branching ratio and the cross-section in such parameter points. However, we simply omit such particular parameter points in the following analysis to extract general features in the 2HDMs.

152

7 Higgs Strahlung Process in Electron-Positron Colliders

h Z in the Type-I (red), Type-II (blue), Type-X (green), Fig. 7.8 Correlation between RτhτZ and Rbb Type-Y (purple) 2HDMs. The left panel shows the results with cβ−α < 0 in the 2HDMs, and the right panel shows those with cβ−α > 0. The ranges of the parameters are explained in the text. The lighter color corresponds to the lighter mass scale of the additional Higgs bosons, m  ≥ 400, 800, 1200, and 1600 GeV in order

Before moving on to the numerical results, we discuss the general property of R hXZY . We can rewrite the ratio R hXZY as R hXZY = R Z h + R X Y + R Z h R X Y ,

(7.87)

with R Z h defined in Eq. (7.79). We define R X Y as R X Y =

BRNP (h → X Y ) − 1. BRSM (h → X Y )

(7.88)

The order of loop expansion of R Z h R X Y is O(2 ), and higher than R Z h and R X Y . Therefore, the qualitative behavior of R hXZY can be understood by independently analyzing R Z h and R X Y . The behaviors of R X Y have been discussed in Chap. 6. hZ in the four types In Fig. 7.8, we show the correlations between RτhτZ and Rbb of 2HDMs. We take the color codes where red, blue, green, and purple correspond to Type-I, Type-II, Type-X, and Type-Y, respectively. The lighter color corresponds to the lighter mass scale of the additional Higgs bosons, m  ≥ 400, 800, 1200, and 1600 GeV in order. In the left (right) panel, we show the results with cβ−α < 0 (cβ−α > 0). As we have discussed in Sec. 7.2, R Z h takes a negative value in most cases independently of the types of 2HDMs. On the other hand, a sign of R X Y depends on the types of 2HDMs. In Type-I with cβ−α < 0, both of Rbb and Rτ τ are hZ and RτhτZ . In Type-II with negative, and R Z h increase the magnitudes of Rbb cβ−α < 0, both of Rτ τ and Rbb are positive, while Rτ τ and Rbb are positive in Type-X and Type-Y, respectively. Therefore, R Z h decreases the magnitudes of

7.3 Numerical Results for the Cross Section Times Decay Branching Ratios

153

h Z . Color codes and the ranges of the parameters are Fig. 7.9 Correlation between RτhτZ and Rcc the same as in those of Fig. 7.8

hZ Rbb and/or RτhτZ in Type-II, Type-X and Type-Y. However, a typical size of R X Y is larger than R Z h , and we can distinguish the four-types of 2HDMs by analyzing hZ and RτhτZ . the correlation between Rbb hZ In the case with cβ−α > 0, the directions of the deviations in RτhτZ and Rbb are the same in Type-I and Type-II, while Type-X and Type-Y show the different correlations. This is because the typical size of R Z h is larger than R X Y in Type-I. While Rτ τ and Rbb tend to be positive in Type-I, R Z h makes both of RτhτZ hZ and Rbb tend to be negative. hZ . The color codes In Fig. 7.9, we show the correlations between RτhτZ and Rcc and gradations are the same as those in Fig. 7.8. In the left (right) panel, we show the results with cβ−α < 0 (cβ−α > 0). The qualitative behavior of the deviations in each model is the same as in Fig. 7.8. In Type-II, Type-X, and Type-Y, a typical size of R X Y is larger than R Z h , and the pattern of the deviation is mainly determined by R X Y . On the other hand, we also have sizable deviations in Type-I, and they reach about 10%. In Type-I with cβ−α > 0, R X Y takes a positive value. However, the typical size of R X Y is smaller than R h Z , and R XZ Yh takes a negative value in most of the parameter regions. hZ If cβ−α is negative, the directions of the deviations in RτhτZ and Rcc are the same in Type-I and Type-Y. This overlap can be resolved by looking at the correlation hZ , where Type-Y shows a different correlation with Typebetween RτhτZ and Rbb I. On the other hand, if cβ−α is positive, the directions of the deviations in RτhτZ hZ and Rbb are the same in Type-I and Type-II. This overlap can also be resolved hZ where the Type-II 2HDM by looking at the correlation between RτhτZ and Rcc shows a different correlation with others.

154

7 Higgs Strahlung Process in Electron-Positron Colliders

References 1. Masashi A, Shinya K, Kentarou M (2021) Next-to-leading-order corrections to the Higgs strahlung process from electron-positron collisions in extended Higgs models. Eur Phys J C 81(11):1000 2. Denner A, Kublbeck J, Mertig R, Bohm M (1992) Electroweak radiative corrections to e+ e− HZ. Z Phys C 56:261–272 3. Fujii K et al (2017) Physics case for the 250 GeV stage of the international linear collider 4. Fujii K et al (2018) The role of positron polarization for the inital 250 GeV stage of the international linear collider 5. Zyla PA et al (2020) Review of particle physics. PTEP 2020(8):083C01 6. Bohm M, Spiesberger H, Hollik W (1986) On the one loop renormalization of the electroweak standard model and its application to leptonic processes. Fortsch Phys 34:687–751 7. Hollik WFL (1990) Radiative corrections in the standard model and their role for precision tests of the electroweak theory. Fortsch Phys 38:165–260 8. Kniehl Bernd A (1992) Radiative corrections for associated Z H production at future e+ e− colliders. Z Phys C 55:605–618 9. Fleischer J, Jegerlehner F (1983) Radiative corrections to Higgs production by e+ e− → ZH in the Weinberg-Salam model. Nucl Phys B 216:469–492 10. Xie W, Benbrik R, Habjia A, Taj S, Gong B, Yan Q-S (2021) Signature of 2HDM at Higgs factories. Phys Rev D 103(9):095030 11. Shinya K, Mariko K, Kodai S, Kei Y (2018) H-COUP: a program for one-loop corrected Higgs boson couplings in non-minimal Higgs sectors. Comput Phys Commun 233:134–144 12. Shinya K, Mariko K, Kentarou M, Kodai S, Kei Y (2020) H-COUP version 2: a program for one-loop corrected Higgs boson decays in non-minimal Higgs sectors. Comput Phys Commun 257:107512 13. Belanger G, Boudjema F, Fujimoto J, Ishikawa T, Kaneko T, Kato K, Shimizu Y (2006) Automatic calculations in high energy physics and Grace at one-loop. Phys Rept 430:117–209 14. Sun Q-F, Feng F, Yu J, Sang W-L (2017) Mixed electroweak-QCD corrections to e+ e− → HZ at Higgs factories. Phys Rev D 96(5):051301 15. Gong Y, Li Z, Xu X, Yang LL, Zhao X (2017) Mixed QCD-EW corrections for Higgs boson production at e+ e− colliders. Phys Rev D 95(9):093003 16. Appelquist T, Carazzone J (1975) Infrared singularities and massive fields. Phys Rev D 11:2856 17. Baer H (2013) The international linear collider technical design report. Physics 2:6 18. Yan J, Watanuki S, Fujii K, Ishikawa A, Jeans D, Strube J, Tian J, Yamamoto H (2016) Measurement of the Higgs boson mass and e+ e− → ZH cross section using Z → μ+ μ− and Z → e+ e− at the ILC. Phys Rev D 94(11):113002. [Erratum: Phys Rev D 103, 099903 (2021)] 19. Masashi A, Shinya K, Mariko K, Kentarou M, Kodai S, Kei Y (2021) Probing extended Higgs sectors by the synergy between direct searches at the LHC and precision tests at future lepton colliders. Nucl Phys B 966:115375 20. Misiak M, Rehman A, Steinhauser M (2020) Towards B → X s γ at the NNLO in QCD without interpolation in mc . JHEP 6:175

Chapter 8

One-Loop Calculations for Decays of the Charged Higgs Bosons

In this chapter, we discuss the decay rates of charged Higgs bosons for various decay modes in the 2HDMs. Decay branching ratios of charged Higgs bosons are evaluated including NLO EW corrections, as well as QCD corrections up to NNLO. We have newly implemented them into the H-COUP program [1, 2]. We comprehensively study the impacts of the NLO EW corrections on the branching ratios in approximate alignment scenarios. We find that the H ± → W ± h decay modes can be dominant decay modes even in the case where deviation in h Z Z coupling is quite small and cannot be detected at the ILC. Thus, we can extract the information on the mixing angle by studying the H ± → W ± h decay modes at future collider experiments.

8.1 Decay Rates with Higher Order Corrections In this section, we first define renormalization vertex functions of the charged Higgs bosons. The decay rates for H ± → f f¯ and H ± → W ± φ (φ = h, H, A) with NLO EW corrections are given in terms of form factors of the vertex functions. The decay rates for the loop-induced processes, H ± → W ± V (V = Z , γ), are given in Chap. 4. The QCD corrections on the decay into a pair of quarks, H ± → Q Q¯  , are also given in Chap. 4. We perform the NLO calculations based on the on-shell renormalization scheme discussed in Chap. 5. For the H ± → Q Q¯  decay, we neglect quark-mixing contributions and do not perform a renormalization of the CKM matrix. We regularize the IR divergences, which appear in one-loop diagrams containing a virtual photon, by introducing the finite photon mass. The photon-mass dependences are canceled by adding the decay rates of the real-photon emission processes.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 M. Aiko, Theoretical Studies on Extended Higgs Sectors Towards Future Precision Measurements, Springer Theses, https://doi.org/10.1007/978-981-99-1324-4_8

155

156

8 One-Loop Calculations for Decays of the Charged Higgs Bosons

Fig. 8.1 Momentum assignment for the renormalized H + f f¯ vertex

p1

f

p2

f

q H+

8.1.1 Form Factors for Vertex Functions of the Charged Higgs Bosons 8.1.1.1

H ± f f  Vertex

The renormalized H ± f f  vertex functions can be expressed as H ± f f  ( p1 , p2 , q) = HS ± f f  + γ5HP ± f f  + /p V1±  + /p V2±  1 H ff 2 H ff + /p γ5 A1±  + /p γ5 A2±  1

H ff

2

H ff

T + /p 1 /p 2H p 1 /p 2 γ5HP T± f f  , ± ff + /

(8.1)

where p1 ( p2 ) is the incoming four-momentum of the fermion f  (= D, L) (the SU (2) partner fermion f (= U, N )), and q is the outgoing four-momentum of the charged Higgs boson (see Fig. 8.1). The renormalized form factors are composed of the tree-level and the one-loop level parts as X,loop HX ± f f  =  X,tree H ± f f  +  H ± f f  , (X = S, P, V1 , V2 , A1 , A2 , T, P T ),

(8.2)

where the tree-level couplings for H ± f f  vertex are given by m f ζ f − m f ζ f  m f ζ f + m f ζ f  ,  HP,tree , √ √ ± f f  = ∓V f f  2v 2v = 0, (X = S, P).

 HS,tree ± f f  = ±V f f 

(8.3)

 HX,tree ± ff

(8.4)

The one-loop parts are further decomposed into contributions from 1PI diagrams and counterterms, X,loop

X  H ± f f  =  HX,1PI ± f f  + δ H ± f f  .

(8.5)

The 1PI diagrams contributions  HX,1PI ± f f  are given in Appendix D.3.5. The counterterms, δ HX ± f f  , are given by

8.1 Decay Rates with Higher Order Corrections S/P

δ H ± f f  =

157

 1 R L X δ H δ H ± f f  ± δ H (X = S, P). ± ff , ± f f  = 0, 2

(8.6)

with   f f δζ f  2m f  ζ f  δm f  δZR + δZL δ Z H+ H− δv 1 δ Z G+ H − + + + − + v m f v ζf ζf 2 2 2 √   2m f  ζ f  δm f  δv 1 δ Z G+ H − − =∓ − ζ f  δβ + − δβ v m f v ζf 2   f f δZR + δZL δ Z H+ H− + (8.7) + , 2 2   √ f f δZR + δZL 2m f ζ f δm f δζ f δv 1 δ Z G+ H − δ Z H+ H− =± − + + + + v mf v ζf ζf 2 2 2 √   2m f ζ f δm f δv 1 δ Z G+ H − =± − − ζ f δβ + − δβ v mf v ζf 2  f f δZR + δZL δ Z H+ H− . (8.8) + + 2 2 √

R δ H ± ff = ∓

L δ H ± ff

When we neglect the effects of the CP violation, the renormalized H ± f f  vertex functions satisfy the following relations due to the CP invariance. S + ¯  (m 2f  , m 2f , m 2H ± ) = +S − ¯ (m 2f  , m 2f , m 2H ± ), H f f H f f

(8.9)

 P + ¯  (m 2f  , m 2f , m 2H ± ) = − P − ¯ (m 2f  , m 2f , m 2H ± ), H f f H f f

(8.10)

V1+ ¯  (m 2f  , m 2f , m 2H ± ) = −V2− ¯ (m 2f  , m 2f , m 2H ± ), H f f H f f

(8.11)

V2+ ¯  (m 2f  , m 2f , m 2H ± ) H f f  A1+ ¯  (m 2f  , m 2f , m 2H ± ) H f f  A2+ ¯  (m 2f  , m 2f , m 2H ± ) H f f

−HV1− f¯ f (m 2f  , m 2f , m 2H ± ), −HA2− f¯ f (m 2f  , m 2f , m 2H ± ), −HA1− f¯ f (m 2f  , m 2f , m 2H ± ),

(8.12)

T + ¯  (m 2f  , m 2f , m 2H ± ) = +T − ¯ (m 2f  , m 2f , m 2H ± ), H f f H f f

(8.15)

 P T+ ¯  (m 2f  , m 2f , m 2H ± ) H f f

(8.16)

= = = =

−HP T− f¯ f (m 2f  , m 2f , m 2H ± ).

(8.13) (8.14)

We note that the moemntum arguments are p1 = p f  and p2 = p f in the H + f¯ f  vertex, while p1 = p f and p2 = p f  in the H − f¯ f vertex.

8.1.1.2

H ± W ∓ φ Vertex

The renormalized H ± W ∓ φ vertex functions (φ = h H A) can be expressed as μ ± ∓ ( p1 , p2 , q) = ( p1 + q)μH ± W ∓ φ , H W φ

(8.17)

158

8 One-Loop Calculations for Decays of the Charged Higgs Bosons

Fig. 8.2 Momentum assignment for the renormalized H + W −μ φ vertex

p1

φ

p2

Wµ+

q H

+

where p1 and p2 denote the incoming four-momentum of the scalar boson φ and the W boson, respectively. The momentum q is the outgoing four-momentum of the charged Higgs boson (see Fig. 8.2). Since we assume that the external W boson is onshell, the term proportional to p2 vanishes due to the orthogonality of the polarization vector. The renormalized form factors are composed of the tree-level and the one-loop level parts as loop H ± W ∓ φ =  tree H ± W ∓φ + H ± W ∓φ,

(8.18)

where  tree H ± W ∓ φ are given by mW cβ−α , v mW sβ−α , = −ig H H ± W ∓ = ± v mW . = −ig AH ± W ∓ = i v

 tree H ± W ∓ h = −igh H ± W ∓ = ∓

(8.19)

 tree H±W ∓ H

(8.20)

 tree H±W ∓ A

(8.21)

The one-loop level parts are further decomposed into contributions from 1PI diagrams and counterterms, loop

 H ± W ∓ φ =  1PI H ± W ∓ φ + δ H ± W ∓ φ .

(8.22)

The 1PI diagram contributions to these vertex functions are given in Appendix D.3.6. The counterterms δ H ± W ∓ φ are given as 

δm 2W 1 δv + (δ Z W + δ Z H + H − + δ Z hh ) − 2 v 2 2m W

 δ Z Hh δ Z G+ H − − δβ − + δα , + tan (β − α) 2 2  2 δm W 1 δv + (δ Z W + δ Z H + H − + δ Z H H ) =  tree − H±W ∓ H v 2 2m 2W

 δ Zh H δ Z G+ H − − δβ + + δα , − cot (β − α) 2 2

δ H ± W ∓ h =  tree H ±W ∓h

δ H ± W ∓ H

(8.23)

(8.24)

8.1 Decay Rates with Higher Order Corrections

 δ

H±W ∓ A

=

 tree H±W ∓ A

δm 2W δv 1 + − − + (δ Z W + δ Z H H + δ Z A A ) . v 2 2m 2W

159

(8.25)

When we neglect the effects of the CP violation, the renormalized H ± W ∓ φ vertex functions satisfy the following relations due to the CP invariance. H + W − h (m 2h , m 2W , m 2H ± ) = −H − W + h (m 2h , m 2W , m 2H ± ), H + W − H (m 2H , m 2W , m 2H ± ) = −H − W + H (m 2H , m 2W , m 2H ± ),

(8.27)

H + W − A (m 2A , m 2W , m 2H ± ) = −H − W + A (m 2A , m 2W , m 2H ± ).

(8.28)

(8.26)

8.1.2 Decay Rates for H ± → f f  The decay rates for H ± → f f  are given by 2

2  2 √2G m ± m m  f f F H   λ1/2 (H ± → f f¯ ) = Ncf V f f   , 8π m 2H ± m 2H ±

  m 2f + m 2f   2 2  RR EW m 1 +  ζ +  × 1−   f f QCD RR m 2H ±   LL + m 2f ζ 2f 1 + QCD + EW LL  

m f m f RL EW   + 4 2 m f m f ζ f ζ f 1 + QCD +  R L m H± + (H ± → f f¯ γ) (8.29) The QCD corrections QCD X X (X X = L L , R R, R L) are discussed in Sect. 4.4.1. The (X X = L L , R R, R L) are given by EW corrections EW XX    H + G − (m 2 ± ) 2 Re S,loop P,loop H Re G H ± f f  + G H ± f f  − − r, ζf m 2H ± 2m f  ζ f  V f f  (8.30)   2  2 Re H + G − (m H ± ) 2v S,loop P,loop =√ Re G H ± f f  − G H ± f f  − − r, ζf m 2H ± 2m f ζ f V f f  (8.31) v = −√ 2m f m f  ζ f ζ f  V f f       S,loop P,loop S,loop P,loop m f ζ f Re G H ± f f  + G H ± f f  − m f  ζ f  Re G H ± f f  − G H ± f f 

EW RR = − √

EW LL

EW RL

2v

160

8 One-Loop Calculations for Decays of the Charged Higgs Bosons

 −

1 1 + ζf ζf S,loop



 H + G − (m 2 ± ) Re H − r. m 2H ±

(8.32)

P,loop

The functions G H ± f f  and G H ± f f  are given in terms of the form factors for the H ± f f  vertex as S,loop

S,loop

V 1,loop

V 2,loop

G H ± f f  = H ± f f  + m f  H ± f f  − m f H ± f f 

2 2 + m m  − m f m f f f T,loop + m 2H ± 1 − H ± f f  , 2 m H± P,loop

P,loop

A1,loop

(8.33)

A2,loop

G H ± f f  = H ± f f  − m f  H ± f f  − m f H ± f f 

m 2f + m 2f  + m f m f  P T,loop 2 + m H± 1 − H ± f f  . 2 m H±

(8.34)

 H + G − (m 2 ± ) comes from H + W − The contribution of the renormalized self-energy  H + − and H G mixings, which are derived by using the Slavnov-Taylor identity [3],  H + W − (m 2H ± ) = − 

mW  H + G − (m 2H ± ),  m 2H ±

(8.35)

μ

 H + W − ( p 2 ).  + − ( p) = p μ  where  H W + The decay rate for H → f f¯  γ can be written as (H + → f f¯  γ) e2 Nc   2  2  2 LL LL LL LL (c L  + c R  ) Q H + 00 + Q 2f 11 + Q 2f  22 + Q H + Q f 01 = (4π)3 m H ±   LL LL LR LR LR + Q H + Q f  02 + Q f Q f  12 + Q 2f 11 + Q 2f  22 + (c L c∗R + c∗L c R ) Q 2H + 00   LR LR LR + Q H + Q f 01 + Q H + Q f  02 + Q f Q f  12 (8.36) +

+

+

+

with c L = g SH − g PH and c R = g SH + g PH . We have also introduced Q H ± as the electric charge of the H ± . The functions iLjL and iLj R are given by LL 00 = −4m 2H ± I0 − 4m 2H ± (m 2H ± − m 2f − m 2f  )I00 ,

(8.37)

LL 11 LL 22 LL 01

(8.38)

= −2I + 2(m 2H ± + m 2f − m 2f  )I1 − 4m 2f (m 2H ± − m 2f − m 2f  )I11 , = −2I + 2(m 2H ± − m 2f + m 2f  )I2 − 4m 2f  (m 2H ± − m 2f − m 2f  )I22 , = −2I10 − 2(3m 2H ± − m 2f − 3m 2f  )I1 + 4(m 2f + m 2f  )I0 − 4(m 2H ± − m 2f − m 2f  )(m 2H ± + m 2f − m 2f  )I01 , LL 02 = 2I20 + 2(3m 2H ± − 3m 2f − m 2f  )I2 − 4(m 2f + m 2f  )I0 + 4(m 2H ± − m 2f − m 2f  )(m 2H ± − m 2f + m 2f  )I02 ,

(8.39) (8.40) (8.41)

8.1 Decay Rates with Higher Order Corrections

161

LL 12 = 8I + 2I12 + 2I21 − 2(3m 2H ± − m 2f − 3m 2f  )I1 − 2(3m 2H ± − 3m 2f − m 2f  )I2

+ 4(m 2H ± − m 2f − m 2f  )2 I12 ,

(8.42)

LR 00 = 8m 2H ± m f m f  I00 ,

(8.43)

LR 11 LR 22 LR 01 LR 02 LR 12

(8.44)

= = = = =

8m 3f m f  I11 , 8m f m 3f  I22 , 



8m f m f  I0 + I1 + (m 2H ± + m 2f − m 2f  )I01 ,   −8m f m f  I0 + I2 + (m 2H ± − m 2f + m 2f  )I02 ,   8m f m f  I1 + I2 − (m 2H ± − m 2f − m 2f  )I12 ,

(8.45) (8.46) (8.47) (8.48)

j ,··· j

where Ii11,··· ,imn (m 0 , m 1 , m 2 ) are given in [4], and we list them in Appendix C.2.

8.1.3 Decay Rates for H ± → W ± φ The decay rates for H ± → W ± φ (φ = h, H, A) with NLO EW corrections are given by Refs. [5–8]   + (H ± → W ± φγ). (8.49) (H ± → W ± φ) = LO (H ± → W ± φ) 1 + EW φ The LO decay rate LO (H ± → W ± φ) are given in Eq. (4.144). The NLO corrections EW φ are given by EW φ

  loop∗ tree 2Re  tree   H + G − (m 2 ± ) φG ± W ∓ Re H ± W ∓φ H ± W ∓φ H  W W (m 2W ), = − 2 tree − r − Re  2  tree  φH ± W ∓ m 2H ±  H ± W ∓ φ 

(8.50) tree where the φG ± W ∓ couplings are given by

mW sβ−α , v mW cβ−α , =∓ v

tree hG ± W ∓ = −ighG ± W ∓ = ∓

 tree H G ± W ∓ = −ig H G ± W ∓  tree AG ± W ∓

= 0.

The decay rate for H ± → W ± φγ is given by (H ± → W ± φγ)  2 2

   e2 gφW ∓ H ±  m 3H ±  m φ m 2W m2 ± I22 + H2 I00 λ =− , 2 2 3 16π m H± m H± mW

(8.51) (8.52) (8.53)

162

8 One-Loop Calculations for Decays of the Charged Higgs Bosons

+ 1+

m H ± − m 2φ m 2W

I02 +



1 11 (I + I ) − 2I 2 0 22 . m 2W

(8.54)

8.2 Next-to-Leading Order Electroweak Corrections to the Decay Rates In this section, we study the NLO EW corrections to the decay rates of the charged Higgs bosons in the 2HDM. We only show the results in Type-I and Type-II because the results in Type-X and Type-Y are almost similar to those in Type-I or Type-II. We introduce the following quantity to parametrize the size of the NLO EW correction. EW (H ± → X Y ) =

NLOEW (H ± → X Y ) − 1, LO (H ± → X Y )

(8.55)

where NLOEW (H ± → X Y ) is the decay rate for H ± → X Y including NLO EW corrections but no QCD corrections. For the calculation of LO decay rates, LO (H ± → X Y ), we employ the quark running masses, not the pole masses. We study the alignment case, sβ−α = 1, and the non-alignment case, sβ−α = 0.99 with cβ−α < 0 and cβ−α > 0. We assume that the masses of additional Higgs √ bosons are degenerate. For each scenario, we take tan β = 1, 5, and 10, and scan M 2 under the perturbative unitarity bound, the vacuum stability bound and the constraints from S and T parameters. In order to discuss the theoretical behavior of NLO EW corrections, we dare to omit the constraint from the direct and indirect search and flavor experiment. In Fig. 8.3, we show EW (H ± → X Y ) as a function of the common mass scale of the additional Higgs bosons, m  ≡ m H ± = m H = m A with sβ−α = 1. The red, blue, and green colored regions correspond to tan β = 1, 3, and 10, respectively. The solid √ (dashed) lines correspond to the results with maximum (minimum) values of M 2 under the constraints. The charged Higgs bosons mainly decay into a pair of fermions. For the H + → t b¯ decay with tan β = 1, there are kinks at m  m t + m b , 2m t , and 600 GeV. The first kink at m  m t + m b is the threshold of the (t, b) loop diagram in the H + –H − self-energy. The second kink at m  2m t is the threshold of the top loop diagram in the A–G 0 mixing self-energy, which appears in the counterterms for the H + f f  vertex. The third kink at m  600 GeV corresponds to the points where the value of √ M 2 changes from zero to non-zero due to the theoretical bounds. At this point, the scalar couplings λ H + H − φ (φ = h, H, A) take maximal values, and non-decoupling effects of scalar loops in the H + –H − self-energy become dominant. The size of ¯ reaches almost 10% for all the types of 2HDMs. EW (H ± m → t b) √ For the results with tan β = 3, and 10, the possible values of M 2 are almost constants due to the strict theoretical constraints, i.e. M ∼ m  . While for Type-I, one does not see a large difference between tan β = 3, and tan β = 10, for Type-II, ¯ the corrections can be sizable in the case of tan β = 10, e.g., EW (H ± → t b)

8.2 Next-to-Leading Order Electroweak Corrections to the Decay Rates

163

Fig. 8.3 Sizes of the NLO corrections to the decay rates for the charged Higgs bosons in the alignment limit sβ−α = 1 with tan β = 1 (red), 3 (blue), 10 (green). Masses of additional Higgs bosons are √ degenerate, m  = m H ± = m A = m H . Mmax (Mmin ) denotes the maximum (minimum) value of M 2 under the perturbative unitarity bound, the vacuum stability bound and the constraints from S and T parameters

−25 % at m  = 2 TeV. We find that these behaviors can be explained by large T TP negative contributions from the tensor form factors  H ± f f  and  H ± f f  , which give the contributions proportional to the square of the charged Higgs boson mass in the decay rate (see Eqs. 8.33 and 8.34). For other decay modes, one can see similar behaviors described above. On the other hand, the remarkable thing is that the correction EW (H ± → c¯s ) can be over −100% at m  = 600 GeV. We note that EW (H ± → c¯s ) tends to be larger than the other decay modes for the following reason. The decay rate with NLO EW corrections (H ± → c¯s ) is evaluated by using the pole masses for the charm quark and the strange quark while the LO decay rates are evaluated with the running masses at the scale of μ = m H ± . Consequently, the difference between the pole masses and the running masses enhances EW (H ± → c¯s )1 . For instance, the ratios of the running masses and pole masses are estimated as m c /m¯ c (m H ± ) = 1.67 GeV/0.609 GeV = 2.74, m s /m¯ s (m H ± ) = 0.1 GeV/0.0491 GeV = 2.03. Magnitude of EW (H ± → c¯s ) is enlarged by these factors. The discussion does not depend on sβ−α , so that the same holds in Figs. 8.4 and 8.5. Namely EW (H ± → c¯s ) can also be over − 100% in case of sβ−α = 0.99.

1

164

8 One-Loop Calculations for Decays of the Charged Higgs Bosons

Fig. 8.4 Magnitudes of NLO corrections to the decay widths for charged Higgs bosons in the case of sβ−α = 0.99 with cβ−α < 0. The masses of the additional Higgs bosons are degenerate, m  ≡ m H ± = m A = m H . The dimensionful parameter Mmax (Mmin ) is the maximum (minimum) value of M under the theoretical constraints

In Fig. 8.4, the results in the nearly alignment scenario, sβ−α = 0.99 with cβ−α < 0, are shown as a function of the degenerate mass m  . In the non-alignment case, an upper bound of m  is given for each value of tan β because of the theoretical constraints. However, the maximum magnitudes of NLO EW corrections for the case of tan β = 1 are almost unchanged from the scenario of the alignment limit. Apart from that, the charged Higgs bosons can decay into W + h in the nearly alignment scenario. For this decay mode, peaks appear at m H ± m h + m W , which correspond to the thresholds of 1PI diagrams in the H ± W − h vertex function such as (W, H ± m/G ± , h) and the (h, h/H, W ) loop diagrams. The maximum value of the corrections is 26% in the case of tan β = 1 for all the types of 2HDMs. In the Fig. 8.5, we also show the results with sβ−α = 0.99 and cβ−α > 0. The remarkable difference from the results with cβ−α < 0 is that the allowed regions for tan β = 10 are broader than those for tan β = 3. Hence, compared with cβ−α < 0, the corrections EW for tan β = 10 can be larger. In addition, for H ± → W + h, direction of the threshold peak at m φ m h + m W is opposite from cβ−α < 0 because the contributions from the H + W − h vertex function depends on the tree-level coupling gh H + W − (see Eq. 8.50). On the other hand, one can see that the sign of EW (H ± → W + h) in the region m   500 with tan β = 1 are positive in both

8.3 Next-to-Leading Order Electroweak Corrections …

165

Fig. 8.5 Magnitudes of NLO corrections to the decay widths for charged Higgs bosons in the case of sβ−α = 0.99 with cβ−α > 0. The masses of the additional Higgs bosons are degenerate, m  ≡ m H ± = m A = m H . The dimensionful parameter Mmax (Mmin ) is the maximum (minimum) value of M under the theoretical constraints

cases of cβ−α < 0 and cβ−α > 0. The dominant contributions in this region mainly come from non-decoupling effects of additional Higgs bosons, i.e., pure scalar loop diagrams in δC h and δC H ± , which are proportional to the square of the scalar couplings λφi φ j φk λφi  φ j  φk  . Among them, there are contributions that are not proportional to cβ−α , so they do not depend on the sign of cβ−α .

8.3 Next-to-Leading Order Electroweak Corrections to the Decay Branching Ratios In this section, we discuss the size of the NLO EW corrections to the decay branching ratios of the charged Higgs bosons. We study the following two distinct scenarios for the mass of the charged Higgs boson, Scenario A: m H ± = 400 GeV, Scenario B: m H ± = 1000 GeV.

(8.56) (8.57)

166

8 One-Loop Calculations for Decays of the Charged Higgs Bosons

In order to avoid the constraint from the T parameter, we assume that the masses of the charged and CP-odd Higgs bosons are degenerate, m H ± = m A . For Scenario A, we study Type-I and Type-X because Type-II and Type-Y are already excluded by the flavor constraints [9]. We scan the mass of the heavier CPeven Higgs boson as Scenario A: 250 GeV < m H < 800 GeV.

(8.58)

The lower bound on m H comes from the direct search for H → Z Z , by which the parameter regions with m H  250 GeV and tan β  6 (5) are excluded in the case of sβ−α = 0.995 with cβ−α < 0 for Type-I (Type-X) [10]. The remaining parameters are scanned as 0.995 < sβ−α < 1, 2 < tan β < 10, 0 < M < m H ± + 500 GeV.

(8.59)

with both of cβ−α < 0 and cβ−α > 0. The lower bound on tan β comes from the constraint of Bd → μμ [9]. For Scenario B, we study all types of 2HDMs. We scan the mass of the heavier CP-even Higgs boson as Scenario B: 800 GeV < m H < 1200 GeV.

(8.60)

The scan regions of the remaining parameters are the same as those in Eq. (8.59). The direct search for H → hh excludes the parameter regions with tan β  2 in the case of sβ−α = 0.995 with cβ−α < 0 [10]. We impose the theoretical constraints and the constraint from S and T parameters. We also impose the constraint from the measurements of  the Higgs signal strengths

2HDM SM at the LHC [11]. We evaluate the scaling factors κ X = h→X X / h→X X including NLO EW and NNLO QCD corrections. We then remove parameter points that are not consistent with the values presented in Table 11 (a) in [11] at 95 % CL. In order to study the size of the NLO EW corrections to the decay branching + ratios, we introduce BR EW (H → X Y ) as + BR EW (H → X Y ) =

BRNLO EW+QCD (H + → X Y ) − 1, BRLO+QCD (H + → X Y )

(8.61)

where BRNLO EW+QCD (H + → X Y ) denotes the branching ratios with NLO EW and higher-order QCD corrections, while BRLO+QCD (H + → X Y ) denotes the branching ratios only with higher-order QCD corrections. At the one-loop level, we can + approximate BR EW (H → X Y ) as tot

+ + BR EW (H → X Y ) ≈ EW (H → X Y ) − EW ,

(8.62)

8.3 Next-to-Leading Order Electroweak Corrections …

167

Fig. 8.6 Decay branching ratios for the charged Higgs bosons as a function of κ Z (≡ κ Z − 1) in Scenario A, where colored points denote different values of tan β. Predictions on Type-I (Type-X) are shown in the first and third columns (the second and fourth columns)

tot

where EW (H + → X Y ) and EW are given by NLO EW+QCD (H + → X Y ) − 1, LO EW+QCD (H + → X Y ) tot NLO EW+QCD = − 1. tot LO+QCD

EW (H + → X Y ) = tot

EW

(8.63) (8.64)

We use the quark running masses for the charged Higgs decays into quarks.

8.3.1 Scenario A In Fig. 8.6, we show the decay branching ratios of the charged Higgs bosons including the NLO EW corrections for Scenario A as a function of κ Z . The plots in the first and third columns are the results in Type-I, while those in the second and fourth columns are the results in Type-X. The color differences correspond to the values of tan β. The ¯ reaches almost 100% without depending on the value of κ Z size of BR(H + → t b) ¯ becomes sizable in the low as well as types of 2HDMs. This is because BR(H + → t b) tan β region, where the top Yukawa coupling is dominant. Thus, the results in Type-I and Type-X are almost the same in the low tan β region. The size of BR(H + → τ + ν) in Type-X is larger than that of Type-I because of the tan β enhancement of the τ Yukawa coupling in Type-X. The size of BR(H + → W + h) exceeds 20% in Type-I, while that in Type-X is maximally around 11%. We note that this signature mostly appears in the regions of κ Z  0.76%. If κ Z  −1%, BR(H + → W + h) is less than 5% in both Type-I and Type-X. In addition, we comment on the behavior of the branching ratio for H + → W +(∗) H . This decay mode kinematically opens when the

168

8 One-Loop Calculations for Decays of the Charged Higgs Bosons

Fig. 8.7 The total decay width of the charged Higgs bosons as a function of the mass difference m H − m H ± in Scenario A for Type-I and Type-X. The colored dots correspond to different values of tan β

Fig. 8.8 Decay branching ratios of the charged Higgs bosons as a function of the mass difference m H − m H ± in Scenario A for Type-I and Type-X. The colored dots correspond to different values of tan β

heavier CP-even Higgs boson is lighter than the charged Higgs boson. The maximal size of BR(H + → W +(∗) H ) reaches almost 90% in both Type-I and Type-X. In Fig. 8.7, we show the size of the higher-order corrections to the total width in Scenario A as a function of the mass difference for the additional Higgs bosons. The color differences denote the values of tan β. The behaviors of the higher-order corrections in Type-I and Type-X are similar in the regions with tan β  4, while these are different in tan β  6. When m H − m H ± −80 GeV, threshold effects appear, in which H + → W + H opens and the correction EW (total width) reaches −7%.

8.3 Next-to-Leading Order Electroweak Corrections …

169

+ + ¯ In Fig. 8.8, we show the size of BR EW (H → X Y ). For H → t b, the kink appears when m H − m H ± 80 GeV as similar to the correction for the total width. The size BR + + ¯ ¯ of BR EW (H → t b) takes −2.75%  EW (H → t b)  +1.25% in Type-I, while BR + ¯  +2.3% in Type-X. Since the size of EW (H + → t b) ¯ −1.9  EW (H → t b) is close to EW (total width) in the bulk of parameter points, they are canceled in BR + + the definition of BR EW . On the other hand, the behavior of EW (H → W H ) is BR + + + ¯ The size of EW (H → W H ) is relatively different from that of EW (H → t b). + small, i.e., −2.5%  EW  0%, for both Type-I and Type-X, so that BR EW (H → + W H ) is dominated by the EW (total width). Since we have not implemented the + + NLO EW corrections for the off-shell H + → W +∗ H decay, BR EW (H → W H ) is determined only by the EW (total width) in the range of m W < m H − m H ± < 0 GeV. + + The size of BR EW (H → W h) takes a remarkably large value, and it exceeds +100% + especially in the low tan β region. For the parameter points where BR EW (H → + −2 W h)  +100%, cβ−α is close to 0, i.e., |cβ−α |  2.5 × 10 . We note that such + + large BR EW (H → W h) does not mean a breakdown of the perturbation, since it is caused by the smallness of the tree-level coupling. The one-loop amplitude of H + → W + h does not vanish even if cβ−α=0 . The counterterms δC h and δC H ± give such non-zero contributions, which can be enhanced by the non-decoupling effect of the additional Higgs bosons in case of M ∼ v. In this case, the one-loop + + amplitude can overcome the tree-level amplitude, and it gives BR EW (H → W h)  BR + + 100%. Furthermore, we found that in some parameter points EW (H → W h) can be smaller than −100%. This is caused by the truncation of the squared one-loop amplitude. The effect of the squared one-loop amplitude is discussed in Sect. 8.3.3.

8.3.2 Scenario B In Fig. 8.9, we show the decay branching ratios of the charged Higgs bosons including the NLO EW corrections for Scenario B as a function of κ Z . The results in Type-I, Type-II, Type-X, and Type-Y are shown from left to right panels. The color ¯ differences correspond to the values of tan β. The behavior of the BR(H + → t b) are similar to that in Scenario A. The size of the branching ratio becomes large in the low tan β region for all types of 2HDMs. The behavior of the BR(H + → W + H ) is also simiar to that in Scenario A. In Scenario B, BR(H + → W + h) in Type-I is considerably larger than the other types of 2HDMs. The size of BR(H + → τ + ν) reaches about 30% only in Type-X. In Scenario B, a sizable BR(H + → W + h) is only realized in the case of cβ−α > 0 differently from Scenario A. The branching ratio BR(H + → W + h) is enhanced in large tan β regions, which does not occur in the case of cβ−α < 0 due to the theoretical constraints. We note that all points with BR(H + → W + h)  10% correspond to cβ−α > 0 for all types of 2HDMs. In Fig. 8.10, we show the size of the higher-order corrections to the total width in Scenario B as a function of the mass difference for the additional Higgs bosons. The color differences denote the values of tan β. Similar to Scenario A, behavior for tan β  4 is almost the same for all types of 2HDMs. The clear difference appears

170

8 One-Loop Calculations for Decays of the Charged Higgs Bosons

Fig. 8.9 Decay branching ratios for the charged Higgs bosons as a function of κ Z (≡ κ Z − 1) in Scenario B, where colored points denote different values of tan β. Predictions on Type-I, II, X, and Y are shown from the left panels to the right panels

Fig. 8.10 The total decay width of the charged Higgs bosons as a function of the mass difference m H − m H ± in Scenario B for Type-I, II, X, and Y. The colored dots correspond to different values of tan β

for tan β  4. For Type-I, the allowed region of the mass difference m H − m H ± is wider than the other types of 2HDMs, and the correction becomes positive when m H − m H ±  30 GeV. On the other hand, for Type-II and Type-Y, the bulk of points with tan β  5 show large negative corrections, compared with those for tan β  4. This is because the bottom Yukawa coupling is enhanced by large tan β. The correction reaches −20 (−25) % when m H − m H ± −80 GeV for Type-II (TypeY) due to the effect of the threshold of the mode H + → W + H . For Type-X, the results in the large tan β regions are almost similar to those in the low tan β regions.

8.3 Next-to-Leading Order Electroweak Corrections …

171

Fig. 8.11 Decay branching ratios of the charged Higgs bosons as a function of the mass difference m H − m H ± in Scenario B for Type-I, II, X, and Y. The colored dots correspond to different values of tan β

+ + ¯ In Fig. 8.11, we show the size of BR EW (H → X Y ). For H → t b, we have the almost same picture in Scenario A for small tan β regions. When tan β 2, + ¯ is almost cancel with EW (total width), and −2.5%  BR EW (H + → t b) EW (H → ¯ t b)  +1.5% for all types of 2HDMs. For large tan β, the size of the correction + ¯ becomes much large. In Type-I, the size of BR EW (H → t b) exceeds 12% near the BR + ¯ becomes negative threshold region, m H − m H ± −80 GeV, while EW (H → t b) in the case of m H − m H ±  50 GeV due to the effect of EW (total width). In Type+ ¯ II and Type-Y, BR EW (H → t b) takes a negatively large vale. It reaches −15.5% ¯ due to the tan β enhancement of the bottom Yukawa coupling for EW (H + → t b), + + which can be seen in Fig. 8.5. For the H → τ ν decay, we can see that the + + behavior of BR EW (H → τ ν) with small tan β is similar in all types of 2HDMs, while the difference appears in large tan β region. For H + → W + H , we note that the size of EW (H + → W + H ) monotonically decreases as the mass difference m H − m H + becomes negatively large, e.g., EW (H + → W + H ) ∼ −8% (−4%) when m H − m H + =200 GeV (100 GeV) for all types of 2HDMs. The maximum + + value of the BR EW (H → W H ) is +13% and +9% for Type-I and Type-X, respectively, while those for Type-II and Type-Y are +22%. Finally, we comment on the results in the other case of the degenerate mass of the additional Higgs bosons, i.e., m H = m H ± , where the constraint from T parameter

172

8 One-Loop Calculations for Decays of the Charged Higgs Bosons

Fig. 8.12 The decay branching ratio for H + → W + h for Scenario A of Type-I in the alignment limit, sβ−α = 1, as a function of the mass difference m H − m H ± , where the squared one-loop amplitude for H + → W + h is included in the evaluation. The colored dots correspond to different values of M

can be satisfied when sβ−α 1. We have performed the same analysis in this case and obtained qualitatively similar results for magnitudes of the branching ratios, while the allowed parameter regions under the constraints from theoretical bounds and the electroweak oblique parameters are more strict than the case of m H ± = m A . For Scenario A, EW (total width) shows a cusp structure at m A = 2m t , which is realized by the threshold of the top loop diagrams in δβ. Therefore, the behavior of EW (total width) is different from the case the m A = m H ± . However, the maximum and minimum values of EW (total width) are similar to the results of m A = m H ± . Similarly, the behavior of BR EW is different from the case of m A = m H ± due to the threshold effects and the difference of allowed parameter regions. However, values of BR EW distribute in the almost similar region to the case of m A = m H ± . For Scenario B, we also note that the size of BR EW for all the processes in the case of m H = m H ± tend to be smaller than that of the case m A = m H ± except for Type-I. For Type-I, the BR + + + ¯ maximum value of BR EW (H → t b) and EW (H → τ ν) is +12.7% and +2.6%, BR + + respectively. The maximum value of EW (H → W h) is similar to the case of m A = m H±.

8.3.3 Size of the Squared One-Loop Amplitude for H + → W + h In this section, we discuss the size of the squared one-loop amplitude for H + → W + h. The squared amplitude can be expressed as 2    M(H + → W + h) =

 2 

   m 4H ±  loop 2   tree loop tree   . + λ(μ , μ ) + 2Re      + − + − + − + − h W H W h H W h H W h H W h m 2W (8.65)

8.3 Next-to-Leading Order Electroweak Corrections to the Decay Branching Ratios

173

The first and second terms correspond to LO and NLO contributions, respectively. The third term is in the same order as contributions from the tree-level amplitude times two-loop amplitude, namely NNLO contributions. We note that the third term loop does not vanish even if cβ−α = 0. Therefore, the | H + W − h |2 term gives a leading 2    contribution when we expand M(H + → W + h) into a power series of cβ−α . Especially in the alignment limit, H + → W + h is prohibited at the LO, and it becomes a loop-induced decay process. As we have discussed in Sects. 8.3.1 and 8.3.2, the sum of the first and second terms can be negative in the nearly alignment regions, and the decay rate becomes negative. Therefore, we need to include the third term to obtain a physically meaningful result. loop If sβ−α = 1, the | H + W − h |2 term contains the IR divergence, and evaluation of real photon emissions at NNLO are required to obtain an IR finite result. The discussion loop of the size of | H + W − h |2 in the region of 0.995  sβ−α < 1 is beyond the scope of this thesis. In the alignment limit, the third term is IR finite, and contributions from the tree-level amplitude times two-loop amplitude vanish. Therefore, we discuss the NNLO corrections due to the third term in Eq. (8.65) to the decay branching ratio for H + → W + h in the case of sβ−α = 1. In Fig. 8.12, we show the decay branching ratio for H + → W + h as a function of the mass difference m H − m H ± in the alignment limit. We include the NNLO corrections only for H + → W + h and evaluate the decay branching √ ratios in Type-I for Scenario A. The color differences correspond to the√value of M 2 . The branching ratio reaches 0.1% at most, and it is maximized M 2 is small. We have also calculated the branching ratio in Scenario A for Type-X and Scenario B for all types of 2HDMs. For Type-X in Scenario A, the result is the almost same as that in Type-I. √ For Scenario B, M 2 cannot be small under the theoretical consistencies such as the perturbative unitarity and the vacuum stability, and the branching ratio is small, as compared with Scenario A. The maximum value of the branching ratio is 0.004% in Scenario B for all types of 2HDMs.

References 1. Shinya K, Mariko K, Kodai S, Kei Y (2018) H-COUP: a program for one-loop corrected Higgs boson couplings in non-minimal Higgs sectors. Comput Phys Commun 233:134–144 2. Shinya K, Mariko K, Kentarou M, Kodai S, Kei Y (2020) H-COUP version 2: a program for one-loop corrected Higgs boson decays in non-minimal Higgs sectors. Comput Phys Commun 257:107512 3. Williams Karina E, Heidi R, Georg W (2011) Higher order corrections to Higgs boson decays in the MSSM with complex parameters. Eur Phys J C 71:1669 4. Ansgar D (1993) Techniques for calculation of electroweak radiative corrections at the one loop level and results for W physics at LEP-200. Fortsch Phys 41:307–420 5. Santos R, Barroso A, Brucher L (1997) Top quark loop corrections to the decay H + → h 0 W + in the two Higgs doublet model. Phys Lett B 391:429–433 6. Akeroyd AG, Arhrib A, Naimi El-M (2000) Yukawa coupling corrections to the decay H + → W + A0. Eur Phys J C 12:451–460. [Erratum: Eur Phys J C 14:371 (2000)]

174

8 One-Loop Calculations for Decays of the Charged Higgs Bosons

7. Akeroyd AG, Arhrib A, Naimi E (2001) Radiative corrections to the decay H + → W + A0. Eur Phys J C 20:51–62 8. Marcel K, Robin L, Margarete M, Rui S, Hanna Z (2016) Gauge-independent renormalization of the 2-Higgs-doublet model. JHEP 09:143 9. Johannes H, Andreas H, Roman K, Klaus M, Thomas P, Jörg S (2018) Update of the global electroweak fit and constraints on two-Higgs-doublet models. Eur Phys J C 78(8):675 10. Masashi A, Shinya K, Mariko K, Kentarou M, Kodai S, Kei Y (2021) Probing extended Higgs sectors by the synergy between direct searches at the LHC and precision tests at future lepton colliders. Nucl Phys B 966:115375 11. Georges A et al (2020) Combined measurements of√Higgs boson production and decay using up to 80 fb−1 of proton-proton collision data at s = 13 TeV collected with the ATLAS experiment. Phys Rev D 101(1):012002

Chapter 9

One-Loop Calculations for Decays of the CP-Odd Higgs Boson

In this chapter, we discuss the decay rates of the CP-odd Higgs boson for various decay modes in the 2HDMs. Decay branching ratios of the CP-odd Higgs boson are evaluated including NLO EW corrections, as well as QCD corrections up to NNLO. We have newly implemented them into the H-COUP program [1, 2]. We comprehensively study the impacts of the NLO EW corrections on the branching ratios in approximate alignment scenarios. We find that the A → Z h decay modes can be dominant decay modes even in the case where deviation in h Z Z couplings is quite small and cannot be detected at the ILC. Thus, we can extract the information on the mixing angle by studying the A → Z h decay mode in future collider experiments. In addition, we find that the types of 2HDMs can be classified by studying the decay pattern of the CP-odd Higgs boson even in the alignment scenario. In the alignment scenario, it is difficult to distinguish the types of 2HDMs from the precision measurement of the SM-like Higgs boson. Thus, the study of the CP-odd Higgs boson is quite useful to investigate not only the approximate but also the exact alignment scenario.

9.1 Decay Rates with Higher-Order Corrections 9.1.1 Form Factors for Vertex Functions of CP-Odd Higgs Boson 9.1.1.1

A f f¯ Vertex

The renormalized A f f¯ vertex functions can be expressed as

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 M. Aiko, Theoretical Studies on Extended Higgs Sectors Towards Future Precision Measurements, Springer Theses, https://doi.org/10.1007/978-981-99-1324-4_9

175

176

9 One-Loop Calculations for Decays of the CP-Odd Higgs Boson

f¯

p1

p1

φ

p2

V

q

q A

A p2

f

Fig. 9.1 Momentum assignment for the renormalized A f f¯ and AV φ vertices

  A f f¯ ( p1 , p2 , q) =   AP f f¯ + /p 1  VA1f f¯ + /p 2  VA2f f¯ + /p 1 γ5  AA1f f¯ + /p 2 γ5  AA2f f¯  AS f f¯ + γ5  TA f f¯ + /p 1 /p 2 γ5  AP fT f¯ , + /p 1 /p 2

(9.1)

where p1 ( p2 ) is the incoming four-momentum of the anti-fermion (fermion), and q μ (= p1 + p2 ) is the outgoing four-momentum of the CP-odd Higgs boson (see Fig. 9.1). The renormalized form factors are composed of the tree-level and the one-loop parts as X,loop   AX f f¯ =  AX,tree +  A f f¯ , (X = S, P, V1 , V2 , A1 , A2 , T, P T ), f f¯

(9.2)

where the tree-level couplings for A f f¯ vertex are given by = i2I f  AP,tree f f¯

m fζf ,  AX,tree = 0, (X = P). f f¯ v

(9.3)

The one-loop parts are further decomposed into contributions from 1PI diagrams and counterterms, X,loop

+ δ AX f f¯ .  A f f¯ =  AX,1PI f f¯

(9.4)

The 1PI diagrams contributions  AX,1PI are given in Appendix D.3.8. The counterterms f f¯

δ AX f f¯ are given by δ P ¯ =  P,tree Af f A f f¯



δm f 1 δv δZ AA f − ζ f δβ f + δ Z V + + − mf v 2 ζf



  δ Z G0 A − δβ − ζ f δβ , 2

(9.5) δ X ¯ = 0, Af f

(X  = P).

(9.6)

The AZ mixing contribution is related to AG 0 mixing contribution through the Slavnov-Taylor identity [3],

9.1 Decay Rates with Higher-Order Corrections

 AG 0 (m 2A ) + i 

m 2A  AZ (m 2A ) = 0,  mZ

177

(9.7)

 AZ ( p 2 ). Since   AG 0 (m 2A ) = 0 by the on-shell renormaliza μAZ ( p) = p μ  where  tion condition, the AZ mixing contribution vanishes in the decay amplitude. When we neglect the effects of the CP violation, the renormalized A f f¯ vertex functions satisfy the following relations,   AS f

f¯

9.1.1.2

=  TA f

f¯

= 0,   VA1f

f¯

=  VA2f f¯ ,   AA1f f¯ =   AA2f f¯ , (q 2 = m 2A , p12 = p22 = m 2f ).

(9.8)

AV φ Vertex

The renormalized AV φ vertex functions (V, φ) = (Z , h), (Z , H ) and (W ± , H ∓ ) can be expressed as μ   AV φ ( p1 , p2 , q) = ( p1 + q)μ  AV φ ,

(9.9)

where p1 and p2 denote the incoming four-momentum of the scalar boson φ and the gauge boson V , respectively. The momentum q is the outgoing four-momentum of the CP-odd Higgs boson (see Fig. 9.1). Since we assume that the external gauge boson is on-shell, the term proportional to p2 vanishes due to the orthogonality of the polarization vector. For the A → W ± H ∓ decays, we use the renormalized H ± W ∓ A vertex function given in Sect. 8.1.1.1,   μAW ± H ∓ ( p1 , p2 , q) = −(q + p1 )μ  H ± W ± A (q 2 , p22 , p12 ),

(9.10)

where the additional minus sign comes from the change of momentum assignment. The renormalized form factors are composed of the tree-level and the one-loop parts as loop   AV φ =  tree AV φ +  AV φ ,

(9.11)

where  tree AZ φ are given by mZ cβ−α , v mZ sβ−α , = −ig H AZ = −i v mW . = −i(−g AH ± W ∓ ) = −i v

 tree AZ h = −igh AZ = i

(9.12)

 tree AZ H

(9.13)

 tree AW ∓ H ±

(9.14)

The one-loop parts are further decomposed into contributions from 1PI diagrams and counterterms,

178

9 One-Loop Calculations for Decays of the CP-Odd Higgs Boson loop

 AV φ =  1PI AV φ + δ AV φ .

(9.15)

The 1PI diagrams contributions  1PI AV φ are given in Appendix D.3.9. The counterterms δ AZ φ are given by 

δm 2Z δv 1 − + (δ Z hh + δ Z A A + δ Z Z ) v 2 2m 2Z   δ Z Hh δ Z G0 A − δβ − + δα , + tan (β − α) 2 2  2 δm Z 1 δv + (δ Z H H + δ Z A A + δ Z Z ) =  tree − AZ H v 2 2m 2Z   δ Zh H δ Z G0 A − δβ + + δα , − cot (β − α) 2 2   2 δm W 1 δv + H− + δ Z AA + δ ZW ) . + =  tree − Z (δ ∓ ± H AW H v 2 2m 2W

δ AZ h =  tree AZ h

δ AZ H

δ AW ∓ H ±

(9.16)

(9.17) (9.18)

9.1.2 Decay Rates for A → f f¯ The decay rates of the CP-odd Higgs boson into a pair of fermions with NLO EW and QCD corrections are given by   f f (A → f f¯) = LO (A → f f¯) 1 + EW + QCD + (A → f f¯γ ),

(9.19)

where the LO decay rate is given by Eq. (4.95) and the QCD corrections are discussed f in Sect. 4.3.1. The EW correction EW is given by   2 f P,tree P,loop∗ − r,

EW = 2 Re  A f f¯ G A f f¯ P,tree  A f f¯

(9.20)

with

P,loop G A f f¯

=

P,loop  A f f¯

− 2m

A1 ,loop f  A f f¯

+

m 2A

1−

3m 2f m 2A

 P T,loop

 A f f¯

,

(9.21)

where we have used Eq. (9.8). The IR divergences in the NLO EW correction are regularized by introducing the finite photon mass. The photon mass dependence is canceled by adding the decay rates of real photon emission (A → f f¯γ ). The analytic expression of (A → f f¯γ ) is given by

9.1 Decay Rates with Higher-Order Corrections

(A → f f¯γ ) = Ncf

2αem Q 2f 16π 2 m A



2I f m f ζ f v

179

2 [ 11 + 22 + 12 ] .

(9.22)

This expression can be obtained from the decay rate of H ± → f f¯  γ given in Eq. (8.36) by replacing c L → −i

2I f m f ζ f 2I f m f ζ f , cR → i , v v

(9.23)

and H ± → A, f  → f . The functions i j are defined by i j = iLjL − iLj R ,

(9.24)

where the functions iLjL and iLj R are given in Eqs. (8.37) to (8.48).

9.1.3 Decay Rates for A → V φ The decay rate for the CP-odd Higgs boson decays into a Z boson and a CP-even Higgs boson φ (φ = h, H ) with NLO EW corrections is given by   φ (A → Z φ) = LO (A → Z φ) 1 + EW ,

(9.25) φ

where the LO decay rates are given by Eq. (4.103). The EW correction EW is given by φ

EW

  loop∗ 2Re  tree AZ φ  AZ φ  Z Z (m 2Z ). = − r − Re tree 2  AZ φ

(9.26)

The decay rate for the CP-odd Higgs boson decays into a W ± boson and a charged Higgs boson H ± with NLO EW corrections is given by   H± + (A → H ± W ∓ γ ), (A → W ± H ∓ ) = LO (A → W ± H ∓ ) 1 + EW (9.27) ±

H where the LO decay rates are given by Eq. (4.109). The EW correction EW is given by

180

9 One-Loop Calculations for Decays of the CP-Odd Higgs Boson

±

H

EW

  loop∗ 2Re  tree AW ∓ H ±  AW ∓ H ±  W W (m 2W ). = − r − Re 2 tree  AW ∓ H ±

(9.28)

The IR divergences in the NLO EW correction are regularized by introducing the finite photon mass. The photon mass dependence is canceled by including the decay rates of real photon emission A → H ± W ∓ γ as similar to A → f f¯. The analytic expression of (A → H ± W ∓ γ ) is given by  √ m2 2 m2 ± αem 2G F m 2W m 3A H± , mW I22 + H2 I11 λ 2 2 2 2π mA mA mW 

 2 2  m ± − mA 1 2 11 I12 + 2 (I1 + I2 ) − 4 I + 2I21 + I22 + 1+ H 2 . mW mW mA

(A → H ± W ∓ γ ) = −

(9.29)

9.2 Next-to-Leading-Order Electroweak Corrections to the Decay Rates In this section, we examine the impact of NLO EW corrections on the decay rates of the CP-odd Higgs boson in Type-I and Type-II. The results in Type-X and TypeY are almost similar to those in Type-I or Type-II. We parameterize the NLO EW corrections to the decay rates as

EW (A → X Y ) =

LO+EW (A → X Y ) − 1, LO (A → X Y )

(9.30)

where LO+EW (A → X Y ) is the decay rate of A → X Y including the NLO EW correction but no QCD correction. For the calculation of LO decay rates, LO (A → X Y ), we use the quark pole masses in Yukawa couplings. We parameterize the NLO EW corrections to the total decay width as

tot EW =

tot LO+EW − 1, tot LO

(9.31)

tot where LO+EW is the total decay width of the CP-odd Higgs boson including the NLO EW corrections but no QCD correction. Similar to EW (A → X Y ), we use tot . the quark pole masses for the calculation of LO total decay width, LO tot We evaluate EW (A → X Y ) and EW in the alignment limit, sβ−α = 1, and the nearly alignment scenario, sβ−α = 0.995 with cβ−α < 0 and cβ−α > 0. We assume that the masses of additional Higgs √ bosons are degenerate. For each scenario, we take tan β = 1, 3, and 10, and scan M 2 under the constraints of perturbative unitarity,

9.2 Next-to-Leading-Order Electroweak Corrections to the Decay Rates

181

vacuum stability and S and T parameters. In order to discuss the theoretical behavior of NLO EW corrections, we dare to omit the constraint from the direct and indirect searches and flavor experiments. In Fig. 9.2, we show EW (A → X Y ) as a function of m  with sβ−α = 1. The red, blue, and green colored regions correspond to tan β=1, 3, and 10, respectively. The solid √ and dashed lines correspond to the results with maximum and minimum values of M 2 under the constraints, respectively. For tan β = 1, there are two kinks at m   350 GeV and m   600 GeV. The kink at m   350 GeV is the threshold of the top quark. For A → t t¯, threshold effects appear both in 1PI triangle diagrams and top-quark loop diagrams in the twopoint function of the neutral Higgs bosons. The latter contributions also apper in A → bb¯ and A → τ τ¯ . √ The kink at m   600 GeV corresponds to the point where the minimum value of M 2 changes from zero to non-zero due to the theoretical bounds. The behaviors of EW (A → t t¯) and EW (A → τ τ¯ ) in Type-I are almost the same as those in Type-II. The size of non-decoupling effects for EW (A → t t¯) reaches about −3%, while that for EW (A → τ τ¯ ) reaches about −19%. On the other ¯ in Type-I is different from that in Type-II. This hand, the behavior of EW (A → bb) is because of the contributions from 1PI diagrams including the virtual G ± , H ± and W ± bosons, that proportional to ζt ζb . Due to the large top Yukawa coupling and ¯ depends on the types of the type dependence of ζt ζb , the behavior of EW (A → bb) ¯ reaches about −15% 2HDMs. The size of non-decoupling effects for EW (A → bb) and −20% in Type-I and Type-II, respectively. √ For tan β = 3 and 10, M 2 is almost degenerate with m  due to the theoretical constraints. EW (A → t t¯) is almost constant above the top threshold, and it is about ¯ is about −4% (−6%) for Type−2% for m  = 1 TeV. The size of EW (A → bb) I (Type-II) with m  = 1 TeV, while that of EW (A → τ τ¯ ) is about −5% both in Type-I and Type-II. The behavior of tot EW in Type-I is different from that in Type-II, especially with ¯ in Type-II. Above tan β = 10. This is because of tan β enhancement of BR(A → bb) the top-quark threshold, tot can be expanded as EW ¯ ¯

tot EW = EW (A → t t¯)BRLO (A → t t¯) + EW (A → bb)BRLO (A → bb) + other channels.

(9.32)

As we can see from Figs. 4.6 and 4.8, BR(A → t t¯) ≈ 1 in Type-I, and tot EW ≈ ¯ is also dominant in Type-II with

EW (A → t t¯). On the other hand, BR(A → bb) ¯ also contributes to tot large tan β. Therefore, EW (A → bb) EW . In Type-I (Type-II), is about −2 (−5)% for m = 1 TeV with tan β = 10.

tot  EW In Fig. 9.3, we show EW (A → X Y ) and tot EW as a function of m  with sβ−α = 0.995 and cβ−α < 0. The main differences between the case with sβ−α = 1 and sβ−α = 1 are upper bound on m  and the new decay mode A → Z h. The upper bound on m  is about 1 TeV due to the theoretical constraints. The qualitative

182

9 One-Loop Calculations for Decays of the CP-Odd Higgs Boson

Fig. 9.2 NLO EW corrections to the partial decay widths of the CP-odd Higgs boson with sβ−α = 1 and tan β=1 (red), 3 (blue), 10 (green). Masses of additional Higgs bosons are√degenerate, m  = m H ± = m A = m H . Mmax (Mmin ) denotes the maximum (minimum) value of M 2 satisfying the theoretical constraints and S, T parameters

Fig. 9.3 NLO corrections for the decay widths of the CP-odd Higgs boson with sβ−α = 0.995, cβ−α < 0 and tan β=1 (red), 3 (blue), 10 (green)

9.2 Next-to-Leading-Order Electroweak Corrections to the Decay Rates

183

Fig. 9.4 NLO corrections for the decay widths of the CP-odd Higgs boson with sβ−α = 0.995, cβ−α > 0 and tan β=1 (red), 3 (blue), 10 (green)

behaviors of EW (A → f f¯) is almost similar to those with sβ−α = 1, except for the region with large m  . The behavior of EW (A → Z h) in Type-I is almost the same as that in Type-II. There are two kinks at m   200 GeV and 350 GeV. The first one corresponds to the threshold effects of Z h, and the second one corresponds to those of the top quark. Near the top-quark threshold, the magnitude of the NLO EW correction is above 50%, while it decreases as m  becomes large. When tan β = 1, EW (A → Z h) is positive, and the non-decoupling effects of scalar loops enhance its magnitude. On the other hand, non-decoupling effects decrease the size of EW (A → Z h) when tan β = 3, and it can be negative for m  ≤ 300 GeV. The behavior of tot EW can be understood by Eq. (9.32), while we have an additional contribution from A → Z h. When tan β = 1, the behavior of tot EW in Type-I is almost the same as that in Type-II, except for below the Z h threshold. Below the ¯ Z h threshold, A → bb¯ and A → gg are the main decay modes, and EW (A → bb) causes the difference between Type-I and Type-II. Above the Z h threshold, A → Z h and A → gg are the main decay modes, while A → tt ∗ also contribute to the decay branching ratio, especially for m   2m t . Above the top-quark threshold, A → t t¯ ¯ mainly determine tot EW . When tan β = 10, A → t t is suppressed by cot β, and A → Z h can be dominant, while A → bb¯ also contributes to tot EW in Type-II. In Fig. 9.4, we show EW (A → X Y ) and tot EW as a function of m  with sβ−α = 0.995 and cβ−α > 0. The main differences from the case with cβ−α < 0 are the

184

9 One-Loop Calculations for Decays of the CP-Odd Higgs Boson

possible value of tan β and the sign of EW (A → Z h). The allowed regions for tan β = 10 are broader than those in cβ−α < 0, and the upper bound on m  is about 850 GeV. The sign of the NLO EW corrections to A → Z h is opposite to those in cβ−α < 0, and higher-order corrections mainly decrease the partial decay width. ¯ Above the top-threshold, the behavior of tot EW is mainly determined by A → t t and A → Z h in Type-I similar to the case with cβ−α < 0, while A → bb¯ also contributes to tot EW when tan β is large in Type-II.

9.3 Next-to-Leading-Order Electroweak Corrections to the Decay Branching Ratios In this section, we discuss the impact of NLO EW corrections on the decay branching ratios of the CP-odd Higgs boson in the nearly alignment scenario including the higher-order corrections. In addition to the theoretical constraints, we take into account the constraint from the S and T parameters, the signal strength of the SM-like Higgs boson, the direct searches of the additional Higgs bosons, and flavor experiments explained in Sect. 3.4. Since the branching ratios of A → Z h and H → hh are sensitive to the value of sβ−α , the excluded region would be changed when we include the higher-order corrections. Therefore, we dare to omit the constraints from the Higgs-to-Higgs decay modes in this thesis. We consider the following two scenarios for the mass spectrum of the additional Higgs bosons. Scenario A :

m A = m H ± = 300 GeV, m A ≤ m H ,

(9.33)

Scenario B :

m A = m H ± = 800 GeV, m H ≤ m A − m Z .

(9.34)

These scenarios satisfy the constraint from the T parameter due to the custodial symmetry in the Higgs potential (m A = m H ± ). In Scenario A, A → t t¯ does not ¯ τ τ¯ and gg as open, and the CP-odd Higgs boson mainly decays into Z h, tt ∗ , bb, shown in Figs. 4.5 and 4.7. In Scenario B, the CP-odd Higgs boson mainly decays into Z h, Z H, t t¯, bb¯ and τ τ¯ as shown in Figs. 4.6 and 4.8. For Scenario A, we study Type-I with tan β = 2 and 5. Type-II and Type-Y are excluded by the flavor constraints [4], and Type-X is also excluded by the constraint from A → τ τ¯ [5] (See Figs. 4.13 and 4.14). The lower bound on tan β comes from the constraint of Bd → μμ [4]. In addition, the region with tan β  2 is excluded by the direct searches for A → τ τ¯ and H ± → tb [5] (See Figs. 4.13 and 4.14). The mass of the additional CP-even Higgs boson is scanned as m A ≤ m H ≤ m A + 500 GeV.

(9.35)

9.3 Next-to-Leading-Order Electroweak Corrections to the …

185

The remaining parameters are scanned as 0.995 ≤ sβ−α ≤ 1, 0 ≤

√

M 2 ≤ m A + 500GeV,

(9.36)

with both of cβ−α < 0 and cβ−α > 0. For Scenario B, we study all types of 2HDMs. We scan tan β as 2 < tan β < 10.

(9.37)

For m H ± = 800 GeV, the region with tan β  1.2 is excluded for all types of 2HDM by Bd → μμ [4]. While there are allowed parameter regions with tan β ≤ 2, we take tan β = 2 as a minimum value conservatively. We also scan the mass of the additional CP-even Higgs bosons as m A − 500 GeV < m H < m A − m Z .

(9.38)

The scan regions of the remaining parameters are the same as those in Eqs. (9.35) and (9.36). We parametrize the NLO EW corrections on the decay branching ratios as

BR EW (A → X Y ) =

BRLO+EW+QCD (A → X Y ) − 1, BRLO+QCD (A → X Y )

(9.39)

where BRLO+EW+QCD (A → X Y ) includes the NLO EW and the higher-order QCD corrections, while BRLO+QCD (A → X Y ) includes only the higher-order QCD corrections. We use the running quark masses for the calculations of the LO parts. At the one-loop level, we can approximate BR EW (A → X Y ) as tot

BR EW (A → X Y ) ≈ EW (A → X Y ) − EW ,

(9.40)

tot

where EW (A → X Y ) and EW are given by LO+EW+QCD (A → X Y ) − 1, LO+QCD (A → X Y ) tot LO+EW+QCD = − 1. tot LO+QCD

EW (A → X Y ) = tot

EW

(9.41) (9.42)

¯ (Q = q, t) For the decays into a pair of quarks, we can approximate EW (A → Q Q) as ¯ ≈

EW (A → Q Q)

m 2Q m 2Q (m A )

¯

EW (A → Q Q),

(9.43)

186

9 One-Loop Calculations for Decays of the CP-Odd Higgs Boson

where EW (A → X Y ) given in Eq. (9.30). The ratio of the pole and running mass of ¯ For instance, m b /m b (m A ) ≈ 2 for the quarks enhances the size of EW (A → Q Q). ¯ becomes four times larger than m A = 500 GeV, and the size of EW (A → bb) ¯ For A → ¯ and A → V φ, EW (A → X Y ) = EW (A → X Y )

EW (A → bb). because we have no QCD correction at the one-loop level. We parametrize the deviation in the SM-like Higgs boson coupling as κ Z = κ Z − 1, where κ Z is defined in Eq. (3.95) with the partial decay rate of h → Z Z ∗ . In the 2HDM, κ Z is mostly negative independently of the sign of cβ−α [6] (See Fig. 6.5). On the other hand, allowed parameter regions depend on the sign of cβ−α . Therefore, we take Sign(cβ−α ) κ Z as the horizontal axis in some figures shown below to exhibit the difference due to the sign of cβ−α .

9.3.1 Scenario A In Fig. 9.5, we show the branching ratio of A → Z h including the NLO EW corrections in Type-I for Scenario A. The plots in the first column are the results with tan β = 2, while those in the second column are the results √ with tan β = 5. The color differences correspond to the values of cβ−α and M 2 in the top and bottom panels, respectively. When cβ−α  0.1, BR(A → reaches almost 80% Z h) (100%) for tan β = 2 (5). Since κ Z is proportional to cβ−α at LO, we expect that BR(A → Z h) becomes large when κ Z is large. However, this is not necessarily true when we include NLO corrections. When tan β = 2, we have the parameter points, where κ Z sizably deviates from the SM value, while the BR(A → Z h) √ is small. This is because of the non-decoupling scalar-loop effects in κ Z . When M 2 ≈ 0, the non-decoupling effects make κ Z large even with cβ−α ≈ 0. When tan β = 5, √ M 2 is almost degenerate with m A due to the theoretical bounds, and the size of the higher-order corrections is smaller than the tree-level contributions. Therefore, the correlation between BR(A → Z h) and κ Z follows the LO expectation. We note that BR(A → Z h) can reach about 100% even with κ Z ≤ 0.6, where it is difficult to observe the deviation in the h Z Z coupling at the ILC250 [7]. In this sense, A → Z h is useful to explore the nearly alignment scenario. In Fig. 9.6, we show the size of NLO EW corrections to the decay rates EW (A → tot Z h) and the total decay width EW . The plots in the first and third columns are the results with tan β = 2, while those in the second and fourth columns are the results √ with tan β = 5. The color differences correspond to the values of cβ−α and M 2 in the left and right panels, respectively. The sign of EW (A → Z h) is opposite √ with the sign of cβ−α , except for M 2  150 GeV with tan β = 2. The size of

EW (A → Z h) becomes quite large when cβ−α  0 since the LO contribution is close to zero. We note that such large EW (A → Z h) does not mean a breakdown of the perturbation, since it is caused by the smallness of the tree-level coupling. √ 2 When M  150 GeV with tan β = 2, there are two kinks around κ Z  0.8 with

9.3 Next-to-Leading-Order Electroweak Corrections to the …

187

Fig. 9.5 Decay branching ratios of A √ → Z h in Type-I 2HDM in Scenario A. The color differences correspond to the values of cβ−α and M 2 , in the top and bottom panels, respectively

Fig. 9.6 NLO corrections to the decay rates of A → Z h and the total decay width of the CP-odd Higgs boson √ in Type-I 2HDM in Scenario A. The color differences correspond to the values of cβ−α and M 2 , in the left and right panels, respectively

188

9 One-Loop Calculations for Decays of the CP-Odd Higgs Boson

Fig. 9.7 NLO corrections to the decay branching ratios of the CP-odd Higgs boson in Type-I √ 2HDM in Scenario A. The color differences correspond to the values of cβ−α and M 2 , in the top and bottom panels, respectively

cβ−α  0. Due to the smallness of the LO contribution, EW (A → Z h) becomes large, and the non-decoupling effects of scalar loops change the sign of EW (A → Z h) as shown Figs. 9.3 and 9.4. In addition, κ Z takes a relatively large value even if cβ−α  0 due to the non-decoupling effects. tot As similar to Eq. (9.32), we can expand EW as tot ¯ ¯

EW = EW (A → bb)BR LO (A → bb) + EW (A → Z h)BR LO (A → Z h)

+ other channels.

(9.44) tot

When cβ−α  0.1, the behavior of EW is mainly determined by EW (A → Z h) since BRLO (A → Z h) is dominant. On the other hand, when cβ−α  0, both ¯ and EW (A → Z h) contribute to tot

EW (A → bb) EW . While EW (A → Z h) takes quite large value with cβ−α  0, BRLO (A → Z h) is close to zero. Therefore, we have tot no singular behavior on EW even at κ Z  0.

9.3 Next-to-Leading-Order Electroweak Corrections to the …

189

Fig. 9.8 Decay branching ratios of the CP-odd Higgs boson in Scenario B. Predictions on TypeI, Type-II, Type-X, and Type-Y are shown from the left to the right panels in order. The color differences correspond to the values of m = m A − m H

In Fig. 9.7, we show the size of NLO EW corrections to the decay branching BR ratios BR EW (A → Z h). The behavior of EW (A → Z h) can be understood from tot

EW (A → Z h) and EW as shown in Eq. (9.40). When cβ−α  0.1, BRLO (A → tot Z h)  1 and tot EW  EW (A → Z h). Therefore, EW (A → Z h) and EW are canBR celed, and BR EW (A → X Y ) is close to zero. When cβ−α  0, EW (A → Z h) can be larger than 100% since BRLO (A → Z h)  0.

9.3.2 Scenario B In Fig. 9.8, we show the branching ratios of the CP-odd Higgs boson including the NLO EW corrections in Scenario B. The results in Type-I, Type-II, Type-X, and Type-Y are shown from the left to the right panels in order. The color differences correspond to the values of m = m A − m H . In Scenario B, A → Z H is open in addition to A → Z h, t t¯, bb¯ and τ τ¯ , and it can be the dominant decay mode. The value of tan β is approximately 2 independently of the types of 2HDM for large, both cβ−α > 0 and cβ−α < 0. The exceptional regions, where tan β can be √ are cβ−α  0 in all types of 2HDMs and cβ−α > 0 in Type-I. The value of M 2 is almost degenerate with m H . As similar to Scenario A with tan β = 2, we have parameter regions where non-decoupling effects enlarge κ Z even with cβ−α  0 in all types of 2HDMs. The maximal size of κ Z with cβ−α  0 is about 1%. If cβ−α = 0, κ Z can be larger than 1%, and it reaches about 1.5% independently of the types of 2HDM for both cβ−α > 0 and cβ−α < 0. For A → Z h, the branching ratio reaches about 80% (20%) in Type-I with cβ−α > 0 (cβ−α < 0). In Type-II, Type-X, and Type-Y, it reaches about 20% indepen-

190

9 One-Loop Calculations for Decays of the CP-Odd Higgs Boson

Fig. 9.9 NLO corrections to the decay rates and the total decay width of the CP-odd Higgs boson as a function of m A − m H in Scenario B. Predictions on Type-I, Type-II, Type-X, and Type-Y are shown from the left to the right panels in order. The color differences correspond to the values of cβ−α

dently of the sign of cβ−α . When m  300 GeV, the branching ratio of A → Z H reaches about 100% independently of the types of 2HDMs. Since the tree-level AZ H coupling is proportional to sβ−α , we expect that BR(A → Z H ) can be large with

κ Z  0. However, BR(A → Z H ) tends to be small when κ Z  0. This is because of the theoretical bounds such as perturbative unitary and vacuum stability. If m is so large that BR(A → Z H )  1, cβ−α = 0 is favored by the theoretical bounds, and κ Z becomes large due to the both of tree-level mixing and loop effects. tot In Fig. 9.9, we show the size of EW (A → X Y ) and EW as a function of m. The color differences correspond to the values of cβ−α . For the illustration purpose, we take the range of EW (A → Z h) as [−50, 50]%, while EW (A → Z h) can be larger than 100% if cβ−α is quite small. We find that both EW (A → Z h) and EW (A → tot Z H ) are the almost same among all types of 2HDMs, while EW shows typedependent behavior, especially for m  200 GeV. When cβ−α < 0, the magnitude of EW (A → Z h) is almost independent of m. On the other hand, it monotonically increases as m becomes large when cβ−α > 0. The magnitude of EW (A → Z H ) increases as m becomes large due to the non-decoupling effects of scalar loops. tot When m  200 GeV, EW is mainly determined by A → Z H , and its magnitude also increases as m becomes large in all types of 2HDMs. In Type-II and Typetot tot Y, A → bb¯ contributes to EW when m  200 GeV with large tan β, and EW reaches about −20%. In Fig. 9.10, we show the size of BR EW (A → X Y ) as a function of m. The color differences correspond to the values of cβ−α . BR EW (A → Z h) is negative when

9.3 Next-to-Leading-Order Electroweak Corrections to the …

191

Fig. 9.10 NLO corrections to the decay branching ratios for the CP-odd Higgs boson as a function of m A − m H in Scenario B. Predictions on Type-I, Type-II, Type-X, and Type-Y are shown from the left to the right panels in order. The color differences correspond to the values of cβ−α

tot

cβ−α > 0, while it can be positive when cβ−α < 0 due to the contribution of EW .

BR EW (A → Z H ) is close to zero in large m region due to the cancellation between BR

EW (A → Z H ) and tot EW . When m  150 GeV and cβ−α  0, EW (A → Z H ) tot becomes large in Type-II and Type-Y, where A → bb¯ contributes to EW .

9.3.3 Discrimination of Types of the Yukawa Interaction in the 2HDMs In this subsection, we discuss the discrimination of the types of 2HDMs by the decays is proportional to ζ f , we can discriminate the of the CP-odd Higgs boson. Since  AP,tree f f¯ types of 2HDMs by the correlations among BR(A → f f¯). We study the correlation ¯ including the NLO EW and higher-order between BR(A → τ τ¯ ) and BR(A → bb) QCD corrections in Scenario B. ¯ in In Fig. 9.11, we show the correlation between BR(A → τ τ¯ ) and BR(A → bb) Scenario B. The color differences correspond to the values of tan β and m in the left and right panels, respectively. In Scenario B, there is A → Z H , and it can be dominant depending on m. From the left panel of Fig. 9.11, we can see that both ¯ become small in Type-I, while both of them can BR(A → τ τ¯ ) and BR(A → bb) take several dozens of percent due to the tan β enhancement in Type-II. On the other ¯ becomes large in hand, BR(A → τ τ¯ ) can be large in Type-X, while BR(A → bb) Type-Y. From the right panel of Fig. 9.11, we can see that both of BR(A → τ τ¯ ) and ¯ can be several dozens of percent if m  150 GeV, while they are BR(A → bb) below 1% when m  400 GeV. Therefore, we can discriminate the types of 2HDMs by examining the decay pattern of the CP-odd Higgs boson if m is not so large.

192

9 One-Loop Calculations for Decays of the CP-Odd Higgs Boson

¯ in Type-I, Type-II, Type-X and Fig. 9.11 Correlation between BR(A → τ τ¯ ) and BR(A → bb) Type-Y for Scenario B. The color differences correspond to the values of tan β and m in the left and the right panels, respectively

We note that A → bb¯ and A → τ τ¯ can be dominant when cβ−α ≈ 0. In the case with cβ−α  0, it would be difficult to measure the deviation in the SM-like Higgs boson couplings. Therefore, the decay pattern of the CP-odd Higgs boson is useful to discriminate the types of 2HDMs if m is not so large. When m is large, we have sizable deviations in the SM-like Higgs boson couplings as discussed in Sect. 9.3.2. Therefore, the measurement of SM-like Higgs boson couplings can be used to study the types of 2HDMs [8, 9]. Thus, the decay pattern of the CP-odd Higgs boson and the deviation in the SM-like Higgs boson couplings play a comprehensive role, and we can determine the types of 2HDMs by using them.

References 1. Kanemura Shinya, Kikuchi Mariko, Sakurai Kodai, Yagyu Kei (2018) H-COUP: a program for one-loop corrected Higgs boson couplings in non-minimal Higgs sectors. Comput Phys Commun 233:134–144 2. Kanemura Shinya, Kikuchi Mariko, Mawatari Kentarou, Sakurai Kodai, Yagyu Kei (2020) H-COUP version 2: a program for one-loop corrected Higgs boson decays in non-minimal Higgs sectors. Comput Phys Commun 257:107512 3. Williams Karina E, Rzehak Heidi, Weiglein Georg (2011) Higher order corrections to Higgs boson decays in the MSSM with complex parameters. Eur Phys J C 71:1669 4. Haller Johannes, Hoecker Andreas, Kogler Roman, Mönig Klaus, Peiffer Thomas, Stelzer Jörg (2018) Update of the global electroweak fit and constraints on two-Higgs-doublet models. Eur Phys J C 78(8):675 5. Aiko Masashi, Kanemura Shinya, Kikuchi Mariko, Mawatari Kentarou, Sakurai Kodai, Yagyu Kei (2021) Probing extended Higgs sectors by the synergy between direct searches at the LHC and precision tests at future lepton colliders. Nucl Phys B 966:115375 6. Kanemura Shinya, Kikuchi Mariko, Mawatari Kentarou, Sakurai Kodai, Yagyu Kei (2018) Loop effects on the Higgs decay widths in extended Higgs models. Phys Lett B 783:140–149

References

193

7. Barklow Tim, Fujii Keisuke, Jung Sunghoon, Karl Robert, List Jenny, Ogawa Tomohisa, Peskin Michael E, Tian Junping (2018) Improved formalism for precision Higgs coupling fits. Phys Rev D 97(5):053003 8. Kanemura Shinya, Tsumura Koji, Yagyu Kei, Yokoya Hiroshi (2014) Fingerprinting nonminimal Higgs sectors. Phys Rev D 90:075001 9. Kanemura Shinya, Kikuchi Mariko, Mawatari Kentarou, Sakurai Kodai, Yagyu Kei (2019) Full next-to-leading-order calculations of Higgs boson decay rates in models with nonminimal scalar sectors. Nucl Phys B 949:114791

Chapter 10

Conclusion and Discussion

Although the SM successfully describes the nature of fundamental particles, there are phenomena that cannot be explained within the SM, such as the existence of dark matter, baryon asymmetry of the universe, and tiny but non-zero neutrino masses. In addition, there are conceptual problems in the SM such as the hierarchy problem, the strong CP problem, no unified description for the gauge group, and so on. Therefore, we are convinced that the SM is not a fundamental theory and it must be replaced by a more fundamental theory. While the Higgs boson was found, the structure of the Higgs sector is still a mystery. We note that there is no theoretical principle to insist on the minimal structure of the Higgs sector as introduced in the SM, and models with extended Higgs sectors can be compatible with current experimental data. Such extended Higgs sector often appears in various models, where the conceptual problems in the SM are tried to be solved. In addition, new physics models with non-minimal Higg sectors can solve the above-mentioned phenomenological problems. Therefore, the scenario of new physics can be narrowed down by unraveling the nature of the Higgs sector, and the determination of its structure is one of the central interests of current and future high-energy physics. In this thesis, we have investigated the 2HDM as a representative of extended Higgs models. In Chap. 4, we have discussed the direct searches of the additional Higgs bosons at current and future hadron colliders. The direct search for new particles is the key program to study new physics models, and the LHC and HL-LHC will make remarkable progress in this direction. Especially, Higgs-to-Higgs decays such as H → hh and A → Z h are quite useful channels to explore the nearly alignment scenario. If we have no signal, they lead to a lower bound on the mass scale of the additional Higgs bosons. We have pointed out that indirect searches through the precision measurement of the SM-like Higgs boson couplings play a complemental role by imposing the upper bound on the mass scale of the additional Higgs bosons through the theoretical argument such as perturbative unitarity and vacuum stability. By utilizing the synergy between them, we can explore the wide range of the parameter space of the 2HDM in future collider experiments. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 M. Aiko, Theoretical Studies on Extended Higgs Sectors Towards Future Precision Measurements, Springer Theses, https://doi.org/10.1007/978-981-99-1324-4_10

195

196

10 Conclusion and Discussion

The tree-level study in Chap. 4 leads us to investigate higher-order EW corrections. The precision of measurements of the SM-like Higgs boson couplings at future linear colliders such as ILC reaches a per-mile level, and theoretical predictions at the lowest order of perturbation are not enough. Especially, the current SM-like situation makes the study of higher-order corrections further important because higher-order corrections would sizably change the theoretical predictions from the lowest-order analysis. After reviewing the higher-order EW corrections to the SM-like Higgs boson decays in Chap. 6, we have discussed the production cross sections for the Higgs strahlung process at electron-proton colliders including the higher-order corrections in Chap 7. The production cross-section of this process takes the maximal value at the center of beam energy around 250 GeV, and most of the future lepton collider experiments are planned to perform the precision measurement of the SMlike Higgs boson at this beam energy. In the 2HDMs, we have analyzed the deviation in the total cross-section from its SM prediction. We have found that the magnitude of higher-order corrections via additional Higgs bosons loop diagrams is comparable with that of the tree-level contribution, especially in the nearly alignment scenario. In addition, we have examined the deviation in the cross-section times branching ratio, and it is found that we can distinguish the type of Yukawa interactions in 2HDMs by future precision measurements. In Chaps. 8 and 9, we have discussed decays of the charged and CP-odd Higgs bosons including the higher-order corrections. We have found that higher-order corrections to the Higgs-to-Higgs decays are not negligible, especially In the nearly alignment scenario. This is because the Higgs-to-Higgs decays are suppressed at the lowest order in the alignment scenario. Therefore, it is important to include these effects for direct searches of the additional Higgs bosons at future hadron colliders. From the series of these studies, we have found that the nature of the Higgs sector can be widely investigated by combining the precision measurement of the SM-like Higgs boson and the direct search of the additional Higgs bosons. By unraveling the nature of the Higgs sector, we can narrow down the scenario of new physics and approach the physics beyond the SM.

Appendix A

Input Parameters and Basics of Quantum Chromodynamics

A.1 Input Parameters We summarize the input parameters, which are used for numerical studies in this thesis. These values are taken from Ref. [1].

A.1.1 Lepton Masses The lepton masses are m e = 0.510998918 × 10−3 GeV, m μ = 0.105658367 GeV,

(A.1) (A.2)

m τ = 1.77686 GeV.

(A.3)

A.1.2 Electroweak Parameters The fine structure constant is −1 = 137.035999139. αem

(A.4)

G F = 1.1663787 × 10−5 GeV−2 .

(A.5)

The Fermi constant is

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 M. Aiko, Theoretical Studies on Extended Higgs Sectors Towards Future Precision Measurements, Springer Theses, https://doi.org/10.1007/978-981-99-1324-4

197

198

Appendix A: Input Parameters and Basics of Quantum Chromodynamics

The mass of the Z boson is m Z = 91.1876 GeV.

(A.6)

A.1.3 QCD Parameters The strong coupling constant at the scale of Z boson mass is αs (m Z ) = 0.1181.

(A.7)

The pole masses of the quarks are m t = 173.1 GeV,

(A.8)

m b = 4.78 GeV, m c = 1.27 GeV.

(A.9) (A.10)

A.1.4 Higgs Boson Mass The mass of the discovered Higgs boson is m h = 125.1 GeV.

(A.11)

A.2 Running of QCD Parameters We summarize the basics of QCD corrections. This part is based on Refs. [2, 3].

A.2.1 The Strong Coupling Constant The beta function of the αs is defined by (n )    (n f ) d αs f (μ) (n f ) (n f ) = β α =− βi μ s 2 dμ π i=0 2



(n f )

αs

π

(μ)

i+2 ,

(A.12)

Appendix A: Input Parameters and Basics of Quantum Chromodynamics

199

where n f is the number of active flavors. The coefficients are given by (n ) β0 f

=

(n f )

=

(n f )

=

(n f )

=

β1 β2 β3

  2 1 (n f ) 1 (A.13) β 11 − n f , = 4 0,Djouadi 4 3   38 1 (n f ) 1 (A.14) β1,Djouadi = 102 − n f , 8 16 3   1 (n f ) 1 2857 5033 325 2 (A.15) β − nf + n , = 128 2,Djouadi 64 2 18 54 f 

1,078,361 6508 1 149,753 + 3564ζ(3) + − − ζ(3) n f 256 6 162 27 

50,065 6472 1093 3 (A.16) + ζ(3) n 2f + n , + 162 81 729 f

where ζ(n) is Riemann’s zeta function, with value ζ(3) ≈ 1.202057. It is convenient to introduce the following notation (n f )

bi

(n f )

=

βi

, a (n f ) (μ) = (n f )

(n f )

αs

π

β0

(μ)

,

(A.17) (n )

f In the following, we drop flavor index n f . Integrating Eq. (A.12) from QCD to μ leads to   −1 da μ2   = da = ln 2 β QCD β a2 1 + b a + b a2 + b a3 + · · ·



0

1

2

3

 b3 b13 −1 2 2 2 1 − b − b a a − (b − b )a − b + + · · · da 1 2 1 2 1 β0 a 2 2 2   3

b3 b 1 1 + b1 ln a + (b2 − b12 )a + − b1 b2 + 1 a 2 + const, ≈ β0 a 2 2 (A.18) 



=

where we made Taylor expansion around a = 0. By iteratively solving Eq. (A.18), we obtain

1 b1 ln L αs (μ) 1 2 2 = − b + (ln L − ln L − 1) + b 2 1 π β0 L (β0 L)2 (β0 L)3 

 1 5 2 1 b3 3 3 b − ln ln − 3b , + L + L + 2 ln L − b ln L + 1 2 (β0 L)4 1 2 2 2 (A.19) where L = ln (μ2 /2QC D ) and terms of O(1/L 5 ) have been neglected. QCD is defined in MS scheme so that integration constant becomes (b1 /β0 ) ln β0 . In this

200

Appendix A: Input Parameters and Basics of Quantum Chromodynamics

definition, Eq. (A.19) does not contain a term proportional to (const/L 2 ). There are (n ) (n ) three way to compute αs f (μ2 ) for given αs f (μ1 ). 1. Solve the differential equation Eq. (A.12) numerically using αs (μ)|μ=μ1 = αs (μ1 ) as initial condition. We may use Runge-Kutta method to evaluate (n ) αs f (μ2 ) in this way. This method is less efficient than the following method because we need to make numerical integration. 2. Setting μ = μ1 and αs (μ1 ) in Eq. (A.18), one can obtain ln

 1 μ21 1 + b1 ln a(μ1 ) + (b2 − b12 )a(μ1 ) = 2 β0 a(μ1 ) QCD

 b3 b13 2 − b1 b2 + a (μ1 ) + 2 2 b1 ln β0 . + β0

(A.20)

we can regard Eq. (A.20) as the equation of QCD and determine its value by solving this equation numerically. We can evaluate αs (μ2 ) by substituting determined QCD into Eq. (A.19). 3. Setting μ = μ1 and αs (μ1 ) in Eq. (A.19), one can obtain

αs (μ1 ) 1 b1 ln L 1 1 2 2 = b − + (ln L − ln L − 1) + b 1 1 2 π β0 L 1 (β0 L 1 )2 (β0 L 1 )3 1 

1 5 2 1 3 3 b − ln L 1 + ln L 1 + 2 ln L 1 − + (β0 L 1 )4 1 2 2  b3 , (A.21) − 3b1 b2 ln L 1 + 2 where L 1 = ln (μ21 /2QC D ). we can regard Eq. (A.21) as the equation of QCD and determine its value by solving this equation numerically. We can evaluate αs (μ2 ) by substituting determined QCD into Eq. (A.19). Let we compare the values of αs(5) (m b ) evaluated by using above three method. We use μ1 = m Z = 91.18 GeV, αs(5) (m Z ) = 0.118 and μ2 = m b = 4.7 GeV. We assume the input value of αs(5) (m b ) is determined from the experiment with threeloop accuracy, which means that in the β function Eq. (A.12) only the coefficients up to β2 are considered and β3 is neglected. Then, we have (see Ref. [2]) 1. αs(5) (m b ) = 0.216712. 2. αs(5) (m b ) = 0.216610, 3. αs(5) (m b ) = 0.216712,

(5) QC D = 0.208905 GeV. (5) QC D = 0.208348 GeV.

The values of αs(5) (m b ) differ from each other at 4 digit level.

Appendix A: Input Parameters and Basics of Quantum Chromodynamics

201

A.2.2 Masses of Quarks In general, the masses, which are relevant for Higgs decay processes are not the quark pole masses but the running quark masses evaluated at the Higgs mass. For the given on-shell mass, the corresponding MS quantity can be computed by  m q (m q ) α2 (m q ) 4 αs (m q ) + (1.0414N f − 14.3323) s 2 = 1− mq 3 π π + (−0.65269N 2f + 26.9239N f − 198.7068)

 αs3 (m q ) . π2

(A.22)

In the MS scheme, the running of the quark masses is governed by the anomalous dimension γm (αs ).    (n f ) d (n f ) (n ) (n ) m (μ) = m (n f ) (μ)γm f αs f = −m (n f ) (μ) γm,i μ 2 dμ i=0 2



(n f )

αs

π

(μ)

i+1 , (A.23)

where the coefficients γm,i are known up to the four-loop order, (n )

γm,0f = 1, (n )



(A.24)



1 202 20 (A.25) − nf , 16 3 9  

2216 160 1 140 2 (A.26) 1249 + − − ζ(3) n f − n , = 64 27 3 81 f  1 4603055 135680 + ζ(3) − 8800ζ(5) = 256 162 27

18400 91723 34192 − ζ(3) + 880ζ(4) + ζ(5) n f + − 27 9 9

 160 5242 800 332 64 2 + ζ(3) − ζ(4) n f + − + ζ(3) n 3f , + 243 9 3 243 27 (A.27)

γm,1f = (n )

γm,2f

(n )

γm,3f

where ζ(3) ≈ 13.202057, ζ( 4) = π 4 /90 and ζ(5) ≈ 1.036928. It is convenient to introduce (n ) ci f

(n )

=

γm,if

(n f )

β0

.

(A.28)

202

Appendix A: Input Parameters and Basics of Quantum Chromodynamics

The solution of Eq. (A.23) is given by   c αs (μ)/π  m q (m q ), m q (μ) =  c αs (m q )/π

(A.29)

where c(x) function is defined as   1 (c1 − b1 c0 )2 + c2 − b1 c1 + b12 c0 − b2 c0 x 2 c(x) = x 1 + (c1 − b1 c0 )x + 2  1 1 + (c1 − b1 c0 )3 + (c1 − b1 c0 )(c2 − b1 c1 + b12 c0 − b2 c0 ) 6 2   1 2 3 (A.30) + (c3 − b1 c2 + b1 c1 − b2 c1 − b1 c0 + 2b1 b2 c0 − b3 c0 ) x 3 . 3 c0

We neglect terms of O(x 4 ). We list the c(x) coefficients for each active flavor number cs (x) = x 4/9 (1 + 0.895062x + 1.37143x 2 + 1.95168x 3 )

(N f = 3), (A.31)

cc (x) = x

12/25

(1 + 1.01413x + 1.38921x + 1.09054x )

(N f = 4), (A.32)

cb (x) = x

12/23

(1 + 1.17549x + 1.50071x + 0.172478x )

(N f = 5), (A.33)

ct (x) = x

4/7

2

3

2

3

(1 + 1.39796x + 1.79348x + 0.683433x ) 2

3

(N f = 6). (A.34)

A.2.3 Decoupling at Flavor Threshold A.2.3.1

Matching of αs (μ)

In MS or MS renormalization scheme, the decoupling theorem in general does not apply to the quantities which do not represent physical observables such as beta functions. So, quarks with masses much larger than the considered energy scale do not automatically decouple. We need to consider matching condition when we want (n f )

to evaluate αs

(n f )

(μ2 ) from the input αs (n f

)

(μ1 ) where n f = n f .

To evaluate αs (μ2 ), we first construct effective field theory by integrating out heavy fields by hand. Starting from the full theory which has n f flavors composed of one heavy quark flavor f heavy with mass m heavy and n light = n f − 1 massless flavors. We may choose the heavy quark MS mass m heavy (μ) which is evaluated at pole mass m heavy as heavy quark threshold (where we integrated out the heavy flavor), μ(n f ) = m heavy (m heavy ).

(A.35)

The connection between the strong coupling constant on the effective and full theory is given by,

Appendix A: Input Parameters and Basics of Quantum Chromodynamics (n f −1)

αs

(n f )

(μ) = ζg2 αs

203

(μ),

(A.36)

where ζg is known up to three-loop order 2

 (n f ) (n ) 2  αs (μ) αs f (μ) 1 μ2 MS ζg =1− + ln π 6 (μ(n f ) )2 π

2 2 11 11 μ 1 2 μ − ln (n f ) 2 + ln 12 25 (μ ) 36 (μ(n f ) )2  (n ) 3  αs f (μ) 564731 82043 μ2 955 − ζ3 − ln (n f ) 2 + π 124416 27648 576 (μ ) μ2 53 2 μ2 1 ln ln3 (n f ) 2 − (n f − 1) + − (n ) 2 f 576 216 (μ ) (μ )

 2633 67 μ2 1 2 μ2 . − ln (n f ) 2 + ln 31104 576 (μ ) 36 (μ(n f ) )2 (n f −1)

We use Eq(A.36) to evaluate αs is valid up to NLO.

A.2.3.2

(n f −1)

(μ(n f ) ) and in this case, αs

(A.37) (n f )

(μ) = αs

(μ)

Matching of m q (μ)

If we want to evaluate lighter quark masses at the scale where RG evaluation crosses the flavor threshold, we need to take into account the matching condition of quark masses. The connection between the quark mass in the effective and full theory is given by, (n f −1)

mq

(n f )

= ζm m q

,

(A.38)

the matching constant ζm is given by 

ζmMS

2

89 5 μ2 1 2 μ2 − ln (n f ) 2 + ln π 432 36 (μ ) 12 (μ(n f ) )2  (n ) 3 

2951 407 5 1 5 αs f (μ) 311 − ζ(3) + ζ(4) − B4 + − − ζ(3) + π 2916 864 4 36 2592 6

=1+

(n f )

αs

μ2

(μ)

175 2 μ2 29 3 μ2 ln ln + + (n ) 2 f 432 216 (μ ) (μ(n f ) )2

 2 53 μ2 μ2 1327 1 , − ζ(3) − ln (n f ) 2 − ln3 (n f ) 2 + (n f − 1) 11,664 27 432 (μ ) 108 (μ ) (A.39)

ln

(μ(n f ) )2

204

Appendix A: Input Parameters and Basics of Quantum Chromodynamics

where B4 = 16Li4 (n f −1)

Note that m q



13 1 − ζ(4) − 4ζ(2) ln2 2 ≈ −1.762800. 2 2 (n f )

(μ(n f ) ) = m q

(μ(n f ) ) is valid up to NLO.

(A.40)

Appendix B

Feynman Rules

B.1 Standard Model The interaction terms among the Higgs bosons and the weak gauge bosons are given by μ

Lkin = gφV1 V2 g μν φV1μ V2μ + gφ1 φ2 V (∂ μ φ1 φ2 − φ1 ∂ μ φ2 )Vμ + gφ1 φ2 V1 V2 gμν φ1 φ2 V1 V2ν .

(B.1)

where coefficients of the Scalar-Gauge-Gauge vertex gφV1 V2 are listed in Table. B.1, while those of the Scalar-Scalar-Gauge vertex gφ1 φ2 V and the Scalar-Scalar-GaugeGauge vertex gφ1 φ2 V1 V2 are listed in Table. B.2. φ1

p1 Vμ

φ2 φ1

φ2

=

gφ1 φ2 V ( p1 − p2 )μ ,

(B.2)

=

igφ1 φ2 V1 V2 g μν .

(B.3)

p2 μ

V1

V2ν

The Yukawa interaction terms are given in terms of the Higgs bosons as LSM Y ukawa = −

 mf   f¯ f h − 2i I f f¯γ5 f G 0 v f =u,d,e

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 M. Aiko, Theoretical Studies on Extended Higgs Sectors Towards Future Precision Measurements, Springer Theses, https://doi.org/10.1007/978-981-99-1324-4

205

206

Appendix B: Feynman Rules

Table B.1 Coefficients of Scalar-Gauge-Gauge vertices in the SM Vertices gφV1 V2 g2 2 v g 2Z 4 v − gg2Z eg 2 v

hW + W − hZ Z G± Z W ∓ G ± γW ∓

2 vsW

Table B.2 Coefficients of Scalar-Scalar-Gauge and Scalar-Scalar-Gauge-Gauge vertices in the SM

Vertices hG ± W ∓ G0G± W ∓ hG 0 Z G+G− Z G+ G−γ

gφ1 φ2 V ∓i g2 − g2 − g2Z i g2Z c2W ie

Vertices

gφ1 φ2 V1 V2

Vertices

gφ1 φ2 V1 V2

hhW + W −

g2 4 g2 4 g2 2 g 2Z 8 g 2Z 8 g 2Z 2 4 c2W

G + G − γγ

e2

G ± hW ∓ Z

2 s − gg2Z sW β−α

G0G0 W + W − G+G− W + W − hh Z Z G0G0 Z Z G+G− Z Z

G±G0 W ∓ Z

2 ±i gg2Z sW

G ± hW ∓ γ G± G0 W ∓γ

eg 2 sβ−α ∓i eg 2

G+ G−γ Z

eg Z c2W

√ +

 2 Vud u(m ¯ u ζu PL − m d ζd PR )dG + − m e ζe ν¯ PR eG + + h.c. . v (B.4)

f hf f

h

=

ig S

0

=

ig P

 m  f , =i − v

(B.5)

2I f m f γ5 . γ5 = i i v

(B.6)

f f G f

G0 f f

Appendix B: Feynman Rules

207

u G

+

=

  + √  mu md  + PL − PR Vud = i g SG + g GP γ5 , i 2 v v

d (B.7) d G−

=

  − √  mu md  ∗ − PR − PL Vud = i g SG + g GP γ5 , i 2 v v

u (B.8) with +

mu − md mu + md + Vud , g GP = − √ Vud , √ 2v 2v mu − md ∗ mu + md ∗ − = √ Vud , g GP = √ Vud . 2v 2v

g SG = g SG

−

(B.9) (B.10)

u Wμ+

=

igW γμ PL ,

(B.11)

Wμ−

=

∗ igW γμ PL ,

(B.12)

d d

u with g gW = √ Vud . 2

(B.13)

208

Appendix B: Feynman Rules

f Zμ

=

ig Z γμ (v f − a f γ5 ) = ig Z γμ (a Z − b Z γ5 ), (B.14)

γμ

=

ieQ f γμ = igγ γμ (aγ − bγ γ5 ),

f f (B.15)

f with g 1 1 2 , a Z = v f = I f − Q f sW , bZ = a f = I f , cW 2 2 gγ = e, aγ = Q f , bγ = 0.

gZ =

(B.16) (B.17)

We list the expression of the scalar couplings. We use the following notation for these couplings, L ⊃ +λφi φ j φk φi φ j φk + λφi φ j φk φ φi φ j φk φ .

(B.18)

The Higgs trilinear couplings are given by λhG + G − = −λv, λhG 0 G 0 =

1 1 λhG + G − , λhhh = − λv. 2 2

(B.19)

The Higgs quartic couplings are given by 1 1 1 λG + G − G + G − = − λ, λG + G − G 0 G 0 = − λ, λG 0 G 0 G 0 G 0 = − λ, 2 2 8 1 1 1 λhhG + G − = − λ, λhhG 0 G 0 = − λ, λhhhh = − λ. 2 4 8

(B.20) (B.21)

B.2 Two-Higgs Doublet Model Similar to the SM, the interaction terms among the Higgs bosons and the weak gauge bosons are given by Eq. (B.1). Coefficients of the Scalar-Gauge-Gauge vertex gφV1 V2 are listed in Table. B.3, while those of the Scalar-Scalar-Gauge vertex gφ1 φ2 V and the Scalar-Scalar-Gauge-Gauge vertex gφ1 φ2 V1 V2 are listed in Table. B.4.

Appendix B: Feynman Rules

209

Table B.3 Coefficients of scalar-gauge-gauge vertices in the 2HDM Vertices gφV1 V2 g2 2 vsβ−α g2 2 vcβ−α g 2Z 4 vsβ−α g 2Z 4 vcβ−α 2 − gg2Z vsW eg 2 v

hW + W − H W +W − hZ Z HZZ G± Z W ∓ G ± γW ∓

Table B.4 Coefficients of Scalar-Scalar-Gauge and Scalar-Scalar-Gauge-Gauge vertices in the 2HDM

Vertices hG ± W ∓ H G± W ∓ G0G± W ∓ h H±W ∓ H H±W ∓ AH ± W ∓ hG 0 Z H G0 Z h AZ H AZ G+G− Z H+H−Z G+ G−γ H + H −γ

gφ1 φ2 V ∓i g2 sβ−α ∓i g2 cβ−α − g2 ∓i g2 cβ−α ±i g2 sβ−α − g2 − g2Z sβ−α − g2Z cβ−α − g2Z cβ−α gZ 2 sβ−α i g2Z c2W i g2Z c2W ie ie

Vertices

gφ1 φ2 V1 V2

Vertices

gφ1 φ2 V1 V2

hhW + W −

g2 4 g2 4 g2 4 g2 4 g2 2 g2 2 g 2Z 8 g 2Z 8 g 2Z 8 g 2Z 8 g 2Z 4 g 2Z 4 e2

G±G0 W ∓ Z

2 ±i gg2Z sW

H H W +W − A AW + W − G0G0 W + W − G+G− W + W − H+ H−W +W − hh Z Z HHZZ A AZ Z G0G0 Z Z G+G− Z Z H+H−Z Z G + G − γγ H + H − γγ

e2

H ± AW ∓ Z G± H W ∓ Z H ± hW ∓ Z G ± hW ∓ Z

2 ±i gg2Z sW

2 c − gg2Z sW β−α 2 c − gg2Z sW β−α 2 s − gg2Z sW β−α

G± H W ∓γ

gg Z 2 2 sW sβ−α ∓i eg 2 ∓i eg 2 eg c 2 β−α eg 2 cβ−α

2 c2W

G+ G−γ Z

eg Z c2W

2 c2W

H + H −γ Z G ± hW ∓ γ H ± H W ∓γ

eg Z c2W eg 2 sβ−α − eg 2 sβ−α

H±HW∓Z H ± AW ∓ γ G± G0 W ∓γ H ± hW ∓ γ

The Yukawa interaction terms are given in terms of the mass eigenstates of the Higgs bosons as   mf  f f ζh f¯ f h + ζ H f¯ f H − 2i I f ζ f f¯γ5 f A v f =u,d,e √  2 Vud u(m + ¯ u ζu PL − m d ζd PR )d H + − m e ζe ν¯ PR eH + + h.c. , v (B.22)

L2HDM Y ukawa = −

210

Appendix B: Feynman Rules

with f

ζh = sβ−α + ζ f cβ−α ,

(B.23)

f ζH

(B.24)

= cβ−α − ζ f sβ−α .

The Yukawa interactions of the NG bosons are the same as those in the SM.

f

m f ζh f f , =i − v

(B.25)

m f ζH f f , =i − v

(B.26)

h

=

hf f ig S

H

=

ig S

=

Af f ig P γ5

f f Hff

f f A

2I f m f ζ f γ5 . =i i v

(B.27)

f

u H+

=



  + √ m u ζu m d ζd + i 2 PL − PR Vud = i g SH + g PH γ5 , v v

d

(B.28) d H−

=

  − √ m u ζu m d ζd − ∗ i 2 = i g SH + g PH γ5 , PR − PL Vud v v

u

(B.29)

Appendix B: Feynman Rules

211

with +

m u ζu − m d ζd m u ζu + m d ζd + Vud , g PH = − Vud , √ √ 2v 2v m u ζu − m d ζd ∗ m u ζu + m d ζd ∗ − = Vud , g PH = Vud . √ √ 2v 2v

g SH = g SH

−

(B.30) (B.31)

We list the expression of the scalar couplings in terms of the coefficients of the Higgs potential in the Higgs basis. We use the notation in Eq. (B.18). In terms of the masses of Higgs bosons and mixing angles, the coefficients Z 1 , ..., Z 7 are given as Z 1 v 2 = m 2H cos2 (β − α) + m 2h sin2 (β − α), Z2v = 2

m 2H

+

cos (β − α) + 2

m 2h

(B.32)

sin (β − α) 2

− − α) sin (β − α) cot 2β  2  + 4 (m H − M ) sin (β − α) + (m 2h − M 2 ) cos2 (β − α) cot2 2β, (B.33) 4(m 2h

m 2H ) cos (β 2 2

Z 3 v 2 = m 2H cos2 (β − α) + m 2h sin2 (β − α) + 2(m 2h − m 2H ) cos (β − α) sin (β − α) cot 2β + 2(m 2H ± − M 2 ), Z 4 v = m 2H sin2 (β − α) + m 2h cos2 (β − α) + m 2A − 2m 2H ± , Z 5 v 2 = m 2H sin2 (β − α) + m 2h cos2 (β − α) − m 2A , Z 6 v 2 = (m 2h − m 2H ) cos (β − α) sin (β − α),  Z 7 v 2 = 2 (m 2H − M 2 ) sin2 (β − α) + (m 2h − M 2 ) cos2 (β − + (m 2h − m 2H ) cos (β − α) sin (β − α).

(B.34)

2

(B.35) (B.36) 

(B.37)

α) cot 2β (B.38)

We use the short-hand notation Z 345 = Z 3 + Z 4 + Z 5 . The Higgs trilinear couplings are given by λhG + G − = −(Z 1 sβ−α + Z 6 cβ−α )v, 1 λhG 0 G 0 = λhG + G − , 2 λ H G + G − = −(Z 1 cβ−α − Z 6 sβ−α )v, 1 λH G0 G0 = λH G+ G− , 2 λh H + H − = −(Z 3 sβ−α + Z 7 cβ−α )v, λ H H + H − = −(Z 3 cβ−α − Z 7 sβ−α )v,  1 λh A A = − (Z 3 + Z 4 − Z 5 )sβ−α + Z 7 cβ−α v, 2  1 λ H A A = − (Z 3 + Z 4 − Z 5 )cβ−α − Z 7 sβ−α v, 2   1 λhG ± H ∓ = − Z 6 sβ−α + (Z 4 + Z 5 )cβ−α v, 2

(B.39) (B.40) (B.41) (B.42) (B.43) (B.44) (B.45) (B.46) (B.47)

212

Appendix B: Feynman Rules   1 λ H G ± H ∓ = − Z 6 cβ−α − (Z 4 + Z 5 )sβ−α v, 2   λhG 0 A = − Z 6 sβ−α + Z 5 cβ−α v,   λ H G 0 A = − Z 6 cβ−α − Z 5 sβ−α v, i λ AG ± H ∓ = ∓ (Z 4 − Z 5 )v, 2

1 3 2 2 3 v, λhhh = − Z 1 sβ−α + Z 345 cβ−α sβ−α + 3Z 6 sβ−α cβ−α + Z 7 cβ−α 2 1 2 3 2 λhh H = − [3Z 1 sβ−α cβ−α + Z 345 (cβ−α − 2sβ−α cβ−α ) 2 3 2 2 − 3Z 6 (sβ−α − 2sβ−α cβ−α ) − 3Z 7 cβ−α sβ−α ]v, λh H H

λH H H

1 2 3 2 = − [3Z 1 cβ−α sβ−α + Z 345 (sβ−α − 2cβ−α sβ−α ) 2 3 2 2 ]v, + 3Z 6 (cβ−α − 2cβ−α sβ−α ) + 3Z 7 cβ−α sβ−α

1 3 2 2 3 v. = − Z 1 cβ−α + Z 345 sβ−α cβ−α − 3Z 6 cβ−α sβ−α − Z 7 sβ−α 2

(B.48) (B.49) (B.50) (B.51) (B.52)

(B.53)

(B.54) (B.55)

The Higgs quartic couplings are given by 1 λG + G − G + G − = − Z 1 , 2 1 λG + G − G 0 G 0 = − Z 1 , 2 1 λG 0 G 0 G 0 G 0 = − Z 1 , 8 1 + − + − λH H H H = − Z2, 2 1 λH + H − A A = − Z2, 2 1 λA A A A = − Z2, 8 λG + G − H + H − = −(Z 3 + Z 4 ), 1 λG + G − A A = − Z 3 , 2 1 λH + H − G0 G0 = − Z 3, 2 1 λG ± H ∓ G 0 A = − (Z 4 + Z 5 ), 2 1 λG 0 G 0 A A = − Z 345 , 4 1 λG ± H ∓ G ± H ∓ = − Z 5 , 2 λG ± G ∓ G ± H ∓ = −Z 6 , λG ± G ∓ G 0 A = −Z 6 , 1 λG 0 G 0 G ± H ∓ = − Z 6 , 2 1 λG 0 G 0 G 0 A = − Z 6 , 2

(B.56) (B.57) (B.58) (B.59) (B.60) (B.61) (B.62) (B.63) (B.64) (B.65) (B.66) (B.67) (B.68) (B.69) (B.70) (B.71)

Appendix B: Feynman Rules λ H ± H ∓ H ± G ∓ = −Z 7 , λ H ± H ∓ G 0 A = −Z 7 , 1 λ A AG ± H ∓ = − Z 7 , 2 1 λ A AG 0 A = − Z 7 , 2  1 2 2 Z 1 cβ−α λH H G+ G− = − + Z 3 sβ−α − 2Z 6 sβ−α cβ−α , 2  1 2 2 λhhG + G − = − + Z 3 cβ−α + 2Z 6 sβ−α cβ−α , Z 1 sβ−α 2   2 2 λh H G + G − = − (Z 1 − Z 3 ) sβ−α cβ−α + Z 6 cβ−α − sβ−α ,   1 2 2 Z 1 cβ−α λH H G0 G0 = − + (Z 3 + Z 4 − Z 5 )sβ−α − 2Z 6 sβ−α cβ−α , 4  1 2 2 λhhG 0 G 0 = − + (Z 3 + Z 4 − Z 5 )cβ−α + 2Z 6 sβ−α cβ−α , Z 1 sβ−α 4   1 2 2 , λh H G 0 G 0 = − (Z 1 − Z 3 − Z 4 + Z 5 ) sβ−α cβ−α + Z 6 cβ−α − sβ−α 2   1 2 2 Z 2 sβ−α λH H H + H − = − + Z 3 cβ−α − 2Z 7 sβ−α cβ−α , 2  1 2 2 λhh H + H − = − + Z 3 sβ−α + 2Z 7 sβ−α cβ−α , Z 2 cβ−α 2   2 2 , λh H H + H − = − − (Z 2 − Z 3 ) sβ−α cβ−α + Z 7 cβ−α − sβ−α

1 2 2 λ H H A A = − Z 2 sβ−α + (Z 3 + Z 4 − Z 5 ) cβ−α − 2Z 7 sβ−α cβ−α , 4

1 2 2 λhh A A = − Z 2 cβ−α + (Z 3 + Z 4 − Z 5 ) sβ−α + 2Z 7 sβ−α cβ−α , 4   1 2 2 , λh H A A = − − (Z 2 − Z 3 − Z 4 + Z 5 ) sβ−α cβ−α + Z 7 cβ−α − sβ−α 2

1 2 2 λhhG ± H ∓ = − Z 7 cβ−α + Z 6 sβ−α + (Z 4 + Z 5 ) cβ−α sβ−α , 2

1 2 2 λ H H G ± H ∓ = − Z 7 sβ−α + Z 6 cβ−α − (Z 4 + Z 5 ) cβ−α sβ−α , 2

1 2 2 λ H hG ± H ∓ = − (Z 4 + Z 5 )(cβ−α − sβ−α ) + 2 (Z 6 − Z 7 ) cβ−α sβ−α , 2  1 2 2 Z 7 sβ−α λH H G0 A = − + Z 6 cβ−α − 2Z 5 sβ−α cβ−α , 2  1 2 2 Z 7 cβ−α λhhG 0 A = − + Z 6 sβ−α + 2Z 5 sβ−α cβ−α , 2   2 2 , λ H hG 0 A = − (Z 6 − Z 7 ) cβ−α sβ−α + Z 5 cβ−α − sβ−α i λ H ± G ∓ AH = ± (Z 4 − Z 5 )cβ−α , 2 i λ H ± G ∓ Ah = ± (Z 4 − Z 5 )sβ−α , 2 i λ H ± G ∓ G 0 H = ± (Z 4 − Z 5 )sβ−α , 2 i λ H ± G ∓ G 0 h = ∓ (Z 4 − Z 5 )cβ−α , 2

213

(B.72) (B.73) (B.74) (B.75) (B.76) (B.77) (B.78) (B.79) (B.80) (B.81) (B.82) (B.83) (B.84) (B.85) (B.86) (B.87) (B.88) (B.89) (B.90) (B.91) (B.92) (B.93) (B.94) (B.95) (B.96) (B.97)

214

Appendix B: Feynman Rules 1 4 4 2 2 Z 1 sβ−α + Z 2 cβ−α + 2Z 345 cβ−α sβ−α 8  3 3 +4Z 6 cβ−α sβ−α + 4Z 7 cβ−α sβ−α ,  1 4 4 2 2 =− + Z 2 sβ−α + 2Z 345 cβ−α sβ−α Z 1 cβ−α 8  3 3 −4Z 6 cβ−α sβ−α − 4Z 7 cβ−α sβ−α ,

λhhhh = −

λH H H H

λH H H h

λ H H hh

λ H hhh

(B.98)

(B.99)

  1 3 3 2 2 = − Z 1 cβ−α sβ−α − Z 2 cβ−α sβ−α − Z 345 cβ−α sβ−α cβ−α − sβ−α 2  2 2 2 2 2 2 + Z 6 cβ−α (cβ−α − 3sβ−α ) − Z 7 sβ−α (sβ−α − 3cβ−α ), (B.100)   1 2 2 2 2 4 2 2 4 = − 3Z 1 cβ−α sβ−α + 3Z 2 cβ−α sβ−α + Z 345 cβ−α − 4cβ−α sβ−α + sβ−α 4  2 2 − sβ−α ), (B.101) + 6 (Z 6 − Z 7 ) cβ−α sβ−α (cβ−α   1 3 3 2 2 = − Z 1 cβ−α sβ−α − Z 2 cβ−α sβ−α + Z 345 cβ−α sβ−α cβ−α − sβ−α 2  2 2 2 2 2 2 (B.102) − Z 6 sβ−α (sβ−α − 3cβ−α ) + Z 7 cβ−α (cβ−α − 3sβ−α ).

Appendix C

Loop Functions

C.1 Passarino-Veltman Functions Following Ref. [4], we define A, B, C and D functions as i A0 (m 1 ) = μ4−D 16π 2



d Dk 1 , (2π) D N1

(C.1) 

d D k [1, k μ , k μ k ν ] , (C.2) (2π) D N1 N2  i d D k [1, k μ , k μ k ν ] μ μν 2 2 2 4−D [C , C , C ]( p , p , ( p + p ) ; m , m , m ) = μ , (C.3) 0 1 2 1 2 3 1 2 16π 2 (2π) D N1 N2 N3 i [D0 , D μ , D μν ]( p12 , p22 , p32 , ( p1 + p2 + p3 )2 , 16π 2 ( p 1 + p 2 )2 , ( p 2 + p 3 )2 ; m 1 , m 2 , m 3 , m 4 )  d D k [1, k μ , k μ k ν ] = μ4−D , (C.4) (2π) D N1 N2 N3 N4 i [B0 , B μ , B μν ]( p12 ; m 1 , m 2 ) = μ4−D 16π 2

where D = 4 − 2 , and μ is a dimensionful parameter introduced to keep the mass dimension four in the k-integral. The propagators are defined by N1 = k 2 − m 21 + iε, N2 = (k +

(C.5)

p1 ) − m 22 + iε, p1 + p2 )2 − m 23 +

iε,

N4 = (k + p1 + p2 + p3 ) −

m 24

N3 = (k +

2

2

(C.6) (C.7) + iε.

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 M. Aiko, Theoretical Studies on Extended Higgs Sectors Towards Future Precision Measurements, Springer Theses, https://doi.org/10.1007/978-981-99-1324-4

(C.8)

215

216

Appendix C: Loop Functions

The vector and tensor functions for B, C and D are reduced to scalar functions as μ

B μ = p1 B1 , μ B μν = p1 p1ν B21 + g μν B22 , μ

μ

μ

μ

(C.9) (C.10)

C μ = p1 C11 + p2 C12 , μ μ μ μ C μν = p1 p1ν C21 + p2 p2ν C22 + ( p1 p2ν + p1ν p2 )C23 + g μν C24 ., μ

D μ = p1 D11 + p2 D12 + p3 D13 , μ μ μ D μν = p1 p1ν D21 + p2 p2ν D22 + p3 p3ν D23 μ

μ

μ

(C.11) (C.12) (C.13)

μ

+ ( p1 p2ν + p1ν p2 )D24 + ( p1 p3ν + p1ν p3 )D25 μ μ + ( p2 p3ν + p2ν p3 )D26 + g μν D27 .

(C.14)

The UV divergent part is contained in the A, B0 , B1 , B21 , B22 and C24 functions. They are given by Div[A0 (m 1 )] = m 21 ,

(C.15)

Div[B0 ( p12 ; m 1 , m 2 )]

= , 1 Div[B1 ( p12 ; m 1 , m 2 )] = − , 2 1 Div[B21 ( p12 ; m 1 , m 2 )] = , 3 1 2 Div[B22 ( p1 ; m 1 , m 2 )] = (3m 21 + 3m 22 − p12 ), 12 1 Div[C24 ( p12 , p22 , ( p1 + p2 )2 ; m 1 , m 2 , m 3 )] = , 4

(C.16) (C.17) (C.18) (C.19) (C.20)

where the UV divergent part  is given by ≡

1 − γ E + ln 4π + ln μ2 ,

(C.21)

with γ E being the Euler constant. It is convenient to define the following functions [5] B2 ( p 2 , m 1 , m 2 ) = B21 ( p 2 , m 1 , m 2 ),

(C.22)

B3 ( p , m 1 , m 2 ) = −B1 ( p , m 1 , m 2 ) − B21 ( p , m 1 , m 2 ),

(C.23)

−m 21 B1 ( p 2 , m 2 , m 1 )

(C.24)

2

B4 ( p , m 1 , m 2 ) = 2

2

2

B5 ( p , m 1 , m 2 ) = A(m 1 ) + A(m 2 ) − 2

For numerical evaluations use LoopTools [6].

of

the

− m 22 B1 ( p 2 , m 1 , m 2 ), 4B22 ( p 2 , m 1 , m 2 ).

Passarino-Veltman

(C.25)

functions,

we

Appendix C: Loop Functions

217

C.2 Bremsstrahlung Integral For the decay width of a massive particle with p02 = m 20 into two massive particles with p12 = m 21 and p22 = m 22 and a photon with q 2 = μ2 , we need the following phase space integrals [7], 

j ,··· j

Ii11,··· ,imn (m 0 , m 1 , m 2 ) = 32π 3

d3

(±2q · p j1 ) · · · (±2q · p jm ) , (±2q · pi1 ) · · · (±2q · pin )

(C.26)

where jk , i  = 0, 1, 2 and the plus signs belong to p1 , p2 , the minus signs to p0 . We introduce the abbreviations λ = λ(m 20 , m 21 , m 22 ), −

(C.27)

− +λ , 2m 1 m 2 m 2 − m 21 + m 22 − λ1/2 β1 = 0 , 2m 0 m 2 m 2 + m 21 − m 22 − λ1/2 β2 = 0 , 2m 0 m 1

β0 =

m 20

m 21

m 22

1/2

(C.28) (C.29) (C.30)

with β0 β1 β2 = 1.

(C.31)

From Eq. (C.26), we can see that the integrals with the indices 1 and 2 interchanged are obtained by interchanging m 1 and m 2 . The IR-singular integrals are

 

λ β1 1 λ1/2 log − λ1/2 − (m 21 − m 22 ) log − m 20 log (β0 ) , (C.32) 4 μm 0 m 1 m 2 β2 4m 0 



 1 λ β0 1/2 1/2 2 2 2 λ − λ − m = log − (m − m ) log log (β ) , 1 0 2 1 μm 0 m 1 m 2 β2 4m 21 m 20

I00 = I11



I22 =

I01

I02



1 λ λ1/2 log μm 0 m 1 m 2 4m 22 m 20



− λ1/2 − (m 20 − m 21 ) log

β0 β1





(C.33)

− m 22 log (β2 ) ,

(C.34)

λ 1 log (β2 ) + 2 log2 (β2 ) − log2 (β0 ) − log2 (β1 ) 2 log = I10 = μm 0 m 1 m 2 4m 20

+ 2Sp(1 − β22 ) − Sp(1 − β02 ) − Sp(1 − β12 ) , (C.35)

1 λ = I20 = 2 log log (β1 ) + 2 log2 (β1 ) − log2 (β0 ) − log2 (β2 ) μm 0 m 1 m 2 4m 20

+ 2Sp(1 − β12 ) − Sp(1 − β02 ) − Sp(1 − β22 ) , (C.36)

I12 = I21 = −I01 − I02

218

Appendix C: Loop Functions =



1 λ 2 log log (β0 ) + 2 log2 (β0 ) − log2 (β1 ) − log2 (β2 ) μm 0 m 1 m 2 4m 20

+ 2Sp(1 − β02 ) − Sp(1 − β12 ) − Sp(1 − β22 ) .

(C.37)

For the IR finite integrals, we obtain

I0 I1 I2 I01



 λ1/2  2 m 0 + m 21 + m 22 + 2m 20 m 21 log (β2 ) + 2m 20 m 22 log (β1 ) 2

2 2 +2m 1 m 2 log (β0 ) , (C.38)

1 −2m 21 log (β2 ) − 2m 22 log (β1 ) − λ1/2 , = (C.39) 4m 20

1 −2m 20 log (β2 ) − 2m 22 log (β0 ) − λ1/2 , = (C.40) 2 4m 0

1 −2m 20 log (β1 ) − 2m 21 log (β0 ) − λ1/2 , = (C.41) 2 4m 0     λ1/2  2 1 4 2 2 2 2 2 2 m 2m log − m − = log − 2m + m − 3m + 5m m , (β ) ) (β 2 1 1 2 0 1 2 0 1 2 4 4m 20

I =

1 4m 20

(C.42)

    λ1/2  2 1 4 2 2 2 2 2 2 m m 2m log , − m − I02 = log − 2m + m − 3m + 5m (β ) ) (β 1 2 2 1 0 2 1 0 2 1 4 4m 20

(C.43)



I10 =

λ1/2

1 m 40 log (β2 ) − m 22 (2m 21 − 2m 20 + m 22 ) log (β0 ) − (m 21 − 3m 20 + 5m 22 ) , 4 4m 20

(C.44)



I20 =



λ1/2



1 m 40 log (β1 ) − m 21 (2m 22 − 2m 20 + m 21 ) log (β0 ) − (m 22 − 3m 20 + 5m 21 ) , 4 4m 20

(C.45)

I21 = −I − I20   λ1/2 2 1 4 2 2 2 2 2 2 − m − = log (2m − 2m + m ) log − 3m + 5m ) , m (m ) ) (β (β 0 1 1 0 2 1 0 2 1 0 4 4m 20

(C.46)

I12 = −I − I10   λ1/2 2 1 m 42 log (β0 ) − m 20 (2m 21 − 2m 22 + m 20 ) log (β2 ) − = (m 1 − 3m 22 + 5m 20 ) , 2 4 4m 0 

12 21 I00 = I00 =−

λ1/2

   3m 21 + 3m 22 − m 20 ,

λ3/2

λ1/2



1 m 41 log (β2 ) + m 42 log (β1 ) + + 4 4m 20 6m 20 

02 20 I11 = I11 =−

(C.47)

λ3/2

1 m 40 log (β2 ) + m 42 log (β0 ) + + 4 4m 20 6m 21

3m 20 + 3m 22 − m 21

 

(C.48) ,

(C.49)

Appendix C: Loop Functions 01 10 I22 = I22 =−

   λ3/2 1 λ1/2  2 4 4 2 2 3m , m + m + log log + + 3m − m ) ) (β (β 1 0 0 1 0 1 2 4 4m 20 6m 22 

00 02 I11 = −I10 − I11 =

00 I22

=

−I20

−

219

01 I22

=

11 12 I00 = −I01 − I00 =

22 21 I00 = −I02 − I00 =

22 02 I11 = −I12 − I11 =

11 01 I22 = −I21 − I22 =





1 2m 22 m 21 + m 22 − m 20 log (β0 ) + 4m 20    1 2m 21 m 22 + m 21 − m 20 log (β0 ) + 4m 20    1 2m 22 m 20 + m 22 − m 21 log (β1 ) + 2 4m 0    1 2m 21 m 20 + m 21 − m 22 log (β2 ) + 2 4m 0    1 2m 20 m 21 + m 20 − m 22 log (β2 ) + 2 4m 0    1 2m 20 m 22 + m 20 − m 21 log (β1 ) + 2 4m 0

λ3/2 6m 21

 + 2λ1/2 m 22 ,

 λ3/2 1/2 2 + 2λ m 1 , 6m 22  λ3/2 1/2 2 + 2λ m 2 , 6m 20  λ3/2 1/2 2 + 2λ m 1 , 6m 20  λ3/2 1/2 2 + 2λ m 0 , 6m 21  λ3/2 1/2 2 + 2λ m 0 . 6m 22

(C.50) (C.51) (C.52) (C.53) (C.54) (C.55) (C.56)

When m 21 = 0, collinear singularity would appear. However, if that massless particle doesn’t have an electromagnetic charge, relevant integrals only have an index of 0 or 2, and these are collinear finite. This is the case of H + → ν¯  decay. We have λ(m 20 , m 21 , m 22 ) = m 20 − m 22 −

m 20 + m 22 2 m + · · · → m 20 − m 22 , m 20 − m 22 1

m 20 − m 22 → ∞, m2m1 m2 β1 = , m0 m2m1 β2 = 2 → 0. m 0 − m 22 β0 =

(C.57) (C.58) (C.59) (C.60)

The IR-singular integrals are  1/2

2  1 λ m2 1/2 1/2 2 λ − λ , ln + m ln 2 μm 0 4m 40 m 20  1/2

2  1 λ m2 1/2 1/2 2 2 λ ln − λ + (m 0 + m 2 ) ln , = 2 2 μm 4m 0 m 2 m 20 0  1/2 2





 λ m2 m 22 1 2 m2 ln ln + ln + 2Sp 1 − . = μm 0 m0 4m 20 m 20 m 20

I00 =

(C.61)

I22

(C.62)

I02

(C.63)

220

Appendix C: Loop Functions

For the IR finite integrals, we obtain 2  λ1/2 2 m2 2 2 2 , (m 0 + m 2 ) + m 0 m 2 ln 2 m 20   2

1 m2 1/2 I2 = − 2 m 20 ln + λ , 4m 0 m 20

  m2 1 λ1/2 2 4 2 I20 = ln − 3m ) . (m m − 0 2 0 m0 4 4m 20 1 I = 4m 20



(C.64) (C.65) (C.66)

Appendix D

One Particle Irreducible Diagrams in the Two-Higgs Doublet Model

In this section, we give the analytic expressions for the 1PI diagram contributions to one, two, and three-point functions. We also list the pinch terms for the scalar mixing contributions and box diagram contributions for e+ e− → h Z . We calculate 1PI diagrams in the t’ Hooft-Feynman gauge in which the masses of NambuGoldstone bosons m G ± and m G 0 and those of Fadeev-Popov ghosts m c± and m c Z are the same as corresponding masses of the gauge bosons; i.e., m G ± = m c± = m W and m G 0 = m c Z = m Z . 1PI diagrams with bosonic external lines are separately calculated by the fermion-loop and boson-loop contributions. We denote the fermionicand bosonic-loop contributions by the subscript of F and B, respectively.

D.1 One-Point Functions The 1PI tadpole diagrams for h and H are given by [8] 1PI 16π 2 h,F =−



Ncf

f

16π 2  1PI H,F = −

 f

16π

2

1PI h,B

16π 2  1PI H,B



Ncf

4m 2f v 4m 2f v

f

(D.1)

f

(D.2)

ζh A( f ), ζ H A( f ),

 3 3 3 = sβ−α 3gm W A(W ) + g Z m Z A(Z ) − 2gm W − g Z m Z 2 − λ H + H − h A(H ± ) − λ A Ah A(A) − λ H H h A(H ) − 3λhhh A(h) − λG + G − h A(G ± ) − λG 0 G 0 h A(G 0 ), (D.3)   3 = cβ−α 3gm W A(W ) + g Z m Z A(Z ) − 2gm 3W − g Z m 3Z 2 − λ H + H − H A(H ± ) − λ A AH A(A) − 3λ H H H A(H ) − λ H hh A(h)

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 M. Aiko, Theoretical Studies on Extended Higgs Sectors Towards Future Precision Measurements, Springer Theses, https://doi.org/10.1007/978-981-99-1324-4

221

222

Appendix D: One Particle Irreducible Diagrams in the Two-Higgs Doublet Model

− λG + G − H A(G ± ) − λG 0 G 0 H A(G 0 ).

(D.4)

D.2 Two-Point Functions D.2.1 Scalar Boson The fermionic contributions to the scalar boson two-point functions are given by [8, 9] 16π

2

2 1PI hh ( p ) F

=−



Ncf

f

2 16π 2 1PI H H ( p )F = −



Ncf

f

2 16π 2 1PI H h ( p )F = −



Ncf

f





4m 2f

f (ζ )2 v2 h

4m 2f v2 4m 2f v2 4m 2f

 f

(ζ H )2  f

f

ζh ζ H



 p2 2 2 A( f ) + 2m f − B0 ( p ; f, f ) , 2 (D.5)

 p2 2 2 A( f ) + 2m f − B0 ( p ; f, f ) , 2 (D.6)

 p2 2 2 A( f ) + 2m f − B0 ( p ; f, f ) , 2





p2 B0 ( p 2 ; f, f ) , 2 v 2 f    4m 2f p2 2 f 2 16π 2 1PI A( f ) − B ( p ) = − N ζ ( p ; f, f ) , F f 0 c AG 0 v2 2 f  2 2  m u ζu + m 2d ζd2 2 1PI 2 2Nc 16π  H + H − ( p ) F = − v2 u,d   A(u) + A(d) + (m 2u + m 2d − p 2 )B0 ( p 2 ; u, d)  m u ζu m d ζd 2 B0 ( p ; u, d) − 4m u m d v2  m2ζ 2 − 2 2  [A() + (m 2 − p 2 )B0 ( p 2 ; 0, )], v   2  m u ζu + m 2d ζd 2 ( p ) = − 2N 16π 2 1PI + − c H G F v2 u,d 2 16π 2 1PI A A( p )F = −

Ncf

ζ 2f A( f ) −

[A(u) + A(d) + (m 2u + m 2d − p 2 )B0 ( p 2 ; u, d)]  m u m d (ζu + ζd ) 2 B ( p ; u, d) − 2m u m d 0 v2

(D.7) (D.8)

(D.9)

(D.10)

Appendix D: One Particle Irreducible Diagrams in the Two-Higgs Doublet Model

−

 m2ζ 2 2 2  [A() + (m 2 − p 2 )B0 ( p 2 ; 0, )]. v 

223

(D.11)

The bosonic contributions to the scalar boson two-point functions are given by [8, 9] 2 16π 2 1PI hh ( p ) B

 g2  2 4 − sβ−α A(W ) 2  g2  g2 2 2 A(Z ) (3m 2Z − p 2 )B0 ( p 2 ; Z , Z ) + Z 4 − sβ−α + Z sβ−α 2 4  g2 2  2 A(W ) − A(H ± ) + (2m 2H ± − m 2W + 2 p 2 )B0 ( p 2 ; W, H ± ) − cβ−α 2  g2 2  2 A(Z ) − A(A) + (2m 2A − m 2Z + 2 p 2 )B0 ( p 2 ; Z , A) − Z cβ−α

4 1 2 (2g 2 m 2W + g 2Z m 2Z ) + − sβ−α 2 − 2λ H + H − hh A(H ± ) − 2λ A Ahh A(A) − 2λ H H hh A(H ) − 12λhhhh A(h)

2 = g 2 sβ−α (3m 2W − p 2 )B0 ( p 2 ; W, W ) +

− 2λG + G − hh A(G ± ) − 2λG 0 G 0 hh A(G 0 ) + λ2H + H − h B0 ( p 2 ; H ± , H ± ) + λ2G + G − h B0 ( p 2 ; G ± , G ± ) + 2λ2H + G − h B0 ( p 2 ; H ± , G ± ) + 2λ2A Ah B0 ( p 2 ; A, A) + 2λ2G 0 G 0 h B0 ( p 2 ; G 0 , G 0 ) + λ2AG 0 h B0 ( p 2 ; A, G 0 ) + 2λ2H H h B0 ( p 2 ; H, H ) + 18λ2hhh B0 ( p 2 ; h, h) + 4λ2H hh B0 ( p 2 ; h, H ), (D.12) 2 16π 2 1PI H H ( p )B

g2 2 [4 − cβ−α ]A(W ) 2  g2  g2 2 2 A(Z ) (3m 2Z − p 2 )B0 ( p 2 ; Z , Z ) + Z 4 − cβ−α + Z cβ−α 2 4  g2 2  2 A(W ) − A(H ± ) + (2m 2H ± − m 2W + 2 p 2 )B0 ( p 2 ; W, H ± ) − sβ−α 2  g 2Z 2  sβ−α 2 A(Z ) − A(A) + (2m 2A − m 2Z + 2 p 2 )B0 ( p 2 ; Z , A) −  4 1 2 (2g 2 m 2W + g 2Z m 2Z ) + − cβ−α 2 − 2λ H + H − H H A(H ± ) − 2λ A AH H A(A) − 12λ H H H H A(H ) − 2λ H H hh A(h)

2 = g 2 cβ−α (3m 2W − p 2 )B0 ( p 2 ; W, W ) +

− 2λG + G − H H A(G ± ) − 2λG 0 G 0 H H A(G 0 ) + λ2H + H − H B0 ( p 2 ; H ± , H ± ) + λ2G + G − H B0 ( p 2 ; G ± , G ± ) + 2λ2H + G − H B0 ( p 2 ; H ± , G ± ) + 2λ2A AH B0 ( p 2 ; A, A) + 2λ2G 0 G 0 H B0 ( p 2 ; G 0 , G 0 ) + λ2AG 0 H B0 ( p 2 ; A, G 0 )

224

Appendix D: One Particle Irreducible Diagrams in the Two-Higgs Doublet Model

+ 18λ2H H H B0 ( p 2 ; H, H ) + 2λ2H hh B0 ( p 2 ; h, h) + 4λ2H H h B0 ( p 2 ; h, H ), 16π

2

(D.13)

2 1PI H h ( p )B

 g2 = sβ−α cβ−α g 2 (3m 2W − p 2 )B0 ( p 2 ; W, W ) − A(W ) 2 2 g 2Z g (3m 2Z − p 2 )B0 ( p 2 ; Z , Z ) − Z A(Z ) + 2 4 g2 + [2 A(W ) − A(H ± ) + (2m 2H ± − m 2W + 2 p 2 )B0 ( p 2 ; W, H ± )] 2 g 2Z [2 A(Z ) − A(A) + (2m 2A − m 2Z + 2 p 2 )B0 ( p 2 ; Z , A)] + 4  − (2g 2 m 2W + g 2Z m 2Z )

− λ H + H − H h A(H ± ) − λ A AH h A(A) − 3λ H H H h A(H ) − 3λ H hhh A(h) − λG + G − H h A(G ± ) − λG 0 G 0 H h A(G 0 ) + λ H + H − h λ H + H − H B0 ( p 2 ; H ± , H ± ) + λG + G − h λG + G − H B0 ( p 2 ; G ± , G ± ) + 2λ H + G − h λ H + G − H B0 ( p 2 ; H ± , G ± ) + 2λ A Ah λ A AH B0 ( p 2 ; A, A) + 2λhG 0 G 0 λG 0 G 0 H B0 ( p 2 ; G 0 , G 0 ) + λ AG 0 h λ AG 0 H B0 ( p 2 ; A, G 0 ) + 6λ H H h λ H H H B0 ( p 2 ; H, H ) + 6λhhh λ H hh B0 ( p 2 ; h, h) + 4λ H hh λ H H h B0 ( p 2 ; H, h), 16π

2

(D.14)

2 1PI A A( p )B

1 = 2g 2 A(W ) + g 2Z A(Z ) − (2g 2 m 2W + g 2Z m 2Z ) 2  g2  ± 2 A(W ) − A(H ) + (2m 2H ± − m 2W + 2 p 2 )B0 ( p 2 ; W, H ± ) − 2  g 2Z 2  cβ−α 2 A(Z ) − A(h) + (2m 2h − m 2Z + 2 p 2 )B0 ( p 2 ; Z , h) − 4  g 2Z 2  s 2 A(Z ) − A(H ) + (2m 2H − m 2Z + 2 p 2 )B0 ( p 2 ; Z , H ) − 4 β−α − 2λ H + H − A A A(H ± ) − 12λ A A A A A(A) − 2λ A AH H A(H ) − 2λ A Ahh A(h) − 2λG + G − A A A(G ± ) − 2λ A AG 0 G 0 A(G 0 ) + 2|λ H + G − A |2 B0 ( p 2 ; H ± , G ± ) + 4λ2A Ah B0 ( p 2 ; A, h) + 4λ2A AH B0 ( p 2 ; A, H ) + λ2AG 0 h B0 ( p 2 ; h, G 0 ) + λ2AG 0 H B0 ( p 2 ; H, G 0 ), 2 16π 1PI AG 0 ( p ) B  g2 = sβ−α cβ−α Z 2

4

−

  2 A(Z ) − A(H ) + (2m 2H − m 2Z + 2 p 2 )B0 ( p 2 ; Z , H )

 g 2Z [2 A(Z ) − A(h) + (2m 2h − m 2Z + 2 p 2 )B0 ( p 2 ; Z , h)] 4

(D.15)

Appendix D: One Particle Irreducible Diagrams in the Two-Higgs Doublet Model

225

− λ H + H − AG 0 A(H ± ) − 3λ A A AG 0 A(A) − λ AG 0 H H A(H ) − λ AG 0 hh A(h) − λG + G − AG 0 A(G ± ) − 3λ AG 0 G 0 G 0 A(G 0 ) + 2λ A Ah λ AG 0 h B0 ( p 2 ; A, h) + 2λ A AH λ AG 0 H B0 ( p 2 ; A, H ) + 2λ AG 0 h λG 0 G 0 h B0 ( p 2 ; G 0 , h) + 2λ AG 0 H λG 0 G 0 H B0 ( p 2 ; G 0 , H ),

(D.16)

2 16π 1PI H + H − ( p )B g2 2 = − cβ−α [2 A(W ) 2

− A(h) + (2 p 2 + 2m 2h − m 2W )B0 ( p 2 ; W, h)] 4 g2 2 [2 A(W ) − A(H ) + (2 p 2 + 2m 2H − m 2W )B0 ( p 2 ; W, H )] − sβ−α 4 g2 − [2 A(W ) − A(A) + (2 p 2 + 2m 2A − m 2W )B0 ( p 2 ; W, A)] 4 g 2Z 2 c [2 A(Z ) − A(H ± ) + (2 p 2 + 2m 2H ± − m 2Z )B0 ( p 2 ; Z , H ± )] − 4 2W − e2 [2 A(γ) − A(H ± ) + (2 p 2 + 2m 2H ± − m 2γ )B0 ( p 2 ; γ, H ± )]       1 1 1 2 A(Z ) − m 2Z + 4e2 A(γ) − m 2γ + 2g 2 A(W ) − m 2W + g 2Z c2W 2 2 2 + λ2H + H − h B0 ( p 2 ; H ± , h) + λ2H + G − h B0 ( p 2 ; G ± , h) + λ2H + H − H B0 ( p 2 ; H ± , H ) + λ2H + G − H B0 ( p 2 ; G ± , H )

+ λ H + G − A λ H − G + A B0 ( p 2 ; G ± , A) − λ H + H − hh A(h) − λ H + H − G 0 G 0 A(G 0 ) − λ H + H − G + G − A(G ± ) − λ H + H − H H A(H ) − λ H + H − A A A(A) − 4λ H + H − H + H − A(H ± ), 16π

2

(D.17)

2 1PI H + G− ( p )B 2

g sβ−α cβ−α [2 A(W ) − A(h) + (2 p 2 + 2m 2h − m 2W )B0 ]( p 2 ; W, h) 4 g2 + sβ−α cβ−α [2 A(W ) − A(H ) + (2 p 2 + 2m 2H − m 2W )B0 ]( p 2 ; W, H ) 4 + λ H − G + h λ H + H − h B0 ( p 2 ; H ± , h) + λG + G − h λ H − G + h B0 ( p 2 ; G ± , h)

=−

+ λG + H − H λ H + H − H B0 ( p 2 ; H ± , H ) + λG + G − H λ H − G + H B0 ( p 2 ; G ± , H ) − λG + H − hh A(h) − λG + H − G 0 G 0 A(G 0 ) − 2λG + H − G + G − A(G ± ) − λG + H − H H A(H ) − λG + H − A A A(A) − 2λG + H − H + H − A(H ± ).

(D.18)

The pinch terms for the two-point functions of the scalar bosons are given by [10] 2 2 2 2 2 2 2 2 ± 16π 2 PT hh ( p ) = −g ( p − m h )[sβ−α B0 ( p ; W, W ) + cβ−α B0 ( p ; H , W )]

g 2Z 2 2 2 B0 ( p 2 ; Z , Z ) + cβ−α B0 ( p 2 ; A, Z )], ( p − m 2h )[sβ−α 2 2 2 2 2 2 2 2 2 ± 16π 2 PT H H ( p ) = −g ( p − m H )[cβ−α B0 ( p ; W, W ) + sβ−α B0 ( p ; H , W )] −

(D.19)

226

Appendix D: One Particle Irreducible Diagrams in the Two-Higgs Doublet Model −

g 2Z 2 2 2 ( p − m 2H )[cβ−α B0 ( p 2 ; Z , Z ) + sβ−α B0 ( p 2 ; A, Z )], 2

(D.20)

g2 (2 p 2 − m 2h − m 2H )sβ−α cβ−α [B0 ( p 2 ; H ± , W ) − B0 ( p 2 ; W, W )] 2 g2 + Z (2 p 2 − m 2h − m 2H )sβ−α cβ−α [B0 ( p 2 ; A, Z ) − B0 ( p 2 ; Z , Z )], 4

2 16π 2 PT Hh( p ) =

(D.21)

16π

2

2 PT AA( p )

= −g ( p 2

−

g 2Z

2

− m 2A )B0 ( p 2 ; W,

±

H )

2 2 B0 ( p 2 ; Z , h) + sβ−α B0 ( p 2 ; Z , H )], ( p 2 − m 2A )[cβ−α

2

g 2Z (2 p 2 − m 2A )sβ−α cβ−α [B0 ( p 2 ; Z , H ) − B0 ( p 2 ; Z , h)]. 4

 g2  2 2 2 2 16π 2 PT B0 ( p 2 ; W, H ) + cβ−α B0 ( p 2 ; W, h) p − m 2H ± sβ−α H+ H− ( p ) = − 2  g2  2 p − m 2H ± B0 ( p 2 ; W, A) − 2   g 2Z 2 2 − ) p 2 − m 2H ± B0 ( p 2 ; Z , H ± ) (1 − 2sW 2  e2  − 2 p 2 − m 2H ± B0 ( p 2 ; γ, H ± ), 8π  

g2 2 PT 2 16π  H + G − ( p ) = sβ−α cβ−α 2 p 2 − m 2H ± B0 ( p 2 ; W, H ) − B0 ( p 2 ; W, h) . 2 16π 2 PT ( p2 ) = AG 0

(D.22) (D.23)

(D.24) (D.25)

D.2.2 Gauge Boson The 1PI diagram contributions to the gauge boson two-point functions are given as [8] 2 16π 2 1PI W W ( p )F =



  f Nc g 2 2 p 2 B3 − B4 ( p 2 ; f, f  ),

(D.26)

f, f 

2 16π 2 1PI γγ ( p ) F =

 f

2 16π 2 1PI Z γ ( p )F =

 f

2 16π 2 1PI Z Z ( p )F =



f

8Nc e2 Q 2f p 2 B3 ( p 2 ; f, f ),

(D.27)



f 2 Nc eg Z 2 p 2 (2I f Q f − 4sW Q 2f )B3 ( p 2 ; f, f ),

(D.28)



f 4 2 Nc g 2Z 2 p 2 (4sW Q 2f − 4sW Q f I f + 2I 2f )B3 − 2I 2f m 2f B0 ( p 2 ; f, f ),

f



(D.29)

1 1 2 1 2 B5 ( p 2 ; H, H ± ) + cβ−α B5 ( p 2 ; h, H ± ) B5 ( p 2 ; A, H ± ) + sβ−α 4 4 4



1 1 2 2 m 2W B0 + B5 ( p 2 ; h, W ) + cβ−α m 2W B0 + B5 ( p 2 ; H, W ) + sβ−α 4 4

2 2 16π 2 1PI W W ( p )B = g

Appendix D: One Particle Irreducible Diagrams in the Two-Higgs Doublet Model

227

 1 2 2 2 B5 + (m 2W − 4sW m 2W + m 2Z − 8 p 2 cW )B0 ( p 2 ; Z , W ) + 2cW 4 

2 2 B5 + (2m 2W − 4 p 2 )B0 ( p 2 ; 0, W ) − p 2 , + 2sW (D.30) 3   2 2 2 2 ± ± 2 2 16π 2 1PI (D.31) 12B3 + 5B0 ( p 2 ; W, W ) + , γγ ( p ) B = e B5 ( p ; H , H ) − e p 3

eg Z 11 2 2 16π 2 1PI B5 ( p 2 ; H ± , H ± ) − eg Z p 2 10B3 + B0 + ( p 2 ; W, W ) Z γ ( p )B = 2 2 3 sW 1PI 2 −  ( p )B , (D.32) cW γγ  1 2 1 2 2 2 1 16π 2 1PI B5 ( p 2 ; H, A) + cβ−α B5 ( p 2 ; h, A)] B5 ( p 2 ; H ± , H ± ) + sβ−α Z Z ( p )B = gZ 4 4 4



1 1 2 2 m 2Z B0 + B5 ( p 2 ; h, Z ) + cβ−α m 2Z B0 + B5 ( p 2 ; H, Z ) + sβ−α 4 4 

23 2 + (2m 2W − p 2 )B0 − 9 p 2 B3 ( p 2 ; W, W ) − p 2 4 3 

+

−

s 2 1PI 2 2sW 1PI 2 Z γ ( p )B − W γγ ( p ) B . 2 cW cW

(D.33)

D.2.3 Fermion The 1PI diagram contributions to the fermion two-point functions are given by [8] 2 2 2 2 2 2 2 16π 2 1PI f f,V ( p ) = −e Q f (2B1 + 1)( p ; f, γ) − g Z (v f + a f )(2B1 + 1)

( p 2 ; f, Z ) g2 (2B1 + 1)( p 2 ; f  , W ) 4

m 2f f f − 2 (ζh )2 B1 ( p 2 ; f, h) + (ζ H )2 B1 ( p 2 ; f, H ) v

m 2f − 2 B1 ( p 2 ; f, G 0 ) + ζ 2f B1 ( p 2 ; f, A) v m 2f + m 2f  m 2f ζ 2f + m 2f  ζ 2f  2  ± − B ( p ; f , G ) − B1 ( p 2 ; f  , H ± ), 1 v2 v2 g2 2 2 2 (2B1 + 1)( p 2 ; f  , W ) 16π 2 1PI f f,A ( p ) = −2g Z v f a f (2B1 + 1)( p ; f, Z ) − 4 m 2f − m 2f  m 2f ζ 2f − m 2f  ζ 2f  2  ± + B ( p ; f , G ) + B1 ( p 2 ; f  , H ± ), 1 v2 v2 (D.34) −

2 2 2 2 2 2 2 16π 2 1PI f f,S ( p ) = −2e Q f (2B0 − 1)( p ; f, γ) − 2g Z (v f − a f )

228

Appendix D: One Particle Irreducible Diagrams in the Two-Higgs Doublet Model

(2B0 − 1)( p 2 ; f, Z )

m 2f f f + 2 (ζh )2 B0 ( p 2 ; f, h) + (ζ H )2 B0 ( p 2 ; f, H ) v

m 2f − 2 B0 ( p 2 ; f, G 0 ) + ζ 2f B0 ( p 2 ; f, A) v m 2f    − 2 2 B0 ( p 2 ; f  , G ± ) + ζ f ζ f  B0 ( p 2 ; f  , H ± ) . v

(D.35)

D.3 Three-Point Functions D.3.1 hhh Vertex The 1PI diagrams for the hhh coupling are given by [8] 1PI 16π 2 hhh ( p12 , p22 , q 2 ) F

=−

 f

f

Nc

8m 4f v3

f (ζh )3 B0 ( p12 , f, f ) + B0 ( p22 , f, f ) + B0 (q 2 , f, f )

+ (4m 2f − q 2 + p1 · p2 )C0 ( f, f, f ) , 1PI 16π 2 hhh ( p12 , p22 , q 2 ) B

g3 3 3 = m W sβ−α 16C0 (W, W, W ) − C0 (c± , c± , c± ) 2

g3 2 SV V 2 SV V C hhh (G ± , W, W ) + cβ−α C hhh (H ± , W, W ) − m W sβ−α sβ−α 2

g 3Z 3 3 m Z sβ−α 16C0 (Z , Z , Z ) − C0 (c Z , c Z , c Z ) + 4

g 3Z m Z 2 SV V 2 SV V − C hhh (G 0 , Z , Z )cβ−α C hhh (A, Z , Z ) sβ−α sβ−α 4 g2 g2 2 V SS 2 V SS C hhh (W, G ± , G ± ) + λ H + H − h cβ−α C hhh (W, H ± , H ± ) + λG + G − h sβ−α 2 2 g2 V SS V SS + λ H + G − h sβ−α cβ−α [C hhh (W, G ± , H ± ) + C hhh (W, H ± , G ± )] 2 g2 g2 2 V SS 2 V SS + Z λG 0 G 0 h sβ−α C hhh (Z , G 0 , G 0 ) + Z λ A Ah cβ−α C hhh (Z , A, A) 2 2 g2 V SS V SS (Z , A, G 0 ) + C hhh (Z , G 0 , A)] + Z λ AG 0 h sβ−α cβ−α [C hhh 4 + 2g 3 m W sβ−α [B0 ( p12 , W, W ) + B0 ( p22 , W, W ) + B0 (q 2 , W, W )] − 3g 3 m W sβ−α

3 3 g m Z sβ−α 2 Z + 2λ H + H − h λ H + H − hh [B0 ( p12 , H ± , H ± ) + B0 ( p22 , H ± , H ± ) + B0 (q 2 , H ± , H ± )] + g 3Z m Z sβ−α [B0 ( p12 , Z , Z ) + B0 ( p22 , Z , Z ) + B0 (q 2 , Z , Z )] −

+ 2λhG + G − λhhG + G − [B0 ( p12 , G ± , G ± ) + B0 ( p22 , G ± , G ± ) + B0 (q 2 , G ± , G ± )]

(D.36)

Appendix D: One Particle Irreducible Diagrams in the Two-Higgs Doublet Model

229

+ 4λ H + G − h λ H + G − hh [B0 ( p12 , H ± , G ± ) + B0 ( p22 , H ± , G ± ) + B0 (q 2 , H ± , G ± )] + 4λ A Ah λ A Ahh [B0 ( p12 , A, A) + B0 ( p22 , A, A) + B0 (q 2 , A, A)] + 4λG 0 G 0 h λG 0 G 0 hh [B0 ( p12 , G 0 , G 0 ) + B0 ( p22 , G 0 , G 0 ) + B0 (q 2 , G 0 , G 0 )] + 2λ AG 0 h λ AG 0 hh [B0 ( p12 , A, G 0 ) + B0 ( p22 , A, G 0 ) + B0 (q 2 , A, G 0 )] + 4λ H H h λ H H hh [B0 ( p12 , H, H ) + B0 ( p22 , H, H ) + B0 (q 2 , H, H )] + 12λ H hh λ H hhh [B0 ( p12 , h, H ) + B0 ( p22 , h, H ) + B0 (q 2 , h, H )] + 72λhhh λhhhh [B0 ( p12 , h, h) + B0 ( p22 , h, h) + B0 (q 2 , h, h)] − 2λ3H + H − h C0 (H ± , H ± , H ± ) − 2λ3G + G − h C0 (G ± , G ± , G ± ) − 8λ3G 0 G 0 h C0 (G 0 , G 0 , G 0 ) − 8λ3A Ah C0 (A, A, A) − 8λ3H H h C0 (H, H, H ) − 216λ3hhh C0 (h, h, h) − 2λ H + H − h λ2H + G − h [C0 (G ± , H ± , H ± ) + C0 (H ± , G ± , H ± ) + C0 (H ± , H ± , G ± )] − 2λG + G − h λ2H + G − h [C0 (H ± , G ± , G ± ) + C0 (G ± , H ± , W ) + C0 (G ± , G ± , H ± )] − 2λ A Ah λ2AG 0 h [C0 (G 0 , A, A) + C0 (A, G 0 , A) + C0 (A, A, G 0 )] − 2λG 0 G 0 h λ2AG 0 h [C0 (A, G 0 , G 0 ) + C0 (G 0 , A, G 0 ) + C0 (G 0 , G 0 , A)] − 8λ H H h λ2H hh [C0 (h, H, H ) + C0 (H, H, h) + C0 (H, h, H )] − 24λhhh λ2H hh [C0 (h, h, H ) + C0 (H, h, h) + C0 (h, H, h)],

(D.37)

where SV V (X, Y, Z ) C hhh

1 ≡ p12 C21 + p22 C22 + 2 p1 p2 C23 + 4C24 − − (q + p1 )( p1 C11 + p2 C12 ) + qp1 C0 (X, Y, Z ) 2 1 + p12 C21 + p22 C22 + 2 p1 p2 C23 + 4C24 − + (3 p1 − p2 )( p1 C11 + p2 C12 ) 2

+ 2 p1 ( p1 − p2 )C0 (Z , X, Y ) 1 + p12 C21 + p22 C22 + 2 p1 p2 C23 + 4C24 − + (3 p1 + 4 p2 )( p1 C11 + p2 C12 ) 2

(D.38) + 2q(q + p2 )C0 (Y, Z , X ), V SS (X, Y, Z ) C hhh 1 ≡ p12 C21 + p22 C22 + 2 p1 p2 C23 + 4C24 − + (4 p1 + 2 p2 )( p1 C11 + p2 C12 ) 2

+ 4 p1 · qC0 (X, Y, Z ) 1 + p12 C21 + p22 C22 + 2 p1 p2 C23 + 4C24 − + 2 p2 ( p1 C11 + p2 C12 ) 2

− p1 ( p1 + 2 p2 )C0 (Z , X, Y ) 1 + p12 C21 + p22 C22 + 2 p1 p2 C23 + 4C24 − − 2 p2 ( p1 C11 + p2 C12 ) 2

− q( p1 − p2 )C0 (Y, Z , X ).

(D.39)

230

Appendix D: One Particle Irreducible Diagrams in the Two-Higgs Doublet Model

D.3.2 h f f¯ Vertex The h f f¯ vertex can be decomposed into the eight form factors as shown in Eq .(6.1). The assignment for external momentum is given in the left panel of Fig. 6.1. The 1PI contributions are given by  m −1 f

16π 2  S,1PI h f f¯ v 4 2 2 = −2g Z v (v f − a 2f )sβ−α C0 (Z , f, Z )  f − 4ζh e2 Q 2f [m 2f C0 + p12 (C11 + C21 ) + p22 (C12 + C22 ) + p1 · p2 (2C23 − C0 ) + 4C24 − 1]( f, γ, f ) + g 2Z (v 2f − a 2f )[m 2f C0 + p12 (C11 + C21 ) + p22 (C12 + C22 )  + p1 · p2 (2C23 − C0 ) + 4C24 − 1]( f, Z , f )

m 2f f f (ζh )2 C hFfSfF ( f, h, f ) + (ζ H )2 C hFfSfF ( f, H, f ) − C hFfSfF ( f, G 0 , f ) − ζ 2f C hFfSfF ( f, A, f ) 2 v

2m 2f   f − ζh C hFfSfF ( f  , G ± , f  ) + ζ f ζ f  C hFfSfF ( f  , H ± , f  ) 2 v m 2f  f f − 6(ζh )2 λhhh C0 (h, f, h) + 2(ζ H )2 λ H H h C0 (H, f, H ) v f

+ ζh

f

f

+ 2ζh ζ H λ H hh [C0 (h, f, H ) + C0 (H, f, h)] − 2λG 0 G 0 h C0 (G 0 , f, G 0 )  − 2ζ 2f λ A Ah C0 (A, f, A) − ζ f λ AG 0 h [C0 (A, f, G 0 ) + C0 (G 0 , f, A)] 2m 2f  

λG + G − h C0 (G ± , f  , G ± ) + ζ f ζ f  λ H + H − h C0 (H ± , f  , H ± )  1 + λ H + G − h (ζ f + ζ f  )[C0 (G ± , f  , H ± ) + C0 (H ± , f  , G ± )] 2

g2 s − C V F S (W, f  , G ± ) + C hS fFfV (G ± , f  , W ) 4 β−α h f f

g2 ζ f cβ−α C hVfFf S (W, f  , H ± ) + C hS fFfV (H ± , f  , W ) − 4

g 2Z sβ−α C hVfFf S (Z , f, G 0 ) + C hS fFfV (G 0 , f, Z ) − 8

g 2Z ζ f cβ−α C hVfFf S (Z , f, A) + C hS fFfV (A, f, Z ) , − 8  m −1 f 16π 2  P,1PI h f f¯ v 2 m f (ζ f  − ζ f )[C0 (G ± , f  , H ± ) − C0 (H ± , f  , G ± )] = λH + G−h v

g2 C V F S (W, f  , G ± ) − C hS fFfV (G ± , f  , W ) s − 4 β−α h f f

g2 ζ f cβ−α C hVfFf S (W, f  , H ± ) − C hS fFfV (H ± , f  , W ) − 4

− g 2Z v f I f sβ−α C hVfFf S (Z , f, G 0 ) − C hS fFfV (G 0 , f, Z )

− g 2Z v f I f ζ f cβ−α C hVfFf S (Z , f, A) − C hS fFfV (A, f, Z ) , +

v

(D.40)

(D.41)

Appendix D: One Particle Irreducible Diagrams in the Two-Higgs Doublet Model

231

V1 ,1PI h f f¯

2m 2f f ζh g 2Z (v 2f + a 2f )(C0 + 2C11 )( f, Z , f ) + e2 Q 2f (C0 + 2C11 )( f, γ, f ) = v

16π 2 

+ g2

m 2f 

f ζ (C0 + 2C11 )( f  , W, f  ) 2v h

g4 − sβ−α g 4Z v(v 2f + a 2f )(C0 + C11 )(Z , f, Z ) − sβ−α v(C0 + C11 )(W, f  , W ) 4 4 m f f f f + ζh 3 (ζh )2 (C0 + 2C11 )( f, h, f ) + (ζ H )2 (C0 + 2C11 )( f, H, f ) v

+ (C0 + 2C11 )( f, G 0 , f ) + ζ 2f (C0 + 2C11 )( f, A, f ) + −

m 2f 

f

ζ v3 h

m 2f  v2



(m 2f + m 2f  )(C0 + 2C11 )( f  , G ± , f  ) + (m 2f ζ 2f + m 2f  ζ 2f  )(C0 + 2C11 )( f  , H ± , f  )

f f 6(ζh )2 λhhh (C0 + C11 )(h, f, h) + 2(ζ H )2 λ H H h (C0 + C11 )(H, f, H ) f

f

+ 2ζh ζ H λ H hh [(C0 + C11 )(H, f, h) + (C0 + C11 )(h, f, H )] + 2λG 0 G 0 h (C0 + C11 )(G 0 , f, G 0 ) + 2ζ 2f λ A Ah (C0 + C11 )(A, f, A)  + ζ f λ AG 0 h [(C0 + C11 )(A, f, G 0 ) + (C0 + C11 )(G 0 , f, A)] λ + − − G 2G h (m 2f + m 2f  )(C0 + C11 )(G ± , f  , G ± ) v λH + H −h 2 2 − (m f ζ f + m 2f  ζ 2f  )(C0 + C11 )(H ± , f  , H ± ) v2 λ + − − H 2G h (m 2f ζ f + m 2f  ζ f  )[(C0 + C11 )(G ± , f  , H ± ) + (C0 + C11 )(H ± , f  , G ± )] v m 2f  s − g2 (2C0 + C11 )(W, f  , G ± ) + sβ−α (−C0 + C11 )(G ± , f  , W ) 4v β−α

− ζ f  cβ−α (2C0 + C11 )(W, f  , H ± ) − ζ f  cβ−α (−C0 + C11 )(H ± , f  , W ) − g 2Z

m 2f s (2C0 + C11 )(Z , f, G 0 ) + sβ−α (−C0 + C11 )(G 0 , f, Z ) 8v β−α

− ζ f cβ−α (2C0 + C11 )(Z , f, A) − ζ f cβ−α (−C0 + C11 )(A, f, Z ) ,

V ,1PI 16π 2  2 ¯ hf f

f ζh g 2Z (v 2f + a 2f )(C0 + 2C12 )( f, Z , f ) + e2 Q 2f (C0 + 2C12 )( f, γ, f ) v m 2f   f + g2 ζ (C0 + 2C12 )( f  , W, f  ) 2v h g4 − sβ−α g 4Z v(v 2f + a 2f )C12 (Z , f, Z ) − sβ−α vC12 (W, f  , W ) 4 m 4f f f f + ζh 3 (ζh )2 (C0 + 2C12 )( f, h, f ) + (ζ H )2 (C0 + 2C12 )( f, H, f ) v

+ (C0 + 2C12 )( f, G 0 , f ) + ζ 2f (C0 + 2C12 )( f, A, f )

=

2m 2f

(D.42)

232

Appendix D: One Particle Irreducible Diagrams in the Two-Higgs Doublet Model f

+ ζh

m 2f 

(m 2f + m 2f  )(C0 + 2C12 )( f  , G ± , f  ) + (m 2f ζ 2f + m 2f  ζ 2f  )(C0 + 2C12 )( f  , H ± , f  )

v3 m 2f  f f − 2 6(ζh )2 λhhh C12 (h, f, h) + 2(ζ H )2 λ H H h C12 (H, f, H ) v f

f

+ 2ζh ζ H λ H hh [C12 (H, f, h) + C12 (h, f, H )] + 2λG 0 G 0 h C12 (G 0 , f, G 0 ) + 2ζ 2f λ A Ah C12 (A, f, A)  + 2ζ f λ AG 0 h [C12 (G 0 , f, A) + C12 (A, f, G 0 )] λ + − − G 2G h (m 2f + m 2f  )C12 (G ± , f  , G ± ) v λH + H −h 2 2 − (m f ζ f + m 2f  ζ 2f  )C12 (H ± , f  , H ± ) v2 λ + − − H 2G h (m 2f ζ f + m 2f  ζ f  )[C12 (G ± , f  , H ± ) + C12 (H ± , f  , G ± )] v 2 2 m g f sβ−α (2C0 + C12 )(W, f  , G ± ) + sβ−α (−C0 + C12 )(G ± , f  , W ) − 4 v

− ζ f  cβ−α (2C0 + C12 )(W, f  , H ± ) − ζ f  cβ−α (−C0 + C12 )(H ± , f  , W )

2 g2 m f sβ−α (2C0 + C12 )(Z , f, G 0 ) + sβ−α (C12 − C0 )(G 0 , f, Z ) − Z 8 v

+ ζ f cβ−α (2C0 + C12 )(Z , f, A) + ζ f cβ−α (C12 − C0 )(A, f, Z ) ,

(D.43)

A ,1PI 16π 2  1 ¯ hf f

m 2f

m 2f 

f

ζ (C0 + 2C11 )( f  , W, f  ) 2v h g4 + 2sβ−α g 4Z v f a f v(C0 + C11 )(Z , f, Z ) + sβ−α v(C0 + C11 )(W, f  , W ) 4

m 2f   f + 3 ζh (m 2f − m 2f  )(C0 + 2C11 )( f  , G ± , f  ) + (m 2f ζ 2f − m 2f  ζ 2f  )(C0 + 2C11 )( f  , H ± , f  ) v λG + G − h 2 − (m f − m 2f  )(C0 + C11 )(G ± , f  , G ± ) v2 λ + − − H 2H h (m 2f ζ 2f − m 2f  ζ 2f  )(C0 + C11 )(H ± , f  , H ± ) v λH + G−h 2 − (m f ζ f − m 2f  ζ f  )[(C0 + C11 )(G ± , f  , H ± ) + (C0 + C11 )(H ± , f  , G ± )] v2 2 g2 m f  sβ−α (2C0 + C11 )(W, f  , G ± ) + sβ−α (−C0 + C11 )(G ± , f  , W ) + 4 v

− ζ f  cβ−α (2C0 + C11 )(W, f  , H ± ) − ζ f  cβ−α (−C0 + C11 )(H ± , f  , W )

= −4g 2Z v f a f

v

f

ζh (C0 + 2C11 )( f, Z , f ) − g 2

m 2f sβ−α (2C0 + C11 )(Z , f, G 0 ) + sβ−α (−C0 + C11 )(G 0 , f, Z ) v

+ ζ f cβ−α (2C0 + C11 )(Z , f, A) + ζ f cβ−α (−C0 + C11 )(A, f, Z ) ,

+ g 2Z I f v f

16π 2 

A2 ,1PI h f f¯

m 2f  m 2f f f (C0 + 2C12 )( f, Z , f ) − ζh g 2 (C0 + 2C12 )( f  , W, f  ) = −4ζh g 2Z v f a f v 2v

(D.44)

Appendix D: One Particle Irreducible Diagrams in the Two-Higgs Doublet Model + 2sβ−α g 4Z v f a f vC12 (Z , f, Z ) + sβ−α f

+ ζh

m 2f  v3

233

g4 vC12 (W, f  , W ) 4

(m 2f − m 2f  )(C0 + 2C12 )( f  , G ± , f  ) + (m 2f ζ 2f − m 2f  ζ 2f  )(C0 + 2C12 )( f  , H ± , f  )

λ + − λ + − − G 2G h (m 2f − m 2f  )C12 (G ± , f  , G ± ) − H 2H h (m 2f ζ 2f − m 2f  ζ 2f  )C12 (H ± , f  , H ± ) v v λH + G−h 2 2 ±  ± − (m f ζ f − m f  ζ f  )[C12 (G , f , H ) + C12 (H ± , f  , G ± )] v2 2 g2 m f  + sβ−α (2C0 + C12 )(W, f  , G ± ) + sβ−α (−C0 + C12 )(G ± , f  , W ) 4 v

− ζ f  cβ−α (2C0 + C12 )(W, f  , H ± ) − ζ f  cβ−α (−C0 + C12 )(H ± , f  , W ) m 2f sβ−α (2C0 + C12 )(Z , f, G 0 ) + sβ−α (−C0 + C12 )(G 0 , f, Z ) v

+ ζ f cβ−α (2C0 + C12 )(Z , f, A) + ζ f cβ−α (−C0 + C12 )(A, f, Z ) ,

+ g 2Z I f v f  m −1 f

v

(D.45)

16π 2  T,1PI ¯ hf f

m 2f f f f = ζh 2 (ζh )2 (C11 − C12 )( f, h, f ) + (ζ H )2 (C11 − C12 )( f, H, f ) v

− (C11 − C12 )( f, G 0 , f ) − ζ 2f (C11 − C12 )( f, A, f )

2m 2f  (C11 − C12 )( f  , G ± , f  ) + ζ f ζ f  (C11 − C12 )( f  , H ± , f  ) 2 v g2 s (−2C0 − 2C11 + C12 )(W, f  , G ± ) + sβ−α (−C0 − C11 + 2C12 )(G ± , f  , W ) − 4 β−α f

− ζh

+ ζ f cβ−α (−2C0 − 2C11 + C12 )(W, f  , H ± ) + ζ f cβ−α (−C0 − C11 + 2C12 )(H ± , f  , W )

g2 − Z sβ−α (−2C0 − 2C11 + C12 )(Z , f, G 0 ) + sβ−α (−C0 − C11 + 2C12 )(G 0 , f, Z ) 8

+ ζ f cβ−α (−2C0 − 2C11 + C12 )(Z , f, A) + ζ f cβ−α (−C0 − C11 + 2C12 )(A, f, Z ) ,

(D.46)  m −1 f

16π 2  P T,1PI h f f¯

v g2 = (2C0 + 2C11 − C12 )(W, f  , G ± ) − sβ−α (C0 + C11 − 2C12 )(G ± , f  , W ) s 4 β−α

− ζ f cβ−α (−2C0 − 2C11 + C12 )(W, f  , H ± ) − ζ f cβ−α (C0 + C11 − 2C12 )(H ± , f  , W ) − g 2Z I f v f sβ−α (−2C0 − 2C11 + C12 )(Z , f, G 0 ) + sβ−α (C0 + C11 − 2C12 )(G 0 , f, Z )

+ ζ f cβ−α (−2C0 − 2C11 + C12 )(Z , f, A) + ζ f cβ−α (C0 + C11 − 2C12 )(A, f, Z ) , (D.47)

where C hFfSfF (X, Y, Z ) = [m 2f C0 + p12 (C11 + C21 ) + p22 (C12 + C22 ) + 2 p1 · p2 (C12 + C23 ) + 4C24 ](X, Y, Z ) −

1 , 2

(D.48)

234

Appendix D: One Particle Irreducible Diagrams in the Two-Higgs Doublet Model

C hVfFf S (X, Y, Z ) = [ p12 (2C0 + 3C11 + C21 ) + p22 (2C12 + C22 ) + 2 p1 · p2 (2C0 + 2C11 + C12 + C23 ) + 4C24 ](X, Y, Z ) −

1 , 2

(D.49)

C hS fFfV (X, Y, Z ) 1 = [ p12 (C21 − C0 ) + p22 (C22 − C12 ) + 2 p1 · p2 (C23 − C12 ) + 4C24 ](X, Y, Z ) − . 2

(D.50)

D.3.3 hV V Vertex The 1PI diagram contributions to the form factors of the h Z Z vertices are given by [8] 2 2 2 16π 2 h1,1PI Z Z ( p 1 , p2 , q ) F f  16m 2f m 2Z Nc  (v 2f + a 2f ) B0 ( p12 , f, f ) + B0 ( p22 , f, f ) + 2B0 (q 2 , f, f ) 3 v f

+ (4m 2f − p12 − p22 )C0 ( f, f, f ) − 8C24 ( f, f, f )

 − (v 2f − a 2f ) B0 ( p22 , f, f ) + B0 ( p12 , f, f ) + (4m 2f − q 2 )C0 ( f, f, f ) ,

=

(D.51)

2 2 2 16π 2 h2,1PI Z Z ( p 1 , p2 , q ) F f  32m 2f m 4Z Nc (v 2f + a 2f )(4C23 + 3C12 + C11 + C0 ) v3 f

+ (v 2f − a 2f )(C12 − C11 ) ( f, f, f ),

=−

f  64m 2f m 4Z Nc 2 , p2 , q 2 ) = 16π 2 h3,1PI ( p v f a f (C11 + C12 + C0 )( f, f, f ), F 1 2 ZZ v3 f

(D.52) (D.53)

2 2 2 16π 2 h1,1PI Z Z ( p 1 , p2 , q ) B 4 C (G ± , W, G ± ) = 2g 2Z λG + G − h m 2W sW 0  V V V (W, W, W ) − 2c2 C (c± , c± , c± ) + s 2 C SV V (G ± , W, W ) + g 3 m W sβ−α 2c2W C hV W 24 W hV V 1 V1 4 2 C V V S (W, W, G ± ) − 2 sW m 2 s ± + sW W β−α C 0 (W, G , W ) hV V 1 2 cW 2  2 ) sW [C (W, G ± , G ± ) + C (G ± , G ± , W )] − (c2W − sW 24 24 2 cW

 g3 2 2 C0 (Z , h, Z ) + cβ−α C0 (Z , H, Z ) + Z m Z sβ−α − 2m 2Z sβ−α 2 2 2 + sβ−α [C24 (G 0 , h, Z ) + C24 (Z , h, G 0 )] + cβ−α [C24 (G 0 , H, Z ) + C24 (Z , H, G 0 )]

Appendix D: One Particle Irreducible Diagrams in the Two-Higgs Doublet Model

235

 2 + cβ−α [C24 (A, h, Z ) + C24 (Z , h, A) − C24 (A, H, Z ) − C24 (Z , H, A)]  2 2 C0 (h, Z , h) + λ H H h cβ−α C0 (H, Z , H ) + 2g 2Z m 2Z 3λhhh sβ−α  + λ H hh sβ−α cβ−α [C0 (H, Z , h) + C0 (h, Z , H )] 2 )2 λ ± ± ± ± ± ±  − 2g 2Z (c2W − sW G + G − h C 24 (G , G , G ) + λ H + H − h C 24 (H , H , H ) 2 − 2g 2Z sβ−α 3λhhh C24 (h, G 0 , h) + λGGh C24 (G 0 , h, G 0 )

+ λ H H h C24 (H, A, H ) + λ A Ah C24 (A, H, A) 2 3λhhh C24 (h, A, h) + λ A Ah C24 (A, h, A) − 2g 2Z cβ−α

+ λ H H h C24 (H, G 0 , H ) + λGGh C24 (G 0 , H, G 0 ) − 2g 2Z sβ−α cβ−α λ H hh [C24 (h, G 0 , H ) + C24 (H, G 0 , h) − C24 (h, A, H ) − C24 (H, A, h)] − 2g 2Z sβ−α cβ−α λ AGh [C24 (A, h, G 0 ) + C24 (G 0 , h, A) − C24 (A, H, G 0 ) − C24 (G 0 , H, A)] 2 g2 2 )2 B (q 2 , G ± , G ± ) + g Z λ 2 2 2 2 ± ± + Z λG + G − h (c2W − sW + − (c − sW ) B0 (q , H , H ) 0 2 2 H H h W g2 g2 + Z λGGh B0 (q 2 , G 0 , G 0 ) + Z λ A Ah B0 (q 2 , A, A) 2 2 2 g 2Z 3g λ H H h B0 (q 2 , H, H ) + Z λhhh B0 (q 2 , h, h) + 2 2 4 s − g3 W m W sβ−α [B0 ( p22 , W, G ± ) + B0 ( p12 , G ± , W )] c2W

g3 − Z m Z sβ−α [B0 ( p12 , h, Z ) + B0 ( p22 , h, Z )] 2 − 6 g 3 c2W m W sβ−α B0 (q 2 , W, W ) + 4 g 3 c2W m W sβ−α , 2 2 2 (g 2Z m 2Z )−1 16π 2 h2,1PI Z Z ( p1 , p2 , q ) B V V V (W, W, W ) − 2gc4 m s ± ± ± = 2gm W c4W sβ−α C hV W W β−α C 1223 (c , c , c ) V2 2 c2 s SV V ± VVS ± + gm W sW W β−α [C hV V 2 (G , W, W ) + C hV V 2 (W, W, G )] 2 2 2 SSV ± ± V SS − gm W (cW − sW )sW sβ−α [C hV V 2 (G , G , W ) + C hV V 2 (W, G ± , G ± )] gZ V SS (Z , h, G 0 ) + C V SS (G 0 , h, Z )] m Z [C hV + V2 hV V 2 2

gZ 3 V SS (Z , h, G 0 ) + C SSV (G 0 , h, Z )] m Z sβ−α [C hV V2 hV V 2 2 gZ 2 V SS V SS (Z , H, G 0 ) − C V SS (Z , H, A) + m Z sβ−α cβ−α [C hV V 2 (Z , h, A) + C hV V2 hV V 2 2 SSV SSV 0 SSV + C hV V 2 (A, h, Z ) + C hV V 2 (G , H, Z ) − C hV V 2 (A, H, Z )] 2 )2 λ ± ± ± ± ± ±  − 2(c2W − sW G + G − h C 1223 (G , G , G ) + λ H + H − h C 1223 (H , H , H ) 2 − 2sβ−α 3λhhh C1223 (h, G 0 , h) + λGGh C1223 (G 0 , h, G 0 )

+ λ H H h C1223 (H, A, H ) + λ A Ah C1223 (A, H, A) 2 3λhhh C1223 (h, A, h) + λ A Ah C1223 (A, h, A) − 2cβ−α

+ λ H H h C1223 (H, G 0 , H ) + λGGh C1223 (G 0 , H, G 0 ) +

(D.54)

236

Appendix D: One Particle Irreducible Diagrams in the Two-Higgs Doublet Model − 2sβ−α cβ−α λ H hh [C1223 (h, G 0 , H ) + C1223 (H, G 0 , h) − C1223 (h, A, H ) − C1223 (H, A, h)] − 2sβ−α cβ−α λ AGh [C1223 (A, h, G 0 ) + C1223 (G 0 , h, A) − C1223 (A, H, G 0 ) − C1223 (G 0 , H, A)],

2 2 2 h3,1PI Z Z ( p1 , p2 , q ) B = 0.

(D.55) (D.56)

The 1PI diagram contributions to the form factors of the hW W vertices are given by [8] 1,1PI 2 2 2 16π 2 hW W ( p 1 , p2 , q ) F

=

f  4m 2W m 2f Nc  1 1 B0 ( p22 , f, f  ) + B0 (q 2 , f, f ) + B0 ( p12 , f, f  ) 3 2 2 v  f, f

 1 − 4C24 ( p12 , p22 , q 2 , f, f  , f ) + (2m 2f + 2m 2f  − p12 − p22 )C0 ( f, f  , f ) + (m f ↔ m f  ), 2

(D.57) f −4m 4W m 2f Nc 2,1PI 2 2 2 16π 2 hW (4C23 + 3C12 + C11 + C0 ) ( f, f  , f ) + (m f ↔ m f  ), W ( p 1 , p2 , q ) F = v3

(D.58)

f −4m 4W m 2f Nc 3,1PI 2 , p2 , q 2 ) = 16π 2 hW ( p (C11 + C12 + C0 ) ( f, f  , f ) + (m f ↔ m f  ), F W 1 2 v3

(D.59)

1,1PI 2 2 2 16π 2 hW W ( p 1 , p2 , q ) B V V V (Z , W, Z ) + c2 C V V V (W, Z , W ) + s 2 C V V V (W, γ, W ) = g 3 m W sβ−α [C hV W hV V 1 W hV V 1 V1 2 C (c± , c , c± )] − C24 (c Z , c± , c Z ) − c2W C24 (c± , c Z , c± ) − sW γ 24

g3 2 s SV V ± SV V ± m W sW β−α [C hV V 1 (G , Z , W ) − C hV V 1 (G , γ, W ) 2 V V S (W, Z , G ± ) − C V V S (W, γ, G ± )] + C hV V1 hV V 1

−

s4 3 2 s C0 (Z , G ± , Z ) − g 3 m 3W sβ−α C0 (W, h, W ) − gm 3W sβ−α cβ−α C0 (W, H, W ) − g 3 m 3W W c4W β−α s4 2 2 m2 λ ± ± + g2 W m W λG + G − h C0 (G ± , Z , G ± ) + sW W G + G − h C 0 (G , γ, G ) c2W 2 2 C0 (h, W, h) + 2g 2 λ H H h m 2W cβ−α C0 (H, W, H ) + 6g 2 λhhh m 2W sβ−α

+ 2g 2 λ H hh m 2W cβ−α sβ−α [C0 (h, W, H ) + C0 (H, W, h)]  g3 2 m W sβ−α sβ−α [C24 (W, h, G ± ) + C24 (G ± , h, W )] 2 2 + cβ−α [C24 (W, H, G ± ) + C24 (G ± , H, W ) + C24 (W, h, H ± ) + C24 (H ± , h, W )  − C24 (W, H, H ± ) − C24 (H ± , H, W )]

+

s2 g3 mW W s [C24 (G 0 , G ± , Z ) + C24 (Z , G ± , G 0 )] 2 c2W β−α − g 2 λG + G − h C24 (G ± , G 0 , G ± ) + λ H + H − h C24 (H ± , A, H ± )

+

Appendix D: One Particle Irreducible Diagrams in the Two-Higgs Doublet Model

237

+ 2λGGh C24 (G 0 , G ± , G 0 ) + 2λ A Ah C24 (A, H ± , A) 2 6λhhh C24 (h, G ± , h) + 2λ H H h C24 (H, H ± , H ) − g 2 sβ−α

+ λG + G − h C24 (G ± , h, G ± ) + λ H + H − h C24 (H ± , H, H ± ) 2 − g 2 cβ−α 6λhhh C24 (h, H ± , h) + 2λ H H h C24 (H, G ± , H )

+ λG + G − h C24 (G ± , H, G ± ) + λ H + H − h C24 (H ± , h, H ± ) − g 2 λ H + G − h sβ−α cβ−α C24 (G ± , h, H ± ) + C24 (H ± , h, G ± )

− C24 (G ± , H, H ± ) − C24 (H ± , H, G ± )

− 2g 2 λ H hh sβ−α cβ−α [C24 (h, G ± , H ) + C24 (H, G ± , h) − C24 (h, H ± , H ) − C24 (H, H ± , h)]

− g 3 m W sβ−α 3B0 (q 2 , W, W ) + 3B0 (q 2 , Z , Z ) − 4 g2 g2 3g 2 λ + − B0 (q 2 , G ± , G ± ) + λGGh B0 (q 2 , G 0 , G 0 ) + λhhh B0 (q 2 , h, h) 2 G G h 2 2 g2 g2 g2 λ H + H − h B0 (q 2 , H ± , H ± ) + λ A Ah B0 (q 2 , A, A) + λ H H h B0 (q 2 , H, H ) + 2 2 2  s4 g3 − m W sβ−α B0 ( p12 , W, h) + B0 ( p22 , W, h) + W [B0 ( p12 , Z , G ± ) + B0 ( p22 , Z , G ± )] 2 c2W  2 [B ( p 2 , γ, G ± ) + B ( p 2 , γ, G ± )] , + sW 0 1 0 2

+

2,1PI 2 2 2 (g 2 m 2W )−1 16π 2 hW W ( p1 , p2 , q ) B V V V (Z , W, Z ) + c2 C V V V (W, Z , W ) + s 2 C V V V (W, γ, W ) = gm W sβ−α C hV W hV V 2 W hV V 2 V2

2 C ± ± − C1223 (c Z , c± , c Z ) − c2W C1223 (c± , c Z , c± ) − sW 1223 (c , cγ , c )

g 2 SV V (G ± , Z , W ) + C V V S (W, Z , G ± ) s m W sβ−α [C hV V2 hV V 2 2 W SV V ± V V S − C hV V 2 (G , γ, W ) − C hV V 2 (W, γ, G ± )]

g 3 V SS (W, h, G ± ) + C SSV (G ± , h, W ) + m W sβ−α C hV V2 hV V 2 2 g 2 V SS (W, H, G ± ) + C V SS (W, h, H ± ) − C V SS (W, H, H ± ) + m W sβ−α cβ−α C hV V2 hV V 2 hV V 2 2

SSV (G ± , H, W ) + C SSV (H ± , h, W ) − C SSV (H ± , H, W ) + C hV V2 hV V 2 hV V 2

−

2

g sW 3 V SS (Z , G ± , G 0 ) + C SSV (G 0 , G ± , Z ) C hV m W sβ−α V 2 hV V 2 2 2c W − λG + G − h C1223 (G ± , G 0 , G ± ) + λ H + H − h C1223 (H ± , A, H ± )

+ 2λGGh C1223 (G 0 , G ± , G 0 ) + 2λ A Ah C1223 (A, H ± , A) 2 6λhhh C1223 (h, G ± , h) + 2λ H H h C1223 (H, H ± , H ) − sβ−α

+ λG + G − h C1223 (G ± , h, G ± ) + λ H + H − h C1223 (H ± , H, H ± ) 2 − cβ−α 6λhhh C1223 (h, H ± , h) + 2λ H H h C1223 (H, G ± , H )

+ λG + G − h C1223 (G ± , H, G ± ) + λ H + H − h C1223 (H ± , h, H ± )

+

(D.60)

238

Appendix D: One Particle Irreducible Diagrams in the Two-Higgs Doublet Model

− λ H + G − h sβ−α cβ−α [C1223 (G ± , h, H ± ) + C1223 (H ± , h, G ± ) − C1223 (G ± , H, H ± ) − C1223 (H ± , H, G ± )] − 2λ H hh sβ−α cβ−α [C1223 (h, G ± , H ) + C1223 (H, G ± , h) − C1223 (h, H ± , H ) − C1223 (H, H ± , h)], 3,1PI 2 2 2 16π 2 hW W ( p1 , p2 , q ) B = 0,

(D.61) (D.62)

where VVV (X, Y, Z ) = 18C24 + p12 (2C21 + 3C11 + C0 ) + p22 (2C22 + C12 ) C hV V1

+ p1 · p2 (4C23 + 3C12 + C11 − 4C0 ) (X, Y, Z ) − 3, (D.63) SV V C hV (X, Y, Z ) = 3C24 + p12 (C21 − C0 ) + p22 (C22 − 2C12 + C0 ) V1

1 (D.64) + 2 p1 · p2 (C23 − C11 ) (X, Y, Z ) − , 2 VVS 2 2 C hV V 1 (X, Y, Z ) = 3C 24 + p1 (C 21 + 4C 11 + 4C 0 ) + p2 (C 22 + 2C 12 )

1 + 2 p1 · p2 (C23 + 2C12 + C11 + 2C0 ) (X, Y, Z ) − , 2 (D.65) VVV C hV V 2 (X, Y, Z ) = (10C 23 + 9C 12 + C 11 + 5C 0 ) (X, Y, Z ),

(D.66)

SV V C hV V 2 (X, Y, VVS C hV V 2 (X, Y, V SS C hV V 2 (X, Y, SSV C hV V 2 (X, Y,

Z ) = (4C11 − 3C12 − C23 ) (X, Y, Z ),

(D.67)

Z ) = (2C11 − 5C12 − 2C0 − C23 ) (X, Y, Z ),

(D.68)

Z ) = (C23 + C12 + 2C11 + 2C0 )(X, Y, Z ),

(D.69)

Z ) = (C23 − C12 )(X, Y, Z ),

(D.70)

C1223 (X, Y, Z ) = (C12 + C23 )(X, Y, Z ).

(D.71)

D.3.4 V f f¯ Vertex The 1PI diagram contributions to these vertices are calculated as = v f e2 Q 2f FF V F ( f, γ, f ) + g 2Z v f (v 2f + 3a 2f )FF V F ( f, Z , f ) 16π 2  ZV,1PI ff +

g2 2 (v f  + a f  )FF V F ( f  , W, f  ) + I f g 2 cW FV F V (W, f  , W ) , 4 (D.72)

2 2 2 2 2 16π 2  ZA,1PI f f = a f e Q f FF V F ( f, γ, f ) + g Z a f (3v f + a f )FF V F ( f, Z , f )

Appendix D: One Particle Irreducible Diagrams in the Two-Higgs Doublet Model

+

V,1PI 16π 2 W ff =

239

g2 2 (v f  + a f  )FF V F ( f  , W, f  ) + I f g 2 cW FV F V (W, f  , W ) , 4 (D.73)

 e2

Q f Q f  FF V F ( f, γ, f  ) +

g 2Z (v f + a f )(v f  + a f  )FF V F ( f, Z , f  ) 2

2 g2 + [FV F V (W, f  , Z ) + FV F V (Z , f, W )] 2 + 2e2 Q f I f [FV F V (γ, f, W ) − FV F V (Z , f, W )]

 + 2e2 Q f  I f  [FV F V (W, f  , γ) − FV F V (W, f  , Z )] ,

WA,1PI ff

=

V,1PI W ff ,

(D.74) (D.75)

where FF V F (X, Y, Z ) = 2q 2 [C11 + C23 ]( p12 , p22 , q 2 ; m X , m Y , m Z ) + 4C24 ( p12 , p22 , q 2 ; m X , m Y , m Z ) − 2, FV F V (X, Y, Z )

= q [C0 + C11 + C23 ]( p12 , p22 , q 2 ; m X , m Y , m Z ) + 6C24 ( p12 , p22 , q 2 ; m X , m Y , m Z ) − 1.

H + f f  Vertex

D.3.5

The 1PI diagrams for the H + f f  couplings are calculated as S 16π 2  H + ff +

= e2 Q f  Q f g SH C+F V F ( f  , γ, f ) +

+ + g 2Z g SH (v f v f  − a f a f  )C+F V F + g PH (a f v f  − v f a f  )C−F V F ( f  , Z , f )

g2 + √ − g hf cβ−α C SV F S (W, f, h) + g Hf sβ−α C SV F S (W, f, H ) − ig Af C SV F S (W, f, A) 4 2

g2 − √ − g hf  cβ−α C SS F V (h, f  , W ) + g Hf sβ−α C SS F V (H, f  , W ) + ig Af C SS F V (A, f  , W ) 4 2

+ + e2 g SH Q f C SS F V (H ± , f, γ) − Q f  C SV F S (γ, f  , H ± )

g 2Z + + + + c2W (v f g SH + a f g PH )C SS F V (H ± , f, Z ) − (v f  g SH − a f  g PH )C SV F S (Z , f  , H ± ) 2 + − g SH g hf g hf  C SF S F ( f  , h, f ) + g Hf g Hf C SF S F ( f  , H, f )

0 0 + g Af g Af C SF S F ( f  , A, f ) + g Gf g Gf  C SF S F ( f  , G 0 , f )  + − m f g SG [λ H + G − h g hf C0 (G ± , f, h) + λ H + G − H g Hf C0 (G ± , f, H )] +

+

(D.76)

2

+ g SH [λ H + H − h g hf C0 (H ± , f, h) + λ H + H − H g Hf C0 (H ± , f, H )]  + A ± + gG P λ H + G − A g f C 0 (G , f, A)

(D.77)

240

Appendix D: One Particle Irreducible Diagrams in the Two-Higgs Doublet Model  + − m f  g SG [λ H + G − h g hf  C0 (h, f  , G ± ) + λ H + G − H g Hf C0 (H, f  , G ± )] +

+ g SH [λ H + H − h g hf  C0 (h, f  , H ± ) + λ H + H − H g Hf C0 (H, f  , H ± )]  + A  ± + gG P λ H + G − A g f  C 0 (A, f , G ) , 16π

2

(D.78)

 HP + f f  +

= e2 Q f  Q f g PH C−F V F ( f  , γ, f ) +

+ + g 2Z g SH (a f v f  − v f a f  )C+F V F + g PH (v f v f  − a f a f  )C−F V F ( f  , Z , f )

g2 − √ − g hf cβ−α C SV F S (W, f, h) + g Hf sβ−α C SV F S (W, f, H ) − ig Af C SV F S (W, f, A) 4 2

g2 − √ − g hf  cβ−α C SS F V (h, f  , W ) + g Hf sβ−α C SS F V (H, f  , W ) + ig Af C SS F V (A, f  , W ) 4 2

+ + e2 g PH Q f C SS F V (H ± , f, γ) − Q f  C SV F S (γ, f  , H ± )

g 2Z + + + + c2W (a f g SH + v f g PH )C SS F V (H ± , f, Z ) − (v f  g PH − a f  g SH )C SV F S (Z , f  , H ± ) 2 + + g PH g hf g hf  C PF S F ( f  , h, f ) + g Hf g Hf C PF S F ( f  , H, f )

0 0 + g Af g Af C PF S F ( f  , A, f ) + g Gf g Gf  C PF S F ( f  , G 0 , f )  + h ± H ± − m f gG P [λ H + G − h g f C 0 (G , f, h) + λ H + G − H g f C 0 (G , f, H )] +

+

+ g PH [λ H + H − h g hf C0 (H ± , f, h) + λ H + H − H g Hf C0 (H ± , f, H )]  + + g SG λ H + G − A g Af C0 (G ± , f, A)  + h  ± H  ± − m f  gG P [λ H + G − h g f  C 0 (h, f , G ) + λ H + G − H g f  C 0 (H, f , G )] +

+ g PH [λ H + H − h g hf  C0 (h, f  , H ± ) + λ H + H − H g Hf C0 (H, f  , H ± )]  + + g SG λ H + G − A g Af C0 (A, f  , G ± ) , 16π

2

V1 H + ff +

= 2e2 Q f  Q f g SH [m f C11 + m f  (C11 + C0 )]( f  , γ, f ) + + 2g 2Z g SH (v f v f  + a f a f  )[m f C11 + m f  (C11 + C0 )]

+ − g PH (v f a f  + a f v f  )[m f C11 − m f  (C11 + C0 )] ( f  , Z , f ) g2 + √ m f − g hf cβ−α (C11 + 2C0 )(W, f, h) + g Hf sβ−α (C11 + 2C0 )(W, f, H ) 4 2

+ ig Af (C11 + 2C0 )(W, f, A) g2 − √ m f  − g hf  cβ−α (C11 − C0 )(h, f  , W ) + g Hf sβ−α (C11 − C0 )(H, f  , W ) 4 2

− ig Af (C11 − C0 )(A, f  , W )

+ + e2 g SH Q f m f (C11 − C0 )(H ± , f, γ) − Q f  m f  (C11 + 2C0 )(γ, f  , H ± ) g 2Z + + c2W m f (v f g SH − a f g PH )(C11 − C0 )(H ± , f, Z ) 2

+ + − m f  (v f  g SH + a f  g PH )(C11 + 2C0 )(Z , f  , H ± )

+

(D.79)

Appendix D: One Particle Irreducible Diagrams in the Two-Higgs Doublet Model +

241



g hf g hf  [m f  (C11 + C0 ) + m f C11 ]( f  , h, f ) + g Hf g Hf [m f  (C11 + C0 ) + m f C11 ]( f  , H, f )

0 0 − g Af g Af [m f  (C11 + C0 ) + m f C11 ]( f  , A, f ) − g Gf g Gf  [m f  (C11 + C0 ) + m f C11 ]( f  , G 0 , f ) + g SH

+

− g SG [λ H + G − h g hf  (C0 + C11 )(h, f  , G ± ) + λ H + G − H g Hf (C0 + C11 )(H, f  , G ± )] +

− g SH [λ H + H − h g hf  (C0 + C11 )(h, f  , H ± ) + λ H + H − H g Hf (C0 + C11 )(H, f  , H ± )] G+

+ g P λ H + G − A g Af (C0 + C11 )(A, f  , G ± ) +

− g SG [λ H + G − h g hf (C0 + C11 )(G ± , f, h) + λ H + G − H g Hf (C0 + C11 )(G ± , f, H )] +

− g SH [λ H + H − h g hf (C0 + C11 )(H ± , f, h) + λ H + H − H g Hf (C0 + C11 )(H ± , f, H )] +

A ± + gG P λ H + G − A g f (C 0 + C 11 )(G , f, A),

(D.80)

V2 16π 2  H + ff +

= 2e2 Q f  Q f g SH [m f C12 + m f  (C12 + C0 )]( f  , γ, f ) + + g 2Z 2g SH (v f v f  + a f a f  )[m f C12 + m f  (C12 + C0 )]

+ − 2g PH (v f a f  + a f v f  )[m f C12 − m f  (C12 + C0 )] ( f  , Z , f ) g2 + √ m f − g hf cβ−α (C12 + 2C0 )(W, f, h) + g Hf sβ−α (C12 + 2C0 )(W, f, H ) 4 2

+ ig Af (C12 + 2C0 )(W, f, A) g2 − √ m f  − g hf  cβ−α (C12 − C0 )(h, f  , W ) + g Hf sβ−α (C12 − C0 )(H, f  , W ) 4 2

− ig Af (C12 − C0 )(A, f  , W )  +  + e2 g SH Q f m f (C12 − C0 )(H ± , f, γ) − Q f  m f  (C12 + 2C0 )(γ, f  , H ± ) g 2 c2W + + − m f  (v f  g SH + a f  g PH )(C12 + 2C0 )(Z , f  , H ± ) + Z 2

+ + + m f (v f  g SH − a f  g PH )(C12 − C0 )(H ± , f, Z ) + + g SH g hf g hf  [m f  (C12 + C0 ) + m f C12 ]( f  , h, f ) + g Hf g Hf [m f  (C12 + C0 ) + m f C12 ]( f  , H, f )

0 0 − g Af g Af [m f  (C12 + C0 ) + m f C12 ]( f  , A, f ) − g Gf g Gf  [m f  (C12 + C0 ) + m f C12 ]( f  , G 0 , f ) +

− g SG [λ H + G − h g hf  C12 (h, f  , G ± ) + λ H + G − H g Hf C12 (H, f  , G ± )] +

− g SH [λ H + H − h g hf  C12 (h, f  , H ± ) + λ H + H − H g Hf C12 (H, f  , H ± )] +

A  ± + gG P λ H + G − A g f  C 12 (A, f , G ) +

− g SG [λ H + G − h g hf C12 (G ± , f, h) + λ H + G − H g Hf C12 (G ± , f, H )] +

− g SH [λ H + H − h g hf C12 (H ± , f, h) + λ H + H − H g Hf C12 (H ± , f, H )] +

A ± + gG P λ H + G − A g f C 12 (G , f, A),

16π 2  HA1+ f f  +

= 2e2 Q f Q f  g PH [m f C11 − m f  (C11 + C0 )]( f  , γ, f ) + − g 2Z 2g SH (a f v f  + v f a f  )[m f C11 + m f  (C11 + C0 )]

+ − 2g PH (v f v f  + a f a f  )[m f C11 − m f  (C11 + C0 )] ( f  , Z , f )

(D.81)

242

Appendix D: One Particle Irreducible Diagrams in the Two-Higgs Doublet Model g2 − √ m f − g hf cβ−α (C11 + 2C0 )(W, f, h) + g Hf sβ−α (C11 + 2C0 )(W, f, H ) 4 2

+ ig Af (C11 + 2C0 )(W, f, A) g2 + √ m f  − g hf  cβ−α (C11 − C0 )(h, f  , W ) + g Hf sβ−α (C11 − C0 )(H, f  , W ) 4 2

− ig Af (C11 − C0 )(A, f  , W )  +  + e2 g PH Q f  m f  (C11 + 2C0 )(γ, f  , H ± ) + Q f m f (C11 − C0 )(H ± , f, γ) g 2 c2W + + + Z m f  (v f  g PH + a f  g SH )(C11 + 2C0 )(Z , f  , H ± ) 2

+ + + m f (v f g PH − a f g SH )(C11 − C0 )(H ± , f, Z ) + + g PH g hf g hf  [m f  (C11 + C0 ) − m f C11 ]( f  , h, f ) + g Hf g Hf [m f  (C11 + C0 ) − m f C11 ]( f  , H, f )

0 0 − g Af g Af [m f  (C11 + C0 ) − m f C11 ]( f  , A, f ) − g Gf g Gf  [m f  (C11 + C0 ) − m f C11 ]( f  , G 0 , f ) +

h  ± H  ± + gG P [λ H + G − h g f  (C 0 + C 11 )(h, f , G ) + λ H + G − H g f  (C 0 + C 11 )(H, f , G )] +

+ g PH [λ H + H − h g hf  (C0 + C11 )(h, f  , H ± ) + λ H + H − H g Hf (C0 + C11 )(H, f  , H ± )] +

− g SG λ H + G − A g Af (C0 + C11 )(A, f  , G ± ) +

h ± H ± − gG P [λ H + G − h g f (C 0 + C 11 )(G , f, h) + λ H + G − H g f (C 0 + C 11 )(G , f, H )] +

− g PH [λ H + H − h g hf (C0 + C11 )(H ± , f, h) + λ H + H − H g Hf (C0 + C11 )(H ± , f, H )]  + + g SG λ H + G − A g Af (C0 + C11 )(G ± , f, A) , 16π

2

(D.82)

 HA2+ f f  +

= 2e2 Q f Q f  g PH [m f C12 − m f  (C12 + C0 )]( f  , γ, f ) + − g 2Z 2g SH (a f v f  + v f a f  )[m f C12 + m f  (C12 + C0 )]

+ − 2g PH (v f v f  + a f a f  )[m f C12 − m f  (C12 + C0 )] ( f  , Z , f ) g2 − √ m f − g hf cβ−α (C12 + 2C0 )(W, f, h) + g Hf sβ−α (C12 + 2C0 )(W, f, H ) 4 2

+ ig Af (C12 + 2C0 )(W, f, A) g2 + √ m f  − g hf  cβ−α (C12 − C0 )(h, f  , W ) + g Hf sβ−α (C12 − C0 )(H, f  , W ) 4 2

− ig Af (C12 − C0 )(A, f  , W )

+ + e2 g PH Q f  m f  (C12 + 2C0 )(γ, f  , H ± ) + Q f m f (C12 − C0 )(H ± , f, γ) , g 2Z c2W



+

+

(D.83)

m f (v f g PH − a f g SH )(C12 − C0 )(H ± , f, Z )

+ + + m f  (v f  g PH + a f  g SH )(C12 + 2C0 )(Z , f  , H ± ) + + g PH g hf g hf  [m f  (C12 + C0 ) − m f C12 ]( f  , h, f ) + g Hf g Hf [m f  (C12 + C0 ) − m f C12 ]( f  , H, f )

0 0 − g Af g Af [m f  (C12 + C0 ) − m f C12 ]( f  , A, f ) − g Gf g Gf  [m f  (C12 + C0 ) − m f C12 ]( f  , G 0 , f ) +

2

+

h  ± H  ± gG P [λ H + G − h g f  C 12 (h, f , G ) + λ H + G − H g f  C 12 (H, f , G )]

Appendix D: One Particle Irreducible Diagrams in the Two-Higgs Doublet Model

243

+

+ g PH [λ H + H − h g hf  C12 (h, f  , H ± ) + λ H + H − H g Hf C12 (H, f  , H ± )] +

− g SG λ H + G − A g Af C12 (A, f  , G ± ) +

h ± H ± − gG P [λ H + G − h g f C 12 (G , f, h) + λ H + G − H g f C 12 (G , f, H )] +

− g PH [λ H + H − h g hf C12 (H ± , f, h) + λ H + H − H g Hf C12 (H ± , f, H )]  + − g SG λ H + G − A g Af C12 (G ± , f, A) , 16π

2

(D.84)

 TH + f f 

g2 = √ − g hf cβ−α (2C0 + 2C11 − C12 )(W, f, h) + g Hf sβ−α (2C0 + 2C11 − C12 )(W, f, H ) 4 2

− ig Af (2C0 + 2C11 − C12 )(W, f, A) g2 − √ − g hf  cβ−α (C0 + C11 − 2C12 )(h, f  , W ) + g Hf sβ−α (C0 + C11 − 2C12 )(H, f  , W ) 4 2

+ ig Af (C0 + C11 − 2C12 )(A, f  , W )

+ + e2 g SH Q f (C0 + C11 − 2C12 )(H ± , f, γ) − Q f  (2C0 + 2C11 − C12 )(γ, f  , H ± ) g 2Z c2W + + (v f g SH + a f g PH )(C0 + C11 − 2C12 )(H ± , f, Z ) 2

+ + − (v f  g SH − a f  g PH )(2C0 + 2C11 − C12 )(Z , f  , H ± ) + + g SH g hf g hf  (C11 − C12 )( f  , h, f ) + g Hf g Hf (C11 − C12 )( f  , H, f )

0 0 + g Af g Af (C11 − C12 )( f  , A, f ) + g Gf g Gf  (C11 − C12 )( f  , G 0 , f ) , +

16π

2

(D.85)

 HP T+ f f 

g2 = − √ − g hf cβ−α (2C0 + 2C11 − C12 )(W, f, h) + g Hf sβ−α (2C0 + 2C11 − C12 )(W, f, H ) 4 2

− ig Af (2C0 + 2C11 − C12 )(W, f, A) g2 − √ − g hf  cβ−α (C0 + C11 − 2C12 )(h, f  , W ) + g Hf sβ−α (C0 + C11 − 2C12 )(H, f  , W ) 4 2

+ ig Af (C0 + C11 − 2C12 )(A, f  , W )

+ + e2 g PH Q f (C0 + C11 − 2C12 )(H ± , f, γ) − Q f  (2C0 + 2C11 − C12 )(γ, f  , H ± ) g 2Z c2W + + (v f g PH + a f g SH )(C0 + C11 − 2C12 )(H ± , f, Z ) 2

+ + − (v f  g PH − a f  g SH )(2C0 + 2C11 − C12 )(Z , f  , H ± ) + − g PH g hf g hf  (C11 − C12 )( f  , h, f ) + g Hf g Hf (C11 − C12 )( f  , H, f )

0 0 + g Af g Af (C11 − C12 )( f  , A, f ) + g Gf g Gf  (C11 − C12 )( f  , G 0 , f ) , +

where C SS F V (X, Y, Z )

= B0 ( p22 ; Y, Z ) + (m 2X − q 2 + p22 )C0 − (q 2 − p12 − p22 )C11

(D.86)

244

Appendix D: One Particle Irreducible Diagrams in the Two-Higgs Doublet Model

+ (q 2 − p12 − 2 p22 )C12 ( p12 , p22 , q 2 ; X, Y, Z ), C SV F S (X, Y,

(D.87)

Z)

= B0 ( p22 ; Y, Z ) + (m 2X + 2 p12 )C0 + 3 p12 C11 + 2(q 2 − p12 )C12 ( p12 , p22 , q 2 ; X, Y, Z ),

C SF S F (X, Y,

(D.88)

Z)

= B0 ( p22 ; Y, Z ) + (m 2X + m f  m f )C0 + (q 2 − p22 )C11 + p22 C12 ( p12 , p22 , q 2 ; X, Y, Z ),

C PF S F (X, Y, =

Z)

B0 ( p22 ; Y,

(D.89)

Z ) + (m 2X − m f  m f )C0 + (q 2 − p22 )C11 + p22 C12

( p12 , p22 , q 2 ; X, Y, Z ), Z) = 4 p12 C21 + 4 p22 C22 + 4(q 2 − p12 − p22 )C23 + 16C24 − 4

(D.90)

C+F V F (X, Y,

+ 2(q 2 + p12 − p22 )C11 + 2(q 2 − p12 + p22 )C12 + 4m f  m f C0

( p12 , p22 , q 2 ; X, Y, Z ), Z) = 4 p12 C21 + 4 p22 C22 + 4(q 2 − p12 − p22 )C23 + 16C24 − 4

(D.91)

C−F V F (X, Y,

+ 2(q 2 + p12 − p22 )C11 + 2(q 2 − p12 + p22 )C12 − 4m f  m f C0 ( p12 , p22 , q 2 ; X, Y, Z ).

D.3.6

(D.92)

H + W − φ Vertex

The 1PI diagrams for the H + W − h couplings are calculated as 2 16π 2  1PI H + W −h ( p )F   m u ζhu F F F m d ζhd F F F = −Nc g C H + W − φ (u, u, d) + C H + W − φ (d, d, u) v v

m  ζh F F F (D.93) C H + W − φ (, , ν ), v 2 1PI 2 16π  H + W − h ( p ) B   g3 SV V SV V ± 2 SV V ± = − cβ−α C H + W − φ (A, Z , W ) + c2W C H + W − φ (H , W, Z ) + 2s W C H + W − φ (H , W, γ) 8 −g

Appendix D: One Particle Irreducible Diagrams in the Two-Higgs Doublet Model

245

 g3 2 V SS ± 2 V SS ± CH cβ−α cβ−α + W − φ (W, H , h) + sβ−α C H + W − φ (W, G , h) 8 2 V SS ± 2 V SS ± + sβ−α CH + W − φ (W, H , H ) − sβ−α C H + W − φ (W, G , H )  c 2W V SS V SS ± C H + W − φ (Z , A, H ± ) − CH + W − φ (W, H , A) − 2 cW  s2 g3 c (B0 − B1 )(q 2 ; H ± , Z ) + cβ−α (B0 − B1 )(q 2 ; h, W ) − W 2 2W 8 cW

−

2 + 2sW (B0 − B1 )(q 2 ; H ± , γ) + (B0 − B1 )( p12 ; H ± , W ) +

2 sW 2 cW



(B0 − B1 )( p12 ; A, Z )

  g3 2 2 m W sβ−α cβ−α (C11 + 2C0 )(W, W, h) − (C11 + 2C0 )(W, W, H ) 4  s2 g2 SV S ± − mW i W λ H + G − A cβ−α C H + W − φ (A, Z , G ) 2 4 cW +

2 SV S ± SV S ± CH + λ H + G − h sβ−α + W − φ (G , W, h) + λ H + H − h sβ−α cβ−α C H + W − φ (H , W, h) SV S ± 2 SV S ± + λ H + G − H sβ−α cβ−α C H + W − φ (G , W, H ) + λ H + H − H cβ−α C H + W − φ (H , W, H )



 g2 SSV 2 SSV m W 6λhhh sβ−α cβ−α C H + W − φ (h, h, W ) − 2λ H hh sβ−α C H + W − φ (H, h, W ) 4 SSV 2 SSV − 2λ H H h sβ−α cβ−α C H + W − φ (H, H, W ) + 2λ H hh cβ−α C H + W − φ (h, H, W )

−

−

2 sW

c λ + − C SSV (H ± , G ∓ , 2 2W H G h H + W − φ cW

2 SSV ± ∓ Z ) + 2sW λH + G− h C H + W − φ (H , G , γ)



g SSS ± SSS ± − 6λhhh λ H + H − h cβ−α C H + W − φ (h, h, H ) − 2λ H hh λ H + H − H cβ−α C H + W − φ (H, h, H ) 2 SSS ± SSS ± + 2λ H hh λ H + H − h sβ−α C H + W − φ (h, H, H ) + 2λ H H h λ H + H − H sβ−α C H + W − φ (H, H, H ) −

SSS ± SSS ± − 6λhhh λ H + G − h sβ−α C H + W − φ (h, h, G ) − 2λ H hh λ H + G − H sβ−α C H + W − φ (H, h, G ) SSS ± SSS ± − 2λ H hh λ H + G − h cβ−α C H + W − φ (h, H, G ) − 2λ H H h λ H + G − H cβ−α C H + W − φ (H, H, G ) SSS 0 ± − 2λ AG − H + λhG 0 A C H + W − φ (A, G , G ) SSS ± ± SSS ± ± + λ2H + H − h cβ−α C H + W − φ (H , H , h) − λ H + H − H λ H + H − h sβ−α C H + W − φ (H , H , H ) SSS ± ± SSS ± ± + λ2H + G − h cβ−α C H + W − φ (G , H , h) − λ H + G − H λ H + G − h sβ−α C H + W − φ (G , H , H ) SSS ± ± − iλ H + G − A λ H + G − h C H + W − φ (G , H , A) SSS ± ± + λ H + G − h λ H + H − h sβ−α C H + W − φ (H , G , h) SSS ± ± + λ H + G − h λ H + H − H cβ−α C H + W − φ (H , G , H ) SSS ± ± + λG + G − h λ H + G − h sβ−α C H + W − φ (G , G , h)

 SSS ± ± + λG + G − h λ H + G − H cβ−α C H + W − φ (G , G , H ) ,

(D.94)

where we defined m ζ f f [B0 ( p22 ; f f  ) + m 2f (2C11 + 3C0 ) + p12 (C11 − C12 ) v  m f ζ f  + q 2 C12 ] − m f m f  (2C11 + C0 ) ( f, f, f  ), v

F  C HF +F W − φ ( f, f, f ) =

(D.95)

246

Appendix D: One Particle Irreducible Diagrams in the Two-Higgs Doublet Model

SV V 2 2 2 2 2 2 CH + W − φ (X, Y, Z ) = 4B0 ( p2 ; Y, Z ) − 4C 24 − (3 p1 − p2 + q )C 21 − 2(q − p1 )C 23 + (q 2 − p12 − 2 p22 )C11 − 2(q 2 − p12 )C12

+ ( p22 + 4m 2X )C0 (X, Y, Z ), V SS 2 2 2 2 2 CH + W − φ (X, Y, Z ) = 4C 24 + (3 p1 − p2 + q )C 21 + 2(q − p1 )C 23

(D.96)

+ (5 p12 − 3 p22 + 3q 2 + m 2X )C11

+ 2(q 2 − p12 )C12 + (2 p12 − 2 p22 + 2q 2 + m 2X )C0 (X, Y, Z ), SV S CH + W − φ (X, Y, SSS C H + W − φ (X, Y,

Z) =

SSV CH + W − φ (X, Y,

(D.97)

Z ) = [C0 − C11 ](X, Y, Z ),

(D.98)

Z ) = [C0 + C11 ](X, Y, Z ).

(D.99)

The 1PI diagrams for the H + W − H couplings are calculated as 2 16π 2  1PI H + W − H ( p )F   u d  mu ζH md ζH mζH F FFF F = −Nc g (u, u, d) + (d, d, u) −g C HF +F W C C HF +F W −φ + W −φ − φ (, , ν ), H v v v

(D.100) 2 16π 2  1PI H + W − H ( p )B  g3 SV V = sβ−α C H + W − φ (A, 8

SV V ± 2 SV V ± Z , W ) + c2W C H + W − φ (H , W, Z ) + 2s W C H + W − φ (H , W, γ)

 g3 2 V SS ± 2 V SS ± CH sβ−α cβ−α + W − φ (W, H , h) − cβ−α C H + W − φ (W, G , h) 8 2 V SS ± 2 V SS ± CH + sβ−α + W − φ (W, H , H ) + cβ−α C H + W − φ (W, G , H )  c2W V SS V SS ± C H + W − φ (Z , A, H ± ) − CH + W − φ (W, H , A) − 2 cW  s2 g3 c (B0 − B1 )(q 2 ; H ± , Z ) − sβ−α (B0 − B1 )(q 2 ; H, W ) − W 2 2W 8 cW



+

2 (B0 − B1 )(q 2 ; H ± , γ) + (B0 − B1 )( p12 ; H ± , W ) + + 2sW

2 sW 2 cW



(B0 − B1 )( p12 ; A, Z )

  g3 2 2 m W cβ−α sβ−α (C11 + 2C0 )(W, W, h) − (C11 + 2C0 )(W, W, H ) 4  s2 g2 SV S ± λ H + G − A sβ−α C H − mW − i W + W − φ (A, Z , G ) 2 4 cW +

SV S ± 2 SV S ± + λ H + G − h sβ−α cβ−α C H + W − φ (G , W, h) − λ H + H − h sβ−α C H + W − φ (H , W, h) 2 SV S ± SV S ± + λ H + G − H cβ−α CH + W − φ (G , W, H ) − λ H + H − H sβ−α cβ−α C H + W − φ (H , W, H )



 g2 SSV 2 SSV m W 2λ H hh sβ−α cβ−α C H + W − φ (h, h, W ) − 2λ H H h sβ−α C H + W − φ (H, h, W ) 4 SSV 2 SSV − 6λ H H H sβ−α cβ−α C H + W − φ (H, H, W ) + 2λ H H h cβ−α C H + W − φ (h, H, W )

−

− −

2 sW

c λ + − C SSV (H ± , G ∓ , 2 2W H G H H + W − φ cW



2 SSV ± ∓ Z ) + 2sW λH + G− H C H + W − φ (H , G , γ)

g SSS ± SSS ± − 2λ H hh λ H + H − h cβ−α C H + W − φ (h, h, H ) − 2λ H H h λ H + H − H cβ−α C H + W − φ (H, h, H ) 2

Appendix D: One Particle Irreducible Diagrams in the Two-Higgs Doublet Model

247

SSS ± SSS ± + 2λ H H h λ H + H − h sβ−α C H + W − φ (h, H, H ) + 6λ H H H λ H + H − H sβ−α C H + W − φ (H, H, H ) SSS ± SSS ± − 2λ H hh λ H + G − h sβ−α C H + W − φ (h, h, G ) − 2λ H H h λ H + G − H sβ−α C H + W − φ (H, h, G ) SSS ± SSS ± − 2λ H H h λ H + G − h cβ−α C H + W − φ (h, H, G ) − 6λ H H H λ H + G − H cβ−α C H + W − φ (H, H, G ) SSS ± ± 2 SSS ± ± + λ H + H − h λ H + H − H cβ−α C H + W − φ (H , H , h) − λ H + H − H sβ−α C H + W − φ (H , H , H ) SSS ± ± 2 SSS ± ± + λ H + G − h λ H + G − H cβ−α C H + W − φ (G , H , h) − λ H + G − H sβ−α C H + W − φ (G , H , H ) SSS 0 ± − 2λ AG − H + λ H G 0 A C H + W − φ (A, G , G ) SSS ± ± − iλ H + G − A λ H + G − H C H + W − φ (G , H , A) SSS ± ± + λ H + G − H λ H + H − h sβ−α C H + W − φ (H , G , h) SSS ± ± + λ H + G − H λ H + H − H cβ−α C H + W − φ (H , G , H ) SSS ± ± + λG + G − H λ H + G − h sβ−α C H + W − φ (G , G , h)

 SSS ± ± + λG + G − H λ H + G − H cβ−α C H + W − φ (G , G , H ) . +

(D.101)

−

The 1PI diagrams for the H W A couplings are calculated as 2 16π 2  1PI H + W − A ( p )F   m u ζu F F F m d ζd F F F m  ζ F F F = i Nc g C H + W − φ (u, u, d) + C H + W − φ (d, d, u) + ig C H + W − φ (, , ν ), v v v

(D.102) 2 16π 2  1PI H + W − A( p )B  3 g c2 C SV+V − (h, =i 8 β−α H W φ

2 SV V Z , W ) + sβ−α CH + W − φ (H, Z , W )  SV V ± 2 SV V ± + c2W C H + W − φ (H , W, Z ) + 2s W C H + W − φ (H , W, γ)

g3  2 2 V SS ± V SS ± C V +SS − (W, H ± , h) + sβ−α CH c + W − φ (W, H , H ) − C H + W − φ (W, H , A) 8 β−α H W φ  c2W 2 c2W 2 V SS ± V SS ± + 2 cβ−α CH sβ−α C H + W − φ (Z , h, H ) + + W − φ (Z , H, H ) 2 cW cW 2  3 s g 2 c (B0 − B1 )(q 2 ; H ± , Z ) + 2sW (B0 − B1 )(q 2 ; H ± , γ) (B0 − B1 )(q 2 ; A, W ) − W −i 2 2W 8 cW 2  s2 2 sW 2 2 2 + (B0 − B1 )( p12 ; H ± , W ) + W c (B − B )( p ; h, Z ) + s (B − B )( p ; H, Z ) 0 1 0 1 1 1 β−α β−α 2 2 cW cW 2 2  2 s sW g SV S ± SV S ± λ H + G − h cβ−α C H + i mW W + W − φ (h, Z , G ) − 2 λ H + G − H sβ−α C H + W − φ (H, Z , G ) 2 4 cW cW  SV S ± SV S ± + λ H + H − h sβ−α C H + W − φ (H , W, h) + λ H + H − H cβ−α C H + W − φ (H , W, H ) −i

 g2 SSV SSV m W 2λ A Ah sβ−α C H + W − φ (A, h, W ) + 2λ A AH cβ−α C H + W − φ (A, H, W ) 4  s2 SSV ± ∓ 2 SSV ± ∓ −i W c2W λ H + G − A C H + W − φ (H , G , Z ) + 2is W λ H + G − A C H + W − φ (H , G , γ) 2 cW g SSS ± SSS ± − i 2λh A A λ H + H − h C H + W − φ (h, A, H ) + 2λ H A A λ H + H − H C H + W − φ (H, A, H ) 2 SSS ± SSS ± + 2iλ A Ah λ H + G − A sβ−α C H + W − φ (A, h, G ) + 2iλ A AH λ H + G − A cβ−α C H + W − φ (A, H, G ) +i

248

Appendix D: One Particle Irreducible Diagrams in the Two-Higgs Doublet Model SSS ± ± − iλ H − G + A λ H + G − h cβ−α C H + W − φ (G , H , h)

SSS ± ± + iλ H − G + A λ H + G − H sβ−α C H + W − φ (G , H , H ) SSS ± ± − λH − G+ A λH + G− A C H + W − φ (G , H , A) SSS 0 ± SSS 0 ± + λ AGh λ H + G − h C H + W − φ (h, G , G ) + λ AG H λ H + G − H C H + W − φ (H, G , G ) SSS ± ± − iλ H + G − A λ H + H − h sβ−α C H + W − φ (H , G , h)

 SSS ± ± − iλ H + G − A λ H + H − H cβ−α C H + W − φ (H , G , H ) .

D.3.7

(D.103)

H + V W − Vertex

The 1PI diagrams for the H + Z W − couplings are calculated as 16π 2  1,1PI H + Z W − ,F   1 f gg Z − m 2f ζ f (v f  + a f  ) B0 ( p22 ; f  , f  ) − 2C24 + (q 2 + p12 − p22 + 2m 2f )C0 = 4I f Nc v 2 f

1 2 (q + 3 p12 − p22 )C11 + (q 2 − p12 )C12 + m 2f m 2f  ζ f (v f  − a f  )C0 2

1 + m 2f  ζ f  (v f  + a f  ) B0 ( p22 ; f  , f  ) − 2C24 + m 2f C0 + p12 C11 + (q 2 − p12 − p22 )C12 2 1 2 2 2   2 2 2 − m f  ζ f  (v f  − a f  ) B0 ( p2 ; f , f ) + m f C0 + (q + p1 − p2 )C11 2

 1 2 2 2    + (q − p1 + p2 )C12 ( f, f , f ) + ( f ↔ f ), (D.104) 2 +

16π 2  2,1PI H + Z W − ,F     f gg Z m 2f ζ f (v f  + a f  ) C11 + 2C12 + C0 + 2C23 = 4I f Nc v f

    − m 2f  ζ f  (v f  + a f  ) C12 + 2C23 + m 2f  ζ f  (v f  − a f  ) C11 − C12 ( f, f  , f  ) + ( f ↔ f  ), 16π 2  3,1PI H + Z W − ,F =

 f

f

2Nc

gg Z v

(D.105) 

  − m 2f ζ f (v f  + a f  ) C0 + C11 + m 2f  ζ f  (v f  + a f  )C12

  + m 2f  ζ f  (v f  − a f  ) C11 − C12 ( f, f  , f  ) + ( f ↔ f  ), 16π 2  1,1PI H + Z W − ,B gg Z = λ H + H − h vcβ−α C24 (H ± , A, h) − λ H + H − H vsβ−α C24 (H ± , A, H ) v − λ H + H − h vcβ−α c2W C24 (h, H ± , H ± ) + λ H + H − H vsβ−α c2W C24 (H, H ± , H ± )

(D.106)

Appendix D: One Particle Irreducible Diagrams in the Two-Higgs Doublet Model

249

− sβ−α cβ−α (m 2H ± − m 2A )C24 (G ± , h, A) + sβ−α cβ−α (m 2H ± − m 2A )C24 (G ± , H, A) + sβ−α cβ−α (m 2H ± − m 2h )C24 (G ± , G 0 , h) − sβ−α cβ−α (m 2H ± − m 2H )C24 (G ± , G 0 , H ) − sβ−α cβ−α (m 2H ± − m 2h )c2W C24 (h, G ± , G ± ) + sβ−α cβ−α (m 2H ± − m 2H )c2W C24 (H, G ± , G ± ) − m 2W sβ−α cβ−α C24 (W ± , h, A) + m 2W sβ−α cβ−α C24 (W ± , H, A) + m 2Z c2W sβ−α cβ−α C24 (W ± , h, Z ) − m 2Z c2W sβ−α cβ−α C24 (W ± , H, Z ) 2 2 + m 2W sW sβ−α cβ−α C24 (h, G ± , W ± ) − m 2W sW sβ−α cβ−α C24 (H, G ± , W ± ) 2 (m 2H ± − m 2h )sβ−α cβ−α C0 (h, W ± , G ± ) − m 2W sW 2 + m 2W sW (m 2H ± − m 2H )sβ−α cβ−α C0 (H, W ± , G ± ) 2 − m 2Z sW (m 2H ± − m 2h )sβ−α cβ−α C0 (G ± , Z , h) 2 + m 2Z sW (m 2H ± − m 2H )sβ−α cβ−α C0 (G ± , Z , H )

+ m 2W sβ−α cβ−α FV V S (W ± , Z , h) − m 2W sβ−α cβ−α FV V S (W ± , Z , H ) 2 2 − m 2W cW sβ−α cβ−α FSV V (h, W ± , W ± ) + m 2W cW sβ−α cβ−α FSV V (H, W ± , W ± ) 1 2 1 2 λ H + H − h vcβ−α B0 (q 2 ; h, H ± ) + sW λ H + H − H vsβ−α B0 (q 2 ; H, H ± ) − sW 2 2 1 2 1 2 sβ−α cβ−α (m 2H ± − m 2h )B0 (q 2 ; h, G ± ) + sW sβ−α cβ−α (m 2H ± − m 2H )B0 (q 2 ; H, G ± ) − sW 2 2 2 2 + m 2W sW sβ−α cβ−α B0 ( p12 ; W ± , h) − m 2W sW sβ−α cβ−α B0 ( p12 ; W ± , H )

2 2 + m 2Z sW sβ−α cβ−α B0 ( p22 ; Z , h) − m 2Z sW sβ−α cβ−α B0 ( p22 ; Z , H ) , (D.107)

16π 2  2,1PI H + Z W − ,B gg Z λ H + H − h vcβ−α (C12 + C23 )(H ± , A, h) − λ H + H − H vsβ−α (C12 + C23 )(H ± , A, H ) = v − λ H + H − h vcβ−α c2W (C12 + C23 )(h, H ± , H ± ) + λ H + H − H vsβ−α c2W (C12 + C23 )(H, H ± , H ± ) − sβ−α cβ−α (m 2H ± − m 2A )(C12 + C23 )(G ± , h, A) + sβ−α cβ−α (m 2H ± − m 2A )(C12 + C23 )(G ± , H, A) + sβ−α cβ−α (m 2H ± − m 2h )(C12 + C23 )(G ± , G 0 , h) − sβ−α cβ−α (m 2H ± − m 2H )(C12 + C23 )(G ± , G 0 , H ) − sβ−α cβ−α (m 2H ± − m 2h )c2W (C12 + C23 )(h, G ± , G ± ) + sβ−α cβ−α (m 2H ± − m 2H )c2W (C12 + C23 )(H, G ± , G ± ) − m 2W sβ−α cβ−α (2C0 + 2C11 + C12 + C23 )(W ± , h, A) + m 2W sβ−α cβ−α (2C0 + 2C11 + C12 + C23 )(W ± , H, A) + m 2Z c2W sβ−α cβ−α (C23 − C12 )(H ± , h, Z ) − m 2Z c2W sβ−α cβ−α (C23 − C12 )(H ± , H, Z ) 2 2 + m 2W sW sβ−α cβ−α (C23 − C12 )(h, G ± , W ± ) − m 2W sW sβ−α cβ−α (C23 − C12 )(H, G ± , W ± )

+ m 2W sβ−α cβ−α (2C0 − 2C11 + 5C12 + C23 )(W ± , Z , h) − m 2W sβ−α cβ−α (2C0 − 2C11 + 5C12 + C23 )(W ± , Z , H ) 2 − m 2W cW sβ−α cβ−α (C23 + 3C12 − 4C11 )(h, W ± , W ± )

2 + m 2W cW sβ−α cβ−α (C23 + 3C12 − 4C11 )(H, W ± , W ± ) ,

(D.108)

250

Appendix D: One Particle Irreducible Diagrams in the Two-Higgs Doublet Model

16π 2  3,1PI = 0, H + Z W − ,B

(D.109)

where FV V S (V1 , V2 , S) = −B0 ( p22 ; V2 , S) + C24 − (q 2 + 3 p12 − p22 )C11 − 2(q 2 − p12 )C12 − (2q 2 + 2 p12 − 2 p22 + m 2V1 )C0 , FSV V (S, V1 , V2 )

= −B0 ( p22 ; V1 , V2 ) + C24 + ( p12 − p22 − m 2S )C0 .

+ (q − 2

(D.110) p12

−

p22 )C11

+

2 p22 C12 (D.111)

The 1PI diagrams for the H + γW − couplings are calculated as 16π 2  1,1PI H + γW − ,F =



f

4I f Nc Q f 

f

eg v



1 − m 2f ζ f B0 ( p22 ; f  , f  ) − 2C24 + (q 2 + p12 − p22 + 2m 2f − 2m 2f  )C0 2

1 2 1 (q + 3 p12 − p22 )C11 + (q 2 − p12 )C12 2 2

 1 2 2 + m f  ζ f  − 2C24 − (q − p12 − p22 )C11 − p22 C12 ( f, f  , f  ) + ( f ↔ f  ), 2

+

(D.112)

16π 2  2,1PI H + γW − ,F

   eg f m 2f ζ f C11 + 2C12 + C0 + 2C23 4I f Nc Q f  v f   + m 2f  ζ f  C11 − 2C12 − 2C23 ( f, f  , f  ) + ( f ↔ f  ),

=



16π 2  3,1PI H + γW − ,F =

 f

eg f 2Nc Q f  v



   − m 2f ζ f C0 + C11 + m 2f  ζ f  C11 ( f, f  , f  ) + ( f ↔ f  ),

(D.113)

(D.114)

16π 2  1,1PI H + γW − ,B 2eg − λ H + H − h vcβ−α C24 (h, H ± , H ± ) + λ H + H − H vsβ−α C24 (H, H ± , H ± ) = v − sβ−α cβ−α (m 2H ± − m 2h )C24 (h, G ± , G ± ) + sβ−α cβ−α (m 2H ± − m 2H )C24 (H, G ± , G ± ) − + − −

m 2W 2 m 2W 2 m 2W 2 m 2W 2

sβ−α cβ−α C24 (h, G ± , W ± ) +

m 2W 2

sβ−α cβ−α C24 (H, G ± , W ± )

(m 2H ± − m 2h )sβ−α cβ−α C0 (h, W ± , G ± ) (m 2H ± − m 2H )sβ−α cβ−α C0 (H, W ± , G ± ) sβ−α cβ−α FSV V (h, W ± , W ± ) +

m 2W 2

sβ−α cβ−α FSV V (H, W ± , W ± )

1 1 B0 (q 2 ; h, H ± ) − λ H + H − H vsβ−α B0 (q 2 ; H, H ± ) λ + − vc 4 H H h β−α 4 1 1 2 2 2 + sβ−α cβ−α (m H ± − m h )B0 (q ; h, G ± ) − sβ−α cβ−α (m 2H ± − m 2H )B0 (q 2 ; H, G ± ) 4 4

+

Appendix D: One Particle Irreducible Diagrams in the Two-Higgs Doublet Model −

m 2W 2

sβ−α cβ−α B0 ( p12 ; W ± , h) +

m 2W 2

sβ−α cβ−α B0 ( p12 ; W ± , H ) ,

251

(D.115)

16π 2  2,1PI H + γW − ,B 2eg − λ H + H − h vcβ−α (C12 + C23 )(h, H ± , H ± ) = v + λ H + H − H vsβ−α (C12 + C23 )(H, H ± , H ± ) − sβ−α cβ−α (m 2H ± − m 2h )(C12 + C23 )(h, G ± , G ± ) + sβ−α cβ−α (m 2H ± − m 2H )(C12 + C23 )(H, G ± , G ± ) m2 s c (C23 − C12 )(h, G ± , W ± ) + W sβ−α cβ−α (C23 − C12 )(H, G ± , W ± ) 2 β−α β−α 2 m 2W + s c (4C11 − 3C12 − C23 )(h, W ± , W ± ) 2 β−α β−α

m2 − W sβ−α cβ−α (4C11 − 3C12 − C23 )(H, W ± , W ± ) , (D.116) 2

−

m 2W

16π 2  3,1PI = 0. H + γW − ,B

D.3.8

(D.117)

A f f¯ Vertex

The 1PI contributions to the form factor of A f f¯ vertex are given by 16π 2  AS,1PI f f¯ m f m 2f 

  (ζ f + ζ f  ) C0 (H ± , f  , G ± ) − C0 (G ± , f  , H ± ) v2

g2 m f ζh f f S F V C A f f¯ (h, f, Z ) − C AV fFfS¯ (Z , f, h) + i Z cβ−α v f 2 v

g2 m f ζH f f S F V C A f f¯ (H, f, Z ) − C AV fFfS¯ (Z , f, H ) − i Z sβ−α v f 2 v

g2 I f m f ζ f S F V ±  C A f f¯ (H , f , W ± ) − C AV fFfS¯ (W ± , f  , H ± ) , +i 2 v 16π 2  AP,1PI f f¯

2I f m f ζ f 2 2 g Z (v f − a 2f )C AF fV fF¯ ( f, Z , f ) + e2 Q 2f C AF fV fF¯ ( f, γ, f ) =i v  2I f m 3f ζ f 2 2 FSF ζh f f C AF fS Ff¯ ( f, h, f ) + ζ H +i f f C A f f¯ ( f, H, f ) v2  − C AF fS Ff¯ ( f, G 0 , f ) − ζ 2f C AF fS Ff¯ ( f, A, f ) = 2I f λ H + G − A

4I f  m f m 2f  ζ f  C AF fS Ff¯ ( f  , G ± , f  ) + ζ f ζ f  C AF fS Ff¯ ( f  , H ± , f  ) v3 2

m f ζh f f 2I f m f C0 (h, f, G 0 ) + C0 (G 0 , f, h) + iλ AG 0 h v v −i

(D.118)

252

Appendix D: One Particle Irreducible Diagrams in the Two-Higgs Doublet Model 2

m f ζ H f f 2I f m f C0 (H, f, G 0 ) + C0 (G 0 , f, H ) v v 2 m f ζh f f 2I f m f ζ f + i2λ A Ah [C0 (h, f, A) + C0 (A, f, h)] v v 2 m f ζ H f f 2I f m f ζ f + i2λ A AH [C0 (H, f, A) + C0 (A, f, H )] v v 2 m f m f   (ζ f − ζ f  ) C0 (H ± , f  , G ± ) + C0 (G ± , f  , H ± ) + 2I f λ H + G − A v2

g2 m f ζh f f S F V C A f f¯ (h, f, Z ) + C AV fFfS¯ (Z , f, h) + i Z cβ−α a f 2 v

g2 m f ζH f f S F V C A f f¯ (H, f, Z ) + C AV fFfS¯ (Z , f, H ) − i Z sβ−α a f 2 v

g2 I f m f ζ f S F V ±  C A f f¯ (H , f , W ± ) + C AV fFfS¯ (W ± , f  , H ± ) , +i 2 v

+ iλ AG 0 H

16π

2

(D.119)

 VA1f,1PI f¯

= i4g 2Z v f a f +i

2 g 2 2I f  m f  ζ f  C0 ( f  , W, f  ) 2 v  m 2f − m 2f  m 2f ζ 2f − m 2f  ζ 2f   ±   ±  C ( f , G , f ) + C ( f , H , f ) 0 0 v2 v2

2I f m 2f ζ f

2I f  m 2f  ζ f 

v 

v

C0 ( f, Z , f ) + i

m 2f ζ f + m 2f  ζ f    (C0 + C11 )(H ± , f  , G ± ) − (C0 + C11 )(G ± , f  , H ± ) v2 m 2f ζh f f g2 + i Z cβ−α v f [(C11 − C0 )(h, f, Z ) − (C11 + 2C0 )(Z , f, h)] 2 v 2 2 m f ζH f f g − i Z sβ−α v f [(C11 − C0 )(H, f, Z ) − (C11 + 2C0 )(Z , f, H )] 2 v 2  g2 I f  m f  ζ f   +i (C11 − C0 )(H ± , f  , W ± ) − (C11 + 2C0 )(W ± , f  , H ± ) , 2 v

− 2I f λ H + G − A

(D.120)

16π 2  VA2f,1PI f¯

= i4g 2Z v f a f +i

2 g 2 2I f  m f  ζ f  C0 ( f  , W, f  ) v 2 v  2  m f − m 2f  m 2f ζ 2f − m 2f  ζ 2f   ±   ±  C ( f , G , f ) + C ( f , H , f ) 0 0 v2 v2

2I f m 2f ζ f

2I f  m 2f  ζ f  v

C0 ( f, Z , f ) + i

m 2f ζ f + m 2f  ζ f    C12 (H ± , f  , G ± ) − C12 (G ± , f  , H ± ) v2 m 2f ζh f f g2 + i Z cβ−α v f [(C12 − C0 )(h, f, Z ) − (C12 + 2C0 )(Z , f, h)] 2 v 2 2 m f ζH f f g − i Z sβ−α v f [(C12 − C0 )(H, f, Z ) − (C12 + 2C0 )(Z , f, H )] 2 v 2 2  g I fm fζf  (C12 − C0 )(H ± , f  , W ± ) − (C12 + 2C0 )(W ± , f  , H ± ) , −i 2 v − 2I f λ H + G − A

(D.121)

16π 2  AA1f ,1PI f¯ = −i

2I f m 2f ζ f v

2

g 2 2I f  m f  ζ f  2g 2Z (v 2f + a 2f )C0 ( f, Z , f ) + 2e2 Q 2f C0 ( f, γ, f ) − i C0 ( f  , W, f  ) 2 v

Appendix D: One Particle Irreducible Diagrams in the Two-Higgs Doublet Model +i +i

2I f m 4f ζ f v3

253

  2 0 2 C ( f, H, f ) + C ( f, G , f ) + ζ C ( f, A, f ) ζh2 f f C0 ( f, h, f ) + ζ H 0 0 0 ff f

2I f  m 2f  ζ f  v



m 2f + m 2f  v2

C0 ( f  , G ± , f  ) +

m 2f ζ 2f + m 2f  ζ 2f 

 C0 ( f  , H ± , f  )

v2

m f ζh f f 2I f m f (C0 + C11 )(h, f, G 0 ) − (C0 + C11 )(G 0 , f, h) − iλ AG 0 h v v

m f ζ H f f 2I f m f (C0 + C11 )(H, f, G 0 ) − (C0 + C11 )(G 0 , f, H ) − iλ AG 0 H v v m f ζh f f 2I f m f ζ f − i2λ A Ah [(C0 + C11 )(h, f, A) − (C0 + C11 )(A, f, h)] v v m f ζ H f f 2I f m f ζ f − i2λ A AH [(C0 + C11 )(H, f, A) − (C0 + C11 )(A, f, H )] v v 2 2 m f ζf − m fζ f   − 2I f λ H + G − A (C0 + C11 )(H ± , f  , G ± ) − (C0 + C11 )(G ± , f  , H ± ) v2 m 2f ζh f f g2 + i Z cβ−α a f [(−C11 + C0 )(h, f, Z ) + (C11 + 2C0 )(Z , f, h)] 2 v 2 2 m f ζH f f g − i Z sβ−α a f [(−C11 + C0 )(H, f, Z ) + (C11 + 2C0 )(Z , f, H )] 2 v 2  g2 I f  m f  ζ f   +i (−C11 + C0 )(H ± , f  , W ± ) + (C11 + 2C0 )(W ± , f  , H ± ) , 2 v

(D.122)

16π 2  AA2f ,1PI f¯ = −i

2I f m 2f ζ f

2g 2Z (v 2f + a 2f )C0 ( f, Z , f ) + 2e2 Q 2f C0 ( f, γ, f )

v 2 g 2 2I f  m f  ζ f  C0 ( f  , W, f  ) −i 2 v   2I f m 4f ζ f 2 2 0 2 ζ C ( f, h, f ) + ζ C ( f, H, f ) + C ( f, G , f ) + ζ C ( f, A, f ) +i 0 0 0 0 h f f H f f f v3   m 2f ζ 2f + m 2f  ζ 2f  2I f  m 2f  ζ f  m 2f + m 2f   ±   ±  C ( f , G , f ) + C ( f , H , f ) +i 0 0 v v2 v2

m f ζh f f 2I f m f C12 (h, f, G 0 ) − C12 (G 0 , f, h) − iλ AG 0 h v v

m f ζ H f f 2I f m f C12 (H, f, G 0 ) − C12 (G 0 , f, H ) − iλ AG 0 H v v m f ζh f f 2I f m f ζ f − i2λ A Ah [C12 (h, f, A) − C12 (A, f, h)] v v m f ζ H f f 2I f m f ζ f − i2λ A AH [C12 (H, f, A) − C12 (A, f, H )] v v 2 2 m f ζf − m fζ f   C12 (H ± , f  , G ± ) − C12 (G ± , f  , H ± ) − 2I f λ H + G − A v2 m 2f ζh f f g2 + i Z cβ−α a f [(−C12 + C0 )(h, f, Z ) + (C12 + 2C0 )(Z , f, h)] 2 v m 2f ζ H f f g2 − i Z sβ−α a f [(−C12 + C0 )(H, f, Z ) + (C12 + 2C0 )(Z , f, H )] 2 v 2  g2 I f  m f  ζ f   (−C12 + C0 )(H ± , f  , W ± ) + (C12 + 2C0 )(W ± , f  , H ± ) , −i 2 v 16π 2  T,1PI A f f¯

(D.123)

254

Appendix D: One Particle Irreducible Diagrams in the Two-Higgs Doublet Model g 2Z m f ζh f f cβ−α v f [(C0 + C11 − 2C12 )(h, f, Z ) − (2C0 + 2C11 − C12 )(Z , f, h)] 2 v 2 g m f ζH f f − i Z sβ−α v f [(C0 + C11 − 2C12 )(H, f, Z ) − (2C0 + 2C11 − C12 )(Z , f, H )] 2 v  g2 I f m f ζ f  (C0 + C11 − 2C12 )(H ± , f  , W ± ) − (2C0 + 2C11 − C12 )(W ± , f  , H ± ) , +i 2 v

=i

(D.124)

16π

2

 AP fT,1PI f¯

=i

2I f m 3f ζ f v3

 2 ζh2 f f (−C11 + C12 )( f, h, f ) + ζ H f f (−C 11 + C 12 )( f, H, f )

 − (−C11 + C12 )( f, G 0 , f ) − ζ 2f (−C11 + C12 )( f, A, f ) −i

 4I f  m f m 2f  ζ f   (−C11 + C12 )( f  , G ± , f  ) + ζ f ζ f  (−C11 + C12 )( f  , H ± , f  ) 3 v

g 2Z m f ζh f f cβ−α a f [(C0 + C11 − 2C12 )(h, f, Z ) + (2C0 + 2C11 − C12 )(Z , f, h)] 2 v g2 m f ζH f f − i Z sβ−α a f [(C0 + C11 − 2C12 )(H, f, Z ) + (2C0 + 2C11 − C12 )(Z , f, H )] 2 v 2 g If m f ζf [(C0 + C11 − 2C12 )(H ± , f  , W ± ) + (2C0 + 2C11 − C12 )(W ± , f  , H ± )]. +i 2 v

+i

(D.125)

where C AS Ff Vf¯ (X, Y, Z ) = B0 ( p22 ; Y, Z )   + (m 2X − q 2 + p22 )C0 − (q 2 − p12 − p22 )C11 + (q 2 − p12 − 2 p22 )C12 ( p12 , p22 , q 2 ; X, Y, Z ),

(D.126) C AV fFfS¯ (X, Y, 

Z) =

B0 ( p22 ; Y,

Z)

 + (m 2X + 2 p12 )C0 + 3 p12 C11 + 2(q 2 − p12 )C12 ( p12 , p22 , q 2 ; X, Y, Z ),

(D.127)

C AF fV fF¯ (X, Y, Z ) = 4B0 ( p22 ; Y, Z ) − 2   + 4m X (m X − m Z )C0 + 2(q 2 + p12 − p22 )C11 + 2(q 2 − p12 + p22 )C12 ( p12 , p22 , q 2 ; X, Y, Z ),

(D.128) C AF fS Ff¯ (X, Y,

Z) =

B0 ( p22 ; Y,

Z)

 + m X (m X − m Z )C0 + (q 2 − p22 )C11 + p22 C12 ( p12 , p22 , q 2 ; X, Y, Z ). 

(D.129)

The loop functions satisfy the following relations C AS Ff Vf¯ (X, Y, Z ) = C hSfFfV¯ (X, Y, Z ),

(D.130)

C AV fFfS¯ (X, Y, C AF fV fF¯ (X, Y,

(D.131)

Z) = Z) =

C hVfFf¯S (X, Y, Z ), C−F V F (X, Y, Z ),

(D.132)

Appendix D: One Particle Irreducible Diagrams in the Two-Higgs Doublet Model

C AF fS Ff¯ (X, Y, Z ) = C PF S F (X, Y, Z ),

255

(D.133)

where C hSfFfV¯ and C hVfFf¯S are given in Eq. (C.44) in Ref. [8], while C−F V F and C PF S F are given in Eqs. (B.23) and (B.22) in Ref. [9].

D.3.9

AV φ Vertex

The 1PI contributions to the form factor of AZ h vertex are given by 2 2 2 16π 2  1PI AZ h ( p1 , p2 , q ) F

=i



f

8I f Nc g Z a f

m 2f ζ f ζh f f 

f

v2

 B0 ( p22 ; f, f ) + 2m 2f C0 + p12 C11 + (q 2 − p12 )C12 ( f, f, f ),

(D.134) 2 2 2 16π 2  1PI AZ h ( p1 , p2 , q ) B

g3 SV V (H ± , W ± , W ± ) c cW C AZ φ 2 β−α 3 g 2 SS 2 V SS + i Z cβ−α cβ−α CV AZ φ (Z , A, h) + sβ−α C AZ φ (Z , A, H ) 8

=i

2 SS 0 2 V SS 0 2 2 2 V SS ± ± ± CV + sβ−α AZ φ (Z , G , h) − sβ−α C AZ φ (Z , G , H ) − 2cW (cW − sW )C AZ φ (W , H , H ) g3 2 c2 (B − B )(q 2 ; H ± , W ± ) − i Z cβ−α (B0 − B1 )(q 2 ; h, Z ) + 2sW 1 W 0 8

2 c2 (B − B )( p 2 ; H ± , W ± ) + (B0 − B1 )( p12 ; A, Z ) + 2sW 1 1 W 0

g3 m 2 2 − i Z Z sβ−α cβ−α (2C0 + C11 )(Z , Z , h) − (2C0 + C11 )(Z , Z , H ) 4 g 2Z m Z 2 6λhhh sβ−α cβ−α (C0 − C11 )(h, h, Z ) + 2λhh H cβ−α +i (C0 − C11 )(h, H, Z ) 4 2 (C0 − C11 )(H, h, Z ) − 2λh H H sβ−α cβ−α (C0 − C11 )(H, H, Z ) − 2λh H h sβ−α

2 c2 λ ± ± ± + 2sW W h H + G − (C 0 − C 11 )(H , G , W ) g2 m Z 2 +i Z 2λ A Ah sβ−α cβ−α (C0 − C11 )(A, Z , h) + 2λ A AH cβ−α (C0 − C11 )(A, Z , H ) 4 2 (C0 − C11 )(G 0 , Z , h) + λ AG 0 H sβ−α cβ−α (C0 − C11 )(G 0 , Z , H ) + λ AG 0 h sβ−α 2 c2 λ ± ± ± − isW W AG + H − cβ−α (C 0 − C 11 )(H , W , G )

2 c2 λ 2 2 ± ± ± + isW W AG − H + cW cW (C 0 − C 11 )(H , W , G ) gZ +i − 12λ Ah A λhhh cβ−α (C0 + C11 )(h, h, A) + 4λ Ah A λhh H sβ−α (C0 + C11 )(h, H, A) 2 − 4λ AH A λh H h cβ−α (C0 + C11 )(H, h, A) − 4λ AH A λh H H sβ−α (C0 + C11 )(H, H, A) + 4λ2A Ah cβ−α (C0 + C11 )(A, A, h) − 4λ A AH λh A A sβ−α C0 + C11 )(A, A, H ) − 6λ AhG 0 λhhh sβ−α (C0 + C11 )(h, h, G 0 ) − 2λ AhG 0 λhh H cβ−α (C0 + C11 )(h, H, G 0 )

256

Appendix D: One Particle Irreducible Diagrams in the Two-Higgs Doublet Model − 2λ AH G 0 λh H h sβ−α (C0 + C11 )(H, h, G 0 ) − 2λ AH G 0 λh H H cβ−α (C0 + C11 )(H, H, G 0 ) + 2λ A Ah λh AG 0 sβ−α (C0 + C11 )(A, G 0 , h) + λ2

c (C + C11 )(G 0 , A, h) AG 0 h β−α 0

+ 2λ A AH λh AG 0 cβ−α (C0 + C11 )(A, G 0 , H ) − λ AG 0 H λhG 0 A sβ−α (C0 + C11 )(G 0 , A, H ) + 2λ AG 0 h λhG 0 G 0 sβ−α (C0 + C11 )(G 0 , G 0 , h) + 2λ AG 0 H λhG 0 G 0 cβ−α (C0 + C11 )(G 0 , G 0 , H ) 2 )λ ± ± ± + i(c2W − sW AG − H + λhG + H − (C 0 + C 11 )(G , H , H ) 2 )λ ± ± ± − i(c2W − sW AG + H − λhG − H + (C 0 + C 11 )(G , H , H ) 2 )λ ± ± ± + i(c2W − sW AH − G + λh H + G − (C 0 + C 11 )(H , G , G )

2 )λ ± ± ± − i(c2W − sW AH + G − λh H − G + (C 0 + C 11 )(H , G , G ) .

(D.135)

where  SV V (X, Y, Z ) = 2B ( p 2 ; Y, Z ) + − 1 (3 p 2 − p 2 + q 2 )C + ( p 2 − q 2 )C − 2C C AZ 0 21 23 24 2 1 2 1 φ 2  1 1 + (− p12 − 2 p22 + q 2 )C11 + ( p12 − q 2 )C12 + ( p22 + 4m 2X )C0 ( p12 , p22 , q 2 ; X, Y, Z ), 2 2

(D.136)

 SS 2 2 2 2 2 2 2 2 2 CV AZ φ (X, Y, Z ) = (q + 3 p1 − p2 )C 21 + 2(q − p1 )C 23 + 4C 24 + (3q + 5 p1 − 3 p2 + m X )C 11  + 2(q 2 − p12 )C12 + (2q 2 + 2 p12 − 2 p22 + m 2X )C0 ( p12 , p22 , q 2 ; X, Y, Z ). (D.137)

The 1PI contributions to the form factor of AZ H vertex are given by 2 2 2 16π 2  1PI AZ H ( p1 , p2 , q ) F

=i



f

8I f Nc g Z a f

m 2f ζ f ζ H f f

f

v2



 B0 ( p22 ; f, f ) + 2m 2f C0 + p12 C11 + (q 2 − p12 )C12 ( f, f, f ),

(D.138) 2 2 2 16π 2  1PI AZ H ( p1 , p2 , q ) B

g3 SV V ± ± ± cW sβ−α C AZ φ (H , W , W ) 2 g3 2 V SS 2 V SS C AZ + i Z sβ−α − cβ−α φ (Z , A, h) − sβ−α C AZ φ (Z , A, H ) 8

= −i

2 V SS 0 2 V SS 0 2 2 2 V SS ± ± ± + cβ−α C AZ φ (Z , G , h) − cβ−α C AZ φ (Z , G , H ) + 2cW (cW − sW )C AZ φ (W , H , H )

g 3Z 2 2 sβ−α (B0 − B1 )(q 2 ; H, Z ) + 2sW cW (B0 − B1 )(q 2 ; H ± , W ± ) 8

2 2 cW (B0 − B1 )( p12 ; H ± , W ± ) + (B0 − B1 )( p12 ; A, Z ) + 2sW

+i



g 3Z m 2Z 2 sβ−α cβ−α (2C0 + C11 )(Z , Z , h) − (2C0 + C11 )(Z , Z , H ) 4 g2 m Z 2 2λ H hh sβ−α cβ−α (C0 − C11 )(h, h, Z ) + 2λ H h H cβ−α (C0 − C11 )(h, H, Z ) +i Z 4 2 − 2λ H H h sβ−α (C0 − C11 )(H, h, Z ) − 6λ H H H sβ−α cβ−α (C0 − C11 )(H, H, Z )

2 2 cW λ H H + G − (C0 − C11 )(H ± , G ± , W ± ) + 2sW

−i

Appendix D: One Particle Irreducible Diagrams in the Two-Higgs Doublet Model

257

g 2Z m Z 2 2λ A Ah sβ−α (C0 − C11 )(A, Z , h) + 2λ A AH cβ−α sβ−α (C0 − C11 )(A, Z , H ) 4 2 − λ AG 0 h cβ−α sβ−α (C0 − C11 )(G 0 , Z , h) − λ AG 0 H cβ−α (C0 − C11 )(G 0 , Z , H ) −i

2 2 cW λ AG + H − sβ−α (C0 − C11 )(H ± , W ± , G ± ) − isW

2 2 + isW cW λ AG − H + sβ−α (C0 − C11 )(H ± , W ± , G ± ) gZ − 4λ Ah A λ H hh cβ−α (C0 + C11 )(h, h, A) + 4λ Ah A λ H h H sβ−α (C0 + C11 )(h, H, A) +i 2 − 4λ AH A λ H H h cβ−α (C0 + C11 )(H, h, A)12λ AH A λ H H H sβ−α (C0 + C11 )(H, H, A)

+ 4λ A Ah λ H A A cβ−α (C0 + C11 )(A, A, h) − 4λ2A AH sβ−α (C0 + C11 )(A, A, H ) − 2λ AhG 0 λ H hh sβ−α (C0 + C11 )(h, h, G 0 ) − 2λ AhG 0 λ H h H cβ−α (C0 + C11 )(h, H, G 0 ) − 2λ AH G 0 λ H H h sβ−α (C0 + C11 )(H, h, G 0 ) − 6λ AH G 0 λ H H H cβ−α (C0 + C11 )(H, H, G 0 ) + 2λ A Ah λ H AG 0 sβ−α (C0 + C11 )(A, G 0 , h) + λ AG 0 h λ H G 0 A cβ−α (C0 + C11 )(G 0 , A, h) + 2λ A AH λ H AG 0 cβ−α (C0 + C11 )(A, G 0 , H ) − λ2AG 0 H sβ−α (C0 + C11 )(G 0 , A, H ) + 2λ AG 0 h λ H G 0 G 0 sβ−α (C0 + C11 )(G 0 , G 0 , h) + 2λ AG 0 H λ H G 0 G 0 cβ−α (C0 + C11 )(G 0 , G 0 , H ) 2 2 − sW )λ AG − H + λ H G + H − (C0 + C11 )(G ± , H ± , H ± ) + i(cW 2 2 − i(cW − sW )λ AG + H − λ H G − H + (C0 + C11 )(G ± , H ± , H ± ) 2 2 − sW )λ AH − G + λ H H + G − (C0 + C11 )(H ± , G ± , G ± ) + i(cW

2 2 − i(cW − sW )λ AH + G − λ H H − G + (C0 + C11 )(H ± , G ± , G ± ) .

D.3.10

(D.139)

AV1 V2 Vertex

The 1PI contributions to the form factor of AW + W − vertex are given by 16π 2  3,1PI ( p 2 , p22 , q 2 ) F = ig 2 AW + W − 1



f

Nc

  C0 + C11 − C12 ( f, f  , f ) + ( f ↔ f  ),

2I f m 2f ζ f v

f

(D.140) where f  is the SU (2) L partner of f . The 1PI contributions to the form factor of AZ Z vertex are given by 2 2 2 2 16π 2  3,1PI AZ Z ( p1 , p2 , q ) F = ig Z



f

Nc

16I f m 2f ζ f v

f



 v 2f C0 + a 2f (C0 + 2C11 − 2C12 ) ( f, f, f ).

(D.141) The 1PI contributions to the form factor of AZ γ vertex are given by 2 2 2 16π 2  3,1PI AZ γ ( p1 , p2 , q ) F = ieg Z

 f

Ncf Q f

16I f m 2f ζ f v

v f C0 ( f, f, f ).

(D.142)

258

Appendix D: One Particle Irreducible Diagrams in the Two-Higgs Doublet Model W

e+ ν

ν

W W

e−

W

Z

ν

W

h

(1)

W G±

W

W

(2)

(3)

W ν

G±

ν

ν

W

W

(4)

(5)

e

W

e Z

Z

(6)

Fig. D.1 Box diagrams for e+ e− → h Z

The 1PI contributions to the form factor of Aγγ vertex are given by 2 2 2 2 16π 2  3,1PI Aγγ ( p1 , p2 , q ) F = ie



Ncf Q 2f

16I f m 2f ζ f

f

v

C0 ( f, f, f ).

(D.143)

D.4 Four-Point Functions D.4.1 Box Diagrams for e+ e− → hZ We here give the analytic expressions for the box diagram contributions denoted by Fik (s, t) in Eq. (7.66) and Eq. (7.70). The analytic expressions are given as follows (Fig. D.1). 1 4 g m W cW , 2 Fe1 = 4(D0 + D11 + D13 + D25 )(0, 0, m 2Z , m 2h , s, u; W, 0, W, W ),

C1 = Fe1 1

F

= 2(D13 − D12 + 2D26 )(0, 0, m 2Z , m 2h , s, u; W, 0, W, W ), = −2C0 (s, m 2Z , m 2h ; W, W, W ) − [4D27 + (s − t + m 2Z )(D0 + + u D13 + 2(u − m 2Z )D12 ](0, 0, m 2Z , m 2h , s, u; W, 0, W, W ),

(D.144) (D.145) (D.146) D11 ) (D.147)

1 C 2 = g 4 m W cW , 2 Fe2 = 2(D13 − D12 + 2D26 )(0, 0, m 2Z , m 2h , s, t; W, 0, W, W ), Fe2 2

=

F =

4(D0 + D11 + D13 + D25 )(0, 0, m 2Z , m 2h , s, t; W, 0, W, W ), −2C0 (s, m 2Z , m 2h ; W, W, W ) − [4D27 + (m 2Z + s − u)(D0 +

(D.148) (D.149) (D.150) D11 )

Appendix D: One Particle Irreducible Diagrams in the Two-Higgs Doublet Model

+ 2(t − m 2Z )D12 + t D13 ](0, 0, m 2Z , m 2h , s, t; W, 0, W, W ), 1 C 3 = − g3 g Z m W , 4 Fe3 = 0, Fe3 3 F

259

(D.151) (D.152) (D.153)

= −4(D12 − D13 )(0, 0, m 2Z , m 2h , s, u; W, 0, W, G ± ), (D.154) 2 2 = C0 (s, m Z , m h ; W, W, W ) + 2[−(u − m 2h )(D0 + D11 ) + u D13 ](0, 0, m 2Z , m 2h , s, u; W, 0, W, G ± ), (D.155)

1 2 C 4 = − g 3 g Z m W sW , 4 Fe4 = −4(D12 − D13 )(0, 0, m 2Z , m 2h , s, t; W, 0, W, G ± ), Fe4 4

F

= 0,

+ t D13 ](0, 0, m 2Z , m 2h , s, t; W, 0, W, G ± ), (D.159)

2m 2W 2 g gZ , v Fe5 = −2(D0 + D11 + D12 + D24 )(0, m 2Z , 0, m 2h , t, u; W, 0, 0, W ), F

(D.157) (D.158)

= C0 (s, m 2Z , m 2h ; W, W, W ) + 2[−(t − m 2h )(D0 + D11 )

C 5 = −(vν + aν ) Fe5 5

(D.156)

(D.160) (D.161)

= −2D26 (0, m 2Z , 0, m 2h , t, u; W, 0, 0, W ), (D.162) 2 = −C0 (t, 0, m h ; W, 0, W ) − [−2D27 + (t − m 2Z )(D0 + D11 ) + m 2Z D12 ](0, m 2Z , 0, m 2h , t, u; W, 0, 0, W ), (D.163) 4m 2Z

g3 , v Z F6 = −2(D0 + D11 + D12 + D24 )(0, m 2Z , 0, m 2h , t, u; Z , 0, 0, Z ),

C6 = − F6 6

F

= −2D26 (0, m 2Z , 0, m 2h , t, u; Z , 0, 0, Z ), = −C0 (t, 0, m 2h ; Z , 0, Z ) − [−2D27 + (t − m 2Z )(D0 + D11 ) + m 2Z D12 ](0, m 2Z , 0, m 2h , t, u;

(D.164) (D.165) (D.166) Z , 0, 0, Z ). (D.167)

References 1. Zyla PA et al (2020) Review of particle physics. PTEP 8:083C01 2. Chetyrkin KG, Kuhn JH, Steinhauser M (2000) RunDec: a mathematica package for running and decoupling of the strong coupling and quark masses. Comput Phys Commun 133:43–65

260

Appendix D: One Particle Irreducible Diagrams in the Two-Higgs Doublet Model

3. Xing Z-z, Zhang H, Zhou S (2008) Updated values of running quark and lepton masses. Phys Rev D 77:113016 4. Passarino G, Veltman MJG (1979) One loop corrections for e+ e- annihilation into mu+ muin the weinberg model. Nucl Phys B 160:151–207 5. Hagiwara K, Matsumoto S, Haidt D, Kim CS (1995) A novel approach to confront electroweak data and theory. Z Phys C 64:559–620. [Erratum: Z Phys C 68:352 (1995)] 6. Hahn T, Perez-Victoria M (1999) Automatized one loop calculations in four-dimensions and D-dimensions. Comput Phys Commun 118:153–165 7. Ansgar D (1993) Techniques for calculation of electroweak radiative corrections at the one loop level and results for W physics at LEP-200. Fortsch Phys 41:307–420 8. Shinya K, Mariko K, Kei Y (2015) Fingerprinting the extended Higgs sector using one-loop corrected Higgs boson couplings and future precision measurements. Nucl Phys B 896:80–137 9. Masashi A, Shinya K, Kodai S (2021) Radiative corrections to decays of charged Higgs bosons in two Higgs doublet models. Nucl Phys B 973:115581 10. Shinya K, Mariko K, Kodai S, Kei Y (2017) Gauge invariant one-loop corrections to Higgs boson couplings in non-minimal Higgs models. Phys Rev D 96(3):035014