Relativistic Quantum Invariance 9789811979484, 9789811979491

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Lecture Notes in Physics

Chueng-Ryong Ji

Relativistic Quantum Invariance

Lecture Notes in Physics Founding Editors Wolf Beiglböck Jürgen Ehlers Klaus Hepp Hans-Arwed Weidenmüller

Volume 1012 Series Editors Roberta Citro, Salerno, Italy Peter Hänggi, Augsburg, Germany Morten Hjorth-Jensen, Oslo, Norway Maciej Lewenstein, Barcelona, Spain Angel Rubio, Hamburg, Germany Wolfgang Schleich, Ulm, Germany Stefan Theisen, Potsdam, Germany James D. Wells, Ann Arbor, MI, USA Gary P. Zank, Huntsville, AL, USA

The series Lecture Notes in Physics (LNP), founded in 1969, reports new developments in physics research and teaching - quickly and informally, but with a high quality and the explicit aim to summarize and communicate current knowledge in an accessible way. Books published in this series are conceived as bridging material between advanced graduate textbooks and the forefront of research and to serve three purposes: • to be a compact and modern up-to-date source of reference on a well-defined topic; • to serve as an accessible introduction to the field to postgraduate students and non-specialist researchers from related areas; • to be a source of advanced teaching material for specialized seminars, courses and schools. Both monographs and multi-author volumes will be considered for publication. Edited volumes should however consist of a very limited number of contributions only. Proceedings will not be considered for LNP. Volumes published in LNP are disseminated both in print and in electronic formats, the electronic archive being available at springerlink.com. The series content is indexed, abstracted and referenced by many abstracting and information services, bibliographic networks, subscription agencies, library networks, and consortia. Proposals should be sent to a member of the Editorial Board, or directly to the responsible editor at Springer: Dr Lisa Scalone Springer Nature Physics Tiergartenstrasse 17 69121 Heidelberg, Germany [email protected]

Chueng-Ryong Ji

Relativistic Quantum Invariance

Chueng-Ryong Ji Department of Physics North Carolina State University Raleigh, NC, USA

ISSN 0075-8450 ISSN 1616-6361 (electronic) Lecture Notes in Physics ISBN 978-981-19-7948-4 ISBN 978-981-19-7949-1 (eBook) https://doi.org/10.1007/978-981-19-7949-1 © 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

To my parents.

Acknowledgements

I would like to thank my collaborators for the papers that led to this monograph. This work was supported in part by the DOE Contract No. DE-FG02-03ER41260.

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About This Book

This book describes the invariant nature of the relativistic quantum field theories utilizing the idea of interpolating the instant form dynamics and the light-front dynamics. While the light-front dynamics (LFD) based on the light-front time was proposed by Dirac in 1949, there has not yet been a salient review on the connection between the LFD and the instant form dynamics (IFD) based on the ordinary time. Reviewing the connection between LFD and IFD using the idea of interpolating the two different forms of the relativistic dynamics, one can learn the distinguished features of each form and how one may utilize those distinguished features in solving the complicate relativistic quantum field theoretic problems more effectively. With the ongoing 12-GeV Jefferson Lab experiments and the forthcoming electron-ion collider (EIC) facilities, the internal structures of the nucleon and nuclei are vigorously investigated in particular using the physical observables defined in the LFD rather than in the IFD. This book offers a clear demonstration on why and how the LFD is more advantageous than the IFD for the study of hadron physics, illustrating the differences and similarities between these two distinguished forms of the dynamics. It aims at presenting the basic first-hand knowledge of the relativistic quantum field theories, describing why and how the different forms of dynamics (e.g. IFD and LFD) can emerge in them, connecting the IFD and the LFD using the idea of the interpolation, and demonstrating explicit examples of the interpolation in quantum electrodynamics and other field theories. While the level of presentation is planned mainly for the advanced undergraduate students and the beginning graduate students, the topics of the interpolation between the IFD and the LFD are innovative enough for even the experts in the field to appreciate its usefulness.

ix

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Special Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Four-Vectors and Relativistic Collisions . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Symmetry Properties of Special Relativity . . . . . . . . . . . . . . . . . . . . . . . 1.4 Poincaré Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Interpolation Between Instant Form Dynamics and Light-Front Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Dirac’s Proposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Interpolating Scattering Amplitudes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Kinematic Transformations of Particle Momenta . . . . . . . . . . . . . . . . . 2.4 Application of Transformations on Interpolating Scattering Amplitudes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Interpolating Scattering Amplitudes in Infinite Momentum Frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 3 8 11 16 17 17 20 26 38 41 45 46

3 Interpolation of Quantum Electrodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3.2 Formal Derivation of the Interpolation of QED . . . . . . . . . . . . . . . . . . . 54 3.2.1 Scattering Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 3.2.2 Canonical Field Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 3.3 Toy Calculation of e+ e− Annihilation Producing Two Scalar Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 3.3.1 Collinear Scattering/Annihilation, θ = π . . . . . . . . . . . . . . . . . . 75 3.3.2 Summary of e+ e− → Two Scalar Particles . . . . . . . . . . . . . . . 81 3.4 Interpolating Helicity Scattering Probabilities . . . . . . . . . . . . . . . . . . . . 84 3.4.1 e+ e− Pair Annihilation into Two Photons . . . . . . . . . . . . . . . . . 84 3.4.2 Compton Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 3.5 Summary and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

xi

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Contents

4 Interpolation of Quantum Chromodynamics in 1+1 Dimension . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 The Mass Gap Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 The Hamiltonian Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 The Feynman Diagram Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.3 Behavior of the Gap Equation When Approaching the Light-Front . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 The Mass Gap Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Chiral Condensate and Constituent Quark Mass . . . . . . . . . . . . . . . . . . 4.4.1 The Chiral Condensate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2 The Fermion Propagator and Constituent Mass . . . . . . . . . . . . 4.5 The Bound-State Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 The Bound-State Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.1 Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.2 Wavefunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.3 Quasi-PDFs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7 Conclusion and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

111 112 115 116 122 125 128 134 134 137 146 151 152 158 162 168 170

QED Appendix A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 QCD1+1 Appendix B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221

About the Author

Prof. Chueng-Ryong Ji at Department of Physics, North Carolina State University (NCSU), joined the faculty in 1987 and served as the Director of the Graduate Program for 2013–2015. He is a Fellow and Outstanding Referee of American Physical Society (APS) and has served for many years since 2010 as the Chair of the International Light Cone Advisory Committee. His research work with the Jefferson Lab Angular Momentum (JAM) collaborators including his graduate student published in Phys. Rev. Lett.121,152001(2018) “First Monte Carlo Global QCD Analysis of Pion Parton Distributions” was highlighted with a short summary (Synopsis) on the APS website as well as the IOP’s “Physics World”. Professor Ji pioneered the idea of connecting the instant form dynamics and the light-front dynamics and applied this idea in solving relativistic bound state and scattering problems. His seminal work with his graduate students and visiting scholar “Interpolating quantum electrodynamics between instant and front forms” was published in Phys. Rev. D98, 036017(2018). His subsequent work with his graduate student “Interpolating ’t Hooft model between instant and front forms” was published in Phys. Rev. D104, 036004(2021). Professor Ji wrote the book entitled Pedestrian Approach to Particle Physics as the first monograph published by the Asia Pacific Center for Theoretical Physics in 2007. Before he joined the faculty at NCSU, he was a visiting scholar at the theory group of the Stanford Linear Accelerator Center (SLAC) for 1982–84, a postdoctoral fellow at Department of Physics, Stanford University for 1984–86 and a research associate at Brooklyn College of the City University of New York in for 1986–97. Professor Ji received his Ph.D. in 1982 from Korea Advanced Institute of Science and Technology and his Bachelor’s degree with Honor in 1976 from Seoul National University.

xiii

1

Introduction

In this chapter, we will discuss special relativity [1–3] established by Albert Einstein in 1905. Special relativity is one of the backbones of modern particle physics. We will first introduce the main ideas of special relativity and then present the physical effects of special relativity that can be measured from experiments. We will also discuss the symmetry properties of space and time described by the Poincaré group.

1.1

Special Relativity

A key of the special relativity is that the events occurring simultaneously in one frame may not be simultaneous in another frame that is moving with a speed v relative to the frame S. As shown in Fig. 1.1, suppose the frame S is the frame fixed to the car and the frame is fixed to the ground. In Fig. 1.1, if the flash light in the middle of the ceiling is turned on, the signal transfers to the two photo-tubes at each end A and B. In the frame S, because the light is in the middle between A and B, the two photo-tubes receive the signal simultaneously, i.e. t A = t B . However, the same event would be shown differently to the observer in the frame S  ; the photo-tube at B receives the signal before the photo-tube at A receives, i.e. t A > t B . The difference t A − t B can be calculated using the Lorentz transformation which relates the space-time of two frames shown in Fig. 1.2. For your exercise of Lorentz transformation, verify that t A − t B = √vL2AB 2 , where c c =v

L AB is the distance between the two photo-tubes and c(= 3 × 108 m/s) is the speed of light. As we see from the Lorentz transformation shown in Fig. 1.2, the equations of the Lorentz transformation are exactly symmetric under the exchange of z and ct or space and time. This means that the space and time in special relativity are interchangible or symmetric. However, as you may have heard often, the time is dilated or elongated while the length is contracted or shortened. Thus, an immediate question arises; how can the time be dilated and the length be contracted from the Lorentz transformation although the transformation is perfectly symmetric under the interchange of space and time? The answer comes from the consideration of causality, i.e. any signal cannot move faster than c. To be more specific, let’s draw © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 C.-R. Ji, Relativistic Quantum Invariance, Lecture Notes in Physics 1012, https://doi.org/10.1007/978-981-19-7949-1_1

1

2

1

Introduction

Fig. 1.1 Reference frame

Fig. 1.2 Lorentz transformation

the Minkowski space diagram of the two frames S and S  in Fig. 1.3. In Fig. 1.3, the causality requires that any measurable events must lie within the light-cone or the time-like region. Note, then, that the axis of ct  is in the time-like region while the axis of z  is in the space-like region. This means that any scale change in the time axis can be measured while the same scale change in the space (z) axis cannot be measured. In fact, the Lorentz transformation given by Fig. 1.2 shows the expansion of both scales of ct  and z  because ct  and z  are interchangible. The only difference in the scale expansion of ct  and z  is that ct  can be measured but z  cannot be measured. By substituting z = 0 and ct = 1 in the Lorentz transformation in Fig. 1.2, one obtains ct  = γ and z  = γβ, which indicates that the scale expansion is given by the factor γ . Thus, while the time dilation of the factor γ can be measured, the space expansion cannot be measured. In fact, the length contraction can be understood by measuring the same length with the expanded scale. In order to see more clearly the phenomenon of the length contraction, consider the diagram in Fig. 1.4. In Fig. 1.4, the measured ruler is shown by the shaded bar at each different time t = t A , t B and tC . Now, the point is that one needs some time to recognize the entire length of the ruler. Imagine in Fig. 1.4 that the eye which measures the length of the ruler is sitting at z = 0. At t = t A , the eye just receives the signal from the position A. In order to receive the signal from the position D, one has to wait up to the moment t = tC . By measuring tC − t A , one can obtain the length of the ruler, AD. Let us consider that the length AD = c(tC − t A ) is given by a unit length 1. Then, the question is the length observed in S  . Again in order to measure the length, one has to wait from the position A, i.e. z  = 0. Now, imagine that the signal from D is received by an observer’s eye in S  frame at the position C  as shown in Fig. 1.4. Then, the time taken from A to C  is given by c(tC − t A ) = γ (1 + β), which is not

1.1 Special Relativity

3

Fig. 1.3 Minkowski space

Fig. 1.4 Length contraction

1/γ . What is going on? The reason why we did’nt get 1/γ is because the position of the ruler is also changed during this time period of γ (1 + β). Note that the ruler is moving to +z direction with the speed of βc. Thus, to find the length of the ruler observed in S  , one has to subtract the distance that the ruler moved during γ (1 + β): AD  = c(tC − t A ) − βc(tC − t A ) = γ (1 + β) − βγ (1 + β) 1 = . γ

(1.1)

This shows the length contraction as expected.

1.2

Four-Vectors and Relativistic Collisions

We have discussed physical meaning of Minkowski space in the last section using the time dilation and length contraction. In this section, we introduce a convenient

4

1

Introduction

four-vector notation to discuss the four-dimensional space-time structure and discuss an application to the collision problems. The four-vector notation for the space and time is given by x μ = (x 0 , x 1 , x 2 , x 3 ) = (x 0 , x),

(1.2)

where x 0 = ct and x = (x 1 , x 2 , x 3 ) = (x, y, z). The Greek index μ runs from 0 to 3, and the four-vector with the superscript Greek index as shown in Eq. (1.2) is called contravariant vector. If the Greek index is used as a subscript, xμ , then the four-vector is called covariant vector. The contravariant vector x μ and the covariant vector xμ are needed to use a convenient summation convention when the two quantities with the Greek indices are multiplied. For example, the Lorentz transformation introduced in the last section can be simply written as ν x μ = μ νx ,

(1.3)

μ

where ν x ν is a second rank tensor of the Lorentz transformation given by ⎡

γ ⎢ ⎢ 0 μ ν =⎣ 0 γβ

0 1 0 0

0 0 1 0

⎤ γβ 0 ⎥ ⎥, 0 ⎦ γ

(1.4)

so that 00 = 33 = γ , 30 = 03 = γβ, 11 = 22 = 1, and all other components are zero. The summation convention is that the repeated Greek index used as both subscript and superscript (e.g. ν in Eq. (1.3) is to be summed from 0 to 3. With the four-vector notation and the summation convention, the four separate equations for the Lorentz transformation are summarized simply by one equation as shown in Eq. (1.3). The contravariant and covariant vectors are related to each other by a Minkowski metric tensor gμν : xμ = gμν x ν or x μ = g μν xν ,

(1.5)

where ⎡

gμν = g μν

1 ⎢0 =⎢ ⎣0 0

0 −1 0 0

⎤ 0 0 0 0 ⎥ ⎥. −1 0 ⎦ 0 −1

(1.6)

1.2 Four-Vectors and Relativistic Collisions

5

As we have seen above, a four-vector needs four components. However, to be a four-vector, it is not sufficient to have four components. In order to be a four-vector, the four components must follow the same transformation as the space-time fourvector x μ . The examples of four-vector are the four-momentum p μ = (E/c, p), the  etc. These four-current J μ = (cρ, J) and the four-vector-potential Aμ = (c, A), μ four-vectors follow the Lorentz transformation exactly same as x . As an example, if the four-momentum p μ of a particle measured in S frame is seen as p μ in S  frame, then ν p μ = μ ν p ,

(1.7)

μ ν

where is given by Eq. (1.4). Thus, if the particle in S frame has the energy and momentum given by p μ = (E/c, 0, 0, pz ), then the energy and momentum of the particle in S  frame are given by p μ = (E  /c, 0, 0, pz ), where E  /c = γ (E/c + β pz )

(1.8)

pz = γ ( pz + β E/c).

(1.9)

and If the particle of mass m is at rest in the frame S, then pz = 0 and p μ is given by  using the Einstein’s equation E = mc2 . Now, in the S  frame, using p μ = (mc, 0) Eqs. (1.8) and (1.11), one finds E  = γ mc2

(1.10)

p  = γ mcβ.

(1.11)

and Equation (1.10) can be expanded in the orders of β = v/c as follows: E  = γ mc2 mc2 =  2 1 − vc2

1 v2 3 4 4 = mc2 1 + + c + · · · v 2 c2 8 1 3 v4 = mc2 + mv 2 + m 2 + · · · 2 8 c = R + T,

(1.12)

where the rest energy and kinetic energy are denoted by the R = mc2 and T = 1 3 v4 2 2 mv + 8 m c2 + · · · , respectively. Thus, one can find that the relativistic corrections to the non-relativistic kinetic energy, 21 mv 2 , appear as higher orders of vc .

6

1

Introduction

Fig. 1.5 Before and after an inelastic collision of two bodies

Another interesting point to mention is the scalar product of four-vectors. If a and b are four-vectors, then the scalar product a · b = aμ bμ is Lorentz invariant, i.e. a · b = a  · b . Note that the Lorentz transformation given by Eq. (1.4) satisfies the μ μ μ relation ν νλ = gλ , where gλ can be represented as a 4 × 4 identity matrix. Thus, in the above example of four-momenta, one can find the on-mass-shell condition; 2

p =p

μ

·

pμ

μ

= p = p · pμ = 2



E c

2 − p2 = mc2 .

(1.13)

We now apply the above discussion to a simple collision problem and show that a hot object is heavier than a cold object. Consider a head-on collision of two equal masses as shown in Fig. 1.5. The four-momenta of two particles are given by μ

p1 = (E 1 /c, p1 ) = (γ mc, 0, 0, γ mv) μ p2 = (E 2 /c, p2 ) = (γ mc, 0, 0, −γ mv),

(1.14)

using Eqs. (1.10) and (1.11). Now, suppose the two masses stick together after the collision, then the total mass would be at rest and the total four-momentum p μ is given by μ

μ

p μ = p1 + p2 = (2γ mc, 0, 0, 0).

(1.15)

Note that the total rest mass M is given by M = 2γ m 2m > 2 m. =  2 1 − vc2

(1.16)

Thus, one finds that the total mass is bigger than the original two masses after they stick together. In non-relativistic physics, one would have understood the conservation of energy in this collision as a generation of heat energy in the process of sticking together. However, in relativistic understanding, the generation of heat energy is realized by the mass increase. Thus, one may say that a hot potato is heavier than a cold potato [4].

1.2 Four-Vectors and Relativistic Collisions

7

Fig. 1.6 Elastic collisions in COM and Lab

A little more elaborate example of collision problem may be shown in Fig. 1.6, where two equal-mass particles make a perfect elastic collision and the two particles scatter with a center-of-momentum (COM) angle θ . The problem is to find the scattering angle of the same event in the laboratory frame (Lab) where the target particle is at rest. In COM, if the particle 1 has the velocity v1 = v zˆ , then obviously the particle 2 has the velocity v2 = −v zˆ . Thus, we can write the four-momenta of the initial two particles 1 and 2 as follows: ⎡

⎤ γc ⎢ 0 ⎥ μ ⎥ p1 = m ⎢ ⎣ 0 ⎦ and γβc where β =

v c

and γ = √ 1

1−β 2

⎡

⎤ γc ⎢ 0 ⎥ μ ⎥ p2 = m ⎢ ⎣ 0 ⎦, −γβc

(1.17)

.

Similarly, from Fig. 1.6, we can find the four-momenta of final particles 3 and 4 in COM: ⎡ ⎡ ⎤ ⎤ γc γc ⎢ ⎢ ⎥ ⎥ 0 0 μ μ ⎢ ⎥ ⎥ p3 = m ⎢ (1.18) ⎣ γβc sin θ ⎦ and p4 = m ⎣ −γβc sin θ ⎦ , γβc cos θ −γβc cos θ Now, in the Lab frame, the initial particle 2 is at rest as a target as shown in Fig. 1.5. Thus, it is rather easy to find the Lorentz transformation between COM and Lab. We note that this transformation is precisely given by Eq. (1.4). As a check, you can μ μ transform p2 in the COM to p  2 in the Lab as follows:

8

1

Introduction

μ

ν p  2 = μ ν p2 ⎡ γ 0 ⎢ 0 1 = m⎢ ⎣ 0 0 γβ 0 ⎡ ⎤ mc ⎢ 0 ⎥ ⎥ =⎢ ⎣ 0 ⎦. 0

0 0 1 0

⎤⎡ ⎤ γβ γc ⎢ ⎥ 0 ⎥ ⎥⎢ 0 ⎥ ⎦ ⎣ 0 0 ⎦ γ −γβc (1.19)

Thus, the particle 2 is at rest in the Lab as expected. Using the four-vector property, we can now transform all other four-momenta using the same Lorentz transformation given by Eq. (1.4). Applying to the particle 3, we get μ

ν p  3 = μ ν p3 ⎤ ⎡ 2 γ c(1 + β 2 cos θ ) ⎥ ⎢ 0 ⎥. = m⎢ ⎦ ⎣ γβc sin θ γ 2 βc(1 + cos θ )

(1.20)

From Fig. 1.6, we note that the Lab scattering angle θ can be obtained as tan ϑ =

( p 3 ) y γβc sin θ sin θ = 2 = . ( p  3 )z γ βc(1 + cos θ ) γ (1 + cos θ )

(1.21)

This yields the relation between the scattering angles θ and θ . In the non-relativistic limit, γ ≈ 1 and thus tan ϑ =

1.3

sin θ θ θ = tan , or ϑ ≈ . (1 + cos θ ) 2 2

(1.22)

Symmetry Properties of Special Relativity

If one considers the Lorentz transformation more deeply [5], one finds an interesting symmetry. To see this, consider the Lorentz transformation given by Eq. (1.3) discussed in the last section: ⎡ 0 ⎤ ⎡ ⎤⎡ 0⎤ x x γ 0 0 γβ ⎢ x 1 ⎥ ⎢ 0 1 0 0 ⎥ ⎢ x 1 ⎥ ⎢ ⎥=⎢ ⎥⎢ ⎥ ⎣ x 2 ⎦ ⎣ 0 0 1 0 ⎦ ⎣ x 2 ⎦ , γβ 0 0 γ x 3 x3 (1.23)

1.3 Symmetry Properties of Special Relativity

9

and now let’s introduce a new variable called “rapidity” ω representing the boosting velocity v = βc as v β = = tanh ω , (1.24) c so that the γ factor is then given by 1 = cosh ω . γ = 1 − β2

(1.25)

Using the rapidity variable ω, we rewrite the Lorentz transformation in Eq. (1.23) as ⎡

cosh ω ⎢ 0 (ω) = ⎢ ⎣ 0 sinh ω

0 1 0 0

⎤ 0 sinh ω 0 0 ⎥ ⎥. 1 0 ⎦ 0 cosh ω

(1.26)

From this, one can find the following properties of (ω): 1. 2. 3. 4.

(ω)(ω ) = (ω + ω ); closure (0) = I ; idenity −1 (ω) = (−ω); inverse (ω){(ω )(ω )} = {(ω)(ω )}(ω );

associativity

(1.27)

These four properties are exactly the properties of a group in mathematics. Thus, one finds that the Lorentz transformation satisfies the group properties and can be given by a group representation. There are two distinguished groups in mathematics. One group is called a discrete group, and the other is called a continuous group. Since the above example of Lorentz transformation has continuous group elements represented by a continuous variable “rapidity” ω, the above example belongs to the continuous group. The continuous group has been investigated thoroughly by a Norwegian mathematician Sophus Lie (1842–1899), and thus people often call the continuous group as Lie group [6–8]. The basic Lie groups of n × n matrices M (d is the dimension of the group) are listed in Table 1.1. The most general transformation of space and time is given by ν μ x μ = μ νx +a , μ

(1.28)

where ν includes the Lorentz transformation discussed above (i.e. the boost) and the rotation and a μ is the space and time translation. As we will discuss in the next section, the group representing the most general transformation of space and time given by Eq. (1.28) is called the Poincaré group. We will find that the Poincaré group belongs to the inhomogeneous SL(2, C) group in Table 1.1. Thus, the Poincaré group is sometimes referred as ISL(2, C) group. Before we discuss ISL(2, C), let me mention briefly the operations involved in the Poincaré group. In Eq. (1.28), one

10

1

Introduction

Table 1.1 Basic Lie groups General (G) linear (L) group of complex (C) regular (det M = 0) matrices ; d = 2n 2 Special (S: det M=1) linear group, subgroup of GL(n, C); d = 2(n 2 − 1) General linear group of real (R) regular matrices; d = n 2 Special linear group of real matrices, a subgroup of GL (n, R); d = n2 − 1 Unitary group of unitary (U : M M † = M † M = 1, where M † is the Hermitian conjugate of M) matrices; d = n 2 Special unitary group, a subgroup of U(n); d = n 2 − 1 Orthogonal (O) group of complex orthogonal matrices (M M T = 1, where M T is the transposed M); d = n(n − 1) Orthogonal group of real orthogonal matrices, d = n(n − 1)/2 Special orthogonal group or group of rotations in n-dimensional space, a subgroup of O(n); d = n(n − 1)/2 Symplectic (Sp) group of unitary n × n matrices, where n is even, satisfying the condition M T J M = J , where J is a nonsingular antisymmetrical matrix Pseudo-unitary group of complex matrices satisfying the condition Mg M † = g, where g is a diagonal matrix with elements gkk = 1 for 1 ≤ k ≤ m and gkk = −1 for m + 1 ≤ k ≤ n; d = n 2 Pseudo-orthogonal group of real matrices satisfying the condition Mg M T = g; d = n(n − 1)/2

GL( n , C) SL( n , C) GL(n, R) SL(n, R) U(n) SU(n) O(n, C) O(n) ≡ O(n, R) SO(n) Sp(n)

U(m, n − m)

O(m, n − m)

finds the total ten operations of transformation which are consisted of three Lorentz transformations (or boosts), three rotations, three space translations and one time translation. As we will see, each operation can be represented by the corresponding physical quantity: 3 3 3 1

boosts → 3 boost momenta, K rotations → 3 angular momenta, J space translations → 3 linear momenta, P time translations → 1 energy, E

These ten physical quantities would be conserved if the system is symmetric (or invariant) under the transformation given by the Poincaré group or ISL(2, C). Also,  E) are the quantities conjugate to the ten note that these ten quantities ( K , J, P,  transformation variables (ω,  θ , x, t), respectively, where ω  is the three-dimensional rapidity variable, θ is the three-dimensional orientation angle variable, x is the threedimensional space variable and t is the time variable.

1.4 Poincaré Group

1.4

11

Poincaré Group

As we introduced in the last section, the Poincaré group belongs to the Lie Group ISL(2, C). To see this, let’s represent the space and time by 2 × 2 matrix X :

0 x + x3 x1 − i x2 , (1.29) X = x μ σμ = x1 + i x2 x0 − x3 where



10 σ0 = 01



01 , σ1 = 10



0 −i , σ2 = i 0



1 0 , σ3 = 0 −1

.

(1.30)

The three matrices σ = (σ1 , σ2 , σ3 ) are the Pauli matrices. The four-vector x μ can be obtained from X by the following relation: xμ =

1 Tr(σμ X ) . 2

(1.31)

Now, let’s consider the 2 × 2 matrix representation of the Lorentz transformation and the rotation given by i  · σ ) ≡  (θ, ω)  = Exp(− (θ + i ω) 2

(1.32)

Then the elements of the Poincaré group are ordered pairs of 2 × 2 matrices given by (, a) and (−, a),

(1.33)

det = 1,

(1.34)

1 Tr(† ) = 00 ≥ 1, 2

(1.35)

where

and a† = a (1.36) Here, the dagger (†) notation in Eqs. (1.35) and (1.36) represents the adjoint operation to the matrix, and Eq. (1.34) can be shown by the traceless condition of the Pauli matrices

i  det  = det Exp − (θ + i ω)  · σ 2

i   · σ = Exp Tr − (θ + i ω) 2

i   · (Tr σ ) = Exp − (θ + i ω) 2 = Exp(0) = 1. (1.37)

12

1

Introduction

μ

Also, the second rank tensor ν of the Lorentz transformation and the rotation is given by  1  † μ (1.38) ν = Tr σμ σν  , 2 and the four-vector a μ is given by aμ =

 1  Tr σμ a . 2

(1.39)

Then, the ISL(2,C) transformation of the space-time space is given by X  = X † + a,

(1.40)

where there are ten parameters of the transformation given by ω,  θ and a μ . We note that the homogeneous part of the Poincaré group is called the proper orthochronous Lorentz group because of the properties given by Eqs. (1.34) and (1.35). One may consider other possible conditions of  such as det = −1, det = −1, det = +1,

00 ≥ 1, 00 ≤ −1, 00 ≤ −1.

(1.41) (1.42) (1.43)

One finds interestingly that these three other conditions lead to the discrete groups. Equation (1.41) corresponds to the space inversion or Parity (P). Equation (1.42) corresponds to the time reversal (T). Equation (1.43) corresponds to the combination of P and T, which one may call as the charge conjugation (C) by assuming the CPT invariance of the theory. Thus, we find that the most general groups of the space and time are the (continuous) Poincaré group and the discrete groups of C, P, T. For the Poincaré group, one can show that the group elements given by Eq. (1.34) satisfy the four group properties as follows: 1. 2. 3. 4.

(2 , a 2 )(1 , a 1 ) = (2 1 , 2 a 1 †2 + a 2 ); closure (I , 0); idenity −1 † −1 −1 −1 (, a) = ( , − a( ) ; inverse (3 , a 3 ){(2 , a 2 )(1 , a 1 )} = {(3 , a 3 )(2 , a 2 )}(1 , a 1 ); associativity

(1.44)

Now, in order to introduce infinitesimal generators of the Poincaré group, let’s use the unitary representation of ISL(2, C) given by  ω),  a) = Exp(−i( J · θ + K · ω  + P μ · aμ )). U ((θ,

(1.45)

1.4 Poincaré Group

13

Then, the infinitesimal generators are given by ∂ U (, a)|ω=  θ=a μ =0 , ∂ω j ∂ J j = i j U (, a)|ω=  μ =0 ,  θ=a ∂θ ∂ P μ = ig μν ν U (, a)|ω=  θ=a μ =0 . ∂a

Kj =i

(1.46) (1.47) (1.48)

Among these 10 infinitesimal generators, there are 45 commutation relations followed from ISL(2, C) group properties. In order to show the derivation of these commutation relations, let me give an example for the derivation of the following commutation relation: [J 2 , K 1 ] = −i K 3 .

(1.49)

First, from the definition of  given by Eq. (1.32), one can consider the following two infinitesimal transformations: i 1 ≈ 1 − (iω1 )σ1 + O(ω12 ) 2 i 2 ≈ 1 − (θ2 )σ2 + O(θ22 ). 2

(1.50)

Then, one can obtain the following product given by 1 2 1 −1 2 −1 1 = 1 − ω1 θ2 [(iσ1 )(σ2 ) + (iσ1 )(−σ2 ) + (σ2 )(−iσ1 ) + (−iσ1 )(−σ2 )] + · · · 4 i (1.51) = 1 − ω1 θ2 (iσ3 ) + · · · 2 Next, one can obtain the corresponding unitary representation using Eqs. (1.32) and (1.45); U (1 2 1 −1 2 −1 )

i = U 1 − ω1 θ2 (iσ3 ) + · · · 2 = 1 − iω1 θ2 K 3 + · · ·

(1.52)

On the other hand, we know from Eqs. (1.32), (1.45) and (1.50) that U (1 , 0) = 1 − iω1 K 1 + O(ω12 ) U (2 , 0) = 1 − iθ2 J 2 + O(θ22 ).

(1.53)

14

1

Introduction

Therefore, we obtain U (1 2 1 −1 2 −1 ) = U (1 )U (2 )U (1 −1 )U (2 −1 ) = 1 + ω1 θ2 [(−i K 1 )(−i J 2 ) +(−i K 1 )(i J 2 ) + (−i J 2 )(−i K 1 ) + (i K 1 )(i J 2 )] + · · ·

(1.54)

By comparing Eqs. (1.52) and (1.54), we find that J 2 K 1 − K 1 J 2 = [J 2 , K 1 ] = −i K 3 ,

(1.55)

which is the commutation relation given by Eq. (1.49). In this way, one can find all 45 commutation relations among 10 generators as summarized below: [J j , J k ] = −iε jkl J l

(1.56)

[K j , K k ] = −iε jkl J l

(1.57)

[J j , K k ] = iε jkl K l

(1.58)

[P μ , P ν ] = 0

(1.59)

[K j , P 0 ] = −i P j

(1.60)

[J j , P 0 ] = 0

(1.61)

[K k , P j ] = −iδ jk P 0

(1.62)

[J j , P k ] = iε jkl P l ,

(1.63)

where j, k, l run from 1 to 3, μ, ν run from 0 to 3 and ε jkl is the Levi-Civita symbol. Some of these commutation relations have significant physical meanings. Equation (1.56) shows the closure among the angular momentum operators, meaning that the rotation group is subgroup of the Poincaré group. Likewise, Eq. (1.59) shows that the translation group is also a subgroup. The subgroup of translation is indeed an invariant subgroup due to Eqs. (1.60)–(1.63). Furthermore, Eq. (1.61) shows that the Hamiltonian (or energy) and the angular momentum can be simultaneously identified. Equation (1.57) has an important implication to the Thomas precession because it shows that a boost in one direction combined with another boost in another direction yield a rotation. More physical implications are followed by the consideration of quantum theory. However, note that we have not yet introduced a quantum theory. The derivation of the above commutation relations is not based on the quantum theory but based on just the group properties of the space-time. Thus, it is remarkable that the above commutation relations are valid regardless of classical or quantum level. The concept of quantization is established only after we identify the ten infinitesimal generators as physical quantities such as angular, linear and boost momenta and

1.4 Poincaré Group

15

energy. Until then, we emphasize that the above commutation relations are equally valid in classical and quantum levels. A final remark on these commutation relations is that one can further simplify the expressions by introducing covariant notations. By introducing a covariant angular momentum tensor J μν given by ⎤ 0 K1 K2 K3 ⎢ −K 1 0 J 3 −J 2 ⎥ ⎥ =⎢ ⎣ −K 2 −J 3 0 J 1 ⎦ , −K 3 J 2 −J 1 0 ⎡

J μν

(1.64)

where J 0 j = K j , J jk = ε jkl J l and J μν = −J νμ , one can summarize Eqs.(1.56)– (1.63) as [J αβ , J ρσ ] = −i(g βσ J αρ − g βρ J ασ + g αρ J βσ − g ασ J βρ ),

(1.65)

[P μ , P ν ] = 0,

(1.66)

[J ρσ , P μ ] = −i(g μρ P σ − g μσ P ρ ).

(1.67)

J μν

Note here that is an antisymmetric second rank tensor and has a very similar  and structure as the electromagnetic field strength tensor F μν in which electric ( E)  field correspond to the boost ( K ) and the angular momentum ( J) genmagnetic ( B) erators. Likewise, the four-vector potential Aμ corresponds to the four-momentum P μ. Again, the quantum commutation relations are exactly same as Eqs. (1.65)–(1.67) by assigning the four-momentum P μ and the angular momentum J μν as P μ = i

∂ , ∂ xμ

(1.68)

and J μν = x μ P ν − x ν P μ

∂ ∂ . = i x μ − xν ∂ xν ∂ xμ

(1.69)

In another words, one can derive Eqs. (1.65)–(1.67) using only the quantum mechanical relations given by Eqs. (1.68) and (1.69). This confirms our previous assertion that the quantum interpretation of the Poincaré algebra comes only after one identifies the Poincaré generators as physical operators that measure the corresponding physical quantities.

16

1

Introduction

References 1. 2. 3. 4. 5. 6. 7. 8.

Smith, J.: Introduction to Special Relativity. Benjamin, New York (1967) Taylor, E., Wheeler, J.: Spacetime Physics. Freeman, San Francisco (1966) Hagedorn, R.: Relativistic Kinematics. Benjamin, New York (1964) Griffiths, D.J.: Introduction to Elementary Particles. Wiley, Canada (1987) Jackson, J.D.: Classical Electrodynamics, 2nd edn. Wiley (1975) Stillwell, J.: Naive Lie Theory. Springer, New York (2008) Gilmore, R.: Lie Groups, Physics, and Geometry. Cambridge University Press (2008) Hall, B.C.: Lie Groups, Lie Algebras and Representations, 2nd edn. Springer, New York (2015)

2

Interpolation Between Instant Form Dynamics and Light-Front Dynamics

Having introduced the special relativity, we now apply it to quantum field theories. The form of the Hamiltonian dynamics which respects the relativistic causality is not unique but dependent on the evolution parameter taken for the description of the physical system. Typically, the ordinary time t is taken as the evolution parameter. Such form of the Hamiltonian dynamics is well known as the instant form dynamics (IFD). However, the IFD is not the only form available for the description of the dynamics but other well qualified forms √ are available on par with the IFD. In particular, the light-front time (t + z/c)/ 2 can be taken as the evolution parameter as we discuss in the present chapter adopted from our published work Phys. Rev. D87065015(2013).

2.1

Dirac’s Proposition

When the particle systems have the characteristic momenta which are of the same order or even much larger than the masses of the particles involved, it is part of nature that a relativistic treatment is called for in order to describe those systems properly. In particular, relativistic effects are most essential to describe the lowlying hadron systems in terms of strongly interacting quarks–antiquarks and gluons in quantum chromodynamics (QCD). For the study of relativistic particle systems, Dirac proposed the three different forms of the relativistic Hamiltonian dynamics √ in 1949 [1], i.e. the instant (x 0 = 0), front (x + = (x 0 + x 3 )/ 2 = 0) and point (xμ x μ = a 2 > 0, x 0 > 0) forms. While the instant form dynamics (IFD) of quantum field theories is based on the usual equal time t = √ x 0 quantization (c = 1 unit is taken here), the equal light-front time τ ≡ (t + z/c)/ 2 = x + quantization yields the front form dynamics, more commonly called light-front dynamics (LFD), correspondingly. Although the point form dynamics has also been explored [2–4], the

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 C.-R. Ji, Relativistic Quantum Invariance, Lecture Notes in Physics 1012, https://doi.org/10.1007/978-981-19-7949-1_2

17

18

2

Interpolation Between Instant Form Dynamics and Light-Front Dynamics

most popular choices were thus far the equal-t (instant form) and equal-τ (front form) quantizations. A crucial difference between the instant form and the front form may be attributed to their energy–momentum dispersion relations. When a particle has the mass m and the four-momentum k = (k 0 , k 1 , k 2 , k 3 ), the relativistic energy–momentum dispersion relation of the particle at equal-t is given by  k0 =

k2 + m 2 ,

(2.1)

where the energy k 0 is conjugate to t and the three-momentum vector k is given by k = (k 1 , k 2 , k 3 ). However, the corresponding energy–momentum dispersion relation at equal-τ is given by k− =

2 + m2 k⊥ , k+

(2.2)

√ where the light-front energy k − conjugate to√τ is given by k − = (k 0 − k 3 )/ 2 and the light-front momenta k + = (k 0 + k 3 )/ 2 and k⊥ = (k 1 , k 2 ) are orthogonal to k − . While the instant form (Eq. (2.1)) exhibits an irrational energy–momentum relation, the front form (Eq. (2.2)) yields a rational relation and thus the signs of k + and k − are correlated, e.g. the momentum k + is always positive when the system evolve to the future direction (i.e. positive τ ) so that the light-front energy k − is positive. In the instant form, however, no sign correlations for k 0 and k exist. Such a dramatic difference in the energy–momentum dispersion relation makes the LFD quite distinct from other forms of the relativistic Hamiltonian dynamics. The light-front quantization [1,5] has already been applied successfully in the context of current algebra [6–9] and the parton model [10–12] in the past. With further advances in the Hamiltonian renormalization program [12,13,13–16], LFD appears to be even more promising for the relativistic treatment of hadrons. In the work of Brodsky, Hiller and McCartor [17–20], it is demonstrated how to solve the problem of renormalizing light-front Hamiltonian theories while maintaining Lorentz symmetry and other symmetries. The genesis of the work presented in [17–20] may be found in [19,20], and additional examples including the use of LFD methods to solve the bound-state problems in field theory can be found in the review of QCD and other field theories on the light-cone [21]. These results are indicative of the great potential of LFD for a fundamental description of non-perturbative effects in strong interactions. This approach may also provide a bridge between the two fundamentally different pictures of hadronic matter, i.e. the constituent quark model (CQM) (or the quark parton model) closely related to experimental observations and the QCD based on a covariant non-abelian quantum field theory. Again, the key to possible connection between the two pictures is the rational energy–momentum dispersion relation given by Eq. (2.2) that leads to a relatively simple vacuum structure. There is no spontaneous creation of massive fermions in the LF quantized vacuum. Thus, one can immediately obtain a constituent-type picture [22] in which all partons in a hadronic state are connected directly to the hadron instead of being simply disconnected excitations

2.1 Dirac’s Proposition

19

(or vacuum fluctuations) in a complicated medium. A possible realization of chiral symmetry breaking in the LF vacuum has also been discussed in the literature [23,24]. Moreover, the Poincaré algebra in the ordinary equal-t quantization is drastically changed in the light-front equal-τ quantization. In LFD, the maximum number (seven) of the ten Poincaré generators is kinematic (i.e. interaction independent), and they leave the state at τ = 0 unchanged [13]. However, the transverse rotation whose direction is perpendicular to the direction of the quantization axis z at equal τ becomes a dynamical problem in LFD because the quantization surface τ is not invariant under the transverse rotation and the transverse angular momentum operator involves the interaction that changes the particle number [25]. Leutwyler and Stern showed that the angular momentum operators can be redefined to satisfy the SU(2) spin algebra and the commutation relation between mass operator and spin operators [26]: [Ji , J j ] = iεi jk Jk ,

(2.3)

[M, J] = 0.

(2.4)

Nonetheless, in LFD, there are two dynamic equations to solve:

and

    2 2 = S H (S H + 1)  H ; p + , p⊥ J 2  H ; p + , p⊥

(2.5)

    2 2 = m H 2  H ; p + , p⊥ , M 2  H ; p + , p⊥

(2.6)

where the total angular momentum (or spin) and the mass eigenvalues of the hadron (H ) are given by S H and m H . Thus, it is not a trivial matter to specify the total angular momentum of a specific hadron state. As a step toward understanding the conversion of the dynamical problem from boost to rotation, we constructed the Poincaré algebra interpolating between instant and light-front time quantizations [27]. We used an orthogonal coordinate system which interpolates smoothly between the equal time and the light-front quantization hypersurface. Thus, our interpolating coordinate system had a nice feature of tracing the fate of the Poincaré algebra at equal time as the hypersurface approaches to the light-front limit. The same method of interpolating hypersurfaces has been used by Hornbostel [28] to analyze various aspects of field theories including the issue of nontrivial vacuum. The same vein of application to study the axial anomaly in the Schwinger model has also been presented [29], and other related works [30–33] can also be found in the literature. In the present work, we introduce the interpolating scattering amplitude that links the corresponding time-ordered amplitudes between the two forms of dynamics. We exemplify the physical meaning of the kinematic transformations in contrast to the dynamic transformations by means of checking the invariance of each individual time-ordered amplitude for an arbitrary interpolation angle. Our analysis further clarifies why and how the longitudinal boost is kinematical only in the LFD but not in any other interpolation angle dynamics including IFD. In particular, we show

20

2

Interpolation Between Instant Form Dynamics and Light-Front Dynamics

the disappearance of the connected contributions to the current arising from the vacuum when the interpolation angle is taken to yield the LFD. Since we don’t need any infinite momentum frame (IMF) to show this disappearance and our proof is completely independent of reference frames, it resolves the confusion between the LFD and the IMF that sometimes appears in the discussion on related topics. The well-known utility of IMF usually discussed in the instant form dynamics is now also extended to any other interpolation angle dynamics using our interpolating scattering amplitudes.

2.2

Interpolating Scattering Amplitudes

We begin by adopting the following convention of the space-time coordinates to define the interpolating angle: 

 x+ − x





cos δ sin δ = sin δ − cos δ



 x0 , x3

(2.7)

and 

x0 x3



 =

cos δ sin δ sin δ − cos δ



 x+  , x−

(2.8)

in which the interpolating angle is allowed to run from 0 through 45◦ , 0 ≤ δ ≤ π4 . All the indices with the widehat notation signify the variables with the interpolation   = x 0 and x − = −x 3 so that we recover the angle δ. For the limit δ → 0 we have x + covariant, z-axis reflected space-time while for the other extreme limit, δ → π4 we √  = (x 0 ± x 3 )/ 2 ≡ x ± which leads to the standard light-front coordinates. have x ± Of course, the same interpolation applies to the momentum variables: 

 p+  p−



 =

cos δ sin δ sin δ − cos δ



 p0 . p3

(2.9)

μ and b μ , the scalar product a b μ For any two interpolating four-vector variables a   μ μ must be identical to aμ b and is given by  

 

 

 

 

 

1 1 2 2 + + − − + − − +  μ a μ b = (a b − a b ) cos 2δ + (a b + a b ) sin 2δ − a b − a b . (2.10)

We may define C = cos 2δ, S = sin 2δ, 

(2.11) 

1 ˆ + a 2 yˆ , a⊥  =a x

2.2 Interpolating Scattering Amplitudes

21

Fig. 2.1 Scattering amplitude of spinless particles

for shorthand notations and convenience, so that the Minkowski space-time metric μ ν = (+ , − ,  1,  2) with interpolating angle may be written as g ⎡

⎤ C S 0 0 ⎢ S −C 0 0 ⎥ μ ν ⎥ g =⎢ μ ν. ⎣ 0 0 −1 0 ⎦ = g 0 0 0 −1

(2.12)

Thus, the covariant and contravariant indices are related by 





 +

 −

 −

+ − + a+  = Ca + Sa ; a = Ca+  + Sa− 

(2.13)

a−  = Sa − Ca ; a = Sa+  − Ca−   aj = −a j , ( j = 1,  2).

As the coordinate variable x + plays the role of the time evolution parameter and  to be the canonical conjugate energy variable is p+ = p − in LFD, we also take x + the evolution parameter and the conjugate energy variables with the corresponding subscripts, e.g. q+ . Now, we discuss the scattering amplitude of two spinless particles, e.g. an analog of the well-known QED annihilation/production process e+ e− → μ+ μ− in a toy φ 3 model theory, as depicted in Fig. 2.1. In this work, we do not involve spins and any other degrees of freedom except the fundamental degrees of freedom, i.e. particle momenta, for the simplest possible illustration. Although we discuss here just this simple scattering amplitude, the bare-bone structure that we demonstrate in this analysis will be commonly applicable to any further complicated amplitudes including other degrees of freedom. In particular, the main points that we discuss from the energy denominator structure in this simple amplitude will prevail in other complicated amplitudes since not only the basic structure of the amplitudes but also the fundamental degrees of freedom to describe the scattering process are applicable to any quantum field theories. Further complications from other degrees of freedom beyond the particle momenta would appear separately without modifying the energy denominator structure that we discuss in this work. For example, the terms associated with the spin degrees of freedom in QED would appear as the matrix elements in the numerator but not in the denominator of the amplitude. The extension of the present work to the gauge field theories

22

2

Interpolation Between Instant Form Dynamics and Light-Front Dynamics

Fig. 2.2 Time-ordered amplitudes in IFD for the Feynman amplitude depicted in Fig. 2.1

involving other degrees of freedom such as QED and QCD is presented in the forthcoming chapters, Chaps. 3 and 4, respectively. In this section, we will focus on the basic structure of the scattering amplitudes, i.e. the energy denominators, considering only the fundamental degrees of freedom, i.e. particle momenta. Modulo inessential factors including the square of the coupling constant, the lowest-order tree-level Feynman diagram shown in Fig. 2.1 is proportional to the propagator of the intermediate particle, that is =

1 s − m2

(2.14)

where s = ( p1 + p2 )2 is the Mandelstam variable which is invariant under any Poincaré transformations and m is the mass of the intermediate boson. Of course, the physical process can take place only above the threshold s > 4M 2 , where M is the mass of the final particle and antiparticle that are produced, e.g. like the muon mass in the e+ e− → μ+ μ− scatterring process. In the instant form dynamics (IFD), where the initial conditions are set on the hyperplane t = 0 and the system evolves with the ordinary time t > 0, this manifestly covariant Feynman amplitude is decomposed into the corresponding two time-ordered amplitudes, graphically represented in Fig. 2.2a and b. These two time-ordered amplitudes correspond respectively to the following analytic expressions:   1 1 a , (2.15) IFD = 0 2q p10 + p20 − q 0 and b IFD

1 =− 0 2q



1 0 p1 + p20 + q 0

 .

(2.16)

2.2 Interpolating Scattering Amplitudes

23

It is not difficult to show that the sum of the time-ordered amplitudes is identical to the manifestly covariant Feynman amplitude: IFD = aIFD + bIFD   1 1 1 = 0 − 0 2q p10 + p20 − q 0 p1 + p20 + q 0 =

1 , s − m2

(2.17)

where the conservation of the three momentum p1 + p2 = q as well as the energy–  momentum dispersion relation q 0 = q 2 + m 2 in IFD is used to get the covariant denominator s − m 2 in the last step. To obtain the corresponding time-ordered amplitudes in an arbitrary interpolating angle δ, we just need to change the superscript 0 of the IFD energy variables in ˆ and multiply an overall factor C to the the energy denominators to the superscipt + amplitudes, i.e.   C 1 a , (2.18) δ =     2q + p1+ + p2+ − q + and δb

1 =−  2q +



C 



 p1+ + p2+ + q +

 .

(2.19)

The overall factor C is necessary because the physical energy of the particle with μ in an arbitrary interpolation angle is given by p and as the four-momentum p  +  + + S p should be used to evaluate the interpolating shown in Eq. (2.13) p = C p+   − amplitudes. Note here that the factor S in front of the longitudinal momentum p−  is irrelevant because the longitudinal momenta of the initial particles must be canceled by the longitudinal momentum of the intermediate particle due to the conservation of longitudinal momentum. Again, it is not so difficult to show that the sum of the timeordered amplitudes for any angle δ is identical to the manifestly covariant Feynman amplitude: δ = δa + δb   C 1 C = −        2q + p1+ + p2+ − q + p1+ + p2+ + q + 1 = , s − m2

(2.20)

where we used the relation between the covariant and contravariant indices (see Eq.  = Cq+ (2.13)) such as q +  + Sq−  and the conservation of momenta p1−  + p2 −  =  1⊥  2⊥ ⊥ q−  and p  as well as the four-momentum scalar product relation (see +p  =q Eq. (2.10)) to get the covariant denominator s − m 2 in the last step. It is also rather

24

2

Interpolation Between Instant Form Dynamics and Light-Front Dynamics

easy to see that Eq. (2.20) becomes Eq. (2.17) as C goes to the unity. In LFD a , i.e. however, i.e. as C goes to zero, the denominator in the first amplitude δ=π/4 



 ) = 1/( p1+ + p2+ − q + ) goes to infinity due to the conservation 1/( p1+ + p2+ − q + + + + p1 + p2 = q but the multiplication of C = 0 with this infinity makes the finite b is wiped out due to C = 0. result 1/(s − m 2 ), while the second amplitude δ=π/4 This result is akin to the very well-known result from the work “Dynamics at Infinite Momentum” [7]. However, we would like to make it clear that the disappearance of b in LFD is different from what has been known from the second amplitude δ=π/4 IMF, i.e. when Pz → ±∞ with P ≡ p1 + p2 for a shorthand notation (e.g. P 2 = s). As far as any correlation between the interpolation angle δ and the total longitudinal momentum Pz is avoided, our derivation is completely independent of the frame and the only relevant parameter to show this disappearance is the interpolation angle δ which has nothing to do with the choice of reference frame. In Sect. 2.4, we will discuss the special case with a particular correlation between δ and Pz and the associated treacherous point similar to the zero-mode issue in LFD. For the rest of this section, we elaborate more details of our derivations discussed above. The dispersion relation q 2 = m 2 in terms of interpolating angle variables results in a quadratic equation in q+  and q−  that can be solved for the energy variable  ⊥ as well as mass m: in terms of momentum components q q+   and q −

q+  =

−Sq−  ± ωq , C

(2.21)

in which we introduced the notation  ωq =

  2 +C q 2 + m2 .  q−   ⊥

(2.22)

For the physical solution with positive energy in Eq. (2.21), we must take q+  =

−Sq−  + ωq , C

(2.23)

which identifies ωq as 

+ ωq = Cq+ (2.24)  + Sq−  =q .  For δ = 0 and δ = π4 , ωq becomes q 0 = q 2 + m 2 and q + = q 0 + q 3 , respectively. Using this variable ωq , we may rewrite Eqs. (2.18) and (2.19) as follows:

1 2ωq D+ 1 δb = , 2ωq D−

δa =

(2.25)

2.2 Interpolating Scattering Amplitudes

25

where Sq−  − ωq C Sq−  + ωq D− = P+ , + C D+ = P+ +

(2.26)

in which we used the momentum conservation P−  = ( p1 )−  + ( p2 )−  = q−  . The sum of both contributions given by Eq. (2.20) can then be expressed as δ = δa + δb   1 1 1 = − , Sq−  +ωq 2ωq P + Sq− −ωq P+ + + C C

(2.27)

which is identical to the second line of Eq. (2.20). In Eq. (2.27), we can confirm δ = 1/(s − m 2 ): δ =  = = =

1 C

P+ +

2 Sq−  C

−

 ωq 2 C

1 2 CP+  q− +  + 2SP+

2 S2 q−  C

−

ωq2 C

1 2 CP+ 

2  2 − m2 + 2SP+  P−  − CP−  − P⊥ 

1 , s − m2

(2.28)

2 2 + m 2 ), P = q and P   = q  . Using Eq. (2.27), where we used ωq2 = q− q⊥   − − ⊥ ⊥  + C(  we may now recapture the instant form and light-front limits, as follows. For the instant form limit (IFD), we have δ → 0 (i.e. C → 1 and S → 0) and ωq → q+  . In this limit, it is apparent that Eq. (2.27) becomes

 1 1 , − P0 − q0 P0 + q0 (2.29) where δ = 0 is taken in the interpolating angle variables given by Eq. (2.9). For the light-front limit (LFD), δ → π4 (i.e. C → 0 and S → 1), we expand ωq given by Eq. (2.22) in the orders of C and get δ→0 ≡  I F D =

1 2q+ 



1 1 − P+ − q P + q+     + +

ωq → q− +



  2 + m2 C q ⊥  2q− 

=

1 2q0



+ O (C2 ).

(2.30)

26

2

Interpolation Between Instant Form Dynamics and Light-Front Dynamics

Substituting this expansion of ωq in the denominator of the first term in Eq. (2.27), we get q 2 + m 2 Sq−  − ωq + O (C) →− ⊥ C 2q−  →−

2 + m2 q ⊥ 

2q− 

as C → 0.

(2.31)

For the second denominator in Eq. (2.27), however, we get q 2 + m 2 Sq− 2  + ωq + O (C) → − ⊥ C C 2q−  → ∞ as C → 0.

(2.32)

Thus, in the light-front limit (C → 0), the contribution from the second diagram vanishes and δ→ π4 =

1  2q− 

1 P+ −

2 +m 2 ) ( q⊥  2q− 

=

1  P+

1 P−

−

 2 +m 2 ) (P ⊥ 2P +

,

(2.33)

+  ⊥ , respectively, due ⊥  ⊥ are same with P + and P where q−  → q− = q and q  →q to the momentum conservation in LFD. Again, we would like to make it clear that b in LFD is different from what has the disappearance of the second amplitude δ=π/4 been known from IMF, i.e. when Pz → ±∞. As we will discuss in the next section, Sect. 2.3, the longitudinal boost is kinematic in LFD so that the disappearance of the b connected contribution δ→ π to the current arising from the vacuum is independent 4 of Pz or the IMF. This is certainly not the case for any other interpolation case, i.e. δ = π/4. The longitudinal boost becomes dynamic for δ = π/4, the contributions from δa and δb depend on Pz (or the reference frames), and the well-known utility of IMF can be extended to an arbitrary interpolating angle 0 ≤ δ < π4 . We will discuss more on this point in Sect. 2.4 after we present the physical meaning of the kinematic transformations in Sect. 2.3.

2.3

Kinematic Transformations of Particle Momenta

As we presented in the previous section, Sect. 2.2, the sum of all the time-ordered amplitudes (just two in our example discussed in Sect. 2.2) must be independent of

2.3 Kinematic Transformations of Particle Momenta

27

the interpolation angle δ and identical to the manifestly covariant Feynman amplitude. Although the total amplitude is Poincaré invariant, the individual time-ordered amplitude is neither invariant in general nor independent of δ. Thus, one may ask a question if the individual time-ordered amplitude can be invariant at least under some subset of Poincaré generators. The answer is yes and this issue is what we would like to address in this section. The point is that the individual time-ordered ampli does’nt change tude would not change as far as the time evolution parameter x + so that the individual time-ordered amplitude would be invariant under a certain  . To the extent transformation which does’nt alter the time evolution parameter x +  + that the time evolution parameter x does’nt change, all the momentum components   such as q + would not change because the same transformation rules apply with + to both the space-time coordinates and the four-momenta of the particles involved. Such subset of the Poincaré group that does’nt alter the time evolution parameter  is known as the stability group. Since the transformations that belong to the x+  , each time-ordered stability group do not modify the time evolution parameter x + amplitude must be invariant under these transformations. Individual time-ordered amplitudes represent the dynamics given at each instant of time defined by the time  in the given form of the relativistic quantum field theory. For evolution parameter x + this reason, it may be appropriate for the transformations that leave each individual time-ordered amplitudes invariant to be called as the kinematic transformations and the generators of those transformations belong to the stability group deserve to be distinguished from the other Poincaré group generators. All other Poincaré group generators besides the kinematic generators are dynamical and change the contributions from each individual time-ordered amplitudes. In this section, we discuss the kinematic transformations for an arbitrary interpolation angle δ. In particular, we take the limits to δ = 0 and π/4 to discuss the fates of the kinematic transformations in the two distinguished forms of the relativistic dynamics, IFD and LFD, respectively. Since we focus mainly on the fundamental dynamic variables not involving any other degrees of freedom such as spins in this section, our results of the kinematic transformations apply explicitly only to the particle momenta. The matrix of the homogeneous part of Poincaré group in the interpolating angle basis may be written [27] as ⎡



0 K 3 D1 ⎢  3 0 K1 ⎢ −K M μ ν =⎢   ⎣ −D 1 −K 1 0   −D 2 −K 2 −J 3

⎤  D2 ⎥ K 2⎥ ⎥ J3 ⎦ 0

(2.34)

where K

 1  1

= −K 1 sin δ − J 2 cos δ ; K

 2

 2

= J 1 cos δ − K 2 sin δ

D = −K 1 cos δ + J 2 sin δ ; D = −J 1 sin δ − K 2 cos δ .

(2.35)

28

2

Interpolation Between Instant Form Dynamics and Light-Front Dynamics

  The kinematic generators K j ,  j = (1, 2) and the dynamic ones D j ,  j = (1, 2)   j j can also be written as the combinations of E and F :

K



 1









= CF 1 − SE 1 ; K

 2







= CF 2 − SE 2 



D 1 = −SF 1 − CE 1 ; D 2 = −SF 2 − CE 2 ,

(2.36)

where 







E 1 = J 2 sin δ + K 1 cos δ ; E 2 = K 2 cos δ − J 1 sin δ F 1 = K 1 sin δ − J 2 cos δ ; F 2 = J 1 cos δ + K 2 sin δ . 

(2.37)



The interpolating operators E j and F j coincide with the usual E j and F j of LFD in the limit δ = π/4. As discussed in Ref. [27], the transverse boosts (K 1 , K 2 ) are dynamic whereas the transverse rotations (J 1 , J 2 ) are kinematic in IFD (δ = 0), while the LF transverse boosts (E 1 , E 2 ) are kinematic whereas the LF transverse rotations (F 1 , F 2 ) are dynamic in LFD (δ = π4 ). One may note the swap of the roles between “boosts” and “rotations” in the two forms of relativistic dynamics, IFD and LFD, and utilize it for some hadron phenomenology [34].  j We may check  that the generators K given above satisfy the commu explicitly    + j tation relation K , P = 0 with the momentum operator P + using Eq. (2.36) and the interpolating Poincaré algebra presented in Ref. [27] : 

           K j , P+ = C F j , P+ − S E j , P+       = C −i P j S − S −i P j C = 0.

(2.38)

 This means that each transformation of the form exp (−iωK j ), ( j = 1,  2) leaves   + is an the momentum operator P invariant. As a consequence if the momentum P +  + eigenvalue of the operator P , it remains invariant under the cited transformations. ) component of any four-vector is invariant under such transLikewise, the plus (+  remains invariant as well. It verifies that the generators formations, and the time x +  j K are kinematic.  j In asimilar way,  for the generators D , we may check explicitly that the commu  j + tators D , P are now non-vanishing:

            D j , P + = −S F j , P + − C E j , P +        = −S −i P j S − C −i P j C = i P j .

(2.39) 

Since commutators above are not only non-vanishing but also proportional to P j ,  j = 1,  2) no longer leaves the each transformation of the form exp (−iωD j ), (

2.3 Kinematic Transformations of Particle Momenta

29

 momentum P + invariant developing transverse components of the momentum and  j thus D are dynamic generators. Among the elements involved in the matrix given by Eq. (2.34), it is interesting to note that the rotation around the longitudinal direction, i.e. J 3 , is unique because  and thus kinematic for any interpolation angle δ. However, it does’nt change x + the longitudinal boost K 3 has a quite different characteristic compared to any other  and operators in Eq. (2.34). To see this, let’s look at the commutator between P + 3 K in the Poincaré algebra:



  P + , K 3 = i P−      = i SP + − CP − ,

(2.40)

 + 3 P , K = iP+

(2.41)

which leads to

in the limit δ → π4 . This shows that the longitudinal boost has a distinguished property in the limit δ → π4 , namely it becomes kinematic in this limit. Although the right-hand side of Eq. (2.41) does’nt vanish but yield the same P + operator in the commutation relation. This means that the eigenvalues of P + operator, or the LF longitudinal momentum P + , are just scaled by the factor eβ3 when it is boosted in the longitudinal direction by the rapidity β3 . By the same token, the LF energy P − is scaled by the factor e−β3 under the same transformation due to the commutation relation in LFD, 

 P − , K 3 = −i P − .

(2.42)

It may be interesting to note that the algebra among P + , P − and K 3 works just the same way as the algebra among the creation, annihilation and number operators in one-dimensional simple harmonic oscillator. Due to the conservation of three momenta (P + , P⊥ ) as well as the compensating scale factors of e−β3 and eβ3 between the LF energy (P − ) and the LF longitudinal momentum (P + ), one can show that each individual LF time-ordered amplitudes is invariant under the longitudinal boost K 3 . This may be also understood from the intactness of the LF time x + modulo the same β scaling factor e3 for the LF longitudinal momentum under the K 3 operation. With this reasoning, one may understand that K 3 becomes the kinematic generator in LFD although it is dynamical for any other interpolation angle 0 ≤ δ < π/4. As the boost problem in IFD is one of the most difficult problems to deal with in the relativistic many-body calculations, all of the boost operators (K 1 , K 2 , K 3 ) have been known as difficult operators in IFD. Since at least K 3 can change its difficult characteristic to a good one, i.e. from dynamical to kinematical, and joins the stability group in LFD, one may regard such dramatic change of K 3 nature in LFD as a kind of “return of a prodigal son” and the community of LFD welcomes the addition of K 3 in the stability group. For this reason, the number of kinematic generators in LFD is one

30

2

Interpolation Between Instant Form Dynamics and Light-Front Dynamics

Table 2.1 Kinematic and dynamic generators for different angles Angle

Kinematic

δ=0

 K1

P1,

0, then the particle momentum state is changed to the state  P  , where |P and  P  are the eigenstates of the operator  P μ with the eigenvalues of P μ and P μ , respectively. From this, one can find that the  operation of T3† P μ T3 and P μ to the state |P yields the eigenvalues P μ, μ and P respectively. Thus, the results given in Eq. (2.44) can be translated into  P+  + C sinh β3 P−   = (cosh β3 − S sinh β3 ) P+

 P−  + C sinh β3 P+   = (cosh β3 + S sinh β3 ) P−

  P  j = P j , ( j = 1,  2) .

(2.45)

32

2

Interpolation Between Instant Form Dynamics and Light-Front Dynamics

This result satisfies the energy–momentum dispersion relation as it should: 2

  μ ν        P P −  − CP −  −P  + 2SP + ⊥ μ g P ν = CP + 2

2

2 2 2  = CP+  P−  − CP−  + 2SP+  − P⊥

= M2

(2.46)

Taking the limit δ → 0 in Eq. (2.45), we get P 0 = cosh β3 P 0 + sinh β3 P 3 P 3 = cosh β3 P 3 + sinh β3 P 0 P  j = P j , ( j = 1, 2) ,

(2.47)

which are the usual Lorentz transformations along the z-direction in IFD. Taking the limit δ → π4 , on the other hand, we get P − = e−β3 P − P + = eβ3 P + P  j = P j , ( j = 1, 2) ,

(2.48)

which are the expected results in LFD since P + and P − are decoupled with the corresponding scaling factors. This result confirms that T3 is kinematical in LFD. After the T3 (longitudinal) transformation, we now take the T12 (transverse) transformation following the Jacob and Wick’s procedure as mentioned above. In Ref.[27], the effect of T12 transformation on the momentum operator P μ was obtained as follows:  (1 − cos α) sin α    1 2 β P − S P + β P  1 2 − α2 α   sin α   1 β1 P + β2 P 2 = P−  cos α + C α  sin α (cos α − 1)     1 2 β , = P j − βj P− P + β P  + Cβ j 1 2 α α2 ( j = 1, 2) (2.49)

† 2 P+ T12  T12 = P+  + Sβ⊥ † T12 P−  T12 

† T12 P j T12

  2 . It is interesting to note that this where we have defined α = C(β12 + β22 ) = Cβ⊥ result indicates a dramatic difference in the outcome of the particle momentum after   1 2 the application of the kinematic transformation T12 = e−i(β1 K +β2 K ) to the particle in the rest frame between IFD (δ = 0) and LFD (δ = π/4). The particle of mass M in the rest frame (i.e. P 0 = M, P = 0) has the interpolating momentum compo  = 0. If we write the interpolating nents given by P+  = M cos δ, P−  = M sin δ, P ⊥

2.3 Kinematic Transformations of Particle Momenta

33

momentum components with the prime notation after the T12 transformation, we get    2 (1 − cos α)  P+ = M cos δ + S β sin δ  ⊥ α2  P−  = M sin δ cos α sin α  P  j = −Mβ j sin δ ( j = 1,  2) , α

(2.50)

which shows that the particle can gain some longitudinal momentum although the transformation T12 is transverse and the amount of the gained longitudinal momentum depends on the interpolating angle δ. In IFD (δ = 0), the particle in the rest frame remains in the rest frame since T12 is just a transverse rotation, i.e. P 0 = M, P  = 0. However, in LFD (δ = π4 ), the result given by Eq. (2.49) can be written as P

−

P + P j

  2 β⊥ M = √ 1+ 2 2 M = √ 2 M = −√ βj ( j = 1, 2). 2

(2.51)

From this, we find the energy and longitudinal √ momentum components are related to the transverse momentum P ⊥ = −M β⊥ / 2, i.e. P 0 = M + P

3

2 P ⊥ 2M

2 P ⊥ =− 2M

(2.52) 2 P

which shows that the particle gains the longitudinal momentum − 2M⊥ while the par1 2 = ei(β1 E +β2 E ) . One should ticle is transversely boosted by T12 √ √ note that the LF transverse boosts E 1 = (J 2 + K 1 )/ 2 and E 2 = (K 2 − J 1 )/ 2 involve not only K 1 , K 2 (ordinary transverse boosts) but also J 1 , J 2 (ordinary transverse rotation) so that the particle’s moving direction cannot be kept just in the transverse direction while the particle is transversely boosted. This yields the momentum in the longitudinal direction as well as in the transverse direction. It is also interesting to note that the relativistic energy–momentum dispersion relation works although the particle energy takes a non-relativistic form: 

P

 0 2

 2

−P =

2 P M+ ⊥ 2M



2 2 ⊥ −P

2 P − − ⊥ 2M

2 = M2 .

(2.53)

34

2

Interpolation Between Instant Form Dynamics and Light-Front Dynamics

This may be regarded as another distinguished feature of the LFD. We now apply the T12 transformation subsequently after we do the T3 transformation in order to combine the longitudinal boost and the transverse kinematic 

−i β K 1 +β K

 2

1 2 transformations, i.e. TK = T3 T12 = e−iβ3 K e . This allows not only  to primed P but also the subsequent transforthe transformation of the unprimed P μ  μ  mation from the primed four-momentum P μ to the double-primed four-momentum  of the particle that we consider. Under the T transformation, we get P K μ 3

†  P μ TK μ = TK P   † = T12 T3† P μ T3 T12 †  = T12 P μ T12 .

(2.54)

From this, we get the following general transformation relations:  P+   = (cosh β3 − S cos α sinh β3 ) P+

 β2   + 1 − S2 cos α sinh β3 + S (1 − cos α) cosh β3 ⊥2 P−  α   sin α   β1 P 1 + β2 P 2 −S α  P−  + cos α (cosh β3 + S sinh β3 ) P−   = C cos α sinh β3 P+   sin α   β1 P 1 + β2 P 2 +C α sin α sin α   P  j = P j − Cβ j sinh β3 P+ (cosh β3 + S sinh β3 ) P−  − βj  α α   (cos α − 1)   β1 P 1 + β2 P 2 , (2.55) +Cβ j α2 which of course satisfy the dispersion relation as expected:  2    2  2 M 2 = CP+  + 2SP+  P−  − CP−  −P ⊥ 2 2 2  = CP+ .  P−  − CP−  + 2SP+  − P⊥

(2.56)

2 and get In the IFD limit, δ → 0, we note that α 2 → (β12 + β22 ) = β⊥

P 0 = cosh β3 P 0 + sinh β3 P 3

 sin β⊥  β1 P 1 + β2 P 2 β⊥  3

P 3 = cos β⊥ sinh β3 P 0 + cos β⊥ cosh β3 P 3 + sin β⊥  sinh β3 P 0 + cosh β3 P β⊥  (cos β⊥ − 1)  β1 P 1 + β2 P 2 , +β j 2 β⊥

P  j = P j − β j

(2.57)

2.3 Kinematic Transformations of Particle Momenta

35

 2 . Here, the transverse vector β = (β , β ) can be represented by where β⊥ = β⊥ ⊥ 1 2 β⊥ = θ (ˆz × nˆ ⊥ ) defining the angle θ and the rotation axis as the unit transverse vector   nˆ ⊥ = (n 1 , n 2 ) because the kinematic transformations K 1 and K 2 are nothing but the ordinary transverse rotations −J 2 and J 1 , respectively, in IFD. Since zˆ × nˆ ⊥ = −n 2 xˆ + n 1 yˆ = (−n 2 , n 1 ), one may identify β1 = −θ n 2 and β2 = θ n 1 to rewrite Eq. (2.57) as P 0 = cosh β3 P 0 + sinh β3 P 3     ⊥ P 3 = cos θ sinh β3 P 0 + cosh β3 P 3 + sin θ zˆ × nˆ ⊥ · P    ⊥ − (ˆz × nˆ ⊥ ) sin θ sinh β3 P 0 + cosh β3 P 3 P ⊥ = P ⊥ . + (ˆz × nˆ ⊥ )(cos θ − 1)(ˆz × nˆ ⊥ ) · P

(2.58)

Taking nˆ ⊥ = yˆ (i.e. zˆ × nˆ ⊥ = −ˆx), we have P 0 = P 0 = cosh β3 P 0 − sinh β3 P 3

  P 1 = − sin θ P 3 + cos θ P 1 = − sin θ sinh β3 P 0 + cosh β3 P 3 + cos θ P 1

P 2 = P 2 = P 2

  P 3 = cos θ P 3 + sin θ P 1 = cos θ sinh β3 P 0 + cosh β3 P 3 + sin θ P 1 ,

(2.59)

where the boost in zˆ direction and the subsequent rotation around yˆ axis are manifest. Next, we consider the other extreme that corresponds to the LFD, δ = π4 . As δ → π4 , α → 0 and it leads to the following limits for the expressions that appear in the different components of momentum given by Eq. (2.55): 

2 β⊥ α2 2 1 (1 − cos α) → α2 2 sin α → 1. α

1 − S2 cos α

2  β⊥

→

(2.60)

Using the usual LFD notations, we thus get P − = e−β3 P − +

2 eβ3 β⊥ ⊥ P + − β⊥ · P 2

P + = eβ3 P +  ⊥ − eβ3 β⊥ P + , P ⊥ = P

(2.61)

which satisfies the LF dispersion relation as expected 2

2 ⊥ 2P + P − − P ⊥ = 2P + P − − P = M2 .

(2.62)

36

2

Interpolation Between Instant Form Dynamics and Light-Front Dynamics

In the case that the particle is at rest in the unprimed frame, i.e. M P+ = P− = √ 2  P⊥ = 0 ,

(2.63)

we obtain   2 M −β3 β3 β⊥ P = √ e +e 2 2 M P + = √ eβ3 2 M  ⊥ = − √ β⊥ eβ3 , P 2 −

(2.64)

which can be translated into M 2 β3 β e 4 ⊥ M 2 β3 = M sinh β3 − β⊥ e 4 M = − √ β⊥ eβ3 . 2

P 0 = M cosh β3 + P 3 P ⊥

(2.65)

From this, we may extend the relation between the energy and the transverse momentum (as well as between the longitudinal momentum and the transverse momentum) given by Eq. (2.52) as P

0

P 3

2 P ⊥ −β3 = M cosh β3 + e 2M 2 P = M sinh β3 − ⊥ e−β3 . 2M

(2.66)

For β3 = 0, this equation is reduced to Eq. (2.52). As we explained about Eq. (2.52), the gained longitudinal momentum is correlated with the transverse momentum due 1 2 to the kinematic transformation T12 = ei(β1 E +β2 E ) in such a way that a paraboloid 2 P

shape of surface (note P 3 = − 2M⊥ for β3 = 0) can be drawn for the gained momentum components in the momentum space as shown in Ref. [27]. In the case β3 = 0, we find that the similar shapes of paraboloids can be drawn. However, the corresponding paraboloids are shifted in the longitudinal direction as β3 gets more positive values and the curvatures of the corresponding paraboloids get modified as shown in Fig. 2.3. This plot shows three surfaces corresponding to three different values of β3 = 0, 1, 2, with the momenta scaled by the mass of the particle, i.e. p = P  /M,

2.3 Kinematic Transformations of Particle Momenta

37

Fig. 2.3 General kinematic transformation on a fixed interpolating front

in the range −4 < ( p 1, p 2 ) < 4 and −12 < p 3 < 4. For the positive values of β3 as shown in Fig. 2.3, the paraboloid of β3 = 0 is shifted to upward in p 3 and gets flattened due to the factors given by sinhβ3 and e−β3 in Eq. (2.66), respectively. The top 3 point of each paraboloid corresponds to the momentum gained by the T3 = e−iβ3 K transformation in IFD (see Eq. (2.47)). Although the particle at rest stays at rest in IFD when only the kinematic transformation T12 (i.e. the ordinary transverse rotation in IFD) is applied, the longitudinal boost T3 is dynamical in IFD so that it can generate the longitudinal momentum of the particle. However, in LFD, both T12 and T3 are kinematic transformations, and the entire momentum region of p can be covered by these kinematic transformations.

38

2.4

2

Interpolation Between Instant Form Dynamics and Light-Front Dynamics

Application of Transformations on Interpolating Scattering Amplitudes

In the previous sections, we discussed that the scattering amplitude in Fig. 2.1 has two non-vanishing time-ordered contributions in an arbitrary interpolating angle for the range 0 ≤ δ < π4 including IFD (δ = 0) while in LFD (δ = π4 ) only the contribution of the first diagram Fig. 2.2a survives. We now apply the transformations of the particle momenta that we obtained in the last section, Sect. 2.3, to the scattering amplitudes and discuss a quantitative measure on the invariance of the individual time-ordered amplitudes under the kinematic transformations. In order to see this in an arbitrary interpolating angle, let us first consider the expression for D+ found in Eq. (2.26) under the transverse kinematic boost T12 , i.e.   = P+ D+ +

  Sq−  − ωq

C

,

(2.67)

†  where the prime indicates the transformed frame variables via P+  T12 , etc.  = T12 P+  expresses the difference between the interpolating angle energies This quantity D+   of P+  and q+  for the first diagram Fig. 2.2a. Under T12 (see Eq. (2.49)), we get

(β12 + β22 )

sin α   = P+ (1 − cos α)P− (β1 P 1 + β2 P 2 ) +S −S α2 α   ωq S sin α   1 2 − q− cos α + C P + β P ) − (β    1 2 C α C  ωq S = P+ q− , (2.68) + − C C  where we used α = C(β12 + β22 ) and the momentum conservation P−  = q−  . This  D+

 = D and the first term by means that if ωq = ωq as defined by Eq. (2.22), then D+ + itself is invariant under T12 . We may use the solution in terms of q+  of the quadratric equation for the dispersion relation and show ωq = ωq , i.e.

q+  =

ωq − Sq−  ⇒ ωq = Cq+  + Sq−  C

(2.69)

so that   ωq = Cq+  + Sq−  = Cq+  + Sq−  = ωq ,

(2.70)

according to Eq. (2.49). It is now manifest that D+ by itself is invariant under T12 . Similar manifestation can be done for D− for the second diagram Fig. 2.2b. Now, we apply the longitudinal boost T3 to the interpolating time-ordered amplitudes. As we have already discussed in Sect. 2.3, the longitudinal boost K 3 is

2.4 Application of Transformations on Interpolating Scattering Amplitudes

39

dynamical for any δ in the range 0 ≤ δ < π4 and becomes kinematical only at δ = π4 . To exhibit this feature quantitatively, we show Fig. 2.4 which plots δa and δb as functions of the initial particle total momentum ( p1 + p2 ) · zˆ = Pz while ( p1 + p2 ) · xˆ = 0 and ( p1 + p2 ) · yˆ = 0 for convenience, as well as the interpolation angle δ. The ranges of δ and Pz are taken as 0 ≤ δ < π4 and −4 ≤ Pz ≤ 4 in some unit of energy, e.g. GeV, respectively. For illustrative purpose, we took s = 2 and m = 1 using the same energy unit. As clearly shown in Fig. 2.4, the contributions from δa and δb are such that the sum of them yields a constant, independent of Pz and δ. For δ = 0, δa and δb have the maximum and the minimum, respectively, at Pz = 0. For δ = π4 , δa is the whole answer and δb = 0. For positive values of momentum, Pz > 0, the amplitudes δa and δb show a smooth behavior (see also the next section, Sect. 2.5), while for negative values of Pz we observe the presence of a J -shaped peak in δa matched by a similar J -shaped valley in δb . We find that this J -shaped line of maximum/minimum is given by  the following function for momentum in terms of the interpolating angle as Pz = −

s(1−C) 2C . This

J -shaped track is plotted in Fig. 2.5.

On this J -shaped track, a stable maximum and minimum of δa and δb , respectively, are present throughout the negative values of momentum Pz , i.e. 1 , √ 2m( s − m) 1 δb = − , √ 2m( s + m) 1 δa + δb = . s − m2 δa =

(2.71)

One interesting point to observe in this J -shaped graph for negative values of momentum Pz is that it is stable in the peak as well as in the valley as it is independent of the mass and does not vanish as the momentum goes to the negative infinity. Thus, if the limit δ → π4 is taken in the exact correlation with Pz given by the J  C→0 −→ −∞ , then the connected contribution to the shaped track, i.e. Pz = − s(1−C) 2C b current arising from the vacuum δ→ π does not vanish but remains as a nonzero

constant, i.e. − 2m(√1s+m) = −

√1 2( 2+1)

4

≈ −0.207, although this nonzero constant

b (i.e. the minimum of δ→ π ) is canceled by the same magnitude of the constant (i.e. 4 a ≈ 1.207 to yield the total the maximum of δ→ π ) given by 2m(√1s−m) = √1 2( 2−1) 4 1 amplitude s−m 2 = 1.

This may clarify the prevailing notion of the equivalence between IFD and LFD in the IMF since it works for the limit of Pz → ∞ but requires a great caution in the limit of Pz → −∞. Although the IFD in IMF is entirely symmetric between Pz = ∞ and Pz = −∞, there is treacherous point Pz = −∞ in LFD. As far as the limit of Pz = −∞ is taken off the J -shaped track, i.e. without the specific correlation

40

2

Interpolation Between Instant Form Dynamics and Light-Front Dynamics

Fig. 2.4 Interpolating amplitudes

2.5 Interpolating Scattering Amplitudes in Infinite Momentum Frame

41

Fig. 2.5 J -shaped line of maximum/minimum for δa and δb

 C→0 b Pz = − s(1−C) −→ −∞, then our result of δ= π = 0 is valid. However, if the 2C 4 limit of Pz = −∞ is taken exactly with this particular correlation, then the result b δ= π = 0 is not correct but should be modified to be the nonzero minimum value 4

b √1

= 0. In this sense, the J -shaped track which we find in this of δ= π = − 2m( s+m) 4 work is singular. Nevertheless, even in this case, the sum of the two amplitudes a b remains invariant as it should be. δ= π + δ= π 4

2.5

4

Interpolating Scattering Amplitudes in Infinite Momentum Frame

As we discussed in Sect. 2.2, we can rewrite the interpolating time-ordered amplitudes in the same form as in the IFD by changing the superscript 0 (i.e. the energy) to  as well as multiplying an overall factor C. Then, it follows that interposuperscript + lating amplitudes become IFD amplitudes as C → 1. In the LFD case as C → 0, the 1 + = q + but the multiplication of fraction P + −q + → ∞ due to the conservation of P

1 zero and infinity makes the finite s−m 2 just from the first diagram alone, while the second diagram vanishes since the denominator P + + q + is nonzero. The disappearance of the connected contributions to the current arising from the vacuum at C = 0 b = 0, should be distinguished from the similar disappearance of (LFD), i.e. δ=π/4 Z -graph in the IMF at C = 1 (IFD). In this section, we apply the longitudinal boost T3 (see Eq. (2.44)) and take a specific limit to an infinite momentum frame, viz. (Pz , qz ) ≡ (P 3 , q 3 ) → ∞, in order to discuss more details of the disappearance of the contributions connected to the vacuum for the entire range of the interpolation angle 0 ≤ δ ≤ π4 .

42

2

Interpolation Between Instant Form Dynamics and Light-Front Dynamics

First of all, let us consider the case of the IFD (see Eq. (2.17)), where the longi3 tudinal component of interest is P−  = Pz ≡ P , etc. The time-ordered diagram of Fig. 2.1 is dependent on the reference frame: a IFD =

1 2q 0



1 0 P − q0

 .

(2.72)

From the dispersion relation q 2 = m 2 , the expansion of q 0 for the IMF is given by   2 + m2, q 2 + m 2 = qz2 + q ⊥    2 + m2 q ⊥ 1 = qz 1 + +O . 2qz2 qz4

q0 =

(2.73)

Similarly, from the dispersion relation P 2 = s, the expansion of P 0 for the IMF is given by   2  2 + s,  P = P + s = Pz2 + P ⊥    2  +s P 1 ⊥ = Pz 1 + +O . 2 2Pz Pz4 0

(2.74)

Substituting Eqs. (2.73) and (2.74) into Eq. (2.72), we get a = IFD

1





2 +m 2 q ⊥ 2qz2



2qz 1 + +O ⎧ ⎪ ⎨ 1 ×  2 2 2  ⎪ ⎩ Pz − qz + P⊥ +s − q ⊥ +m + O 13 , 2Pz 2qz q 1 qz4

z

1 Pz3



⎫ ⎪ ⎬ ⎪ ⎭

.

(2.75)

 ⊥ = q ⊥ , the result (2.75) Due to the three-momentum conservation, Pz = qz and P in the IMF limit yields a IFD

1 = 0 2q



1 P0 − q0



Pz =qz →∞

−−−−−−→

1 . s − m2

(2.76)

Likewise, for the diagram of Fig. 2.2b, we get b = IFD

1 2q 0



1 P0 + q0



Pz =qz →∞

−−−−−−→ 0.

This reveals that the results (2.76) and (2.77) are frame-dependent.

(2.77)

2.5 Interpolating Scattering Amplitudes in Infinite Momentum Frame

43

Next, we consider what happens in the LFD case, where we have P− = P + and q− = q + . Independent of reference frames, i.e. regardless of the Pz value, the result is given by a LFD ≡ LFD

=

⎛ ⎞ 1 1 ⎝ ⎠ = + 2 +m 2 q ⊥ 2q − P − 2q +

1 . 2 + m2) 2q + P − − ( q⊥

(2.78)

Since q + = P + , q ⊥ = P⊥ , we get a = LFD ≡ LFD

=

1 2P + P −

 2 − m2 −P ⊥

1 . s − m2

(2.79)

This result is frame independent and thus valid even in the IMF limit, or Pz → ∞. Finally, let us consider the case of an arbitrary interpolating angle in the range of 0 < δ < π4 . The contribution of diagram of Fig. 2.2a is given by δa

1 = 2ωq





1 P+ +

Sq−  −ωq C

,

(2.80)

  2  2   = q  , we can rewrite  ⊥ + m 2 . Since P− where ωq = q−  = q−  and P ⊥ ⊥ +C q these expressions as δa =

1 2ωq



C CP+  + SP−  − ωq



 ; ωq =

  2 2 2 P−  + C P⊥ + m .

(2.81)

Using Eq. (2.13), we can further reduce the time-ordered amplitude of Fig. 2.2a as δa =

2ωq

C . − 2ωq2

 P+

(2.82)

 Since P + = P 0 cos δ + P 3 sin δ as in Eq. (2.7), we can express P 0 in terms of P 3 using the dispersion relation P 2 = s as

  2 + s P 1 ⊥ P =P + +O 2P 3 (P 3 )3   2  +s P 1 . = Pz + ⊥ +O 2Pz Pz3 0

3

(2.83)

44

2

Interpolation Between Instant Form Dynamics and Light-Front Dynamics

Thus, we get P

 +

  2 + s P 1 ⊥ , = Pz (sin δ + cos δ) + cos δ + O 2Pz Pz3

(2.84)

  2 + s P 1 ⊥ . sin δ + O 2Pz Pz3

(2.85)

and similarly P−  = Pz (sin δ + cos δ) +

The result given by Eq. (2.85) is used to evaluate ωq2 :     2 2 ⊥ ⊥ + s sin δ(sin δ + cos δ) + C P + m2 ωq2 = Pz2 (sin δ + cos δ)2 + P   1 , (2.86) +O Pz2 which leads to  ωq = Pz (sin δ + cos δ) +   1 , +O Pz2

2 + s P ⊥ 2Pz



 sin δ +

 2 + m2 P ⊥ 2Pz

 (cos δ − sin δ) (2.87)

where we used the identity C ≡ cos 2δ = cos2 δ − sin2 δ = (cos δ + sin δ)(cos δ − sin δ). Putting all the ingredients to calculate the denominator, we obtain    2 ⊥ + s (sin δ + cos δ)2 2ωq P + − 2ωq2 = 2Pz2 (sin δ + cos δ)2 + P   2 ⊥ +C P + m2   2 ⊥ −2Pz2 (sin δ + cos δ)2 − 2 P + s (sin2 δ + sin δ cos δ)     1 2 2  −2C P⊥ + m + O Pz2   1 . (2.88) = C(s − m 2 ) + O Pz2 This leads to δa =

2ωq

C 1 Pz →∞ −−−−→ . 2 s − m2 − 2ωq

 P+

(2.89)

2.6 Conclusions

45

For the diagram of Fig. 2.2b, since 

2ωq P + + 2ωq2 = 4Pz2 (sin δ + cos δ)2   2 ⊥ + P + s (3 sin2 δ + cos2 δ 2 + 4 sin δ cos δ)   1 2 2  , (2.90) +3C(P⊥ + m ) + O Pz2 we get δb =

2.6

2ωq

C Pz →∞ −−−−→ 0. 2 + 2ωq

 P+

(2.91)

Conclusions

In the present work, we discussed the fundamental aspects of the time-ordered scattering amplitudes in relativistic Hamiltonian dynamics. Using the interpolating angle between IFD and LFD, we presented a simple but clear example of interpolating scattering amplitudes and demonstrated a physical meaning of kinematical transformations introduced often formally in the stability group of Poincaré transformations. We confirmed the well-known IMF result [7] for the IFD and extended it for any arbitrary interpolating angle 0 ≤ δ < π4 . We also showed that the disappearance of the connected contributions to the current from the vacuum in LFD is independent of the reference frame and should be distinguished from the IMF result. We demonstrated that the longitudinal boost K 3 joins the stability group only in the LFD. We did this not only using explicit expressions of kinematic transformation effects on the fundamental dynamical variables of physical momenta but also discussing the interpolating time-ordered scattering amplitudes. The addition of K 3 in the stability group is a great advantage of LFD in hadron phenomenology [34]. Computing the individual time-ordered amplitudes for the whole range of total momentum Pz and the interpolating angle δ, we showed not only the invariance of the sum of time-ordered amplitudes but also the behavior of each individual timeordered amplitudes (see Fig. 2.4). Our work demonstrates a rather clear distinction between the well-known IMF result in IFD and the LFD result on the disappearance of the connected contribution to the  current from the vacuum. Our result exhibits the which reminds a treacherous zero-mode J -shaped track given by Pz = − s(1−C) 2C issue in LFD. The J -shaped track provides a correlation between the total momentum Pz and the interpolation angle δ. It traces the maximum of the time-ordered amplitude a b . Thus, if 0≤δ< π as well as the minimum of the time-ordered amplitude  0≤δ< π4 4 π one takes the interpolating angle to the limit of 4 in an exact correlation with the limit Pz → −∞ following the J -shaped track, then one should be careful not to miss b the contribution from the minimum value of 0≤δ< π which must be canceled by the 4

46

2

Interpolation Between Instant Form Dynamics and Light-Front Dynamics

a maximum value of 0≤δ< π . Although our work in this chapter is limited to a simple 4 example without spins or any other degrees of freedom except the particle momenta, the results offer interesting and significant aspects of the relativistic Hamiltonian dynamics which interpolates between IFD and LFD.

References 1. Dirac, P.A.M.: Forms of relativistic dynamics. Rev. Mod. Phys. 21, 392–399 (1949) 2. Glozman, L.Y., et al.: Unified description of light- and strange-baryon spectra. Phys. Rev. D 58(9), 094030 (1998) 3. Wagenbrunn, R., et al.: Covariant nucleon electromagnetic formfactors from the Goldstoneboson—exchange quark model. Phys. Lett. B 511(1), 33–39 (2001) 4. Melde, T., et al.: Electromagnetic nucleon form factors in instant and point form. Phys. Rev. D 76(7), 074020 (2007) 5. Steinhardt, P.J.: Problems of quantization in the infinite momentum frame. Ann. Phys. 128(2), 425–447 (1980) 6. Fubini, S., Furlan, G.: Renormalization effects for partially conserved currents. Phys. Phys. Fizika 1(4), 229–247 (1965) 7. Weinberg, S.: Dynamics at infinite momentum. Phys. Rev. 150(4), 1313–1318 (1966) 8. Jersák, J., Stern, J.: Relativistic non-invariance symmetries generated by local currents. Nucl. Phys. B 7, 413–431 (1968) 9. Leutwyler, H.: Current algebra and lightlike charges. In: Fries, D., Zeitnitz, B. (ed.) Springer Tracts in Modern Physics, vol. 50, pp. 29–41 (1969) 10. Bjorken, J.D.: Asymptotic sum rules at infinite momentum. Phys. Rev. 179(5), 1547–1553 (1969) 11. Drell, S.D., et al.: A theory of deep inelastic lepton nucleon scattering and lepton pair annihilation processes. 2. Deep inelastic electron scattering. Phys. Rev. D 1, 1035–1068 (1970) 12. Pauli, H.-C., Brodsky, S.J.: Solving field theory in one space and one time dimension. Phys. Rev. D 32(8), 1993–2000 (1985) 13. Brodsky, S.J., Pauli, H.C.: Light-cone quantization of quantum chromodynamics. In: Mitter, H., Gausterer, H. (ed.) Recent Aspects of Quantum Fields, pp. 51–121. Springer, Berlin Heidelberg (1991) 14. Perry, R.J., et al.: Light-front Tamm-Dancoff field theory. Phys. Rev. Lett. 65(24), 2959–2962 (1990) 15. Harindranath, A., Perry, R.J.: Lowest-order mass corrections for a (1+1)-dimensional Yukawa model in light-front perturbation theory. Phys. Rev. D 43(2), 492–498 (1991) 16. Mustaki, D., et al.: Perturbative renormalization of null-plane QED. Phys. Rev. D 43(10), 3411– 3427 (1991) 17. Brodsky, S.J. et al.: Pauli-Villars regulator as a nonperturbative ultraviolet regularization scheme in discretized light-cone quantization. Phys. Rev. D 58(2), 025005 (1998) 18. Hiller, J.R.: Pauli-Villars regularization in a discrete light cone model. In: 3rd Workshop on Continuous Advances (1998). arXiv: hep-ph/9807245 19. Robertson, D.G., McCartor, G.: An equal time quantized field theory on the light cone. Z. Phys. C 53, 661–672 (1992) 20. McCartor, G., Robertson, D.G.: Bosonic zero modes in discretized light cone field theory. Z. Phys. C 53, 679–686 (1992) 21. Brodsky, S.J., et al.: Quantum chromodynamics and other field theories on the light cone. Phys. Rept. 301, 299–486 (1998). arXiv: hep-ph/9705477 22. Gubankova, E., et al.: Flow equations for quark-gluon interactions in light-front QCD. Phys. Rev. D 62(12), 125012 (2000)

References

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23. Susskind, L., Burkardt, M.: A model of mesons based on CSB in the light-front frame. In: 4th International Workshop on Light Cone Quantization and Non-Perturbative Dynamics (1994) 24. Wilson, K.G., Robertson, D.G.: Light front QCD and the constituent quark model. In: 4th International Workshop on Light Cone Quantization and Non-perturbative Dynamics (1994). arXiv: hep-th/9411007 25. Ji, C.-R., Surya, Y.: Calculation of scattering with the light-cone two-body equation in ϕ 3 theories. Phys. Rev. D 46(8), 3565–3575 (1992) 26. Leutwyler, H., Stern, J.: Relativistic dynamics on a null plane. Ann. Phys. 112, 94 (1978) 27. Ji, C.-R., Mitchell, C.: Poincaré invariant algebra from instant to light-front quantization. Phys. Rev. D 64(8), 085013 (2001) 28. Hornbostel, K.: Nontrivial vacua from equal time to the light cone. Phys. Rev. D 45(10), 3781– 3801 (1992) 29. Ji, C.-R., Rey, S.-J.: Light-front view of the axial anomaly. Phys. Rev. D 53(10), 5815–5820 (1996) 30. Chen, T.W.: Almost-infinite-momentum frame, light-cone commutators, and scaling laws. Phys. Rev. D 3(8), 1989–1991 (1971) 31. Elizalde, E., Gomis, J.: Quasi light cone frame and approximate Galilean symmetries. Nuovo Cim. A 35, 367–376 (1976) 32. Frishman, Y., et al.: Novel inconsistency in two-dimensional gauge theories. Phys. Rev. D 15(8), 2275–2281 (1977) 33. Sawicki, M.: Light-front limit in a rest frame. Phys. Rev. D 44(2), 433–440 (1991) 34. Carlson, C.E., Ji, C.-R.: Angular conditions, relations between the Breit and light-front frames, and subleading power corrections. Phys. Rev. D 67(11), 116002 (2003) 35. Jacob, M., Wick, G.C.: On the general theory of collisions for particles with spin. Ann. Phys. 7, 404–428 (1959)

3

Interpolation of Quantum Electrodynamics

The instant form and the front form of relativistic dynamics proposed by Dirac in 1949 can be linked by an interpolation angle parameter δ spanning between the instant form dynamics (IFD) at δ = 0 and the front form dynamics which is now known as the light-front dynamics (LFD) at δ = π/4. We present the formal derivation of the interpolating quantum electrodynamics (QED) in the canonical field theory approach which is adopted from our published work Phys. Rev. D98, 036017(2018) and discuss the constraint fermion degree of freedom which appears uniquely in the LFD. The constraint component of the fermion degrees of freedom in LFD results in the instantaneous contribution to the fermion propagator, which is genuinely distinguished from the ordinary equal-time forward and backward propagation of relativistic fermion degrees of freedom. As discussed in our previous work Phys. Rev. D92, 105014(2015), the helicity of the on-mass-shell fermion spinors in LFD is also distinguished from the ordinary Jacob-Wick helicity in the IFD with respect to whether the helicity depends on the reference frame or not. To exemplify the characteristic difference of the fermion propagator between IFD and LFD, we compute the helicity amplitudes of typical QED processes such as e+ e− → γ γ and eγ → eγ and present the whole landscape of the scattering amplitudes in terms of the frame dependence or the scattering angle dependence with respect to the interpolating angle dependence. Our analysis clarifies any conceivable confusion in the prevailing notion of the equivalence between the infinite momentum frame approach and the LFD.

3.1

Introduction

As we have presented in the previous chapter, Chap. 2, Dirac [1] proposed three differ0 ent forms of the relativistic √ Hamiltonian dynamics in 1949, i.e. the instant (x = 0), front (x + = (x 0 + x 3 )/ 2 = 0) and point (xμ x μ = a 2 > 0, x 0 > 0) forms for the study of relativistic particle systems. The instant form dynamics (IFD) of quantum field theories is based on the usual equal-time t = x 0 quantization (units such © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 C.-R. Ji, Relativistic Quantum Invariance, Lecture Notes in Physics 1012, https://doi.org/10.1007/978-981-19-7949-1_3

49

50

3

Interpolation of Quantum Electrodynamics

that c = 1 are taken here), which provides a traditional approach evolved from the non-relativistic dynamics. The IFD makes a close contact with the Euclidean space, developing temperature-dependent √ quantum field theory, lattice QCD, etc. The equal light-front time τ ≡ (t + z/c)/ 2 = x + quantization yields the front form dynamics, nowadays more commonly called light-front dynamics (LFD), which provides an innovative approach to the study of relativistic dynamics. The LFD works strictly in the Minkowski space, developing useful frameworks for the analyses of deep inelastic scattering (DIS), parton distribution functions (PDFs), deeply virtual Compton scattering (DVCS), generalized parton distributions (GPDs), etc. The quantization in the point form (x μ xμ = a 2 > 0, x 0 > 0) is called radial quantization, and this quantization procedure has been much used in string theory and conformal field theories [2] as well as in hadron physics [3–5]. Among these three forms of relativistic dynamics proposed by Dirac, however, the LFD carries the largest number (seven) of the kinematic (or interaction independent) generators leaving the least number (three) of the dynamics generators while both the IFD and the point form dynamics carry six kinematic and four dynamic generators within the total ten Poincaré generators. Indeed, the maximum number of kinematic generators allowed in any form of relativistic dynamics is seven, and the LFD is the only one which possesses this maximum number of kinematic generators. Effectively, the LFD maximizes the capacity to describe hadrons by saving a lot of dynamical efforts in obtaining the QCD solutions that reflect the full Poincaré symmetries. We discuss here the correspondence between the LFD and the IFD which has been the traditional approach, using an interpolation angle parameter spanning between the IFD and LFD. There have been earlier works closely related on this subject, using a tilted coordinate near the light-front time axis to quantize the fields slightly away from the light-front [6–8]. In particular, the work of Ref. [8] was concerning the role of light-front zero-modes in QED and QCD with respect to the confinement and chiral symmetry of QCD in contrast to QED. We maintain the orthogonality of the coordinate system in the process of interpolation to correspond the QED between the IFD and the LFD. Although we want ultimately to obtain a general formulation for the QCD, we start from a simpler theory to discuss first the bare-bone structure that will persist even in the more complicated theories. Starting from the scalar field theory [9] to discuss the interpolating scattering amplitude with only momentum degree of freedom, we have extended the discussion to the electromagnetic gauge degree of freedom [10] and the on-mass-shell fermion [11]. In particular, we discussed the link between the Coulomb gauge in IFD and the light-front gauge in LFD [10] and the chiral representation of the helicity spinors interpolating between the IFD and the LFD [11]. In this chapter, we entwine the fermion propagator interpolation with our previous works of the electromagnetic gauge field [10] and the helicity spinors [11] and fasten the bolts and nuts necessary to launch the interpolating QED. As we have already discussed the prototype of QED scattering processes “eμ → eμ” and “e+ e− → μ+ μ− ” involving a photon propagator in our previous work [11], we present in this work the two-photon production amplitude in the pair annihilation of fermion and antifermion process “e+ e− → γ γ ” as well as the Compton scattering amplitude “eγ → eγ ” involving a fermion propagator. Since the effects of external fermions and bosons have already been studied in our previous works [10,11], we will focus on the intermediate fermion propagator in this work.

3.1 Introduction

51

To trace the forms of relativistic quantum field theory between IFD and LFD, we take the same convention of the space-time coordinates as introduced in Eq. (2.7) to define the interpolation angle [9–13]: 

 x+ − x



 =

cos δ sin δ sin δ − cos δ



 x0 , x3

(3.1)

in which the interpolation angle is allowed to run from 0 through 45◦ , 0 ≤ δ ≤ The lower index variables x+  and x −  are related to the upper index variables     + − − +  μ  μ as x+  = g+   = g−  μ x = Cx + Sx and x − μ x = −Cx + Sx , denoting C = cos2δ and S = sin2δ and realizing g+ +  = −g− −  = cos2δ = C and g+ −  = g− +  = sin2δ = S. All the indices with the widehat notation signify the variables with the   = x 0 and x − = −x 3 so that interpolation angle δ. For the limit δ → 0 we have x + we recover usual space-time coordinates although the z-axis √ is inverted while for  = (x 0 ± x 3 )/ 2 ≡ x ± which leads to the other extreme limit, δ → π4 , we have x ± the standard light-front coordinates. Since the perpendicular components remain the  same (x j = x j , xj = x j , j = 1, 2), we will omit the “^” notation unless necessary from now on for the perpendicular indices j = 1, 2 in a four-vector. Of course, the same interpolation applies to the four-momentum variables too as it applies to all four-vectors. The details of the relationship between the interpolating variables and the usual space-time variables can be seen in our previous works Ref. [9–11]. In Ref. [10], we developed the electromagnetic gauge field propagator interpolated between the IFD and the LFD and found that the light-front gauge A+ = 0 in the LFD is naturally linked to the Coulomb gauge ∇ · A = 0 in IFD. We identified the dynamical degrees of freedom for the electromagnetic gauge fields as the transverse photon fields and clarified the equivalence between the contribution of the instantaneous interaction and the contribution from the longitudinal polarization of the virtual photon. Our results for the gauge propagator and time-ordered diagrams clarified whether one should choose the two-term form [14] or the three-term form [15–17] for the gauge propagator in LFD. There has been a sustained interest and discussion on this issue of the two-term versus three-term gauge propagator in LFD [18]. Our transverse photon propagator in LFD assumes the three-term form, but the third term cancels the instantaneous interaction contribution. Thus, one can use the two-term form of the gauge propagator for effective calculation of amplitudes if one also omits the instantaneous interaction from the Hamiltonian. But if one wants to show equivalence to the covariant theory, all three terms should be kept because the instantaneous interaction is a natural result of the decomposition of Feynman diagrams, and the third term in the propagator is necessary for the total amplitudes to be covariant. We also see that the photon propagator was derived according to the generalized gauge that links the Coulomb gauge to light-front gauge and thus the three-term form appears appropriate in order to be consistent with the appropriate gauge. In Ref. [11], we derived the generalized helicity spinor that links the instant form helicity spinor to the light-front helicity spinor. For a given generalized helicity spinor, the spin direction does not coincide with the momentum direction in general. π 4.

52

3

Interpolation of Quantum Electrodynamics

Thus, we studied how the spin orientation angle θs changes in terms of both δ and the angle θ that defines the momentum direction of the particle. In particular, the helicity in IFD depends on the reference frame. If the observer moves faster than the positive helicity spinor, then the direction of the momentum becomes opposite to the spin direction and the helicity of the spinor flips its sign. In contrast, the helicity defined in LFD is independent of the reference frame. We have detailed the increment of the angle difference θ − θs with the increment of the interpolation angle δ in Ref. [11], which bifurcates at a critical  interpolation angle δc . We found this critical interpolation angle δc = arctan |P| E , where |P| and E are the magnitude of the three-momentum and the energy of the particle under investigation. The IFD and the LFD belong to separately the two different branches bifurcated and divided out at the critical interpolation angle δc . This bifurcation indicates the necessity of the distinction in the spin orientation between the IFD and the LFD and clarifies any conceivable confusion in the prevailing notion of the equivalence between the LFD and the infinite momentum frame (IMF) approach [19] formulated in the IFD. Now that the spinor has been interpolated between IFD and LFD, we show in q +m this work that the covariant Feynman propagator  = q/2 −m 2 of the intermediate virtual fermion with the four-momentum q and the mass m can also be decomposed into the two interpolating time-ordered processes, one with the “forward moving” intermediate fermion in the sense that its interpolating longitudinal momentum q−  is > 0 and the other with the “backward moving” intermediate fermion positive, i.e. q−  carrying the opposite sign of −q−  , i.e. −q−  < 0. The corresponding “forward” and “backward” amplitudes are given by F =

QF + m − Q B + m 1 1 , B = ,  q −Q  −q − Q + + 2Q 2Q   F+ B+ + +  

(3.2)

where Q F+  = Q B+  = and 

Q+ =

 −SqF− + Q +

C

 −SqB− + Q +

C

,

(3.3)

,

(3.4)

 2 2 2 q−  + C(q⊥ + m ),

(3.5)

with the four-momenta qF = q and qB = −q which are those of the off-shell fermion and antifermion, while Q F and Q B are the corresponding on-shell four-momenta. Only the interpolating energies of the “forward” and “backward” moving intermediate fermions, i.e. Q F+  and Q B+  are different from qF and qB , respectively, as given by Eqs. (3.3) and (3.4). In the light-front limit δ → π4 , i.e. C → 0, we get F,δ→ π4 =

q on + m , q2 − m2

B,δ→ π4 =

γ+ , 2q +

(3.6)

3.1 Introduction

53

where qon is the on-shell momentum four-vector with its spacial part equal to that of q while it satisfies the Einstein energy–momentum relationship. Here, B,δ→ π4 turns out to be the instantaneous contribution in the light-front propagator. This proves the usual light-front decomposition of the fermion propagator given by [20] 1 = q − m



¯ s) s u(q, s)u(q, q2 − m2

+

γ+ , 2q +

(3.7)

where the numerator q on + m of F,δ→ π4 in Eq. (3.6) is replaced by the spin sum of  the on-shell spinor product s u(q, s)u(q, ¯ s). In the next section, Sect. 3.2, we present the formal derivation of the interpolat -ordered ing QED. We outline two different derivations of the Feynman rules for x + diagrams formulated at any interpolation angle. The first approach directly decomposes the covariant Feynman diagram, and the second one utilizes the canonical field theory and the old-fashioned perturbation theory. We notice in particular the constraint fermion degree of freedom which appears uniquely in the LFD, resulting in the instantaneous contribution to the fermion propagator. The canonical field theory is studied for the entire range of the interpolation angle 0 ≤ δ ≤ π/4. Equations of motion, free fields, gauge condition, momentum and angular momentum tensor are  is found. In Sect. 3.3, we study the examined, and the Hamiltonian at constant x +  + x -ordered fermion propagator in more detail. Taking a simple example, the annihilation of fermion and antifermion into two scalar particles, we show the characteristic behavior of the amplitudes as the form interpolates between IFD and LFD, and also examine the angular momentum conservation. In Sect. 3.4, we present the results for the e+ e− → γ γ process and the Compton scattering eγ → eγ . We compute all 16 helicity amplitudes and discuss the frame dependence and/or the scattering angle dependence with respect to the interpolation angle dependence. For the e+ e− → γ γ amplitudes, the symmetry between the forward and backward angle dependence is discussed. The limit to the LFD (δ = π/4) is analyzed and the comparison with the well-known analytic results from the manifestly covariant calculation is presented. Summary and conclusions follow in Sect. 3.5. In Appendix A.1, we derive Eq. (3.2) and present the fermion propagator in the position space which supplements the discussion in Sect. 3.2.1. In Appendix A.2, we present the derivation of interpolating QED Hamiltonian which supplements the discussion in Sect. 3.2.2. In Appendix A.3, the manifestly covariant fermion propagator is explicitly derived from the sum of the interpolating time-ordered fermion propagators. In Appendix A.4, we discuss the non-collinear case of the annihilation of fermion and antifermion into two scalar particles and provide the relation between the center-of-mass scattering angle and the apparent scattering angle in a boosted frame and correspond the angular distributions for the center-of-mass frame to the apparent angle distributions in boosted frames. The angular distribution and the frame dependence of the e+ e− → γ γ helicity amplitudes are summarized in in Appendices A.5 and A.6, respectively.

54

3

3.2

Interpolation of Quantum Electrodynamics

Formal Derivation of the Interpolation of QED

In our previous works [10,11], we studied in great detail the interpolation of the photon polarization vectors, the gauge propagator and the on-mass-shell helicity spinors. Here, we complete the interpolation of the QED theory by providing the final piece of the entity: the interpolating fermion propagator. The form of this interpolating fermion propagator is derived. In Sect. 3.2.1, we decompose the covariant Feynman  -ordered diagrams, from which a general set of Feynman rules for diagrams into x +  + any x -ordered scattering theory is obtained. In Sect. 3.2.2, the canonical field theory approach is studied and the corresponding Hamiltonian for the old fashioned perturbation theory is derived.

3.2.1

Scattering Theory

Following what Kogut and Soper did in their light-front QED paper [21],1 we regard the perturbative expansion of the S matrix in Feynman diagrams as the foundation of quantum electrodynamics. In this section, we decompose the covariant Feynman  -ordered diagrams. We shall not be concerned with the diagram into a sum of x + convergence of the perturbation series, or convergence and regularization of the integrals here as those issues in QED are common regardless of the interpolation between IFD and LFD.

3.2.1.1 Propagator Decomposition In Ref. [10], we obtained the decomposition of the photon propagator given by

∞

 T  μ  μ μ ν   (q− dq− (x + )e−iqμ x + (−x + )eiqμ x  )  2 2 2 q−  + Cq⊥ −∞ 2  n d q⊥ ∞ ⊥ − μ n ν  + iδ(x + ) dq 2 e−i(q− x +q⊥ x ) (3.8) 2 (2π )3 −∞ − q− + Cq  ⊥

DF (x) μ ν =

d2 q⊥ (2π )3

   2 + Cq2 /C is the interpolating on-mass-shell energy where q+  = −Sq−  + q−  ⊥ and the explicit form of T μ ν is given by T μ ν ≡

λ=±

∗  ν (λ) μ (λ)

= −g μ ν +

(q · n)(q Cq q 2 n μ n ν + q ν n μ) μ q ν μ n ν − 2 − 2 2 2 2 2 q⊥ C + q− q C + q q C + q    ⊥ ⊥ − −

(3.9)

1 Although Kogut and Soper represented their work in Ref. [21] as the QED in the infinite momentum

frame, it actually was the formulation of QED in the light-front dynamics (LFD).

3.2 Formal Derivation of the Interpolation of QED

55

 μ 2  μ with the obvious familiar notation q · n = q μ n and q = q μ q . Here, the polariza μ tion vectors  ( p, ±) are explicitly given in Ref. [10], and T μ ν given by Eq. (3.9)  = 0 and is obtained in the radiation gauge for any interpolating angle, i.e. A+ ∂−  A−  + ∂ ⊥ · A⊥ C = 0. As discussed in Ref. [10], our interpolating radiation gauge links naturally the Coulomb gauge in IFD (C = 1) and the light-front gauge in LFD  (q− (C = 0). One should also note that  ) in Eq. (3.8) is the interpolating step function given by

(q−  ) = (q−  ) + (1 − δC0 ) (−q− )

1 (C = 0) = (q + ) (C = 0)

(3.10)

which was introduced to combine the results of C = 0 and C = 0. Similarly, the manifestly covariant Klein–Gordon propagator F (x) in the position space given by

F (x) ≡

d4 q i  μ exp(−iq μx )  4 μ 2 (2π ) q q μ − m + i

d2 q⊥ dq−  dq+    + − ⊥ exp [−i(q+  x + q−  x + q⊥ · x )] (2π )4 i × 2 2 2 2 Cq+  q+  − Cq−  + 2Sq−  − q⊥ − m + i =

(3.11)

can also be obtained by combining the results of C = 0 and C = 0 with the interpo (q− lating step function  ):

d2 q⊥ (2π )2



∞

dq− 1  (q− )  2 2 2 −∞ 2π 2 q−  + C(q⊥ + m ) 

 μ  μ   × (x + )e−iqμ x + (−x + )eiqμ x ,

F (x) =

(3.12)

where the value of q+  in the exponent is taken to be the interpolating on-mass-shell energy, i.e. ⎧    2 2 + 2 ⎨ −Sq−  + q−  + C(q⊥ + m ) /C, for x > 0,    q+  =  ⎩ −Sq − q 2 + C(q2 + m 2 ) /C, for x + < 0. −  ⊥ − The detailed derivation of Eqs. (3.11) and (3.12) will be given in Appendix A.1, where the pole integration is done explicitly.

56

3

Interpolation of Quantum Electrodynamics

 (q− The result for C = 0, i.e.  ) = 1, in Eq. (3.12) can be obtained by noting the two poles for q+  in Eq. (3.11) given by     2 + C(q2 + m 2 ) /C − i  , A+ (3.13)  − i = −Sq−  + q−  ⊥      2 2 2 (3.14) −B+  + i = −Sq−  − q−  + C(q⊥ + m ) /C + i , where   > 0. In order not to involve any contribution from the arc in the contour integration, we evaluate the q+  integral in Eq. (3.11) by closing the contour in the lower   > 0 (x + < 0). This produces the desired decomposition for (upper) half plane if x + (q−

F (x) with  ) = 1 in Eq. (3.12) given by

F (x) =

d2 q⊥ (2π )3

∞ dq−  −∞

 1

 μ  μ   (x + )e−iqμ x + (−x + )eiqμ x ,  2Q +

 where we denoted the denominator factor in Eq. (3.12) by Q + , i.e.   2 2 2 Q + ≡ q−  + C(q⊥ + m ).

(3.15)

Note here that the integration measure in Eq. (3.12) is the invariant differential surface element on the mass shell, i.e. 2 d4 q d q⊥ dq−  = 2π δ(q 2 − m 2 ). (3.16)  (2π )3 2Q + (2π )4 + (q− The result for C = 0, i.e.  ) = (q ), in Eq. (3.12) can also be obtained by − noting the single pole for q+  = q in Eq. (3.11) given by

q− =

2 + m2 q⊥  −i + 2q + 2q

(3.17)

which should be taken in the contour integration of the light-front energy q − variable without involving the arc contribution in Eq. (3.11). Note here that this single pole corresponds to A+  in Eq. (3.13) in the limit of C → 0. This requires to close the contour in the lower (upper) half plane of the complex q − space if x + > 0 (x + < 0), as we explained essentially the same procedure for C = 0 case. Due to the rational relation between q − and q + given by Eq. (3.17), the value of q + must be positive to keep the q − pole in the lower half plane for x + > 0, while the value of q + must be negative to keep the q − pole in the upper half plane for x + < 0. This leads to the result given by

F (x) =

d2 q⊥ (2π )3 +

× (x )e

∞

dq +

0

−iq·x

1 2q +

 + (−x + )eiq·x ,

(3.18)

3.2 Formal Derivation of the Interpolation of QED

57

q2 +m 2

where q · x = q + x − + ( ⊥2q + )x − − q⊥ · x⊥ noting x⊥ = −x⊥ . This result is identical to Eq. (3.12) for C = 0. Thus, our result in Eq. (3.12) covers both C = 0 and C = 0 cases together. As the fermion propagator in the position space can be obtained by  μ SF (x) = (i∂ μ γ + m) F (x),

(3.19)

we can now use Eqs. (3.12) and (3.19) to derive a decomposition for the fermion propagator given by SF (x) =

d2 q⊥ (2π )3 

+ iγ +



∞

(q− dq−  )

−∞



d2 q⊥ (2π )3

∞ −∞

 1

 μ  μ   (x + )(q + m)e−iqμ x + (−x + )(−q + m)eiqμ x  + 2Q

(q− dq−  )

 1 +  μ  μ   δ(x )e−iqμ x − δ(x + )eiqμ x ,  + 2Q

(3.20)

” component of q takes the corresponding pole values, as mentioned where the “+   before. Here, the differentiation of (x + ) and (−x + ) in Eq. (3.12) with respect to  μ  μ    x + gives us two terms: δ(x + )e−iqμ x and −δ(x + )eiqμ x in C = 0 case, and these two  is performed will cancel each other exactly when an integration with respect to x + as we show explicitly in the next subsection, so that they don’t contribute to the Feynman rules. Therefore, we can drop them from the decomposition. Thus, when C = 0, the second line in Eq. (3.20) automatically drops off, and the first line is the whole result. However, in the C = 0 case, the integration over q− = q + (note that q−  is just q− without hat when C = 0) goes from 0 to ∞ instead of −∞ to + (q− ∞ as denoted by the interpolating step function  ). Thus, the two δ(x ) terms + resulting from differentiating the (x ) function do not cancel each other, and the term proportional to δ(x + ) is left over. This term is the instantaneous contribution unique to the LF. Thus, when we take C = 0, our fermion propagator result given by Eq. (3.20) coincides with the LF propagator previously derived by Kogut and Soper [21]: SF (x)LF =

d2 q⊥ (2π )3 +

∞

+ iδ(x )γ

 dq +

+ −iq·x + iq·x (x )( q + m)e + (−x )(− q + m)e 2q +

0 +



d2 q⊥ (2π )3

∞ −∞

dq + −iq + x − +iq⊥ ·x⊥ e . 2q +

(3.21)

Note here that the interpolating widehat notations are switched to the usual lightfront notations.

58

3

Interpolation of Quantum Electrodynamics

Fig. 3.1 Lowest-order tree level covariant annihilation diagram in (a) position space and (b) momentum space; Lowest-order tree level covariant scattering diagram in (c) position space and (d) momentum space  3.2.1.2 Rules for x + -Ordered Diagrams  To find the rules for x + -ordered diagrams, we start with the Feynman diagrams in coordinate space. The amplitude for the diagram shown in Fig. 3.1a for the process of e+ e− → γ γ can be written as ∗  μ ¯ i M = (−ie)2 d4 xd4 y  μ (y)[ψ2 (y)γ

× SF (y − x)γ ν ψ1 (x)]ν∗ (x).

(3.22)

Here, we use the plane wave solution of the Dirac equation for the electron and the charge conjugate plane wave solution for the positron: ψ1 (x) = e−i p·x u( p, s),

(3.23)

i p ·y

(3.24)

ψ2 (y) = e

v( p  , s  ),

3.2 Formal Derivation of the Interpolation of QED

59

where p and s are the momentum and spin of the fermion. The photon wavefunction is −ik·x   μ (x) = e μ (k, λ),

(3.25)

where  μ (k, λ) is the polarization vector with momentum k and helicity λ, the explicit form of which was given in Ref. [10]. With the change of variables x → x,

y → T = y − x,

(3.26)

Equation (3.22) becomes

   ∗   μ d4 xd4 T ei(k − p )·T  ¯ p  , s  )γ  μ (k , λ ) v(    × SF (T )γ ν u( p, s) ν∗ (k, λ)ei(k+k − p− p )·x .

i M = (−ie)

2

(3.27)

The x integration immediately gives the total energy–momentum conservation con dition. After we plug in the decomposed SF given by Eq. (3.20), we finish the T + integration using the following relations ∞





 +



 +

dT + (T + )ei P+ T =

−∞

∞ −∞



i , P+  + i

dT + (−T + )ei P+ T = −

i , P+  − i

(3.28)

(3.29)

where the causality of the relativistic quantum field theory is assured with the ±i  region, respectively. Thus, we get the interpolating energy denomfactor for the ±T + inator factor of P  −Pi  +i for each intermediate state. For the momentum assignini+ inter+  ment shown in Fig. 3.1b, Pini+  + p+  = p+  is the total “energy” of the initial particles, and Pinter+ +  gives the total “energy” of the intermediate particles, which is k+       + + + + q+  + p+  − q+  + k+  when y > x and p+  when y < x . On the other hand, the  2 ⊥ in out 2 in out d T integration gives straightforwardly (2π )3 δ(P− dT −  − P−  )δ (P⊥ − P⊥ ) at each vertex. Lastly, the δ(T + ) term in Eq. (3.20) gives an extra instantaneous contribution at the light-front (C = 0) and is easy to calculate. Similar analysis can be done for the process of eμ → eμ shown in Fig. 3.1c, d, with the decomposition equation of the photon propagator given by Eq. (3.8).

60

3

Interpolation of Quantum Electrodynamics

 Fig. 3.2 Vertices that appear in the x + -ordered diagrams. When C = 0, only two kinds of vertices (a) and (b) exist. When C = 0, all three vertices (a–c) are present

After the above analysis, with a little thought, one can summarize and write down  -ordered diagrams as the following: the rules for x + 1. u( p, s), u( ¯ p, s), v( p, s), v( ¯ p, s), μ ( p, λ) and μ∗ ( p, λ) for each incoming and outgoing external lines; ¯ p, s) for electron propagators; (− p + m) = −s v( p, s) 2. ( p + m) = s u( p, s)u( ∗ ≡ v( ¯ p, s) for positron propagators; T μ ν ν (λ) for photon propagaλ=±  μ (λ) tors; μ (2π )3 δ(P in − P out )δ 2 (Pin − Pout ) for each vertex as shown in Fig. 3.2a; 3. −ieγ    ⊥ ⊥ − − − e2

in μ n ν in out 2 in out (2π )3 δ(P−  − P−  )δ (P⊥ − P⊥ ) 2 + Cq⊥ μ · · · γ ν ··· ×···γ

2 q− 

for each vertex as shown in Fig. 3.2b, where q−  , q⊥ are the total momentum transferred; −ie2 γ μ γ + γ ν

1 + 2 in out (2π )3 δ(Pin+ − Pout )δ (P⊥ − P⊥ ), 2q +

for each vertex as shown in Fig. 3.2c appearing only if C = 0, i.e. only in LFD, where q + = k + − p + ; 4. P  −Pi  +ı for each internal line, where Pint+  and Pinter+  are the sums of ini+ inter+ energies for the initial and intermediate particles; in out 5. an overall factor of (2π )δ(P+  − P+  ) for the interpolating energy conservation; 6. an integration

dq⊥ (2π )3

∞ −∞

dq−   (q− )  2Q +

for every internal propagating line, with m in Eq. (3.15) being the mass of the exchanged particle.

3.2 Formal Derivation of the Interpolation of QED

61

The rules for x + -ordered diagrams on the light-front, first derived by Kogut and Soper [21], are reproduced by taking C = 0 in the above rules. For instance, in rule + 6, when C = 0, the integration limits of q−  = q− = q change to (0, ∞), i.e.

dq⊥ (2π )3

∞

dq + 2q +

0

for every internal line. In the next Sect. 3.2.2, we develop the canonical field theory of quantum electrodynamics in any interpolating angle. And we will see that it reproduces the Feynman rules we obtained here.

3.2.2

Canonical Field Theory

3.2.2.1 Equations of Motion The Lagrangian density for QED is 1  μ ν  μ ¯ L = − F + ψ(iγ D μ νF μ − m)ψ, 4

(3.30)

where D μ = ∂ μ + ie A μ , and F μ ν = ∂ μ A ν − ∂ ν A μ . The equations of motion are therefore  μ ν ¯ ν ψ, = e J ν = eψγ ∂ μF  μ

(3.31)

 μ

(iγ ∂ μ − eγ A μ − m)ψ = 0.

(3.32)

By converting the upper index components into lower index components, Eq. (3.31) can be written as 2 (C∂ 2⊥ + ∂−   )A+



+ 2 =(C∂+  + S∂−  )∂ ⊥ · A⊥ + (∂+  ∂−  − S∂ ⊥ )A−  − eJ .

(3.33)

Next, we apply the generalized transverse gauge condition [10]: ∂−  A−  + C∂ ⊥ · A⊥ = 0,

(3.34)

and Eq. (3.33) simplifies to 



+ + 2 2 2 (C∂ 2⊥ + ∂−  + SA−  ) = (C∂ ⊥ + ∂−  )(CA+  )A = −e J C.

(3.35)

From Eqs. (3.34) and (3.35), we see that we can regard A1 and A2 as the two  , A− independent free components, while at any given “time” x +  can be determined

62

3

Interpolation of Quantum Electrodynamics

by A1 , A2 and A+  determined by A1 , A2 and ψ. We may take the boundary condi 1 2  1 2 + + tion, A−  (x , x , x , +∞) = −A−  (x , x , x , −∞), which is consistent with the choice made by Kogut and Soper for the light-front QED [21]. Then, the solution to Eq. (3.34) is found as 



+ 1 2 − A−  (x , x , x , x ) 1      = − C d x − (x − − x − )∂ ⊥ · A⊥ (x + , x 1 , x 2 , x − ) 2 1  1 2 −     + = C d x − |x − − x − |∂−  ∂ ⊥ · A⊥ (x , x , x , x ), 2

using the integration by parts and noting (x) =

d|x| dx ,

(3.36)

i.e.

1, x > 0, (x) = −1, x < 0. √     By a simple change of variables X + ≡ x+ , X⊥ ≡ x⊥ / C, X − ≡ x− , Eq. (3.35) becomes   ∂2 ∂2    A+ = −e J + C (i = 1, 2), + (3.37) ∇¯ 2 A+ ≡  )2 ∂(X i )2 ∂(X − which has the solution 

A+ = e



⊥



d2 X d X −



 J+ (X  )C

 − X −  )2 4π (X⊥ − X ⊥ )2 + (X −

,

(3.38)

   μ ≡ (X + where the argument of J + (X  ) denotes the four-vector X  , X 1 , X 2 , X − )= 1 2   √  + 3 0 (x + , x , √x , x − ). In the instant form limit (C → 1), A−  → A , A → A , C

C

 → J 0 and the above solutions given by Eqs. (3.36) and (3.38) agree with the J+  + instant form results. In the light-front limit (C → 0), both A−  and A in Eqs. (3.36) and (3.38), respectively, can also be easily shown to be consistent with the light-front gauge A+ = 0 due to the apparent C factor in the numerator.    + − + However, we note that both A+  = −SA−  /C + A /C and A = SA /C − − A−  /C carry overall 1/C factor, and thus A in LFD, i.e. the C → 0 limit of A+  or  − A , does not vanish. In fact, the A+  component satisfies the following constraint equation without containing any time derivatives:

SA− SA−    2 ) = (C∂ 2⊥ + ∂− ) = −e J + , ∇¯ 2 (A+ + +  )(A+ C C

(3.39)

where the three-dimensional Laplace operator reduces to a one-dimensional operator when C = 0.

3.2 Formal Derivation of the Interpolation of QED

63

From Eq. (3.36), we can find that the term −SA−  /C in the C → 0 (or S → 1) limit becomes   1 dx − |x − − x − |∂− ∂ ⊥ · A⊥ (x + , x 1 , x 2 , x − ). −SA− (3.40)  /C → − 2  /C in the C → 0 limit becomes Also, from Eq. (3.38), we can see that the term A+

 e  dx − |x − − x − |J + (x + , x 1 , x 2 , x − ), A+ /C → − (3.41) 2

where the X ⊥ integration can be made straightforwardly by realizing the suppression  (X  ) and assigning Y = of X ⊥ component in the light-front (C → 0) limit of J +  X − X to use 1 d2 Y⊥   )2 ⊥ 4π (Y )2 + (Y − 1 = d(Y⊥ )2   )2 4 (Y⊥ )2 + (Y − 1  = − |Y − | (3.42) 2 that becomes − 21 |x − − x − | in LFD with the current J + (x + , x 1 , x 2 , x − ) vanishing in the limit |x⊥ | → ∞. Combining Eqs. (3.40) and (3.41), we thus get the LFD result A− (x + , x 1 , x 2 , x − )

1 dx − |x − − x − | ∂− ∂ ⊥ · A⊥ (x + , x 1 , x 2 , x − ) =− 2  +e J + (x + , x 1 , x 2 , x − )) ,

(3.43)

which was also derived in Ref. [21] except for some superficial differences in the conventions used. Eq. (3.43) was noted in Ref. [22] as well. To simplify the notations and make the derivations easier to follow, we may write A−  (x) = −C 

A+ (x) = −

∂ ⊥ · A⊥ (x) , ∂− 

 (x)C eJ+

2 C∂ 2⊥ + ∂− 

,

(3.44) (3.45)

instead of the explicit integral forms shown in Eqs. (3.36) and (3.38). We also write A+  as A+  (x) = S

 (x) ∂ ⊥ · A⊥ (x) eJ+ − , 2 2 ∂− C∂ + ∂−  ⊥ 

(3.46)

64

3

Interpolation of Quantum Electrodynamics

  + + which represents A+  = −SA−  /C + A /C with A−  and A given by Eqs. (3.36) and (3.38), respectively. Written in this way, Eqs. (3.44)–(3.46) also show very clearly, that only the A1 and A2 components of A μ are dynamical variables. For the fermion fields, Eq. (3.32) can be written as

    − ⊥ i γ + ∂+  + γ ∂−  + γ · ∂⊥      − ⊥ − e γ + A+  + γ A−  + γ · A⊥ − m ψ = 0,

(3.47)

μ, γ  ν} = where the interpolating gamma matrices satisfy the usual Clifford algebra {γ   μ  ν 2g and the interpolating metric is given by

⎛

C 0 0 ⎜ 0 −1 0  μ ν ⎜ g = g μ ν =⎝ 0 0 −1 S 0 0

⎞ S 0 ⎟ ⎟. 0 ⎠ −C

(3.48)

If C = 0, Eq. (3.47) contains the interpolating time derivative ∂+  and thus all four components of ψ field are dynamical. However, if C = 0, then one may notice a rather dramatic change of two components of ψ field from being dynamical to the 2 2 constrained components due to {γ + , γ + } = γ + = 0 as well as {γ − , γ − } = γ − = 0, while {γ + , γ − } = 2. This may be shown explicitly by writing Eq. (3.47) for C = 0,

  i γ + ∂+ + γ − ∂− + γ ⊥ · ∂ ⊥    −e γ + A+ + γ − A− + γ ⊥ · A⊥ − m ψ = 0,

(3.49)

and splitting ψ into ψ+ = P+ ψ and ψ− = P− ψ with the projection operators P+ = 1 − + 1 + − 2 γ γ and P− = 2 γ γ , i.e. ψ = ψ+ + ψ− = P+ ψ + P− ψ.

(3.50)

Then, because γ + P− = 0, ψ− can be determined at any light-front time x + through the following constraint equation

 2 (i∂− − e A− ) ψ− = (i∂ ⊥ − eA⊥ ) γ ⊥ + m γ + ψ+ ,

(3.51)

which reduces in the light-front gauge A− = A+ = 0 to 

2 (i∂− ψ− ) = (i∂ ⊥ − eA⊥ ) γ ⊥ + m γ + ψ+ .

(3.52)

Thus, the two components of ψ given by ψ− in LFD become constrained in the sense that the time dependence of ψ− is provided by the other fields that satisfy the dynamic equation with the light-front time derivative ∂+ such as A⊥ and ψ+ . No

3.2 Formal Derivation of the Interpolation of QED

65

new time-dynamic information can be provided by the constrained field ψ− . As done in Ref. [21], we may split this constrained field ψ− into the “free” part ψ˜ − and the “interaction” part ϒ, i.e. ψ− = ψ˜ − + ϒ, identifying from Eq. (3.52): ψ˜ − =

(iγ ⊥ · ∂ ⊥ + m)γ + ψ+ , 2i∂−

(3.53)

−eγ ⊥ · A⊥ γ + ψ+ . 2i∂−

(3.54)

and ϒ=

Then, as shown in Ref. [21], the light-front fermion instantaneous diagram depicted in − ∂ )ϒ. ¯ Fig. 3.2c corresponds to the interaction Hamiltonian density given by ϒ(iγ − This reveals that the instantaneous contribution to the fermion propagator given by Eq. (3.21) is obtained through the “interaction” part of the constraint field ψ− . We may define ψ = ψ˜ + δC0 ϒ.

(3.55)

When C = 0, ψ = ψ˜ is the free fermion field. When C = 0, ψ can be split into ψ = ψ+ + ψ− , where only ψ+ = ψ˜ + is independent. The constraint field ψ− can be further split into ψ− = ψ˜ − + ϒ, where ϒ is the “interaction” part of the field. We write ψ˜ = ψ˜ + + ψ˜ − , then ψ˜ is the free part of the field in any interpolating angle 0 ≤ δ ≤ π4 . We discuss this unique feature of the fermion propagator in LFD further illustrating the old-fashioned perturbation theory in Sect. 3.2.2.4 and presenting the physical processes such as the electron–positron annihilation to the pair production of two photons (e+ e− → γ γ ) and the Compton scattering (eγ → eγ ) in Sects. 3.3 and 3.4.

3.2.2.2 Free Fields The Fourier expansion of the free fermion field ψ(x) takes the form



 d2 p⊥ d p− − ⊥  u (s) e−i x p− −ix ·p⊥  + 3 (2π ) (2 p ) s=±1/2   ⊥   + (s) i x − p−  +ix ·p⊥ d † (p , p ; s; x + × b(p⊥ , p− ) ,  ; s; x ) + v e  ⊥ − 



ψ(x + , x⊥ , x − ) =

(3.56)

where the spinors of the particle (u) and the antiparticle (v) satisfy the Dirac equation: μ p (γ  μ − m)u = 0,  μ

(γ p μ + m)v = 0.

(3.57) (3.58)

Here, we take u and v to be the generalized helicity spinors u H and v H whose explicit expressions in the chiral basis have been given in Ref. [11]. For simplicity, we will omit the subscript “H” here.

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Plugging ψ given by Eq. (3.56) to the free Dirac equation, 



+ ⊥ − μ (iγ  ∂ μ − m)ψ(x , x , x ) = 0,

(3.59)

 + and using the relations in Eqs. (3.57) and (3.58), we find that b(p⊥ , p−  ; s; x ) and  + d † (p⊥ , p−  ; s; x ) satisfy the following differential equations: 





+ + [iγ + ∂+  − γ p+  ] b(p⊥ , p−  ; s; x ) = 0,  +

 +

 +

† [iγ ∂+  + γ p+  ] d (p⊥ , p−  ; s; x ) = 0.

(3.60) (3.61)

Solving these equations, we get 

+ −i x b(p⊥ , p−  ; s; x ) = e 

 + p+ 

 i x+ p+ 

+ d † (p⊥ , p−  ; s; x ) = e

b(p⊥ , p−  ; s; 0),

(3.62)

d † (p⊥ , p−  ; s; 0).

(3.63)

Since the time dependence decouples from the rest of the operator, we may drop the time labels and define b(p⊥ , p−  ; s) ≡ b(p⊥ , p−  ; s; 0),

(3.64)

d (p⊥ , p−  ; s) ≡ d (p⊥ , p−  ; s; 0).

(3.65)

†

†

Then, the free fermion field can be summarized as ψ(x) =



d2 p⊥ d p−  μ  u (s) e−i x pμ b(p⊥ , p−  ; s)  + 3 (2π ) 2 p s=±1/2 +v (s) ei x

 μp  μ

 d † (p⊥ , p−  ; s) .

(3.66)

Following a similar procedure, we can also find the free photon field as μ A (x) =



d2 p⊥ d p−  μ   μ  ( p, λ) e−i x pμ a(p⊥ , p−  ; s)  (2π )3 2 p + λ=± +ei x

 μp  μ

 a † (p⊥ , p−  ; s) ,

(3.67)

μ ( p, ±) are explicitly given in Ref. [10]. where again the polarization vectors  

3.2 Formal Derivation of the Interpolation of QED

67

3.2.2.3 Energy–Momentum and Angular Momentum Tensors Using Noether’s theorem, the conserved energy–momentum tensor and angular momentum tensor can be written as 

μ μ μλ μ ¯  ∂ν ψ − F  ∂ν Aλ − g  T  ν = i ψγ  νL ,

 J λ μ ν

=

  λ λ x ν − x μ μT  νT 

+

 Sλ μ ν,

(3.68) (3.69)

where 1 λ    λ λ ¯ [γ Sλ μ ν = i ψγ μ A ν A μ , γ ν ]ψ + F  ν −F  μ. 4

(3.70)

In particular, the total four-momentum and total angular momentum given by   = d2 x⊥ dx − T +  (3.71) P μ, μ   d2 x⊥ dx − J +  (3.72) M μ ν μ ν = are constants of motion. In particular, the kinematic generators which do not alter the  , such as P1 , P2 , P− interpolating time x +  , M12 , M2−  , M1−  , are provided by their corresponding densities given by     ¯ + T + i = i ψγ ∂i ψ − ∂i A j (∂ + A j − ∂ j A+ ),  +

 +

 +

 +

¯ ∂− = i ψγ  ψ − ∂−  A j (∂ A − ∂ A ), 1     ¯ + J + 12 = x1 T + 2 − x2 T + 1 + i ψγ γ1 γ2 ψ 2     + A2 ∂ + A1 − A1 ∂ + A2 + A1 ∂ 2 A+ − A2 ∂ 1 A+ , 1     + + ¯ + J + 1− γ1 γ−  − x−  T 1 + i ψγ ψ  = x1 T − 2     + + + + + A−  ∂ A1 − A1 ∂ A−  + A1 ∂−  A − A−  ∂1 A , 1     + + ¯ + J + 2− γ2 γ−  = x2 T −  − x−  T 2 + i ψγ ψ 2     + + + + + A−  ∂ A2 − A2 ∂ A−  + A2 ∂−  A − A−  ∂2 A ,

T

 −

j

j

(3.73) (3.74)

(3.75)

(3.76)

(3.77)

 + where A−  and A are given by Eqs. (3.44) and (3.45), and thus these operators involve only independent dynamical fields ψ and A j ( j = 1, 2). Finally, the most important operator of the theory is of course the interpolating Hamiltonian density:     μ ¯  ¯ −iγ j ∂ j − iγ − T ++ ∂− ψ  =ψ  + m ψ + e A μ ψγ

+

1  j −  + + ∂+ ∂+ F μν F Aj − F  A− , μ ν −F 4

(3.78)

where the transverse index j is summed over according to the summation convention.

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3.2.2.4 Old-Fashioned Perturbation Theory  With Eqs. (3.44)–(3.45), as well as Eqs. (3.53)–(3.55), we can rewrite T +  in terms + 1 2 ˜ and separate out the interaction of the independent degrees of freedom A , A , ψ, part of the Hamiltonian density from the free part. The detailed derivation is given in Appendix A.2. Eq. (3.78) becomes 

H ≡ T ++  = H0 + V

(3.79)

 ¯˜ j ˜ H0 = ψ(−iγ ∂ j − iγ − ∂−  + m)ψ 1 μν ˜ j −  + ˜ j − F˜ + ˜− + F˜  ∂+ ∂+ F A A , μ ν − F˜ 4 1  μ ˜ − ˜¯  ¯ ψ + δC0 ϒ(iγ V = e A˜  ∂− )ϒ + eφ J + , μ ψγ 2

(3.80)

with

(3.81)

where we have defined A˜  μ as ∂ ⊥ · A⊥ ˜ 1 , A˜ 2 , A˜ − , A1 , A2 , A− ( A˜ + , A  ) ≡ (S  ), ∂− 

(3.82)

and  A+ ˜ φ(x) ≡ A+ (x) − A (x) =   + C  + e J (x) =− 2 2 C∂ ⊥ + ∂−   J+ (X  ) ⊥  = e d2 X dX −  ,  − X −  )2 4π (X⊥ − X ⊥ )2 + (X −

(3.83)

where we switched the simplified notation in the second line into the expression 1 2  √  μ ≡ (x + of integration in the third one. The capital X  , x , √x , x − ) is introduced C C previously above Eq. (3.37). Equation (3.83) may be considered as a generalization of Eq. (4.58) in Ref. [21] for the quantization interpolating between IFD and LFD. We can then calculate the scattering matrix element S f i =< f |S|i > between the initial and final states |i > and | f > with the “old-fashioned” perturbation theory expansion S f i = δ f i − i2π δ(P+ i − P+ f)   −1 × < f | V + V (P+  − P+ 0 + i) V + · · · |i >,

(3.84)

3.2 Formal Derivation of the Interpolation of QED

69

 2 ⊥ −  2 ⊥ −   where P+ 0 = d x dx H0 and V = d x dx V . This leads to the same rules  + for x -ordered diagrams which we obtained in Sect. 3.2.1.2 by directly decomposing the covariant Feynman diagrams. This can be seen by calculating a few matrix elements of the interaction Hamiltonian V . The first term in Eq. (3.81) after volume integration gives the interaction at equal  = 0: interpolating time x + −i V1 = −ie

  ¯˜  μ ˜  ⊥ − x⊥ , x − )γ  ψ(0, x⊥ , x − ), d2 x⊥ dx − A˜  μ (0, x , x )ψ(0,

(3.85)

which is the “ordinary” vertex interaction as demonstrated in Fig. 3.2a. With Eq. (3.54), the second term in Eq. (3.81) can be shown to provide the fermion instantaneous interaction 1 ¯˜ x⊥ , x − )γ i A˜ i (0, x⊥ , x − ) −i V2 = − e2 δC0 d2 x⊥ dx − ψ(0, 2 γ+ ˜ ˜ x⊥ , x − ) × A j (0, x⊥ , x − )γ j ψ(0, ∂− 1 2 ¯˜ x⊥ , x − )γ i A˜ i (0, x⊥ , x − )γ + = − e δC0 d2 x⊥ dx − ψ(0, 4 ˜ x⊥ , x − ). × dx − (x − − x − ) A˜ j (0, x⊥ , x − )γ j ψ(0, (3.86) Using 1 2



dx − (x − − x − ) e−iq

+ (x − −x − )

=

i , q+

(3.87)

Eq. (3.86) can be shown to yield the vertex of Fig. 3.2c, as discussed in Ref. [21]. The third term in Eq. (3.81) written out in full is 1 −i V3 = ie2 2 ×



 ¯    x⊥ , x − )γ + ψ(0, x⊥ , x − ) d2 x⊥ dx − ψ(0,

1 2 ∂ 2⊥ C + ∂− 

   ¯ ψ(0, x⊥ , x − )γ + ψ(0, x⊥ , x − )

1 2  ¯    x⊥ , x − )γ + ψ(0, x⊥ , x − ) d2 x⊥ dx − ψ(0, = − ie 2   +  ⊥ − ⊥ − ¯  ψ(0, X , X )γ ψ(0, X , X ) . × d2 X⊥ dX −   − X −  )2 4π (X⊥ − X⊥ )2 + (X −

(3.88)

⊥ In the scaled transverse space with the variable √ of X , one should note that the corresponding √ transverse momentum becomes Cq⊥ due to the equality given by q⊥ · x⊥ = Cq⊥ · X⊥ . Using

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Fig. 3.3 Feynman diagram for e+ e− → γ γ process. While this figure is drawn for the t-channel Feynman diagram, the crossed channel (or u-channel) can be drawn by crossing the two final state particles



=



d2 X⊥ dX −

1 , 2 + q2 Cq⊥  −

e

−i

√

   − − Cq⊥ ·(X⊥ −X⊥ )+q−  (X −X )

  − X −  )2 4π (X⊥ − X⊥ )2 + (X − (3.89)

where q is the momentum transfer at the vertex, as depicted in Fig. 3.2b. We find that the interaction −i V3 yields the “Coulomb” vertices of Fig. 3.2b. We note that this interpolation result coincides with the IFD result for C = 1, where ψ = ψ˜ is the free fermion field, while for C = 0, the ψ field in Eq. (3.88) changes naturally to ψ˜ due to the γ +2 = 0 property of the LF, so that the ϒ field does not contribute to the + component of the current. The transverse components of the momentum in Eq. (3.89) also drop off naturally due to the C factor in front, reproducing smoothly Kogut and Soper’s result in Ref. [21]. Thus, when we calculate the scattering matrix formally in the interpolating QED, we get the same rules as we summarized in Sect. 3.2.1.2 when we decompose the covariant Feynman diagrams directly [23].

3.3

Toy Calculation of e+ e− Annihilation Producing Two Scalar Particles

Having laid out the foundation of interpolating QED, we can now make some calculations. The first simple heuristic example we consider is e+ e− annihilation producing two scalar particles. In the next section, we will consider the typical QED process of e+ e− → γ γ , as well as eγ → eγ , but for now we don’t consider the photon polarization to make things simpler. While the Feynman diagram of e+ e− → γ γ is shown in Fig. 3.3, the photon (γ ) line should be understood as the scalar particle line for the production process of two scalar particles.

3.3 Toy Calculation of e+ e− Annihilation Producing Two Scalar Particles

71

Fig. 3.4 Time-ordered diagrams (a) and (b) for e+ e− → γ γ annihilation process. The u-channel amplitudes can be obtained by crossing the two final state particles

As mentioned in the Introduction, the covariant propagator of the intermediate virtual fermion is given by =

q + m . q2 − m2

(3.90)

In the instant form where the system evolves with ordinary time t, this covariant Feynman amplitude can be decomposed into two time-ordered ones, as shown in Figs. 3.4a, b, where again the photon (γ ) line should be understood as the scalar particle line for the production process of two scalar particles. Figures 3.4a, b correspond to the following time-ordered amplitudes 1 0 2qon 1 = 0 2qon

aIFD = bIFD

q a + m , 0 q 0 − qon −q b + m . 0 −q 0 − qon

(3.91) (3.92)

Here, qon is the momentum four-vector with its spacial part equal to that of q (= p1 − p3 = qa ) but satisfies the Einstein energy–momentum relationship, and qb corresponds to the negative energy (antiparticle) contribution with qb = −qa = −q. The sum of the two propagators can easily be verified to be equal to the covariant one, Eq. (3.90): aIFD

+ bIFD

1 = 0 2qon



q + m q + m − 0 0 0 0 q − qon q + qon

0 (q + m) 1 2qon 0 0 0 )2 2qon (q )2 − (qon q + m = 2 , q − m2



=

(3.93)

 0 = q2 + m 2 is used. where the on-shell condition qon Such time-ordering also exists in the interpolating dynamics, whose “time” means  . The interpolating time-ordered diagrams are also Figs. the interpolating time x + 3.4a, b, and the propagators of the intermediate virtual fermion for each time-ordering

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Interpolation of Quantum Electrodynamics

are given by Qa + m 1  q −Q 2Q +   + a+ − Q b + m 1 , b =  −q − Q 2Q +   + b+

a =

(3.94) (3.95)

where Q a +  and Q b+  are the interpolating on-mass-shell energy of the intermediate propagating fermion as mentioned in the introduction and again their expressions are explicitly given by  + −Sqa − +Q , C  + −Sqb− +Q = , C

Q a+  =

(3.96)

Q b+ 

(3.97)

  denoting the on-mass-shell value of q + as given by Eq. (3.15). If the interwith Q + polating longitudinal momentum q−  is positive, i.e. q−  > 0, then the intermediate propagating fermion in the time-ordered amplitude in Fig. 3.4a is “forward” moving and the corresponding time-ordered amplitude a is equivalent to F given by Eq. (3.2), while the time-ordered amplitude in Fig. 3.4b with the “backward” moving (−q−  < 0) intermediate fermion corresponds to B in the same equation, Eq. (3.2). Using Eqs. (3.96)–(3.97), the sum of these two interpolating propagators can also be verified to be equal to Eq. (3.90) as shown in Appendix A.3. When we take the limit to the LFD, i.e. δ → π4 or C → 0, the expressions in Eq. (3.94) and (3.95) change to the so-called on-mass-shell propagating contribution and the instantaneous fermion contribution, respectively, if and only if q + > 0. For q + > 0, the time-ordered diagram shown in Fig. 3.4b has the “backward” moving + intermediate fermion (for C = 0, −q−  = −q < 0), and the LF energy for the intermediate virtual fermion, Q b+  , goes to infinity; however, the existence of the spin sum on the numerator makes it altogether a finite result. The finite result turns out to be the instantaneous fermion contribution unique in the LFD as formally discussed in Sect. 3.2. This can be now shown explicitly as follows: ⎛ ⎞ 1 Qb − m ⎠ b,δ→ π4 = lim ⎝   + Sq− C→0 2Q +  +Q q+ + C    +  Sq−   +Q + − ⊥ C γ − γ q−  − γ · q⊥ − m C 1 = lim   + C→0 2Q + Cq+  + Sq− +Q   γ + q− + Q +  = + 2q q− + Q +

=

γ+ . 2q +

(3.98)

3.3 Toy Calculation of e+ e− Annihilation Producing Two Scalar Particles

73

At the same time, the first diagram shown in Fig. 3.4a turns out to be the on-mass-shell contribution as shown explicitly in the following:   1 Qa + m a,δ→ π4 = lim  q −Q C→0 2Q +   + a+ 1 Qa + m = 2q− q − − Q a− +m q = + on− − 2q q − qon +m q . (3.99) = 2on q − m2 This proves the decomposition of the covariant fermion propagator in LFD [20] given by Eq. (3.7) as discussed in the introduction (Sect. 3.1) as well as in the formal derivation (Sect. 3.2). Let’s now compute the time-ordered amplitudes for the e+ e− annihilation into two scalar particles using the interpolating formulation, which are given by Maλ1 ,λ2 = v¯λ2 ( p2 ) · a · u λ1 ( p1 )

(3.100)

Mbλ1 ,λ2 = v¯λ2 ( p2 ) · b · u λ1 ( p1 ),

(3.101)

and represent the helicities of the initial e−

and e+

where λ1 and λ2 spinors, respectively, and the overall factor such as the coupling constant e, etc., is taken to be 1. Here, a and b are given by Eqs. (3.94) and (3.95). If q = p1 − p3 , then these amplitudes λ1 ,λ2 λ1 ,λ2 and Ma,t . Similarly, are the t-channel amplitudes which we may denote as Ma,t λ1 ,λ2 if q = p1 − p4 , then we may denote them as the u-channel amplitudes Ma,u and λ1 ,λ2 Mb,u , respectively. The spinors in the interpolation form were studied in Ref. [11], and the results were given by  ⎛  ⎞ (+1/2) (P) uH

⎜ ⎜ ⎜ =⎜ ⎜ ⎜ ⎝

 +P P−  +P P+ 2P sin δ+cos δ ⎟   δ+cos δ  + P⎟ + P R sin P ⎟ 2P(P+P− )  ⎟,   ⎟  −P P−  +P P+ ⎟ cosδ−sin δ ⎠  2P δ−sin δ  + P R cos P − P 2P(P+P− ) 

  ⎞ δ−sin δ  + −P L cos 2P(P+P− ) P −P  ⎟ ⎜    −P ⎟ ⎜ P−  +P P+ ⎟ ⎜ (−1/2) 2P cos δ−sin δ ⎟, ⎜   uH (P) = ⎜ sin δ+cos δ  L + + P⎟ P ⎟ ⎜ −P ⎠ ⎝  2P(P+P  − ) ⎛

P−  +P 2P

 +P P+ sin δ+cos δ

74

3

  )2 − m 2 C = where P = (P +



Interpolation of Quantum Electrodynamics

− →2 2 R 1 2 L 1 2 P−  + P ⊥ C, P = P + i P , P = P − i P .

Here, the antiparticle spinors are obtained by charge conjugation. To make the numerical calculations, we need to specify the kinematics for the process, as shown in Fig. 3.5. We choose the initial reference frame to be the e+ e− center-of-mass frame (CMF), and study the whole landscape of the amplitude change under the boost operation in the zˆ -direction as well as the change of the interpolation angle δ. The moving direction of the incoming electron is chosen as the +ˆz -direction. Then, the four-momenta of the initial and final particles can be written as p1 p2 p3 p4

= (E 0 , 0, 0, Pe ) = (E 0 , 0, 0, −Pe ) = (E 0 , E 0 sin θ, 0, E 0 cos θ ) = (E 0 , −E 0 sin θ, 0, −E 0 cos θ ).

(3.102)

In the kinematics given by Eq. (3.102), we note that the intermediate propagating fermion momentum in the time-ordered process depicted in Fig. 3.4a is given by q = p1 − p3 = (0, −E 0 sin θ, 0, 0, Pe − E 0 cos θ ) and thus its light-front plus component q + = Pe − E 0 cos θ can be negative as well as positive depending on the scattering angle θ . Thus, for the kinematic region of q + < 0 in LFD, the tchannel process in Fig. 3.4a corresponds to the “backward” process B although it corresponds to the “forward” process F for the kinematic region of q + > 0. The critical scattering angle which separates the kinematic region between q + > 0 and q + < 0 can of course be obtained by q + = 0 in the corresponding process. In the present kinematics given by Eq. (3.102), the critical scattering angles for t-channel with q = p1 − p3 and u-channel with q = p1 − p4 are respectively given by  θc,t = arccos

Pe E0

 ,

  Pe . θc,u = arccos − E0

(3.103)

(3.104)

λ1 ,λ2 One should realize that the same amplitude, e.g. Ma,t given by Eq. (3.100) with q = p1 − p3 , can correspond to either the “on-mass-shell propagating contribution” or the “instantaneous fermion contribution” in LFD depending on the scattering angle, e.g. θ > θc,t or θ < θc,t , respectively. Detailed discussions on the angular distribution of each and every helicity amplitude are presented in Appendix A.4, contrasting the results between the IFD and the LFD. In the next Sect. 3.3.1, we focus on the collinear scattering/annihilation case, θ = π , where the correspondence to the “on-mass-shell propagating contribution” or the “instantaneous fermion conλ1 ,λ2 in the δ → π4 limit corresponds tribution” in LFD is fairly obvious, e.g. Ma,t only to the “on-mass-shell propagating contribution” in LFD as π > θc,t . We then summarize the results of e+ e− → two scalar particles in the subsequent Sect. 3.3.2.

3.3 Toy Calculation of e+ e− Annihilation Producing Two Scalar Particles

75

Fig. 3.5 e+ e− pair annihilation process at angle θ in center-of-mass frame

3.3.1

Collinear Scattering/Annihilation, θ = π

We consider here the collinear amplitude taking the the center-of-mass angle θ between the moving direction of incoming electron (particle 1) and outgoing photon (particle 3) as π , i.e. the collinear back-to-back scattering/annihilation process. The purpose is to exhibit the essential landscape of the helicity amplitudes depending on the reference frame, i.e. the center-of-mass momentum in the zˆ -direction P z , and the interpolation angle δ. In this collinear kinematics, the two time-ordered t-channel processes depicted in Fig. 3.4a, b correspond to the “forward” moving and “backλ1 ,λ2 ward” moving processes without any complication. Thus, the amplitudes Ma,t λ1 ,λ2 and Mb,t in the δ → π4 limit readily correspond to the “on-mass-shell propagating contribution” and “instantaneous fermion contribution” in LFD, respectively. In order not to concern ourselves with the absolute values, we also scale all the energy and momentum values by the electron mass m e and take the scalar particles as massless. For the simple illustration, we take the initial energy of each particle as 2m e , √ i.e. E 0 = 2m e and Pe = 3m e . The results of the collinear back-to-back scattering/annihilation, i.e. θ = π , are shown in Figs. 3.6 and 3.7, where we use “+” and “−” to denote the helicity of the initial fermions. For example, “+−” means a right-handed electron and a lefthanded positron annihilation. As the final state particles are scalars, they don’t have any designation of helicities. Here, t(a) means the first time-ordering of t-channel, corresponding to the diagram Fig. 3.4a, t(b) means Fig. 3.4b, etc. There is also the u-channel, which can be obtained by swapping the two outgoing particles, and the two time-ordering of u-channel can be drawn in a similar way. The results of the u-channel are shown in Figs. 3.8 and 3.9. The amplitudes are plotted as a function of P z and δ. When δ → 0, i.e. the back ends of the figures, the IFD results are obtained, while δ → π/4, i.e. the front ends of the figures, the LFD results are obtained. The red solid line in the middle of all the figures is given by

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Interpolation of Quantum Electrodynamics

Fig. 3.6 Annihilation amplitudes for e+ e− to two scalars t-channel time-ordering process-a : for (a) helicity ++, (b) helicity +− , (c) helicity −+ and (d) helicity −−

 P =− z

√

s(1 − C) , 2C

(3.105)

where s is the center-of-mass energy. This characteristic curve called “J-curve” has been discussed in Sect. 2.4 and also extensively in our previous works [9–11] in conjunction with the zero-mode in P z → −∞ limit where the plus component of the light-front momentum for all the particles involved in the scattering/annihilation process vanishes, i.e. pi+ → 0 (i = 1, 2, 3, 4). We note here that this characteristic “J-curve” corresponds to the zero of the interpolating total longitudinal momentum, P−  = 0. As discussed in Ref. [11], “J-curve” sits in between the two boundaries indicated by blue dashed lines in all of the figures, Figs. 3.6, 3.7, 3.8 and 3.9, across which the amplitude changes abruptly. The reason for this abrupt change, as we also discussed in our previous spinor work [11], is because the electron and positron

3.3 Toy Calculation of e+ e− Annihilation Producing Two Scalar Particles

77

Fig. 3.7 Annihilation amplitudes for e+ e− to two scalars t-channel time-ordering process-b: for (a) helicity ++, (b) helicity +−, (c) helicity −+ and (d) helicity −−

moving along the zˆ -direction have the speed less than the speed of light c so that the direction of the particle motion can be swapped to the opposite direction in the frame which moves faster than the particle. Namely, the helicity defined in IFD is not invariant but dependent on the reference frame. For a given helicity amplitude in IFD, the particle’s spin must flip when its moving direction flips to maintain the given helicity. This results in a sudden abrupt change in each helicity amplitude. In other words, a different spin configuration appears going across the boundary. For example, the left and right boundaries drawn in all the panels of Fig. 3.6 correspond respectively to p1−  = 0 (zero longitudinal interpolating momentum for the electron) and p2−  =0 (zero longitudinal interpolating momentum for the positron). The change of the helicity depending on the reference frame has been extensively discussed in our previous spinor work [11]. In particular, the LF helicity of the particle moving in the −ˆz direction is opposite to the Jacob-Wick helicity defined in the IFD. Such swap of the helicity between the IFD and LFD for the particle moving in the −ˆz direction

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Fig. 3.8 Annihilation amplitudes for e+ e− to two scalars u-channel time-ordering process-a: for (a) helicity ++, (b) helicity +−, (c) helicity −+ and (d) helicity −−

has been extensively discussed in Ref. [11], and the application in the deeply virtual Compton scattering has been reviewed in Ref. [24]. We find indeed that the behavior of the angle between the momentum direction and the spin direction bifurcates at a critical interpolation angle and the IFD and the LFD separately belong to the two different branches bifurcated at this critical interpolation angle. The details of the discussion on the boundaries in the helicity amplitudes, similar to the left and right boundaries in Fig. 3.6, can be found in Ref. [11] with the examples of eμ → eμ and e+ e− → μ+ μ− processes. Solving the equation p1−  = 0, we get  E 0 P z + Pe (2E 0 )2 + (P z )2  tan δ = − Pe P z + E 0 (2E 0 )2 + (P z )2

(3.106)

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Fig. 3.9 Annihilation amplitudes for e+ e− to two scalars u-channel time-ordering process-b: for (a) helicity ++, (b) helicity +−, (c) helicity −+ and (d) helicity −−

for the electron and similarly from p2−  = 0 we get  E 0 P z − Pe (2E 0 )2 + (P z )2 tan δ = −  E 0 (2E 0 )2 + (P z )2 − Pe P z

(3.107)

for the positron. These two boundaries are depicted in Fig. 3.10. At P z = 0, the critical interpolating angle δc corresponding to the boundary due to the√positron’s helicity swap is given by δc = tan−1 (P√ e /E 0 ). For E 0 = 2m e and Pe = 3m e , this critical value is given by δc = tan−1 ( 3/2) ≈ 0.713724 as one can see from Fig. 3.10. The bifurcation of the two helicity branches, one belongs to the IFD side and the other belongs to the LFD side, occurs exactly at δ = δc in the CMF (P z = 0) and the abrupt change of the helicity amplitudes crossing from one branch to another branch, e.g. 0 ≤ δ < δc ≈ 0.713724 and δc ≈ 0.713724 < δ ≤ π/4, can be understood in the example presented in Figs. 3.6, 3.7, 3.8 and 3.9 as well as in our previous

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Pz 15 10 5

0.2

0.4

0.6

0.8

5 10 15

Fig. 3.10 Two boundaries

works [11]. One should note that this bifurcation of the two helicity branches is independent of the scattering angle θ and thus persists even in the non-collinear helicity amplitudes that we discuss in Appendix A.4. However, one should note that the LFD result is completely outside of these boundaries, as it appears as a straight line on the LF end. This is due to the boost invariance of the helicity in LFD as we emphasize in the present work as well as in our previous works [9–11]. In LFD, we note that the results depicted in Figs. 3.7 and 3.9 correspond to the instantaneous fermion contribution as shown in Eq. (3.98). One may note [25] that the amplitude vγ ¯ + u vanishes for the helicity non-flip case, i.e. ++ and −−, while it survives for the helicity flip case, i.e. +− and −+. This demonstrates that the LFD (δ = π/4) results of ++ and −− helicity amplitudes, +,+ −,− +,+ −,− Mb,t , Mb,t , Mb,u and Mb,u , respectively, are zero while the LFD results of +,− −,+ +,− −,+ , Mb,t , Mb,u and Mb,u , respectively, +− and −+ helicity amplitudes, Mb,t are nonzero as shown in Figs. 3.7 and 3.9. For this collinear back-to-back scattering/annihilation process, the apparent angular momentum conservation can be rather easily seen in all of Figs. 3.6, 3.7, 3.8 and 3.9. Because the initial electron and positron are spin 21 particles and the final state particles are spinless, only the spin singlet system of the two spin-half particles can annihilate and produce two scalar particles in the center-of-mass frame (i.e. P z = 0) due to the angular momentum conservation. Thus, only when the initial particles have their spins in opposite direction, the amplitude can be nonzero. In Figs. 3.6, 3.7, 3.8 and 3.9, we note that the +− and −+ helicity amplitudes between the two blue line boundaries vanish as they correspond to the spin triplet configuration not satisfying the angular momentum conservation. Also, the relative sign between the non-vanishing ++ and −− helicity amplitudes in the same kinematic region is opposite revealing the nature of spin singlet configuration. Moreover, these results are consistent with the the well-known symmetry based on parity conservation that

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the amplitudes in helicity basis must satisfy [24] M (−λ1 , −λ2 ) = (−1)−λ1 −λ2 M (λ1 , λ2 )

(3.108)

where λ1 and λ2 are the helicities of the incoming electron and positron. The sum of t-channel and u-channel amplitude of each initial helicity state is shown in Fig. 3.11. All the symmetry that each channel and time-ordered amplitude individually satisfy of course work in the sum of the individual amplitude as well. Thus, again in Fig. 3.11, the angular momentum conservation and the spin singlet nature of the system are also manifest, i.e. +− and −+ helicity amplitudes between the two blue line boundaries vanish and the non-vanishing ++ and −− helicity amplitudes in the same kinematic region have opposite sign to each other. They are again consistent with Eq. (3.108). In Figs. 3.6, 3.7, 3.8, 3.9 and 3.11, we note that the IFD results in P z → +∞ appear to yield the corresponding LFD results as one can see the smooth connection of each and every amplitude in the right region outside the right boundary. This may suggest that the IFD result in the infinite momentum frame (IMF) yields the LFD result. However, one should note that the IFD results in P z → −∞ are not only different from the corresponding LFD results but also incapable of achieving the LFD results as they are apart by the two blue boundaries in between. Thus, the IMF in the left region outside the left boundary in IFD cannot yield the desired LFD result although the IMF in the right region outside the right boundary may do the job. One should be cautious in the prevailing notion of the equivalence between the IFD at the IMF and the LFD. Of course, if each helicity amplitude shown in Fig. 3.11 is squared and summed over all four helicity states, then the result is completely independent of P z and δ as a flat constant in the entire region of P z and δ space.

3.3.2

Summary of e+ e− → Two Scalar Particles

As we have shown in all of these results, the LFD results are completely independent of the reference frame due to the boost invariance while the IFD results are dependent on the reference frame. As discussed in the previous subsection (see Figs. 3.6, 3.7, 3.8, 3.9 and 3.11), the LFD results are outside the spin-flip boundary and the LF helicity of the particle moving in the −ˆz direction is opposite to the Jacob-Wick helicity defined in the IFD. The same characteristic behavior can be seen also in the angular distribution of helicity amplitudes for the non-collinear case presented in Appendix A.4. With the helicity swap between the IFD and the LFD for the particle moving in the −ˆz direction, one can see the angular momentum conservation and the spin-singlet nature in the LFD results. Namely, the ++ and −− LF helicity amplitudes vanish at θ = 0 and θ = π (see e.g. Fig. A.7 in Appendix A.4), and they are opposite in the angular dependence while the relative sign between the +− and −+ LF helicity amplitudes is equal to each other in accordance with Eq. (3.108). As discussed in the collinear case, the IFD results in P z → −∞ do not yield the LFD results. Likewise, in the non-collinear case shown in Appendix A.4, we also note

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Fig. 3.11 Total annihilation amplitudes for e+ e− to two scalars: for (a) helicity ++, (b) helicity +−, (c) helicity −+ and (d) helicity −−

that the angular distribution of the IFD amplitudes in P z → −∞ is opposite in sign with respect to the corresponding angular distribution of the LFD amplitudes (see Figs. A.7a and A.7b in Appendix A.4), let alone that each time-ordered amplitudes of IFD in P z → −∞ yields far different angular distribution from the corresponding LFD results (see e.g. Figs. A.11 and A.12b in Appendix A.4). Although the angular distribution of each IFD helicity amplitude in P z → +∞ is supposed to yield the identical corresponding angular distribution of the LFD helicity amplitude, one would need to boost the P z value much higher than +15m e (see e.g. Figs. A.11 and A.12 in Appendix A.4) in order to get indeed the very similar profile of “onmass-shell propagating contribution” and the “instantaneous fermion contribution” in LFD. In light of this discussion on the P z dependence, we may comment on the validity of ongoing efforts to compute the so-called quasi-parton distributions in an infinite momentum frame. While the idea to circumvent the difficulties with the continuation of parton distribution functions (PDFs) to the Euclidean region, namely the large

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83

momentum effective theory (LaMET) [26], surged the lattice QCD community to work on quasi-PDFs, it has been noted that this approach encounters some problems related to the power divergent mixing pattern of deep inelastic scattering operators when implemented within the lattice regularization [27]. Besides this power divergent mixing issue on the lattice, our work indicates an importance of tracking the direction dependence, P z versus −P z , in particular for the spin-dependent amplitudes in the LaMET approach. The numerical differences of the helicity amplitudes in IFD appear significant between P z > 0 and P z < 0 for the large |P z | values, let alone the difference between the Jacob-Wick helicity and the LF helicity. In the LaMET application of the lattice QCD computation, the gauge link direction may be taken both in the original direction and the direction opposite to the original direction to check numerically if the results are consistent to each other between the two directions. If enough boost is achieved in each of the positive and negative directions, then the numerical results are not only consistent to each other but also close to the LFD result. Our work indicates an importance of checking the direction dependence, e.g. gauge link, as numerical differences between P z and −P z results would reveal whether the LaMET achieved enough boost or not. In the helicity amplitude square (or probability) level, we see the built-in t − u symmetry in the e+ e− annihilation process regardless of IFD or LFD as manifested in the θ → π − θ symmetry of the angular distributions presented in Appendix A.4 (see Figs. A.7c and A.7d as well as Fig. A.14 in Appendix A.4). We have verified that the result of Fig. A.14 in Appendix A.4 is in exact agreement with the analytic result of the total amplitude square for the scalar particle pair production in e+ e− annihilation given by |M |2scalar ≡ 



λ1 ,λ2 λ1 ,λ2 λ1 ,λ2 2 λ1 ,λ2 |Ma,t + Mb,t + Ma,u + Mb,u |

λ1 ,λ2 + m 2 (4s

 − 5t + 3u) − 15m 4 (t − m 2 )2   2 2 tu + m (4s − 5u + 3t) − 15m 4 + (u − m 2 )2   2 (s + u)u + (s + t)t + 2m 2 (3s − t − u) − 30m 4 + , (t − m 2 )(u − m 2 ) =

2 ut

(3.109)

where m = m e and the Mandelstam variables s = ( p1√+ p2 )2 , t = ( p1 − p3 )2 and u√ = ( p1 − p4 )2 are given by s = 16m 2e , t = (−7 + 4 3 cos θ )m 2e and √ u = −(7 + 2 4 3 cos θ )m e in CMF given by Eq. (3.102) with E 0 = 2m e and Pe = 3m e for our numerical calculation. In fact, the built-in t − u symmetry in each and every helicity amplitude square is completely independent of the interpolation angle δ as shown in Appendix A.4 (see Figs. A.14c, d, A.10c, d, A.13c, d in Appendix A.4). Essentially the same kind of t − u symmetry can be found in the e+ e− → γ γ QED process which we now discuss in the next section, Sect. 3.4.

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3.4

Interpolating Helicity Scattering Probabilities

3.4.1

e+ e− Pair Annihilation into Two Photons

Having discussed all the helicity amplitudes of the pair production of scalar particles in e+ e− annihilation, we now look into the two photon production process in the same initial state of e+ e− annihilation. While there must be some similarity inherited from the same initial state, there must be also some difference in the helicity amplitudes due to the change of the final state from the spinless pair of scalar particles to the two real photons in QED. The identification of the real photon helicity would require a particular attention as it doesn’t carry any rest mass and moves invariantly with the speed of light. The lowest-order t-channel QED Feynman diagram is already in place as Fig. 3.3 and the corresponding u-channel diagram can be attained by swapping the two final photons in Fig. 3.3. The two time-ordered diagrams in the t-channel are also displayed in Figs. 3.4a, b, and the kinematics is the same with the previous calculation illustrated in Fig. 3.5 and written in the previous section in Eq. (3.102). The QED helicity amplitudes Mtλ1 ,λ2 ,λ3 ,λ4 and Muλ1 ,λ2 ,λ3 ,λ4 with the two initial lepton helicities λ1 and λ2 and the final two photon helicities λ3 and λ4 in t and u channels, respectively, are now expressed with the interpolating Lorentz indices  μ and  ν as μ λ3 ∗ λ1 Mtλ1 ,λ2 ,λ3 ,λ4 = v¯ λ2 ( p2 )νλ4 ( p4 )∗ γ ν t γ   μ ( p3 ) u ( p1 ),

(3.110)

λ3 ∗  μ  ν λ4 ∗ λ1 ( p2 ) μ ( p3 ) γ u γ  ν ( p4 ) u ( p1 ),

(3.111)

Muλ1 ,λ2 ,λ3 ,λ4

= v¯

λ2

where t and u are t =

 p1 −  p3 + m , t − m2

u =

 p1 −  p4 + m u − m2

(3.112)

λ with t = ( p1 − p3 )2 and u = ( p1 − p4 )2 , and the polarization vector  μ (P) is given by [10]

  1 P1 P−  − i P2 P P2 P−  + i P1 P +  (P) = − |, | , S|P , , −C|P √ ⊥ ⊥ μ |P⊥ | |P⊥ | 2P   1 P1 P−  + i P2 P P2 P−  − i P1 P − (P) = |, | , S|P  , , −C|P √ ⊥ ⊥ μ |P⊥ | |P⊥ | 2P    P+ M2 0  P+ (3.113) , P1 , P2 , P− − +  , μ (P) = MP P with P =



2 2 P−  + P⊥ C =



 2 (P + ) − M 2 C. Note that this interpolating polarization

 λ + vector   A−  + ∂ ⊥ · A⊥ C = 0, μ (P) respects the gauge condition A = 0 and ∂− which links the light-front gauge A+ = 0 in the LFD and the Coulomb gauge ∇ · A = 0 in IFD as discussed in Ref. [10]. For the sake of generality, we kept here the generic

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85

fermion and gauge boson mass as m and M, respectively. The real photon helicity λ takes only + or − but not 0 as M → 0 limit and thus there is no issue involved in taking the massless limit. One should note that not only the final state momenta p3 and p4 are swapped but also the QED vertices with the γ matrices are exchanged between the t-channel amplitude and the u-channel amplitude given by Eqs. (3.110) and (3.111), respectively. The symmetry of the helicity amplitudes based on the parity conservation [24] given by Eq. (3.108) for the pair production of the scalar particles is also now extended for the two-photon production as M −λ1 ,−λ2 ,−λ3 ,−λ4 = (−1)λ3 +λ4 −λ1 −λ2 M λ1 ,λ2 ,λ3 ,λ4 ,

(3.114)

where λ3 and λ4 are the helicities of the outgoing photons while λ1 and λ2 are the incoming electron and positron helicities, respectively. As this symmetry works identically for both t and u channels, the subscripts t and u in the helicity amplitude above are suppressed in Eq. (3.114). Now, recalling Eqs. (3.94) and (3.95), the time-ordered amplitudes in t-channel can be written in short-hand notations without specifying the helicities as Ma,t = v( ¯ p2 ) ( p4 )∗

 Q a,t + m q +  − Q a,t +  2Q t t + 1

!

and ∗

Mb,t = v( ¯ p2 ) ( p4 )

− Q b,t + m  −q +  − Q b,t +  2Q t t+ 1

 ( p3 )∗ u( p1 ),

(3.115)

!  ( p3 )∗ u( p1 ),

(3.116)

where qa,t = qt ≡ p1 − p3 , qb,t = −qa,t = −qt , and Q a,t +  and Q b,t +  are the interpolating on-mass-shell energy of the intermediate propagating fermion given by 

Q a,t +  Q b,t +  

+ −Sqa,t −  + Qt = , C  + −Sqb,t −  + Qt = , C

(3.117) (3.118)



+ with Q + t denoting the on-mass-shell value of qt as 

Q+ t ≡

 2 2 qt2−  + C(qt⊥ + m ).

(3.119)

The kinematics here is of course identical to the ones given in the last section, Sect. 3.3, despite the explicit notations to specify t and u channels which now involve the swap of not only the final state particle momenta but also the QED photon and fermion vertices. Thus, we elaborate the notations to designate the t and u channels more explicitly for this section.

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As the final state photons with momentum p3 and p4 must be swapped for the u-channel amplitudes, we denote the intermediate fermion momentum between the two photon vertices as qu = p1 − p4 and correspondingly designate all other timeordered variables replacing qa,t and qb,t in the t-channel time-ordered amplitudes by qa,u = qu = p1 − p4 and qb,u = −qa,u = −qu , respectively. Consequently, the interpolating on-mass-shell energy of the intermediate propagating fermion Q a,u +  and Q b,u +  for the two time-ordered amplitudes are also given by replacing qa,t and qb,t by qa,u and qb,u , respectively, in Eqs. (3.117) and (3.118) together with the   + replacement of Q + t in Eq. (3.119) by Q u as 

Q+ u ≡

 2 2 qu2−  + C(qu⊥ + m ).

(3.120)

While the notations are more elaborated in this section as described here, there’s no change in the kinematics from the ones provided in the last section, Sect. 3.3. To make the numerical calculations, we take the same initial energy of each √ particle (i.e. E e = 2m e and Pe = 3m e ) and three different reference frames (i.e. CMF given by Eq. (3.102) and boosted frames with P z = 15m e and P z = −15m e ) for the angular distribution analysis of the interpolating helicity amplitudes. While we focus on the CMF result in this section, the results in the boosted frames (P z = 15m e and P z = −15m e ) are summarized in the Appendix A.5 and the P z dependence of the interpolating helicity amplitudes for a particular scattering, e.g. θ = π/3 case, is shown in the Appendix A.6. In Fig. 3.12, we show the whole landscape of the interpolation angle (δ) dependence for the angular distributions of the helicity amplitudes with the notation λ1 λ2 → λ3 λ4 for ++ → ++, ++ → +−, ++ → −+, ++ → −−, as well as +− → ++, +− → +−, +− → −+, +− → −− at CMF (i.e. P z = 0) in (a) λ1 ,λ2 ,λ3 ,λ4 λ1 ,λ2 ,λ3 ,λ4 Ma,t and (b) Mb,t . The far most left two columns of Fig. 3.12 show

+,+,λ3 ,λ4 +,−,λ3 ,λ4 and Ma,t with the final four helicity conthe helicity amplitudes Ma,t figurations of the photon pairs {λ3 , λ4 } = {+, +}, {+, −}, {−, +}, {−, −} but with the initial ++ and +− helicity configurations of e+ e− pair annihilation. While the results here are shown for the nonzero fermion mass m = m e , one may check first the consistency with chiral symmetry by taking the massless limit m → 0. We did this check for the IFD (δ = 0) amplitudes which might be more accessible for intuitive understanding due to the more familiar Jacob-Wick helicity used in the IFD. In the massless limit, the chirality coincides with the helicity. Consistency check with the chiral symmetry here is thus equivalent to the helicity conservation of the massless fermion fields in the electromagnetic vector coupling. For the illustration, the IFD (δ = 0) profiles of the left two columns in Fig. 3.12 are shown in Fig. 3.13 and the corresponding profiles for the massless limit of the fermion, m → 0 limit, +,+,λ3 ,λ4 and in Fig. 3.14. While the helicity amplitudes are nonzero both for Ma,t +,−,λ3 ,λ4 +,+,λ3 ,λ4 Ma,t as shown in Fig. 3.13 for m = m e , the helicity amplitudes Ma,t all vanish for m = 0 as shown in Fig. 3.14. One may understand this result as a consequence of chiral symmetry and the helicity conservation in the m = 0 limit. As the electromagnetic interaction preserves the chirality/helicity in the massless limit,

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87

Fig. 3.12 Angular distribution of the helicity amplitudes for (a) t-channel time-ordering process-a and (b) t-channel time-ordering process-b

Fig. 3.13 Helicity amplitudes for the nonzero electron mass. As the chirality is not conserved, the upper four amplitudes are not zero

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Fig. 3.14 Helicity amplitudes for the massless electron. When setting electron mass equal to zero, chirality is conserved. The upper four amplitudes are zero as the initial helicity of the electron and positron are the same +,+,λ3 ,λ4 one may understand why all the helicity amplitudes Ma,t vanish for m = 0 as shown in Fig. 3.14. Having checked the consistency of our results with respect to the chiral symmetry and helicity conservation in the massless limit, we now turn to each individual helicity amplitudes and examine their individual characteristics of the angular distribution in the whole landscape of the interpolating helicity amplitudes. The far most left column +,+,λ3 ,λ4 with the final four helicity of Fig. 3.12 shows the helicity amplitudes Ma,t configurations of the photon pairs {λ3 , λ4 } = {+, +}, {+, −}, {−, +}, {−, −} but with the same initial ++ helicity configuration of e+ e− pair annihilation. This column may be compared with Fig. A.5a in Appendix A.4 which shows the helicity amplitude of the pair production of scalar particles with the same initial ++ helicity configuration of e+ e− pair annihilation. A thin boundary sheet at δ = δc ≈ 0.713724 in CMF (P z = 0) is shown in each and every figure of Fig. 3.12 to denote the critical interpolation angle δc which separates the IFD side and the LFD side of helicity branches. Although there are four final helicity configurations in the photon pair production, the basic structure of the initial ++ helicity configuration of e+ e− pair annihilation is inherited as one can see the clear separation of the “instantaneous fermion contribution” from the “on-mass-shell propagating contribution” in LFD with the critical angle θc,t given by Eq. (3.103). As discussed in Sect. 3.3, the √ critical angle θc,t turns out to be θc,t = π/6 ≈ 0.523599 for E 0 = 2m e and Pe = 3m e and it’s apparent that the unique feature of LFD with respect to separation of the “instantaneous fermion contribution” from the “on-mass-shell propagating contribution” is persistent whether the final states are the pair of the scalar particles or the pair of two photons. For the photon pair production, however, a dramatic new feature appears λ due to the photon polarization given by Eq. (3.113). In particular,  μ in Eq. (3.113) reveals a singular feature as δ → π/4. For λ = + as an example, at δ = π/4, i.e. in + + = − √|P⊥ |+ behaves as + = − √ sin θ so LFD, the polarization component + 2P 2(1+cos θ ) 2 ≈ − ε for θ = π − ε with small ε. This explains the singular behavior near +,+,+,+ in LFD shown in the top far left figure of Fig. 3.12. For θ ≈ π , for Ma,t

+ that +

θ =π

√

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89

one should note that p3+ ≈ 0 and the corresponding photon’s polarization compo+,+,+,+ + yields the singular behavior exhibited in the LFD result of Ma,t . This nent + +,+,+,+ light-front singularity in Ma,t turns out to be canceled by the same with the +,+,+,+ opposite sign in Mb,u as one may see in Fig. 3.15. Similarly, the light-front +,+,+,+ for θ ≈ 0 due to p4+ ≈ 0 in the (b) time-ordered singularity appearing in Mb,t +,+,+,+ . Thus, the process is canceled by the same with the opposite sign in Ma,u total helicity amplitude summing all the t- and u-channel time-ordered amplitudes, +,+,+,+ +,+,+,+ +,+,+,+ +,+,+,+ + Mb,t + Ma,u + Mb,u , is free from any singular i.e. Ma,t behavior as shown in Fig. 3.16. One may also notice that the effect of the overall + − + ∗ − sign change between  μ and  μ , i.e. ( μ ) = − μ , in Eq. (3.113) is reflected in the negative versus positive sign difference of the helicity amplitudes and ultimately +,+,+,+ +,+,−,+ and Ma,t as shown in Fig. 3.12. the light-front singularity between Ma,t +,+,+,+ Similar to the cancellation of the light-front singularity between Ma,t and +,+,+,+ +,+,−,+ Ma,t , the light-front singularity in Ma,t turns out to be canceled by the +,+,−,+ same with the opposite sign in Mb,u as shown in Fig. 3.15. Again, the total helicity amplitude summing all the t- and u-channel time-ordered amplitudes, i.e. +,+,−,+ +,+,−,+ +,+,−,+ +,+,−,+ Ma,t + Mb,t + Ma,u + Mb,u , is completely free from any singular behavior as shown in Fig. 3.16. However, one should also note that the survival of this singular behavior depends on the time-ordering of the process as well as the helicities of the particles in the process as not only the longitudinal component but also the transverse component of the polarization vector also matters in affecting the removal or survival of the singular behavior in the helicity amplitude. As an example, one can see that the singular behavior from the zero-mode +,+,+,− +,+,+,+ while it shows up in Ma,t and p3+ ≈ 0 for θ ≈ π is removed in Ma,t +,+,−,+ Ma,t . The reason why all of the four t-channel (a) time-ordered helicity ampli+,+,+,+ +,+,+,− +,+,−,+ +,+,−,− tudes (Ma,t , Ma,t , Ma,t , Ma,t ) with the initial ++ helic+ − ity configuration of e e pair annihilation vanish for θ < θc,t = π/6 ≈ 0.523599 is because the region θ < θc,t = π/6 ≈ 0.523599 belongs to the “instantaneous fermion contribution” and v¯ ↑ γ + u ↑ = 0, i.e. the γ + operator of the instantaneous contribution in LFD cannot link between the initial electron and positron pair with the same helicity. Conversely, in the t-channel (b) time-ordered process, the region of “instantaneous fermion contribution” is for θ > θc,t = π/6 ≈ 0.523599 and all +,+,+,+ +,+,+,− , Mb,t , of the four t-channel (b) time-ordered helicity amplitudes (Mb,t +,+,−,+ +,+,−,− + − Mb,t , Mb,t ) with the initial ++ helicity configuration of e e pair annihilation vanish in the region θ > θc,t = π/6 ≈ 0.523599 as shown in Fig. 3.12. In the region θ < θc,t = π/6 ≈ 0.523599, however, these amplitudes are non-vanishing +,+,+,+ +,+,+,− and Mb,t exhibit even the singular behavior near θ ≈ 0 due to and Mb,t the p4+ ≈ 0 zero-mode as depicted in Fig. 3.12b. +,+,λ3 ,λ4 In the left column of Fig. 3.16, we show the helicity amplitudes Ma,t + +,+,λ3 ,λ4 +,+,λ3 ,λ4 +,+,λ3 ,λ4 Mb,t + Ma,u + Mb,u . We note here the IFD/LFD profile correspondence. Namely, for the outgoing photon helicities λ3 and λ4 , the IFD profile of the incident ++ helicity amplitude corresponds to the LFD profile of the incident +− helicity amplitude modulo overall signs of the helicity amplitudes, and vice versa.

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Fig. 3.15 Angular distribution of the helicity amplitudes for (a) u-channel time-ordering process-a and (b) u-channel time-ordering process-b

While the reason for this correspondence is partly due to the swap of the helicity between the IFD and LFD for the incident positron moving in the −ˆz direction, we should also note the interesting characteristic of the outgoing real photon helicities λ3 and λ4 . The relationship between the LF helicity and the Jacob-Wick helicity defined in the IFD is generally given by a Wigner rotation [28]. For the massless particle such as the real photon, the relationship gets particularly simplified as unity unless the massless particle is moving in the −ˆz direction. Thus, for the region 0 < θ < π without involving exact boundary values of θ = 0 and θ = π , the LF helicity and the Jacob-Wick helicity coincide so that there is no difference between the LF helicity and the Jacob-Wick helicity for the real photons. For this reason, the helicity amplitude +,+,λ3 ,λ4 +,+,λ3 ,λ4 +,+,λ3 ,λ4 +,+,λ3 ,λ4 Ma,t + Mb,t + Ma,u + Mb,u in IFD/LFD corresponds +,−,λ3 ,λ4 +,−,λ3 ,λ4 +,−,λ3 ,λ4 +,−,λ3 ,λ4 + Mb,t + Ma,u + Mb,u in LFD/IFD, respecto Ma,t tively, for the region 0 < θ < π . As an example, in Fig. 3.16, the correspondence +,−,+,− +,−,+,− +,−,+,− + Mb,t + Ma,u + between the profile of the total amplitude Ma,t +,−,+,− +,+,+,− +,+,+,− Mb,u in LFD and the profile of the total amplitude Ma,t + Mb,t +

3.4 Interpolating Helicity Scattering Probabilities

Fig. 3.16 Angular distribution of the t + u helicity amplitudes

91

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Interpolation of Quantum Electrodynamics

+,+,+,− +,+,+,− Ma,u + Mb,u in IFD is manifest. Likewise, the IFD/LFD profile correspondence of the probability for each and every {λ3 , λ4 } pair of photon helicities is self-evident as shown in Fig. 3.17. For the exact boundary values θ = 0 and θ = π , one of the outgoing real photons moves in the −ˆz direction and thus the only care that one has to take is to swap the values of the LF helicity amplitudes according to the correspondence between the LF helicity and the Jacob-Wick helicity defined in the IFD as discussed above for the particle moving in the −ˆz direction. For θ = 0, p3 = E 0 (1, 0, 0, 1) and p4 = E 0 (1, 0, 0, −1) in the CMF kinematics given by Eq. (3.102). Thus, the Jacob-Wick helicity pair {λ3 , λ4 } in IFD corresponds to the LF helicity pair {λ3 , −λ4 } in LFD at exact θ = 0. Likewise, the Jacob-Wick helicity pair {λ3 , λ4 } in IFD corresponds to the LF helicity pair {−λ3 , λ4 } in LFD at exact θ = π . This treacherous point of the LF helicity identification at the exact boundary values of θ = 0 and π can be analyzed with the care of procedure in taking massless limit (M → 0) for the gauge boson polarization vector given by Eq. (3.113) and the details of analysis will be presented elsewhere. In this work, although we keep in mind of the treacherous LF helicity identification at the exact boundary values, we present our work focusing on the region 0 < θ < π without involving the exact boundary values of θ = 0 and θ = π. +,−,λ3 ,λ4 +,−,λ3 ,λ4 +,−,λ3 ,λ4 + Mb,t + Ma,u + The result of the total amplitude Ma,t

+,−,λ3 ,λ4 Mb,u may be further analyzed by taking a look at each channel and timeordered process separately shown in Figs. 3.12 and 3.15 for the region 0 < θ < π . +,−,λ3 ,λ4 with the initial e+ e− helicity pair {+−} and the final four helicFor Ma,t ity configurations of the photon pairs {λ3 , λ4 } = {+, +}, {+, −}, {−, +}, {−, −} depicted in the second column of Fig. 3.12, one may compare the result with Fig. A.6a in Appendix A.4 which shows the helicity amplitude of the pair production of scalar particles with the same initial +− helicity configuration of e+ e− pair annihilation. Not all of the four t-channel (a) time-ordered helicity amplitudes with the initial +− helicity configuration of e+ e− pair annihilation vanish for θ < θc,t = π/6 ≈ 0.523599 in LFD although the region θ < θc,t = π/6 ≈ 0.523599 belongs to the light-front “instantaneous fermion contribution”, because v¯ ↓ γ + u ↑ = 0, i.e. the γ + operator of the instantaneous contribution in LFD can link between the ini+,−,+,− tial electron and positron pair with the opposite helicity. For example, Ma,t is clearly nonzero as shown in the second figure from the top of the second column of Fig. 3.12. Depending on the final photon helicities, however, the amplitude can +,−,−,+ +,−,−,− and Ma,t . Moreover, it is interesting still vanish as in the case of Ma,t +,−,+,+ to note the dramatic rise of the amplitude Ma,t as the scattering/annihilation process becomes collinear (θ ≈ 0) due to the light-front zero-mode p4+ ≈ 0 yield+,−,+,+ appears to vanish ing nonzero finite amplitude although the amplitude Ma,t for the region 0 < θ < θc,t = π/6 ≈ 0.523599 as depicted in the top figure of the +,−,+,− +,−,+,− and Mb,t second column of Fig. 3.12. In particular, the profile of Ma,t in LFD appears as shown in Fig. 3.18. In LFD, as discussed previously, the regions 0 < θ < θc,t = π/6 ≈ 0.523599 and θc,t = π/6 ≈ 0.523599 < θ < π provide the “instantaneous fermion contribution” and the “on-mass-shell propagating contribu-

3.4 Interpolating Helicity Scattering Probabilities

93

Fig. 3.17 Angular distribution of the t + u helicity probabilities. The figures in the last row are the results of summing over all the figures above each of them

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Fig. 3.18 (a) Profile of the t-channel a time-ordered annihilation amplitude in LFD for “+− → +−”, (b) Profile of the t-channel b time-ordered annihilation amplitude in LFD for “+− → +−”

Fig. 3.19 Time-ordered annihilation amplitudes (a) on-shell and (b) instantaneous contributions at LF for “+− → +−”

tion” for the light-front (a) time-ordered amplitude, while the regions are swapped for the light-front (b) time-ordered amplitude. As shown in Fig. 3.19, one may collect the “instantaneous fermion contribution” and the “on-mass-shell propagating contribution” by themselves separately to show the combined (a) + (b) time-ordered amplitude. Whichever way we present the result, both Figs. 3.18 and 3.19 manifest the cancellation of the light-front singular features and yield the finite total t-channel amplitude as shown in Fig. 3.20. We note that the total t-channel amplitude at θ = θc,t = π/6 ≈ 0.523599 is zero. As θ → θc,t , qt+ = p1+ − p3+ → 0 and the interaction behaves as if a contact interaction while the propagator shrinks to a point. For the case of contact interaction, squaring the diagram can yield either a fermion loop or a boson loop. Due to the (−1) factor difference between the fermion loop and the boson loop, the only consistent value of the amplitude square must be zero for the contact interaction. This reasoning may offer the understanding of zero amplitude at θ = θc,t = π/6 ≈ 0.523599 in Fig. 3.20. +,−,λ3 ,λ4 +,−,λ3 ,λ4 and Mb,u shown Likewise, the u-channel helicity amplitudes Ma,u in Fig. 3.15 can be understood by realizing the symmetry under the exchange of the outgoing pair of the photons as well as the forward–backward correspondence θ ↔ π − θ . It may not be too difficult to see the θ ↔ π − θ correspondence between +,−,±,± +,−,±,± +,−,±,± +,−,±,± Ma,u and Ma,t as well as Mb,u and Mb,t modulo overall

3.4 Interpolating Helicity Scattering Probabilities

95

Fig. 3.20 Sum of time-ordered annihilation amplitudes in LFD for “+− → +−”

Fig. 3.21 e+ e− → γ γ (a) sum of helicity probabilities for P z = 0 (CMF), (b) result from the manifestly Lorentz invariant formula given by Eq. (3.121)

sign change of the amplitudes in the IFD side (0 < δ < δc ≈ 0.713724). The sim+,−,±,∓ +,−,∓,± +,−,±,∓ and Ma,t as well as Mb,u and ilar correspondence between Ma,u +,−,∓,± Mb,t can be observed without much difficulty comparing Figs. 3.12 and 3.15. It is evident that the same symmetry is inherited in the sum of the amplitudes presented in Fig. 3.16 as one can see the θ ↔ π − θ symmetry in the θ ↔ π − θ cor+,±,λ3 ,λ4 +,±,λ3 ,λ4 +,±,λ3 ,λ4 +,±,λ3 ,λ4 + Mb,t + Ma,u + Mb,u and respondence between Ma,t +,±,λ4 ,λ3 +,±,λ4 ,λ3 +,±,λ4 ,λ3 +,±,λ4 ,λ3 Ma,t + Mb,t + Ma,u + Mb,u for any photon helicity λ3

−,±,λ3 ,λ4 and λ4 . Due to Eq. (3.114), the same correspondence applies to Ma,t + −,±,λ3 ,λ4 −,±,λ3 ,λ4 −,±,λ3 ,λ4 −,±,λ4 ,λ3 −,±,λ4 ,λ3 Mb,t + Ma,u + Mb,u and Ma,t + Mb,t + −,±,λ4 ,λ3 −,±,λ4 ,λ3 Ma,u + Mb,u as well.

+,±,λ3 ,λ4 +,±,λ3 ,λ4 +,±,λ3 ,λ4 + Mb,t + Ma,u + The corresponding probabilities |Ma,t

+,±,λ3 ,λ4 2 +,±,λ4 ,λ3 +,±,λ4 ,λ3 +,±,λ4 ,λ3 +,±,λ4 ,λ3 2 Mb,u | and |Ma,t + Mb,t + Ma,u + Mb,u | shown

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in Fig. 3.17 of course exhibit the same symmetry with the definite positive sign everywhere. The bottom two figures in Fig. 3.17 summing the final helicities,  +,±,λ3 ,λ4 +,±,λ3 ,λ4 +,±,λ3 ,λ4 +,±,λ3 ,λ4 2 |Ma,t + Mb,t + Ma,u + Mb,u | , exhibit the swap λ3 ,λ4

of the helicity between the IFD and LFD for the particle moving in the −ˆz direction  +,±,λ3 ,λ4 |Ma,t + which we have discussed previously. Namely, the IFD result of λ3 ,λ4

+,±,λ3 ,λ4 +,±,λ3 ,λ4 +,±,λ3 ,λ4 2 Mb,t + Ma,u + Mb,u | is identical to the LFD result of  +,∓,λ3 ,λ4 +,∓,λ3 ,λ4 +,∓,λ3 ,λ4 +,∓,λ3 ,λ4 2 |Ma,t + Mb,t + Ma,u + Mb,u | and vice versa. By λ3 ,λ4

adding the two initial helicity states as well, we may now compare our total result with the well-known manifestly Lorentz invariant result given by

" " "M (e+ e− → γ γ )"2 ≡

λ1 ,λ2 ,λ3 ,λ4 λ1 ,λ2 ,λ3 ,λ4 λ1 ,λ2 ,λ3 ,λ4 2 λ1 ,λ2 ,λ3 ,λ4 |Ma,t + Mb,t + Ma,u + Mb,u |

λ1 ,λ2 ,λ3 ,λ4



=2

+,λ2 ,λ3 ,λ4 +,λ2 ,λ3 ,λ4 +,λ2 ,λ3 ,λ4 2 +,λ2 ,λ3 ,λ4 |Ma,t + Mb,t + Ma,u + Mb,u |

λ2 ,λ3 ,λ4

 =8

um tm + + 2m 2 tm um



sm 1 1 − − tm u m tm um



 − 4m 4

1 1 + 2 tm2 um

 ,

(3.121)

where sm = s − 4m 2 , tm = t − m 2 , u m = u − m 2 and the electric charge factor is√taken to be one. Taking the√specific values, m = m e , s = 16m 2e , t = (−7 + 4 3 cos θ )m 2e and u = −(7 + 4 3 cos θ )m 2e , given just below Eq. (3.109) for our numerical calculation in CMF, we find that the two results, (a) the twice of summing the bottom two figures in Fig. 3.17 and (b) the analytic result given by Eq. (3.121) coincide each other as shown in Fig. 3.21. The result shown in the left panel of Fig. 3.21 is of course completely independent of the interpolation angle δ as it should be. The analytic result in Eq. (3.121) is apparently symmetric under t ↔ u exchange as it must be and gets reduced to the well-known textbook result [29] in the massless limit (m → 0) given by   " " "M (e+ e− → γ γ )"2 = 8 u + t . (3.122) t u It may be interesting to compare this result with the massless limit of Eq. (3.109) for the pair production of spinless particles (or “scalar photons”) given by   u t 2 |M |scalar = 2 + −2 , (3.123) t u where the normalization is reduced by the factor 4 due to the lack of final spin (or helicity) degrees of freedom. When t = u, i.e. θ = π/2 in the massless limit of the initial fermions, we may note that the probability of producing two “scalar photons” is zero while the probability of producing two real photons is nonzero. This may be understood from the fact that the two final “scalar photons” do not carry enough

3.4 Interpolating Helicity Scattering Probabilities

97

Fig. 3.22 s-channel and u-channel Feynman diagrams for Compton scattering

number of degrees of freedom while the real photon carries the transverse spin-1 polarization to offer the matching of the number of degrees of freedom between the initial and final states involving both spin singlet and triplet configurations in the annihilation/production process. As we have now shown that the square of the sum of all the individual channel and time-ordered helicity amplitudes in CMF (P z = 0) is identical to the completely Lorentz-invariant expression in terms of the Mandelstam variables (s, t, u), we are assured that our CMF result in Eq. (3.121) must be reproduced even if each individual channel and time-ordered helicity amplitudes are computed in other boosted frames, e.g. P z = 15m e or P z = −15m e . Nevertheless, each individual amplitudes are not boost invariant except the LFD (δ = π/4) profiles. The IFD (δ = 0) profiles in the P z = −15m e are vastly different not only from the corresponding IFD (δ = 0) profiles in the P z = 15m e but also from the corresponding LFD (δ = π/4) profiles. As we have already discussed in Sect. 3.3, it requires a great caution in the prevailing notion of the equivalence between the IFD in IMF and the LFD. The results in the boosted frames (P z = 15m e and P z = −15m e ) are summarized in the Appendix A.5. We have also shown the P z dependence of the interpolating helicity amplitudes for a particular scattering, e.g. θ = π/3 case in the Appendix A.6.

3.4.2

Compton Scattering

Another important physical scattering process in QED which involves the fermion propagator in the lowest order is the Compton scattering eγ → eγ . Similar to the e+ e− → γ γ process shown in Fig. 3.3 which we have extensively discussed in the previous subsection, the lowest Feynman diagrams for the Compton scattering process is shown in Fig. 3.22. For the obvious reason from the Compton kinematics and the corresponding Mandelstam variables given by s = ( p1 + p2 )2 = 2 p1 · p2 + m 2

(3.124)

t = ( p1 − p3 ) = −2 p1 · p3 + 2m 2

2

(3.125)

u = ( p1 − p4 ) = −2 p1 · p4 + m ,

(3.126)

2

2

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Fig. 3.23 Time-ordered diagrams for s- and u-channel Compton scattering

we call the diagram shown in Fig. 3.22a as the s-channel diagram and the crossed diagram in Fig. 3.22b as the u-channel diagram. The s-channel Feynman diagram is then equivalent to the sum of the top two timeordered diagrams (a) and (b) shown in Fig. 3.23. Similarly, the two time-ordered diagrams for the u-channel (c) and (d) are shown in the bottom of Fig. 3.23. For clarity and simplicity, we call these two u-channel time-ordered diagrams as the u-channel (a) and (b) time-ordered diagrams for the rest of presentation. Now, the s-channel and u-channel Compton amplitudes are given by μ λ2 λ1 Msλ1 ,λ2 ,λ3 ,λ4 = u¯ λ3 ( p3 )νλ4 ( p4 )∗ γ ν s γ   μ ( p2 )u ( p1 ), λ2 λ4  μ ∗  ν λ1 Muλ1 ,λ2 ,λ3 ,λ4 = u¯ λ3 ( p3 ) μ ( p2 )γ u  ν ( p4 ) γ u ( p1 ),

(3.127)

where s and u are s =

q s + m q + m and u = u 2 s−m u − m2

(3.128)

with qs = p1 + p2 and s = qs2 , while qu = p1 − p4 and u = qu2 . Then, the timeordered amplitudes of the s-channel Compton scattering can be written in short-hand

3.4 Interpolating Helicity Scattering Probabilities

99

notations without specifying the helicities as  Q a,s + m q +  − Q a,s +  2Q s s + 1

∗

Ma,s = u( ¯ p3 ) ( p4 )

!

and ∗

Mb,s = u( ¯ p3 ) ( p4 )

− Q b,s + m  −q +  − Q b,s +  2Q s s+ 1

 ( p2 )u( p1 ),

(3.129)

!  ( ps )u( p1 ),

(3.130)

where Q a,s +  and Q b,s +  are the interpolating on-mass-shell energy of the intermediate propagating fermion given by  + −Sqa,s −  + Qs , C  + −Sqb,s −  + Qs = , C

Q a,s +  =

(3.131)

Q b,s + 

(3.132)

  + with qa,s = qs , qb,s = −qs and Q + s denoting the on-mass-shell value of qs as 

Q+ s ≡

 2 2 qs2−  + C(qs⊥ + m ).

(3.133)

Similarly, the time-ordered amplitudes of the u-channel Compton scattering can be written by replacing the s-channel variables by the corresponding u-channel variables. In contrast to the time-ordered processes in e+ e− → γ γ (see Fig. 3.4), the schannel time-ordered processes (a) and (b) in Compton scattering, eγ → eγ , involve one-particle and five-particle Fock states, respectively, while both of the u-channel time-ordered processes (c) and (d) in Compton scattering involve three-particle Fock states as one can see in Fig. 3.23. In particular, the one-particle intermediate state in the s-channel time-ordered process (a) in Compton scattering provides immediately 2 )/( the positivity of qs+ = (s + qs⊥

p21⊥ +m 2 p1+

+

p22⊥ ) p2+

> 0 no matter what the kinemat-

ics are chosen. There is no need to figure out the critical scattering angles as we have obtained in the case of the e+ e− → γ γ process such as Eqs. (3.103) and (3.104). Regardless of kinematics the Compton scattering, the positivity of qs+ > 0 allows the use of Eqs. (3.98) and (3.99) to identify immediately the “on-mass-shell propagating contribution” and the “instantaneous contribution” in LFD as corresponding to the s-channel time-ordered processes (a) and (b) in Fig. 3.23, respectively. For the u-channel Compton scattering, however, the identification of the “on-massshell propagating contribution” and the “instantaneous contribution” in LFD depends on the kinematics similar to the e+ e− → γ γ case. Nevertheless, we note that the CMF kinematics in the Compton scattering allows the identification of the “onmass-shell propagating contribution” and the “instantaneous contribution” in LFD as corresponding to the u-channel time-ordered processes (c) and (d) in Fig. 3.23,

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respectively, regardless of the scattering angle. For the immediate identification of the “on-mass-shell propagating contribution” and the “instantaneous contribution” in LFD both for the s and u channels with the correspondence to the time-ordered processes shown in Fig. 3.23, we choose the CMF in this work for the rest of the discussion on the Compton scattering. The well-known Klein–Nishina formular [30] in the target rest frame and the Thomson limit in the low energy Compton scattering, etc. are also assured. The kinematics pictured in Fig. 3.5 can be applied in the Compton scattering and written as the following: p1 p2 p3 p4

= (E 0 , 0, 0, Pe ) = (Pe , 0, 0, −Pe ) = (E 0 , Pe sin θ, 0, Pe cos θ ) = (Pe , −Pe sin θ, 0, −Pe cos θ ),

(3.134)

 where Pe = E 02 − m 2e . In this section, we discuss the whole landscape of Compton scattering with respect to the interpolation angle δ and the C.M. momentum P z to show the frame dependence of each and every time-ordered scattering amplitudes in both s and u channels. For the numerical calculation of the interpolating helicity amplitudes, we scale all the energy and momentum values by the electron mass as done previously and take m = m e , E 0 = 2m e and θ = π/3. The results of s-channel (a) and (b) as well as u-channel (a) and (b)time-ordered helicity amplitudes are shown in Figs. 3.24, 3.25, 3.26 and 3.27, respectively. The probabilities, or the square of the sum of each and every helicity amplitudes, are also shown in Fig. 3.28. In all of these figures, the boundary of bifurcated helicity branches between IFD and LFD due to the initial electron moving in zˆ direction given by Eq. (3.106) (i.e. p1− = 0) is denoted by the blue curve while the characteristic z “J-curve” given by Eq. (3.105) (i.e. P−  = 0) existing in the frames boosted in -ˆ direction is depicted as the red curve. It is also apparent that the relationship between different helicity amplitudes given by Eq. (3.114) is satisfied by noting that λ2 and λ4 are now the helicities of the incoming and outgoing photons while λ1 and λ3 are the incoming and outgoing electrons’ helicities, respectively, in Eq. (3.114). This relationship holds as one can see that the upper left block of 2 by 2 figures are identical to the lower right block of 2 by 2 figures while the upper right block of 2 by 2 figures and the lower left bock of 2 by 2 figures are same but with the opposite sign to each other. For the square of amplitudes shown in Fig. 3.28, the same correspondence holds without any sign difference as it should be. Computing the s-channel (a) time-ordered diagram shown in Fig. 3.23a, we obtain λ1 ,λ2 ,λ3 ,λ4 for the results presented in Fig. 3.24 for all 16 helicity amplitudes Ma,s λi = ± (i = 1, 2, 3, 4). All of the LFD profiles (δ = π/4) appear as straight lines indicating the P z independence or the frame independence of the light-front helicity amplitudes as they should be, while the results for all other interpolation angles

3.4 Interpolating Helicity Scattering Probabilities

101

Fig. 3.24 Compton scattering amplitudes—s-channel, time-ordering (a)

0 ≤ δ < π/4 depend on P z , i.e. frame dependent. As discussed earlier, the s-channel (a) time-ordered diagram shown in Fig. 3.23a corresponds to the “on-mass-shell propagating contribution” in LFD. However, it is remarkable that the “on-mass-shell propagating contribution” in LFD turned out to be absent as the values of the LFD profiles shown in Fig. 3.23a are identically zero regardless of the initial and final helicities. We note that this triviality of the LFD results here is due to the fact that the initial photon is incident in the −ˆz direction in the kinematics chosen for this calculation (see Eq. (3.134)) and thus gets only the zero-mode p2+ = 0 and p2⊥ = 0. The zero-mode contributions are apparently absent in the “on-mass-shell propagating contribution” in LFD. The question is then where the non-trivial LFD result can be realized. It turns out that the non-trivial LFD result is realized in the “instantaneous contribution” corresponding to the process depicted in Fig. 3.23b for the kinematics given by Eq. (3.134) which we use in the present calculation.

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Fig. 3.25 Compton scattering amplitudes—s-channel, time-ordering (b) λ1 ,λ2 ,λ3 ,λ4 Figure 3.25 shows the results of all 16 helicity amplitudes Mb,s for λi = ± (i = 1, 2, 3, 4) which were obtained by computing the s-channel (b) time-ordered diagram shown in Fig. 3.23b. The “instantaneous contribution” in LFD corresponds the process shown in Fig. 3.23b with the understanding of the correspondence given by ! − Q b,s + m 1 γ+ (3.135) = +, lim  C→0 2Q + −qs + 2qs  − Q b,s +  s √ where qs+ = (E 0 + Pe )/ 2 in the kinematics provided by Eq. (3.134). As mentioned above, due to the absence of the “on-mass-shell propagating contribution” in LFD for the s-channel in the present kinematics, the entire s-channel contribution in LFD should be obtained from the “instantaneous contribution”. Due to 2 {γ + , γ + } = γ + = 0, the only non-vanishing “instantaneous contribution” to the s-channel helicity amplitudes in the light-front gauge A+ = 0 is provided by only the transverse components of the photon polarization vectors for the helicity non-flip

3.4 Interpolating Helicity Scattering Probabilities

103

Fig. 3.26 Compton scattering amplitudes—u-channel, time-ordering (a)

matrix elements between the initial and final electron spinors generically given by u¯ λ3 ( p3 ) ( p4 )∗

γ+  ( ps )u λ1 ( p1 ) 2qs+

∼ u¯ λ3 ( p3 )γ⊥i γ + γ⊥ u λ1 ( p1 )  = 4δλ1 ,λ3 p1+ p3+ (δ i j + i i j ), j

(3.136)

where δ i j and  i j are the two-dimensional (i, j = 1, 2) Kronecker delta and LeviCivita symbol, respectively. As one can see in Fig. 3.23b, only the two helicity +,+,+,− −,−,−,+ and Mb,s which are equal to each other appear to be amplitudes Mb,s nonzero for δ ≈ π/4. Besides the caveat in assigning the light-front helicity for the real photon moving in the −ˆz direction, which was discussed earlier, it is remarkable that the “instantaneous contribution” of effectively only one helicity amplitude in LFD provides the entire s-channel Compton amplitude.

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Fig. 3.27 Compton scattering amplitudes—u-channel, time-ordering (b) λ1 ,λ2 ,λ3 ,λ4 Likewise, the u-channel (a) and (b) time-ordered helicity amplitudes, Ma,u λ1 ,λ2 ,λ3 ,λ4 and Mb,u , for λi = ± (i = 1, 2, 3, 4) are shown in Figs. 3.26 and 3.27, respectively. As mentioned earlier, the u-channel time-ordered processes (c) and (d) in Fig. 3.23 correspond to the “on-mass-shell propagating contribution” and the “instantaneous contribution” in LFD for the CMF kinematics as we takein this work. √ From Eq. (3.134), p4+ = Pe (1 − cos θ ) = 3m e /2 = 0 for Pe = E 02 − m 2e = √ 3m e with E 0 = 2m e and θ = π/3 and the “on-mass-shell propagating contribution” in LFD corresponding to the u-channel (a) time-ordered process shown in Fig. 3.23c is non-trivial in contrast to the trivial s-channel (a) time-ordered result. However, the “instantaneous contribution” in LFD corresponding to the u-channel (b) time-ordered process shown in Fig. 3.23d gets again effectively only one helicity 2 amplitude in LFD due to {γ + , γ + } = γ + = 0 and the light-front gauge A+ = 0 as discussed in the s-channel “instantaneous contribution”. As one can see in Fig. 3.27, +,−,+,+ −,+,−,− and Mb,u which only nonzero helicity amplitudes for δ ≈ π/4 are Mb,u are equal to each other.

3.4 Interpolating Helicity Scattering Probabilities

105

Fig. 3.28 Compton scattering probabilities in center-of-mass frame. The last row is sum over all final states for each initial state

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Interpolation of Quantum Electrodynamics

Now, summing all the s-channel and u-channel time-ordered amplitudes shown in Fig. 3.23 and squaring the total amplitude, we obtain the Compton scattering probabilities for each and every helicities shown in Fig. 3.28. As these results are the helicity amplitude squares, one may regard them as the polarization observables exhibiting the change of the predicted magnitudes depending on the reference frames in the range of total center of momentum −15m e < P z < 15m e from the lowestorder interpolating QED computation for the Compton scattering process in the range of interpolation angle between IFD (δ = 0) and LFD (δ = π/4). These results again alert the caution in the prevailing notion of the equivalence between the IMF formulated in IFD and the LFD as the IFD results in P z → −∞ appear incapable of achieving the LFD results although the IFD results in large P z > 0 seem to yield the corresponding LFD results. As the clear differences between the P z → −∞ IFD and the LFD show up in level of physical observables, one should be cautious in the prevailing notion of the equivalence between the IFD at the IMF and the LFD. The sum of the probabilities over all final helicity states for each initial helicities is shown in the last row of Fig. 3.28, and the sum over initial helicity states (i.e. the sum over all sixteen total helicity amplitude squares) turns out to be completely independent of δ and P z as it must be (see Fig. 3.29). Indeed, this result is in complete agreement with the well-known manifestly Lorentz invariant result given by |M (eγ → eγ )|2 ≡



λ1 ,λ2 ,λ3 ,λ4 λ1 ,λ2 ,λ3 ,λ4 2 λ1 ,λ2 ,λ3 ,λ4 λ1 ,λ2 ,λ3 ,λ4 |Ma,s + Mb,s + Ma,u + Mb,u |

λ1 ,λ2 ,λ3 ,λ4



= −8

sm um + + 2m 2 sm um



1 1 tm − − sm u m sm um



 − 4m 4

1 1 + 2 sm2 um

 ,

(3.137)

where sm = s − m 2 , tm = t − 4m 2 , u m = u − m 2 and the electric charge factor is taken to be one. For the kinematics given by Eq. (3.134) with E 0 = 2m e , θ = π/3 and m = m e used in our numerical computation, the value from the√analytic result given by Eq. (3.137) yields |M (eγ → eγ )|2 = (4/169)(991 − 186 3) ≈ 15.8305 which is in precise agreement with the total probability obtained in Fig. 3.29. In the high energy limit, Eq. (3.137) in the massless limit (m → 0) reduces to the well-known textbook [29] Compton result given by |M (eγ → eγ )|2 = −8

u s

+

s . u

(3.138)

The crossing symmetry between the eγ → eγ process and the e+ e− → γ γ process is reflected by the s ↔ t symmetry between Eqs. (3.137) and (3.121) as well as Eqs. (3.138) and (3.122) with the overall sign consistent to each other for the positivity of the amplitude square.

3.5 Summary and Conclusion

107

Fig. 3.29 Total probability of Compton scattering in center-of-mass frame

3.5

Summary and Conclusion

In this chapter, we have completed the interpolation of quantum electrodynamics between the instant form and the front form proposed by Dirac [1] in 1949. We started from the QED Lagrangian and presented the interpolating Hamiltonian formulation introducing a parameter δ which corresponds between the instant form dynamics (IFD) at δ = 0 and the front-form dynamics which we call the light-front dynamics (LFD) at δ = π/4. Not only have we summarized the interpolating timeordered diagram rules for the computation of QED processes in terms of the interpolation angle parameter 0 ≤ δ ≤ π/4 as presented in Sect. 3.2, but also we have applied these rules to the typical QED processes such as e+ e− → γ γ and eγ → eγ which involve the fermion propagator beyond what we have already presented in our previous works [10,11]. Entwining the fermion propagator interpolation with our previous works of the interpolating helicity spinors and the electromagnetic gauge field interpolation, we have now fastened the bolts and nuts necessary in launching the interpolating QED. Our interpolating formulation reveals that there exists the constraint fermion degree of freedom in LFD (δ = π/4) distinguished from the ordinary equal-time fermion degrees of freedom. The constraint component of the fermion degrees of freedom in LFD results in the instantaneous contribution to the fermion propagator distinguished from the ordinary equal-time forward and backward propagation of relativistic fermion degrees of freedom. It is interesting to note that the manifestly covariant fermion propagator decouples to the “on-mass-shell propagating contribution” and the “instantaneous fermion contribution” only in LFD but not in any other

108

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interpolating dynamics (0 ≤ δ < π/4). The helicity of the on-mass-shell fermion spinors in LFD is also distinguished from the ordinary Jacob-Wick helicity in the IFD with respect to whether the helicity depends on the reference frame or not [11]. To exemplify these distinguished features of the fermion degrees of freedom in LFD, we have computed the annihilation process of the fermion and antifermion pair interpolating the fermion degrees of freedom between the IFD and the LFD. We presented the leading order QED processes (e+ e− → γ γ and eγ → eγ ), providing the whole landscape of helicity amplitudes from the IFD to the LFD. In the cross-section level, we showed the precise agreement of our result with the textbook formula. The helicity conservation in the chiral limit was discussed, and the angular momentum conservation was checked in each case. Our analysis clarifies any conceivable confusion in the prevailing notion of the equivalence between the IMF approach in the IFD and the LFD. By investigating the dependence of the helicity amplitudes on the reference frame, i.e. P z -dependence, we find that in IFD, P z → +∞ and P z → −∞ yield very different results from each other, and that one has to be very cautious about the direction of boost in approaching to the IMF when one tries to obtain the equivalent LFD result. In this respect, it would be important to check the direction dependence, e.g. gauge link, in the LaMET [26] approach as numerical difference between P z and −P z results may reveal whether the LaMET achieved enough boost or not. We have shown that although in some cases one can indeed reproduce the LFD result by boosting the system to the correct direction, in some other cases a finite, large momentum boost yields only qualitatively similar result. On the other hand, all the helicity amplitudes in LFD are independent of the reference frame, and certain simplifications to the theory (e.g. suppression of vacuum fluctuations, vanishing of a number of diagrams, etc. ) can be realized even in the rest frame of the system. Since the helicity definition in LFD is frame-independent, no boundaries exist for the lightfront helicity amplitude. One should also note that for the massless particle moving in the −ˆz -direction the helicity defined in the LFD is opposite to the Jacob-Wick helicity defined in the IFD. Further treacherous correspondence between IFD and LFD can be explored, extending the interpolation to the the loop-level computation and ultimately to the QCD.

References 1. Dirac, P.A.M.: Forms of relativistic dynamics. Rev. Mod. Phys. 21, 392–399 (1949) 2. Fubini, S., et al.: New approach to field theory. Phys. Rev. D 7, 1732–1760 (1973) 3. Glozman, L.Y., et al.: Unified description of light- and strange-baryon spectra. Phys. Rev. D 58, 094030 (1998) 4. Wagenbrunn, R.F., et al.: Covariant nucleon electromagnetic form-factors from the Goldstone boson exchange quark model. Phys. Lett. B 511, 33–39 (2001). arXiv: nucl-th/0010048 5. Melde, T. et al.: Electromagnetic nucleon form factors in instant and point form. Phys. Rev. D 76, 074020 (2007) 6. Prokhvatilov, E.V., Franke, V.A.: Limiting transition to lightlike coordinates in the field theory and QCD Hamiltonian (in Russian). Sov. J. Nucl. Phys. 49, 688–692 (1989)

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7. Lenz, F., et al.: Hamiltonian formulation of two-dimensional gauge theories on the light cone. Ann. Phys. 208, 1–89 (1991) 8. Naus, H.W.L., et al.: QCD near the light cone. Phys. Rev. D 56, 8062–8073 (1997) 9. Ji, C.-R., Suzuki, A.T.: Interpolating scattering amplitudes between the instant form and the front form of relativistic dynamics. Phys. Rev. D 87, 065015 (2013) 10. Ji, C.-R., et al.: Electromagnetic gauge field interpolation between the instant form and the front form of the Hamiltonian dynamics. Phys. Rev. D 91, 065020 (2015) 11. Li, Z., et al.: Interpolating helicity spinors between the instant form and the light-front form. Phys. Rev. D 92, 105014 (2015) 12. Hornbostel, K.: Nontrivial vacua from equal time to the light cone. Phys. Rev. D 45, 3781–3801 (1992) 13. Ji, C.-R., Mitchell, C.: Poincaré invariant algebra from instant to light-front quantization. Phys. Rev. D 64, 085013 (2001) 14. Perry, R.J., et al.: Light-front Tamm-Dancoff field theory. Phys. Rev. Lett. 65, 2959–2962 (1990) 15. Srivastava, P.P., Brodsky, S.J.: Light-front-quantized QCD in the light-cone gauge: The doubly transverse gauge propagator. Phys. Rev. D 64, 045006 (2001) 16. Suzuki, A.T., Sales, J.H.O.: Light front gauge propagator reexamined. Nucl. Phys. A 725, 139–148 (2003). arXiv:nucl-th/0303016 17. Leibbrandt, G.: Light-cone gauge in Yang-Mills theory. Phys. Rev. D 29, 1699–1708 (1984) 18. Mantovani, L., et al.: Revisiting the equivalence of light-front and covariant QED in the lightcone gauge. Phys. Rev. D 94, 116005 (2016) 19. Weinberg, S.: Dynamics at infinite momentum. Phys. Rev. 150, 1313–1318 (1966) 20. Chang, S.-J., Yan, T.-M.: Quantum field theories in the infinite-momentum frame. II. Scattering matrices of scalar and Dirac fields. Phys. Rev. D 7, 1147–1161 (1973) 21. Kogut, J.B., Soper, D.E.: “Quantum electrodynamics in the infinite-momentum frame. Phys. Rev. D 1, 2901–2914 (1970) 22. Zhao, X., et al.: Scattering in time-dependent basis light-front quantization. Phys. Rev. D 88, 065014 (2013) 23. Li, Z.: Interpolation from instant form dynamics to light-front dynamics and update on lightfront NCSU. Ph.D. Dissertation (2015) 24. Ji, C.-R., Bakker, B.L.G.: Conceptual issues concerning generalized parton distributions. Int. J. Mod. Phys. E 22(2), 1330002 (2013) 25. Lepage, G.P., Brodsky, S.J.: Exclusive processes in perturbative quantum chromodynamics. Phys. Rev. D 22, 2157–2198 (1980) 26. Ji, X.: Parton physics on a Euclidean lattice. Phys. Rev. Lett. 110, 262002 (2013) 27. Rossi, G.C., Testa, M.: Note on lattice regularization and equal-time correlators for parton distribution functions. Phys. Rev. D 96, 014507 (2017) 28. Carlson, C.E., Ji, C.-R.: Angular conditions, relations between the Breit and light-front frames, and subleading power corrections. Phys. Rev. D 67, 116002 (2003) 29. Halzen, F., Martin, A.: Quarks and Leptons: An Introductory Course in Modern Particle Physics. Wiley, New York (1984) 30. Klein, O., Nishina, Y.: AIJber die Streuung von Strahlung durch freie Elektronen nach der neuen relativistischen Quantendynamik von Dirac. Z. Physik 52, 853–868 (1929)

4

Interpolation of Quantum Chromodynamics in 1+1 Dimension

Having interpolated the quantum electrodynamics in the previous chapter, we now present the interpolation of the ’t Hooft model, i.e. the two-dimensional quantum chromodynamics in the limit of infinite number of colors which is adopted from our published work Phys. Rev. D104, 036004(2021). With an angle parameter δ between δ = 0 for the instant form dynamics (IFD) and δ = π /4 for the light-front dynamics (LFD) we formulate the interpolating mass gap equation which takes into account the non-trivial vacuum effect on the bare fermion mass to find the dressed fermion mass. Our interpolating mass gap solutions not only reproduce the previous IFD result at δ = 0 as well as the previous LFD result at δ = π /4 but also link them together between the IFD and LFD results with the δ parameter. We find the interpolation angle independent characteristic energy function which satisfies the energy–momentum dispersion relation of the dressed fermion, identifying the renormalized fermion mass function and the wavefunction renormalization factor. The renormalized fermion condensate is also found independent of δ, indicating the persistence of the non-trivial vacuum structure even in the LFD. Using the dressed fermion propagator interpolating between IFD and LFD, we derive the corresponding quark–antiquark bound-state equation in the interpolating formulation verifying its agreement with the previous bound-state equations in the IFD and LFD at δ = 0 and δ = π /4, respectively. The mass spectra of mesons bearing the feature of the Regge trajectories are found independent of the δ-parameter reproducing the previous results in LFD and IFD for the equal mass quark and antiquark bound-states. The Gell-Mann–Oakes–Renner relation for the pionic ground state in the zero fermion mass limit is confirmed indicating that the spontaneous breaking of the chiral symmetry occurs in the ’t Hooft model regardless of the quantization for 0 ≤ δ ≤ π /4. We obtain the corresponding bound-state wavefunctions and discuss their reference frame dependence with respect to the frame independent LFD result. Applying them for the computation of the so-called quasi-parton distribution functions now in the interpolating formulation between IFD and LFD, we note a possibility of utilizing not only the reference frame dependence but also the interpolation angle dependence to get an alternative effective approach to the LFD-like results. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 C.-R. Ji, Relativistic Quantum Invariance, Lecture Notes in Physics 1012, https://doi.org/10.1007/978-981-19-7949-1_4

111

112

4.1

4

Interpolation of Quantum Chromodynamics in 1+1 Dimension

Introduction

The two-dimensional quantum chromodynamics (QCD2 ) with the number of colors Nc → ∞ has served as a theoretical laboratory for the study of strong interactions. In ’t Hooft’s seminal paper in 1974 [1], the power of 1/Nc expansion [2] was demonstrated in solving QCD2 in the limit of Nc → ∞, which was then widely studied also in relation to the string model and dual theories with the idea of 1/N c expansion as a topological expansion in the motion of physical strings (e.g. by Witten [3]). Under the large Nc approximation, non-planar diagrams are negligible and thus, for example, only the rainbow diagrams need to be summed over for the computation of the quark’s self-mass. The two other parameters in QCD2 besides Nc are the dimensionful coupling constant g and the quark mass m. Sharing the same mass dimension, g and m play an important role in determining the phase of QCD2 [4,5]. Depending on the value of the dimensionless coupling g 2 Nc /m 2 , it is known that there are at least two phases in QCD2 [6]. While the regime of the strong coupling phase which doesn’t require the finiteness condition on the dimensionless coupling g 2 Nc /m 2 [7] can be studied by the bosonization method [8], the regime of the weak coupling phase which keeps the so-called ’t Hooft coupling λ ∼ g 2 Nc finite in the limit of not only Nc → ∞ but also g → 0 is investigated typically in QCD2 . Although the strong coupling regime of QCD2 is interesting and deserves further study, the scope of the present work is limited to the weak coupling regime of QCD2 . Yet, we notice that solving QCD2 in the weak coupling regime, i.e. ’t Hooft model, is still highly nontrivial as the theory captures the property of quark confinement and involves the infrared-cutoff procedures discussed in two-dimensional gauge field theories [9,10]. In particular, we notice that the ’t Hooft model was originally formulated and solved in the light-front dynamics (LFD) [1] well before it was re-derived and discussed in the instant form dynamcis (IFD) [11]. The numerical solution of the ’t Hooft model in the IFD was presented in the rest frame of the meson [12] and more recently also in the moving frames [13]. While a particular family of the axial gauges interpolated between the IFD and the LFD was explored, the principal value prescription for regulating the infrared divergences was shown to be inconsistent with the interpolated general axial gauges [10]. Since then, however, this issue involving the interpolation between the IFD and the LFD has not yet been examined any further although the nontrivial vacua in two-dimensional models including QED2 have been extensively discussed [14]. It thus motivates us to explore the interpolation of the ’t Hooft model between the IFD and the LFD fully beyond the gauge sector and discuss the outcome of the full interpolation which naturally remedies the previous issue on novel inconsistency with the interpolating gauge [10]. Before we get into the details and specific discussions on the 1+1 dimensional nature of the ’t Hooft model, we again briefly summarize the general remarks on distinguished features of the IFD and the LFD proposed originally by Dirac in 1949 [15] and the efforts of interpolating them together [14,16–22]. The LFD has the advantage of having the maximum number (seven) of the kinematic operators among the ten Poincaré operators. More kinematic operators provide more symmetries that effectively save the efforts of solving dynamic equations. The conversion of the dynamic

4.1 Introduction

113

operator in one form of the dynamics into the kinematic operator in another form of the dynamics can be traced by introducing an interpolation angle parameter spanning between the two different forms of the dynamics. This in fact motivates the study of the interpolation between the IFD and the LFD. In our previous works, we have applied the interpolation method to the scattering amplitude of two scalar particles [17], the electromagnetic gauge fields [18], as well as the helicity spinors [19] and established the interpolating QED theory between the IFD and LFD [20]. In the previous chapter, Chap. 3, we presented [20] the formal derivation of the interpolating QED in the canonical field theory approach and discussed the constraint fermion degrees of freedom, which appear uniquely in the LFD. The constraint component of the fermion degrees of freedom in LFD results in the instantaneous contribution to the fermion propagator, which is genuinely distinguished from the ordinary equal-time forward and backward propagation of the relativistic fermion degrees of freedom. The helicity of the on-mass-shell fermion spinors in LFD is also distinguished from the ordinary Jacob-Wick helicity in the IFD with respect to whether the helicity depends on the reference frame or not. Our analyses clarified any conceivable confusion in the prevailing notion of the equivalence between the infinite momentum frame (IMF) approach and the LFD. To link the 1 + 1 dimensional IFD space-time coordinates with the LFD ones, we again introduce the “hat notation” for the interpolating variables, as we have done in Chaps. 2 and 3: 

 x+ − x



 =

cos δ sin δ sin δ − cos δ



x0 x1

 ,

(4.1)

where the interpolation angle δ is allowed to be in the region of 0 ≤ δ ≤ π4 . When δ = 0, we recover the IFD coordinates (x 0 , −x 1 ), and when√δ = π /4, we arrive at the LFD coordinates denoted typically by x ± = (x 0 ± x 1 )/ 2 without the “hat.” In this coordinate system, the metric becomes g

μˆ νˆ

 = gμˆ νˆ =

 C S , S −C

(4.2)

where we use the shorthand notation C = cos 2δ  and S = sin 2δ. Apparently, the 1 0 μ ˆ ν ˆ interpolating g goes to the IFD metric when δ = 0, and the LFD metric 0 −1   01 when δ = π /4. 10 The components of covariant and contravariant two-vector a are then related with each other by a+ˆ = Ca + + Sa − ;

ˆ

ˆ

a + = Ca+ˆ + Sa−ˆ ;

ˆ

ˆ

a − = Sa+ˆ − Ca−ˆ ,

a−ˆ = Sa + − Ca − ;

ˆ ˆ

(4.3)

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Interpolation of Quantum Chromodynamics in 1+1 Dimension

and the inner product of two vectors a and b can be written as     ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ a μˆ bμˆ = C a + b+ − a − b− + S a + b− + a − b+ .

(4.4)

The same transformation as shown in Eq. (4.1) applies to momentum variables as well, i.e. ˆ

p + = p 0 cos δ + p 1 sin δ; ˆ

p − = p 0 sin δ − p 1 cos δ.

(4.5)

According to Eq. (4.3), we also have p+ˆ = p 0 cos δ − p 1 sin δ; p−ˆ = p 0 sin δ + p 1 cos δ.

(4.6)

A useful relationship for the energy–momentum of the on-mass-shell particle with mass m and two-momentum vector p μˆ can be found as follows ˆ

( p + )2 = ( p−ˆ )2 + Cm 2 .

(4.7)

With the “hat notation,” the theory of QCD1+1 in the interpolating quantization is then given by the Lagrangian density1 1 ˆ νa μˆ ¯ + ψ(iγ Dμˆ − m)ψ, L = − Fμaˆ νˆ F μˆ 4

(4.9)

Dμˆ = ∂μˆ − ig Aaμˆ ta

(4.10)

Fμaˆ νˆ = ∂μˆ Aaνˆ − ∂νˆ Aaμˆ + g f abc Abμˆ Acνˆ .

(4.11)

where

and

In this work, we start from the interpolating Lagrangian density, Eq. (4.9), derive the corresponding Hamiltonian and solve the mass gap equation which interpolates between the IFD and the LFD. We then apply the solutions of mass gap equation to the calculations of chiral condensates and the quark–antiquark bound-states to find

1 It is worth noting that our definition of

g is the same with that of Ref. [11], but differs with that of Ref. [1] by a factor of √1 , i.e. when δ → π4 , (quantities with a superscript “t” denote the notation 2 used in Ref. [1], and the ones without are ours). √ 1 Atμ = −i 2 Aaμ ta , g t = √ g, and γμt = −iγ μ . 2 .

(4.8)

4.2 The Mass Gap Equation

115

the meson mass spectra and the corresponding wavefunctions. As expected for any physical observables, the meson mass spectra are found to be independent of the interpolation angle parameter. Since we obtain the meson wavefunctions in terms of the interpolation angle parameter δ, we use these δ-dependent wavefunctions to compute the corresponding parton distribution functions (PDFs), comparing them with the PDFs in the LFD and the so-called quasi-PDFs based on the IMF approach in IFD [23,24]. This chapter is organized as follows. In Sect. 4.2, we derive the fermion mass gap equation in QCD1+1 (Nc → ∞) in the quantization interpolating between the IFD and the LFD, using a couple of different methods, namely, the Hamiltonian method and the Feynman diagram method. In Sect. 4.3, we present the solutions of the mass gap equation numerically spanning the interpolation angle between δ = 0 (IFD) and δ = π /4 (LFD). In Sect. 4.4, we apply the mass gap solutions to the calculations of the chiral condensates and the constituent quark mass defined in the full fermion propagator. In Sect. 4.5, we derive the quark–antiquark boundstate equations in the interpolating dynamics and present their solutions in Sect. 4.6, including the meson mass spectra, wavefunctions and (quasi-)PDFs in Sects. 4.6.1, 4.6.2 and 4.6.3, respectively. The summary and conclusions follow in Sect. 4.7. In Appendix B.1, we describe in detail the derivation of the interacting quark–antiquark spinor representation using the Bogoliubov transformation. In Appendix B.2, we show the method of minimizing the vacuum energy with respect to the Bogoliubov angle in getting the mass gap equation. In Appendix B.3, we discuss the interpolating mass gap equation √ and solution in terms of the rescaled variables with respect to the mass dimension 2λ and its treatment associated with the λ = 0 (Free) case vs. the λ = 0 (Interacting) case. In Appendix B.4, we present additional numerical solutions of the mesonic wavefunction for a few different quark masses beyond the ones presented in Sect. 4.6.2. The corresponding quasi-PDFs are discussed in Appendix B.5. In Appendix B.6, we present the quark–antiquark bound-state equations and solutions in the rest frame of the meson.

4.2

The Mass Gap Equation

In this section, we will derive the quark self-energy equation in QCD1+1 (Nc → ∞) in the interpolating dynamics between the IFD and the LFD. While we use two different methods, i.e. the Hamiltonian method in Sect. 4.2.1 and the Feynman diagram method in Sect. 4.2.2, we show that both methods provide exactly the same set of equations. When δ → 0, C → 1 and p−ˆ → p 1 , these equations become the IFD mass gap equations presented in Ref. [11] (i.e. Eqs. (3.18) and (3.19) of Ref [11]). The agreement to Ref. [1] of the δ → π4 limit is discussed in Sect. 4.2.3.

116

4.2.1

4

Interpolation of Quantum Chromodynamics in 1+1 Dimension

The Hamiltonian Method

Before we start, we need to choose a gauge as in the case of any gauge field theory. We adopt here the interpolating axial gauge, i.e. Aa−ˆ = 0, as explored previously in Ref. [10]. In this gauge, the gluon self-couplings are absent. With the gauge condition, Eq. (4.9) reduces to 2 1 ˆ ˆ + ¯ ∂−ˆ Aa+ˆ + ψ(iγ L = D+ˆ + iγ − ∂−ˆ − m)ψ. (4.12) 2 As no interpolation-time derivative of Aa+ˆ , i.e. ∂+ˆ Aa+ˆ , appears in Eq. (4.12), Aa+ˆ is not a dynamical variable but a constrained degree of freedom. We substitute this constrained degree of freedom using the equation of motion for the gluon field Aa+ˆ given by ˆ

ˆ

+a 2 a † 0 + a a ∂− . ˆ A+ ˆ = ψ γ γ gt ψ ≡ ρ = J

(4.13)

The general solution of Eq. (4.13) is given by  1 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ dy − |x − − y − |ρ a (x + , y − ) − x − F a (x + ) + B a (x + ), (4.14) Aa+ˆ (x + , x − ) = 2 where F a and B a are constants. While B a is irrelevant as it can always be eliminated by a gauge transformation, F a is a background electric field which can provide some interesting physical effect such as the axial anomaly in Abelian gauge field theory [25]. For the color-singlet sector in the non-Abelian gauge field theory, however, the background field F a has no effect, e.g. on the spectrum of hadrons in the q q¯ channel. For this reason, we drop the background field F a and take the first term of Eq. (4.14) as the solution of Aa+ˆ (x +ˆ , x −ˆ ) in this work. More details of the discussion on the effect from dropping the background field in the’t Hooft model can be found in Ref. [11]. The energy–momentum tensor in the interpolation form is ˆ ¯ μˆ Dνˆ ψ − g μˆ νˆ L . T μˆ νˆ = −F μˆ λa Fνˆ λˆ a + i ψγ

Thus, the interpolating Hamiltonian is  ˆ ˆ H ≡ P+ˆ = d x − T + +ˆ      1 ˆ ˆ ˆ ˆ = dx− (∂−ˆ Aa+ˆ )2 + ψ † (x − ) −iγ 0 γ − ∂−ˆ + γ 0 m ψ(x − ) . 2

(4.15)

(4.16)

As we have shown in [20], all components of the ψ field are dynamical degrees of freedom for 0 ≤ δ < π /4, while half of the components become constrained for δ = π /4. The field operator conjugate to ψ(x) is

(x) =

∂L = iγ 0 γ +ˆ ψ † (x). ∂ ∂+ˆ ψ(x) 

(4.17)

4.2 The Mass Gap Equation

117

The anticommutation relation at x +ˆ = x +ˆ is





ˆ ˆ ˆ

(x), ψ(x ) x +ˆ =x +ˆ = iγ 0 γ + ψ † (x − ), ψ(x − )

ˆ

ˆ

= iδ(x − − x − ).

(4.18)

Consequently, 

 ˆ ˆ ˆ ˆ −1 ˆ δ(x − − x − ). ψ † (x − ), ψ(x − ) = γ 0 γ +



(4.19)

The Dirac field ψ can be expanded in terms of the quark creation and annihilation operators ˆ −

ψ(x ) =



dp−ˆ ˆ −  [b( p−ˆ )u( p−ˆ ) + d † (− p−ˆ )v(− p−ˆ )] e−i p−ˆ x . 2π 2 p +ˆ

(4.20)

When the form goes to the limit of IFD, i.e. δ → 0, p−ˆ → − p1 = p 1 , and x −ˆ → −x 1 , and Eq. (4.20) becomes the ordinary field operator expansion in IFD2 . The non-trivial vacuum, | >, which is defined by bi | >= 0, d i | >= 0,

(4.21)

is different from the trivial vacuum |0 > defined by bi(0) |0 >= 0, d i(0) |0 >= 0.

(4.22)

The trivial and non-trivial sets of creation and annihilation operators are related by a Bogoliubov transformation 

bi ( p−ˆ ) i† d (− p−ˆ )



 =

cos ζ ( p−ˆ ) − sin ζ ( p−ˆ ) sin ζ ( p−ˆ ) cos ζ ( p−ˆ )

  i(0)  b ( p−ˆ ) · . d i(0)† (− p−ˆ )

(4.23)

The non-trivial set of operators, just like the trivial ones, satisfy the canonical anticommutation relations at x +ˆ = x +ˆ

ˆ ˆ + ij bi ( p−ˆ , x + ), b† j ( p− = 2π δ( p−ˆ − p− (4.24) ˆ , x ) +ˆ ˆ )δ , ˆ + x =x



ˆ ˆ + d i (− p−ˆ , x + ), d † j (− p− , x ) ˆ

x +ˆ =x +ˆ

ij = 2π δ( p−ˆ − p− ˆ )δ ,

(4.25)

and all others are zero.

2 It

differs from the expression in Ref. [11] by a normalization factor √ 1

order to be consistent with standard textbook [26].

2 p +ˆ

, which was inserted in

118

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Interpolation of Quantum Chromodynamics in 1+1 Dimension

The spinors can be defined through a combination of boost and Bogoliubov transformation, which can be represented by θ ( p−ˆ ) = θ f ( p−ˆ ) + 2ζ ( p−ˆ ),

(4.26)

where θ f ( p−ˆ ) is the boost part given by pˆ θ f ( p−ˆ ) = arctan √ − , Cm

(4.27)

and ζ ( p−ˆ ) is the Bogoliubov angle defined in Eq. (4.23). While the details of the derivation are given in Appendix B.1, the results of the spinors are given by ⎛ ⎞ 1−sin θ ( p−ˆ )  δ−sin δ) ⎠ u( p−ˆ ) = 2 p +ˆ ⎝  2(cos 1+sin θ ( p ) ˆ −

(4.28)

2(cos δ+sin δ)

and

⎛  ⎞ 1+sin θ ( p−ˆ )  δ−sin δ) ⎠ v(− p−ˆ ) = 2 p +ˆ ⎝ 2(cos . 1−sin θ ( p−ˆ ) − 2(cos δ+sin δ)

(4.29)

In the case of free particles, Eqs. (4.28) and (4.29) simplify with θ ( p−ˆ ) = θ f ( p−ˆ ) given by Eq. (4.27) as ⎞ ⎛ p+ˆ − p ˆ

u

and

(0)

⎜ cos δ−sin− δ ⎟ ⎟  ( p−ˆ ) = ⎜ ⎠ ⎝ ˆ + p + p−ˆ cos δ+sin δ

⎛ 

p+ˆ + p−ˆ cos δ−sin δ

⎜  v (0) (− p−ˆ ) = ⎜ ⎝ −

p+ˆ − p−ˆ

(4.30)

⎞ ⎟ ⎟. ⎠

(4.31)

cos δ+sin δ

Now, plugging the gluon field solution Eq. (4.14) without the background field into Eq. (4.16), we obtain the interpolating Hamiltonian as 

  ˆ ˆ ˆ ˆ d x − ψ † (x − ) −iγ 0 γ − ∂−ˆ + γ 0 m ψ(x − )   1 ˆ ˆ ˆ ˆ ˆ ˆ d x − dy − ρ a (x − )|x − − y − |ρ a (y − ) − 4 = T + V,

H=

(4.32)

4.2 The Mass Gap Equation

119

where the kinetic energy 

  ˆ ˆ ˆ ˆ d x − ψ † (x − ) −iγ 0 γ − ∂−ˆ + γ 0 m ψ(x − ),

T =

(4.33)

and the potential energy   1 ˆ ˆ ˆ ˆ ˆ ˆ d x − dy − ρ a (x − )|x − − y − |ρ a (y − ) 4   g2 ˆ ˆ ˆ ˆ ˆ ˆ ˆ d x − dy − |x − − y − |ψ † (x − )γ 0 γ + t a ψ(x − ) =− 4

V =−

ˆ

ˆ

ˆ

× ψ † (y − )γ 0 γ + t a ψ(y − ).

(4.34)

Now, if we define the’t Hooft coupling as λ=

g 2 (Nc − 1/Nc ) , 4π

(4.35)

then λ has the dimension of mass squared in 1+1 dimension. By normal-ordering the Hamiltonian, we can write it in three pieces H = L Nc Ev + : H2 : + : H4 : .

(4.36)

Here, the vacuum energy density Ev for the one-dimensional volume L is given by 

d p−ˆ

Ev =

Tr

   ˆ −γ 0 γ − p−ˆ + mγ 0 v(− p−ˆ )v † (− p−ˆ )

(2π )(2 p+ˆ )     dk−ˆ d p−ˆ 1 λ ˆ ˆ 0γ + † (k )γ 0 γ + † (− p ) , Tr γ u(k )u v(− p )v + ˆ ˆ ˆ ˆ − − − − 4π 2 p+ˆ 2k +ˆ ( p−ˆ − k−ˆ )2

(4.37)

the two-body interaction term including the kinetic energy is given by : H2 :=: T : + : V2 : ,

with : T :=



dp−ˆ

(2π )(2 p +ˆ )2

(4.38)

   ˆ Tr −γ 0 γ − p−ˆ + mγ 0 u( p−ˆ )u † ( p−ˆ ) b† ( p−ˆ )b( p−ˆ )   ˆ −γ 0 γ − p−ˆ + mγ 0 v(− p−ˆ )u † ( p−ˆ ) b† ( p−ˆ )d † (− p−ˆ )    ˆ + Tr −γ 0 γ − p−ˆ + mγ 0 u( p−ˆ )v † (− p−ˆ ) d(− p−ˆ )b( p−ˆ )  

 ˆ −Tr −γ 0 γ − p−ˆ + mγ 0 v(− p−ˆ )v † (− p−ˆ ) d † (− p−ˆ )d(− p−ˆ ) ,

+ Tr



(4.39)

120

4

Interpolation of Quantum Chromodynamics in 1+1 Dimension

and : V2 : =

λ 2





d p−ˆ

(2π )(2 p +ˆ )2

dk−ˆ 2  (2k +ˆ ) p−ˆ − k−ˆ

    ˆ ˆ × Tr γ 0 γ + u(k−ˆ )u † (k−ˆ ) − v(−k−ˆ )v † (−k−ˆ ) γ 0 γ + u( p−ˆ )u † ( p−ˆ ) b† ( p−ˆ )b( p−ˆ )     ˆ ˆ + Tr γ 0 γ + u(k−ˆ )u † (k−ˆ ) − v(−k−ˆ )v † (−k−ˆ ) γ 0 γ + v(− p−ˆ )u † ( p−ˆ ) b† ( p−ˆ )d † (− p−ˆ )     ˆ ˆ + Tr γ 0 γ + u(k−ˆ )u † (k−ˆ ) − v(−k−ˆ )v † (−k−ˆ ) γ 0 γ + u( p−ˆ )v † (− p−ˆ ) d(− p−ˆ )b( p−ˆ )   

 ˆ ˆ −Tr γ 0 γ + u(k−ˆ )u † (k−ˆ ) − v(−k−ˆ )v † (−k−ˆ ) γ 0 γ + v(− p−ˆ )v † (− p−ˆ ) d † (− p−ˆ )d(− p−ˆ ) ,

(4.40) and the four-body interaction term is given by : H4 := −

g2 4



ˆ

dx−



ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

dy − |x − − y − | : ψ † (x − )γ 0 γ + t a ψ(x − )ψ † (y − )γ 0 γ + t a ψ(y − ) : .

(4.41) Although the mass gap equation can be obtained either by minimizing Ev with respect to the Bogoliubov angle or by requiring : H2 : to be diagonal in the quark–antiquark creation and annihilation operator basis, both methods provide the same resulting equations. While we present the derivation of minimizing Ev in Appendix B.2, we derive here the mass gap equations by requiring : H2 : to be diagonal. The requirement of : H2 : to be diagonal means that it must take the form 

dp−ˆ

(2π )(2 p +ˆ )

[E u ( p−ˆ )b† ( p−ˆ )b( p−ˆ ) − E v ( p−ˆ )d † (− p−ˆ )d(− p−ˆ )].

(4.42)

The divergent piece that comes out during the normal-ordering process is regulated removing the infinite energy [11], and using the principal value prescription as was done in Ref. [12] 

dy f (y) → (x − y)2



dy (x − y)2

 f (y) − f (x) − (y − x)

  dy d f (x) ≡− f (y). dx (x − y)2

(4.43) Thus, the eigenvalue conditions on the spinors are given by E u ( p−ˆ ) = Tr



 ˆ −γ 0 γ − p−ˆ + mγ 0  + ( p−ˆ )

  0 +ˆ + dk−ˆ λ ˆ  + 0 + − + −  γ γ (k ) −  (k ) γ γ  ( p ) , ˆ ˆ ˆ − − − 2 ( p−ˆ − k−ˆ )2 (4.44a)

4.2 The Mass Gap Equation

E v ( p−ˆ ) = Tr



121



ˆ

−γ 0 γ − p−ˆ + mγ 0  − ( p−ˆ )

  0 +ˆ − dk−ˆ λ ˆ  + 0 + − + −  γ γ (k ) −  (k ) γ γ  ( p ) , ˆ ˆ ˆ − − − 2 ( p−ˆ − k−ˆ )2 (4.44b)

where  ± is defined by  + ( p−ˆ ) ≡

 − ( p−ˆ ) ≡

u( p−ˆ

)u † ( p

2 p +ˆ

ˆ) −

⎛ =⎝

v(− p−ˆ )v † (− p−ˆ ) 2 p +ˆ

cos θ ( p−ˆ ) 1−sin θ ( p−ˆ ) √ 2(cos δ−sin δ) 2 C 1+sin θ ( p−ˆ ) cos θ ( p−ˆ ) √ 2(cos δ+sin δ) 2 C

⎛ =⎝

⎞ ⎠,

1+sin θ ( p−ˆ ) cos θ ( p−ˆ ) √ 2(cos δ−sin δ) − 2 C cos θ ( p−ˆ ) 1−sin θ ( p−ˆ ) − √ 2(cos δ+sin δ) 2 C

(4.45)

⎞ ⎠.

(4.46)

By using Eqs. (4.28) and (4.29), one may see that the matrices on the right-hand side of Eqs. (4.45) and (4.46) can be obtained by direct computation. Now, let us define  C E( p−ˆ ) ≡ (4.47) E u ( p−ˆ ) − E v ( p−ˆ ) . 2 Then, by subtracting Eqs. (4.44a) and (4.44b) as well as plugging in Eqs. (4.45) and (4.46), we arrive at √ E( p−ˆ ) = p−ˆ sin θ ( p−ˆ ) + Cm cos θ ( p−ˆ )   dk−ˆ Cλ − + cos θ ( p−ˆ ) − θ (k−ˆ ) . 2 2 ( p−ˆ − k−ˆ )

(4.48)

On the other hand, by adding them, we get E u ( p−ˆ ) + E v ( p−ˆ ) = −

2S pˆ. C −

(4.49)

Also, we know that the off-diagonal elements of : H2 : have to vanish 0 = Tr

 

ˆ

−γ 0 γ − p−ˆ + mγ 0

 v(− p )u † ( p ) ˆ ˆ − − 2 p +ˆ

  0 +ˆ v(− p−ˆ )u † ( p−ˆ ) dk−ˆ λ ˆ  + 0 + − + − γ γ  (k−ˆ ) −  (k−ˆ ) γ γ , 2 ( p−ˆ − k−ˆ )2 2 p +ˆ (4.50a)

122

4

0 = Tr

 

Interpolation of Quantum Chromodynamics in 1+1 Dimension

ˆ

−γ 0 γ − p−ˆ + mγ 0

 u( p )v † (− p ) ˆ ˆ − − 2 p +ˆ

  0 +ˆ u( p−ˆ )v † (− p−ˆ ) dk−ˆ λ ˆ  + 0 + − + − γ γ  (k−ˆ ) −  (k−ˆ ) γ γ . 2 ( p−ˆ − k−ˆ )2 2 p +ˆ (4.50b) From either Eqs. (4.50a) or (4.50b), we get   p−ˆ dk−ˆ m λ sin θ ( p−ˆ ) − θ (k−ˆ ) . cos θ ( p−ˆ ) − √ sin θ ( p−ˆ ) = − 2 C 2 ( p−ˆ − k−ˆ ) C (4.51) Equations. (4.48) and (4.51) are the mass gap equations in the interpolating dynamics. The same set of mass gap equations can be derived using the Feynman diagram method as we present in the following Sect. 4.2.2.

4.2.2

The Feynman Diagram Method

The self-energy equation in the large Nc approximation is drawn pictorially in Fig. 4.1. Following the Feynman rules for the gluon propagator, the free quark prop1 and gγ +ˆ t a , respectively, with the momentum agator and the vertex as k12 , k −m+i ˆ −

assignment shown in Fig. 4.1,we have ( p−ˆ ) = i

 dk ˆ dk ˆ 1 λ ˆ ˆ −  − + 2 γ + γ +. 2π  k − m − (k ) + i ˆ p−ˆ − k−ˆ −

(4.52)

Writing the self-energy as ( p−ˆ ) = =

Fig. 4.1 Self-energy equation

√ √

CA( p−ˆ ) + γ−ˆ B( p−ˆ ) ˆ

ˆ

CA( p−ˆ ) + (Sγ + − Cγ − )B( p−ˆ ),

(4.53)

4.2 The Mass Gap Equation

123

we express the dressed quark propagator as −1  S(k) =  k − m − (k−ˆ ) + i    −1 √  ˆ ˆ  = γ + k+ˆ − SB(k−ˆ ) + γ − k−ˆ + CB(k−ˆ ) − m + CA(k−ˆ ) + i . (4.54) This dressed quark propagator can be obtained from the bare quark propagator with the replacement given by ⎧ ⎨ k+ˆ → k+ˆ − SB(k−ˆ ) k ˆ → k−ˆ +√CB(k−ˆ ) . ⎩ − m → m + C A(k−ˆ )

(4.55)

Then, Eq. (4.52) becomes ( p−ˆ ) = i ×

 dk ˆ dk ˆ λ ˆ −  − + 2 γ + 2π p−ˆ − k−ˆ γ +ˆ

1 ˆ  γ+ √    ˆ − k+ˆ − SB(k−ˆ ) + γ k−ˆ + CB(k−ˆ ) − m + CA(k−ˆ ) + i

 dk ˆ dk ˆ λ −  − + 2 2π p−ˆ − k−ˆ    √   Cγ +ˆ k+ˆ − SB(k−ˆ ) + 2Sγ +ˆ − Cγ −ˆ k−ˆ + CB(k−ˆ ) + C m + CA(k−ˆ ) ×  , 2 √ 2    2  C k+ˆ − SB(k−ˆ ) + 2S k+ˆ − SB(k−ˆ ) k−ˆ +CB(k−ˆ ) − C k−ˆ + CB(k−ˆ ) − m+ CA(k−ˆ ) + i =i

(4.56)  2 where we have used the algebra for the interpolating γ matrices γ +ˆ = C · I2×2 ,  2

γ −ˆ = −C · I2×2 , and γ +ˆ , γ −ˆ = 2S · I2×2 . Now, the two poles of k+ˆ in the denominator of Eq. (4.56) are given by S − k−ˆ ± C

!



2  2 k−ˆ m + B(k−ˆ ) + √ + A(k−ˆ ) ∓ i . C C

(4.57)

After doing the k+ˆ pole integration using Cauchy’s theorem, we get  √ √   C Cm + CA(k ) + γ−ˆ k−ˆ + CB(k−ˆ ) ˆ dk−ˆ − λ ( p−ˆ ) = −  2  2 2  2 √ p−ˆ − k−ˆ Cm + CA(k−ˆ ) k−ˆ + CB(k−ˆ ) +  √  dk−ˆ λ = − C cos θ (k ) + γ sin θ (k ) , (4.58) 2 ˆ ˆ ˆ − − − 2 p−ˆ − k−ˆ

124

4

Interpolation of Quantum Chromodynamics in 1+1 Dimension

Fig. 4.2 Geometrical representation of mass gap equations representing Eqs. (4.62)–(4.64)

where θ (k−ˆ ) is defined by  θ (k−ˆ ) = tan

−1

 k−ˆ + CB(k−ˆ ) . √ Cm + CA(k−ˆ )

(4.59)

By comparing Eq. (4.58) with Eq. (4.53), we can identify A( p−ˆ ) =

 dk−ˆ λ − 2 cos θ (k−ˆ ) 2 p−ˆ − k−ˆ

(4.60)

B( p−ˆ ) =

 dk−ˆ λ − 2 sin θ (k−ˆ ). 2 p−ˆ − k−ˆ

(4.61)

and

From Eq. (4.59), we may geometrically represent the effective mass and longitudinal momentum of the particle moving in non-trivial vacuum by drawing the triangle picture shown in Fig. 4.2. From Fig. 4.2, we identify the energy E( p−ˆ ) as 2  2 √ Cm + CA( p−ˆ ) , E( p−ˆ )2 = p−ˆ + CB( p−ˆ ) +

(4.62)

and find the mass gap equations  dk−ˆ λ − 2 cos θ (k−ˆ ), 2 p−ˆ − k−ˆ  dk−ˆ λ E( p−ˆ ) sin θ ( p−ˆ ) = p−ˆ + C · −  2 sin θ (k−ˆ ), 2 pˆ −kˆ E( p−ˆ ) cos θ ( p−ˆ ) =

√

Cm + C ·

−

(4.63) (4.64)

−

where we used Eqs. (4.60) and (4.61) for A( p−ˆ ) and B( p−ˆ ). Now, multiplying Eq. (4.63) by sin θ ( p−ˆ ) and Eq. (4.64) by cos θ ( p−ˆ ) and subtracting them, we get

4.2 The Mass Gap Equation

125

the exact same equation in Sect. 4.2.1 as given by Eq. (4.51). Also, by multiplying Eq. (4.63) by cos θ ( p−ˆ ) and Eq. (4.64) by sin θ ( p−ˆ ) and adding them, we get the same equation as Eq. (4.48) in Sect. 4.2.1. In Sect. 4.3, we numerically solve Eq. (4.51) to obtain θ ( p−ˆ ) and plug it into Eq. (4.48) to find E( p−ˆ ). We note that the solution of E( p−ˆ ) is not always positive and thus the geometric interpretation of Fig. 4.2 should be regarded as an pictorial device to represent Eqs. (4.62)-(4.64) without assuming that the lengths of the triangle sides are positive. We also recognize Eq. (4.57) as E u and E v mentioned in Sect. 4.2.1, i.e. E( p−ˆ ) E( p−ˆ ) S S E u ( p−ˆ ) = − p−ˆ + , E v ( p−ˆ ) = − p−ˆ − , (4.65) C C C C where we note E u (− p−ˆ ) = −E v ( p−ˆ ) and E v (− p−ˆ ) = −E u ( p−ˆ ) due to the evenness of E( p−ˆ ) under p−ˆ ↔ − p−ˆ , i.e. E( p−ˆ ) = E(− p−ˆ ). When E( p−ˆ ) is positive, E u is the first energy pole corresponding to the plus sign in Eq. (4.57), and E v to the minus sign as written in Eq. (4.65). When E( p−ˆ ) is negative, however, E u is the pole with minus sign, and E v the plus sign. Moreover, one can naturally obtain Eqs. (4.47) and (4.49) by adding and subtracting E u and E v in Eq. (4.65).

4.2.3

Behavior of the Gap Equation When Approaching the Light-Front

We note that the interpolating mass gap equations are greatly simplified in the limit to the LFD, i.e. C → 0. In this limit, Eq. (4.51) becomes p + cos θ ( p + ) = 0,

(4.66)

where one should note p−ˆ = p− = p + as C → 0. The solution of Eq. (4.66) is analytically given by π θ ( p + ) = sgn( p + ). (4.67) 2 Likewise, Eq. (4.48) is simplified as E( p + ) = p + sin θ ( p + ).

(4.68)

Moreover, Eqs. (4.60) and (4.61) are now given by A( p + ) = 0 (except p + = 0),  λ dk + sgn(k + ) , B( p + ) = − + 2 ( p − k + )2

(4.69) (4.70)

126

4

Interpolation of Quantum Chromodynamics in 1+1 Dimension

where the light-front zero-mode p + = 0 contribution should be taken into account separately with the form of A( p + ) = A(0)δ( p + ) solution in mind. Besides the p + = 0 contribution, it is interesting to note a remarkable simplification of the self-energy in the LFD given by ( p + ) = γ + B( p + ) due to the absence of the scalar part, i.e. A( p + ) = 0. For the computation of B( p + ) in Eq. (4.70),’t Hooft [1] didn’t use the principal value but discussed how to make the infrared region finite by introducing the infrared cutoff parameter ε as summaried below for p + > 0 and p + < 0, respectively, i.e. in the limit ε → 0, for p + > 0, "

dk + − + + + 2 −∞ ( p − k )   +∞ dk + + + + 2 p+ +ε ( p − k )   1 1 , − = −λ + p ε

λ B( p ) = 2 +

0



p+ −ε 0

dk + ( p + − k + )2

(4.71)

while for p + < 0, "

p+ −ε

dk + + + ( p − k + )2 −∞   +∞ dk + + ( p + − k + )2 0   1 1 . = −λ + p+ ε

λ B( p ) = 2 +



−

0 p+ +ε

−

dk + − k + )2

( p+

(4.72)

Thus, the’t Hooft’s solution for B( p + ) is given by B( p + ) = λ



sgn( p + ) 1 − + ε p

 .

(4.73)

With these solutions, the replacement given by Eq. (4.55) in the LFD limit (δ → becomes ⎧ ⎨ p+ˆ → p+ˆ − SB p ˆ → p−ˆ + √CB ⎩ − m → m + CA

⎧ ⎪ ⎨ p− → p− + C→0 −→ p + → p + ⎪ ⎩m → m ,

λ p+

− λ sgn(εp

π 4)

+)

(4.74)

4.2 The Mass Gap Equation

127

and it leads to the dressed fermion propagator given by

S( p) =

 γ + p− +

λ p+

− λ sgn(εp

+)



+ γ − p+ + m +

2 p + p − − m 2 + 2λ − 2λ | pε | + i   + − + λ − λ sgn( p ) + γ + p − + γ − p + + m γ + p − − pon on ε p+   = + sgn( p − 2 p + p − − pon + pλ+ − λ ε ) + i =

γ+  p on + m + , + + 2p p 2 − m 2 + 2λ − 2λ | pε | + i

(4.75)

2

− = m and p + = p + . Here, we note the splitting of the instantaneous conwhere pon on 2 p+ tribution (∼ γ + ) and the so-called on-mass-shell part ∼ ( p on + m) of the fermion propagator in LFD. While the interpolating fermion propagator can split into the forward moving part with the energy denominator 1/( p+ˆ − E u ( p−ˆ )) and the backward moving part with the energy denominator 1/( p+ˆ − E v ( p−ˆ )) as we discuss the details of the fermion propagator in Sect. 4.4.2, one can notice the behaviors of E u ( p−ˆ ) m2 and E v ( p−ˆ ) → 2 p+ + 2 +( 2 Cp + B( p + ) + 2mp+ )

and E v ( p−ˆ ) in the limit C → 0 as E u ( p−ˆ ) → B( p + ) + +

−( 2 Cp + B( p + ) +

m2 ) 2 p+

for p + > 0 while E u (− p−ˆ ) → 2

and E v (− p−ˆ ) → −(B( p + ) + 2mp+ ) for p + < 0 using Eqs. (4.62) and (4.65). Taking p + > 0 for the sake of discussion, one can rather easily identify that the on-mass-shell part ∼ ( /p on + m) of the dressed fermion propagator corresponds to the forward moving part from the’t Hooft solution for B( p + ) given by Eq. (4.73) as well as the corre¯ p + ) of the dressed fermion. spondence /p on + m with the spinor biproduct u( p + )u( Likewise, one can identify the instantaneous contribution (∼ γ + ) to the backward + moving part by noting the cancellation of 1/C factor in E v ( p−ˆ ) ∼ − 2 Cp with the + +

¯ p + ) ∼ 2 p Cγ . 1/C factor in the backward moving spinor biproduct v(− p + )v(− However, the instantaneous contribution effectively vanishes in the actual calculation of the rainbow and ladder diagrams as every quark line is multiplied by the   2

vertex factor gγ + from both sides and γ + = 0. Moreover, as γ + , γ − = 2, the dressed quark propagator can be effectively given by S( p) =

p+ +

2 p + p − − m 2 + 2λ − 2λ | pε | + i

,

(4.76)

with the effective vertex factor 2g. We can now see that, due to the 1/ε infrared divergence, the on-mass-shell pole of this dressed quark propagator moves towards − = m 2 . This disappearance of the onthe infinity from the on-mass-shell pole pon 2 p+ mass-shell pole due to the infrared cutoff term was interpreted as the confinement of the fermions in the’t Hooft model [1].

128

4

Interpolation of Quantum Chromodynamics in 1+1 Dimension

As shown in Ref. [1], however, the infrared cutoff terms cancel themselves in the bound-state spectroscopy calculation. Thus, one can use the principal value prescription as defined in Eq. (4.43) to regulate the infinite piece. With the same principal value prescription, B( p + ) in the limit of ε → 0 is given by 

 dk + sgn(k + ) − sgn( p + ) + 2 −k )   + λ sgn( p ) 1 =λ − + − sgn( p + ) ε p 2 " + $  +∞ p −ε + dk dk + + × + + 2 ( p + − k + )2 −∞ p+ +ε ( p − k )

B( p + ) =

λ 2

=−

( p+

λ . p+

(4.77)

The reduced quark propagator without the 1/ε infrared divergence factor in the denominator is then given by S( p) =

2 p+ p−

p+ , − m 2 + 2λ + i

(4.78)

which leads to the same effective planar Feynman rule of the fermion propagator in the light-front gauge presented in Ref. [1] besides the notation difference explained in the footnote in Sect. 4.1.

4.3

The Mass Gap Solution

The mass gap equation given by Eq. (4.51) is solved numerically using a generalized Newton method mentioned in Ref. [12] as well as in Ref. [13], and we use the same numerical method elaborated in Refs. [12] and [13]. Using √ the same 200 grid points and choosing the same quark mass values (in unit of 2λ) as in Ref. [12] and in Ref. [13], we obtain the numerical solutions of Eq. (4.51). To cover the entire interpolating longitudinal momentum range, −∞ < p−ˆ < +∞, we use the variable ξ = tan−1 p−ˆ , where ξ ∈ (− π2 , π2 ). As the solutions of θ ( p−ˆ ) are antisymmetric under the transformation of p−ˆ → − p−ˆ , i.e. θ (− p−ˆ ) = −θ ( p−ˆ ), we present the results for the region 0 < p−ˆ < +∞ only, i.e. 0 < ξ < π2 , for several quark mass values in Figs. 4.3 and 4.4. In Fig. 4.3, the numerical solutions of θ ( p−ˆ ) for several interpolation angles are plotted for the quark mass values presented in Ref. [12]. The profiles of θ (ξ ) for δ = 0 coincide with the IFD results provided in Ref. [12]’s Figure 4.4. Also, the profiles of θ (ξ ) for δ → π /4 approach to the analytic solution given by Eq. (4.67) in LFD. It is interesting to note that the IFD results exhibit a convex profile for the lighter quark mass (m = 0.18), and as the quark mass increases (m = 1.00 and 2.11) the profile gets more and more concave.

4.3 The Mass Gap Solution

129

Fig. 4.3 Numerical solutions of θ( p−ˆ ) for several interpolation angles, corresponding to the quark √ mass values in Fig. 4.4 of Ref. [12]. All the mass values are in the unit of 2λ. Note that θ( p−ˆ ) is an odd function of p−ˆ and only the positive p−ˆ range is plotted with the variable ξ = tan−1 p−ˆ

130 Fig. 4.4 Numerical solutions of θ( p−ˆ ) for several interpolation angles, corresponding to the quark mass values in Fig. 4.2 of Ref. [13]. All the √ mass values are in the unit of 2λ. Note that θ( p−ˆ ) is an odd function of p−ˆ and only the positive p−ˆ range is plotted with the variable ξ = tan−1 p−ˆ

4

Interpolation of Quantum Chromodynamics in 1+1 Dimension

4.3 The Mass Gap Solution

131

For the quark mass values presented in Ref. [13], the numerical solutions of θ ( p−ˆ ) for several interpolation angles are plotted in Fig. 4.4. The authors of Ref. [13] chose the’t Hooft coupling λ as π λ = 0.18 GeV2 in conformity to the value of the string tension in the realistic QCD4 and then determined the light quark√mass m u/d = 0.045 to obtain the physical pion mass Mπ = 0.41 in the unit of 2λ by solving the bound-state equation, which we present in Sect. 4.5. Likewise, the heavy quark mass m c = 4.23 was determined to get the physical J /ψ mass M J /ψ = 9.03 in the same unit. The strange quark mass m s = 0.749 was taken to provide a threshold, below (above) which is called light (heavy) flavor, by minimizing the relative distance between the θ (ξ ) solution and the straight line θ (ξ ) = ξ in IFD as the profile of θ (ξ ) in IFD (i.e. δ = 0) passes from the convex to concave with the increasing quark mass as discussed in Fig. 4.3. The chiral limit with m u = 0 was also considered and the Gell-Mann–Oakes–Renner relation (GOR) was discussed in Ref. [13]. For those quark masses, m = 0, 0.045, 0.749 and 4.23, we obtain the interpolating mass gap solutions for several δ values between δ = 0 (IFD) and δ = π /4 (LFD) as plotted in Fig. 4.4. Our numerical solutions for δ = 0 coincide with the IFD results provided in Ref. [13]’s Fig. 4.2 (left panel). Again, the profiles of θ (ξ ) for δ → π /4 approach to the analytic solution given by Eq. (4.67) in LFD. We then obtain the solutions for E( p−ˆ ) by plugging the mass gap solutions presented in Figs. 4.3 and 4.4 into Eq. (4.48). As the plots of E( p−ˆ ) themselves for different interpolation angles √ are too close to each other to observe clearly, we present the solutions of E( p−ˆ )/ C in Figs. 4.5 and 4.6. The quark mass values in Figs. 4.5 and 4.6 correspond to those in Figs. 4.3 and 4.4, respectively. At δ = 0, the profiles √ of E( p−ˆ )/ C = E( p 1 ) in Figs. 4.5 and 4.6 coincide with the IFD results provided in Ref. [12]’s Fig. 4.6 and Ref. [13]’s Fig. 4.2 (right panel), respectively. Examining the results of E( p−ˆ ) themselves in the limit δ → π /4, we also find that all of them approach to the analytic result of E( p + ) given by Eq. (4.68) independent of quark mass m. Moreover, the LFD result of E( p + ) given by Eq. (4.68) is always positive regardless of p + while the IFD (δ = 0) results of E( p 1 ) for small quark mass values, e.g. m = 0.18 in Fig. 4.5 and m = 0 and 0.045 in Figs. 4.6a and 4.7a, respectively, are negative for small momentum regions. Indeed, we notice that the region of small momentum for the negative value of E( p 1 ) gets shrunken as m gets larger but persists up to m ≈ 0.56. It was argued in Ref. [11] that the existence of negative quark selfenergy does not cause any concern though, due to the lack of observability for the energy of a confined single quark. While similar aspect of the negative quark energy exists for other interpolation angle values unless δ = π /4, the corresponding range of the small momentum gets reduced as δ approaches to π /4 as depicted in Figs. 4.5a, 4.6a and 4.6b. It is interesting to note that our observation of diminishing the negative quark self-energy region in the limit δ → π /4 (LFD) appears consistent with the result that the single particle energies do not change sign due to the additional constant term appearing in converting the light-front longitudinal momentum sum to a principal value integral discussed in Ref. [27] with the formulation of a finite light-front x − coordinate interval.

132

4

Interpolation of Quantum Chromodynamics in 1+1 Dimension

√ Fig. 4.5 Solutions of E( p−ˆ )/ C for several interpolation angles for different choices of quark mass corresponding to the quark mass values in Fig. 4.6 of Ref. [12]. All quantities are in proper √ units of 2λ. E( p−ˆ ) is an even function of p−ˆ . We plot only for positive p−ˆ with the variable ξ = tan−1 p−ˆ

4.3 The Mass Gap Solution Fig. 4.6 √Solutions of E( p−ˆ )/ C for several interpolation angles for different choices of quark mass, corresponding to the quark mass values in Fig. 4.2 of Ref. [13]. All quantities √ are in proper units of 2λ. E( p−ˆ ) is an even function of p−ˆ . We plot only for positive p−ˆ with the variable ξ = tan−1 p−ˆ

133

134

4

Interpolation of Quantum Chromodynamics in 1+1 Dimension

Fig. 4.7 Numerical results of the condensation ¯ > |ren as a function of < ψψ m in comparison with the analytic result in Ref. [29]. All quantities are in proper √ units of 2λ

4.4

Chiral Condensate and Constituent Quark Mass

As we have obtained the mass gap solutions, we now utilize them to discuss the chiral condensate and the constituent quark mass in this section.

4.4.1

The Chiral Condensate

While Coleman’s theorem [28] prohibits the spontaneous breaking of the chiral symmetry (SBCS) in two-dimensional theories with a finite number of degrees of freedom, the large Nc limit of the’t Hooft model does not contradict with the Coleman’s theorem. The exact result for the chiral condensate was found√in the chiral limit (m → 0) for the weak coupling regime of QCD2 (m >> g ∼ 1/ Nc ) [4,5] as √ ¯ >= −Nc / 12 < ψψ

(4.79)

√ in the unit of the mass dimension 2λ with the definition of λ given by Eq. (4.35). This indicates that the SBCS occurs in the’t Hooft model, in contrast to the strong coupling regime of QCD2 in which the SBCS does not occur according to the Coleman’s theorem [28]. Ramifications of the nontrivial chiral condensates with respect to the vacuum in the LFD as well as its non-analytic behavior were discussed for nonzero quark masses and its chiral limit [29]. For nonzero quark masses, the renormalized quark condensation was defined by subtracting the free field expectation value to render the quark condensation finite [29], ¯ > − < ψψ ¯ > |g=0 . ¯ > |ren ≡< ψψ < ψψ

(4.80)

¯ > |g=0 = 0 and thus < ψψ ¯ > |ren =< ψψ ¯ >. In the chiral limit, m → 0, < ψψ The numerical computation verifying Eq. (4.79) was presented in IFD [30] substituting the mass gap solution for θ ( p 1 ) in the chiral limit m → 0 as

4.4 Chiral Condensate and Constituent Quark Mass

¯ > |ren = −Nc < ψψ

+∞ −∞

dp 1 cos θ ( p 1 ) ≈ −0.29Nc . 2π

135

(4.81)

With the interpolating quark field between IFD and LFD given by Eq. (4.20), we find +∞

¯ > |ren = Nc < ψψ

−∞



dp−ˆ

(2π )(2 p +ˆ )



 × Tr v(− p−ˆ )v(− ¯ p−ˆ ) − Tr{v (0) (− p−ˆ )v¯ (0) (− p−ˆ )} Nc =− √ 2π C

+∞   dp−ˆ cos θ ( p−ˆ ) − cos θ f ( p−ˆ ) ,

(4.82)

−∞

where we have used the interpolating spinors given by Eqs. (4.29) and (4.31). We numerically compute Eq. (4.82) in the chiral limit, m → 0, for different interpolation angles, and the results are shown in Table 4.1. We observe that the closer one gets to the LFD (δ = π4 ), the higher accuracy one needs for the numerical computation and thus we list the results obtained by increasing the number of grid points used. One can see that when δ is away from π /4, the coarser grid is already good enough to obtain an accurate result. For δ closer to π /4, the number of grid points should be increased to improve the numerical √ accuary. Table 4.1 indicates an eventual agreement to the analytical value −1/ 12 ≈ −0.29 regardless of the interpolation angle δ with the enhancement of the numerical accuracy. Even if the chiral limit is not taken, we notice that the renormalized chiral condensate must be independent of the interpolating angle δ as it must be the characteristic quantity determining the vacuum property for a given phase of the theory. In fact, the interpolating longitudinal momentum variable p−ˆ can be scaled out by √ the interpolating parameter C and Eq. (4.82) can be given by the rescaled variable √ = p / C, i.e. p− ˆ − ˆ ¯ > |ren = − < ψψ

Nc 2π



+∞

−∞

  cos θ ( p dp− ) − cos θ ( p ) . f ˆ ˆ ˆ − −

(4.83)

The same variable change can be applied to the interpolating mass gap equation given by Eq. (4.51) as we illustrate in Appendix B.3 to obtain the rescaled mass gap equation without any apparent interpolation angle dependence as given by Eq. (B.37). ) being the solution of Eq. (B.37) as well as θ ( p ) = tan−1 ( p /m) With θ ( p− f ˆ ˆ ˆ − − from Eq. (4.27), we can confirm that the chiral condensate is indeed independent of the interpolation angle δ. Thus, the result of the chiral condensate must be identical whichever dynamics is chosen for the computation between δ = 0 (IFD) and ¯ > |ren in Eq. (4.82) for any given interpolation δ = π /4 (LFD). Computing < ψψ

136

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Interpolation of Quantum Chromodynamics in 1+1 Dimension

Table 4.1 Numerically calculated condensation values in the chiral limit with different interpolation angles and computational accuracy δ 0

Number of grid points

200 600 0.4 200 600 0.6 200 600 0.7 200 600 0.75 200 600 0.78 200 600 0.785 600 1000 3000 0.78535 1000 3000 5000 √ All quantities are in proper units of 2λ

¯ > |m→0 /Nc < ψψ –0.285209 –0.287508 –0.285164 –0.287496 –0.284792 –0.287375 –0.283836 –0.287059 –0.281837 –0.286396 –0.296575 –0.291334 –0.298104 –0.294377 –0.290590 –0.304659 –0.294134 –0.291964

angle between δ = 0 and δ = π /4, one can numerically verify the uniqueness of the result for each and every quark mass values that √ one takes. Varying the quark ¯ > |ren masses between m = 0 and m = 4 in the unit of 2λ, we computed < ψψ from Eq. (4.83) and obtained the numerical result shown in Fig. 4.7. Our numerical result for m = 0 in Fig. √ 4.7 confirms the results obtained in Table 4.1 reproducing the analytic value −1/ 12 ≈ −0.29. Likewise, our numerical results for 0 < m ≤ 4 coincide with the analytic result given by Eq. (2.19) of Ref. [29] which was also numerically confirmed in Ref. [13]. In the formulation of a finite light-front x − coordinate interval [27], a phase transition was reported in the weak coupling limit as a dramatic change of the quark condensate value to zero was observed. In the continuum limit, however, there is no such phase transition and the nonzero condensate value is intact regardless of the form of dynamics between δ = 0 (IFD) and δ = π /4 (LFD) as discussed in this subsection. In this respect, it is important to realize that the phase characterized by the SBCS is uniquely viable in the continuum’t Hooft model. While there is a simple analytic step function solution of Eq. (4.51) in IFD (δ = 0) given by θ ( p 1 ) = π2 sgn( p 1 ) in the chiral limit [11], it should be clearly distinguished from the step function solution in LFD given by Eq. (4.67). The step function solution in IFD leads to the zero chiral ¯ >= 0 in contrast to the non-trivial chiral condensate given by condensate < ψψ Eq. (4.79). It was indeed demonstrated in Ref. [31] that this chirally symmetric step function solution in IFD reveals the possession of infinite vacuum energy compared

4.4 Chiral Condensate and Constituent Quark Mass

137

to the vacuum energy of the SBCS solution. Thus, the step function solution in IFD should be clearly distinguished from the physically viable solution characterized by the SBCS [32] discussed in this subsection.

4.4.2

The Fermion Propagator and Constituent Mass

The interpolating free fermion propagator has been discussed at length in Chap. 3 and can be readily applied for the interpolating bare quark propagator in 1+1 dimension with the notation of the on-mass-shell two-momenta Pa μˆ = (Pa +ˆ , p−ˆ ), Pbμˆ = (−Pb+ˆ , − p−ˆ ) taking Pa +ˆ and −Pb+ˆ as the positive and negative on-shell interpolating energies of the bare quark, i.e.  Pa +ˆ

S = − p−ˆ + C

2 + Cm 2 p− ˆ

C

=

−S p−ˆ + p +ˆ , C

(4.84)

and

 2 + Cm 2 p− −S p−ˆ − p +ˆ ˆ S = . − Pb+ˆ = − p−ˆ − C C C The interpolating bare quark propagator is then given by S( p)f =



1 2 p +ˆ

/a + m /b − m P P + p+ˆ − Pa +ˆ + i p+ˆ + Pb+ˆ − i

(4.85)

 ,

(4.86)

  / a + m = u (0) ( p−ˆ )u¯ (0) ( p−ˆ )and P / b − m = v (0) (− p−ˆ )v¯ (0) (− p−ˆ )can be where P easily verified using the interpolating free spinors given by Eqs. (4.30) and (4.31). In Eq. (4.86), we call m as the bare quark mass. As discussed in Sect. 4.2.2, the interpolating dressed quark propagator can be obtained from the interpolating bare quark propagator with the replacement given by Eq. (4.55). Effectively, Pa +ˆ and Pb+ˆ are replaced by E u and E v in Eq. (4.65) as well as the free spinors, u (0) ( p−ˆ ) and v (0) ( p−ˆ ), are replaced by the spinors given by Eqs. (4.28) and (4.29), u( p−ˆ ) and v( p−ˆ ), respectively. The dressed quark propagator is then obtained as S( p) =

1



2 p +ˆ

 u( p−ˆ )u( ¯ p−ˆ ) ¯ p−ˆ ) v(− p−ˆ )v(− , + p+ˆ − E u ( p−ˆ ) + i p+ˆ − E v ( p−ˆ ) − i

(4.87)

where the first and second terms correspond to the forward and backward moving propagators, respectively. The equivalence of Eq. (4.87) to Eq. (4.54) can be verified using the relations ¯ p−ˆ ) u( p−ˆ )u( 2 p +ˆ

⎛ = +γ 0 = ⎝

1−sin θ ( p−ˆ ) cos θ ( p−ˆ ) √ 2(cos δ−sin δ) 2 C cos θ ( p−ˆ ) 1+sin θ ( p−ˆ ) √ 2(cos δ+sin δ) 2 C

⎞ ⎠,

138

4

Interpolation of Quantum Chromodynamics in 1+1 Dimension

v(− p−ˆ )v(− ¯ p−ˆ ) 2 p +ˆ

⎛ = −γ 0 = ⎝

cos θ ( p−ˆ ) 1+sin θ ( p−ˆ ) √ 2(cos δ−sin δ) 2 C cos θ ( p−ˆ ) 1−sin θ ( p−ˆ ) √ 2(cos δ+sin δ) − 2 C

−

⎞ ⎠,

where  + and  − are given by Eqs. (4.45) and (4.46), respectively. As discussed in Sect. 4.2.3, one can verify that the forward and backward moving parts in the limit C → 0 correspond to the on-mass-shell part ∼ ( /p on + m)and the instantaneous contribution (∼ γ + ) in LFD, respectively. To discuss the dressed fermion propagator in more physical terms [33], one can express the dressed quark propagator given by Eq. (4.54) in terms of the mass function M( p−ˆ ) and the wavefunction renormalization factor F( p−ˆ ), i.e. S( p) =

F( p−ˆ ) , p − / M( p−ˆ )

(4.88)

and identify M( p−ˆ ) and F( p−ˆ ), respectively, as

and

√ m + C A( p−ˆ ) pˆ M( p−ˆ ) = p−ˆ = √− cot θ ( p−ˆ ) p−ˆ + CB( p−ˆ ) C

(4.89)

  CB( p−ˆ ) −1 p−ˆ = F( p−ˆ ) = 1 + . p−ˆ E( p−ˆ ) sin θ ( p−ˆ )

(4.90)

We then numerically compute M( p−ˆ ) and F( p−ˆ ) using the mass gap solutions θ ( p−ˆ ) and E( p−ˆ ) obtained in Sect. 4.3. In Figs. 4.8 and 4.9, the results of the mass function M( p−ˆ ) are shown as a function of the variable ξ = tan−1 p−ˆ for the bare quark mass values in Ref. [12] (m = 0.18, 1.00, 2.11) and Ref. [13] (m = 0, 0.045, 0.749, 4.23), respectively. As we can see in Figs. 4.8 and 4.9, the mass function M( p−ˆ ) approaches the respective bare quark mass value m as ξ → π/2 or p−ˆ → ∞ while it gets to the respective characteristic mass value M(0) at ξ = 0 or p−ˆ = 0 regardless of the interpolation angle δ, although the profile of M( p−ˆ ) in 0 < ξ < π/2 does depend on the value of δ. The characteristic mass value M(0) may be regarded as the dressed quark mass acquired from the dynamical self-energy interaction depicted in Fig. 4.1. We note that the profile of M( p−ˆ ) as δ → π /4 approaches to the shape of the step-function which drops from M(0) to m away from p + = 0. In Figs. 4.10 and 4.11, the results of the wavefunction renormalization factor F( p−ˆ ) are plotted with the same variable ξ for the bare quark mass values in Ref. [12] (m = 0.18, 1.00, 2.11) and Ref. [13] (m = 0, 0.045, 0.749, 4.23), respectively. Rather immediately, we notice a dramatic difference in the results of F( p−ˆ ) for the lower values of bare quark mass m = 0.18 in Fig. 4.10 as well as m = 0 and 0.045 in Fig. 4.11 due to the negative values for small ξ = tan−1 p−ˆ region. Interestingly, the appearance of the negative values in F( p−ˆ ) for those bare quark mass

4.4 Chiral Condensate and Constituent Quark Mass

139

Fig. 4.8 Constituent mass as a function of ξ = tan−1 p−ˆ for (a) m = 0.18, (b) m = 1.00 and (c) √ m = 2.11. All quantities are in proper units of 2λ

140 Fig. 4.9 Constituent mass as a function of ξ = tan−1 p−ˆ for (a) m = 0, (b) m = 0.045, (c) m = 0.749 and (d) m = 4.23. All quantities are in proper units √ of 2λ

4

Interpolation of Quantum Chromodynamics in 1+1 Dimension

4.4 Chiral Condensate and Constituent Quark Mass

141

Fig. 4.10 Wavefunction renormalization as a function of ξ = tan−1 p−ˆ for (a) m = 0.18, (b) m = √ 1.0 and (c) m = 2.11. All quantities are in proper units of 2λ

142 Fig. 4.11 Wavefunction renormalization as a function of ξ = tan−1 p−ˆ for (a) m = 0, (b) m = 0.045, (c) m = 0.749 and (d) m = 4.23. All quantities are √ in proper units of 2λ

4

Interpolation of Quantum Chromodynamics in 1+1 Dimension

4.4 Chiral Condensate and Constituent Quark Mass

143

√ values is correlated with the negative values of E( p−ˆ )/ C discussed previously for Figs. 4.5a (m = 0.18), 4.6a (m = 0) and 4.6b (m = 0.045). To comprehend the sign correlation between F( p−ˆ ) and E( p−ˆ ), we write cos θ ( p−ˆ ) and sin θ ( p−ˆ ) in terms of M( p−ˆ ) and F( p−ˆ ) from Eqs. (4.89) and (4.90), as √

cos θ ( p−ˆ ) =

CM( p−ˆ ) , F( p−ˆ )E( p−ˆ )

(4.91)

sin θ ( p−ˆ ) =

p−ˆ , F( p−ˆ )E( p−ˆ )

(4.92)

and

so that we may rewrite Eq. (4.62) associated with the triangle diagram shown in Fig. 4.2 as  2, F( p−ˆ )E( p−ˆ ) = CM( p−ˆ )2 + p− (4.93) ˆ or

 E( p−ˆ ) =

2 CM( p−ˆ )2 + p− ˆ

F( p−ˆ )

.

(4.94)

In contrast to Eq. (4.62), we can now express E( p−ˆ ) itself without squaring it as E( p−ˆ )2 with the support from the wavefunction renormalization factor F( p−ˆ ) as well as the mass function M( p−ˆ ). This is rather remarkable because the issue of E( p−ˆ ) not being always positive for m 0.56, which was discussed in Sect. 4.3, is now resolved by expressing the dressed quark propagator S( p) in terms of F( p−ˆ ) and M( p−ˆ ) as given by Eq. (4.88). While E( p−ˆ ) can be negative, F( p−ˆ )E( p−ˆ ) is always positive due to the sign correlation between E( p−ˆ ) and F( p−ˆ ) as one can see from Eq. (4.90) or equivalently from Eq. (4.92) due to the sign correlation between θ ( p−ˆ ) and p−ˆ . This allows us more physically transparent interpretation of the energy–momentum dispersion relation for the interpolating dressed quark with √ = p / C introduced in its self-energy. Moreover, using the rescaled variable p− ˆ − ˆ Eq. (4.83) for the renormalized chiral condensate, we can assert the interpolation angle independence of the rescaled energy–momentum dispersion given by )E( p )  F( p− ˆ ˆ − )2 + p 2 ≡ E( ˜ p ), = M( p− √ ˆ ˆ ˆ − − C

(4.95)

˜ p ) which where we define the interpolation angle independent energy function E( ˆ − extends the interpolating energy–momentum dispersion relation of the on-mass-shell particle given by Eq. (4.7) to the case of the dressed quark with the rescaled variable, i.e. ˜ p )2 = p 2 + M( p )2 . E( ˆ ˆ ˆ − − −

(4.96)

144

4

Interpolation of Quantum Chromodynamics in 1+1 Dimension

Fig. 4.12 Geometrical representation of mass gap equations with the interpolation angle independent energy function ˜ p ) defined in Eq. (4.95) E( ˆ −

˜ p ) is always positive in contrast to E( p ˆ ), we can now As the solution of E( − ˆ − promote the mere pictorial device of geometric interpretation depicted in Fig. 4.2 ˜ p ), M( p ) to the more physically meaningful geometric interpretation with E( ˆ ˆ − − and p−ˆ as shown in Fig. 4.12 with all the positive lengths of the triangle sides. ) and From Eqs. (4.7) and (4.96), we also note the correspondence m ↔ M( p− ˆ √ ˜ p ) between the bare quark and the dressed quark. As an illustration p +ˆ / C ↔ E( ˆ −

for the two of this correspondence, we plot the profiles of E˜ as a function of p− ˆ √ ˜ p ) → p +ˆ / C cases of m = 0 and m = 0.18 in Fig. 4.13. It is evident that E( ˆ −

→ ∞, which is consistent with the result that the mass function M( p ) as p− ˆ − ˆ approaches the bare quark mass value m as ξ → π/2 or p−ˆ → ∞ (See Figs. 4.8 → 0, however, E( ˜ p ) approaches the characteristic mass value and 4.9). As p− ˆ ˆ − ˜ √ M(0) as shown in Fig. 4.13. Indeed we note E(0) = F(0)E(0) = M(0), confirming C

the sign correlation between E( p−ˆ ) and F( p−ˆ ) mentioned earlier; i.e. the negativity of E( p−ˆ ) for the small p−ˆ region is compensated by the corresponding negativity of F( p−ˆ ) to yield the mass function M( p−ˆ ) positive always for any kinematic region of p−ˆ . In Table 4.2, the numerical values of M(0) and F(0) are tabulated for the quark mass values shown in Figs. 4.8 and 4.9 as well as in Figs. 4.10 and 4.11. As expected, F(0) values are negative for the small bare quark mass values (m 0.56) to compensate the corresponding negative values of E(0) while M(0) values are all positive. ˜ p ) and M( p ) Now, between the two limits ( p −ˆ → 0 and p −ˆ → ∞), both E( ˆ ˆ − − are running with the variable p−ˆ and their profiles of the p−ˆ -dependence are com˜ p ) and M( p ) pletely independent of the interpolation angle δ. The invariance of E( ˆ −

ˆ −

under the interpolation between IFD and LFD indicates their universal nature as ˜ p ) and M( p ) as physically meaningful quantities. In this respect, one may call E( ˆ ˆ − − the effective energy and the constituent mass of the dressed quark moving with the . In principle, these physical quantities can be also scaled longitudinal momentum p− ˆ computed in the well-known Euclidean approaches [34] such as the lattice QCD [35] and the manifestly covariant Dyson–Schwinger formulation. In the Euclidean formulation, which can in principle be applied to the interpolation dynamics as far as

4.4 Chiral Condensate and Constituent Quark Mass

145

˜ p ) for several interpolation angles for two example small quark masses Fig. 4.13 Profiles of E( ˆ − √ of the corresponding (a) m = 0 and (b) m = 0.18, in comparison with p +ˆ / C for a free particle √ . All quantities are in proper units of 2λ same masses, as a function of p− ˆ

0 ≤ δ < π /4 except δ = π /4 due to the light-like nature of LFD, the effective energy and the constituent quark mass would be given in terms of the Lorentz-invariant ˜ P˜ 2 ) and M( P˜ 2 ). In IFD, one may correspond Euclidean variable P˜ 2 < 0, i.e. E( 0 ˜ P˜ 2 ) the Wick rotated energy P˜ with the imaginary effective energy, i.e. P˜ 0 = i E( 1 1 (purely imaginary value), and the longitudinal momentum P˜ with p of the dressed ˜ P˜ 2 )2 + ( p 1 )2 = −( p 1 )2 − M( P˜ 2 )2 + quark, so that P˜ 2 = ( P˜ 0 )2 + ( P˜ 1 )2 = − E( ( p+ˆ )2 −( p )2

ˆ − ( p 1 )2 = −M( P˜ 2 )2 < 0. In the same token, as p 2 = generally in the C ˆ + interpolating dynamics, the Wick rotated interpolating energy P˜ with the imaginary

146

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Interpolation of Quantum Chromodynamics in 1+1 Dimension

Table 4.2 Numerical values of M(0) and F(0) for several different quark mass values m M(0) F(0)

0 0.045 0.18 0.749 0.532778 0.563644 0.659112 1.10105 –0.495173 –0.584175 –0.987673 4.11079 √ All quantities are in proper units of 2λ

1.00 1.31167 2.17976

2.11 2.30969 1.22134

4.23 4.34358 1.05526

√ ˜ p 2 ) (purely imaginary value), and the longitueffective energy, i.e. P˜ +ˆ / C = i E( ˆ − √ ( P˜ +ˆ )2 +( P˜−ˆ )2 dinal momentum P˜−ˆ / C with p−ˆ of the dressed quark, so that P˜ 2 = = C 2 2 2 2 2 2 2 ˜ p ) + p = − p − M( p ) + p = −M( p ) < 0. Thus, it is natu− E( ˆ −

ˆ −

ˆ −

ˆ −

ˆ −

ˆ −

) to the ral to correspond the square of the rescaled longitudinal momentum ( p− ˆ Euclidean variable P˜ 2 for the space-like region P˜ 2 < 0 in the interpolating dynamics, i.e.

p −ˆ ↔ − P˜ 2 . 2

(4.97)

This correspondence is supported not only from the relativistic form invariance of ˜ p ) and M( p ) but also from the matching condition between the Minkowsky E( ˆ ˆ − − space and the Euclidean space. Namely, when the real energy value in the Minkowsky space is converted into the purely imaginary value in the Euclidean space, the matching between the real value and the purely imaginary value occurs precisely where the value itself is zero. For instance, in IFD, the Wick rotation of the energy p 0 → P˜ 0 = i p 0 has the common energy value p 0 = P˜ 0 = 0 so that the Lorentzinvariant momentum-squared value in the Minkowsky space can be matched with the corresponding Euclidean momentum-squared value, P˜ 2 , by taking simultaneously both the real value p 0 = 0 in the Minkowsky space and the purely imaginary value P˜ 0 = i p 0 = 0. Likewise, in the interpolation of the relativistic dynamics for 0 ≤ δ 4 m < ψψ =− = 2 fπ

!

8π 2 m 2 λ , 3

(4.127)

¯ where we used Eq. √ (4.79) for the vacuum condensation < ψψ > and the pion decay constant f π = Nc /π derived [32] from the matrix element of the axial vector current between the pionic ground state and the non-trivial vacuum. Here, the pionic bound-state in the chiral limit corresponds to the zero mass bound-state that occurs when both quarks have mass zero as pointed out by’t Hooft [1]. The corresponding bound-state wavefunction in LFD is given by φ(x) = 1 for x ∈ [0, 1] as we will discuss in the next subsection, Sect. 4.6.2. These results are consistent with the discussions [4,5,13,32,38] on the SBCS in the’t Hooft model (Nc → ∞) which does not contradict with Coleman’s theorem [28] that would prohibit the SBCS in the case of a finite Nc . As the result of M(0) must be zero theoretically for the bare quark mass m = 0, the scrutiny of the numerical sensitivity check depending on the values of δ and r−ˆ as well as the number of computational grid points is maximally enhanced in the m = 0 case. In Table 4.3, the numerical values of M(0) for the bare quark mass m = 0 are listed depending on the values of δ and r−ˆ as well as the number of computational grid points. We take the values of r−ˆ as r−ˆ = 0, 0.2M0.18 , and 2M0.18 , with the scale of M0.18 which is the lowest bound-state mass for m = 0.18, i.e. M0.18 = 0.88, as the lowest bound-state mass for m = 0 is zero, i.e. M0 = 0, and cannot be taken as any reference value of r−ˆ . While it is highly challenging to achieve the typical numerical accuracy mentioned earlier for the bare quark mass m = 0, our numerical results appear to consistently approach the theoretical value M(0) = 0 as the number of grid points is increased from 200 to 600 for the ranges of the δ and r−ˆ values in Table 4.3. As δ gets close to π /4, our numerical computation demands Table 4.3 Numerical results of the ground-state meson mass M(0) depending on the values of δ and r−ˆ as well as the number of computational grid points for the quark bare mass m = 0 case δ

r−ˆ

Number of grid points M(0)

0

0

200 600 200 600 200 600 200 600 200 600 200 600

0.0607101 0.0361024 0 0.2M0.18 0.0632755 0.0395562 0 2M0.18 0.107216 0.0532456 0.6 0 0.0690707 0.0407446 0.6 0.2M0.18 0.0678997 0.0472921 0.6 2M0.18 0.143686 0.0599700 √ All the mass spectra are given in six significant figures with the proper unit of 2λ. The groundstate meson mass M0.18 = 0.88 for m = 0.18 is taken as the reference value of the interpolating longitudinal momentum r−ˆ . The results for r−ˆ = 0 were obtained with the method presented in Appendix B.6

156

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Interpolation of Quantum Chromodynamics in 1+1 Dimension

Table 4.4 Meson mass spectra for the bare quark mass m = 0 with the variation of δ and r−ˆ values δ

r−ˆ

M(0)

M(2)

M(4)

M(6)

0

0 0.2M0.18 2M0.18 0 0.2M0.18 2M0.18 0 0.2M0.18 2M0.18 r−ˆ

0.0361024 0.0395562 0.0532456 0.0407446 0.0472921 0.0599700 0.0884992 0.173979 0.106365

3.76245 3.76266 3.76433 3.76302 3.76285 3.76441 3.76458 3.76292 3.77182

5.68513 5.68573 5.68703 5.68552 5.68542 5.68690 5.68665 5.69002 5.69178

7.15304 7.15381 7.16921 7.15329 7.15361 7.15444 7.15417 7.15655 7.16045

M(1)

M(3)

M(5)

M(7)

0 0.2M0.18 2M0.18 0 0.2M0.18 2M0.18 0 0.2M0.18 2M0.18

2.42728 2.42773 2.42861 2.42772 2.42815 2.43215 2.42906 2.43734 2.43520

4.80619 4.80668 4.80603 4.80668 4.80693 4.80696 4.80807 4.80832 4.81589

6.45726 6.45797 6.45736 6.45756 6.45796 6.45845 6.45857 6.45799 6.46355

7.79124 7.79191 7.79204 7.79145 7.79205 7.79172 7.79239 7.79376 7.79905

0.6

0.78

δ 0

0.6

0.78

√ All the mass spectra are given in six significant figures with the proper unit of 2λ. The groundstate meson mass M0.18 = 0.88 for m = 0.18 is taken as the reference value of the interpolating longitudinal momentum r−ˆ . The results for r−ˆ = 0 were obtained with the method presented in Appendix B.6

much higher numerical accuracy. Although we haven’t increased the number of grid points beyond 600, we anticipate that our numerical results would get closer and closer to zero as we increase the number of grid points even if δ gets close to π /4. In Table 4.4, we list the results of the meson mass spectra M(n) up to n = 7 for the bare quark mass m = 0 including M(0) obtained with the 600 grid points for δ = 0, 0.6 and 0.78. For the mass spectra of excited states, M(n) (n = 1, 2, ..., 7), the results were obtained with the number of grid points just 200 for δ = 0 and 0.6 although the number of grid points for the δ = 0.78 case was still kept as 600. Similarly, we present the results of the meson mass spectra for the bare quark mass m = 0.18 in Table 4.5. Here, all the results M(n) (n = 0, 1, 2, ..., 7) including the ground state were obtained with the 200 grid points for δ = 0 and 0.6 while for δ = 0.78 with the 380 grid points. We note that our numerical results are consistent with each other up to the second digit after the decimal point for the δ values not close to π /4 such as δ = 0 and 0.6 and the r−ˆ values not so large such as r−ˆ = 0 and 0.2M0.18 as listed in Tables 4.4 and 4.5. Such stability persists in all the cases of the bare quark mass values (m = 0, 0.045, 0.18, 1.00, 2.11) that we considered for the computation of the meson mass spectra in this work.

4.6 The Bound-State Solution

157

Table 4.5 Meson mass spectra for the bare quark mass m = 0.18 with the variation of δ and r−ˆ values δ

r−ˆ

M(0)

M(2)

M(4)

M(6)

0

0 0.2M0.18 2M0.18 0 0.2M0.18 2M0.18 0 0.2M0.18 2M0.18 r−ˆ

0.880457 0.880686 0.883080 0.881753 0.881526 0.886997 0.889730 0.856979 0.914992

3.99902 3.99928 3.99921 3.99943 3.99943 4.00048 4.00233 3.99561 4.01176

5.85781 5.85843 5.85774 5.85807 5.85812 5.85938 5.86016 5.85482 5.86721

7.29550 7.29630 7.29676 7.29566 7.29612 7.29641 7.29718 7.29324 7.30555

M(1)

M(3)

M(5)

M(7)

0 0.2M0.18 2M0.18 0 0.2M0.18 2M0.18 0 0.2M0.18 2M0.18

2.73527 2.73565 2.73588 2.73559 2.73595 2.73818 2.73843 2.73213 2.75685

5.00349 5.00401 5.00338 5.00384 5.00406 5.00430 5.00638 5.00048 5.01497

6.61265 6.61336 6.61320 6.61285 6.61329 6.61360 6.61463 6.60966 6.62144

7.92348 7.92401 7.92469 7.92360 7.92419 7.92388 7.92496 7.92062 7.93055

0.6

0.78

δ 0

0.6

0.78

√ All the mass spectra are given in six significant figures with the proper unit of 2λ. The groundstate meson mass M0.18 = 0.88 for m = 0.18 is taken as the reference value of the interpolating longitudinal momentum r−ˆ . The results for r−ˆ = 0 were obtained with the method presented in Appendix B.6 Table 4.6 Summary of meson mass spectra M √(n) (n = 0, 1, 2, ..., 7) for the bare quark mass values m = 0, 0.045, 0.18, 1.00, 2.11 in the unit of 2λ n m m m m m

=0 = 0.045 = 0.18 = 1.00 = 2.11

0 0 0.42 0.88 2.70 4.91

1 2.43 2.50 2.74 4.16 6.17

2 3.76 3.82 4.00 5.21 7.06

3 4.81 4.85 5.00 6.09 7.83

4 5.69 5.73 5.86 6.85 8.51

5 6.46 6.49 6.61 7.53 9.13

6 7.15 7.19 7.30 8.16 9.69

7 7.79 7.82 7.92 8.75 10.23

In Table 4.6, we summarize these stable numerical results up to the second digit after the decimal point. Here, we take M(0) = 0 for the case m = 0 from the theoretical SBCS ground. The values in Table 4.6 are also shown in Fig. 4.16 depicting the feature of “Regge trajectories” for the quark–antiquark bound-states each with the corresponding equal bare mass m [1,9,12,13,41]. It is interesting to note that the Regge trajectory gets slightly modified from just the linear trajectory behavior, developing a bit of curvature in the trajectory for the ground state and the low-lying

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Interpolation of Quantum Chromodynamics in 1+1 Dimension

Fig. 4.16 Feature of “Regge trajectories” for the quark–antiquark bound-states each with the cor√ responding equal bare mass m. All quantities are in proper units of 2λ

excited states. For the small mass, in particular m = 0, the trajectory looks a little concave shape while for the larger mass the curvature turns somewhat into a convex shape. This seems to reflect the fact that the GOR works in the chiral limit as shown in Fig. 4.15 but the chiral symmetry gets broken as the quark mass gets larger. The convex feature of the Regge trajectory for the heavy quarkonia model was shown in Ref. [42].

4.6.2

Wavefunctions

(n) We present here our numerical solutions of the bound-state wavefunctions φˆ ± (r−ˆ , x) interpolating between IFD and LFD for the ground state n = 0 and the first excited state n = 1 with r−ˆ = 0 in terms of the interpolating longitudinal momentum fraction variable x = p−ˆ /r−ˆ . The results for r−ˆ = 0 are presented separately in Appendix B.6 in terms of the variable ξ = tan−1 ( p−ˆ ) without scaling the interpolating momentum variable p−ˆ with respect to r−ˆ . In the chiral limit, where the GOR relation √ M2(0) ∼ m λ → 0 is satisfied, the analytic solution of the pionic ground-state wave(0) function φ± in IFD, i.e. φˆ ± for δ = 0, is known [32] in terms of the IFD longitudinal momentum variables p 1 and r 1 of the quark and the pionic meson, respectively. Corresponding the IFD longitudinal momentum p 1 and r 1 to the interpolating longi-

4.6 The Bound-State Solution

159

(0) (0) Fig. 4.17 Ground-state wavefunctions φˆ + (r−ˆ , x) and φˆ − (r−ˆ , x) for m = 0. All quantities are in √ proper units of 2λ

tudinal momentum p−ˆ and r−ˆ and confirming the consistency with the LFD analytic mass gap solution discussed in Sect. 4.2.3, we note that the corresponding analytic solution in the interpolating formulation is given by  θ (r−ˆ − p−ˆ ) − θ ( p−ˆ ) 1 cos 2 2  θ (r−ˆ − p−ˆ ) + θ ( p−ˆ ) , ± sin 2

(0) φˆ ± (r−ˆ , p−ˆ ) =

(4.128)

where the normalization is taken to satisfy Eq. (4.124) [13]. In Fig. 4.17, our numerical results of the plus and minus components of the bound-state wavefunction for (0) (r−ˆ , x), are shown for the bare quark mass value m = 0 in the ground state, i.e. φˆ ± comparison with the interpolating analytic solution given by Eq. (4.128). The results of δ = 0, 0.6 and 0.78 are shown in the top, middle and bottom panels, respectively. In each panel, the results of r−ˆ = 0.2M0.18 , 2M0.18 and 5M0.18 with the scale of the ground-state meson mass M0.18 = 0.88, i.e. r−ˆ = 0.176, 1.76 and 4.4 (all in units of

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4

Interpolation of Quantum Chromodynamics in 1+1 Dimension

√ 2λ), are depicted by the solid lines for the analytic results and the dashed lines for the numerical results in blue, yellow and green, respectively. Our numerical results coincide with the analytic results given by Eq. (4.128) as shown in Fig. 4.17 except for some wiggle and bulge in the numerical result of δ = 0.78 and r−ˆ = 5M0.18 in Fig. 4.17e due to the numerical sensitivity near the LFD (C → 0) and large longitudinal momentum r−ˆ . Our results in Fig. 4.17 appear to confirm the validity of our numerical results as well as the analytic results. As the longitudinal momen(0) tum r−ˆ , i.e. r 1 for δ = 0, gets large, the numerical results of φˆ + (r−ˆ , x) approach to φ(x) = 1 for x ∈ [0, 1], which is the solution of the ’t Hooft equation given by (0) Eq. (4.118) in LFD [1,41], while φˆ − (r−ˆ , x) results tend to diminish although for very small momentum, e.g. r 1 = 0.2M0.18 , it is still of comparable order of mag(0) nitude to φˆ + (r−ˆ , x) as noted also in Ref. [13]. Ref. [24] also noted that for light mesons the large-momentum IFD numerical results approach the exact light-front solution very slowly. While the LFD solution for m = 0 exhibits an infinite slope at the endpoints x = 0, 1, such feature is not achieved in the IFD large-momentum method [13,24]. As δ gets closer to π /4, however, the resemblance to the LFD solutions is attained even in the smaller longitudinal momentum (e.g. r−ˆ = 2M0.18 ) and thus the large r−ˆ (e.g. r−ˆ = 5M0.18 ) does not need to be taken for the confirmation of the LFD solutions. The similar behavior of resemblance to the LFD results (1) depending on the values of δ and r−ˆ is also found in the first excited states φˆ + (r−ˆ , x) (1) and φˆ − (r−ˆ , x) shown in Fig. 4.18. As dictated by the charge conjugation symmetry, (1) (1) under the exchange of x ↔ 1 − x, φˆ + (r−ˆ , x) and φˆ − (r−ˆ , x) are antisymmetric (0) (0) while φˆ + (r−ˆ , x) and φˆ − (r−ˆ , x) are symmetric. We note also that our results for δ = 0 shown in the top panels of Figs. 4.17, 4.18; i.e. the plots of Figs. 4.17a, b, 4.18a, b appear to be consistent with the results in Ref. [13] although only a qualitative comparison can be made as different momentum values are taken for the moving frames in Ref. [13] compared to what we present here. In contrast to the case of m = 0, there are no known analytic solutions of the bound-state wavefunctions for m = 0. While we present the numerical results of (0) (1) (r−ˆ , x) and φˆ ± (r−ˆ , x) for the cases of m = 0.045, 1.0 and 2.11 in Appendix B.4, φˆ ± we take here m = 0.18 to correspond with the spectroscopy discussion in the last (n) subsection (Sect. 4.6.1) and show its numerical results of φˆ ± (r−ˆ , x) for the ground state n = 0 and the first excited state n = 1. In Fig. 4.19, the numerical results of (0) (0) φˆ + (r−ˆ , x) and φˆ − (r−ˆ , x) for δ = 0, 0.6 and 0.78 are shown in the top, middle and bottom panels, respectively. Similarly, in Fig. 4.20, the numerical results of (1) (1) φˆ + (r−ˆ , x) and φˆ − (r−ˆ , x) for δ = 0, 0.6 and 0.78 are shown in the top, middle and bottom panels, respectively. In each panel, the results of r−ˆ = 0.2M0.18 , 2M0.18 and 5M0.18 with the scale of M0.18 = 0.88, i.e. r−ˆ = 0.176, 1.76 and 4.4 (all in √ units of 2λ), are depicted by the solid lines in blue, yellow and green, respectively. As noted for the case of m = 0, the large-momentum IFD (δ = 0) numerical results approach the LFD results very slowly also for the case of m = 0.18. (n) While the LFD results (φˆ + (r−ˆ , x) = φ (n) (x)) should be constrained in the x-region [0, 1], the IFD results in the top left panel of Fig. 4.19, i.e. Fig. 4.19a, exhibit

4.6 The Bound-State Solution

161

(1) (1) Fig. 4.18 First excited state wavefunctions φˆ + (r−ˆ , x) and φˆ − (r−ˆ , x) for m = 0. All quantities √ are in proper units of 2λ

rather long tails outside the [0, 1] region even for the pretty large longitudinal momentum r 1 = 5M0.18 = 4.4. For δ = 0.78, however, i.e. very close to the LFD (π /4 ≈ 0.785398), shown in Fig. 4.19e, the wavefunctions for the relatively larger momenta r−ˆ = 2M0.18 = 1.76 and 5M0.18 = 4.4 almost coincide with each other while closely fitting in the region [0, 1] although the result with very small longitudinal momentum (r−ˆ = 0.2M0.18 = 0.176) has a long tail outside the [0, 1] region similar to the IFD (δ = 0) result. While we notice some wiggle and bulge (0) in φˆ + (r−ˆ , x)(δ = 0.78) for r−ˆ = 5M0.18 , we didn’t pursue any further numerical accuracy as we understand that it is due to the computational sensitivity arising in the interpolation region where C gets close to 0 in particular as r−ˆ gets very large. (n) Since the LFD results of φˆ − (r−ˆ , x) must vanish as discussed in the derivation of the ’t Hooft’s bound-state equation given by Eq. (4.118), it is manifest in Fig. 4.19b that the large-momentum IFD (δ = 0) results approach to the LFD (δ = π /4) result (0) again very slow for φˆ − (r−ˆ = r 1 , x) while the δ = 0.78 results in Fig. 4.20 are rather immediately close to the LFD result. Figure 4.20 shows the first excited state wave(1) (1) functions φˆ + (r−ˆ , x) and φˆ − (r−ˆ , x) for the input bare quark mass value m = 0.18.

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Interpolation of Quantum Chromodynamics in 1+1 Dimension

(0) (0) Fig. 4.19 Ground-state wavefunctions φˆ + (r−ˆ , x) and φˆ − (r−ˆ , x) for m = 0.18. All quantities are √ in proper units of 2λ

The results for m = 0.18 in Fig. 4.20 look quite similar to the results for m = 0 in Fig. 4.18. They share the same feature of the charge conjugation symmetry, under (1) (1) the exchange of x ↔ 1 − x, i.e. φˆ + (r−ˆ , x) and φˆ − (r−ˆ , x) are antisymmetric while (0) (0) φˆ + (r−ˆ , x) and φˆ − (r−ˆ , x) are symmetric. They also share the similar behavior of resemblance to the LFD results depending on the values of δ and r−ˆ , which we have discussed for the ground state previously.

4.6.3

Quasi-PDFs

Since we obtained the bound-state wavefunctions, we now apply them to compute the so-called quasi-PDFs which have been discussed extensively even in the ’t Hooft model application [24] due to the possibility of computing directly the longitudinal momentum fraction x-dependence of the parton distributions in Euclidean lattice approach using the large-momentum effective field theory (LaMET) program [22].

4.6 The Bound-State Solution

163

(1)

(1)

Fig. 4.20 First excited state wavefunctions φˆ + (r−ˆ , x) and φˆ − (r−ˆ , x) for m = 0.18. All quantities √ are in proper units of 2λ

In our interpolating ’t Hooft model computation, the “quasi-PDFs” may be defined as the following matrix element for the n-th state of the meson with the interpolating longitudinal momentum r−ˆ : q˜(n) (r−ˆ , x) =

 +∞ −ˆ ˆ − dx ˆ + ¯ −ˆ ) · γ ˆ W[x −ˆ , 0] ψ(0) | r +ˆ , r ˆ >C , ei x r−ˆ < r(n) , r−ˆ | ψ(x − (n) − 4π −∞

(4.129)

 ˆ + 2 + CM2 as obtained from Eq. (4.7), and the range of the lonwhere r(n) = r− (n) ˆ gitudinal momentum fraction x = p−ˆ /r−ˆ is unconstrained, −∞ < x < +∞, for 0 ≤ δ < π /4, while bounded, 0 ≤ x ≤ 1, for δ = π /4. One should note that this definition of the “quasi-PDFs” is not unique, e.g. taking γ +ˆ instead of γ−ˆ in front of the interpolating gauge link W [x −ˆ , 0] in Eq. (4.129), but still uniquely approach the PDF defined in the LFD as δ → π /4, whichever definition is taken. While one may consider the so-called universality class [43] of the interpolating “quasi-PDFs,”

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Interpolation of Quantum Chromodynamics in 1+1 Dimension

we note that the definition given by Eq. (4.129) coincides with the canonical definition of the quasi-PDFs in IFD (δ = 0) [24]. While it has been discussed which definition approaches the PDFs in LFD faster for the perspectives of the LaMET program [22,24], we will take the definition given by Eq. (4.129) in this work and discuss our numerical results corresponding to this definition. The interpolating gauge link  ˆ −

"



W [x , 0] = P exp −ig

$

x −ˆ

dx 0

− ˆ

− ˆ

A−ˆ (x )

(4.130)

inserted in Eq. (4.129) assures the gauge invariance of the interpolating “quasiPDFs.” The subscript C in Eq. (4.129) indicates the removal of the disconnected contribution discussed [44] for the forward matrix element computation: ˆ

ˆ

+ + , r−ˆ | r(n) , r−ˆ > < r(n)



+∞

−∞

d x −ˆ i x −ˆ r ˆ ¯ −ˆ ) · γ−ˆ W [x −ˆ , 0] ψ(0) | > . − < | ψ(x e 4π (4.131)

As we adopted the axial gauge in the interpolation form, i.e. Aa−ˆ = 0, the gauge link becomes an identity and the quantization procedure illustrated in Sect. 4.2.1 yields ˆ

q˜(n) (r−ˆ , x) =

+ r(n)

r−ˆ

 (n) (n) sin θ (xr−ˆ ) φˆ + (r−ˆ , x)2 + φˆ − (r−ˆ , x)2

 (n) (n) + φˆ + (r−ˆ , −x)2 + φˆ − (r−ˆ , −x)2 .

(4.132)

For δ = 0, i.e. IFD, Eq. (4.132) coincides with Eq. (74) of Ref. [24]. Based on this formula, we compute the interpolating “quasi-PDFs” for the cases of δ = 0, 0.6 and 0.78. In Figs. 4.21 and 4.22, the interpolating “quasi-PDFs” of the ground state (n = 0) and the first excited state (n = 1) are shown for the case

Fig. 4.21 Quasi-PDFs in IFD (δ = 0) for the ground-state (n = 0) wavefunctions of (a) m = 0, (b) √ m = 0.18. All quantities are in proper units of 2λ

4.6 The Bound-State Solution

165

Fig. 4.22 Quasi-PDFs in IFD (δ = 0) for the first excited state (n = 1) wavefunctions of (a) m = 0, √ (b) m = 0.18. All quantities are in proper units of 2λ

Fig. 4.23 δ = 0.6 interpolating “quasi-PDFs” for the ground-state (n = 0) wavefunctions of (a) √ m = 0, (b) m = 0.18. All quantities are in proper units of 2λ

of δ = 0, respectively. The two panels in each of these figures exhibit our numerical results for the bare quark mass m = 0 and m = 0.18, respectively. Numerical results for other mass cases (m =0.045, 1.0 and 2.11) are summarized in Appendix B.5. In each panel, the results for the meson longitudinal momentum r−ˆ = r 1 = 0.2M0.18 , 2M0.18 , 5M0.18 are depicted by blue, yellow and green solid lines, respectively. The general behaviors of our numerical results with respect to the variation of r 1 values from small (0.2M0.18 ) to large (5M0.18 ) agree with the results presented in Ref. [24], although different longitudinal momentum values were taken between ours and Ref. [24]. As r 1 gets larger, the numerical results of the quasi-PDFs resemble the PDFs in LFD more closely fitting in x ∈ [0, 1] and [−1, 0]. It is our interest to take a look at the rate of achieving the resemblance to the PDFs in LFD as δ gets away from the IFD (δ = 0) and r−ˆ gets larger. The numerical results of the ground state (n = 0) and the first excited state (n = 1) are shown for the case of δ = 0.6 in Figs. 4.23 and 4.24 and for the case of δ = 0.78 in Figs. 4.25 and 4.26, respectively. As noticed previously in Figs. 4.17 and 4.19, the wiggle and bulge in (0) φˆ + (r−ˆ , x)(δ = 0.78) for r−ˆ = 5M0.18 is due to the computational sensitivity arising in the interpolation region where C gets close to 0 in particular as r−ˆ gets very large. The corresponding wiggle and bulge in q˜(0) (r−ˆ , x) is noticed also in Fig. 4.25. Besides such numerical sensitivity for the very large value of r−ˆ , it is apparent that the δ = 0.78 results are rather immediately close to the LFD result.

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Interpolation of Quantum Chromodynamics in 1+1 Dimension

Fig. 4.24 δ = 0.6 interpolating “quasi-PDFs” for the first excited √ state (n = 1) wavefunctions of (a) m = 0, (b) m = 0.18. All quantities are in proper units of 2λ

Fig. 4.25 δ = 0.78 interpolating “quasi-PDFs” for the ground-state (n = 0) wavefunctions of (a) √ m = 0, (b) m = 0.18. All quantities are in proper units of 2λ

Fig. 4.26 δ = 0.78 interpolating “quasi-PDFs” for the first excited √ state (n = 1) wavefunctions of (a) m = 0, (b) m = 0.18. All quantities are in proper units of 2λ

Interestingly, our numerical results of the interpolating “quasi-PDFs” in the moving frames indicate a possibility to utilize both variation of δ and r−ˆ to attain the LFD result more effectively. Namely, one may not need to boost the longitudinal momentum r−ˆ too large but search for a “sweet spot” by varying both δ and r−ˆ together to obtain the “LFD-like” result. In IFD, δ = 0 is fixed and thus the boost to the large longitudinal momentum is necessary for a successful approach to the LFD result. However, in the interpolating formulation between the IFD and the LFD, the LFD result can be approached even at rather small r−ˆ . Moreover, the application to the lattice formulation may be also possible with the existing technique of Wick

4.6 The Bound-State Solution

167

Fig. 4.27 Interpolating “quasi-PDFs” at the fixed value of r−ˆ = 2M0.18 for the (a) ground state and (b) first excited state √ for m = 0 in three cases of δ = 0, δ = 0.6 and δ = 0.78. All quantities are in proper units of 2λ

rotation replacing the ordinary instant form time x 0 by the interpolating time x +ˆ in the process of taking the “imaginary time” in the lattice as far as δ remains in the region 0 ≤ δ < π /4 avoiding the light-like surface δ = π /4. As discussed in the later part of Sect. 4.4.2, one can match the Minkowsky space and the Euclidean space confirming the correspondence given by Eq. (4.97). For an illustration of the δ variation for a given finite r−ˆ , we take r−ˆ = 2M0.18 for the case of m = 0 and show the “quasi-PDFs” of the ground state and the first excited state for the variation of δ parameter as δ = 0, δ = 0.6 and δ = 0.78 in Fig. 4.27. It indicates that a pretty slow approach to the LFD result in the large-momentum IFD can be fairly well expedited by taking δ away from the IFD (δ = 0) and getting closer to the LFD (δ = π /4) while the same value of the longitudinal momentum r−ˆ = 2M0.18 is taken. The numerical sensitivity arising near δ = π /4 for the large r−ˆ , e.g. the “wiggle and bulge” mentioned for r−ˆ = 5M0.18 in Figs. 4.25 and 4.26 for δ = 0.78, is also dodged by taking the smaller value of r−ˆ , e.g. r−ˆ = 2M0.18 for this illustration. We think it would be worthwhile to explore this idea of utilizing the interpolating formulation between IFD and LFD for the application to the lattice computation. For further application of the interpolating bound-state wavefunctions, one can also consider the interpolating “quasi-distribution amplitude (quasi-DA)” which may be written as ˜ (n) (r−ˆ , x) =  ˆ

+ ¯ , r−ˆ |ψ( < r(n)



1 f (n) x −ˆ 2

+∞

−∞ ˆ −

)W [

d x −ˆ i(x− 1 )r ˆ x −ˆ 2 − · e 2π

x x −ˆ x −ˆ , − ]γ−ˆ γ5 ψ(− )| >, 2 2 2

(4.133)

where W is the gauge link introduced in Eq. (4.130), and f (n) is the decay constant mentioned in Sect. 4.6.1. As mentioned in the definition of the interpolating “quasiPDFs” given by Eq. (4.129), the definition of the interpolating “quasi-DAs” is also not unique, e.g. γ +ˆ can be taken instead of γ−ˆ in front of the interpolating gauge ˆ −

ˆ −

link W [ x2 , − x2 ] in Eq. (4.133). Whichever definition is taken, they all uniquely approach the DA defined in the LFD as δ → π /4, belonging to the same universality class [43] as mentioned for the case of interpolating “quasi-PDFs.” Using the

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Interpolation of Quantum Chromodynamics in 1+1 Dimension

definition given by Eq. (4.133) which coincides with the canonical definition of the quasi-DAs in IFD (δ = 0) [24], we note that the interpolating “quasi-DAs” of the even-n mesonic states can be formulated as !   θ xr−ˆ + θ (1 − x)r−ˆ Nc r +ˆ 1 ˜ sin (n) (r−ˆ , x) = (n) πr−ˆ 2 f   (n) (n) × φˆ + (r−ˆ , x) + φˆ − (r−ˆ , x) , (4.134) (n) where φˆ ± (r−ˆ , x) denote the interpolating mesonic wavefunctions associated with the n-th excited mesonic state. The normalization condition of the interpolating “quasi-DAs” given by  +∞ ˜ (2n) (r−ˆ , x) = 1 dx (4.135) −∞

is consistent with the explicit form of the decay constant f (n) given by

f (n)

⎧    θ xr−ˆ +θ (1−x)r−ˆ ⎪ Nc r +ˆ % +∞ ⎪ d x sin ⎪ −∞ πr 2 ⎨  −ˆ  = (n) (n) ˆ + (r−ˆ , x) + φˆ − (r−ˆ , x) even n; × φ ⎪ ⎪ ⎪ ⎩ 0 odd n.

(4.136)

While the corresponding results for the IFD (δ = 0) have been worked out in Refs. [13,24], √ it is interesting to note that the analytic result for the pion decay constant f π = Nc /π [32] can be immediately obtained by taking the LFD solution (0) (0) in the chiral limit, i.e. φˆ + (r−ˆ , x) = φ(x) = 1 for x ∈ [0, 1] and φˆ − (r−ˆ , x) = 0, π as well as θ (xr−ˆ ) = θ (xr + ) = θ ( p + ) = 2 and θ ((1 − x)r−ˆ ) = θ ((1 − x)r + ) = ˆ +

+

θ (r + − p + ) = π2 , noting rr = rr− = 1 in LFD. As the DAs in LFD are directly ˆ − involved with the QCD factorization theorem for the hard exclusive reactions involving hadrons, it would be useful to explore the utility of the interpolating “quasi-DAs” further in the future works.

4.7

Conclusion and Outlook

In this chapter, we interpolated the ’t Hooft model (i.e. QCD2 in the large Nc limit) with the interpolation angle δ between IFD (δ = 0) and LFD (δ = π /4) and analyzed its nontrivial vacuum effects on the quark mass and wavefunction renormalization as well as the corresponding meson mass and wavefunction properties taking the meson as the quark–antiquark bound-state. We derived the interpolating mass gap equation between IFD and LFD using not only the algebraic method based on the Bogoliubov transformation between the trivial and nontrivial vacuum as well as the bare and dressed quark but also the diagrammatic method based on the self-consistent

4.7 Conclusion and Outlook

169

embodiment of the quark self-energy. Our mass gap solutions agree not only with the LFD result in Ref. [1] for δ = π /4 but also with the IFD results in Refs. [12,13] for δ = 0. The renormalized chiral condensate was computed and the agreement of the result in the chiral limit was verified with the exact result in Ref. [4,5]. Its invariance regardless of the δ values between IFD and LFD was also confirmed. Taking into account the wavefunction renormalization factor F( p−ˆ ) as well as the mass function M( p−ˆ ) and expressing the dressed quark propagator S( p) in terms of F( p−ˆ ) and M( p−ˆ ) as given by Eq. (4.88), we resolved the issue of E( p−ˆ ) not being always positive for m 0.56 discussed in Ref. [11]. Extending the interpolating energy– momentum dispersion relation of the on-mass-shell particle given by Eq. (4.7) to the case of the dressed quark with the rescaled variable given by Eq. (4.96), we ˜ p ). Typical profiles obtained the interpolation angle independent energy function E( ˆ − ˜ p ) were exemplified in Fig. 4.13. of E( ˆ −

Utilizing the dressed fermion propagators, we then derived the quark–antiquark bound-state equation interpolating between IFD and LFD for the equal bare quark and antiquark mass m and solved numerically the corresponding bound-state equations. From the numerical solutions of the spectroscopy, we find that the meson mass spectrum is independent of interpolation angle between the IFD and LFD as expected for physical observables. In particular, for the bare quark mass m → 0, we √confirmed the GOR behavior of the pionic ground-state mass square M2(0) ∼ m λ → 0 as shown in Fig. 4.15. Our result is consistent with the discussions [4,5,13,32,38] on the SBCS in the ’t Hooft model (Nc → ∞). Plotting the meson mass spectra M(n) (n = 0, 1, 2, ..., 7) for various m values as summarized in Table 4.6, we also observe the Regge trajectory feature as shown in Fig. 4.16. The corresponding bound-state (n) wavefunctions φˆ ± (r−ˆ , x) were obtained, in particular, for the low-lying states, i.e. n = 0 and 1 states, and were applied to the interpolating formulation of “quasi-PDFs.” The results of the wavefunctions clearly dictate the charge conjugation symmetry, (0) (1) exhibiting the symmetric and antisymmetric behaviors of φˆ ± (r−ˆ , x) and φˆ ± (r−ˆ , x), respectively, under the exchange of x ↔ 1 − x. It is also interesting to note that the massless Goldstone boson cannot exist in the rest frame due to the null normalization (0) (0) from the equivalence between φˆ + ( p−ˆ ) and φˆ − ( p−ˆ ) for the massless ground state in the rest frame as shown in Fig. B.14. (n) Applying the bound-state wavefunctions φˆ ± (r−ˆ , x) for the computation of the interpolating “quasi-PDFs” given by Eq. (4.132), we note the consistency with the observation made in Ref. [24] for the quasi-PDFs at δ = 0 (IFD) that there exists considerable difference between the shapes of the LFD result and the IFD quasiPDF result for the light mesons. Our results indicate that the slow approach to the LFD-like results may be remedied by varying the interpolation parameter δ as well as the interpolating longitudinal meson momentum r−ˆ . For the future work, one may explore such idea to search for the “sweet spot” of δ and r−ˆ to attain most effective computation with the least sensitive numerical errors in getting the LFD result. Extending the Wick rotation technique to the interpolating time x +ˆ , the idea of searching for the “sweet spot” may be applicable to the usual lattice formulation in the Eunclidean space. This would be in good contrast to the recent application

170

4

Interpolation of Quantum Chromodynamics in 1+1 Dimension

of the present interpolating formulation to the two-dimensional φ 4 theory using the discretization technique in Minkowsky space consistent with the discrete lightcone quantization (DLCQ) approach [21,45]. It will be interesting to explore both “Euclidean” and “Minkowsky” numerical approaches implementing the interpolating formulation between IFD and LFD.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24.

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A

QED Appendix

A.1

Fermion Propagator in the Position Space

The Feynman propagator in the position space is given by  F (x) = i

μ

e−iqμ x d4 q .  (2π )4 q 2 − m 2 + iε

(A.1)

In the interpolation form, it can be written as  2 d q⊥ dq−  dq+  F (x) = i (2π )4 e−i(q+ x

 + +q

x −

 − +q⊥ x⊥ )

. × 2 2 2 2 Cq+  q−  − Cq−  + 2Sq+  − q⊥ − m + iε

(A.2)

Solving for the quadratic expression in the denominator of the Feynman propagator in order to separate the two distinct poles, we have the two poles of q+  (a)

 q+  − iε ,  = A+

(b) q+ 

(A.3) 

= −B+  + iε ,

where the real part of the two poles is defined as    2 2 2 q−  + C q⊥ + m Sq−  A+ + ,  ≡− C  C   2 2 2 q−  + C q⊥ + m Sq−  B+ + ,  ≡ C C © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 C.-R. Ji, Relativistic Quantum Invariance, Lecture Notes in Physics 1012, https://doi.org/10.1007/978-981-19-7949-1

(A.4)

(A.5) (A.6) 173

174

QED Appendix A

and the imaginary part of the poles is given by ε ε ≡  .  2 + C q2 + m 2 2 q−  ⊥

(A.7)

For the later purpose, we define the square root part as 

Q+ ≡

   2 2 2 q−  + C q⊥ + m .

(A.8)

From the expressions in Eqs. (A.5) and (A.6), we see that for any sign of q−  , A+  is always positive and corresponds to the positive energy solution, while −B+  is always negative and corresponds to the negative energy solution. Therefore, we see  the pole structure in the q+  complex plane is that A+  − iε , located in the fourth  quadrant, and −B+  + iε , located in the second quadrant. In order to perform the integration in the “energy” variable q+  in Eq. (A.2), we use   > 0, x + < 0, the Cauchy residue theorem. We may consider three possibilities: x +  and x + = 0. We now analyze these three situations case by case.  > 0, this implies that in order to have a converging exponential factor in For x + the integrand, we must have Imq+  < 0. This means that the semicircle C R that closes the contour must be located in the lower half of the complex q+  plane, in a clockwise  direction. A closed contour in this sense encloses the pole q+  = A+  − iε . We thus have for this case:

 +

e−iq+ x dq+      q + B − iε  (2π ) C q+  − A+  + iε   + +

  + +R dq e−iq+ x  +    = lim  q + B − iε  R→∞  − A+  + iε   −R (2π ) C q+ + +    + e−iq+ x dq+    .  +  q + B − iε   − A+  + iε   C R (2π ) C q+ + +

(A.9)

The left-hand side of Eq. (A.9) is (by Cauchy’s theorem) equal to −iRes(A+ − iε ), where the minus sign is due to the clockwise direction of the closed contour. Since the arc contribution in the limit R → ∞ goes to zero, in this limit we have +∞ −∞

 +

e−iq+ x dq+      q + B − iε  (2π ) C q+  − A+  + iε   + +  +

= −i

e−i A+ x ,  C A+  + B+ 



(x + > 0).

(A.10)

 For x + < 0, this implies that in order to have a converging exponential factor in the integrand, we must have Imq+  > 0. This means that the semicircle C R that

QED Appendix A

175

closes the contour must now be located in the upper half of the complex q+  plane, in a counterclockwise direction. A closed contour in this sense encloses now the pole  q+  = −B+  + iε . We thus have for this case:

 +

e−iq+ x dq+      q + B − iε  (2π ) C q+  − A+  + iε   + +

  + +R dq e−iq+ x  +     = lim  q + B − iε  R→∞  − A+  + iε   −R (2π ) C q+ + +    + e−iq+ x dq+    .  +  q + B − iε   − A+  + iε   C R (2π ) C q+ + +

(A.11)

The left-hand side of Eq. (A.11) is (by Cauchy’s theorem) equal to +iRes(−B+ + iε ), where the plus sign now is due to the counterclockwise direction of the closed contour. Since the arc contribution in the limit R → ∞ goes to zero, in this limit we now have +∞ −∞

 +

e−iq+ x dq+      q + B − iε  (2π ) C q+  − A+  + iε   + +  +

= +i

ei B+ x , C −B+  − A+  



(x + < 0).

(A.12)

In this last expression, we have already dropped the ε in the result after the q+  integration and put a reminder that this result is now valid for the specific case of  x+ < 0.  = 0, the main converging factor in the integrand becomes one, that is, For x + 0 e = 1. We have therefore 1 dq+       (2π ) C q+ q+  − A+  + iε  + B+  − iε

 +R dq 1  +    = lim   R→∞ q+  − A+  + iε  + B+  − iε −R (2π ) C q+   1 dq+    .  + (A.13)  q + B − iε   − A+  + iε   C R (2π ) C q+ + + Although for this case the exponential factor in the integrand is absent, the denominator of the integrand has enough powers in q+  to make the arc contribution go to zero when R → ∞. Therefore, closing the contour from below, that is, with C R in  the clockwise direction. This encloses the pole q+  = A+  − iε , and we get

176

QED Appendix A

+∞ −∞

=

1 dq+      q + B − iε  (2π ) C q+ − A + iε     + + +

i −i   = −  , (x + = 0). C A+ 2Q +  + B+  

(A.14)

Closing the contour in the counterclockwise direction, we enclose the other pole,  q+  = B+  + iε , and we obtain +∞ −∞

=

1 dq+      +  q + B − iε  (2π ) 2Q q+  − A+  + iε   + +

i +i   = −  , (x +  = 0). + C −B+ − A 2Q   +

(A.15)

Thus, both circulations yield the same answer, as it should and serve as a check for our results. Finally, the overall result for the Feynman propagator is given by F (x)  =

d2 q⊥ (2π )2 

(x + )e

+∞

dq− 1   (2π ) 2Q +

−∞     ⊥ + − −i A +  x +q−  x +q⊥ ·x



+ (−x + )e

    ⊥ + − i B+  x +q−  x +q⊥ ·x

,

(A.16)

where we have made the variable shifts q⊥ → −q⊥ , q−  → −q− ,

(A.17) (A.18)

in the second term, which are possible because the integration ranges for these variables are from −∞ to +∞. When C = 0, however, the denominator of the Feynman propagator is a linear − expression in q+  = q+ = q instead of a quadratic one, thus it has only one pole − qon =

2 + m2 q⊥ ε −i +. 2q + 2q

(A.19)

Thus when q + > 0, the pole is located in the fourth quadrant of the q − complex plane, and to make sure the arc contribution is zero, when x + > 0, one has to close the contour from below, while when x + < 0, one needs to close the contour from above, and it gives no contribution since there is no pole in the upper half plane.

QED Appendix A

177

Similarly, when q + < 0, the pole is located in the second quadrant of the q − complex plane, and to make sure the arc contribution is zero, when x + > 0, one has to close the contour from below, which again gives no contribution because there is no pole in the lower half plane, while when x + < 0, one needs to close the contour from above, catching the pole there. Thus, the light-front-time- (x + -) ordering imposes clear cut on the signs of q + and consequently on q − due to the sign correlation between them, so that when x + > 0, q + and q − must both be positive, on the other hand when x + < 0, q + and q − must both be negative. As a result, the integration ranges of the momentum variables in the two time-orderings are not both (−∞, +∞) , but are (0, +∞) for the forward time and (−∞, 0) for the backward time. Doing the pole integration, we get in the light-front, F (x)

⎧ +∞   − + + − dq + d2 q⊥ ⎨ + −i qon x +q x +q⊥ ·x⊥ (x = ) e (2π )2 ⎩ (2π )|2q + | 0 ⎫ 0 ⎬ +   dq − + + − ⊥ +(−x + ) e−i qon x +q x +q⊥ ·x + ⎭ (2π )|2q | −∞ ⎧ +∞  2   − + + − dq + d q⊥ ⎨ + −i qon x +q x +q⊥ ·x⊥ (x ) = e (2π )2 ⎩ (2π )|2q + | 0 ⎫ +∞ ⎬ +   dq − x + +q + x − +q ·x⊥ i qon ⊥ . +(−x + ) e ⎭ (2π )|2q + | 

(A.20)

0

Using the interpolating step function given in Eq. (3.10) in Sect. 3.2.1.1, we can combine the results and write as follows  F (x) =

d2 q⊥ (2π )2

+∞ −∞

  1 dq−  μ  μ   −i A  i B + μ x + (−x + μx (q− , ) (x )e )e   (2π ) 2Q + (A.21)

where we have introduced the shorthand notation  C→0  −   + A  , q1 , q2 , q−  → qon , q1 , q2 , q μ ≡ A+  C→0  −   + B  , q1 , q2 , q−  → qon , q1 , q2 , q μ ≡ B+

(A.22) (A.23)

178

QED Appendix A

The explicit form of the Feynman propagator is given by [1]:         1 m δ(x 2 ) + √  x 2 J1 m x 2 − i N1 m x 2 4π 8π x 2      im −  −x 2 K 1 m −x 2 √ 4π 2 −x 2  √ ⎫ ⎧ 2 + iε ⎬ ⎨ m −x K 1   im 1 δ x2 + , (A.24) =− √ ⎭ 4π ⎩ π −x 2 + iε

F (x) = −

the Bessel, Neumann and Hankel where J1 (z), N1 (z) and K√1 (z) are, respectively, √ 2 2 functions of order 1, and −x + iε = i x , for x 2 > 0. Note here that the argument of the Hankel function is imaginary. To derive the fermion propagator we need to apply the Dirac operator on it,     − ⊥ SF (x) = iγ + ∂+  + iγ ∂−  + iγ · ∂ ⊥ + m F (x),

(A.25)

where F (x) is given by Eq. (A.21). We obtain SF (x) 

+∞

 dq−  μ   + −i A μx  (q )  − (x ) (A + m) e  + (2π )(2Q ) −∞    μ  μ  μ    . + (−x + ) (−B + m) ei Bμ x + iγ + δ(x + ) e−i Aμ x − ei Bμ x

=

d 2 q⊥ (2π )2

(A.26)

Then, going back to the the momentum space, we note that when (q−  ) = 1,  + goes from −∞ to +∞, the two δ(x ) terms cancel each i.e. the integration of q−  + ), ) = (q other exactly when a spatial integration is performed, while for (q−  they don’t cancel and an “instantaneous contribution” is leftover. We finally get 

 μ

F (q) ≡ i SF (q) = i d4 x SF (x) eiqμ x   ⎧ 1 Qa + m − Q b + m ⎪ ⎪ , (C = 0), + ⎨ +  q −Q −q+   − Q b+   + a+ = 2Q 1 q +m γ+ ⎪ ⎪ + +, (C = 0), ⎩ + −on − 2q q − qon 2q

(A.27)

where  Q a ≡ A  and  Q b ≡ B  are defined in Eqs. (A.22) and (A.23), respectively, while Q a +  and Q b+  are defined in Eqs. (A.5) and (A.6), respectively.  ≡ A+  ≡ B+ Thus, we get the time-ordered propagators given in the main text.

QED Appendix A

A.2

179

Derivation of Interpolating QED Hamiltonian

In this Appendix, we show how the Hamiltonian in Sect. 3.2.2.4 is derived, and how the consistency with the LFD formulation presented by Kogut and Soper [2] can be seen. We start from the interpolating QED Hamiltonian density, as given in Eq. (3.78),    μ ¯  H = ψ¯ −iγ j ∂ j − iγ − ∂− ψ  + m ψ + e A μ ψγ +

1  j −  + + ∂+ ∂+ F μν F Aj − F  A− . μ ν −F 4

(A.28)

Consider the first two terms of Eq. (A.28), i.e. fermion and fermion—gauge boson interaction terms. According to the definition of free and constrained photon fields, Eqs. (3.82) and (3.83), the first two terms can be written as     − j μ ˜ ¯  Hf = ψ¯ −i∂− ψ + eφ J + .  γ − i∂ j γ + m ψ + e A μ ψγ

(A.29)

Separating ψ = ψ˜ + δC0 ϒ for any general interpolation angle, we write      − j Hf = ψ¯˜ + δC0 ϒ¯ −i∂− ψ˜ + δC0 ϒ  γ − i∂ j γ + m      ¯˜ + δ ϒ¯ γ  μ ˜ + e A˜  ψ + δC0 ϒ + eφ J + . μ ψ C0

(A.30)

The ϒ field exists only in the exact light-front, where we can make use of the identity given in Ref. [2]; ψ¯

     i∂ j − e A j γ j − m ψ = −2ψ¯ i∂− γ − ψ.

(A.31)

Recalling in the light-front we can separate the fermion field into the free one and constrained one ψ = ψ+ + ψ− = ψ˜ + + ψ− with γ + ψ− = γ − ψ+ = 0, and ψ− = ψ˜ − + ϒ with ψ˜ − and ϒ given by Eqs. (3.53) and (3.54), respectively, one realizes that identity (A.31) consists of four different identities     ˜ ψ¯˜ i∂ j γ j − m ψ˜ = −2ψ¯˜ i∂− γ − ψ,

(A.32)

    ¯˜ j ϒ + ϒγ ¯ j ψ˜ = −2ϒ¯ i∂− γ − ϒ, − e A j ψγ

(A.33)

    ψ¯˜ i∂ j γ j − m ϒ + ϒ¯ i∂ j γ j − m ψ˜     ˜ = −ψ¯˜ i∂− γ − ϒ˜ − ϒ¯ i∂− γ − ψ,

(A.34)

180

QED Appendix A

and

¯˜ j ψ˜ = −ψ¯˜ i∂ γ −  ϒ˜ − ϒ¯ i∂ γ −  ψ. ˜ − e A j ψγ (A.35) − −    j The term ϒ¯ i∂ j − e A j γ − m ϒ on the left-hand side vanishes due to γ +2 = 0. Noticing the fact that the transverse and mass components of the ϒ¯ − ϒ terms vanish, we can expand Eq. (A.30) as    − j ˜ Hf = ψ¯˜ −i∂−  γ − i∂ j γ + m ψ      + δC0 ψ¯˜ −i∂− γ − ϒ + ϒ¯ −i∂− γ − ψ˜     + ψ¯˜ −i∂ j γ j + m ϒ + ϒ¯ −i∂ j γ j + m ψ˜   + ϒ¯ −i∂− γ − ϒ  ¯˜ j ϒ + e A˜ ϒγ ¯ j ψ˜ +e A˜ ψγ j

j

 ¯˜  μ ˜ ψ + eφ J + . + e A˜  μ ψγ

(A.36)

   − j ˜ which The first term is the free Hamiltonian Hf,0 = ψ¯˜ −i∂−  γ − i∂ j γ + m ψ,   can be reduced in the LF to Hf,0 = ψ¯˜ i∂− γ − ψ˜ due to identity (A.32). The second and third lines of Eq. (A.36) cancel each other due to identity (A.34). The fifth line of Eq. (A.36) is equal to −2 times the fourth line due to identity (A.33). Thus Eq. (A.36) reduces to    ¯˜  − j μ ˜ ˜ + e A˜  ψ Hf = ψ¯˜ −i∂−  γ − i∂ j γ + m ψ μ ψγ    + δC0 ϒ¯ i∂− γ − ϒ + eφ J + . (A.37) The rest of the Hamiltonian is the gauge boson part Hg =

1  j −  + + ∂+ ∂+ F μν F Aj − F  A− , μ ν −F 4

(A.38)

and similarly we want to separate it into the free part and the constraint part. Hg = Hgfree + Hgconstraint ,

(A.39)

1 ˜ j −  + ˜ j − F˜ + ˜− ∂+ ∂+ F μν F˜ A A , μ ν − F˜ 4

(A.40)

where Hgfree =

μ ν is defined in terms of the free photon fields as given in Eq. (3.82). and F˜    +  μ μ  μ+ ˜ ˜ φ, we find Using A μ = A μ + g μ φ and A = A + g

QED Appendix A

181

Hgconstraint = Hg − Hgfree  1   − j −S∂ + φ∂− =  φ + C∂ φ∂−  φ + C∂ φ∂ j φ 2

(A.41)

with all other terms vanish upon applying the interpolation gauge condition ∂ j A j = − C1 ∂−  A− . Using integration by parts,  1  +   − φ S∂ ∂−  − C∂ ∂−  + C∂ j ∂ j φ 2  1  2 = φ C∂ 2⊥ + ∂−  φ 2 1  = − eφ J + , 2

Hgconstraint =

according to the definition of the constraint photon field φ in Eq. (3.83). Thus 1  Hg = Hgfree − eφ J + , 2

(A.42)

(A.43)

with Hgfree given by Eq. (A.40). Adding two pieces together, we can identify the free and interaction Hamiltonian H = Hf + Hg = H0 + V ,

(A.44)

where    − j ˜ H0 = ψ¯˜ −i∂−  γ − i∂ j γ + m ψ +

1 ˜ j −  + ˜ j − F˜ + ˜− ∂+ ∂+ F μν F˜ A A , μ ν − F˜ 4

(A.45)

and   1  ¯˜  μ ˜ ψ + δC0 ϒ¯ i∂− γ − ϒ + eφ J + . V = e A˜  μ ψγ 2

(A.46)

Thus, we get the interpolating QED Hamiltonian density as shown in the main text Eqs. (3.80) and (3.81).

A.3

Sum of the Interpolating Time-Ordered Fermion Propagators

In this Appendix, we show how the addition of the two time-ordered propagators gives correctly the covariant one. We start with the expressions given in Eqs. (3.94) and (3.95).

182

QED Appendix A

a + b   Qa + m 1 − Q b + m = −  q+ q+ 2Q +  − Q a+  + Q b+     C Q a + Cm 1 −C Q b + Cm = −    + + 2Q + Cq+ Cq+  + Sq−  − Q  + Sq−  + Q

(A.47)

where we have used (3.96) and (3.97). Using the relationship between superscripts and subscripts it can be written as: a + b   C Q a + Cm 1 −C Q b + Cm = −      2Q + q+ − Q+ q+ + Q+       − − ⊥ ⊥ γ+ C Q a+ Q b+ −γ +  + γ · q⊥ + m  + γ · q⊥ + m  + γ q−  + γ q− = −      2Q + q+ − Q+ q+ + Q+ (A.48) where it is worth paying attention to the fact that the sign is different between q and qb and we have replaced all qb ’s with q’s in the second line. The above equation can be further simplified: a + b

               ⊥ + + + + + − Q a+ (q + + Q+ ) − (q + − Q+ )  + γ · q⊥ + m  (q + Q ) + Q b+  (q − Q ) + γ q− C γ =   2  2 2Q + (q + ) − (Q + )          + q + (Q + (Q + γ− ⊥ γ + Q ) + Q − Q ) + 2Q q−  + γ · q⊥ + m     a+ b+ a+ b+ C   =   2 2Q + (q + ) − q 2 + Cq2 + Cm 2  −

⊥

       2Q +  −2Sq−  +    ⊥ + γ+ + 2Q + γ− Cq+ Q q−  + Sq−   + γ · q⊥ + m C C C      =   Cq + Sq  − Cq2 − Cm 2 + − 2Q + q+   − q−  Sq − Cq + − ⊥     − ⊥ Cγ + q+  + C γ q−  + γ · q⊥ + m =   − ⊥ 2 Cq + q+  + Cq−  q + Cq · q⊥ − Cm q + m = 2 q − m2  γ+

(A.49)

Thus, the total result is proved to be consistent with the Feynman propagator.

A.4

Non-collinear Scattering/Annihilation, 0 < θ < π, in e+ e− → Two Scalar Particles

In this Appendix, the non-collinear helicity amplitudes are computed by varying the center-of-mass angle θ for the production of two scalar particles in e+ e− annihilation process. As discussed in Sect. 3.3, for the non-collinear kinematics, the same amplitude can correspond to either the “on-mass-shell propagating contribution” or the

QED Appendix A

183

“instantaneous fermion contribution” in LFD depending on the region of the scatλ1 ,λ2 corresponds to the “instantaneous tering angle. For example, the amplitude Ma,t fermion contribution” in LFD for the region θ < θc,t while it corresponds to the “on-mass-shell propagating contribution” √ for the region θ > θc,t , where θc,t is given by Eq. (3.103). For E 0 = 2m e and Pe = 3m e , θc,t = π/6 and θc,u = 5π/6 from Eqs. (3.103) and (3.104), respectively. To demonstrate the existence of this critical angle only at LFD, we may take a look closely at each light-front helicity amplitude and contrast its behavior with the ones off the value of δ = π/4. As an example, in Fig. A.1, we show the result of the angular distribution for the ++ helicity ampli+,+ +,+ +,+ +,+ +,+ tudes: (a) Ma,t (b) Ma,t +Mb,t and (c) Ma,u +Mb,u at the exact light-front, +,+ i.e. δ = π/4. For Ma,t shown in Fig. A.1, the left side of θc,t = π6 ≈ 0.523599 (i.e. θ < θc,t ) is the “instantaneous fermion contribution” and thus the amplitude is zero as γ+ ¯ ↑ γ + u ↑ = 0 [3]. expected from the light-front instantaneous propagator, 2q + , due to v On the other hand, the right side of the critical angle (i.e. θ > θc,t ) is the “on-mass+,+ . These two distinguished contributions for shell propagating contribution” for Ma,t +,+ θ < θc,t and θ > θc,t yield a dramatic “cliff” feature for Ma,t as shown in Fig. A.1a. Due to the sign change of the intermediate fermion momentum qb = −qa = −q for +,+ the other time-ordered amplitude Mb,t , the angle regions for the “instantaneous fermion contribution” and the “on-mass-shell propagating contribution” swap in +,+ +,+ Mb,t with respect to Ma,t ; i.e. the right side (θ > θc,t ) becomes the “instantaneous fermion contribution” and the left side (θ < θc,t ) becomes the “on-mass-shell +,+ +,+ , while it was the other way around for Ma,t as propagating contribution” for Mb,t +,+ +,+ , discussed above. The addition of the two time-ordered amplitudes, Ma,t +Mb,t is shown in Fig. A.1b. Since the “instantaneous fermion contribution” for the ++ helicity amplitudes in LFD is always zero due to v¯ ↑ γ + u ↑ = 0, the “on-mass-shell +,+ propagating contribution” for Mb,t is rather easily figured out by subtracting the curve depicted in Fig. A.1a from the curve depicted in Fig. A.1b. Essentially the same procedure of obtaining the t-channel amplitude can be applied to the u-channel amplitude by exchanging the two final state scalar particles, i.e. p3 ↔ p4 . Thus, q becomes p1 − p4 in the u-channel while it was p1 − p3 in the t-channel and the +,+ +,+ result of Ma,u +Mb,u is obtained as shown in Fig. A.1c. To exhibit that the “instantaneous fermion contribution” is the unique feature only in LFD (δ = π/4), we take a look at the interpolation angle δ dependence of +,+ the amplitude Ma,t by slightly varying the scattering angle θ around the critical +,+ at (a) θ = θc,t − 0.01 ≈ angle θc,t . In Fig. A.2, we show the δ-dependence of Ma,t 0.513599 (b) θ = θc,t + 0.01 ≈ 0.533599 and (c) θ = θc,t ≈ 0.523599. These three values of the angle θ chosen for Fig. A.2 correspond to slightly left of the “cliff,” at the “cliff,” and slightly right of the “cliff” in Fig. A.1a, respectively. Since the values +,+ of the amplitude Ma,t dramatically change around the critical angle θc,t from 0.0 on the left (θ < θc,t ) to around 2.0 on the right immediately passing the critical angle θc,t as depicted in Fig. A.1a, we should be able to see the corresponding dramatic change also in Fig. A.2. We see indeed this dramatic change in Fig. A.21 on top of the abrupt change of the helicity amplitude due to the bifurcation of two helicity

1 Note

that the scale of Fig. A.2 is doubled from Fig. A.2a, c to fit them all in one collective figure of Fig. A.2.

184

QED Appendix A

+,+ +,+ +,+ +,+ Fig. A.1 ++ annihilation helicity amplitudes for: (a) Ma,t , (b) Ma,t + Mb,t and (c) Ma,u + +,+

Mb,u

branches discussed above in the collinear (θ = π ) helicity amplitudes as well as in our previous work [4] extensively, one in the side of IFD and the other in the side of LFD, divided by the critical interpolating angle δc ≈ 0.713724 discussed below +,+ Eq. (3.107) and depicted in Fig. 3.10. In Fig. A.2a, the value of the amplitude Ma,t at the right end (δ = π4 ) is 0.00 while the value for δc < δ < π4 (not including δ = π4 ) is around 1.0 and falls off to get linked to the smoothly behaving curve for the region δ < δc that belongs to the helicity branch on the IFD side. In Fig. A.2b, however, the +,+ value of the amplitude Ma,t at the right end (δ = π4 ) is around 2.00 while the value π for δc < δ < 4 (again not including δ = π4 ) is still around 1.0 and again falls off to get linked to the smoothly behaving curve for the region δ < δc that belongs to the +,+ helicity branch on the IFD side. Thus, the helicity amplitude Ma,t doesn’t change much except its value at δ = π/4 or at LFD. In the region δc < δ ≤ π4 that belongs to the helicity branch on the LFD side, one can see the dramatic change of the helicity amplitude at the right end point δ = π/4, i.e. only at LFD but not anywhere else. This clearly demonstrates that the “instantaneous fermion contribution” exists only in the LFD.

QED Appendix A

185

+,+ Fig. A.2 Interpolation angle dependence of ++ annihilation helicity amplitudes Ma,t for: (a) θ < θc,t , (b) θ > θc,t , and (c) θ = θc,t

Similarly, in Fig. A.3, we show the δ-dependence of the other time-ordered ++ +,+ helicity amplitude Mb,t at (a) θ = θc,t − 0.01 ≈ 0.513599 (b) θ = θc,t + 0.01 ≈ 0.533599 and (c) θ = θc,t ≈ 0.523599. As discussed earlier, the angle regions for the “instantaneous fermion contribution” and the “on-mass-shell propagating contribu+,+ +,+ tion” swap in Mb,t with respect to Ma,t due to the sign change of the intermediate +,+ , i.e. the right side (θ > θc,t ) and the fermion momentum qb = −qa = −q for Mb,t left side (θ < θc,t ) become the “instantaneous fermion contribution” and the “on+,+ . Since the “instantaneous fermion mass-shell propagating contribution” for Mb,t contribution” for the ++ helicity amplitudes in LFD is always zero (again due to +,+ +,+ v¯ ↑ γ + u ↑ = 0), we now should be able to see Mb,t = 0 for θ > θc,t while Mb,t = 0 for θ < θc,t in LFD. We indeed see this expected LFD result in Fig. A.3 as the value +,+ at δ = π/4 turns out to be exactly 0.0 for θ = θc,t + 0.01 in Fig. A.3b while of Mb,t it is around 2.0 for θ = θc,t − 0.01 in Fig. A.3a. This dramatic change at LFD again clearly demonstrates that the “instantaneous fermion contribution” exists only in the +,+ LFD. For δc < δ < π4 (not including δ = π4 ), however, the value of Mb,t is around 1.0 and rises up to get linked to the smoothly behaving curve for the region δ < δc that belongs to the helicity branch on the IFD side. As shown in Fig. A.3, the helicity +,+ amplitude Mb,t doesn’t change much except its value at δ = π/4 or at LFD.

186

QED Appendix A

+,+ Fig. A.3 Interpolation angle dependence of ++ annihilation helicity amplitudes Mb,t for: (a) θ < θc,t , (b) θ > θc,t , and (c) θ = θc,t

Even more distinct feature of LFD can be noticed in Fig. A.4 where we present the +,− +,− +,− +,− +− helicity amplitudes in LFD (δ = π/4) (a) Ma,t , (b) Mb,t , (c) Ma,t + Mb,t +,− +,+ +,− and (d) Ma,u + Mb,u . In contrast to Ma,t discussed above, the “instantaneous +,− doesn’t vanish due to v¯ ↓ γ + u ↑ = 0[3] and thus the fermion contribution” to Ma,t amplitude shown in Fig. A.4a gets the singularity from the light-front instantaneous γ+ + propagator, 2q = 0 occurs at θ = θc,t . The singularity from the same origin + , as q +,− but with the opposite sign due to qb = −qa = −q for Mb,t shown in Fig. A.4b +,− +,− + Mb,t cancels the singularity shown in Fig. A.4a and the net result of Ma,t is finite and well-behaved as shown in Fig. A.4c. It is interesting to note that the singularities in different light-front time-ordered processes corroborate each other to cancel themselves and make the Lorentz invariant amplitude finite and well-behaved. +,− +,− The crossed channel total amplitude Ma,u + Mb,u is of course also finite and well-behaved with the apparent symmetry θ → π − θ between the t-channel and the u-channel as shown in Figs. A.4c and d.

QED Appendix A

187

+,− +,− +,− +,− Fig. A.4 +− annihilation helicity amplitudes for: (a) Ma,t , (b) Mb,t (c) Ma,t + Mb,t and +,− +,− (d) Ma,u + Mb,u

−,+ −,+ −,+ −,− −,+ The helicity −+ and −− amplitudes, Ma,t , Mb,t , Ma,u , Mb,u , Ma,t , −,− −,− −,− Mb,t , Ma,u , Mb,u , have all been computed as well, and Eq. (3.108) based on the parity conservation has been verified explicitly among all the helicity amplitudes for the present e+ e− scattering/annihilation process. Thus, the helicity −+ and −− amplitudes can be rather easily figured out once the helicity ++ and +− amplitudes are given. In Figs. A.5, A.6 and A.7, we provide the whole landscape of the interpolation angle (δ) dependence for the angular distributions of the helicity ++ and +− amplitudes at CMF (i.e. P z = 0). In each and every figure, the critical interpolation angle δc which separates the IFD side and the LFD side of helicity branches is denoted by a thin boundary sheet at δ = δc ≈ 0.713724 in CMF (P z = 0). In +,+ Fig. A.5, we show the angular distribution of the helicity ++ amplitudes (a) Ma,t +,+ +,+ +,+ (b) Mb,t (c) Ma,u (d) Mb,u . Similarly, in Fig. A.6, we show the angular distri+,− +,− +,− +,− (b) Mb,t (c) Ma,u (d) Mb,u . At bution of the helicity +− amplitudes (a) Ma,t δ = π/4 (LFD), the profiles of the “instantaneous fermion contribution” and the “onmass-shell propagating contribution” depicted in Figs. A.1a and A.4a are visible in Figs. A.5a and A.6a, respectively. Adding both t-channel and u-channel time-ordered amplitudes all together, we get the results shown in Fig. A.7. In Fig. A.7a, b, the sum +,+ +,+ +,+ +,+ of ++ helicity amplitude Ma,t + Mb,t + Ma,u + Ma,u and the sum of +−

188

QED Appendix A

Fig. A.5 Angular distribution of the helicity amplitude ++ for (a) t-channel time-ordering +,+ +,+ process-a, Ma,t (b) t-channel time-ordering process-b, Mb,t (c) u-channel time-ordering +,+ +,+ process-a, Ma,u (d) u-channel time-ordering process-b, Mb,u +,− +,− +,− +,− helicity amplitude Ma,t + Mb,t + Ma,u + Ma,u are, respectively, shown. The corresponding amplitude squares (or probabilities) are also shown in Fig. A.7c, d, respectively. Here, we note a remarkable correspondence between the IFD and LFD profiles of the ++ amplitude in Fig. A.7a and the LFD and IFD profiles of the +− amplitude in Fig. A.7b modulo the overall signs, respectively. This remarkable correspondence between the IFD and LFD profiles is further self-evident in Figs. A.7c and d as the overall sign doesn’t matter in the amplitude square or the probability. As discussed earlier, the LF helicity of the particle moving in the −ˆz direction is opposite to the Jacob-Wick helicity defined in the IFD. Since the incident e− e+ annihilation takes place along the z-axis and the positron (e+ ) is moving in the −ˆz direction, the swap of the helicity between the IFD and LFD for the positron can be understood as we see the IFD/LFD profile correspondence in Fig. A.7.

QED Appendix A

189

Fig. A.6 Angular distribution of the helicity amplitude +− for (a) t-channel time-ordering process+,− +,− a, Ma,t (b) t-channel time-ordering process-b, Mb,t (c) u-channel time-ordering process-a, +,− +,− Ma,u (d) u-channel time-ordering process-b, Mb,u

To examine the frame dependence of the whole landscape, we have computed all the helicity amplitudes discussed above with the nonzero center of momentum (P z = 0) as well. In particular, we took a large enough center of momentum to pass the helicity boundaries given by Eqs. (3.106) and (3.107) that we have discussed extensively in Sect. 3.3.1. In Figs. A.8, A.9 and A.10, we show the results for P z = +15m e while we do for P z = −15m e in Figs. A.11, A.12 and A.13. In these figures, the whole landscapes of the interpolation angle (δ) dependence for the angular distributions of the helicity ++ and +− amplitudes are presented for the boosted frames with P z = +15m e and P z = −15m e . As we have shown in the collinear case presented in Sect. 3.3.1, no helicity boundaries exist between IFD and LFD in the frame with P z = +15m e while there are two distinct helicity boundaries, one from electron and the other from positron [see Eqs. (3.106) and (3.107), respectively], between IFD and LFD for P z = −15m e .

190

QED Appendix A

+,+ +,+ Fig. A.7 Angular distribution of the helicity amplitudes and probabilities: (a) Ma,t + Mb,t + +,+ +,+ +,− +,− +,− +,− +,+ +,+ +,+ +,+ Ma,u + Ma,u (b) Ma,t + Mb,t + Ma,u + Ma,u (c) |Ma,t + Mb,t + Ma,u + Ma,u |2 +,− +,− +,− +,− 2 (d) |Ma,t + Mb,t + Ma,u + Ma,u |

For P z = +15m e , the angular distribution of the helicity ++ amplitudes is shown +,+ +,+ +,+ +,+ in Fig. A.8(a) Ma,t (b) Mb,t (c) Ma,u (d) Mb,u , while the angular distribution +,− +,− +,− (b) Mb,t (c) Ma,u of the helicity +− amplitudes is shown in Fig. A.9(a) Ma,t +,− (d) Mb,u . The profiles of the “instantaneous fermion contribution” and the “onmass-shell propagating contribution” at δ = π/4 (LFD) discussed at CMF (P z = 0) survive invariantly although significant changes for the region 0 ≤ δ < π/4 are apparent in the landscape without any helicity boundaries as expected in this boosted frame with P z = +15m e . The net results adding both t-channel and u-channel timeordered amplitudes all together are shown in Fig. A.10. In Figs. A.10a, b, the sum +,+ +,+ +,+ +,+ of ++ helicity amplitude Ma,t + Mb,t + Ma,u + Ma,u and the sum of +− +,− +,− +,− +,− helicity amplitude Ma,t + Mb,t + Ma,u + Ma,u are, respectively, shown. The corresponding amplitude squares (or probabilities) are also shown in Figs. A.10c and d, respectively.

QED Appendix A

191

Fig. A.8 Angular distribution of the helicity amplitude ++ for (a) t-channel time-ordering process+,+ +,+ a, Ma,t (b) t-channel time-ordering process-b, Mb,t (c) u-channel time-ordering process-a, +,+ +,+ Ma,u (d) u-channel time-ordering process-b, Mb,u in a positive momentum frame

For P z = −15m e , the angular distribution of the helicity ++ amplitudes is shown +,+ +,+ +,+ +,+ in Fig. A.11(a) Ma,t (b) Mb,t (c) Ma,u (d) Mb,u , while the angular distribu+,− +,− (b) Mb,t (c) tion of the helicity +− amplitudes is shown in Fig. A.12(a) Ma,t +,− +,− Ma,u (d) Mb,u . The LFD profiles of the “instantaneous fermion contribution" and the “on-mass-shell propagating contribution” are again invariant regardless of P z values (P z = +15m e , 0, −15m e ) exhibiting the boost invariance of the helicity amplitudes in LFD. For the region 0 ≤ δ < π/4, however, there appear two critical interpolating angles at δ = δc,e− ≈ 0.55062 and δ = δc,e+ ≈ 0.784165, which can be estimated from Eqs. (3.106) and (3.107), respectively. Except the LFD profiles, the whole landscapes of angular distributions are dynamically varied both for 0 ≤ δ < δc,e− ≈ 0.55062 and δc,e− ≈ 0.55062 < δ < δc,e+ ≈ 0.784165 depending on the reference frames (P z = +15m e , 0, −15m e ). The net results adding both t-channel and u-channel time-ordered amplitudes all together are shown in Fig. A.13. In Figs. A.13a +,+ +,+ +,+ +,+ and b, the sum of ++ helicity amplitude Ma,t + Mb,t + Ma,u + Ma,u and the +,− +,− +,− +,− sum of +− helicity amplitude Ma,t + Mb,t + Ma,u + Ma,u are, respectively,

192

QED Appendix A

Fig. A.9 Angular distribution of the helicity amplitude +− for (a) t-channel time-ordering process+,− +,− a, Ma,t (b) t-channel time-ordering process-b, Mb,t (c) u-channel time-ordering process-a, +,− +,− Ma,u (d) u-channel time-ordering process-b, Mb,u in a positive momentum frame

shown. The corresponding amplitude squares (or probabilities) are also shown in Figs. A.13c and d, respectively. Finally, Fig. A.14 shows the sum of the ++ and +− helicity amplitude squares which is the half of the total probability sum including −+ and −− helicity amplitude squares in all three reference frames (P z = +15m e , 0, −15m e ) discussed above. Although the individual helicity amplitude squares in LFD (δ = π/4) are independent of the reference frames, the individual helicity amplitude squares for 0 ≤ δ < π/4 varied depending on the reference frames as we have seen in Figs. A.7c, +,+ +,+ +,+ +,+ 2 A.10c and A.13c for |Ma,t + Mb,t + Ma,u + Ma,u | as well as in Figs. A.7d, +,− +,− +,− +,− 2 A.10d and A.13d for |Ma,t + Mb,t + Ma,u + Ma,u | . For the P z = +15m e frame, there were no helicity boundaries and the individual helicity amplitude squares were same regardless of the δ values as shown in Figs. A.10c and d. However, for the other reference frames with P z = −15m e and P z = 0 (CMF), where there were two (δ = δc,e− ≈ 0.55062 and δ = δc,e+ ≈ 0.784165) boundaries and one (δc ≈ 0.713724) boundary, respectively, each individual helicity amplitude squares

QED Appendix A

193

+,+ +,+ Fig. A.10 Angular distribution of the helicity amplitudes and probabilities: (a) Ma,t + Mb,t + +,+ +,+ +,− +,− +,− +,− +,+ +,+ +,+ +,+ Ma,u + Ma,u (b) Ma,t + Mb,t + Ma,u + Ma,u (c) |Ma,t + Mb,t + Ma,u + Ma,u |2 +,− +,− +,− +,− 2 (d) |Ma,t + Mb,t + Ma,u + Ma,u | in a positive momentum frame

varied significantly across the corresponding helicity boundaries. However, the sum of helicity amplitude squares is completely independent of not only the interpolating angle δ but also the reference frames as it should be. The boost-invariant physical quantity must be of course completely independent of the interpolation angle, regardless of IFD, LFD or any other dynamics in between. While all of these figures (Figs. A.8, A.9, and A.10 for P z = +15m e and Figs. A.11, A.12 and A.13 for P z = −15m e ) were depicted in terms of the CMF angle θ , they all can be also shown in terms of the apparent angle θapp in the boosted frame using the relationship between θapp and θ , i.e. tan θapp =

tan θ sin θ , =  √ γ (β + cos θ ) γ β 1 + tan2 θ + 1

(A.50)

194

QED Appendix A

Fig.A.11 Angular distribution of the helicity amplitude ++ for (a) t-channel time-ordering process+,+ +,+ a, Ma,t (b) t-channel time-ordering process-b, Mb,t (c) u-channel time-ordering process-a, +,+ +,+ Ma,u (d) u-channel time-ordering process-b, Mb,u in a negative momentum frame

 where the γ factor in the boosted frame is given by γ = 1/ 1 − β 2 = 

with β =

 Pz 2E 0  z  P 1+ 2E 0

1+



Pz 2E 0

2

in terms of the total momentum P z in the boosted frame and the

total energy 2E 0 in the CMF. Reversing Eq. (A.50), we get

tan θ =

1+



Pz 2E 0

2

 z| tan θapp + |P tan θ 1 + tan2 θapp app 2E 0 .  z 2 P 2θ 1 − 2E tan app 0

(A.51)

Using Eq. (A.51), we convert Figs. A.8, A.9, A.10, A.11, A.12 and A.13 plotted in the CMF scattering angle θ to Figs. A.15, A.16, A.17, A.18, A.19 and A.20 plotted in terms of the apparent angle viewed from the Lab frame. Likewise, Figs. A.21a and b correspond to Figs. A.14a and b, respectively.

QED Appendix A

195

Fig.A.12 Angular distribution of the helicity amplitude +− for (a) t-channel time-ordering process+,− +,− a, Ma,t (b) t-channel time-ordering process-b, Mb,t (c) u-channel time-ordering process-a, +,− +,− Ma,u (d) u-channel time-ordering process-b, Mb,u in a negative momentum frame

A.5

Boosted e+ e− → γ γ Interpolating Helicity Amplitudes

In this Appendix, similar to what was done in Appendix A.4, we examine the frame dependence of the whole landscape of all the angular distributions of the helicity amplitudes discussed in Sect. 3.4.1 by computing them with nonzero center of momentum (P z = +15m e and P z = −15m e ). In Figs. A.22, A.23, A.24, A.25 and A.26, we show the results for P z = +15m e while we do for P z = −15m e in Figs. A.27, A.28, A.29, A.30 and A.31. As we have seen in Appendix A.4, no helicity boundaries exist between IFD and LFD in the frame with P z = +15m e while there are two distinct helicity boundaries, one from electron and the other from positron (see Eqs. (3.106) and (3.107), respectively) between IFD and LFD for P z = −15m e . The sum of the 16 helicity probabilities for P z = +15m e and P z = −15m e is shown in Fig. A.32, and comparing with Fig. 3.21a shown in Sect. 3.4.1, we can see that the total probability is independent of reference frame, as well as the interpolation angle, as it should be.

196

QED Appendix A

+,+ +,+ Fig. A.13 Angular distribution of the helicity amplitudes and probabilities: (a) Ma,t + Mb,t + +,+ +,+ +,− +,− +,− +,− +,+ +,+ +,+ +,+ 2 Ma,u + Ma,u (b) Ma,t + Mb,t + Ma,u + Ma,u (c) |Ma,t + Mb,t + Ma,u + Ma,u | +,− +,− +,− +,− 2 (d) |Ma,t + Mb,t + Ma,u + Ma,u | in a negative momentum frame

A.6

Boost Dependence in e+ e− → γ γ Interpolating Helicity Amplitudes

In this Appendix, we plot the helicity amplitudes of e+ e− → γ γ , as given by Eq. (3.110) and (3.111), in terms of both the interpolation angle δ and the total momentum P z . As was done in Sect. 3.4.1, we take m = m e , E 0 = 2m e , and instead of looking at the angular distribution, we fix the angle θ to be π/3. The helicity amplitudes for the t-channel, corresponding to Feynman diagram Fig. 3.3, with two time-orderings shown in Fig. 3.4, are presented in Figs. A.33 and A.34, while the u-channel helicity amplitudes are shown in Figs. A.35 and A.36. The probabilities, after summing both time-orderings of both channels, are shown in Fig. A.37, where the last row is the summation over all four final helicity states for each initial state. The total probability, obtained after summing all initial and final helicities, is shown in Fig. A.38 and is independent of boost momentum and interpolation angle.

QED Appendix A

197

Fig. A.14 Sum of ++ and +− helicity probabilities for: (a) P z = +15m e , (b) P z = −15m e , and (c) P z = 0 (CMF)

In these figures, the boundaries of bifurcated helicity branches between IFD and LFD due to the initial electron and positron moving in +ˆz and −ˆz directions given by Eqs. (3.106) and (3.107) are denoted by the blue curves, while the characteristic “J-curve” given by Eq. (3.105) is depicted as the red curve. It is also apparent that the relationship between different helicity amplitudes given by Eq. (3.114) is satisfied, where λ3 and λ4 are the helicities of the outgoing photons while λ1 and λ2 are the incoming electron and positron helicities, respectively. This relationship holds as one can see in Figs. A.33, A.34, A.35 and A.36. Up to an overall sign difference, the upper left 2 by 2 block is the same with the lower right 2 by 2 block, while the upper right block is the same with the lower left block. In the right most column, however, all figures have their signs flipped from their counterparts. For the square of amplitudes shown in Fig. A.37, the same correspondence holds without any sign difference as it should be.

198

QED Appendix A

Fig. A.15 Apparent angular distribution of the helicity amplitude ++ for (a) t-channel a time+,+ +,+ +,+ ordering, Ma,t (b) t-channel b time-ordering, Mb,t (c) u-channel a time-ordering, Ma,u (d) +,+ u-channel b time-ordering, Mb,u in a positive momentum frame

QED Appendix A

199

Fig. A.16 Apparent angular distribution of the helicity amplitude +− for (a) t-channel a time+,− +,− +,− ordering, Ma,t (b) t-channel b time-ordering, Mb,t (c) u-channel a time-ordering, Ma,u (d) +,− u-channel b time-ordering, Mb,u in a positive momentum frame

200

QED Appendix A

Fig. A.17 Apparent angular distribution of the helicity amplitudes and probabilities: +,+ +,+ +,+ +,+ +,− +,− +,− +,− +,+ (a) Ma,t + Mb,t + Ma,u + Ma,u (b) Ma,t + Mb,t + Ma,u + Ma,u (c) |Ma,t + +,+ +,+ +,+ 2 +,− +,− +,− +,− 2 Mb,t + Ma,u + Ma,u | (d) |Ma,t + Mb,t + Ma,u + Ma,u | in a positive momentum frame

QED Appendix A

201

Fig. A.18 Apparent angular distribution of the helicity amplitude ++ for (a) t-channel a time+,+ +,+ +,+ ordering, Ma,t (b) t-channel b time-ordering, Mb,t (c) u-channel a time-ordering, Ma,u (d) +,+ u-channel b time-ordering, Mb,u in a negative momentum frame

202

QED Appendix A

Fig. A.19 Apparent angular distribution of the helicity amplitude +− for (a) t-channel a time+,− +,− +,− ordering, Ma,t (b) t-channel b time-ordering, Mb,t (c) u-channel a time-ordering, Ma,u (d) +,− u-channel b time-ordering,Mb,u in a negative momentum frame

QED Appendix A

203

Fig. A.20 Apparent angular distribution of the helicity amplitudes and probabilities: +,+ +,+ +,+ +,+ +,− +,− +,− +,− +,+ (a) Ma,t + Mb,t + Ma,u + Ma,u (b) Ma,t + Mb,t + Ma,u + Ma,u (c) |Ma,t + +,+ +,+ +,+ 2 +,− +,− +,− +,− 2 Mb,t + Ma,u + Ma,u | (d) |Ma,t + Mb,t + Ma,u + Ma,u | in a negative momentum frame

Fig. A.21 Sum of ++ and +− helicity probabilities for: (a) P z = +15m e and (b) P z = −15m e in terms of apparent angle

204

QED Appendix A

Fig. A.22 Angular distribution of the helicity amplitudes for (a) t-channel a time-ordering and (b) t-channel b time-ordering

QED Appendix A

205

Fig. A.23 Angular distribution of the helicity amplitudes for (a) u-channel a time-ordering and (b) u-channel b time-ordering

206

QED Appendix A

Fig. A.24 Angular distribution of the helicity amplitudes for (a) t-channel and (b) u-channel

QED Appendix A

Fig. A.25 Angular distribution of the helicity amplitudes for t+u amplitudes

207

208 Fig. A.26 Angular distribution of the helicity amplitudes for the probabilities, the figures in the last row are result of summing over all figures above it

QED Appendix A

QED Appendix A

209

Fig. A.27 Angular distribution of the helicity amplitudes for (a) t-channel a time-ordering and (b) t-channel b time-ordering

210

QED Appendix A

Fig. A.28 Angular distribution of the helicity amplitudes for (a) u-channel a time-ordering and (b) u-channel b time-ordering

QED Appendix A

Fig. A.29 Angular distribution of the helicity amplitudes for (a) t-channel and (b) u-channel

211

212

Fig. A.30 Angular distribution of the helicity amplitudes for t + u amplitudes

QED Appendix A

QED Appendix A Fig. A.31 Angular distribution of the helicity amplitudes for the probabilities, the figures in the last row are result of summing over all figures above it

213

214

QED Appendix A

Fig. A.32 e+ e− → γ γ Sum of Helicity Probabilities for: (a) P z = +15m e and (b) P z = −15m e

Fig. A.33 Annihilation amplitudes—t-channel, time-ordering a

QED Appendix A

Fig. A.34 Annihilation amplitudes—t-channel, time-ordering b

215

216

Fig. A.35 Annihilation amplitudes—u-channel, time-ordering a

QED Appendix A

QED Appendix A

Fig. A.36 Annihilation amplitudes—u-channel, time-ordering b

217

218

Fig. A.37 Annihilation probabilities

QED Appendix A

QED Appendix A

Fig. A.38 Total probability for e+ e− → γ γ annihilation process

219

B

QCD1+1 Appendix

B.1

Bogoliubov Transformation for the Interpolating Spinors Between IFD and LFD

In this Appendix, we summarize the interpolating spinors and γ matrices in the 1 + 1 dimensional chiral representation and the Bogoliubov transformation between the free and interacting spinors as well as the corresponding creation/annihilation operators. For the representation of the spinors [4], we adopt the chiral representation (CR), under which the usual γ matrices in IFD for the 1 + 1 dimensions are given by the Pauli matrices:     01 0 1 γ 0 = σ1 = , γ 1 = iσ2 = , 10 −1 0   −1 0 . (B.1) γ 5 = γ5 = −γ0 γ1 = γ 0 γ 1 = −σ3 = 0 1 These can be transformed into standard representation (SR) easily by a transformation matrix   1 1 1 S = S† = √ (B.2) 2 1 −1 through μ

μ

γSR = SγCR S † .

(B.3)

In the SR representation, the free spinors in the rest frame in IFD are typically given by     √ √ 1 0 (0) 1 1 u (0) ( p = 0) = 2 m , v ( p = 0) = 2 m , (B.4) SR SR 0 1 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 C.-R. Ji, Relativistic Quantum Invariance, Lecture Notes in Physics 1012, https://doi.org/10.1007/978-981-19-7949-1

221

222

QCD1+1 Appendix B

(0) where we take the normalization factor u¯ (0) SR u SR = 2m in conformity with the standard textbooks. The corresponding free spinors in the chiral representation are used in this work without denoting the “CR” specification:

  √ 1 u ( p = 0) = S · = 0) = m , 1   √ 1 (0) ( p 1 = 0) = m . v (0) ( p 1 = 0) = S · vSR −1 (0)

1 u (0) SR ( p

1

(B.5)

 The IFD spinors in the moving frame with the energy E p = ( p 1 )2 + m 2 are then obtained as    −η √ 1 e 2 0 η u (0) ( p 1 ) = B(η)u (0) (0) = m 1 0 e2   1 E −p , =  p (B.6) E p + p1 and v (0) (− p 1 ) = B(−η)v (0) (0) =

   1 E p + p , − E p − p1

(B.7) 1

where the usual boost operator B(η) with the rapidity η = tanh−1 Ep p and the longitudinal boost generator K 1 is given by       1 1 ηγ5 . B(η) = exp −iη · K 1 = exp − ησ3 = exp 2 2

(B.8)

In terms of the interpolating momentum variables, the rapidity η can be written [4] as   p +ˆ + p−ˆ η = log (B.9) , m (cos δ + sin δ) where one can note the following equality as well 

 p +ˆ − p−ˆ − η = log . m (cos δ − sin δ)

(B.10)

The boost operator B(η) can then be written in terms of the interpolating momentum variables as ⎛ ⎞ p+ˆ − p−ˆ 0 ⎠, B(η) = ⎝ m(cos δ−sin δ)  (B.11) p+ˆ + p−ˆ 0 m(cos δ+sin δ)

QCD1+1 Appendix B

223

so that the boosted interpolating spinors are given by ⎛ u (0) ( p−ˆ ) = and

p+ˆ − p−ˆ ⎝  cos δ−sin δ p+ˆ + p−ˆ cos δ+sin δ

⎛  (0)

v (− p−ˆ ) =

⎞ ⎠,

p+ˆ + p−ˆ ⎝ cos δ−sin δ p+ˆ − p − cos δ+sin−ˆ δ

(B.12)

⎞ ⎠.

(B.13)

Although Eqs. (B.12)–(B.13) are expressed in terms of the interpolating momentum variables while Eqs. (B.6)–(B.7) are written in terms of the IFD momentum variables, one should note that they are intrinsically the same spinors with respect to each other as we have shown in the above derivation. As δ → π4 , i.e. in the LFD limit, Eqs. (B.12)–(B.13) coincide with the LFD spinors [4] 1 u (0) ( p + ) = √ 2 p+ and 1 v ( p ) = √ 2 p+ (0)

+





√m + 2p



√m − 2 p+

,

(B.14)

 ,

(B.15)

where one may note the correspondence p +ˆ − p−ˆ C→0 m 2 −→ , C 2 p+

(B.16)

with p +ˆ → p + , and p−ˆ → p + . Thus, the interpolating spinors given by Eqs. (B.12)– (B.13) are nothing but the same spinors as given by Eqs. (B.6)–(B.7) in IFD and Eqs. (B.14)–(B.15) in LFD, respectively. There are no differences in the spinors except the expression difference in terms of the momentum variables taken in each different form of the dynamics. Dropping the “CR” specification again for the γ matrices, we follow the link given by Eq. (4.1) to get the following interpolating γ matrices in the chiral representation as   √ 1+S 0 ˆ + √ , γ = 1−S 0   √ 0 − 1−S γ −ˆ = √ . (B.17) 1+S 0

224

QCD1+1 Appendix B

Lowering the indices with the interpolating metric given by Eq. (4.2), we also get  1−S , 0   √ 0 1+S √ = , − 1−S 0

γ+ˆ = Cγ +ˆ + Sγ −ˆ = γ−ˆ = Sγ +ˆ − Cγ −ˆ



√ 0 1+S

√

(B.18)

√ where 1 ± S can be identically given by cos δ ± sin δ with cos δ ≥ sin δ always due to 0 ≤ δ ≤ π4 . One can explicitly check that the interpolating γ matrices satisfy % μˆ νˆ & ˆν · I2×2 , in particular, γ , γ = 2g μˆ (γ +ˆ )2 = (γ+ˆ )2 = C · I2×2 , (γ −ˆ )2 = (γ−ˆ )2 = −C · I2×2 .

(B.19)

As δ → π/4, one gets the usual LFD γ matrices given by  √  0 2 =γ , − 0 0     √ 0 0 = γ+ , = γ0 − γ1 / 2 = √ 20

  √ γ+ = γ0 + γ1 / 2 = γ−

(B.20)

where {γ μ , γ ν } = 2g μν · I2×2 and (γ + )2 = (γ − )2 = 0. Now, using the Bogoliubov transformation given by Eq. (4.23) for the creation/annihilation operators of the quark–antiquark fields as well as the boost operation for the free quark–antiquark fields, we get the following relationship between the interacting spinors and the free spinors at rest: u( p−ˆ ) = T ( p−ˆ )u (0) (0), v(− p−ˆ ) = T ( p−ˆ ) v (0) (0),

(B.21)

where ' T ( p−ˆ ) = '

γ ˆ θ ( p−ˆ ) p +ˆ ] exp[− √− √ Cm C 2

γ ˆ θ f ( p−ˆ ) + 2ζ ( p−ˆ ) p +ˆ ] exp[− √− √ 2 Cm C γˆ = T f ( p−ˆ ) exp[− √− ζ ( p−ˆ )] C =

with γ

e

− √−ˆC ζ ( p−ˆ )

γˆ = cos ζ ( p−ˆ ) · I2×2 − sin ζ ( p−ˆ ) √− C

(B.22)

QCD1+1 Appendix B

225

 ⎞ cos ζ ( p−ˆ ) − sin ζ ( p−ˆ ) 1+S C ⎠,  =⎝ sin ζ ( p−ˆ ) 1−S cos ζ ( p ) ˆ − C ⎛

(B.23)

and '

γ ˆ θ f ( p−ˆ ) p +ˆ ] exp[− √− √ Cm C 2 '   θ f ( p−ˆ ) θ f ( p−ˆ ) γ−ˆ p +ˆ · I2×2 − sin ·√ = √ cos . 2 2 Cm C

T f ( p−ˆ ) =

(B.24) γ

Note here that the free spinors in the moving frame are related to each other as √−Cˆ · γ u (0) ( p−ˆ ) = v (0) (− p−ˆ ) and √−Cˆ · v (0) (− p−ˆ ) = −u (0) ( p−ˆ ), and that the free spinors in the moving frame and the rest frame are related to each other ' u (0) ( p−ˆ ) = ' (0)

v (− p−ˆ ) =

γ ˆ θ f ( p−ˆ ) (0) p +ˆ exp[− √− ]u (0) √ Cm C 2 γ ˆ θ f ( p−ˆ ) (0) p +ˆ ]v (0), exp[− √− √ C 2 Cm

(B.25)

where θ f ( p−ˆ ) is given by Eq. (4.27) and is related in the IFD case to the rapidity sin θ −tan δ η as sin θ f = tanh η, while in general they are related by tanh η = 1−sinf θ f tan δ , or

tanh η+tan δ in other words, sin θ f = 1+tan . From Eq. (B.21), one can get the interacting δ tanh η quark–antiquark spinors given by Eqs. (4.28) and (4.29) in Sect. 4.2.1 as well as Eqs. (4.45) and (4.46).

B.2

Minimization of the Vacuum Energy with Respect to the Bogoliubov Angle

In this Appendix, we show that the mass gap equation given by Eq. (4.51) can also be obtained by minimizing the vacuum energy Ev in Eq. (4.37) with respect to the Bogoliubov angle, θ ( p−ˆ ) in addition to the methods presented in Sect. 4.2. Recall that    dp−ˆ  0 −ˆ Tr −γ γ p−ˆ + mγ 0  − ( p−ˆ ) Ev = 2π     dp−ˆ dk−ˆ λ ˆ + ˆ − 0 + 0 + Tr γ γ  (k )γ γ  ( p ) , (B.26) + ˆ ˆ − − 2 2π ( p−ˆ − k−ˆ )2

226

QCD1+1 Appendix B

where  ± is defined by Eqs. (4.45) and (4.46). Let us now compute the small variation of Ev as    dp−ˆ  0 −ˆ Tr −γ γ p−ˆ + mγ 0 δ − ( p−ˆ ) δEv = 2π     dp−ˆ dk−ˆ λ ˆ ˆ − 0 + + 0 + Tr γ γ δ (k )γ γ  ( p ) + ˆ ˆ − − 2 2π ( p−ˆ − k−ˆ )2     dp−ˆ dk−ˆ λ ˆ + ˆ 0 + 0 + − Tr γ γ  (k )γ γ δ ( p ) . (B.27) + ˆ ˆ − − 2 2π ( p−ˆ − k−ˆ )2 In the second term of the above equation, we are able to swap variables p−ˆ and k−ˆ , i.e.    dp−ˆ dk−ˆ ˆ ˆ − 0 + + 0 + Tr γ γ δ (k )γ γ  ( p ) ˆ ˆ − − ( p−ˆ − k−ˆ )2    dp−ˆ dk−ˆ ˆ ˆ − 0 + + 0 + = Tr γ γ δ ( p )γ γ  (k ) ˆ ˆ − − ( p ˆ − k−ˆ )2  −   dp−ˆ dk−ˆ ˆ ˆ − 0 + − 0 + Tr γ γ δ ( p )γ γ  (k ) , (B.28) =− ˆ ˆ − − ( p−ˆ − k−ˆ )2 where we used the fact that the sum of  + ( p−ˆ ) and  − ( p−ˆ ) is independent of p−ˆ from the second line to the third line. Thus, we obtain 

   ˆ Tr −γ 0 γ − p−ˆ + mγ 0 δ − ( p−ˆ ) 2π     d p−ˆ dk−ˆ λ ˆ ˆ ˆ ˆ + Tr γ 0 γ +  + (k−ˆ )γ 0 γ + δ − ( p−ˆ ) − γ 0 γ + δ − ( p−ˆ )γ 0 γ +  − (k−ˆ ) . 2 2 2π ( p−ˆ − k−ˆ )

δEv =

d p−ˆ

(B.29)

The functional differentiation of Ev relative to θ ( p−ˆ ) for a given p−ˆ is then given by ( )   δ − ( p ) δEv ˆ ˆ − = Tr −γ 0 γ − p−ˆ + mγ 0 δθ ( p−ˆ ) δθ ( p−ˆ ) ( )  − − dk−ˆ λ ˆ) ˆ) 0 + ˆ + ˆ δ ( p− ˆ δ ( p− ˆ − 0γ + 0γ + 0γ + − γ γ + Tr γ  (k )γ γ  (k ) = 0, ˆ ˆ − − 2 δθ ( p−ˆ ) δθ ( p−ˆ ) ( p−ˆ − k−ˆ )2

(B.30)

i.e.

Tr

⎧  ⎪ ⎨ −(cos δ + sin δ) p ⎪ ⎩

m

ˆ −

m (cos δ − sin δ) p−ˆ



⎛

cos θ( p ˆ ) −

⎜ 2(cos δ−sin δ) ⎝ sin θ( p ) ˆ √ − 2 C

⎞

sin θ( p ˆ )  √ − dk−ˆ ⎟ λ 2 C − cos θ( p ˆ ) ⎠ + 2 ( p−ˆ − k−ˆ )2 − 2(cos δ+sin δ)

QCD1+1 Appendix B ⎡⎛

227 ⎞⎛

1−sin θ(k ˆ ) − 2

⎞

cos θ( p ˆ ) (cos δ−sin cos δ−sin − √ δ cos θ(k ˆ ) √ δ) sin θ( p ˆ ) − ⎟⎜ − ⎟ 2 2 C 2 C ⎠⎝ ⎠ 1+sin θ(k ) − cos θ( p ˆ ) ˆ (cos δ+sin cos δ+sin − − √ δ cos θ(k ˆ ) √ δ) sin θ( p ˆ ) − − 2 2 2 C 2 C ⎞⎛ ⎞⎤⎫ ⎛ cos θ( p ˆ ) 1+sin θ(k ˆ ) (cos δ−sin ⎪ δ cos θ(k ) − − ⎬ √ δ) sin θ( p ˆ ) √ − cos δ−sin ˆ ⎟⎥ − ⎟⎜ − 2 2 ⎜ 2 C 2 C − ⎝ ⎠⎝ ⎠⎦ = 0. − cos θ( p ) 1−sin θ(k ) ⎪ ˆ ˆ (cos δ+sin δ) cos δ+sin δ − − ⎭ √ √ sin θ( p−ˆ ) cos θ(k−ˆ ) − 2 2 2 C 2 C

⎢⎜ × ⎣⎝

(B.31)

The computation of the trace leads to the gap equation given by Eq. (4.51).

B.3

Treatment of the λ = 0 (Free) Case Versus the λ = 0 (Interacting) Case with Respect to the Mass Dimension √ 2λ

As the ’t Hooft coupling λ given √ by Eq. (4.35) has the mass square dimension, we scaled out the mass dimension 2λ and used the dimensionless mass m, longitudinal momentum p−ˆ , etc., in presenting all the figures and tables of our work. Of course, we could have explicitly defined the dimensionless variables, e.g. denoted by pˆ E m p¯ −ˆ = √ − , E¯ = √ , m¯ = √ , 2λ 2λ 2λ

(B.32)

and rewrite, for example, Eqs. (4.48) and (4.51) as √ ¯ p¯ −ˆ ) = p¯ −ˆ sin θ ( p¯ −ˆ ) + Cm¯ cos θ ( p¯ −ˆ ) E(    d k¯−ˆ 1 ¯−ˆ ) , cos θ ( p ¯ ) − θ ( k +C· − ˆ − 4 ( p¯ −ˆ − k¯−ˆ )2

(B.33)

and p¯ −ˆ cos θ ( p¯ −ˆ ) −

√

 d k¯−ˆ 1 Cm¯ sin θ ( p¯ −ˆ ) = C · − 4 ( p¯ −ˆ − k¯−ˆ )2   × sin θ ( p¯ −ˆ ) − θ (k¯−ˆ ) ,

(B.34)

respectively. Similarly, we can also scale out the interpolation angle dependence by defining the following rescaled variables p¯ −ˆ E¯  ¯ p¯ − ˆ = √ , E = √ , C C

(B.35)

228

QCD1+1 Appendix B

to reduce Eqs. (B.33) and (B.34) as     ¯− ¯− ¯ cos θ ( p¯ − E¯  ( p¯ − ˆ) = p ˆ sin θ ( p ˆ)+m ˆ)   ¯   dk ˆ 1  ¯ ) , ) − θ ( k + −  −  2 cos θ ( p¯ − ˆ ˆ − 4 ( p¯ ˆ − k¯ ˆ ) − −

(B.36)

and  p¯ − ˆ

 cos θ ( p¯ − ˆ)

− m¯

 sin θ ( p¯ − ˆ)

 d k¯ ˆ 1 = −  − 2 4 ( p¯ ˆ − k¯ ˆ ) − −     × sin θ ( p¯ −ˆ ) − θ (k¯− ˆ) ,

(B.37)

respectively. Indeed, we used such rescaled variables in Eqs. (4.83), (4.95) and (4.96) to present the corresponding results without any dependence of δ ∈ [0, π/4], confirming that the physical results are indeed invariant regardless of the interpolation angle δ as they must be. However, √ one should note the contrast between the scaling 2λ and the scaling by the dimensionless parameby the dimensionful parameter √ ter C. While the rescaling over the dimensionless variable√ C includes the limit of C = 0, i.e. LFD, the rescaling over the dimensionful variable 2λ cannot include the limit to λ = 0. Namely, the free theory without any interaction must be distinguished from the interacting theory and should be discussed separately. For λ = 0, in fact, the mass gap solution θ ( p−ˆ ) = θ f ( p−ˆ ) can be immediately found even analytically by taking the right-hand side of Eq. (4.51) to be zero, i.e.   pˆ , θ f ( p−ˆ ) = arctan √ − Cm

(B.38)

where the dimensionless ratio p−ˆ /m can still be written as p¯ −ˆ /m¯ with the cancelation of the λ = 0 factor in the ratio. In terms of the rescaled variables in Eqs. (B.32) and (B.35), this analytic free solution becomes   θ f ( p¯ − ˆ)

= arctan

 p¯ − ˆ

m¯

 .

(B.39)

 In Fig. B.1, we plot the interacting mass gap solution θ ( p¯ − ˆ ) in comparison with   the free mass gap solution θ f ( p¯ −ˆ ) as functions of rescaled variable ξ¯  = tan−1 p¯ − ˆ.  Note here that the interacting mass gap solution θ ( p¯ −ˆ ) includes the LFD solution  analytically given by Eq. (4.67) while the free analytic solution θ f ( p¯ − ˆ ) is clearly  distinguished from the interacting solution θ ( p¯ −ˆ ), although the difference between the free solution and the interacting solution gets reduced as m¯ gets larger. It confirms that the entire non-trivial contributions from the interaction to the LFD solution  are provided by the zero-mode p + = 0 as the finite p¯ − ˆ can be attained only if √ + p p¯ −ˆ = √2λ = 0 in the limit C → 0 (LFD).

QCD1+1 Appendix B

229

 ) versus the free solutions θ ( p¯  ) for a few choices of quark Fig. B.1 Numerical solutions of θ( p¯ − f ˆ ˆ − mass (a) m¯ = 0 (b) m¯ = 0.18 (c) m¯ = 1.0 and (d) m¯ = 2.11, for several different interpolation angles

B.4

Mesonic√Wavefunctions for m = 0.045, 1.0 and 2.11 in the Unit of 2λ

In Sect. 4.6.2, we discussed the wavefunctions of the √ quark–antiquark bound-states for the cases of m = 0 and m = 0.18 in the unit of 2λ, i.e. m¯ = 0 and 0.18. In this Appendix, we summarize the numerical results of φˆ ±(0) (r−ˆ , x) and φˆ ±(1) (r−ˆ , x) for a few √ other bare quark–antiquark mass cases; m = 0.045, 1.0 and 2.11 in the m¯ = 0.045 corresponds to unit of 2λ, i.e. m¯ = 0.045, 1.0 and 2.11. In particular, √ the physical pion mass Mπ = 0.41 in the unit of 2λ according to the reasoning [5]  mentioned in Sect. 4.3. We also note that the free mass gap solution θ f ( p¯ − ˆ ) given by Eq. (B.39) for m¯ = 1.0 exhibits the straight-line profile in the plot with respect  to ξ¯  = tan−1 p¯ − ¯ > 1.0, e.g. m¯ = 2.11, gets ˆ while the profile of the solution for m bended toward the concave-shaped profile from the convex-shaped profile for m¯ < 1.0 as one can see in Fig. B.1. For the case of m¯ = 0.045, the numerical results of φˆ ±(0) (r−ˆ , x) and φˆ ±(1) (r−ˆ , x) are presented in Figs. B.2 and B.3, respectively. In each figure, the results of δ = 0, 0.6 and 0.78 are shown in the top, middle and bottom panels, respectively. In each panel, the results of r−ˆ = M0.045 , 5M0.045 and 8M0.045 , where M0.045 = 0.42 (see Table 4.6) is the ground-state meson mass for the quark mass value m¯ = 0.045, are depicted by the solid lines in blue, yellow and green, respectively. We note that the IFD (δ = 0) results shown in Figs. B.2a, b and B.3a, b coincide with the corresponding plots in Fig. 9 of Ref. [5]. The results for the case of m¯ = 0.045 exhibit the similar features that we discussed for the case of m¯ = 0 in Sect. 4.6.2. Namely, the large-momentum IFD (δ = 0) numerical results approach to the LFD results quite slowly [6] as the momenta r−ˆ get large (see Figs. B.2a, b and B.3a, b), while the results getting close to δ = π/4

230

QCD1+1 Appendix B

(0) (0) Fig. B.2 Ground-state wavefunctions φˆ + (r−ˆ , x) and φˆ − (r−ˆ , x) for m = 0.045. All quantities are √ in proper units of 2λ

(e.g. δ = 0.78) yield very quickly the essential features of the LFD results fitting in the region [0, 1] regardless of the momenta r−ˆ = M0.045 , 5M0.045 , or 8M0.045 and the minus component disappears (see Figs. B.2e, f and B.3e, f) although the numerical sensitivity gets enhanced with some wiggles or bulges in φˆ +(0) (r−ˆ , x)(δ = 0.78) for r−ˆ = 5M0.045 and 8M0.045 due to the enhanced demand of numerical accuracy as C gets close to zero and r−ˆ gets large. In Figs. B.2 and B.3, the charge conjugation symmetry under the exchange of x ↔ 1 − x is manifest as we have discussed for the ground state and the first excited state previously in Sect. 4.6.2, i.e. φˆ +(1) (r−ˆ , x) and φˆ −(1) (r−ˆ , x) reveal the antisymmetric profiles while φˆ +(0) (r−ˆ , x) and φˆ −(0) (r−ˆ , x) exhibit the symmetric profiles.  For the case of m¯ = 1.0 which we noted above its straight-line profile for θ f ( p¯ − ˆ ) in (0) (1) ˆ ˆ Fig. B.1, the numerical results of φ± (r−ˆ , x) and φ± (r−ˆ , x) are presented in Figs. B.4 and B.5, respectively. The frames for m¯ = 1.0 are chosen as r−ˆ = 0.2M1.0 , 2M1.0

QCD1+1 Appendix B

231

(1) (1) Fig. B.3 First excited state wavefunctions φˆ + (r−ˆ , x) and φˆ − (r−ˆ , x) for m = 0.045. All quantities √ are in proper units of 2λ

and 5M1.0 , where M1.0 = 2.70 (see Table 4.6) is the ground-state meson mass for the quark mass value m¯ = 1.0. The essential features that we discussed for the lower masses (m¯ = 0, 0.045, 0.18) in Sect. 4.6.2 and above remain without much change. As mentioned earlier, we haven’t increased the number of grid points beyond 600 while the numerical accuracy is much more demanded as C gets close to zero and r−ˆ gets large. However, we are not alarmed by the appearance of “rabbit ear” for δ = 0.78 and r−ˆ = 5M1.0 = 13.5 in Fig. B.4 as we are convinced from our numerical analyses that such numerical noise would disappear as we keep pushing the number of grid points even higher. Finally, in Figs. B.6 and B.7, we present our numerical results of φˆ +(0) (r−ˆ , x), (0) φˆ − (r−ˆ , x), φˆ +(1) (r−ˆ , x) and φˆ −(1) (r−ˆ , x) for the case of m¯ = 2.11. In this case, the   θ ( p¯ − ¯− ˆ ) solution gets close to the free mass gap solution θ f ( p ˆ ) as shown in Fig. B.1, which may indicate that the binding effect gets lesser while the quark mass effect gets larger. In fact, the extreme heavy quark mass limit would yield the non-relativistic δ-function type of ground-state meson wavefunction peaked highly at x = 1/2 to

232

QCD1+1 Appendix B

(0) (0) Fig. B.4 Ground-state wavefunctions φˆ + (r−ˆ , x) and φˆ − (r−ˆ , x) for m = 1.0. All quantities are in √ proper units of 2λ

share the longitudinal momentum equally between the two equal mass quark and antiquark. In Fig. B.6, we see a kind of precursor for such tendency toward the heavy quark–antiquark bound-state system. In the case of m¯ = 2.11, we take our frames as r−ˆ = 0.2M2.11 , M2.11 and 2M2.11 , where M2.11 = 4.91 (see Table 4.6). As 2M2.11 is already large enough for our numerical computation, we do not go beyond r−ˆ = 2M2.11 . Besides the tendency toward the heavy quark–antiquark bound-state system, the essential features that we discussed previously including the charge conjugation symmetry under the exchange of x ↔ 1 − x for the ground state and the first excited state appear similar in Figs. B.6 and B.7. We notice some wiggles in the φˆ − component of the wavefunction solution in, e.g. Fig. B.6, but the overall magnitude of the φˆ − wavefunction is always negligible compared to φˆ + whenever this occurs, thus it does not cause concern to us.

QCD1+1 Appendix B

233

(1) (1) Fig. B.5 First excited state wavefunctions φˆ + (r−ˆ , x) and φˆ − (r−ˆ , x) for m = 1.0. All quantities √ are in proper units of 2λ

B.5

“Quasi-PDFs” Corresponding to Mesonic Wavefunctions √ for m = 0.045, 1.0 and 2.11 in the Unit of 2λ

Starting from the definition of the “quasi-PDFs” interpolating between IFD and LFD given by Eq. (4.129), we obtained the “quasi-PDFs” using the mesonic wavefunctions of the quark–antiquark bound-states in the interpolating axial gauge, Aa−ˆ = 0, as given by Eq. (4.132) and discussed the “quasi-PDFs” for the cases of m = 0 and m = 0.18 in Sect. 4.6.3. As the mesonic wavefunctions for m = 0.045, 1.0 and 2.11 were presented in the previous appendix, Appendix B.4, we now discuss the corresponding “quasi-PDFs” in this Appendix B.5. First, the δ = 0 (IFD) results of the ground-state and first excited state mesonic quasi-PDFs are shown in Figs. B.8 and B.9, respectively, for a few different quark mass values, not only m = 0.045 as taken in Ref. [6] but also m = 1.0 and m = 2.11. The results of m = 0.045 shown in Figs. B.8a and B.9a agree very well with the top right panels of Figs. 2 and 3 of Ref. [6]. Due to the charge conjugation symmetry under the exchange of x ↔ 1 − x , φˆ ±(0) (r−ˆ , x) = φˆ ±(0) (r−ˆ , 1 − x) and φˆ ±(1) (r−ˆ , x) = −φˆ ±(1) (r−ˆ , 1 − x), we see the peak and valley at x = 1/2 for q˜0 (r−ˆ , x) and q˜1 (r−ˆ , x),

234

QCD1+1 Appendix B

(0) (0) Fig. B.6 Ground-state wavefunctions φˆ + (r−ˆ , x) and φˆ − (r−ˆ , x) for m = 2.11. All quantities are √ in proper units of 2λ

respectively, as shown in Figs. B.8 and B.9. Although the peak and valley get a little more sharpened as m gets larger, the essential features of the symmetry remain intact regardless of the m values. In each panel, the results of the moving frames with the longitudinal meson momentum r−ˆ = r 1 = M0.045 , 5M0.045 and 8M0.045 , where M0.045 = 0.42 (see Table 4.6) is the ground-state meson mass for the quark mass value m = 0.045, are depicted by the solid lines in blue, yellow and green, respectively. As noted in Ref. [6], the large-momentum IFD (δ = 0) numerical results approach to the LFD results quite slowly as the momentum r−ˆ = r 1 gets large. The corresponding results for the larger m values, m = 1.0 and m = 2.11, are shown in Figs. B.8b, B.9b, B.8c and B.9c for the ground state and the first excited state, respectively. As discussed in Sect. 4.6.3, the variation of the interpolating parameter δ may remedy the slow approach to the LFD results in IFD (δ = 0). To exhibit this feature,

QCD1+1 Appendix B

235

(1)

(1)

Fig. B.7 First excited state wavefunctions φˆ + (r−ˆ , x) and φˆ − (r−ˆ , x) for m = 2.11. All quantities √ are in proper units of 2λ

we show the results of the ground-state and first excited state mesonic quasi-PDFs for different δ values (δ = 0.6 and 0.78) in Figs. B.10, B.11, B.12 and B.13, respectively, with the same arrangement of corresponding m values (m = 0.045, 1.0 and 2.11). For the δ = 0.6 case shown in Figs. B.10 and B.11, one may see already some improvement in the approach to the LFD results by comparing the corresponding q˜0 (r−ˆ , x) and q˜1 (r−ˆ , x) results in Figs. B.8 and B.9 with the corresponding r−ˆ values. For the δ = 0.78 case shown in Figs. B.12 and B.13, the results get improved much more dramatically yielding very quickly the essential features of the LFD results fitting in the region x ∈ [0, 1]. The similar features have been noted earlier in Sects. 4.6.2 and 4.6.3 as well as in Appendix B.4, i.e. taking δ away from the IFD (δ = 0), the resemblance to the PDFs in LFD may appear more swiftly achieved with the boost of the meson longitudinal momentum r−ˆ to the larger value. Nevertheless, one should note here a numerical caveat demanding much higher numerical accuracy as the meson momentum r−ˆ gets larger while the δ value gets close to π/4 (e.g. δ = 0.78). In such situation, the numerical sensitivity kicks in so strongly that the results cannot be trusted unless they get tested for the improvement with much higher numerical accuracy. However, as discussed in Sect. 4.6.3, one doesn’t need to boost the longitudinal momentum r−ˆ too large if the δ value gets close to π/4. As the δ

236

QCD1+1 Appendix B

Fig. B.8 Quasi-PDFs in IFD (δ = 0) for the ground-state (n = 0) wavefunctions of (a) m = 0.045, √ (b) m = 1.00 and (c) m = 2.11 in the unit of 2λ

value gets close to π/4, relatively smaller r−ˆ value can do the job. Thus, it would be worthwhile to search for a “sweet spot” by varying both δ and r−ˆ together to obtain the “LFD-like” result.

B.6

Rest Frame Bound-State Equation and Its Solution

While we presented our numerical solutions of the bound-state wavefunctions φˆ ±(n) (r−ˆ , x) in terms of the interpolating longitudinal momentum fraction variable x = p−ˆ /r−ˆ in Sect. 4.6.2 and Appendix B.4 to discuss the moving frame dependence of the interpolating wavefunctions between IFD and LFD, the rest frame is special and deserves separate description/discussion. In particular, the massless particles can’t exist √ in the rest frame according to the relativity although the GOR relation 2 M(0) ∼ m λ → 0 indicates that the meson mass M(0) → 0 as m → 0 in the chiral limit. As the massless Goldstone boson moves with the speed of light, it can’t exist in the rest frame. We thus devote this final Appendix for the discussion of the rest frame bound-state equation and its solution. Taking r−ˆ = 0 in Eq. (4.116), we get    dk−ˆ ˆ + ˆ −r + 2E( p−ˆ ) φ+ ( p−ˆ ) = λC− ( p−ˆ − k−ˆ )2   × C( p−ˆ , k−ˆ )φˆ + (k−ˆ ) − S( p−ˆ , k−ˆ )φˆ − (k−ˆ ) ,

(B.40a)

QCD1+1 Appendix B

237

Fig. B.9 Quasi-PDFs in IFD (δ = 0) for the first excited √ state (n = 1) wavefunctions of (a) m = 0.045, (b) m = 1.00 and (c) m = 2.11 in the unit of 2λ

Fig. B.10 δ = 0.6 interpolating “quasi-PDFs” for the ground-state (n = 0) wavefunctions of (a) √ m = 0.045, (b) m = 1.00 and (c) m = 2.11 in the unit of 2λ

238

QCD1+1 Appendix B

Fig. B.11 δ = 0.6 interpolating “quasi-PDFs” for the first excited √ state (n = 1) wavefunctions of (a) m = 0.045, (b) m = 1.00 and (c) m = 2.11 in the unit of 2λ

Fig. B.12 δ = 0.78 interpolating “quasi-PDFs” for the ground-state (n = 0) wavefunctions of (a) √ m = 0.045, (b) m = 1.00 and (c) m = 2.11 in the unit of 2λ

QCD1+1 Appendix B

239

Fig. B.13 δ = 0.78 interpolating “quasi-PDFs” for the first excited state (n = 1) wavefunctions of √ (a) m = 0.045, (b) m = 1.00 and (c) m = 2.11 in the unit of 2λ

Fig. B.14 Analytic solutions for the ground-state wavefunctions in the rest frame for the bare quark mass m = 0 for three different interpolation angles as functions of ξ = tan−1 p−ˆ . All quantities are √ in proper units of 2λ

240

QCD1+1 Appendix B

(n) (n) Fig. B.15 Rest frame wavefunctions φˆ + ( p−ˆ ) and φˆ − ( p−ˆ ) for m = 0 (n = 1 and n = 2) as func√ −1 tions of ξ = tan p−ˆ . All quantities are in proper units of 2λ

   dk−ˆ r +ˆ + 2E( p−ˆ ) φˆ − ( p−ˆ ) = λC− ( p−ˆ − k−ˆ )2   × C( p−ˆ , k−ˆ )φˆ − (k−ˆ ) − S( p−ˆ , k−ˆ )φˆ + (k−ˆ ) ,

(B.40b)

where  C( p−ˆ , k−ˆ ) = C( p−ˆ , k−ˆ , r−ˆ = 0) = cos

2

 θ ( p−ˆ ) − θ (k−ˆ ) , 2

(B.41)

 θ ( p−ˆ ) − θ (k−ˆ ) . 2

(B.42)

and  S( p−ˆ , k−ˆ ) = S( p−ˆ , k−ˆ , r−ˆ = 0) = − sin

2

QCD1+1 Appendix B

241

(n) (n) Fig. B.16 Rest frame wavefunctions φˆ + ( p−ˆ ) and φˆ − ( p−ˆ ) for m = 0.045 as functions of ξ = √ tan−1 p−ˆ . All quantities are in proper units of 2λ

The basis wavefunction is also provided without scaling the interpolating momentum variable p−ˆ with respect to r−ˆ [in contrast to Eq. (4.125)] m (α, p−ˆ ) =

3 2   α 1 2 2 α p exp − √ ˆ , ˆ Hm α p− − m 2 2 m! π

(B.43)

and the n-th bound-state wavefunctions φˆ ±(n) ( p−ˆ ) are normalized as 

  dp−ˆ |φˆ +(n) ( p−ˆ )|2 − |φˆ −(n) ( p−ˆ )|2 = 1,

(B.44)

where one should note the caveat of n = 0 solution for m = 0 that becomes null in the rest frame due to the relativity.

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QCD1+1 Appendix B

(n) (n) Fig. B.17 Rest frame wavefunctions φˆ + ( p−ˆ ) and φˆ − ( p−ˆ ) for m = 0.18 as functions of ξ = √ −1 tan p−ˆ . All quantities are in proper units of 2λ ˆ

(r + )2

2 In the frame r−ˆ = 0, the meson mass square M(n) = (n) and the corresponding C (n) ˆ wavefunctions φ± ( p−ˆ ) are obtained by solving the coupled bound-state equations, Eqs. (B.40a) and (B.40b), using essentially the same technique that we described in Sect.4.6.1. √ 2 Due to the GOR relation M(0) ∼ m λ → 0, the ground-state meson mass M(0) → 0 as m → 0. As mentioned earlier, the massless Goldstone boson then moves with the speed of light and can’t exist in the rest frame according to the relativity. How can one understand this distinction of the massless Goldstone boson in the rest frame? To realize it, one may take a look more closely the analytic solution of the ground-state (n = 0) wavefunction (φˆ ±(0) ( p−ˆ )) given by Eq. (4.128) and find that the rest frame r−ˆ = 0 yields φˆ ±(0) ( p−ˆ ) as

φˆ +(0) ( p−ˆ ) = φˆ −(0) ( p−ˆ ) =

1 cos θ ( p−ˆ ). 2

(B.45)

QCD1+1 Appendix B

243

(n) (n) Fig. B.18 Rest frame wavefunctions φˆ + ( p−ˆ ) and φˆ − ( p−ˆ ) for m = 1.0 as functions of ξ = √ tan−1 p−ˆ . All quantities are in proper units of 2λ

Then, indeed, the normalization condition given by Eq. (4.124) becomes null indicating the absence of the massless Goldstone boson in the rest frame. However, the individual interpolating wavefunctions φˆ +(0) ( p−ˆ ) and φˆ −(0) ( p−ˆ ) do not vanish as plotted in Fig. B.14. Here, we use the variable of the horizontal axis ξ = tan−1 ( p−ˆ ). In Fig. B.14, the interpolation angle δ = 0, 0.6, and 0.78 results are depicted in solid, dashed and dotted lines, respectively. In contrast to the ground state, the excited states (n = 0) for m = 0 acquire the nonzero bound-state √ mass as shown in Table 4.6, e.g. M(1) = 2.43, M(2) = 3.76, etc. in the unit of 2λ, and they can take the rest frame r−ˆ = 0. In Fig. B.15, the bound-state wavefunctions φˆ +(n) ( p−ˆ ) and φˆ −(n) ( p−ˆ ) for the first (n = 1) and second (n = 2) excited states obtained by numerically solving Eq. (B.40) are plotted with yellow and green lines, respectively. From Fig. B.15, we can see the odd and even parties, respectively, for n = 1 and n = 2 state wavefunctions φˆ ±(n) ( p−ˆ ) under the exchange of p−ˆ ↔ − p−ˆ . For the δ value close to π/4, the wavefunctions are very sharply peaked and constrained in a relatively small | p−ˆ | region, indicating that

244

QCD1+1 Appendix B

(n) (n) Fig. B.19 Rest frame wavefunctions φˆ + ( p−ˆ ) and φˆ − ( p−ˆ ) for m = 2.11 as functions of ξ = √ −1 tan p−ˆ . All quantities are in proper units of 2λ

ultimately only the zero-mode p + = 0 survives for r + = 0 frame in LFD. Not only does the supporting momentum region ξ = tan−1 ( p−ˆ ) for φˆ ±(n) ( p−ˆ ) get shrunken to the zero-mode ξ = tan−1 ( p + ) = 0 but also the magnitude of φˆ −(n) ( p−ˆ ) gets much more suppressed compare to φˆ +(n) ( p−ˆ ) as δ value close to π/4, indicating the absence of φˆ −(n) ( p + ) solutions in LFD. For the nonzero mass cases, we plot n = 0, 1 and 2 together for the rest frame wavefunctions φˆ +(n) ( p−ˆ ) and φˆ −(n) ( p−ˆ ) as shown in Fig. B.16 for m = 0.045, Fig. B.17 for m = 0.18, Fig. B.18 for m = 1.0, and Fig. B.19 for m = 2.11, respectively. For m = 0.045, although we have compared our results for the moving frames shown in Figs. B.2 and B.3 with the corresponding results in Ref. [5] as discussed in Appendix B.4, we couldn’t compare our results shown in Fig. B.16 with Ref. [5] as the rest frame was not considered in Ref. [5]. However, the results shown in Fig. B.16 are not much different from the corresponding results for m = 0 discussed above as the difference in the mass m is rather marginal.

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245

For m = 0.18, 1.0 and 2.11, all of our δ = 0 results shown in Figs. B.17, B.18 and B.19 can be, respectively, compared with the corresponding rest frame results shown in Figs. 7, 8, 9 and 10 of Ref. [7]. Although the normalization of the bound-state wavefunction was mistaken in Ref. [7] by taking the + sign between |φˆ +(n) ( p−ˆ )|2 |φˆ −(n) ( p−ˆ )|2 in Eq. (B.44), our results look quite consistent with theirs as the magnitude of φˆ +(n) ( p−ˆ ) is much larger than the magnitude of φˆ −(n) ( p−ˆ ) to reveal any sizable difference for the comparison. All of our results look consistent with the characteristics of φˆ +(n) ( p−ˆ ) and φˆ −(n) ( p−ˆ ) discussed previously.

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