Relativistic Quantum Field Theory, Volume 3: Applications of Quantum Field Theory 1643277596, 9781643277592

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Relativistic Quantum Field Theory, Volume 3 Applications of quantum field theory

Relativistic Quantum Field Theory, Volume 3 Applications of quantum field theory Michael Strickland Kent State University, Kent, Ohio, USA

Morgan & Claypool Publishers

Copyright ª 2019 Morgan & Claypool Publishers All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the publisher, or as expressly permitted by law or under terms agreed with the appropriate rights organization. Multiple copying is permitted in accordance with the terms of licences issued by the Copyright Licensing Agency, the Copyright Clearance Centre and other reproduction rights organizations. Rights & Permissions To obtain permission to re-use copyrighted material from Morgan & Claypool Publishers, please contact [email protected]. ISBN ISBN ISBN

978-1-64327-762-2 (ebook) 978-1-64327-759-2 (print) 978-1-64327-760-8 (mobi)

DOI 10.1088/2053-2571/ab3a99 Version: 20191101 IOP Concise Physics ISSN 2053-2571 (online) ISSN 2054-7307 (print) A Morgan & Claypool publication as part of IOP Concise Physics Published by Morgan & Claypool Publishers, 1210 Fifth Avenue, Suite 250, San Rafael, CA, 94901, USA IOP Publishing, Temple Circus, Temple Way, Bristol BS1 6HG, UK

This book is dedicated to my wife, Dr Veronica Antocheviz Dexheimer Strickland, and our amazing daughter Emily.

Contents Preface

xi

Acknowledgements

xiv

Author biography

xv

Units and conventions

xvi

1

1-1

QCD phenomenology

1.1 1.2 1.3 1.4 1.5 1.6 1.7

Electron–muon scattering Form factors Elastic electron–proton scattering and the proton form factors Inelastic electron–proton scattering The parton model and Bjorken scaling Valence partons and sea partons Beyond the naive parton model 1.7.1 Gluon emission cross section 1.7.2 Small-angle approximation 1.7.3 Embedding γ∗-parton scattering in DIS 1.8 DGLAP evolution 1.9 Hadron production in e+e− collisions 1.10 Fragmentation functions 1.11 Solution of the DGLAP equations using Mellin moments 1.12 Drell–Yan scattering References

2

Weak interactions

2.1 2.2 2.3 2.4 2.5

Early models of the weak interaction Muon decay Charged pion decay Electron–neutrino and electron–antineutrino scattering Neutrino–quark scattering 2.5.1 Charge raising current 2.5.2 Charge lowering current 2.5.3 Differential cross section 2.5.4 Embedding

1-1 1-4 1-6 1-11 1-14 1-17 1-21 1-22 1-23 1-24 1-26 1-32 1-35 1-37 1-39 1-41 2-1

vii

2-2 2-3 2-7 2-9 2-11 2-11 2-11 2-12 2-12

Relativistic Quantum Field Theory, Volume 3

2-14 2-15 2-16 2-17

2.6 2.7

Weak neutral currents The Cabibbo angle and the CKM matrix 2.7.1 The Cabibbo–Kobayashi–Maskawa (CKM) matrix References

3

Electroweak unification and the Higgs mechanism

3-1

3.1 3.2

Electroweak Feynman rules Massive gauge fields with local gauge symmetry 3.2.1 Spontaneous symmetry breaking 3.2.2 Breaking of a continuous local symmetry Gauge boson masses in SU(2)L × U(1)Y 3.3.1 The resulting particle spectrum 3.3.2 Fermion masses The discovery of the Higgs boson 3.4.1 The H → γγ decay channel 3.4.2 The H → ZZ → 4l decay channel 3.4.3 The H → τ+τ− decay channel 3.4.4 Other decay channels and the nature of the Higgs References

3-5 3-6 3-7 3-10 3-12 3-12 3-13 3-14 3-16 3-18 3-19 3-19 3-20

3.3

3.4

4

Basics of finite temperature quantum field theory

4-1

4.1

Partition function for a quantum harmonic oscillator 4.1.1 The QHO canonical partition function in the energy basis 4.1.2 Computing the QHO partition function using the path integral formalism The partition function for a free scalar field theory 4.2.1 Fourier representation of the fields 4.2.2 Tricks for evaluating sum-integrals Free scalar thermodynamics 4.3.1 Low temperature limit 4.3.2 High-temperature limit The need for resummation Perturbative expansion of thermodynamics for a scalar field theory 4.5.1 One loop 4.5.2 Two loops 4.5.3 Three loops 4.5.4 Pressure through g5

4-1 4-2 4-3

4.2

4.3

4.4 4.5

viii

4-6 4-7 4-9 4-11 4-15 4-15 4-17 4-21 4-22 4-22 4-23 4-24

Relativistic Quantum Field Theory, Volume 3

4.6

Screened perturbation theory 4.6.1 One-loop contribution 4.6.2 Two-loop contribution 4.6.3 Three-loop contribution 4.6.4 Pressure to three loops 4.6.5 Mass prescription 4.6.6 The tadpole mass prescription 4.6.7 Three-loop SPT Pressure References

5

Hard-thermal-loops for QED and QCD

5.1

Photon polarization tensor 5.1.1 Generalization to d-dimensions 5.1.2 The HTL polarization tensor 5.1.3 Generalization to QCD Fermionic self-energy Collective modes 5.3.1 Gluon modes 5.3.2 Quark modes 5.3.3 Collective modes in an isotropic QGP 5.3.4 Gluon modes 5.3.5 Landau damping 5.3.6 Quark modes Hard-thermal-loop effective action 5.4.1 Minkowski-space HTL gluon propagator 5.4.2 Minkowski-space HTL quark propagator 5.4.3 Three-gluon vertex 5.4.4 Four-gluon vertex 5.4.5 Quark-gluon three-vertex 5.4.6 Quark-gluon four-vertex 5.4.7 Hard thermal loop effective Lagrangian 5.4.8 Euclidean space HTL effective Lagrangian and vertex functions Hard-thermal-loop resummed thermodynamics 5.5.1 Contributions to the HTLpt thermodynamic potential through NNLO 5.5.2 NNLO HTLpt thermodynamic potential 5.5.3 NNLO result for equal chemical potentials

5.2 5.3

5.4

5.5

ix

4-25 4-26 4-27 4-27 4-28 4-28 4-29 4-29 4-30 5-1 5-1 5-4 5-5 5-5 5-6 5-7 5-8 5-9 5-10 5-10 5-12 5-13 5-15 5-15 5-18 5-19 5-19 5-20 5-21 5-21 5-22 5-23 5-25 5-28 5-28

Relativistic Quantum Field Theory, Volume 3

5.5.4 NNLO result—general case 5.5.5 Mass prescription 5.5.6 Thermodynamic functions and susceptibilities 5.5.7 Quark number susceptibilities 5.5.8 Baryon number susceptibilities References

x

5-30 5-32 5-32 5-34 5-35 5-37

Preface In introductory quantum mechanics, one learns how to quantize a system given a fixed number of non-relativistic bosonic or fermionic particles. One uses a formalism in which the Hamiltonian is promoted from being a number to an operator resulting in the Schrödinger equation. The Hamiltonian operator itself is decomposed into kinetic and potential energy contributions and the potential energy form is typically taken as an external input. A question then arises: how does one arrive at the potential in the first place based on first principles? For example, how do we know (beyond experiment) that the Coulomb potential is the appropriate potential for charged particles? Are there quantum corrections to this potential? In addition, since potentials are not well-defined relativistically (instantaneous interactions), how can we generalize quantum mechanics to the relativistic case and satisfy causality? In classical physics, fields are introduced in order to construct physical laws that are causal and local. Classical forces described by, e.g. Coulomb’s law or Newton’s universal gravitation, require action at a distance. As a result, the force felt by a body changes instantaneously if any other body’s position (or charge, etc) changes, no matter how far away the other object is. In the classical field theories of Maxwell (electromagnetic field) and Einstein (gravitational field), the interactions between objects are mediated by a field that acts locally and causality is restored. So it seems that, since they are causal and local, we should figure out how to quantize classical field theories. Another inconsistency in quantum mechanics was our somewhat haphazard treatment of wave–particle duality. For example, both electrons and photons act simultaneously like waves and particles and physically they share many common features. They both undergo wave-like diffraction from obstacles, but can also act like discrete particles (photoelectric effect, Compton scattering, etc). Despite this, in classical theory, electrons are simply postulated to exist as matter, while photons are interpreted as ripples in the electromagnetic field. Is it possible that electrons are themselves ripples in an ‘electron field’? As we will learn over the course of these three volumes, the answer is a definitive yes. In general, the field is the fundamental object and particles are derived concepts that appear only after quantization of the field (e.g. the Higgs field gives birth to the Higgs boson). In quantum mechanics, we take classical number-valued quantities and promote them to operators acting in a Hilbert space. As we will see, at least in ‘canonical quantization’, the rules for quantizing a field are only slightly different. The basic degrees of freedom in quantum field theory (QFT) are operator-valued functions of space and time and, since space and time are continuous, we are dealing with an infinite number of degrees of freedom, so we will need to (re-)learn how to deal with systems with a large number of degrees of freedom (many-body theory). Once we are done, we will be able to properly define QFTs that can be used in a variety of different contexts, e.g. high energy theory, condensed matter, cosmology, quantum gravity, etc.

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Relativistic Quantum Field Theory, Volume 3

Beyond this, Dirac taught us that a consistent theory of relativistic electrons requires the existence of anti-electrons (aka positrons). As a consequence, it is possible to create particle–antiparticle pairs once the energy available exceeds twice the electron rest mass E > 2 mc 2 in a process called pair production. And, of course, the reverse can happen, which is called pair annihilation. And one can easily see both of these types of events using modern particle detectors. The conclusion we must draw from this is that particles are not indestructible objects; they can be created and destroyed and may only live for a short amount of time. They are merely excitations surfing on a quantum field. But the story is more fantastical than this. If pair production has an energy threshold, then one could argue that as long as the energies available do not exceed this threshold (E > 2 mc 2 ), non-relativistic theories would be self-consistent; however, at this point, the Heisenberg uncertainty principle comes into play. Let us say that we wanted to measure the position of a particle with a given spatial resolution L. The Heisenberg uncertainty principle tells us that the uncertainty in the momentum is Δp ≳ 1/L . In a relativistic setting, energy and momentum are connected, therefore, we also have an uncertainty in the energy ΔE ≳ 1/L . However, when the energy plus its uncertainty exceeds E + ΔE > 2mc 2 , then it is possible to create purely quantum-mechanical particle–antiparticle pairs. Equating the two, we obtain a threshold distance L 0 = 1/(2m ) = λ Compton /(4π ) with λ Compton = 2π /m. From this exercise, we learn that the spontaneous production of particle–antiparticle pairs is important when a particle of mass m is localized in space to a distance, which is on the order of less than its Compton wavelength. A similar argument holds if one considers localizing particles in time. The energy uncertainty increases as the time interval is made shorter and one eventually turns on the possibility of pair production. The consequence of this is that as one considers shorter and shorter time (or space) intervals one sees more and more particles! Why is this relevant? Since microscopic particles interacting with one another can resolve the other particles’ microscropic dynamics (they behave like observers since they exchange quanta in order to interact), they ‘see’ their partners as being surrounded by an ensemble of particles and antiparticles that flit in and out of existence. In QFT, these ephemeral particles are called virtual particles. The ‘cloud’ of virtual particles that surround particles can modify its observable properties and must be taken into account to consistently understand them. In figure 0.1, I show a typical Feynman graph for the splitting of an electron (solid line with the arrows) into virtual photons (sinusoidal lines) and virtual electron– positron pairs (closed loops). In this figure, time progresses from the left to the right. Starting from the incoming electron line, we see the radiation of a virtual photon, which then splits into a virtual electron–positron pair, and so forth. In QFT, during this time, all possible configurations of the various intermediate particles are sampled, but some configurations are more probable than others. We will learn how to quantify this over the course of these three volumes. A very similar phenomenon to pair production exists in condensed matter systems. When one considers conduction of electrons in metals, for example, one finds that there is a valence band of electrons that are locally bound to atoms and a xii

Relativistic Quantum Field Theory, Volume 3

Figure 0.1. A typical Feynman graph for the splitting of an electron (solid line with the arrows) into virtual photons (sinusoidal lines) and virtual electron–positron pairs (closed loops).

conduction band in which electrons are able to move around. Because of the spinstatistics theorem, fermions obey the Pauli exclusion principle and ‘stack up’ to an energy called the Fermi energy. It is possible for a photon (or an energetic phonon) to excite one of the electrons that have an energy below the Fermi energy (in the Fermi sea) to above the Fermi sea. In metals, this process requires very little energy; however, in semiconductors, the material is gapped. This introduces a lower limit on the amount of energy required to excite an electron into the conduction band which is called the gap. The processes of electron–hole pair production and annihilation can be described using Feynman graphs similar to the relativistic case and the mathematical machinery used to describe many-body states in materials is very similar to what is encountered in relativistic QFT. This three-volume series began as lecture notes for a two-semester introductory course in quantum field theory and quantum chromodynamics and, as such, is written rather informally. The text is intended to be used in an introductory graduate-level course in quantum field theory or an advanced undergraduate course. Volume 1 introduces classical fields and the method of canonical quantization in order to have a bridge to the language and formalisms used by students focusing on condensed matter physics. Volume 2 builds upon what was learned in Volume 1, but starts anew using the modern path integral formalism and focuses on applications to quantum electrodynamics and chromodynamics. Volume 3 continues with discussions of applications to particle physics phenomenology, the weak interaction, the Higgs mechanism, and finite temperature field theory. Michael Strickland June 30, 2019

xiii

Acknowledgements I thank the students who provided feedback on the lectures notes that formed the basis for this book.

xiv

Author biography Michael Strickland Dr Michael Strickland is a professor of physics at Kent State University. His primary interest is the physics of the quark–gluon plasma (QGP) and high-temperature quantum field theory (QFT). The QGP is predicted by quantum chromodynamics (QCD) to have existed until approximately 10−5 s after the Big Bang. The QGP is currently being studied terrestrially by experimentalists at the Relativistic Heavy Ion Collider (RHIC) at Brookhaven National Laboratory and the Large Hadron Collider (LHC) at CERN. Dr Strickland has published research papers on various topics related to the QGP, quantum field theory, relativistic hydrodynamics, and many other topics. In addition, he has cowritten a classic text on the physics of neural networks.

xv

Units and conventions • I will use natural units in which ℏ = c = 1. In these units, masses, energies, momenta, and temperatures have the same ‘energy-like’ units. I will typically use eV, MeV, or GeV as the standard unit for these energy-like quantities. For example, the mass of an electron is me = 0.511 MeV, where this is understood implicitly to denote me = 0.511 MeV /c2 but, since c = 1, we typically suppress the division by c2 when writing the mass. • Another consequence of using natural units is that the vacuum electric permeability and magnetic permeabilities are both equal to one, ϵ0 = μ0 = 1, which greatly simplifies the presentation of the formulas of electromagnetism. A similar simplification happens in thermodynamic formulas since kB = 1. • In natural units, spatial and temporal separations have units of inverse energy units (e.g. GeV−1). In some places in the notes, we will use a bracket notation to indicate the dimension of a quantity, e.g. [m] = 1 tells us that masses have units of energy to the power one and lengths [L] = −1 indicating that length scales scale like energy to the power minus one. • If the argument of a function, e.g. exp, should be dimensionless, then the argument should be equivalent to an energy times a length scale (E × L), since lengths have units of inverse energies in natural units. • In some cases, you might need to switch from energy scales to explicit distance scales or vice versa. For this purpose, the following combination is very useful:

ℏc = 0.197 326 938 GeV fm, where fm stands for femtometers (1 fm = 10−15 m). For purposes of estimation, one can use the approximate value ℏc ≈ 0.2 GeV fm . For example, if I were to tell you that a length (or time) scale is L = 0.2 fm, you can divide by ℏc to find the equivalent in inverse GeV, e.g. L = (0.2 fm)/(ℏc ) ≃1 GeV−1. Similarly, we can work in reverse to convert energy (mass) scales to inverse distance scales. For example, from the electron mass me = 0.511 MeV we can determine the Compton wavelength λ Compton = 2π /m and then covert from GeV−1 to meters, i.e. λ Compton,e ≃ 12 296GeV−1 × ℏc ≃ 2.426 × 10−12 m. • Finally, in the three volumes in this series, the Minkowski-space metric is assumed to be mostly negative, which means that (g μν )Minkowski ≡ η μν = diag(1, −1, −1, −1).

xvi

IOP Concise Physics

Relativistic Quantum Field Theory, Volume 3 Applications of quantum field theory Michael Strickland

Chapter 1 QCD phenomenology

We will now turn to more practical matters and ask how to compute elastic and inelastic scattering cross sections for hadronic scattering from quantum chromodynamics (QCD) and compare the predictions of QCD with results from high-energy collider experiments. As we proceed, we will need to understand the corresponding processes in quantum electrodynamics (QED) first, so I will jump back to QED from time to time.

1.1 Electron–muon scattering The first process that it will be useful to understand in detail is the relativistic scattering of an electron off a muon depicted in equation (1.1). This will form the template for electron–proton scattering.

.

Using the QED Feynman rules, the invariant amplitude for this process is

M = −e 2u¯(k′)γ μu(k )

1 u¯(p′)γμu(p ), q2

(1.2)

where q = k − k′. As in our examples from volumes 1 and 2, it is useful to separate the sums over the electron and muon spins by writing the spin-averaged squared modulus as

∣M∣2 =

doi:10.1088/2053-2571/ab3a99ch1

e 4 μν muon L e L μν , q4

1-1

(1.3)

ª Morgan & Claypool Publishers 2019

Relativistic Quantum Field Theory, Volume 3

where

L fμν ≡

1 Tr[( k ′ + mf )γ μ( k + mf )γ ν ], 2

(1.4)

where f indicates the type of fermion involved. For the electron, this becomes

1 1 Tr[ k ′γ μ k γ ν ] + m 2 Tr[γ μγ ν ] 2 2 μ ν ν μ = 2[k′ k + k′ k − (k · k′ − m 2 )η μν ],

L eμν =

(1.5)

where m = me is the electron mass and we have used the fact that the trace of three gamma matrices vanishes. Similarly, for the muon, one finds μν L muon = 2[p′μ p ν + p′ν p μ − (p · p′ − M 2 )η μν ],

(1.6)

where M = mμ is the muon mass. Putting the pieces together, we obtain ∣M∣2 =

8e 4 [(k′ · p′)(k · p ) + (k′ · p )(k · p′) − m 2p′ · p − M 2k′ · k + 2m 2M 2 ]. (1.7) q4

Converting to Mandlestam variables (see, e.g. Appendix D of volume 1), we obtain

∣M∣2 =

2e 4 [12m 2M 2 + 6(m 4 + M 4) − 4(m 2 + M 2 )(s + u ) + s 2 + u 2 ], t2

(1.8)

where s = (p + k)2, t = (k − k′)2 = (p − p′)2 = q 2 , and u = (k − p′)2 . The rate for e +e−→μ+ μ− can be obtained by using the crossing relations, interchanging t ↔ s . In the high-energy limit, we can neglect both masses to obtain

lim ∣M∣2 = 2e 4

s →∞

s2 + u2 . t2

(1.9)

Mott cross section At lower/intermediate energies, since M ≫ m, it will be useful to make the approximation that m = 0, while keeping M finite. In this case, one has 8e 4 [(k′ · p′)(k · p ) + (k′ · p )(k · p′) − M 2k′ · k ] q4 ⎤ 8e 4 ⎡ 1 1 = 4 ⎢ − q 2(k · p − k′ · p ) + 2(k′ · p )(k · p ) + M 2q 2⎥ ⎣ ⎦ q 2 2

lim ∣M∣2 =

m e→0

where we have used p′ = k − k′ + p and k 2 = k′2 = 0.

1-2

(1.10)

Relativistic Quantum Field Theory, Volume 3

If we boost into the rest frame of the muon, the scattering process will look like

.

With this, we have k = (E, E,0,0), k′ = (E ′, E ′ cos θ , E ′ sin θ , 0), p = (M,0,0,0), p′ = −q = k′ − k = (E ′ − E , E ′ cos θ − E , E ′ sin θ , 0), where we have used the fact that the electron is assumed to be massless and hence ∣k∣ = E and ∣k′∣ = E ′. This gives

lim ∣M∣2 =

me→0

⎤ 1 8e 4 ⎡ 1 2 − q M (E − E ′) + 2M 2EE ′ + M 2q 2⎥ , 4 ⎢ ⎣ ⎦ 2 2 q

(1.12)

with

q 2 = (k − k′)2 = −2k · k′ = −2EE ′(1 − cos θ ) = −4EE ′ sin2

θ . 2

(1.13)

Using the general elastic scattering relation for AB → CD (derived in volume 1)

dσ =

1 4 (pA · pB )2 − m A2m B2

d Πn ∣Mfi ∣2 ,

(1.14)

we obtain

dσ =

1 ∣M∣2 d 3k′ d 3p′ 4 δ (p + k − p′ − k′) 4ME (2π )2 2E ′ 2p0′

1 d 3p′ 4 2 = ∣ M ∣ E ′ dE ′ d Ω δ (p + q − p′). 32π 2ME 2p0′

(1.15)

Integrating over p′ gives a factor of 3

∫ d2pp′′ δ 4( p + q − p′) = ∫ d 4p′δ 4( p + q − p′)θ ( p0′ )δ( p′2 − M 2) 0

= δ(2ME γ + q 2 ) 1 ⎛ q2 ⎞ δ⎜E γ + = ⎟. 2M ⎝ 2M ⎠

(1.16)

Putting the pieces together, we obtain

⎤ ⎛ (2αE ′)2 ⎡ 2 θ q2 ⎞ dσ q2 2 θ = − δ + cos sin E ⎜ ⎟, ⎥ ⎢ γ dE ′d Ω q4 ⎣ 2 2M 2 2⎦ ⎝ 2M ⎠

1-3

(1.17)

Relativistic Quantum Field Theory, Volume 3

where α = e 2 /(4π ). We can perform the integration over E′ by re-expressing the final delta function as

⎛ 1 ⎛ E⎞ q2 ⎞ δ⎜E γ + ⎟ = δ(E − AE ′) = δ⎜E ′ − ⎟ , ⎝ 2M ⎠ A⎠ A ⎝

(1.18)

where A ≡ 1 + (2E /M )sin2 θ /2 to obtain finally

α2 θ⎤ dσ E′ ⎡ 2 θ q2 = − cos sin2 ⎥ . ⎢ θ 2 dΩ 2 2M 2⎦ 4E 2 sin4 2 E ⎣

(1.19)

This is the Mott cross-section. It is a generalization of the Rutherford scattering cross section, which takes into account the recoil of the muon and its spin. The second term in square brackets is due to scattering from the magnetic moment of the muon. If the muon had no spin, then the second term would vanish. The factor of E′/E accounts for the recoil. Exercise 1.1 Verify that equation (1.19) follows from equation (1.17).

1.2 Form factors The previous example considered scattering off of point-like particles. What if we are instead high-energy scattering of a charge distribution with a finite size? For this, we will need to understand relativistic form factors, but before heading off into this discussion, let us first review the situation of classical electromagnetic scattering off a cloud of charge. In this case, we can determine the structure of an extended charge distribution by scattering an electron beam off of it and measuring the angular distribution of scattered electrons and then comparing our results with the cross section for scattering of electrons from a point like source. Let us imagine that we were given an electric charge distribution described by ρ(x) and we scatter electrons off of it as shown in figure 1.1. We can define the electric form factor F (q) through

⎛ dσ ⎞ dσ ⎟ =⎜ ∣F (q)∣2 , d Ω ⎝ d Ω ⎠point

(1.20)

Figure 1.1. Leading-order perturbative graph for scattering an electron off of a charge distribution ρ(x).

1-4

Relativistic Quantum Field Theory, Volume 3

where q is three-momentum transfer between the incoming electron and the target, q = k − k′. The form factor F (q) encodes the target’s structure and, for a static target, is related to the Fourier transform of the charge distribution

F (q) =

∫ d 3x ρ(x)eiq·x.

(1.21)

For simplicity, let us assume that the target is a static, spinless electric charge distribution with total charge of Ze with ρ normalized such that

∫ d 3x ρ(x) = 1,

(1.22)

which implies that F(0) = 1. The reference cross section in this case would be

⎛ dσ ⎞ (Zα )2 E 2 ⎛ k2 θ⎞ ⎜ ⎟ 1 sin2 ⎟ . = − ⎜ θ 2 4 4 ⎝ d Ω ⎠point E 2⎠ 4k sin 2 ⎝

(1.23)

To see how this arises, we need to compute the scattering amplitude Tfi. At first order in perturbation theory, the transition amplitude can be written as

∫x ψ¯ f V (x)ψi(x) = i . e . ∫ ψγ ¯ μAμ ψi (x ) x = − i ∫ jμfi Aμ , x

Tfi = − i

where

jμfi ≡ −eψ¯ f γμψi = −eu¯f γμuie i (kf −ki )μx . μ

(1.24)

Using Aμ = (ϕ, 0), one obtains

Tfi = −2πiδ(E ′ − E )( −eu¯′γ 0u )

∫ d 3x eiq·xϕ(x),

(1.25)

where ϕ(x) is the electrostatic potential generated by the target. This can be obtained by solving the Poisson equation

∇2 ϕ(x) = −Zeρ(x).

(1.26)

Assuming that ϕ goes to zero at infinity, one has

∫ d 3x ei q·x∇2 ϕ(x) = −∣q∣2 ∫ d 3x ei q·xϕ(x),

(1.27)

which, using the Poisson equation, implies that

∫ d 3x ei q·xϕ(x) = − ∣q1∣2 ∫ d 3x ei q·x∇2 ϕ(x) =

Ze ∣q∣2

∫

d 3x e i q·xρ(x) =

1-5

Ze F (q). ∣q∣2

(1.28)

Relativistic Quantum Field Theory, Volume 3

This gives

Tfi = −2πi

Ze δ(E ′ − E )( −eu¯′γ 0u )F (q). ∣q∣2

(1.29)

Computing ∣Tfi∣2 , we will need the spin-averaged electron trace (summed over final spin and averaged over initial spin) which is

1 2

∑ = ∣u¯f γ 0ui∣2 sf , si

⎛ k2 θ⎞ = 4E 2⎜1 − 2 sin2 ⎟ . ⎝ E 2⎠

(1.30)

Putting the pieces together, we obtain equations (1.20) and (1.23). At low momenta, we can Taylor expand the integrand of equation (1.21)

F (q) =

⎡

⎤

∫ d 3x ρ(x)⎢⎣1 + i q · x − 12 (q · x)2 + ⋯⎥⎦.

(1.31)

If ρ is spherically symmetric, then

F (q) = 1 −

1 2 2 ∣q∣ 〈r 〉 + ⋯. 6

(1.32)

At very low momenta, one is only sensitive to the total charge of the object and then using small angle scattering one can measure the mean squared radius 〈r 2〉 of the distribution. We will only resolve the fine structure of the object at large momentum transfer ∣q∣. Exercise 1.2 Derive equation (1.32) from equation (1.21).

1.3 Elastic electron–proton scattering and the proton form factors Next, we consider electron–proton scattering1. This is different from the case we covered in the last section (static, spinless distribution). We will now take into account (a) the spin of the proton, which results in scattering off of the proton’s magnetic moment and not just its charge and (b) the recoil of the proton. If the proton has mass M and were treated as a point-like scatterer, then the lab-frame scattering section would be the Mott cross section obtained previously

α2 θ⎤ dσ E′ ⎡ 2 θ q2 = − cos sin2 ⎥ , ⎢ θ 2 dΩ 2 2M 2⎦ 4E 2 sin4 2 E ⎣

(1.33)

where E /E ′ = 1 + (2E /M )sin2 θ /2 accounts for the recoil of the target proton. The leading-order amplitude in this case would be given by 1 The same analysis applies for neutrons with minimal modifications. We will mention relevant results for neutrons as we go.

1-6

Relativistic Quantum Field Theory, Volume 3

Tfi = i

∫ d 4x jμe q12 jpμ ,

(1.34)

where jμe and jpμ are the electron and proton currents, respectively. The corresponding diagram would look like

where the white circle encodes our ignorance about the actual charge/magnetic moment distribution of the proton. In this case, the currents would be of the form

jeμ = −eu¯(k′)γ μu(k )e i (k ′−k )·x ,

(1.36)

j pμ = eu¯(p′) Γμ u(p )e i (p ′−p)·x .

(1.37)

The photon–proton vertex Γμ would be γ μ if the proton were a point-like particle. We parameterize the actual vertex, Γμ, using the most general four-vector form constructed from the four vectors pμ, pμ′, and qμ in combination with one or more Dirac matrices (γ μ and γ 5). In the end, the only two independent terms2 that can appear in this decomposition are γ μ and σ μνqν (homework) where we remind you that σ μν = (i /2)[γ μ, γ ν ]. The two spinor-valued quantities γ μ and σ μνqν can each be multiplied by a function which only depend on q2 since this is the only independent scalar entering the photon–proton vertex (homework). Therefore, one has κ Γμ ≡ F1(q 2 )γ μ + F2(q 2 )iσ μνqν , (1.38) 2M where F1 and F2 are two independent form factors and κ is the anomalous magnetic moment. For the proton, κp ≃ 1.79, and for a neutron, one has κn ≃ −1.91 [1]. In the long wavelength limit, q 2 → 0, the substructure of the proton (neutron) should be irrelevant. In this case, one should see scattering from a point-like target with charge e and magnetic moment (1 + κ ) e/ 2M . This implies that, for the proton, one would have F1(0) = 1 and F2(0) = 1. For a neutron, one would have instead F1(0) = 0 and F2(0) = 1. The resulting electron–proton scattering cross section in the lab frame then becomes

q2 dσ E ′ ⎡⎛ 2 α2 κ 2q 2 2⎞ θ θ⎤ ⎢⎜F1 − (F1 + κF2 )2 sin2 ⎥ . (1.39) F2 ⎟ cos2 − = θ 2 2 ⎠ 4M 2 2M 2⎦ dΩ 4E 2 sin4 2 E ⎣⎝

2 In principle, there could also be a term involving γ 5q μ; however, if parity is conserved, then the coefficients of such terms would turn out to be zero.

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Relativistic Quantum Field Theory, Volume 3

This is called the Rosenbluth formula [2]. By construction, if the protons were pointlike and possessed zero anomalous magnetic moment, this reduces to the Mott scattering cross section. In practice, instead of using F1 and F2, it is better to use the following linear combinations:

G E ≡ F1 +

κq 2 F2, 4M 2

(1.40) (1.41)

G M ≡ F1 + κF2,

which are called the electric form factor and magnetic form factor, respectively. Rewriting the cross section in terms of these quantities, one obtains

α2 θ dσ E ′ ⎡ GE2 + τGM2 cos2 + 2τGM2 sin2 = ⎢ θ 2 4 2 dΩ 4E sin 2 E ⎣ 1 + τ

θ⎤ ⎥, 2⎦

(1.42)

where τ = −q 2 /4M . The definitions of G E and G M ensure that there are no cross terms proportional to G EG M in the cross section. This makes it easier to experimentally extract these quantities. Experimental data on the angular dependence of electron–proton scattering can be used to determine G E and G M at different values of q2. Equation (1.42) is the relativistic generalization of the form factor introduced in the last section. The form factors G E and G M are closely related to the proton charge and magnetic-moment distributions in Breit frame where p′ = −p. For ∣q2∣ ≪ M 2 and, in this frame, we can appeal to the Fourier transform interpretation obtained in the non-relativistic case. Note that, in the long wavelength limit, for the proton, we have

lim G E,p = 1,

(1.43)

lim G M,p = 1 + κp ≡ μp ,

(1.44)

q 2→ 0

q 2→ 0

and, for the neutron, we have

lim G E,n = 0,

(1.45)

lim G M,n = κn ≡ μn .

(1.46)

q 2→ 0

q 2→ 0

In figure 1.2, we show some example historical data for the proton electric form factor. At low momentum, the electric form factor is reasonably well described by a fit of the form

⎛ ⎞−2 q2 G E(q 2 ) ≃ ⎜1 − ⎟ . ⎝ 0.71 GeV 2 ⎠

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(1.47)

Relativistic Quantum Field Theory, Volume 3

Figure 1.2. Some historical experimental data for the proton form factor. Figures taken from [3].

This fit can be used to extract the proton radius

⎛ dG ⎞ 〈r 2〉 = 6⎜ 2E ⎟ ≃ (0.81 fm)2 . ⎝ dq ⎠q 2=0

(1.48)

Recent experimental results compare to a reference dipole form factor

⎛ ⎞−2 q2 G D ≡ ⎜1 − ⎟ . ⎝ 0.71 GeV 2 ⎠

(1.49)

In figure 1.3, we plot a collection of the world’s data on the proton electric and magnetic form factors. In both panels, the results are normalized to the reference dipole cross section (1.49) in order to better assess the level of agreement. In figure 1.4, we plot a similar collection for the neutron form factors. We will return to these form factors when we discuss what the QCD predictions for these quantities are. Finally, in figure 1.5, we show the raw data without the normalization factor so that you can appreciate the quality of the fit. Exercise 1.3 Verify that equation (1.38) is the most general form for the elastic proton-photon vertex. Exercise 1.4 Verify that equation (1.42) follows from Eqs (1.39), (1.40), and (1.41).

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Relativistic Quantum Field Theory, Volume 3

Figure 1.3. Recent collection of world data on the proton form factors scaled by the dipole form factor. Figures taken from [4].

Figure 1.4. Recent collection of world data on the neutron form factors scaled by the dipole form factor. Figures taken from [4].

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Relativistic Quantum Field Theory, Volume 3

Figure 1.5. Raw data to illustrate the quality of the dipole fit. Figures taken from [4].

1.4 Inelastic electron–proton scattering Next, we turn our attention to the inelastic scattering of electron with protons. This is a natural extension of the previous exercise, because in principle it would be nice to look at the proton at small scales using large momentum transfers. However, this is not simple because, at high energies the proton will start to breakup. At intermediate Q 2 ≡ −q 2 after being struck by the photon, the proton will transition to excited states such as the Δ(1232) resonance, e.g. ep → eΔ+→epπ 0 , causing extra particles to appear in the final state. At very large Q2, many additional hadrons will be generated in the final state. A general inelastic ep → eX event can be depicted as

where there are N outgoing hadrons with invariant mass W

⎛ N ⎞2 W 2 = ⎜⎜∑ E ⎟⎟ − ⎝ i=1 ⎠

N

∑ pi

2

.

(1.51)

i=1

If W = M, then we have an elastic scattering. If W > M GeV, we have inelastic scattering (excitation of resonances and production of additional hadrons) and finally, if W ≳ 2M , we enter the region of deep inelastic scattering (DIS). These regions are indicated in figure 1.6. Computation of the transition amplitude is now more difficult because, although the electron current jeμ ∼ u¯γ μu flowing along the top of the graph is unaffected, on the bottom of the graph the final state is no longer a single fermion state. As a result, jpμ must have a more complicated form for inelastic scattering. At this point, rather 1-11

Relativistic Quantum Field Theory, Volume 3

Figure 1.6. Total cross section for ep → eX as a function of missing mass W. The elastic peak has been reduced by a factor of 8.5. Figure adapted from [5].

than trying to compute it, we can once again appeal to the idea of a tensor basis to introduce the most general rank-two tensor W μν that will couple to the electron current e Tfi ∼ L μν W μν.

(1.52)

The most general form can be made out of the metric tensor η μν , the independent e momentum p μ and q μ. Additionally, since L μν is symmetric, any antisymmetric μν components of W do not contribute to Tfi. As a result, we can make the decomposition3

W μν = −W1η μν +

W W W2 μ ν p p + 42 q μq ν + 52 (p μ q ν + q μp ν ). 2 M M M

(1.53)

Formally, Wμν is constructed as a sum over all possible many-particle outgoing states X

Wμν =

1 4πM

⎛1 ∑⎜⎜ 2 N ⎝

⎞ ∑⎟⎟ s ⎠

N

∫∏

⎛ × (2π )4δ 4⎜⎜p + q + ⎝

i=1

⎛ d 3p′ ⎞ i ⎜ ⎟ ∑ p , s Jμ† X 〈X ∣Jν∣p , s〉 3 2 (2 ) π E ′ ⎝ i ⎠ si

⎞ ∑ pn′ ⎟⎟. j=1 ⎠ N

(1.54)

In order to have current conservation at the hadronic vertex, it is necessary that qμW μν = qνW μν = 0. These two constraints can be used to express two of the structure functions in terms of the others

3 We have omitted W3, which is a parity violating structure function that comes into play in neutrino–proton scattering.

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Relativistic Quantum Field Theory, Volume 3

⎛ p · q ⎞2 M2 W4 = ⎜ 2 ⎟ W2 + 2 W1, q ⎝ q ⎠ W5 = −

p·q W2, q2

(1.55)

(1.56)

giving finally

W μν = −W1Δμν + W2

(Δ · p ) μ (Δ · p ) ν , M2

(1.57)

where Δμν (q ) = η μν − q μq ν /q 2 projects out the components of a four-vector that are transverse to q μ. It obeys Δ · Δ = Δ and q · Δ = Δ · q = 0. It is straightforward to see that W μν is transverse to q μ in this form. The functions W1 and W2 are functions of all Lorentz-invariant variables that can be constructed from the four momentum at the hadronic vertex. For inelastic scattering, the invariant mass of X is not fixed and, as a result, there are two independent variables which we can choose to be q2 and, e.g. p·q , ν≡ (1.58) M which, in the rest frame of the proton, is the photon energy Eγ. The invariant mass of the system can be written compactly in terms of these variables

W 2 = (p + q )2 = M 2 + 2p · q + q 2 = M 2 + 2Mν + q 2 .

(1.59)

With equation (1.57) in hand we can proceed in the normal way to compute the scattering cross section, with the result being (homework)

⎡ α2 θ θ⎤ dσ = W2(ν, q 2 )cos2 + 2W1(ν, q 2 )sin2 ⎥ . θ⎢ 2 4 dE ′d Ω 2 2⎦ 4E sin 2 ⎣

(1.60)

Finally, looking forward, we note that in place of ν and q2, it is common to use the variables

x=

Q2 Q2 , = 2p · q 2Mν

(1.61)

p·q , p·k

(1.62)

y=

where Q 2 = −q 2 . In terms for these variables, the allowed kinematic region for ep → eX is from 0 ⩽ x ⩽ 1 and 0 ⩽ y ⩽ 1. The variable x is the famous Bjorken x [6]. We will return to this point later. Next, we cover the basics of the naive parton

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Relativistic Quantum Field Theory, Volume 3

model. For this idealization, we will assume that all nucleons can be described as being ‘bags’ of quarks, ignoring, at the first pass, the role of the gluons in the hadrons. Exercise 1.5 Verify that equations (1.55) and (1.56) follow when the transversality of W μν is imposed, i.e qμW μν = qνW μν = 0.

1.5 The parton model and Bjorken scaling If the protons were built from some smaller constituent particles, then there should be evidence of this from high-Q2 scattering that probes the proton’s small-scale structure. In the parton model, we assume that all hadronic processes can be described by scattering off of the quarks (historically partons) which comprise the hadron [7, 8]. We further presume that the quark content of, e.g. the proton is completely specified by assigning a longitudinal momentum fraction ξ to each parton and introducing a function fi (ξ ) which tells us the probability of finding a parton of type i possessing longitudinal momentum fraction ξ of the hadron’s momentum p, i.e. ki = ξp. Of course, the functions fi (ξ ) obey a sum rule that simply states that when all parton momenta are added, they sum up to the total momentum of the hadron

∑ ∫0

1

dξ ξ fi (ξ ) = 1,

(1.63)

i

where the sum over i sums over all partons, which includes valence quarks, ‘sea quarks’ (aka ‘wee partons’), and gluons. For scattering, we assume that the partons are point-like scatterers and that the scattering process can be expressed as an incoherent sum over scattering with each quark flavor. This can be expressed diagrammatically as

where the sum over i only includes electrically charged partons that can interact with the photon. To begin, we will start by assuming that both incoming and outgoing particles have extremely high energies, allowing us to ignore both the electron and proton masses. As before we assume that the electron (proton) has an incoming fourmomentum k(p) and an outgoing four-momentum of k′( p′) such that the momentum transfer q = k − k′. With this setup, we have 1-14

Relativistic Quantum Field Theory, Volume 3

s ≃ 2k · p ,

(1.65)

t = Q 2 ≃ −2k · k′ ,

(1.66)

u ≃ −2k′ · p .

(1.67)

We further assume that all parton momenta are given by pˆ = ξp = (ξE , ξ p) which implies that all partons have only longitudinal momentum (momentum along the direction of the proton itself) and, hence, have no transverse momentum4. With this approximation, the invariant variables for a parton with momentum fraction ξ are

sˆ = (k + ξp )2 = 2ξk · p = ξs ,

(1.68)

tˆ = (k − k′)2 = −2k · k′ = t ,

(1.69)

uˆ = (k′ − ξp )2 = −2ξk′ · p = ξu .

(1.70)

We are now in a position to translate the picture (1.64) into an equation

⎛ dσ ⎞ep→eX ⎜ ⎟ = ⎝ dE ′d Ω ⎠lab

⎛

dσ

⎞eqi→eqi

∑ ∫ dξ fi (ξ )⎜⎝ dE ′d Ω ⎟⎠

.

(1.71)

lab

i

We previously obtained the Mott scattering cross section for scattering from a pointlike spin-1/2 particle. Taking mq → M , we obtain

⎤ ⎛ dσ ⎞eqi→eqi α 2ei2 ⎡ 2 θ Q2 2 θ⎥ ⎢cos ⎜ ⎟ = + sin 2 θ ⎝ dE ′d Ω ⎠lab 2 2 ⎥⎦ 2m q 4E 2 sin4 2 ⎢⎣ ⎛ Q2 ⎞ ⎟, × δ⎜E − E ′ − 2m q ⎠ ⎝

(1.72)

where ei is the parton’s charge in units of e. When integrated over E ′ taking into θ account the fact that Q 2 = 4EE ′ sin2 2 this reproduces equation (1.33). In order to obtain the cross section from this, we have to integrate over the incoming quark momentum. Since pi = ξp and in the lab frame the proton is at rest, this implies mq = ξmp . From equation (1.61), we also have E − E ′ = ν = Q 2 /2mpx . This gives

⎛ Q2 ⎛ Q2 ⎞ Q 2 ⎞ 2mpx 2 ⎟= ⎟ = δ⎜ δ⎜E − E ′ − − δ(ξ − x ). 2m q ⎠ 2mpξ ⎠ Q2 ⎝ 2mpx ⎝

4

(1.73)

This can be formally justified by working in the so-called infinite momentum frame. In reality, however, the proton does not have infinite momentum and one has to include the effect of transverse momentum of the quarks and gluons in hadrons [9, 10].

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Relativistic Quantum Field Theory, Volume 3

We can use this to perform the integration over ξ ⎛ dσ ⎞ep→eX ⎜ ⎟ =∑ ⎝ dE ′d Ω ⎠lab i

× =

⎡

α 2e 2

∫ dξ fi (ξ ) 4E 2 sini 4 θ ⎢⎢cos2 θ2 2

2 mp

x2

Q2

⎣

+

Q2 θ⎤ sin2 ⎥ 2 2 2 ⎥⎦ 2m p ξ (1.74)

δ (ξ − x ) ⎡ 2 mp x 2

α2

∑ ei2fi (x )⎢

θ

4E 2 sin4 2

⎣ Q2

i

cos2

1 θ θ⎤ sin2 ⎥ . + 2 2⎦ mp

Comparing this to the general expression

⎡ α2 dσ 2 2 θ = + 2W1(ν, q 2 )sin2 ⎢W2(ν, q )cos θ 2 4 ⎣ dE ′d Ω 2 4E sin 2

θ⎤ ⎥, 2⎦

(1.75)

we see that the parton model predicts that5

W1(x ) =

1 2mp

∑ ei2fi (x), i

2mpx 2 W2(x ) = Q2

(1.76)

∑

ei2fi

(x ).

i

Defining dimensionless structure functions

F1(x ) = mpW1 =

1 2

∑ ei2fi (x), i 2

F2(x ) = νW2 =

Q W2(x ) = x ∑ ei2fi (x ), 2mpx i

(1.77)

we observe: 1. The structure functions only depend on x. 2. There is a relationship 2x F1 = F2 which will be obeyed if we are indeed scattering off of point-like partons. 3. If the partons had spin zero then F1 = 0 instead. In figure 1.7, we show historical experimental results for F2 = νW2 (left panel) and the ratio 2xF1/F2 (right panel). The left panel demonstrates that F2 does not depend on the momentum transfer, which is indicative of scattering of off point-like particles according to our prior discussions. The right panel demonstrates that for x ≳ 0.2 the scaling relation between F1 and F2 is observed. The right panel also shows arrows 5

We point out that there are different conventions for the normalization of the structure functions W1 and W2 in the literature.

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Relativistic Quantum Field Theory, Volume 3

Figure 1.7. The left panel shows νW2 as a function of Q2. The right panel shows 2xF1/F2 as a function of Q2. Figures taken from [5].

that indicate the value that would be predicted if the partons had spin 1/2 or spin 0. This is clear evidence we are scattering off of partons with spin 1/2. Since the photon only couples to the quarks, this makes perfect sense6.

1.6 Valence partons and sea partons The naive picture that we have of a proton is that it comprised of three valence quarks (uud); however, in reality there are additional sea quarks which are particle– antiparticle pairs that flit in and out of existence. If the energy of the virtual photon is high enough, then the time scale resolved is small τ ∼ 1/Q and during a scattering event one can resolve the quantum fluctuations inside of a hadron. Of course, the proton also contains gluons. The situation faced is sketched in figure 1.8. Ignoring the gluons, for now, one can expand the proton structure function by summing over the different relevant quark flavors to obtain

⎛ 2 ⎞2 ⎛ 1 ⎞2 1 ep F2 (x ) = ⎜ ⎟ [u p(x ) + u¯ p(x )] + ⎜ ⎟ [d p(x ) + d¯ p(x )] ⎝3⎠ ⎝3⎠ x 2 ⎛1 ⎞ + ⎜ ⎟ [s p(x ) + s¯ p(x )], ⎝3⎠

(1.78)

where u p(x ) and u¯ p(x ), for example, are the probability distributions for up quarks and anti-up quarks in the proton. Of course, there could also be charm, bottom, and top components of the proton; however, due to their large mass, they will play a negligible role and can be neglected. Likewise, we could expand the structure function for electron–neutron scattering

⎛ 2 ⎞2 ⎛ 1 ⎞2 1 en F2 (x ) = ⎜ ⎟ [u n(x ) + u¯ n(x )] + ⎜ ⎟ [d n(x ) + d¯ n(x )] ⎝3⎠ ⎝3⎠ x 2 ⎛1 ⎞ + ⎜ ⎟ [s n(x ) + s¯ n(x )]. ⎝3⎠ 6

The virtual photon can couple to gluons only via virtual quark loops.

1-17

(1.79)

Relativistic Quantum Field Theory, Volume 3

Figure 1.8. Schematic representation of the make up of a proton. A subscript ‘v’ indicates a valence quark and a subscript ‘s’ indicates a sea quark. There are also electromagnetic and weak interactions which are not indicated.

If we assume exact isospin invariance, then the proton and neutron are related by u ↔ d which implies that, to good approximation

u p(x ) = d n(x ) ≡ u(x ), d p(x ) = u n(x ) ≡ d (x ), s p(x ) = s n(x ) ≡ s(x ),

(1.80)

and likewise for the anti-particles. As a result, to good approximation, we should only need six structure functions to describe both proton and neutron inelastic scattering. These are not the only constraints on the distribution functions. For each parton type, we can decompose the total distribution function into valence and sea contributions as indicated in figure 1.8, e.g. for the proton one has

u(x ) = u v(x ) + u s(x ), u¯(x ) = u¯ s(x ), d (x ) = d v(x ) + d s(x ), d¯(x ) = d¯s(x ),

(1.81)

s(x ) = ss(x ), s¯(x ) = s¯s(x ). In addition, since as the sea quarks are created in pairs one has us(x ) = u¯s(x ), ds(x ) = d¯s(x ), ss(x ) = s¯s(x ), …. Also, as an approximation, we can assume that the three lightest quarks u, d, and s appear in the sea with roughly the same probability7. As a result of this assumption, one would have 7

The assumption of the equivalence of u, d, and s is not well justified when it comes to the s quark since its mass is dramatically more than the u and d quarks. This approximation will be relaxed further on, but for now it is a decent starting point.

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Relativistic Quantum Field Theory, Volume 3

u s(x ) = u¯ s(x ) = d s(x ) = d¯s(x ) = ss(x ) = s¯s(x ) ≡ S (x ).

(1.82)

Finally, since we know the quark model content of the proton, there are sum rules that must be obeyed by the distribution functions. The proton has charge 1, baryon number 1, and strangeness 0; therefore, it must be the case that

∫ dx [u(x) − u¯(x)] = ∫ dx u v(x) = 2, ∫ dx [d (x) − d¯(x)] = ∫ dx d v(x) = 1, ∫ dx [s(x) − s¯(x)] = 0.

(1.83)

The neutron obeys the same sum rules but with u ↔ d . Combining all of these simplifying assumptions, we can write equations (1.78) and (1.79) as

4 1 ep 1 F2 (x ) = [4u v(x ) + d v(x )] + S (x ), 3 9 x

(1.84)

4 1 en 1 F2 (x ) = [u v(x ) + 4d v(x )] + S (x ). 3 9 x

(1.85)

Since the gluons create quark–antiquark pairs in the sea, S(x) will have a bremsstrahlung-like spectrum which is peaked at very small x ( fs (x ) ∼ 1/x at small x) and we expect that for low-x that the valence quarks will be outnumbered by sea quarks. As a result, one expects that

lim x →0

F2en(x ) = 1. F2ep(x )

(1.86)

In the limit of large x (x ≃ 1), the valence quarks will dominate. In this case, one expects that

F2en(x ) u (x ) + 4d v(x ) . = v ep x → 1 F 2 (x ) 4u v ( x ) + d v ( x ) lim

(1.87)

Experimental fits to the structure functions indicate that u v ≫ d v at large x. Hence, we expect this ratio to approach approximately 1/4. In figure 1.9, we show experimental data for this ratio which shows that these two predictions of the parton model are seen in the data. In this case, I show results from a very highenergy muon scattering experiment which, at these energies, reduces to the same kinematics we have discussed thus far. In this picture, we can also get direct access to the valence quark distributions by computing the difference of the proton and nucleon structure functions

1 ep 1 [F2 (x ) − F2en(x )] = [u v(x ) − d v(x )], 3 x

1-19

(1.88)

Relativistic Quantum Field Theory, Volume 3

Figure 1.9. Experimental data for the ratio F2μn /F2μp as a function of x averaged over Q2 for 90 and 280 GeV muon energies. Figure taken from New Muon Collaboration (NMC) [11].

Figure 1.10. Experimental data for the difference F2μp − F2μn as a function of x. Figure adapted from [12].

since, with the simplifying assumptions we have made, the sea quark distribution cancels. Experimental data for this quantity is shown in figure 1.10. This data suggests that the valence quark distributions are peaked at 1/3, which is what one would expect from the naive quark model. The distribution in x is related to the fact that there are interactions between the quarks that spread the momentum between the valence quarks. In figure 1.11 I plot the most recent extraction of parton

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Relativistic Quantum Field Theory, Volume 3

Figure 1.11. Experimental data for parton distribution functions. The parton distribution functions xuv, xd v, xS = 2x(U¯ + D¯ ) and xg. The left panel shows the results on a logarithmic scale for x and with the gluon and sea quark distributions scaled down by a factor of 20. The right panel shows the data on a linear scale with no scaling. Figures taken from the HERA Collaboration [13].

distribution functions from the HERA collaboration. As we can see from this data, the valence quark distribution functions are peaked at intermediate x whereas the gluon and sea quark distributions are peaked at small x. Exercise 1.6 Verify that equations (1.84) and (1.85) follow from the assumptions made.

1.7 Beyond the naive parton model The parton model considered so far has completely ignored the role of the gluon. We have, for example, ignored the fact that quarks can radiate gluons. Whenever a quark has its momentum changed, there is associated electromagnetic and gluonic radiation (bremsstrahlung). Generally, gluonic radiation will be more important since αs ≫ α . In perturbation theory, gluonic radiation comes in first at O(ααs ) and there are two processes

which map to the possibility that a gluon could be radiated before or after the collision. We also ignored the fact that at O(ααs ) a photon can couple to a gluon through a quark–antiquark pair 1-21

Relativistic Quantum Field Theory, Volume 3

Finally, we have also ignored virtual corrections to the quark and gluon propagators, e.g. self-energy corrections and vertex function corrections. All of these processes need to be taken into account to obtain the correct leading-order perturbative prediction. Let us start by considering each process separately and then we will combine them as we go. 1.7.1 Gluon emission cross section The spin-averaged invariant amplitude squared resulting from the diagrams shown in equation (1.89) can be obtained by a small modification of the Compton scattering cross section obtained in volume 1. Accounting for the color charge and taking u → t one obtains

⎛ tˆ uˆ ⎞ sˆ ∣M∣2 = 32π 2CFei2ααs⎜ − − + 2Q 2 ⎟ , ˆˆ ⎠ ⎝ sˆ st tˆ

(1.91)

where Q2 is the photon virtuality, CF = 4/3 is the quadratic Casimir in the fundamental representation, and the hats on the Mandlestam variables indicate that these are computed from the parton momenta involved in this subprocess. For the kinematics, we work in the center-of-mass frame (COM) of the incoming virtual photon and quark as shown below:

such that

sˆ = (k + q )2 = 2∣k∣2 + 2∣k∣E γ − Q 2 = 4∣k′∣2 , tˆ = (k′ − k )2 = −Q 2 − 2∣k′∣E γ + 2∣k∣∣k′∣cos θ = − 2∣k∣∣k′∣(1 − cos θ ), uˆ = (k′ − q )2 = −2∣k∣∣k′∣(1 + cos θ ),

(1.93)

where q is the four-momentum of the incoming quark, k is the incoming fourmomentum of the virtual photon, k′ is the outgoing four-momentum of the quark, 1-22

Relativistic Quantum Field Theory, Volume 3

and Q 2 = −k 2 = −Eγ2 + ∣k∣2 is the photon virtuality. We have assumed that the quark is massless, i.e. q2 = 0. Note that the transverse momentum of the outgoing quark, defined by pT ≡ k′ sin θ , can be expressed in terms of the Mandelstam variables as

pT2 =

ˆˆ ˆ stu , (sˆ + Q 2 )2

(1.94)

where we have used the fact that 4∣k∣∣k′∣ = −tˆ − uˆ = sˆ + Q 2 . Exercise 1.7 Derive equation (1.94). 1.7.2 Small-angle approximation Since the scattering cross section will be dominated by small angle scattering, we can focus a bit on this case. For small-angle scattering (cos θ ≃ 1), we have −tˆ ≪ sˆ . Using sˆ + tˆ + uˆ = −Q 2 , we have uˆ ≃ −(sˆ + Q 2 ) and (1.94) simplifies to

ˆˆ

(pT2 )small angle = − sˆ +stQ 2 .

(1.95)

Next, we note that, in the small angle limit, the angular phase space for the outgoing quark d Ω = 2π sin θdθ ≃ 2πθdθ which, when written in terms of pT , becomes

(d Ω)small angle =

4π 2 dp . sˆ T

(1.96)

For small-angle (forward) scattering, one has

dσˆ 1 ≃ ∣M∣2 . 2 16πsˆ 2 dpT

(1.97)

Assuming −tˆ ≪ sˆ , we obtain finally

8πei2ααs ⎡ dσˆ 2(sˆ + Q 2 )Q 2 ⎤ ≃ − ⎥. ⎢sˆ + 2 2 ˆˆ ⎣ ⎦ 3ts sˆ dpT

(1.98)

Using equation (1.95) and introducing the variable

z≡

Q2 Q2 Q2 , = = 2 2 2pi · q (k + q ) − k sˆ + Q 2

(1.99)

where pi = k is the incoming parton (quark) momentum, we can write equation (1.98) as

dσˆ 1 α ≃ ei2σˆ 0 2 s Pqq(z ), 2 dpT pT 2π where σˆ0 = 4πα /sˆ and 1-23

(1.100)

Relativistic Quantum Field Theory, Volume 3

⎛1 + z2 ⎞ Pqq(z ) = CF⎜ ⎟. ⎝1−z ⎠

(1.101)

The splitting function Pqq is related to the probability of a quark emitting a gluon and, in doing so, becoming a quark with momentum reduced by a fraction z. This can be visualized as

.

The splitting function (1.101) is divergent when z → 1 which is a divergence associated with the emission of soft gluons. This is an infrared divergence that will be canceled by divergences associated with quark self-energy and vertex corrections (further discussion of this is forthcoming). There is also a collinear divergence in the cross section as pT → 0. The kinematics in this case should be contrasted with leading-order photon– parton scattering where either (a) the pT of the outgoing parton relative to the virtual photon is always zero or (b) is negligible in the limit −tˆ ≪ sˆ . Because of this, the experimental signature of gluon radiation is a quark jet and gluon jet in the final states with neither jet moving along the direction of the virtual photon. When including the possibility of gluonic radiation, the transverse momentum pT of the jets (experimentally the sum of the transverse momentum of the hadrons associated with this jet) is nonzero, with the pT distribution given by equation (1.100). In order to compare this with the experiment, we must write the proton scattering cross section in terms of a sum over virtual photon–parton scattering. Exercise 1.8 Verify equations (1.100) and (1.101).

1.7.3 Embedding γ *-parton scattering in DIS We need to understand how to determine the contribution of γ *-parton scattering diagrams to the DIS cross section beyond the naive parton model. We begin by recalling that the ep → eX cross section is given in terms of the structure functions W1 and W2, or equivalently,

F1 = MW1(ν, Q 2 ), F2 = νW2(ν, Q 2 ).

(1.103)

Since we are going beyond the parton model, we will find that the functions F1,2 no longer scale and now depend on both ν ≡ k · q /M and Q2 = −k2 rather than simply the ratio x = Q 2 /(2Mν ). In the deep inelastic limit, for γ *-proton scattering, one has 1-24

Relativistic Quantum Field Theory, Volume 3

σT , σ0 F2 σT + σL ≃ , x σ0

2F1 ≃

(1.104)

where σT and σL are the γ *p total cross sections for transverse and longitudinal virtual photons, respectively, and

4πα . s

σ0 ≡

(1.105)

Now, we would like to compute the γ *-parton contributions to σT and σL . We first need to translate between the two frames

γ *−proton frame γ *− parton frame p ⟶ pi = ξp , x≡

Q2 x Q2 ⟶z≡ = . 2p · q 2pi · q ξ

We can represent the relations diagrammatically as

.

The cross sections in the two systems can now be related. For example, the ratio entering into F1 is given by

⎛ σ T(x , Q 2 ) ⎞ ⎜ ⎟ = ⎝ ⎠γ *p σ0

∑ ∫0

partons i

1

dz

∫0

1

⎛ σˆ (z , Q 2 ) ⎞ dξ fi (ξ ) δ(x − zξ )⎜ T , (1.107) ⎟ ⎠γ *parton ⎝ σˆ 0 i

where fi are the parton structure functions. Using the delta function to perform the z integration, we obtain

σ T(x , Q 2 ) = σ0

∑ ∫x

partons i

1

dξ σˆ (x / ξ, Q 2 ) fi (ξ ) T , σˆ 0 ξ

(1.108)

where now whether the relevant cross section is γ *p or γ *-parton scattering is indicated solely by the absence or presence of a hat on the symbol. A similar rule applies for F2.

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To see how this works in practice, let us consider the case of no radiation in which case everything should reduce to the naive parton model. For γ *q → q , if we neglect the mass of the outgoing quark, (q + pi )2 = 0, and hence

z=

Q2 = 1, 2pi · q

(1.109)

and σˆ (z, Q 2 ) is proportional to δ (1 − z ) giving

σˆ T(z , Q 2 ) = ei2δ(1 − z ), σˆ 0

(1.110)

σˆ L(z , Q 2 ) = 0.

(1.111)

This follows from the fact that for γ *q → q with incoming photon momentum q, incoming proton momentum p, and outgoing proton momentum p′ one has

∣M∣2 = 2ei2e 2p · q ,

(1.112)

where the average is over the transverse polarization states of the incoming γ *. Using

dσˆ T =

∣M∣2 d 3p′ (2π )3δ 4(p′ − p − q ) , 2p0′ (2π )3 F

(1.113)

where F = 8π 2α /σˆ 0 is the γ *q flux factor. Plugging equations (1.110) and (1.111) into equation (1.108) and using, finally, equation (1.104) we obtain, for example,

F2(x , Q 2 ) = x ∑ ei2 i

∫x

1

⎛ dξ x⎞ fi (ξ )δ⎜1 − ⎟ = x ∑ ei2fi (x ), ⎝ ξ ξ⎠ i

(1.114)

which is precisely the naive parton model relation we previously obtained (1.77). Now, we would like to include gluon radiation using equation (1.100) as the partonic cross section. Exercise 1.9 Fill in all missing steps in the derivation of equation (1.114).

1.8 DGLAP evolution Our next task is to understand how gluon radiation affects the structure functions. The structure functions themselves are related to the total cross sections via equations (1.104), (1.104), and equation (1.107). Integrating equation (1.100), we can obtain the total parton scattering cross section

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Relativistic Quantum Field Theory, Volume 3

σˆ (γ *q → qg ) = =

∫μ

sˆ /4 2

ei2σˆ 0

dpT2

∫μ

sˆ /4 2

dσˆ dpT2 dpT2 αs Pqq(z ) pT2 2π

(1.115)

⎛α Q2 ⎞ = ei2σˆ 0⎜ s Pqq(z )log 2 ⎟ , μ ⎠ ⎝ 2π where the upper limit on the pT2 integration is the maximal transverse momentum squared for γ *q → qg and μ2 is an infrared cutoff necessary to regulate the collinear divergence. Adding the gluon bremstrahlung cross section to the parton model cross sections (1.110) and (1.111), one finds

=∑ eq2 q

∫x

1

⎡ ⎛ ⎛x ⎞ dξ x⎞ Q2 ⎤ α q(ξ )⎢δ⎜1 − ⎟ + s Pqq⎜ ⎟ log 2 ⎥ , ξ ξ ⎠ 2π ⎝ ξ ⎠ μ ⎦ ⎣ ⎝

(1.117)

where q(ξ ) is a general quark structure function. Note that there are no interference terms between the two blocks because each contain different particle content in the final state. The presence of the second term in equation (1.117) results in a breaking of Bjorken-scaling of the structure functions since it indicates that F2 will depend on both x and Q2. Therefore, the violation of Bjorken-scaling of the structure functions can be taken as evidence for gluon radiation8. The result (1.117) represents the first two terms in a power series in αs which relies on αs being small. The strong coupling is only small at asymptotically large momentum transfer and in that case one has αs ∼ α0 / log Q 2 . As a result, in the high-energy limit, we see that both terms in equation (1.117) are of the same order, meaning that the correction is quite important. We can account for this correction by absorbing the log Q 2 term into a modified quark probability distribution by rewriting equation (1.117) in the naive parton model form

8

The non-zero pT of the quark and gluon jets relative to the photon momentum implied when there is gluon radiation and the agreement of the resulting jet spectrum with experimental data is also proof of the existence of gluonic radiation.

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Relativistic Quantum Field Theory, Volume 3

F2(x , Q 2 ) = ∑ eq2 x q

∫x

1

⎛ dξ x⎞ [q(ξ ) + Δq(ξ, Q 2 )]δ⎜1 − ⎟ ⎝ ξ ξ⎠

= ∑ eq2[q(x ) + Δq(x , Q 2 )]

(1.118)

q

= ∑ eq2q(x , Q 2 ), q

with

Δq(x , Q 2 ) ≡

Q2 αs log 2 2π μ

∫x

1

⎛x ⎞ dξ q(ξ )Pqq⎜ ⎟ . ⎝ξ ⎠ ξ

(1.119)

The resulting quark densities q(x , Q 2 ) = q(x ) + Δq(x , Q 2 ) now depend on Q2. This arises from the fact that a photon with a larger Q2 probes a larger range of pT2 within the proton. As Q2 increases, at first one is able to resolve the point-like valence quarks in the proton; however, QCD itself predicts that as we continue to increase Q2 we should see that the quark is itself surrounded by a cloud of other partons. So far, we have only calculated one splitting function from one diagram. In practice, it gets complicated quickly but the end result is that the number of resolved partons increases with increasing Q2. The Q2 evolution of the quark densities can be determined using equation (1.119) by turning this into a differential equation for the evolution of q(x, Q2)

dq(x , Q 2 ) α = s 2 d log Q 2π

∫x

1

⎛x ⎞ dξ q(ξ, Q 2 )Pqq⎜ ⎟ . ⎝ξ ⎠ ξ

(1.120)

This is an example of a DGLAP evolution equation for a quark distribution function. The name DGLAP comes from Dokshitzer, Gribov, Lipatov, Altarelli, and Parisi [14–16]. Equation (1.120) expresses the fact that a quark with momentum fraction x could have come from a parent quark with a larger momentum fraction ξ that has radiated a gluon. The probability that this happens is proportional to αsPqq(x /ξ ). The integral itself is a sum over all possible momentum fractions ξ of the parent. In figure 1.12, I show a figure taken from the Particle Data Group’s QCD review, which shows the kinematic ranges probed by various different experiments. As we can see from this figure, the various experiments have a good coverage in x and Q2 with pp collisions at the LHC probing the highest Q2. The world’s best data currently comes from the HERA (Hadron-Elektron-Ringanlage) at the DESY (Deutsches Elektronen-Synchrotron) collider in Hamburg. Figure 1.13 shows the extraction of F2(x , Q 2 ) obeyed by the HERA experiment. The red lines in the figure are obtained by parameterizing the structure functions at a fixed Q 2 = Q02 = 1.9 GeV 2 as [13]

xf (x ) = Ax B(1 − x )C (1 + Dx + Ex 2 ),

(1.121)

where, as usual, x is the fraction of the proton’s momentum taken by the struck parton in the infinite momentum frame. The PDFs parameterized in this are the

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Figure 1.12. Kinematic domains in x and Q2 probed by fixed-target and collider experiments. Some of the final states accessible at the LHC are indicated in the appropriate regions, where y is the rapidity. At LHC, the incoming partons have x1,2 = (M /14 TeV)exp(±y ) with Q = M where M is the mass of the state shown in blue in the figure. For example, exclusive J /ψ and ϒ production at high ∣y∣ at the LHC may probe the gluon PDF down to x ∼ 10−5. Figure from [1].

gluon distribution xg , the valence quark distributions xu v, xd v, and their corresponding anti-quark distributions. Heavy flavor coefficients were obtained within using the GM VFNS (Generalized Mass Variable Flavor-Number) scheme [17]. Once the fit at Q02 is made, the Q2 evolution of the distributions functions is provided by DGLAP evolution. The full equations solved include the leading-order splitting functions shown in table 1.1. I present these as the final result without derivation. Virtual corrections are responsible for the delta function appearing as the second term in Pqq [1]. Also, we have introduced the notation

1 1 = (1 − x )+ 1−x

∫0

1

dx

f (x ) = (1 − x )+

for 0 ⩽ x < 1,

∫0

1

dx

f (x ) − f (1) . (1 − x )+

(1.122)

(1.123)

In addition, to the splitting functions listed in table 1.1, one has

Pqiqj = Pq¯iq¯j = Pqqδij ,

(1.124)

Pqq¯ = 0,

(1.125)

Pqg ¯ = Pqg

(1.126)

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Figure 1.13. The structure function F2 as a function of Q2 and different values of x. Curves have been scaled on the vertical axis to visually separate the different values of x. Red curves are the predictions of DGLAP evolution. Figure from HERA Inclusive Working Group August 2010.

Pgq¯ = Pgq.

(1.127)

The resulting coupled leading-order DGLAP equations take the form ⎛ q (x , Q 2 ) ⎞ ⎜ i ⎟ d ⎜ q¯i (x , Q 2 )⎟ 2 d log Q ⎜ ⎟ ⎝ g (x , Q 2 ) ⎠ =

αs (Q 2 ) 2π

∫x

1

⎛ Pq q (x / ξ ) 0 Pqig (x / ξ )⎞⎛ q (ξ , Q 2 )⎞ i j ⎟⎜ i ⎟ dξ ⎜ 0 Pqiqj (x / ξ ) Pqig (x / ξ )⎟⎜ q¯i (ξ , Q 2 )⎟ . ⎜ ξ ⎜⎜ ⎟⎟⎜ 2 ⎟ ⎝ Pgqj (x / ξ ) Pgqj (x / ξ ) Pgg (x / ξ ) ⎠⎝ g (ξ , Q ) ⎠

1-30

(1.128)

Relativistic Quantum Field Theory, Volume 3

Table 1.1. Leading order DGLAP parton splitting functions. Here, TR = C (R ) = 1/2 is the Dynkin index and CA,F are the quadratic Casimirs in the adjoint and fundamental representations.

Diagram

Splitting Function

⎡ 1 + z2 ⎤ 3 CF⎢ (1 − z ) + 2 δ (1 − z )⎥ ⎣ ⎦ +

TR[z 2 + (1 − z )2]

⎡ 1 + (1 − z )2 ⎤ CF⎢ ⎥⎦ z ⎣

⎡ z 2CA ⎣ (1 − z ) + (1 − z ) z + +

(

1 z

)⎤⎦ +

11CA − 4Nf T R 6

δ (1 − z )

Note that beyond leading order Pqq ¯ and the off-diagonal Pqiqj no longer vanish due to diagrams of the form

So far, we looked in some detail into deep inelastic scattering. However, in the process we completely ignored the fact that the final states of the partonic level scattering processes were quarks and not hadrons. Due to confinement, outgoing quarks pull additional gluons and quark/anti-quark pairs from the vacuum. The result is at high energy one observes collimated jets of hadrons hitting the detectors. In figure 1.14, we show visualizations of two-jet (dijet) and three-jet events. If we are to make direct connection to experiment, we need to understand how to convert partons in the outgoing channels into the hadrons that are detected experimentally. Exercise 1.10 Verify equations (1.115) and (1.118).

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Figure 1.14. (Left) Visualization of a dijet event recorded by the CMS detector. (Right) Visualization of a three-jet event recorded by the CMS detector. Figures from [18].

Figure 1.15. One of the many possibilities that could occur in e +e−→hadrons. Figure from Bryan Webber, CERN Academic Training Lectures 2008 [19].

1.9 Hadron production in e +e− collisions Prior to the discussion of parton-hadron conversion, let us look at another important process that shows direct evidence for the existence of quarks with different flavors and allows one to experimentally extract their masses. The process is electron/anti-electron scattering to hadrons, e +e−→hadrons. To begin, we need to understand the underlying leading-order partonic process e +e−→qq¯

.

As we will see this leading-order subprocess can account for most of the hadrons produced in high-energy e +e− collisions. The cross section for e +e−→qq¯ is related to the QED process e +e−→μ+ μ− which has a total scattering cross section of 1-32

Relativistic Quantum Field Theory, Volume 3

σ (e+e−→μ+μ−) =

4πα 2 , 3Q 2

(1.131)

where Q2 = s. In the center-of-momentum frame, we have s = (pe− + pe+ )2 = (2Ee )2 = 4Ee2 . For the partonic process, we need to (a) take into account the fractional charges of the quarks in the final state and (b) the color factor associated with the possible color configurations in the final state

σ (e+e−→qi q¯i ) = Ncei2σ (e+e−→μ+μ−).

(1.132)

To obtain the total cross section for producing all types of hadrons, we have to sum over all quark flavors (i = u,d,s, …)

σ (e+e−→hadrons) = ∑ σ (e+e−→qi q¯i ) i

= 3 ∑ ei2σ (e+e−→μ+μ−),

(1.133)

i

where the sum over quark flavors i extends over flavors, which are below the threshold for their production, i.e. for which 2mi < Q. These thresholds will occur at low Q for the light quarks, for the strange quark the threshold is 2mc ≃ 190 MeV , for the charm quark the threshold is 2mc ≃ 2.6 GeV , for the bottom quark the threshold is 2mb ≃ 9.4 GeV , and finally for the top quark the threshold is 2m t ≃ 342 GeV. If we take the ratio of equations (1.133) and (1.131), we can define the ratio R as

R≡

σ (e+e−→hadrons) = Nc ∑ ei2 . σ (e+e−→μ+μ−) i

(1.134)

QCD predicts that R should change as we cross the mass thresholds for the creation of the different flavor quarks with

⎡⎛ 2 ⎞2 ⎛ 1 ⎞2 ⎛ 1 ⎞2 ⎤ R = 3⎢⎜ ⎟ + ⎜ ⎟ + ⎜ ⎟ ⎥ = 2 for u, d, and s, ⎝3⎠ ⎝3⎠ ⎦ ⎣⎝ 3 ⎠ 10 for u, d, s, and c, = 3 11 for u, d, s, c, and b, = 3 = 5 for u, d, s, c, b, and t.

(1.135)

The thresholds quoted above are lower bounds since the energies quoted correspond to states created at rest without any additional gluons necessary to bind them together (binding energy). As a result, the actual thresholds seen in the experiment will be slightly higher than these numbers. In figure 1.16, we show a collection of the world’s data on σ (e +e−→hadrons). As can be seen from this figure, there is quite good agreement between the values of R in each of the plateau regions and the predictions above. The dashed green line shows the prediction of the naive quark

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Relativistic Quantum Field Theory, Volume 3

Figure 1.16. World data on the total cross section of σ (e +e−→hadrons) (top) and the ratio R (bottom). σ (e +e−→hadrons) is the experimental cross section corrected for initial state radiation and electron–positron vertex loops. Data errors are at a total below 2 GeV and statistical above 2 GeV. The curves are a predictive guide: the dashed green curve is a naive quark–parton model prediction, and the solid red curve is a three-loop pQCD prediction [20].

model and the solid red line shows a three-loop pQCD prediction. Breit-Wigner parameterizations of J /ψ , ψ (2s ), and ϒ(nS ), n = 1, 2, 3, and 4, are also shown. The full list of references to the original data and the details of the extraction of R can be found in [20]. Finally, we note that in figure 1.16 we also plotted a three-loop pQCD prediction for R. As we will discuss below, even at leading order in αs , the existence of flat plateaus in R is broken with the result being

⎛ α (Q 2 ) ⎞ R = Nc ∑ ei2⎜1 + s ⎟. ⎝ π ⎠ i

(1.136)

As we can seen from figure 1.16, these corrections are small and become smaller as Q2 increases. The one-loop correction at Q2 = MZ is approximately 0.1/π ∼ 3% .

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1.10 Fragmentation functions With that under our belts, we can now address the question of how the quarks turn into hadrons that hit the detector. So far, it sufficed to say that the quarks must fragment into hadrons with unit probability which resulted in equation (1.133). At leading order in αs , in the center-of-momentum frame, the outgoing quark/anti-quark pair will have equal and opposite momentum and will manifest as two back-to-back jets of hadrons that have total momentum collinear with the original quark/anti-quark. In the stringy picture of confinement, the color potential energy becomes so large that one or more qq¯ pairs are created. As this process continues, eventually all of the energy of the original pair is transferred into two jets of hadrons. To describe the fragmentation of quarks into hadrons, we introduce fragmentation functions which are analogous to the parton distribution functions we have already encountered. In this case, however, the role is reversed and the fragmentation functions are related to the probability of finding a hadron of type h with energy fraction z in the debris formed after the production of a hard quark or anti-quark. In terms of the fragmentation functions, the differential cross section for e +e−→hadrons can be written as

dσ (e+e−→hadrons) = dz

¯ )⎡⎣Dqh(z ) + Dq¯h(z )⎤⎦ ∑ iσ (e+e−→qq

= Ncσ (e+e−→μ+μ−)∑ iei2⎡⎣Dqh(z , Q 2 ) + Dq¯h(z , Q 2 )⎤⎦ ,

(1.137)

where z = Eh /Eq = 2Eh /Q . This can be visualized as

.

One can compute these knowing the wavefunction of the produced hadron, which requires some non-perturbative input. In practice, one can measure the fragmentation functions and then use them in a process-independent manner to make predictions for process other than σ (e +e−→hadrons). Similar to the parton distribution functions, the fragmentation functions obey sum rules implied by conservation of momentum and energy9 The first sum rule assumes that all of the quark energy is converted to hadrons, some energy may also be ‘lost’ to electromagnetic and weak radiation. This will be small compared to the amount that goes to hadrons, so we can ignore it for now. 9

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Relativistic Quantum Field Theory, Volume 3

∑ ∫0

1

dz z Dqh(z , Q 2 ) =

h

∑ ∫0

1

dz z Dq¯h(z , Q 2 ) = 1

(1.139)

h

and

∑ ∫z q

1

min

dz⎡⎣Dqh(z , Q 2 ) + Dq¯h(z , Q 2 )⎤⎦ = nh ,

(1.140)

where zmin is the threshold energy fraction 2mh/Q for producing a hadron with mass mh and nh is the average number of hadrons of type h. The first constraint states that the sum of the energies of all hadrons produced is the energy of the parent quark. The second constraint states that the number of hadrons of type h is given by the sum of probabilities of obtaining h from all possible parent quark flavors. The fragmentation functions also obey a DGLAP equation of the form

∂ Dih(z , Q 2 ) = ∂ log Q 2

∑ ∫z j

1

dz αs Pji (z , Q 2 )D jh(z , Q 2 ), z 2π

(1.141)

where Pij are the splitting functions encountered previously, which have a perturbative expansions of the form

Pji (z , Q 2 ) = P ji(0)(z ) + P ji(1)(z , Q 2 ) + ⋯.

(1.142)

At LO, we can take the ratio of equations (1.137) and (1.133) to obtain

1 dσ (e+e−→hX) = dz σ

∑ei2⎡⎣Dqh(z ) + Dq¯h(z )⎤⎦ i

∑ei2

(1.143)

,

i

which indicates that there should be scaling of the ratio 1/σ dσ /dz; however, as before, the NLO contributions result in predictable scaling violation of the cross sections, which is shown in figure 1.17. The fragmentation functions are, in principle, process independent. Once the fragmentation functions are known at a fixed energy from say e +e−→hX , then then can be used in other processes such as leptoproduction.

.

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Relativistic Quantum Field Theory, Volume 3

Figure 1.17. Collected data for 1/σ dσ /dz at various z as a function of Q2. The lines are the predictions of the DGLAP evolution given a phenomenological fit at fixed Q 2 = Q02 . Figure from Bryan Webber, CERN Academic Training Lectures 2008 [19].

This picture results in a prediction for the scaled differential cross section of the form

1 dσ (ep → hX ) = σ dz

∑eq2fq (x)Dqh(z ) q

∑eq2fq (x)

,

(1.145)

q

where fq(x) are the proton structure functions.

1.11 Solution of the DGLAP equations using Mellin moments Finally, I would like to briefly discuss a bit about how one solves the DGLAP equations for the splitting functions for DIS10. These can be expressed compactly in the form 10

The same method can be used for the fragmentation functions with some small modifications.

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Relativistic Quantum Field Theory, Volume 3

∂fi (x , t ) = ∂ log t

∑ ∫x

1

j

dξ αs α Pij (x / ξ )fi (x , t ) ≡ s Pij ⊗ fi , 2π ξ 2π

(1.146)

where here t ≡ Q 2 is not the Mandelstam variable but a convenient relabeling. Given fi(x, t) at some scale t = t0 we can compute all fi(x, t) at another scale using this equation. One method for doing this is to take moments (Mellin transforms) with respect to x

f˜ (N , t ) ≡

∫0

1

dx x N −1fi (x , t ).

(1.147)

The inverse Mellin transform is

fi (x , t ) =

1 2πi

∫C

dNx −N f˜i (N , t ),

(1.148)

where the contour C is parallel to the imaginary axis and to the right of all singularities of the integrand. After Mellin transformation, the convolution contained in the DGLAP equation becomes a product

∂f˜i (N , t ) = ∂ log t

∑ γij(N , αs )f˜j (N , t ),

(1.149)

j

where moments of the splitting functions result in a perturbative expansion of the anomalous dimensions γij ∞

γij (N , αs ) =

⎛ αs ⎞n+1 ⎟ . 2π ⎠

(1.150)

dz z N −1Pij (z ),

(1.151)

∑ γij(n)(N )⎜⎝ n=0

At leading-order in αs , one has

γij(0)(N ) = P˜ij (N ) =

∫0

1

Using the splitting functions presented in the last lecture, one obtains (homework) (0) (N ) γqq

=

(0) (N ) = γqg

(0) (N ) γgq

=

(0) (N ) = γgg

N ⎡ 1 1⎤ 1 ⎢ CF − + − 2 ∑ ⎥, ⎢⎣ 2 N (N + 1) k ⎥⎦ k=2 ⎡ 2 + N + N2 ⎤ TR⎢ ⎥, ⎣ N (N + 1)(N + 2) ⎦ ⎡2 + N + N2⎤ CF⎢ ⎥, ⎣ N (N 2 − 1) ⎦ ⎡ 1 1 1 2CA⎢ − + + − (N + 1)(N + 2) N (N − 1) ⎢⎣ 12

1-38

(1.152)

N

∑ k=2

1⎤ 2 ⎥ − Nf TR. 3 k ⎥⎦

Relativistic Quantum Field Theory, Volume 3

Figure 1.18. Electromagnetic structure function at small x. Lines are two standard structure function fits used in the literature. Figure from Bryan Webber, CERN Academic Training Lectures 2008 [19].

Larger-N are sensitive to large x. At small x one can take N → 1 which results in γgg → ∞ and γqq → 0. As a result, we expect the structure functions to grow rapidly at small x, which is observed experimentally; see figure 1.18. Note, however, that at small x higher-order corrections also become large and one needs to take these into account. Eventually, the gluon density will become so large that the distribution functions will saturate due to gluon recombination rather than splitting. This is the realm of the colored glass condensate [21–23]. However, prior to saturation, one can have an improved description of the physics using the Balitsky, Fadin, Kuraev, and Lipatov (BFKL) equation11. Exercise 1.11 Verify the results in equation (1.152).

1.12 Drell–Yan scattering At leading-order in αs and α the production of lepton/anti-lepton pairs from scattering of a proton off of another proton can be written in terms of a quark scattering diagram of the form 11

See a nice introduction by Salam and references therein [24].

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Relativistic Quantum Field Theory, Volume 3

.

In addition to DIS and e +e−, this process has played a central role in determining the structure functions and in testing the parton model. To calculate the cross section, we begin with the cross section for the parton-level subprocess

σˆ (qq¯ → ℓ −ℓ +) =

4πα 2 2 eq . 3Q 2

(1.154)

In order to embed this in the hadronic process, we rewrite this as a differential cross section for the production of lepton pairs having invariant mass Q 2 , where

Q 2 = sˆ = (pq + pq¯ )2 = (xp1 + yp2 )2 .

(1.155)

Figure 1.19. Evidence for approximate scaling in the Drell–Yan process. The COM energy s is in GeV. Data is from Fermilab and the figure is taken from [12].

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Relativistic Quantum Field Theory, Volume 3

The resulting differential cross section is

4πα 2 2 dσˆ eq δ(Q 2 − sˆ ). = 3Q 2 dQ 2

(1.156)

The hadronic cross section then becomes

dσ 1 = 2 Nc ∑ 2 dQ Nc q

dσˆ , ∫ dx ∫ dy fq (x)fq¯ (y ) dQ 2

(1.157)

where the factor of 1/Nc2 comes from the averaging over the incoming colors of the q and q¯ and the factor of Nc comes from the sum over qq¯ combinations that result in a colorless state that can couple to the virtual photon. In the high-energy limit, one has

sˆ = (xp1 + yp2 )2 = x 2p12 + y 2 p22 + 2xy p1 · p2 ≃ xy (2p1 · p2 )

(1.158)

s = (p1 + p2 )2 ≃ 2p1 · p2 ,

(1.159)

and

and hence one has sˆ ≃ xys . Putting the pieces together we obtain

4πα 2 dσ = 3NcQ 4 dQ 2

⎛

s ⎞ ⎠

∑ eq2 ∫ dx ∫ dy fq (x)fq¯ (y ) δ⎜1 − xy Q 2 ⎟. ⎝

q

(1.160)

At this lowest order in which we have ignored gluon emission, etc, we expect a scaling result. If we compute Q 4dσ /dQ 2 , we see that the result only depends on the ratio s/Q2. Data for this is shown in figure 1.19.

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]

Tanabashi M et al Particle Data Group 2018 Phys. Rev. D 98 030001 Rosenbluth M N 1950 Phys. Rev. 79 615–19 Janssens T, Hofstadter R, Hughes E B and Yearian M R 1966 Phys. Rev. 142 922–31 Perdrisat C F, Punjabi V and Vanderhaeghen M 2007 Prog. Part. Nucl. Phys. 59 694–764 Friedman J I and Kendall H W 1972 Ann. Rev. Nucl. Part. Sci. 22 203–54 Bjorken J D 1969 Phys. Rev. 179 1547–53 Feynman R P 1969 Phys. Rev. Lett. 23 1415–7 Bjorken J D and Paschos E A 1969 Phys. Rev. 185 1975–82 Tomboulis E 1973 Phys. Rev. D 8 2736–40 Casher A 1976 Phys. Rev. D 14 452 Amaudruz P et al 1992 New Muon Nucl. Phys. B 371 3–31 Halzen F and Martin A 1984 Quarks & Leptons: An Introductory Course in Modern Particle Physics (New York: Wiley) Abramowicz H et al 2015 Eur. Phys. J. C 75 580 Altarelli G and Parisi G 1977 Nucl. Phys. B 126 298–318 Dokshitzer Y L 1977 Sov. Phys. JETP 46 641–53 [Zh. Eksp. Teor. Fiz.73,1216(1977)] Gribov V N and Lipatov L N 1972 Sov. J. Nucl. Phys. 15 438–50 [Yad. Fiz.15,781(1972)] Thorne R S 2006 Phys. Rev. D 73 054019

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[18] CMS Public Wiki https://twiki.cern.ch/twiki/bin/view/CMSPublic/WebHome [19] Weber B 2008 CERN Academic Training Lectures https://www.hep.phy.cam.ac.uk/theory/ webber/ [20] Ezhela V, Lugovsky S B and Zenin O V 2003 (preprint:hep-ph/0312114) [21] McLerran L D and Venugopalan R 1994 Phys. Rev. D 49 2233–41 [22] McLerran L D and Venugopalan R 1994 Phys. Rev. D 49 3352–5 [23] Gelis F, Iancu E, Jalilian-Marian J and Venugopalan R 2010 Ann. Rev. Nucl. Part. Sci. 60 463–89 [24] Salam G P 1999 Acta Phys. Polon. B 30 3679–705

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IOP Concise Physics

Relativistic Quantum Field Theory, Volume 3 Applications of quantum field theory Michael Strickland

Chapter 2 Weak interactions

Having covered QED and QCD, we must now understand the quantum field theory formulation for weak interactions and the unification of the electromagnetic and weak forces. We will begin with a brief historical review and discussion of the basic experimentally observed processes that take place due to weak interactions. Two example weak-decay processes are

π − = μ− + ν¯μ, μ− = e− + ν¯e + νμ,

τ ≃ 2.6 × 10−8 s, τ ≃ 2.2 × 10−6 s,

(2.1) (2.2)

where, to the right of both processes, we have indicated the typical time scales for each process, which are taken from the Particle Data Group Review of Particle Physics [1]. These time scales should be compared to the typical strong and electromagnetic force decay times scales, which are τstrong ∼ 10−23 s and τE&M ∼ 10−16 s, respectively. As we shall see, the long lifetimes for weak decays are related to the strength of the weak coupling relative to the electromagnetic coupling

α weak ≪ α ≪ αs .

(2.3)

Note that we considered the π − but could have equally well considered π + decays; however, the π 0 can decay via quark–antiquark annihilation on the time scale of the electromagnetic force. In all weak-decay processes, we conserve charge, energy-momentum, and lepton numbers (Li) with

Le = 1, Le = −1,

doi:10.1088/2053-2571/ab3a99ch2

e− and νe , e+ and ν¯e ,

2-1

(2.4) (2.5)

ª Morgan & Claypool Publishers 2019

Relativistic Quantum Field Theory, Volume 3

Le = 0 all other particles,

(2.6)

with similar assignments for the μ- and τ-electrons. Such processes are responsible for the β -decay of atomic nuclei, e.g. 10C → 10B * + e + + νe , the decay of the proton inside the nucleus1 p → n + e +νe , and the decay of the neutron n → p + e− + ν¯e .2

2.1 Early models of the weak interaction Inspired by the electromagnetic interaction, in order to describe weak reactions involving protons and neutrons, Fermi postulated [2–4] a local vertex for p + e− → n + νe of the form

which, in analogy to QED, results in an invariant matrix element

M = G ⎡⎣u¯ nγ μup⎤⎦⎡⎣u¯ νeγμu e ⎤⎦ ,

(2.8)

where G is the Fermi constant or ‘weak coupling constant’. This form is local (does not depend on momentum and hence maps to a delta function in space) and is a vector–vector (V–V) interaction which is only one of many possible interactions. Fermi’s model explained some of the experimental results on β-decay but not all. In 1957, Lee, Oehme, and Yang published a survey of all available data including K + → 2π and 3π which concluded that parity is not conserved in weak interactions [5]. Our modern understanding of this is that neutrinos have only one parity state. From observations, it was concluded that the neutrinos are left-handed and antineutrinos are right-handed, e.g. there exist only νL and ν¯ R states in nature. If neutrinos were exactly massless, then their chirality (left or right) can be mapped to two possible alignments of the spin, either parallel or antiparallel to their momentum, which defines their helicity; however, since we now know that neutrinos have small masses and can change flavors due to quantum mechanical state mixing [1], the proper way to understand this is through the chiral projection operators

PL =

1 (1 − γ 5), 2

(2.9)

PR =

1 (1 + γ 5). 2

(2.10)

Note that this decomposition of a spinor is always possible since PL + PR = 1. Also, note that PL2,R = PL,R and PL,RPR,L = 0. 1 2

This process is forbidden in vacuum by energy conservation. This process has a half-life of τ ∼ 900 s in vacuum; however, within the nucleus, most neutrons are stable.

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Relativistic Quantum Field Theory, Volume 3

Using these projection operators, one can introduce charge-lowering

1 j(−μ ) = u¯ eγ μ (1 − γ 5)u v, 2

(2.11)

1 j(+μ ) = u¯ νγ μ (1 − γ 5)u e 2

(2.12)

and charge-raising

weak currents. Note that in the calculations that follow, in order to keep things simple, I will ignore the neutrino masses relative to other large mass scales such as the muon mass.

2.2 Muon decay In this language, Fermi’s V-V model requires only a small modification and for muon decay one has

M(μ− → e− + ν¯e + νμ) =

G ⎡ ⎣u¯ νμγ σ(1 − γ 5)up⎤⎦⎡⎣u¯ eγμ(1 − γ 5)u νe⎤⎦ , 2

(2.13)

where overall factors have been absorbed into the definition of the Fermi weakcoupling constant G.3 At this point, however, the interaction is still point-like and the coupling constant G replaces e 2 /q 2 which would appear if this were QED and hence has units of GeV−2. To turn this into our modern version of the weak-interaction, we replace the point-like VA with one mediated by a massive boson and introduce a dimensionless weak-coupling constant g. For our example of muon decay, we require the exchange of a negatively charged boson, with the corresponding Feynman diagram shown in figure 2.1. Figure 2.1 translates into

⎤ ⎡ g 1 M(μ− → e− + ν¯e + νμ) = ⎢ u¯ νμγ σ (1 − γ 5)up⎥ ⎦ ⎣ 2 2 ⎤ ⎡ g 1 1 × 2 u¯ eγμ (1 − γ 5)u νe⎥ ⎢ 2⎣ ⎦ 2 MW − q 2

Figure 2.1. Leading-order Feynman diagram for muon decay.

3

The square root of two is a convention.

2-3

(2.14)

Relativistic Quantum Field Theory, Volume 3

with MW ≃ 80 GeV [1].4 Compared to equation (2.13), we see that there is a relation between the dimensionless coupling g and the dimensionful Fermi weak-coupling G

lim q →0

G g2 = . 8MW2 2

(2.15)

As a result, we learn that, assuming that g is on the same order as the electromagnetic coupling, the smallness of the Fermi weak-coupling G is due to the largeness of the W-boson mass. In addition, due to the largeness of the weak boson mass, the weak interaction will have a very short range with r ∼ 1/MW . That g should be the same order of magnitude as e is not obvious at this point, but we will show that this is the case in the next chapter. Details for muon decay in the low-energy limit We now work the through details of the low-energy muon decay. Putting momentum labels on the spinors using figure 2.1 and suppressing their type for compactness, one has in this limit

M(μ− → e− + ν¯e + νμ) =

G [u¯(k )γ α(1 − γ 5)u(p )] 2 × [u¯(p′)γα(1 − γ 5)u(k′)],

(2.16)

referring to chapter 3 of volume 1 we learned that the differential decay rate can be expressed in terms of the invariant amplitude using

dΓ =

1 ∣M∣2 dQ , 2E

(2.17)

where the bar over ∣M∣2 indicates the spin-averaged quantity and dQ is the threeparticle phase space for the decay products

⎛ d 3p′ ⎞⎛ d 3k ⎞⎛ d 3k′ ⎞ dQ = ⎜ ⎟⎜ ⎟⎜ ⎟(2π )4δ (4)(p − p′ − k − k′), ⎝ (2π )32E ′ ⎠⎝ (2π )32ω ⎠⎝ (2π )32ω′ ⎠

(2.18)

where we have taken p′ = (E ′, p), k = (ω, k), k′ = (ω′, k′), and the final delta function enforces energy-momentum conservation. Using

∫

d 3k = 2ω

∫ d 4k Θ(ω)δ(k 2),

(2.19)

one obtains

dQ =

4

1 ⎛ d 3p′ ⎞⎛ d 3k′ ⎞ ⎜ ⎟⎜ ⎟Θ(E − E ′ − ω′)δ (4)((p − p′ − k′)2 ). (2π )5 ⎝ 2E ′ ⎠⎝ 2ω′ ⎠

There are also weak neutral currents mediated by the Z boson. We will discuss those shortly.

2-4

(2.20)

Relativistic Quantum Field Theory, Volume 3

Calculation of ∣M∣2 : To proceed, we must calculate the spin-average invariant amplitude squared. Without the spin average, one has

∣M∣2 =

G2 ⎡ ⎤⎡ ⎤ μ 5 5 ⎣ u¯(k )γ (1 − γ )u(p )⎦⎣ u¯(p′)γμ(1 − γ )u(k′)⎦ 2 * * × ⎡⎣ u¯(k )γ ν(1 − γ 5)u(p )⎤⎦ ⎡⎣ u¯(p′)γν(1 − γ 5)u(k′)⎤⎦ .

(2.21)

Performing the complex conjugates in the last two terms and performing the spin averages assuming mν = me = 0 and m = mμ ≠ 0, one obtains ∣M∣2 =

G2 ⎡ μ Tr⎣ k γ (1 − γ 5)( p + m )(1 + γ 5)γ ν ⎤⎦Tr ⎡⎣ p ′γμ(1 − γ 5) k ′(1 + γ 5)γ ν ⎤⎦ . (2.22) 2

To proceed, we must perform the two Dirac traces. This can be done by hand; however, it is quite tedious. At this point, I mention that there exist computer algebra systems that can assist with this task. For example, the Mathematica package FeynCalc is very useful in this regard [6, 7]. Below is an example for how to use this package for this example.

.

Using this result, one obtains

∣M∣2 =

G2 [256(k · p′)(k′ · p )] = 64G 2(k · p′)(k′ · p ). 2

(2.24)

To evaluate the dot products remaining, we work in the muon rest frame, in which one has

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Relativistic Quantum Field Theory, Volume 3

p = (m , 0, 0, 0), p′ = (E ′ , p′), k = (ω, k), k′ = (ω′ , k′).

(2.25)

Using these vectors and the fact that p′ + k = p − k′ by energy conservation, one can find

1 1 (k · p′)(k′ · p ) = (p′ + k )2 (k′ · p ) = (p − k′)2 ω′m 2 2 1 2 = (m − 2 mω′)ω′m , 2

(2.26)

giving, finally

∣M∣2 = 32G 2(m 2 − 2 mω′)ω′m .

(2.27)

Putting the results together: Plugging equation (2.27) into (2.17) with (2.20) gives

dΓ =

G 2 ⎛ d 3p′ ⎞⎛ d 3k′ ⎞ ⎜ ⎟⎜ ⎟ 2 mπ 5 ⎝ 2E ′ ⎠⎝ 2ω′ ⎠ × Θ(m − E ′ − ω′)δ(m 2 − 2mE ′ − 2mω′ + 2E ′ω′(1 − cos θ )) × [(m 2 − 2 mω′)ω′m ],

(2.28)

where θ is the angle between the outgoing electron and anti-electron–neutrino in the muon rest frame. This is the final differential decay rate. If we want to know the total decay rate for decaying into any possible final momenta, we must integrate over p′ and k′ using

d 3p′ = 4π (E ′)2 dE ′ ,

(2.29)

d 3k′ = 2π (ω′)2 dω′d cos θ ,

(2.30)

and use

δ(A + B cos θ ) =

⎞ 1 ⎛A δ⎜ − cos θ ⎟ . ⎝ ⎠ B B

(2.31)

To proceed, we use the delta function to perform the integration over cos θ . For this delta function to have support in the range −1 ⩽ cos θ ⩽ 1, the limits on ω′ and E ′ are

1 1 m − E ′ ⩽ ω′ ⩽ m , 2 2 0 ⩽ E′ ⩽

1 m. 2

2-6

(2.32) (2.33)

Relativistic Quantum Field Theory, Volume 3

Because of these constraints, the Theta function appearing in (2.28) is automatically equal to identify and one obtains

dΓ G 2m = dE ′ 2π 3

1m 2

∫ 1 m−E′ dω′ω′(m − 2ω′) 2

⎛ 4E ′ ⎞ G 2m 2 ⎟. = (E ′)2 ⎜3 − 3 ⎝ 12π m ⎠

(2.34)

Up to the overall scaling (choice of G), this result is in excellent agreement with the experimentally observed electron spectrum resulting from muon decay. To obtain the decay width and associated half-life for this process, we can further integrate over the outgoing electron’s energy to obtain

Γ=

1 = τ

∫0

m /2

dE ′

dΓ G 2m5 . = dE ′ 192π 3

(2.35)

Based on this, we can fit G using the experimentally measured half-life for this process. Once G is fixed in this manner, we should be able to apply the formalism to any other process involved weak decays. Exercise 2.1. Derive equation (2.21) from the previous equation. Exercise 2.2. Use FeynCalc to evaluate the trace Tr⎡⎣( k + m1)γ μ( p + m2 )γ ν ⎤⎦. Exercise 2.3. Verify that the limits specified in equations (2.32) and (2.33) follow from when the delta function has support.

2.3 Charged pion decay In the introduction of this chapter, we mentioned charged pion decay. We would now like to look at this process in more detail. In figure 2.2, I present the leadingorder Feynman diagram for the process π − → μ− + ν¯μ. In the low-energy limit (q ≪ Mweak bosons ), we can once again collapse the weak boson propagator down to a point-like interaction. We have already learned that we can write the right part of the diagram as charge-lowering current and, since the invariant amplitude must be a scalar, this current must be contracted with a vector (or axial-vector) current on the left giving, in general, for the low-energy limit

Figure 2.2. Leading-order Feynman diagram for π − → μ− + ν¯μ .

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Relativistic Quantum Field Theory, Volume 3

M=

G μ⎡ Jπ ⎣u¯(p )γμ(1 − γ 5)v(k )⎤⎦ . 2

(2.36)

To proceed, we need to establish some basic facts about Jπμ. As mentioned already, it could be a vector or an axial vector; however, we know that the π − has spin zero. Therefore, q μ is the only four-vector available from which to construct Jπμ. As with any tensor basis, there is a coefficient function that must be Lorentz invariant and hence can only depend on q 2 = mπ2 , i.e.

( )

Jπμ = f (q 2 )q μ = f m π2 q μ = fπ q μ,    ≡fπ

(2.37)

where we have introduced the pion decay constant, fπ. Using this, one finds that, in the low-energy limit, one has

M=

G fπ (p + k ) μ⎡⎣u¯(p )γμ(1 − γ 5)v(k )⎤⎦ . 2

(2.38)

We can simplify this further by making use of the Dirac equation of motion in momentum space

k v(k ) = 0,

(2.39)

u¯(p )( p − mμ) = 0,

(2.40)

G fπ mμ[u¯(p )(1 − γ 5)v(k )]. 2

(2.41)

to obtain

M=

Differential decay rate: The differential decay rate for this process in the pion rest frame is

dΓ =

⎛ d 3p ⎞⎛ d 3p ⎞ 1 ∣M∣2 ⎜ ⎟⎜ ⎟(2π )4δ (4)(q − p − k ), 2mπ ⎝ (2π )32E ⎠⎝ (2π )32ω ⎠

(2.42)

where p = (E , p) and k = (ω, k) are the muon and antimuon-neutrino fourmomentum, respectively. Spin-averaged ∣M∣2 : the spin-averaged modulus-squared of the invariant amplitude is

∣M∣2 =

G2 2 2 ⎡ f m μ Tr⎣( p + mμ)(1 − γ 5) k (1 + γ 5)⎤⎦ = 4G 2f π2 m μ2(p · k ). 2 π

(2.43)

In the pion rest frame, one has k = −p and

p · k = Eω − k · p = Eω + k2 = ω(E + ω),

(2.44)

where, in the last step, I used the fact that the neutrino is massless (approximately).

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Relativistic Quantum Field Theory, Volume 3

Putting the pieces together: The total decay rate can then be expressed as

Γ= = =

G 2f π2 m μ2

d 3p d 3k δ(mπ − E − ω)δ (3)(k + p)ω(E + ω) E ω

∫ (2π )2 2mπ G 2f π2 m μ2 2πmπ G 2f π2 m μ2 2πmπ

∫

ωdω δ(mπ − E − ω)ω(E + ω) E

∫ ω 2 dω δ ( m π −

(2.45)

⎛ ω⎞ ω 2 + m μ2 − ω ⎜1 + ⎟ . ⎝ E⎠

)

Evaluating the remaining integral, one obtains (homework)

Γ=

2 G 2f π2 mπ m μ2 ⎛ m2⎞ 1 ⎜⎜1 − μ2 ⎟⎟ . = 8π τ mπ ⎠ ⎝

(2.46)

Based on this result and our earlier outline for how to determine G from muon decay, one can determine fπ experimentally. The current experimental value for the charged pion decay constant is [8]

fπ ± = 130.4 ± 0.04 ± 0.2 MeV,

(2.47)

with the first error being due to an error in the determination of the ud element of the Cabibbo–Kobayashi–Maskawa (CKM) matrix (more on this later) and the second being due to higher-order corrections. In addition to extracting fπ, one can also use the ratio of the decay rates to muons and electrons, in which case fπ drops out in the ratio leaving

⎛ m e ⎞2 ⎛ m 2 − m 2 ⎞2 Γe e ⎟⎟ = 1.283 × 10−4 . = ⎜ ⎟ ⎜⎜ π Γμ ⎝ mμ ⎠ ⎝ m π2 − m μ2 ⎠

(2.48)

This parameter-free prediction for the branching ratios is very well confirmed by experiment. In figure 2.3, I show the most common charged pion decay modes [1]. Taking the ratio of the third and first entries, one obtains (1.230 ± 0.004) × 10−4 which is in very good agreement with the calculation presented here considering that we computed only the leading-order contribution. Exercise 2.4. Compute the trace in equation (2.43). Exercise 2.5. Verify equation (2.46).

2.4 Electron–neutrino and electron–antineutrino scattering By exchanging a W ± in the t- or s channels, it is possible to scatter electrons off of neutrinos or antineutrinos, respectively. In figure 2.4, I show the Feynman diagrams for both of these processes. We will now try to understand these a bit more because 2-9

Relativistic Quantum Field Theory, Volume 3

Figure 2.3. Decay modes and branching fractions for π + decays. Taken from the Particle Data Group listings [1].

Figure 2.4. Feynman diagrams for leading-order (left) electron–neutrino and (right) electron–antineutrino scattering.

they will be useful when considering (anti-)neutrino-quark scattering in the next section. Cross section for electron–neutrino: For the t-channel process (left diagram in figure 2.4) the invariant amplitude in the low-energy limit is

M=

G [u¯(k′)γ μ(1 − γ 5)u(p )][u¯(p′)γμ(1 − γ 5)u(k )]. 2

(2.49)

The resulting spin-averaged differential cross section is

dσ eν 1 G 2s 2 = ∣ ∣ = , M dΩ 64π 2s 4π 2

(2.50)

and the total cross section is

σ eν =

G 2s . π

(2.51)

Cross section for electron–antineutrino scattering: For the s-channel process (right diagram in figure 2.4), the spin-averaged differential cross section is

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Relativistic Quantum Field Theory, Volume 3

dσ eν¯ G 2s = (1 − cos θ )2 , dΩ 16π 2

(2.52)

where θ is the angle between the in and outgoing electron momenta in the center-ofmomentum frame. The resulting total cross section is

σ eν¯ =

G 2s . 3π

(2.53)

Prediction: From these two results, we see that a prediction emerges that the total cross-section for low-energy electron antineutrino scattering is one-third that of electron neutrino scattering, i.e. σ eν¯ = 13 σ eν . This prediction has been verified experimentally, see [9] and references therein. For more information about these processes and how to compute them at high-energies and beyond leading order, I refer the reader to [10]. Exercise 2.6. Verify equations (2.50) and (2.52).

2.5 Neutrino–quark scattering We will now consider neutrino–quark scattering. This is used as an additional way to constrain the parton distribution functions introduced in chapter 1 of this volume. Restricting our attention, for now, to the charge raising and lowering interactions mediated by the W ±, one has 2.5.1 Charge raising current The charge raising current is

2.5.2 Charge lowering current The hermitian conjugate of this process gives the charge lowering current

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Relativistic Quantum Field Theory, Volume 3

Figure 2.5. Parton-level embedding of νμ + d → μ− + u . N (p ) on the left is a general nucleon and qi is the incoming quark of flavor i and qf is the outcoming quark of flavor f.

2.5.3 Differential cross section Since, besides their flavor-changing nature, the invariant amplitudes above have the same form as the electron neutrino scattering discussed in the previous subsection so we can immediately write

G 2sˆ dσ (νμ + d → μ− + u ) = dΩ 4π 2

(2.56)

G 2sˆ dσ (ν¯μ + u → μ+ + d ) = (1 + cos θ )2 , dΩ 16π 2

(2.57)

and

where θ is the center-of-momentum frame angle between the incoming up quark and the outgoing down quark and sˆ is the neutrino-parton center-of-mass energy squared. 2.5.4 Embedding Next, we have to embed this into the hadron level description as we did in chapter 1 for QED and QCD processes. The diagrammatic representation of this is shown in figure 2.5. Based on the discussion in chapter 1, we have

dσ (νμ + → μ− + X ) = dxdy

⎛ dσ ⎞

∑ fi (x)⎜ dy ⎟

where we remind the reader that 2-12

i

⎝

⎠sˆ=xs

,

(2.58)

Relativistic Quantum Field Theory, Volume 3

y≡1−

1 p · k′ ≃ (1 + cos θ ). 2 p·k

(2.59)

Using

∣dy∣ =

1 d cos θ , 2

(2.60)

one obtains

d Ω = 2π cos θ = 4πdy ,

(2.61)

G 2xs dσ (νμ + d → μ− + u ) = dy π

(2.62)

G 2sˆ dσ (ν¯μ + u → μ+ + d ) = (1 − y )2 , dy π

(2.63)

allowing us to write

and

with the same holding for ν¯μ + d¯ → μ+ + u¯ and νμ + u¯ → μ− + d¯ , respectively. To proceed, we focus on the case of an isoscalar target in which there are equal numbers of protons and neutrons. So far, we have only considered the interactions of neutrinos with up and down quarks. We will see in the next section that, in general, they can interact with all flavors, but to keep it simple for now, let us focus on this simpler case. In this case, for an isoscalar target the sum of the down and anti-up quark are given by

d p + d n = d (x ) + u(x ) ≡ Q(x ),

(2.64)

u¯ p + u¯ n = u¯(x ) + d¯(x ) ≡ Q¯ (x ),

(2.65)

where we have used the fact that to first approximation isospin invariance allows us to transform a neutron into a proton by interchanging d ↔ u . Summing over quarks and anti-quarks and using equations (2.62) and (2.63), one obtains

G 2xs dσ (νμ + p → μ− + X ) = [d (x ) + (1 − y 2 )u¯(x )], dxdy π

(2.66)

with a similar relation for νμ + n → μ− + X . Averaging over protons and neutrons in our isoscalar target, one obtains

G 2xs dσ (νμ + N → μ− + X ) = [Q(x ) + (1 − y 2 )Q¯ (x )] dxdy 2π and

2-13

(2.67)

Relativistic Quantum Field Theory, Volume 3

Figure 2.6. Example weak scattering process mediated by the neutral weak boson Z0.

G 2xs ¯ dσ (ν¯μ + N → μ+ + X ) = [Q(x ) + (1 − y 2 )Q(x )]. dxdy 2π

(2.68)

By taking a linear combination of the experimental results for each of these differential scattering rates, we can extract Q(x ) and Q¯ (x ). This can be used as an additional constraint for the global determination of parton distributions functions. For more details, we refer to the reader to the ‘Structure Functions’ chapter of [1]. Exercise 2.7. Verify equations (2.67) and (2.68).

2.6 Weak neutral currents As a consequence of electroweak unification, Glashow, Salam, and Weinberg (GSW) predicted the existence of a charge neutral weak boson called the Z0 (figure 2.6) [11–13]. In 1973, experimentalists using the Gargamelle bubble chamber at CERN found evidence for the existence of ‘weak neutral currents’ [14–17]. Example processes are

ν¯μ + e− → ν¯μ + e−,

(2.69)

νμ + N → νμ + X ,

(2.70)

ν¯μ + N → ν¯μ + X.

(2.71)

Unlike the charged weak currents, it was found experimentally that the neutral current is a linear combination of vector–axial-vector (V-A) and and vector–vector (V-V) interactions5. In practice, in the low-energy limit, one has a generalized invariant amplitude of the form

M(ν + q → ν + q ) =

GN [u¯ νγ μ(1 − γ 5)u ν ]⎡⎣u¯ qγμ(cVq − cAqγ 5)u q⎤⎦ , 2

(2.72)

where GN is a new low-energy coupling associated with the weak neutral current. If the interaction were pure V-A, then one would have cVq = cAq . Introducing 5 This was predicted by the GSW model due to mixing of the Z0 and the photon. We discuss further in the next chapter.

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Relativistic Quantum Field Theory, Volume 3

gLq ≡

1 q ( cV + cAq ), 2

(2.73)

gRq ≡

1 q ( cV − cAq ), 2

(2.74)

one finds experimentally [1] that

gLq ≃ 0.30,

(2.75)

gRq ≃ 0.024,

(2.76)

which tells us that the weak neutral current interaction is predominantly V-A. We will return to this shortly when discussing electroweak unification.

2.7 The Cabibbo angle and the CKM matrix So far, we have learned that leptons and quarks participate in weak interactions in pairs called ‘weak isospin doublets’

(eν ), e −

⎛ νμ ⎞ ⎜ ⎟, ⎝ μ−⎠

(du ).

(2.77)

Could there be other doublets to consider such as

(cs ),

(bt )

(2.78)

that must be taken into account? Yes, but that is not all. Experimental observations of K + → μ+ + νμ tell us that there is also a u-s coupling since the valence quark structure of the K + is us¯ [1]. Instead of introducing a separate coupling for each possibility, in 1963 Cabibbo proposed that quarks in the top row are u, c, and b, while the bottom rows of the doublets are linear combinations of the heavier quark states [18]. Taking into account only the u, d, s and c, one has

(du′), ( sc′),

(2.79)

d ′ ≡ d cos θc + s sin θc ,

(2.80)

s′ ≡ −d sin θc + s cos θc ,

(2.81)

with

with θc beginning the Cabibbo angle or quark mixing angle6. The Cabibbo angle can be measured by, for example, comparing strangeness-changing weak interactions with non-strangeness changing weak interactions 6

The original Cabibbo paper had only the first doublet.

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Relativistic Quantum Field Theory, Volume 3

Figure 2.7. Examples of Caibbo favored weak decays.

Figure 2.8. Examples of Caibbo suppressed weak decays.

Γ(K + → μ+ + νμ) ≃ sin θc , Γ(π + → μ+ + νμ)

(2.82)

which gives θc ≃ 13.02° [1]. Since the angle is small, processes that have an overall factor of cos θc are Cabibbo favored and those containing sin θc are said to be Cabibbo suppressed. In figure 2.7 and 2.8, I indicate these rules in diagrammatic form. Taking into account only these quark flavors, the resulting low-energy invariant amplitude for charged neutral currents can be written compactly in terms of a 2 × 2 flavor rotation matrix, U. The result is M = (4G / 2 )Jμ†J μ with

J μ = ( u¯ c¯ )

γ μ(1 − γ 5) U d s 2

()

(2.83)

and

⎛ cos θc sin θc ⎞ U=⎜ ⎟. ⎝−sin θc cos θc ⎠

(2.84)

2.7.1 The Cabibbo–Kobayashi–Maskawa (CKM) matrix This idea can be extended to include all three quark doublets

(du′), ( sc′), (bt′),

2-16

(2.85)

Relativistic Quantum Field Theory, Volume 3

where the top row consists of all quark flavors with electric charge Q = +2/3 and the bottom row are linear combinations of quarks with Q = −1/37. The resulting current has the form ⎛ ⎞ γ μ(1 − γ 5) ⎜ d ⎟ (2.86) J μ = ( u¯ c¯ t¯ ) U ⎜ s ⎟ + h.c., 2 ⎝b⎠ with, up to phases [1],

⎛∣Uud ∣ ∣Uus∣ ∣Uub∣⎞ ⎜ ⎟ ⎜⎜∣Ucd ∣ ∣Ucs∣ ∣Ucb∣⎟⎟ ⎝ ∣Utd ∣ ∣Uts∣ ∣Utb∣ ⎠ . ⎛ 0.9742 ± 0.00021 0.2243 ± 0.00394 ⎞ 0.00036 = ⎜ 0.218 ± 0.004 0.997 ± 0.017 0.0422 ± 0.0008⎟ ⎜ ⎟ ⎝ 0.0081 ± 0.0005 0.0394 ± 0.0023 1.019 ± 0.025 ⎠

(2.87)

As a consequence of the relative size of the different elements in the CKM matrix, one predicts that processes corresponding to diagonal elements in the CKM matrix will dominate heavy quark weak decays, e.g. for weak decays of the top quark the dominate decay chain will be

t → b + e+ + νe → c + 2e+ + 2νe → s + 3e+ + 3νe .

(2.88)

For more information about the CKM matrix, we refer to the reader to the CKM Quark-Mixing Matrix chapter of the PDG [1] and references therein. In the next chapter, we will look with more detail into the emergence of weak neutral currents by exploiting the concept of weak isospin.

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] 7

Tanabashi M et alParticle Data Group 2018 Phys. Rev. D 98 030001 Fermi E 1933 La Ricerca Sci. 2 12 Fermi E 1934 Il Nuovo Cimento (1924–1942) 11 1 Fermi E 1934 Z. Phys. 88 161–77 Lee T D, Oehme R and Yang C N 1957 Phys. Rev. 106 340–5 Mertig R, Bohm M and Denner A 1991 Comput. Phys. Commun. 64 345–59 Shtabovenko V, Mertig R and Orellana F 2016 Comput. Phys. Commun. 207 432–44 Rosner J L and Stone S 2008 (preprint:0802.1043) Radel G and Beyer R 1993 Mod. Phys. Lett. A 8 1067–88 Tomalak O and Hill R 2019 (preprint: 1907.03379) Glashow S L 1959 Nucl. Phys. 10 107–17 Salam A and Ward J C 1959 Il Nuovo Cimento (1955–1965) 11 568–77 Weinberg S 1967 Phys. Rev. Lett. 19 1264–6

The fact that quarks come in pairs is related to quantum chiral anomaly cancellation [19, 20].

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Relativistic Quantum Field Theory, Volume 3

[14] [15] [16] [17] [18] [19] [20]

Hasert F J et al 1973 Phys. Lett. B 46 138–40 Hasert F J et al 1973 Phys. Lett. B 46 121–24 Eichten T et al 1973 Phys. Lett. B 46 274–80 Eichten T et al 1973 Phys. Lett. B 46 281–84 Cabibbo N 1963 Phys. Rev. Lett. 10 531–33 Minahan J A, Ramond P and Warner R C 1990 Phys. Rev. D 41 715–16 Bouchiat C, Iliopoulos J and Meyer P 1972 Phys. Lett. B 38 519–23

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IOP Concise Physics

Relativistic Quantum Field Theory, Volume 3 Applications of quantum field theory Michael Strickland

Chapter 3 Electroweak unification and the Higgs mechanism

Nature tells us that leptons come in pairs, e.g. e and νe and so on, so it is natural to think that there might be an underlying SU(2) symmetry. However, since the weakinteraction only involves left-handed neutrinos/electrons, we call it SU(2)L. This will be referred to as weak isospin. The unification of the electromagnetic and weak forces was formulated by Glashow, Weinberg, and Salam (GSW) in the 1960s [1–3]. Looking at what we have so far

raising:

1 jμ = jμ+ = νγ ¯ μ (1 − γ 5)e = ν¯ Lγμe L , 2 jμ† = jμ− = e¯ Lγμν L

lowering:

(3.1) (3.2)

suggests that we can introduce an SU(2)L doublet of the form

ν χL ≡ e L , L

( )

1

(3.3) 1

with weak isospin projections T3ν = 2 and T3e = − 2 . Additionally, we introduce weak isospin raising and lowering operators

τ± =

1 (τ1 ± iτ2 ), 2

(3.4)

where τi are the Pauli matrices. This allows us to write the charge raising and lowering currents compactly as

doi:10.1088/2053-2571/ab3a99ch3

jμ+ = χ¯L γμτ+χL ,

(3.5)

jμ− = χ¯L γμτ−χL ,

(3.6)

3-1

ª Morgan & Claypool Publishers 2019

Relativistic Quantum Field Theory, Volume 3

where we remind the reader that

⎛ ⎞ τ+ = ⎜ 0 1 ⎟ ⎝0 0⎠

(3.7)

⎛ ⎞ τ− = ⎜ 0 0 ⎟. ⎝1 0 ⎠

(3.8)

and

Viewed in this manner, it is natural to predict that a third current related to the z-projection of weak isospin should exist. This logic was precisely that used to predict the existence of the weak neutral current

1 jμ3 = χ¯L γμ τ3χL 2

1 ⎛⎜1 0 ⎞⎟ ν L 2 ⎝ 0 −1⎠ e L 1 1 = ν¯ Lγμν L − e¯ Lγμe L. 2 2

( )

= ( ν¯ L e¯ L )γμ

(3.9)

Furthermore, we can then combine the three currents into a triplet of weak isospin currents

1 jμi = χ¯L γμ τiχL , 2

(3.10)

with, as usual, [τi , τj ] = iεijkτk . We can compare this to the electromagnetic current (homework)

jμE&M = e¯γμQe , = e¯RγμQeR + e¯ LγμQeL ,

(3.11)

which is invariant under U(1)Q

ψ → e iα(x )ψ ,

(3.12)

and not under SU(2)L because it contains eL instead of χL . We can make the electromagnetic current also invariant under SU(2)L by introducing an SU(2)Linvariant U(1) current

jμY = e¯RγμYRe R + χ¯L γμYLχL ,

(3.13)

where the weak hypercharges YR /L are conserved charge operators associated with U(1)Y symmetry. They are different for right- and left-handed leptons (two different phases). The first term in equation (3.13) is a singlet (only one possible value) and the second term triplet (three possible values). We can write the electromagnetic current jμE&M as a linear combination of jμ3 and jμY/2 (the factor of 2 is a matter of convention) 3-2

Relativistic Quantum Field Theory, Volume 3

jμE&M = e¯RγμQeR + e¯ LγμQeL

(3.14)

1 1 1 1 = ν¯ Lγμ ν L − e¯ Lγμ e L + e¯RγμYR e R + χ¯L γμYL χL , 2  2   2  2  Y 3 jμ /2 jμ

(3.15)

= jμ3 + jμY /2.

(3.16)

Requiring that the first line (3.14) is equal to the second (3.15), we obtain the requiring weak hypercharge relations

YR = 2Q

for e R,

YL = 2Q + 1 YL = −1

(3.17)

for e L ,

(3.18) (3.19)

for ν L,

and assigning eR to a weak isosinglet and νL and eL to the doublet we have

T3(e R) = 0

(singlet that is ‘blind’ to the weak interaction),

(3.20)

1 T3(ν L ) = + , 2

(3.21)

1 T3(e L ) = − . 2

(3.22)

With these assignments, we can combine YL and YR into a single relation:

Y = 2Q − 2T3.

(3.23)

We summarize the weak isospin assignments for electrons, electron–neutrinos, and up and down quarks in tables 3.1 and 3.2. There are similar assignments for the other weak isodoublets. Exercise 3.1. Show that equations (3.11)–(3.19) are correct.

Table 3.1. Weak isospin and hypercharge quantum numbers for electrons and electron neutrinos.

Lepton

T

T3

νe

1 2 1 2

+2

1 −2

−1

0

0

−1

eR− eL−

1

3-3

Q

Y

0

−1 −1 −2

Relativistic Quantum Field Theory, Volume 3

Table 3.2. (Left) Weak isospin and hypercharge quantum numbers for up and down quarks.

Lepton

T

T3

Q

Y

uL dL

1 2 1 2

1 +2 1 −2

+3

0

0

2 +3 1 −3 2 +3 1 −3

uR dR

0

0

1 1

+3 4

+3 2 −3

Unifying the electromagnetic and weak Lagrangians We expect the weak interaction to be mediated by gauge fields similar to QED and QCD. For the electromagnetic interaction, associated with the U(1)Q, the interacting part of the Lagrangian is E&M μ Lint A. QED = − iejμ

(3.24)

Based on this, we construct i iμ Lint EW = − igjμW − i

g′ Y μ j B , 2 μ

(3.25)

which includes four fields, Wi and B, which map to the SU(2)L triplet and singlet. From these, we can construct the massive charged vector bosons

W ±μ ≡

1 (W 1μ ± iW 2μ), 2

(3.26)

and the neutral vector bosons as linear combinations of W 3μ and B μ

Aμ ≡ B μ cos θW + W 3μ sin θW ,

(3.27)

Z μ ≡ −B μ sin θW + W 3μ cos θW ,

(3.28)

where θW is the weak mixing angle or Weinberg angle with

⎛ M ⎞2 sin2 θW ≃ 1 − ⎜ W ⎟ ≃ 0.2223. ⎝ MZ ⎠

(3.29)

The gauge field Aμ maps to the massless photon field, γ, and Z μ is the massive Z0 boson. Using equations (3.27) and (3.28) to rewrite the right-hand side of equation (3.25) in terms of the rotated fields gives

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Relativistic Quantum Field Theory, Volume 3

Lint EW,neutral

⎛ = − i ⎜⎜g sin θw jμ3 + g′ cos θW ⎝ ⎛ − i ⎜⎜g cos θw jμ3 + g′ sin θW ⎝

jμY ⎞ ⎟Aμ 2 ⎟⎠ jμY ⎞ ⎟Z μ. 2 ⎟⎠

(3.30)

The first term above should map to the electromagnetic interaction and since, as we established earlier (3.16),

jμE&M = jμ3 + jμY 2

(3.31)

g sin θW = g′ cos θW = e.

(3.32)

one must have

As a consequence, the three couplings {g, g′, e} become essentially two {e, sin θW }. The second term in equation (3.30) corresponds to the weak neutral current and, using jμY = 2 jμE&M − jμ3 can be written as

(

)

jμNC =

g j 3 − sin2 θW jμE&M , cos θW μ

(

)

(3.33)

where NC indicates the neutral-current weak interactions. Exercise 3.2. Show that equation (3.32) is correct.

3.1 Electroweak Feynman rules With this in hand and knowing that the other components of the electroweak Lagrangian consist of the terms correspond to the relativistic kinetic energy contributions, we are in position to write down the unified electroweak Lagrangian. The interaction terms are QED NC LEW + LCC int = L int int + L int ,

(3.34)

where CC indicates the charged-current weak interactions. These are, respectively, 1. Photon-fermion vertex

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Relativistic Quantum Field Theory, Volume 3

2. W ±-fermion vertex

3. Z0-fermion vertex

where

cVf = T3f − 2 sin2 θW Qf ,

(3.38)

cAf = T3f .

(3.39)

Values of these coefficients can be found in table 3.1. As this demonstrates, within the GSW model, there is no freedom of choice when it comes to the relative vector versus axial coupling for different particles in the neutral current. We note that the model predictions for these coefficients is in very good agreement with experimental observations especially when one takes into account loop corrections (see chapter 5 of the PDG review [4] and reference [5]).

3.2 Massive gauge fields with local gauge symmetry One finds that the W ± and Z0 are massive with M ∼ 100 GeV which, in the end, is the reason for the weak interaction being ‘weak’ in comparison to the Table 3.3. Tree-level values of the weak couplings for particles that undergo electroweak interactions.

Qf

cVf

cAf

νe, νμ, ντ e−, μ− , τ −

0

1 2

1 2

−1

−2

u, c, t

− 2 + 2 sin2 θW ≃ −0.03

2 3

1 2

sin2 θW ≃ 0.19

1 2

d , s, b

1 −3

1 −2

2 3

−2

1

3-6

−

4 3

+

sin2 θW ≃ − 0.34

1

1

Relativistic Quantum Field Theory, Volume 3

Figure 3.1. Cartoon illustrating spontaneous symmetry breaking in a ferromagnetic system.

electromagnetic interaction; however, we are in a bad position because, under normal circumstances, a Lie gauge field theory does not allow for a naive term of the form

LM = −

M2 Aμ Aμ , 2

(3.40)

since it is not gauge invariant under, e.g. a U(1)Q transformation with Aμ → Aμ + ∂ μλ(x ). Our goal is to construct a theory that is manifestly gauge invariant at the Lagrangian level, but for which the physical states less symmetric. The situation is analogous to the condensed matter description of ferromagnetism in materials: as long as no external magnetic field is imposed, the system is rotationally invariant; however, at low temperatures the system can ‘spontaneously’ magnetize as illustrated in figure 3.1. This process is called spontaneous symmetry breaking. This process provided the conceptual and mathematical underpinnings for the Brout–Englert–Higgs mechanism for generating massive gauge fields and resulted in the 2016 Nobel prize in physics after experimental observation of the Higgs boson [6–8]. To understand this at the Lagrangian level, let us first consider a simple real scalar field with Z2 symmetry (invariance under ϕ → −ϕ ). 3.2.1 Spontaneous symmetry breaking Let us first consider a simple real scalar field theory with quartic interaction, aka λϕ4 theory

L=

⎛1 ⎞ 1 1 (∂μϕ)(∂ μϕ) − ⎜ μ2 ϕ 2 + λϕ4⎟ . ⎝ 4 ⎠ 2 2  ≡V (ϕ)

(3.41)

This Lagrangian is symmetric under ϕ → = ϕ (Z2 invariance). For λ > 0, it is bounded from below; however, depending on the sign of μ2 , it can have one or two minima as illustrated in figure 3.2. For μ2 > 0, there is only one stable minimum, and for μ2 < 0, there are two. These can be obtained by looking for points where ∂V /∂ϕ = 0 and ∂ 2V /∂ϕ2 > 0:

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Relativistic Quantum Field Theory, Volume 3

Figure 3.2. Illustration of a real scalar potential in the symmetric phase (left, μ2 > 0 ) and broken-symmetry phase (right, μ2 < 0 ).

∂V ∂ϕ

= ϕ0(μ2 + λϕ02 ) ϕ=ϕ0

⎧ϕ if ⎪ 0 =0→⎨ μ2 ⎪ ϕ0 = ± − if λ ⎩

μ2 > 0 μ2 < 0.

(3.42)

Note that for μ2 < 0 under ϕ → −ϕ the two solutions map into one another ϕ0+ ↔ ϕ0−, so the system is still Z2 symmetric. However, since the vacuum of the theory (lowest energy solution) is no longer at ϕ = 0, we must linearize around ϕ0± if we are to describe the theory in the broken phase (μ2 < 0). To accomplish this, we substitute

ϕ(x ) = ϕ0 + η(x ),

(3.43)

into the original Lagrangian to obtain

1 1 1 L = (∂μ[ϕ0 + η(x )])2 − μ2 [ϕ0 + η(x )]2 − λ[ϕ0 + η(x )] 4 4 2 2 1 3 = (∂μη)2 − (μ2 ϕ0 + λϕ0 ) η  2 =0 ⎛ 1 2 3λ 2⎞ 2 1 −⎜ μ + ϕ0 ⎟ η − λϕ0η3 − λη 4 + constant ⎝ 2 ⎠ 4 2 2 =μ 1 1 1 = (∂μη)2 − m η2η 2 − λϕ0η3 − λη 4 + constant, 2 4 2

(3.44)

where the simplifications on the second line follow from the fact that (ϕ0± )2 = −μ2 /λ and in the last line we have introduced the mass in the symmetry broken phase

m η2 ≡

− 2μ 2 .

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(3.45)

Relativistic Quantum Field Theory, Volume 3

As we can see from this result, in the broken phase we can easily find a stable minimum that maps to m η2 > 0; however, we now have an additional effective cubic interaction with the vacuum condensate describing the ϕ0 . Next, let us extend our example to a complex scalar field with Lagrangian

L = (∂μϕ*)(∂ μϕ) − μ2 ϕ*ϕ −

λ * 2 (ϕ ϕ ) , 2

(3.46)

where ϕ = 1 (ϕ1 + iϕ2 ). This Lagrangian has a continuous symmetry (global U(1)) 2 related to the invariance of the Lagrangian under

ϕ(x ) → e iαϕ.

(3.47)

As with the real scalar field, for λ > 0, the nature of the vacuum in this theory depends on the sign of μ2 . This is illustrated in figure 3.3 with the left panel showing the symmetric phase and the right panel showing the symmetry broken phase. As we can see from the figure, when μ2 < 0 there is a minimum of the potential when

ϕ12 + ϕ22 ≡ v 2 = −

μ2 λ

(3.48)

all points on this circle represent a degenerate vacuum in the broken-symmetry phase. Looking at figure 3.3 we see that in the broken-symmetry phase there will be two normal modes of oscillation: one which runs along the minima of the potential which has zero curvature (zero mass) and one which runs perpendicular to this and which has a finite curvature in this direction (finite mass). This is the key ingredient in how we split the weak gauge bosons from the massless electromagnetic boson. In order to disentangle these two directions, we can choose ϕ0 = v, i.e. ϕ0,1 = v and ϕ0,2 = 0 and consider fluctuations of the form

Figure 3.3. Illustration of a complex scalar potential in the symmetric phase (left, μ2 > 0 ) and brokensymmetry phase (right, μ2 < 0 ). A cut of the region R[ϕ ] > 0 and I[ϕ ] < 0 was performed in order to see inside the potential.

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Relativistic Quantum Field Theory, Volume 3

ϕ(x ) =

1 [v + η(x ) + iξ(x )], 2

(3.49)

as illustrated in figure 3.4. With this decomposition, one has

1 1 1 L = (∂μη)2 + (∂μξ )2 − m η2η 2 2 2 2 3 3 + O(η , ξ ) + constant,

(3.50)

where, as with the real scalar field, m η2 = −2μ2 . As we can see from equation (3.50), there is one massless mode (ξ) and one massive mode (η). Note that we could also have made the decomposition as

ϕ(x ) = e iζ (x )H (x ),

(3.51)

and arrived at the same conclusion. The massless mode that is created in this process is called a Goldstone boson or Goldstone field [9, 10]. 3.2.2 Breaking of a continuous local symmetry With this example under our belt, we can proceed to a more realistic example, namely, a complex gauged scalar field with a Lagrangian of the form

L = [(∂μ − ieAμ )ϕ*][(∂ μ + ieAμ )ϕ ] − μ2 ϕ*ϕ −

λ * 2 (ϕ ϕ ) . 2

(3.52)

This Lagrangian has local U(1) gauge invariance under

Aμ → A μ′ = Aμ − ∂μα(x )/ e ,

(3.53)

ϕ → ϕ′ = e iα(x )ϕ(x ).

(3.54)

Figure 3.4. Sketch of the complex plane indicating the value of the vacuum expectation value v and the decomposition of the field into real and imaginary parts.

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Relativistic Quantum Field Theory, Volume 3

For λ > 0 and μ2 < 0, as before, the vacuum expectation value of the scalar field is given by

v2 = −

μ2 . λ

(3.55)

To proceed, we decompose the complex field into radial and phase oscillations as

ϕ(x ) =

1 [v + H (x )]e iξ(x )/v. 2

(3.56)

In this case, the derivatives of ϕ(ϕ*) will generate additional terms in the resulting Lagrangian; however, these can be absorbed into a field redefinition (gauge transformation)

1 Aμ → A˜μ ≡ Aμ − ∂μξ(x ). ev

(3.57)

Combining the expansion with this gauge transformation, one finds

1 1 2 L = (∂μH )2 − λvH 2 + e 2v 2A˜ μ − λvH 3 2 2 1 4 1 2 ˜2 2 1 2 − λv + e A μ H + ve 2A˜ μ H − F˜μνF˜ μν. 4 2 4

(3.58)

The particle spectrum of this theory contains: 1. A massive scalar field ‘H’ (Higgs field) of mass mH = 2λv2 [second term in (3.58)]. 2. A Goldstone field that has been ‘absorbed’ or ‘eaten’ by A˜μ [no terms involving ξ or ∂μξ remain in (3.58)]. 3. A massive U(1) vector field A˜μ with mass mA = ev [third term in (3.58)]. 4. Higgs field self-interactions [fourth and fifth terms in (3.58)]. 5. Interactions between the Higgs field and the massive gauge field [sixth and seventh terms in (3.58)]. To wrap up, in table 3.4, we enumerate the degrees of freedom (DOF) before and after transforming from (ϕ, A) to (ϕ′, A˜ ). As we can see from this table, the number of independent degrees of freedom is the same before and after the transformation. Table 3.4. Summary of the degrees of freedom (DOF) before and after the transformation of (ϕ, A) to (ϕ′, A˜ ).

L

Fields

DOF

ϕ, A

ϕ = complex scalar Aμ = massless spin-1 vector h = real scalar Aμ = massive spin-1 vector

2 2 1 3

ϕ′, A˜

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Relativistic Quantum Field Theory, Volume 3

Note that the induced mass for A˜μ is similar to what happens to photons as they propagate through a superconductor (Meissner-Ochsenfeld effect). In the case of the superconductor the penetration depth, d, is inversely related to the induced gauge field mass d ∼ 1/mA.

3.3 Gauge boson masses in SU(2)L × U(1)Y To wrap up, we must apply this methodology to the SU(2)L × U(1)Y electroweak theory. To do so, we introduce a gauge covariant derivative of the form

1 1 Dμ ≡ ∂μ − ig τ · Wμ − ig′ YBμ, 2 2

(3.59)

with the corresponding Lagrangian being

L = [iD μϕ ]†[iD μϕ ] − μ2 ϕ†ϕ − λ[ϕ†ϕ ]2 ,

(3.60)

where ϕ is the SU(2) ‘Higgs doublet’

ϕ=

1 ⎛ ϕ1 + iϕ2 ⎞ ⎛ ϕ+ ⎞ ⎜ ⎟ = ⎜ ⎟, 2 ⎝ ϕ3 + iϕ4 ⎠ ⎝ ϕ0 ⎠

(3.61)

which has a weak hypercharge of Y = 1. For λ > 0 and μ2 < 0, one can choose ϕ1 = ϕ2 = ϕ4 = 0 and ϕ3 = v which, up to a phase, gives

ϕ(x ) =

⎞ 1 ⎛0 ⎜ ⎟, 2 ⎝ v + H (x ) ⎠

(3.62)

with, as before, v = −μ2 /λ . This choice of the vacuum breaks SU(2)L × U(1)Y symmetry, but preserves the U(1)Q symmetry of electromagnetism. 3.3.1 The resulting particle spectrum Expanding the relevant part of the Lagrangian (3.60) gives

⎛ 1 1 ⎞ ⎜ −ig τ · Wμ − ig′ Bμ⎟ϕ ⎝ 2 2 ⎠

2

⎛ 1 ⎞2 2 1 → ⎜ vg⎟ W μ+W −μ + v 2 g′Bμ − gW 3μ . ⎝2 ⎠ 8

(

(

)

(3.63)

)

Looking at the final term we recall that Z μ ≡ gW 3μ − g′Bμ / g 2 + (g′)2 and defining

1 vg , 2

(3.64)

1 v g 2 + (g′)2 , 2

(3.65)

MW = MZ = we can write (3.63) compactly as

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Relativistic Quantum Field Theory, Volume 3

⎛ 1 1 ⎞ ⎜ −ig τ · Wμ − ig′ Bμ⎟ϕ ⎝ 2 2 ⎠

2

→ MW2 W μ+W −μ +

1 2 μ MZZ Zμ. 2

(3.66)

Using the fact that g′/g = tan θW , equations (3.64) and (3.65) predict that

MW = cos θW , MZ

(3.67)

such that MW < MZ . Taking the ratio of the experimentally observed masses gives 80.425/91.188 ≃ 0.882 [4]. Taking the experimental value for the Weinberg angle (sin2 θW ≃ 0.222 [4]) gives cos θw = 0.882, which is in quite good agreement with the observed mass ratio. Additionally, using the experimentally measured MW and Fermi’s weak decay constant, we can fix the vacuum expectation value v to be

GF =

2 g2 = 8MW2

1 2v

→

v ≃ 246 GeV.

(3.68)

Using this and the experimentally measured Higgs boson mass mH = 125.18 GeV [4], we can use mH = v 2λ to experimentally constrain the Higgs coupling to be λ ≃ 0.129. Exercise 3.3. Verify equation (3.66).

3.3.2 Fermion masses Now that we have managed to induce masses for the weak gauge bosons we can now couple the Higgs field to the fermions to see how fermion masses are generated. I will focus on T3 = −1/2 fermions like electrons. In QED, the fermion mass terms looks like

Lm = −mψψ ¯ = −m(ψ¯ LψL + ψ¯ RψR).

(3.69)

Next, if we interpret ψL as the bottom component of an SU(2)L-doublet χl and ψR as an SU(2)L singlet, we see that this mass term is not invariant under SU(2)L because ψR transforms trivially whereas ψL changes. We can remedy this by pairing ψL with an adjoint Higgs doublet to give a Lagrangian, which is invariant under SU(2)L. For the electron with T3 = −1/2, one has

⎡ ⎤ ⎛ ϕ+ ⎞ Lm = − G e⎢( ν¯e e¯ )L ⎜ 0 ⎟e R + e¯R ϕ¯ +ϕ¯ 0 ( νe e )L ⎥ ⎢⎣ ⎝ϕ ⎠ ⎦⎥

(

)

G ev G eh =− (e¯ Le R + e¯Re L ) − (e¯ Le R + e¯Re L ), 2 2 where 3-13

(3.70)

Relativistic Quantum Field Theory, Volume 3

⎛ ϕ+ ⎞ ⎞ 1 ⎛0 ⎟, ⎜ ⎜ 0⎟ = H ( x ) + v ⎠ ⎝ϕ ⎠ 2⎝

(3.71)

and Ge is a Yukawa-like coupling of the Higgs to the electron. From this result, we can read off the electron mass term

me =

G ev . 2

(3.72)

and the Higgs–electron coupling

As we can see from the above expression, since the coupling is proportional to the mass, it will be small for electrons, but potentially very large for heavy quarks such as the top quark. For this reason, in the experimental search for the Higgs, heavy quarks play a key role. Finally, we note that one can follow a very similar procedure for quarks; however, in this case, one must introduce another Higgs doublet, which for SU(2) can be taken to be ϕc ≡ iτ2ϕ†.

3.4 The discovery of the Higgs boson The ATLAS and CMS experiments at CERN’s Large Hadron Collider announced on July 4, 2012 that they had observed a particle consistent with the Higgs in the mass region around 125 GeV1. The PDG currently lists the Higgs boson as having a mass of mH = 125.18 ± 0.16 GeV with a full width Γ < 0.013 GeV at the 95% confidence level [4]. A summary of the mass measurements by ATLAS and CMS is provided in figure 3.5. Further on, we will provide more details about the Higgs to diphoton (γγ ), Higgs to four lepton (4l ), and Higgs to ττ decay channels. Due to experimental constraints, e.g. decays to diphotons, etc., the Higgs is listed as a scalar (J = 0) consistent with the picture presented in the previous section. Based on the results to date, it seems that the properties of the Higgs boson are consistent the simplest version of the Brout–Englert–Higgs mechanism [6–8], the so-called ‘standard-model Higgs’. After this discovery, the 2013 Nobel prize in physics was awarded jointly to Englert and Higgs ‘for the theoretical discovery of a mechanism that contributes to our understanding of the origin of mass of subatomic particles, and which recently was confirmed through the discovery of the predicted fundamental particle, by the ATLAS and CMS experiments at CERN’s Large Hadron Collider’. 1

There were prior limits on the Higgs mass using electroweak constraints giving mH < 152 TeV at the 95% confidence level [11, 12]. In addition, LEP experiments constrained the Higgs mass from below mH > 114.4 GeV [12, 13].

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Figure 3.5. Summary of the Higgs mass measurements from the ATLAS and CMS collaborations. Figure from the Particle Data Group ‘Status of Higgs Boson Physics’ section [4].

The production of the Higgs at LHC energies proceeds primarily through the processes illustrated in figure 3.6. The Higgs boson is first produced in high energy proton–proton collisions. The main leading-order diagrams contributing to Higgs production are shown in figure 3.6. These include (a) gluon fusion, (b) vector-boson fusion, (c) associated production with a gauge boson, (d) associated production with a pair of top (or bottom) quarks, (e) and (f) production in association with a single top quark. The cross-sections in the parton frame are folded together with state-ofthe-art parton distribution functions in order to predict the rate of Higgs production in each channel. Once produced, the Higgs boson decays into a variety of decay channels, with the coupling to various standard-model particles depending on the assumed Higgs mass. The search for the Higgs in experimental data relies on matching observations with predictions for the various decay channels, e.g. diphotons, four-leptons (4l ), Z bosons, W bosons, di-τ-leptons, and b (b¯ ) quarks. The branching ratios for the standard model Higgs boson are shown in figure 3.7 as a function of the assumed Higgs mass. As can be seen from this figure, the relevant processes depend on the assumed Higgs boson mass. The ATLAS and CMS results tell us that the left panel of figure 3.7 is the relevant regime. In this regime, one must take into account WW, bb¯ , gg, ττ , etc in order to make predictions about Higgs yields in various channels.

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Relativistic Quantum Field Theory, Volume 3

Figure 3.6. The most important leading-order Feynman diagrams contributing to Higgs production.

Figure 3.7. The standard model Higgs boson branching ratios as a function of the Higgs mass. The left panel shows the low mass region. The right panel shows results at higher masses where, e.g. t¯t production becomes possible. Figures are from [14–16]

3.4.1 The H  γγ decay channel In the H → γγ analysis, one is searching for a narrow peak in the diphoton invariant mass distribution (figure 3.8). Unfortunately, the signal comes with a large background from quantum chromodynamics (QCD) production of two photons (via quark–antiquark annihilation and ‘box’ diagrams) which is irreducible. In addition, there are also background events where some of the photon candidates can be

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Relativistic Quantum Field Theory, Volume 3

Figure 3.8. Leading-order Feynman graphs for H → 2γ .

Figure 3.9. Experimental results for diphoton events from the ATLAS [17] (left) and CMS [18] (right) collaborations.

produced solely due to misidentification of jet fragments. Events are selected that have two isolated photons with the requisite pT and particle identification. In figure 3.9, I show the results from the CMS and ATLAS experiments for the diphoton cross section in the invariant mass m γγ range 110–180 GeV. As can be seen from this figure, there is an excess (small bump) centered around the mass of the Higgs boson. The dashed red line shows the background events from known processes and the solid red line shows the prediction including the Higgs to diphoton channel with an assumed Higgs mass of ≃ 125 GeV. As this figure demonstrates, the excess is well described by production from a Higgs boson with this approximate mass. In addition, we note that, in the bottom left panel, CMS presents the background-subtracted signal along with bands indicating the ±1σ and ±2σ bands for random background fluctuations. From this, we learn that the excess is well beyond the 2σ level.

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Relativistic Quantum Field Theory, Volume 3

3.4.2 The H  ZZ  4l decay channel For the H → ZZ → 4l decay channel, experimentalists look for a narrow fourcharged lepton invariant mass peak. For this channel, backgrounds include an irreducible continuum background from direct ZZ production via quark–antiquark and gluon–gluon processes in the initial hard scatterings and reducible background contributions where the final states contain two isolated leptons and two bottomquark jets which produce secondary leptons. Selected events require two pairs of same-flavor, oppositely charged leptons, and the pair with invariant mass closest to the Z boson mass is required to have a mass in the range 40–120 GeV and the other is required to have a mass in the range 12–120 GeV. The dominant ZZ background is determined using Monte Carlo simulation studies. The resulting four-lepton invariant mass distribution determined by ATLAS and CMS is shown in figure 3.10. As these figures demonstrate, there is clear evidence for Higgs production in this channel. We note that, in the right panel, one can clearly see the Higgs peak along with another one at the Z mass which is due to conversion of a virtual photon simultaneously with a Z-produced dilepton pair. 3.4.3 The H  τ +τ − decay channel The decay channel H → τ +τ − shown in figure 3.11 is detected using ee, eμ, μμ, eτh, μτh , and τhτh final states. The final state electrons and muons come from leptonic

Figure 3.10. Four-lepton invariant mass distribution from the ATLAS [19] (left) and CMS [20] (right) collaborations.

Figure 3.11. Leading-order Feynman graph for H → ττ .

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Relativistic Quantum Field Theory, Volume 3

τ-decays and τh denotes a τ-lepton, which decays into hadrons. Using these final states, a scan is made for an excess in the reconstructed ττ invariant mass distribution. The primary irreducible background is Z → ττ production and the largest reducible backgrounds (W + jets, multijet production, and Z → ee ) are constructed using data-driven methods. Figure 3.12 shows the di-τ mass distributions measured by the CMS collaboration. As can be seen from the inset in this figure, there is an excess over the background contributions, which is consistent with di-τ production from the decay of a Higgs boson. 3.4.4 Other decay channels and the nature of the Higgs Other decay channels include H → bb¯ and H → WW (shown on the right of figure 3.13). Combining all observed decay channels, the ratio between standardmodel predictions and experimental observations is 1.00 ± 0.13 from CMS [22] and 1.30 ± 0.18 from ATLAS [23]. As mentioned previously, the data also indicates that the observed Higgs has spin 0. These findings strongly point to the discovery of the

Figure 3.12. Di-τ invariant mass spectrum as measured by the CMS collaboration [21].

Figure 3.13. Graphs for H → ZZ → 4l (left) and H → WW → 2l 2ν (right).

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Relativistic Quantum Field Theory, Volume 3

simplest incarnation of the Higgs, dubbed the standard model Higgs. For more information about the status of the Higgs, we refer the reader to the Particle Data Group review on the ‘Status of Higgs Boson Physics’ [4].

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]

[12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23]

Glashow S L 1959 Nucl. Phys. 10 107–17 Weinberg S 1967 Phys. Rev. Lett. 19 1264–6 Salam A and Ward J C 1959 Il Nuovo Cimento (1955–65) 11 568–77 Tanabashi M et al Particle Data Group 2018 Phys. Rev. D 98 030001 Erler J and Su S 2013 Prog. Part. Nucl. Phys. 71 119–49 Englert F and Brout R 1964 Phys. Rev. Lett. 13 321–3 Higgs P W 1964 Phys. Lett. 12 132–3 Higgs P W 1964 Phys. Rev. Lett. 13 508–9 Goldstone J 1961 Il Nuovo Cimento (1955–1965) 19 154–64 Goldstone J, Salam A and Weinberg S 1962 Phys. Rev. 127 965–70 Group L E W 2010 (ALEPH, CDF, D0, DELPHI, L3, OPAL, SLD, LEP Electroweak Working Group, Tevatron Electroweak Working Group, SLD Electroweak and Heavy Flavour Groups) (preprint:1012.2367) Dissertori G 2014 Phil. Trans. Roy. Soc. Lond. A373 20140039 Barate R et al 2003 (LEP Working Group for Higgs boson searches, ALEPH, DELPHI, L3, OPAL) Phys. Lett. B 565 61–75 LHC Higgs Cross section Working Group ed S Dittmaier, C Mariotti, G Passarino and Tanaka 2011 R CERN, Geneva, CERN-2011-002 (preprint:1101.0593) LHC Higgs Cross section Working Group ed S Dittmaier, C Mariotti, G Passarino and Tanaka 2012 R CERN, Geneva, CERN-2012-002 (preprint:1201.3084) LHC Higgs Cross section Working Group ed S Heinemeyer, C Mariotti, G Passarino and Tanaka 2013 R CERN, Geneva, CERN-2013-004 (preprint:1307.1347) Aaboud M et al ATLAS 2018 Phys. Rev. D 98 052005 Sirunyan A M et al CMS 2018 JHEP 11 185 Aad G et al ATLAS 2013 Phys. Lett. B 726 88–119 Chatrchyan S et al CMS 2014 Phys. Rev. D 89 092007 Chatrchyan S et al CMS 2014 JHEP 05 104 CMS 2014 CMS-PAS-HIG-14-009 ATLAS 2014 ATLAS-CONF-2014-009

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IOP Concise Physics

Relativistic Quantum Field Theory, Volume 3 Applications of quantum field theory Michael Strickland

Chapter 4 Basics of finite temperature quantum field theory

Normally, quarks and gluons are confined inside of hadrons; however, at temperatures exceeding T ∼ 155 MeV , quarks and gluons can become deconfined from the light hadrons and chiral symmetry is restored (the chiral condensate ‘melts’ and the quark masses decrease to their ’bare’ values). This deconfined state of matter filled the early universe until approximately 10−5 s after the Big Bang, after which quarks and gluons coalesced into hadrons. There are ongoing heavy-ion collision experiments that study the formation of the high-temperature deconfined phase of QCD, which is called the quark–gluon plasma (QGP). For a recent review of the experimental and theoretical work on this front, I refer the reader to [1] and references therein. We will now cover the basics of thermal field theory so that we might calculate the properties of a high-temperature QGP. Before attacking thermal QFT, let us first consider a finite temperature ensemble of quantum harmonic oscillators (QHOs) in order to introduce some basic concepts. For more details on finite temperature field theory, I refer the reader to [2–5].

4.1 Partition function for a quantum harmonic oscillator As we discussed earlier in volume 2, the quantum mechanical partition function can be represented as a Euclidean path integral of the form

Z(β ) =

∫ dq KE (q, q; β ) = N ∫q(0)=q(β ) DqE e−S , E

(4.1)

where

SE[q , q˙ ] =

∫0

β

⎡1 ⎤ dτ ⎢ mq˙2 + V (q )⎥ ⎣2 ⎦

(4.2)

is the Euclidean action, q˙ = dq /dτ , β = 1/T is the inverse temperature, and τ is the Wick-rotated ‘imaginary time’ τ = it . As indicated above, the path integral for the

doi:10.1088/2053-2571/ab3a99ch4

4-1

ª Morgan & Claypool Publishers 2019

Relativistic Quantum Field Theory, Volume 3

partition function integrates over periodic trajectories for which the generalized coordinate variable q(0) = q( β ). 4.1.1 The QHO canonical partition function in the energy basis Before continuing to QFT, let us first consider a finite temperature ensemble of quantum harmonic oscillators in order to introduce some basic concepts. The fundamental quantity of interest is the partition function, Z . In the canonical ensemble, Z is a function of T. The partition function is defined by1

Z = Tr[ ˆρ ],

(4.3)

2

where ρˆ is the density of states operator

ˆ

(4.4)

ρˆ ≡ e−βH .

Once Z is computed, other observables, such as the free energy F, entropy S, and average energy E, can be obtained via standard relations. At finite temperature and zero chemical potential, one has

F = −T log Z , E= TS = −T

(4.5)

1 ˆ −βHˆ ], Tr [He Z

1 ∂F ˆ −βHˆ ] = E − F . = T log Z + Tr [He ∂T Z

(4.6) (4.7)

For future reference, we can explicitly calculate these quantities for the harmonic oscillator. The calculation is straightforward to carry out in the energy basis ∞

Z=

∞

∑ 〈n∣e−βHˆ ∣n〉 = ∑ e−βω(n+1/2) = 1 n=0

n=0

e−βω/2 1 . = −βω 2 sinh(βω /2) −e

From this, we can compute the free energy ω F = T log (e βω/2 − e−βω/2 ) = + T log (1 − e−βω), 2

(4.8)

(4.9)

entropy density

S = −log (1 − e−βω) +

βω , e −1 βω

(4.10)

and average energy The trace in the definition of Z is taken over the full Hilbert space, i.e. Tr[Oˆ ] = ∑n Onn with Onm = 〈n∣Oˆ ∣m〉. ˆ ˆ = Hˆ − ∑ μ Nˆi . In this case, One can generalize this to the grand canonical potential, with ρˆ ≡ e−β Ω and Ω i i ˆ μi is the chemical potential for the i-th conserved charge and Ni is the number operator associated with that charge. 1 2

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Relativistic Quantum Field Theory, Volume 3

ω ω . + βω 2 e −1

E = F + TS =

(4.11)

4.1.2 Computing the QHO partition function using the path integral formalism As an independent crosscheck of the results of the previous section, we now explicitly evaluate the path integral (4.1) for a harmonic oscillator and compare the result with equation (4.8). We will mirror the evaluation of the partition function we made in Minkowski space in volume 2 but we will now carry out the evaluation in Fourier space with respect to the imaginary time coordinate τ. An arbitrary periodic function that obeys q(0) = q(β ) can be represented by a Fourier sum ∞

∑

q(τ ) ≡ T

qne iωnτ ,

(4.12)

n =−∞

where the factor T is a convention. Periodicity requires e iωnβ = 1 and hence

n ∈ ,

ωn = 2πnT ,

(4.13)

which are called Matsubara frequencies. The corresponding amplitudes qn are called Matsubara modes. For the QHO, we also have that q is real-valued, which implies that qn* = q−n . If we write qn = an + ibn , it follows that an = a−n and bn = −b−n . The second condition implies that b0 = 0. As a result, one has

⎧ q(τ ) = T ⎨ a0 + ⎪ ⎩ ⎪

⎫

∞

∑⎡⎣ (an + ibn)e iωτ + (an − ibn)e−iω τ⎤⎦⎬. ⎪

n

⎪

⎭

n=1

(4.14)

The first coefficient, a0, is the amplitude of the Matsubara zero mode. With this setup, we can write a general quadratic structure as

∫0

β

dτ q1(τ )q2(τ ) = T 2 ∑ q1,nq2,m m, n

∫0

β

dτ e i (ωn+ωm )τ (4.15)

= T ∑ q1,nq2,−n. n

Using this, the argument of the exponential in equation (4.1) becomes

−

∫0

β

dτ

⎤ m ⎡ dq(τ ) dq(τ ) + ω 2q(τ )q(τ )⎥ ⎢⎣ ⎦ 2 dτ dτ ∞

1 = − mT ∑ qn−( ωn ω−n + ω 2 )q−n 2 n =−∞

⏟ −ωn

∞

1 = − mT ∑ ωn2 + ω 2 a n2 + bn2 . 2 n =−∞

(

)(

4-3

)

(4.16)

Relativistic Quantum Field Theory, Volume 3

Next, we need to consider the path integration measure. To do this, let us make a change of variables from q(τ ), τ ∈ (0, β ), to the Fourier components an, bn. As we have seen, the independent variables are a0 and {an, bn}, n ⩾ 1. As a result, the path integral measure becomes

⎡ ⎤ ⎡ δq(τ ) ⎤ ⎥ da 0⎢∏ dandbn⎥ . Dq(τ ) = det ⎢ ⎢⎣ n ⩾ 1 ⎥⎦ ⎣ δqn ⎦

(4.17)

The change of bases is independent of the potential V (q ), implying that we can define

⎡ δq(τ ) ⎤ ⎥ , N′ = N det ⎢ ⎣ δqn ⎦

(4.18)

and then regard N′ as an unknown multiplicative normalization to be determined separately. As a result, the partition function becomes

Z = N′

∞

⎡ ⎤ ⎢∏ dandbn⎥ ⎢⎣ n ⩾ 1 ⎥⎦

∞

∫−∞ da0 ∫−∞

⎡ ⎤ 1 × exp ⎢ − mTω 2a 02 − mT ∑( ωn2 + ω 2 )( a n2 + bn2 )⎥ ⎢⎣ 2 ⎥⎦ n⩾1 = N′

2π mTω 2

∞

∏ n=1

π mT ( ωn2 + ω 2 )

(4.19)

.

We now only have to determine N′. As mentioned earlier, it is independent of V (q ), so we can compute in by taking ω → 0 (V → 0) which is a trick we used in the Minkowski-space calculation as well. However, we must regulate the theory before doing this. • The integral over the zero mode a0 in (4.19) is, however, divergent for ω → 0. This is an infrared divergence since the zero mode is the lowest-energy mode. • In order to take the ω → 0, we can temporarily regulate the integration over the zero mode. Noting that from equation (4.14) one has

1 β

∫0

β

dτ q(τ ) = Ta 0,

(4.20)

we learn that a0 represents the average value of q(τ ). As a result, we can regulate the system by putting it in a periodic box, i.e. by restricting the (average) value of q(τ ) to some (large but finite) interval Δq . With this setup, we can now proceed to find N′ via a matching calculation. We will first work with the expression we have with a cutoff in place and then compare that to an exact calculation of the regulated partition function.

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Relativistic Quantum Field Theory, Volume 3

Exercise 4.1. Verify equation (4.19). Effective theory computation: In the ω → 0 limit but in the presence of the regulator, equation (4.19) becomes

lim Z regulated = N′

ω→ 0

∫0

Δq / T

∞

da 0

⎡

⎤

⎣ n⩾1

⎦

∫−∞ ⎢⎢∏ dandbn⎥⎥

⎡ ⎤ × exp ⎢ −mT ∑ ωn2( a n2 + bn2 )⎥ ⎢⎣ ⎥⎦ n⩾1 Δq = N′ T

∞

∏ n=1

(4.21)

π . mTωn2

Exercise 4.2. Verify equation (4.21). Full theory computation: In the ω → 0 limit but in the presence of the regulator, equation (4.19) becomes

lim Z regulated =

∫0

=

∫0

=

∫0

ω→ 0

Δq

pˆ 2

dq 〈q∣e− 2 mT ∣q〉

Δq

∞

dq

∫−∞

dq

∫−∞

Δq

∞

pˆ 2 dp 〈q∣e− 2 mT ∣p 〉〈p∣q〉 2π dp − p 2 〈q ∣p 〉〈 e 2 mT  p∣ q〉 2π

(4.22)

=1

Δq = 2πmT . 2π Matching: Equating the two, we obtain

N′ =

T 2πmT 2π

∞

∏ n=1

mTωn2 . π

Plugging this result into equation (4.19), we obtain

4-5

(4.23)

Relativistic Quantum Field Theory, Volume 3

Z=

T ω

T = ω =

∞

ωn2 ω 2 + ωn2

∏ n=1

∞

∏n=1

1 ⎡ 1 ⎢⎣1 + 2 n

βω 2 ⎤ 2π

( ) ⎥⎦

(4.24)

1 , 2 sinh(βω /2)

where, in going from the second to third lines, we have used

sinh πx = πx

∏ n=1

⎛ x2 ⎞ ⎜1 + 2 ⎟ . ⎝ n ⎠

(4.25)

This is the same result we obtain (just more easily) in the energy basis, however, when including interactions it will be necessary to evaluate loop corrections to the partition function in this manner.

4.2 The partition function for a free scalar field theory A path integral representation for the partition function of a scalar field theory can be derived from this result. In QFT, the form of the theory is most economically defined in terms of the corresponding classical (Minkowski space) Lagrangian LM, rather than the Hamiltonian Hˆ . To transition from the quantum mechanical path integral representation of the partition function to QFT, we re-interpret q as an ‘internal’ degree of freedom ϕ, located at the origin 0 of d-dimensional space. Using ‘QM’ to indicate ‘quantum mechanics’, we can rewrite the quantum mechanical action and Lagrangian density in Minkowski space as

S QM M = LQM M =

∫ dt LQM M ,

(4.26)

m ⎛ ∂ϕ(t , 0) ⎞2 ⎜ ⎟ − V (ϕ(t , 0)). 2 ⎝ ∂t ⎠

(4.27)

We can compare this with the action of a massless scalar field theory defined in (d + 1)-dimensional spacetime

SM = LM =

∫ dt d dx LM ,

(4.28)

1 ∂μϕ∂ μϕ − V (ϕ). 2

(4.29)

In this way, we see that quantum mechanics maps to a (0 + 1)d QFT. Comparing the quantum mechanical and quantum field theory expressions, we see that our scalar field theory looks like a collection of nearly independent harmonic oscillators with 4-6

Relativistic Quantum Field Theory, Volume 3

m = 1, with one oscillator at each point x. These oscillators interact through the derivative term which couples nearest neighbors. With this analogy under our belts, we make use of the fact that, for our scalar field theory, the derivation of the path integral proceeds in an identical manner as in the first chapter of volume 2, i.e. slices in time and sums over all intermediate values of the generalized coordinate. As a result, we can simply translate the quantum mechanical result into an expression for the partition function for a (d + 1)-dimensional scalar QFT3

Z(T ) = N

⎛

∫ϕ(β,x)=ϕ(0,x) Dϕ(τ, x)exp ⎜⎝−∫0

LE = −LM(t → −iτ ) =

1 ⎛ ∂ϕ ⎞2 1 ⎜ ⎟ + 2 ⎝ ∂τ ⎠ 2

d

β

dτ

∫x

⎞ LE ⎟ , ⎠

⎛ ∂ϕ ⎞2 ⎟ + V (ϕ). xi ⎠

∑⎜⎝ ∂ i=1

(4.30)

(4.31)

We can write the Euclidean Lagrangian density compactly as LE = 12 ∂μϕ∂μϕ + V (ϕ ) where the fact that both indices are down visually indicates that we should perform four-vector contractions with a Euclidean metric gEμν = diag(1, 1, 1, 1). 4.2.1 Fourier representation of the fields Since our scalar fields are periodic in imaginary time, it follows that ∞

ϕ(τ , x) =

∑

ϕ˜ (ωn, x)e iωnτ ,

(4.32)

n =−∞

where, as in the case of the QHO, ωn = 2πnT with n ∈  . For the spatial coordinates, it is useful to make each direction finite to begin with, denoting the corresponding extents by Li, and imposing periodic boundary conditions in each direction. Then, one has

ϕ(τ , x) =

T V

∑ ∑ ϕ˜ (ωn, k)e iω τ−ik·x , n

(4.33)

k

ωn

d

where ki = 2πni /L i with ni ∈  and V = ∏i =1 L i . In the infinite volume limit, the sum over k becomes the usual Fourier integral

d 3k . (2π )3

(4.34)

ϕ˜*(ωn, k) = ϕ˜ ( −ωn, −k).

(4.35)

1 L i →∞ V lim

∑→∫ k

Since the field is real-valued, one has

3

For now, we will concentrate on finite temperature and zero chemical potential for simplicity.

4-7

Relativistic Quantum Field Theory, Volume 3

As a result, only half of the Fourier modes are independent in this case. Using the setup above, quadratic forms can be written as

∫0

β

dτ

∫x

ϕ1(τ , x)ϕ2(τ , x) =

T V

∑ ∑ ϕ˜1( −ωn, −k)ϕ˜2(ωn, k). ωn

(4.36)

k

This implies that in the free case, i.e. V (ϕ ) = 12 m2ϕ2 , the exponential in the definition of the partition function can be written as

⎛ β ⎞ exp( −SE ) = exp ⎜ dτ LE ⎟ ⎝ 0 ⎠ x ⎡ 1T ⎤ = exp ⎢ − ωn2 + k2 + m 2 ) ∣ϕ˜ (ωn, k)∣2 ⎥ ∑ ∑ ( ⎢⎣ 2 V n k ⎥⎦ ⎡ T ⎤ = ∏ exp ⎢ − ωn2 + k2 + m 2 ) ∣ϕ˜ (ωn, k)∣2 ⎥ . ∑ ( ⎢⎣ 2V n ⎥⎦ k

∫

∫

(4.37)

This has the same form as equation (4.19), so we can use the result from there with the appropriate translation giving (homework) ∞ ∞ ⎧ ⎫ 2 2 −1/2 2 1/2 T ω + E ω Z =∏ ⎨ ∏ ( n ∏ ( n ′) ⎬⎪, k) ⎪ ⎭ k ⎩ n =−∞ n ′=−∞ ⎡ ⎧ ⎫⎤ 1 1 ⎨log T + ∑ log ω n2′ − ∑ log ( ωn2 + E k2 )⎪ ⎬⎥ , = exp ⎢∑⎪ ⎢⎣ k ⎩ 2 n 2 n′ ⎭⎥⎦ ⎪

⎪

⎪

(4.38)

⎪

where Ek ≡ k2 + m2 and n′ indicate that the zero mode (n = 0) is omitted from the sum/product. An alternative form can be obtained using (homework) ∞

T

−1/2

∏ ( ωn2 + ω2 ) n =−∞

∞

1/2

∏ ( ωn2′) n ′=−∞

⎡ 1 = exp ⎢ − ⎣ T

{ ω2 + T log (1 − e )}⎤⎥⎦, −βω

(4.39)

which gives

Z=

∏ k

⎡ 1 exp ⎢ − ⎣ T

{

⎤ Ek + T log ( 1 − e−βEk ) ⎥ . ⎦ 2

}

(4.40)

Taking the infinite-volume limit and using Z = exp( −F /T ), the free-energy density F/V can be written as

F = V →∞ V lim

d

⎡

∫ (2dπk)d ⎢⎣ E2k

⎤ + T log ( 1 − e−βEk )⎥ . ⎦

4-8

(4.41)

Relativistic Quantum Field Theory, Volume 3

Exercise 4.3. Verify equations (4.38) and (4.39). 4.2.2 Tricks for evaluating sum-integrals Next, I would like to discuss how to evaluate the sum-integrals necessary to go from equation (4.38) to equation (4.39). We would like to evaluate

J (E ) ≡

T 2

∑ log ( ωn2 + E 2 ) − n

T 2

∑ log ωn2 − n′

T log T 2, 2

(4.42)

which, up to the factor of T, appears in equation (4.38). Up to the energyindependent constant, J (E ) can be obtained by integrating

I (E ) ≡

1 dJ (E ) 1 = T∑ 2 . E dE ωn + E 2 n

(4.43)

Let f ( p ) be a generic function, which is analytic in the complex plane (apart from isolated singularities) and regular on the real axis. Now, consider the sum

U ≡ T ∑ f (ωn),

(4.44)

n

where ωn are the bosonic Matsubara frequencies. It is useful to introduce

in B(ip) ≡

i , e iβp − 1

(4.45)

where nB is the Bose distribution since it has poles at p = 2πnT , n ∈  , i.e. at p = ωn . Expanding this function in a Laurent series around any of the poles, we obtain

in B(i [ωn + z ]) =

i i T = iβz ≈ + O(1), e iβ[ωn+z ] − 1 e −1 z

(4.46)

which implies that the residue of each pole is T. This means that we can replace the sum in equation (4.44) by a contour integral of the form +

+∞−i 0 dp dp f (p )inB(ip) ≡ f (p )nB(ip) U= + −∞−i 0 2πi 2π + −∞+i 0 dp + f (p )nB(ip), + +∞+i 0 2π

∮

∫

(4.47)

∫

where the contour runs counter-clockwise around the real axis in the complex pplane. Note that this can be further simplified by substituting p → −p in the second term on the RHS

4-9

Relativistic Quantum Field Theory, Volume 3

n B( −ip) =

1 e−iβp − 1

=

e iβp − 1 + 1 = −1 − n B(ip). 1 − e iβp

(4.48)

Using this, we obtain

U=

∞−i 0 +

∞

=

dp f ( −p ) + ⎡⎣ f (p ) + f ( −p )⎤⎦nB(ip) 2π ∞−i 0 + dp dp ⎡ ⎤ f ( p) + ⎣ f (p ) + f ( −p )⎦nB(ip), + −∞−i 0 2π 2π

∫−∞−i 0 ∫−∞

{

+

}

(4.49)

∫

where we have returned to the real axis in the first term since f ( p ) is regular on the real axis. Looking at this result, we see that the first term is temperature independent and gives the zero-temperature, or vacuum, contribution to U . The second term is the thermal contribution. We note that in the lower half-plane, taking p → x − iy, we have

∣n B(ip)∣ =

1 iβx βy

e e

y ≫T

−1

y ≫x

⎯⎯⎯⎯→ e−βy ⟶ e−β∣p∣.

(4.50)

As a result, if the function f (p ) grows slower than e β∣p∣ at large ∣p∣, e.g. polynomially, the integration contour for the finite-T contribution can be closed in the lower halfplane. In this case, the result is obtained by the poles and residues of the function f ( p ) + f ( −p ). Physically, this is the statement that the thermal contribution to U comes from on-shell particles. Let us apply this general formula to I (E ). Generalizing I (E ) in anticipation of going to finite chemical potential, let us consider

I (E ; c ) ≡ T ∑ ωn

1 , (ω n + c ) 2 + E 2

c ∈ .

(4.51)

In the notation of equation (4.44), one has

f ( p) =

⎤ 1 1 1 i ⎡ = − ⎢ ⎥, ( p + c )2 + E 2 2E ⎣ p + c + iE p + c − iE ⎦

f ( p) + f ( −p) =

i ⎡ 1 1 1 + − ⎢ p − c + iE p + c − iE 2E ⎣ p + c + iE ⎤ 1 − ⎥. p − c − iE ⎦

(4.52)

(4.53)

We need the poles of these functions in the lower half-plane, which for ∣Im c∣ < E are located at p = ±c − iE . From equations (4.52) and (4.53), we see that the residue at each lower half-plane pole is i /(2E ). Therefore, the vacuum term in equation (4.49) gives

4-10

Relativistic Quantum Field Theory, Volume 3

1 i 1 ( −2πi ) , = 2π 2E 2E

(4.54)

and the thermal contribution gives

⎤ 1 i ⎡ 1 1 ( −2πi ) ⎢ β(E −ic ) + β(E +ic ) ⎥. 2π 2E ⎣ e e −1 − 1⎦

(4.55)

Putting the pieces together, we obtain

I (E ; c ) =

1 ⎡ ⎤ ⎣ 1 + n B(E − ic ) + n B(E + ic )⎦ . 2E

(4.56)

This is clearly periodic in c → c + 2πnT with n ∈  . Also, as we can see ic resembles a chemical potential. We now remind the reader that the sum we wanted to evaluate, now generalized to include c, was

J (E ; c ) ≡ T ∑ n

1 log[(ωn + c )2 + E ] − (terms that don′t depend on E). (4.57) 2

Using

dJ (E ; c ) = E I (E ; c ), dE

(4.58)

and integrating with respect to E, one obtains

J (E ; c ) =

E T + log ⎡⎣ 1 − e−β(E −ic )⎤⎦ + log ⎡⎣ 1 − e−β(E +ic )⎤⎦ 2 2 + (constant w.r.t. E ).

{

}

(4.59)

Note that the constant in parenthesis can depend on T and c but not on E. Taking c to zero, we obtain

J (E ; 0) =

E + T log ⎡⎣ 1 − e−βE ⎤⎦ + (constant w.r.t. E ), 2

(4.60)

precisely the form for the partition function given in equation (4.39). Exercise 4.4. Verify equation (4.59).

4.3 Free scalar thermodynamics In the previous section, we obtained the following results for the partition function of a (d + 1)-dimensional free scalar field theory

4-11

Relativistic Quantum Field Theory, Volume 3

⎡ ⎧ 1 ⎨log T + Z = exp ⎢∑⎪ ⎢⎣ k ⎩ 2 ⎪

log ωn2 −

∑ n′

1 2

∑ n

⎫⎤ ⎬⎥ , log ( ωn2 + E k2 )⎪ ⎭⎥⎦ ⎪

(4.61)

where Ek = k2 + m2 and n′ indicates that the zero mode is omitted from the sum. This can be expressed alternatively as

Z=

∏ k

⎡ 1 exp ⎢ − ⎣ T

{

⎤ Ek + T log ( 1 − e−βEk ) ⎥ . ⎦ 2

}

(4.62)

Taking the infinite-volume limit, the free-energy density F/V can be written as

F = V →∞ V

F ≡ lim

d

⎡

⎤

∫ (2dπk)d ⎢⎢⎣T2 ∑ log ( ωn2 + E k2 ) − T2 ∑ log ωn2 − T2 log T 2⎥⎥⎦ (4.63) n

n′

or, equivalently,

F=

⎡

d

∫ (2dπk)d ⎢⎣ E2k

⎤ + T log ( 1 − e−βEk )⎥ . ⎦

(4.64)

Let us now turn to the evaluation of the free energy density, which I will colloquially refer to as simply the free energy. The integrals/sum integrals required are of the form

J (m , T ) ≡

⎡

∫k ⎢⎣ E2k

⎤ + T log ( 1 − e−βEk )⎥ ⎦

⎡1 ⎤ = ∑k⎢ log ( ωn2 + E k2 ) − const.⎥ , ⎣2 ⎦

∫

(4.65)

(4.66)

where we have introduced the notation ∞

∫

∑k ≡ T

∑ ∫ k

,

(4.67)

∫ (2dπk)d ,

(4.68)

n =−∞

and

∫k

d

≡ μ¯ 2ε

where d = 3 − 2ε is the spatial dimensionality, K = (ωn , k) is the Euclidean fourmomentum, and

μ¯ 2 =

e γ μ2 , 4π

with μ being the MS renormalization scale.

4-12

(4.69)

Relativistic Quantum Field Theory, Volume 3

As we discussed in the last section, it is convenient to introduce the following derivative of J:

1 ∂ J (m , T ) m ∂m 1 [1 + nB(Ek )] = k 2Ek 1 = ∑k 2 ωn + E k2 1 = ∑k 2 , K + m2

I (m , T ) =

∫

(4.70)

∫ ∫

where we have used the fact that m−1∂m = E k−1∂ Ek . Let us consider J first. Since in its first form it naturally splits into a vacuum and thermal contribution it makes sense to make this split explicit

J (m , T ) = J0(m) + JT(m),

(4.71)

with

J0(m) ≡

1 2

1 2

∫k

Ek =

∫k

log (1 − e−βEk ).

JT(m) ≡ T

∫k

k2 + m 2 ,

(4.72)

(4.73)

The vacuum contribution J0 is ultraviolet divergent and we need to regulate it using dimensional regularization. The thermal contribution JT is convergent. Let us tackle the vacuum contribution first. For generality, let us consider the following integral from which J0 can be extracted as a specific case: d

Φ(m , d , α ) =

(

∫ (2dπk)d (k2 +1 m2)α

(4.74)

)

with J0(m ) = 12 Φ m, d , − 12 . Since the integrand only depends on the length of the three-momentum. We can immediately perform the angular integrands

Φ( m , d , α ) =

Sd μ¯ 2ε (2π )d

∫0

1 Sd μ¯ 2ε = 2 (2π )d =

∞

∫0

dk ∞

(4.75)

d −2

(k 2 ) 2 dk 2 (k + m 2 ) α

m d −2αμ¯ 2ε (4π )d /2 Γ(d /2)

4-13

k d −1 (k 2 + m 2 ) α 2

∫0

(4.76)

∞

dt t d /2−1(1 + t )−α ,

(4.77)

Relativistic Quantum Field Theory, Volume 3

where Sd = 2π d /2/Γ( d2 ) is the surface area of a d-dimensional sphere and t = k 2 /m2 . The integral that remains can be expressed as a ratio of Euler Gamma functions, with the result being

Φ(m , d , α ) =

Γ(α − d /2) 2ε d −2α μ¯ m . (4π )d /2 Γ(α )

(4.78)

With this general result in hand, we find

Γ( −(d + 1)/2) 2ε d +1 1 μ¯ m d /2 Γ( −1/2) 2(4π ) ⎛ μ¯ ⎞2ε 4 1 1 ⎜ ⎟ m . =− Γ − + ( 2 ) ε ⎝m⎠ 4 π (4π )3/2−ε

J0(m) =

(4.79)

Taylor expanding around ε = 0, we obtain

J0(m) = −

m4 ⎡ 1 μ2 3⎤ + log + ⎥ + O(ε ). ⎢ 64π 2 ⎣ ε m2 2⎦

(4.80)

As expected, this result is divergent in the limit ε → 0. This divergence will be taken care of by the vacuum energy density renormalization counterterm. We will return to this issue soon. Note that, having determined J0, we can determine the corresponding I0(m ), which is the zero temperature limit of the I (m, T ) integral, straightforwardly

I0(m) ≡ lim I (m , T ) = T →0

⎤ μ2 1 dJ0(m) m2 ⎡ 1 =− + + log 1 ⎥ + O(ε ). ⎢ ⎦ 16π 2 ⎣ ε m dm m2

(4.81)

We now consider the finite temperature integrals JT(m ) and IT(m ) which are both finite. As a result, we can set ε = 0. However, I note that if one were considering loop corrections coming from interactions then this integral will reappear and could potentially be multiplied by a pole in ε. In that case, one has to evaluate the subleading terms in ε even though the integral itself is finite. The finite temperature integrals necessary are

JT(m) =

T4 2π 2

∫0

IT(m) =

T2 2π 2

∫0

∞

(

dx x 2 log 1 − e− x2

∞

dx

2

x 2+mˆ 2

),

1 2

ˆ e x +m

x 2+mˆ 2

(4.82)

, −1

(4.83)

where mˆ = m /T . These integrals can be expanded in terms of modified Bessel functions

JT(m) = −

m2T 2 2π 2

∞

ˆ) K2(nm , 2 n n=1

∑

4-14

(4.84)

Relativistic Quantum Field Theory, Volume 3

IT(m) =

mT 2π 2

∞

∑ n=1

ˆ) K1(nm , n

(4.85)

or evaluated numerically. Exercise 4.5. Verify equation (4.78). Exercise 4.6. Verify equations (4.84) and (4.85).

4.3.1 Low temperature limit Next, let us consider the limit mˆ ≫ 1. In this case, one obtains

⎤ ⎛m ⎛T ⎞ ˆ ⎞3/2 ⎡ JT(m) = −T 4⎜ ⎟ e−mˆ ⎢1 + O⎜ ⎟ + O(e−mˆ )⎥ ⎝ 2π ⎠ ⎝m⎠ ⎣ ⎦

(4.86)

⎤ ⎛T ⎞ T2 ⎛ m ˆ ⎟⎞3/2 −mˆ ⎡ ⎜ e ⎢1 + O⎜ ⎟ + O(e−mˆ )⎥ . ⎝ ⎠ ⎝ ⎠ ⎣ ⎦ m m ˆ 2π

(4.87)

and

IT(m) = −

4.3.2 High-temperature limit Next, we consider the limit mˆ ≪ 1. At first sight, it may seem like it may be possible to simply make a Taylor expansion around mˆ = 0; however, as we will see, the function JT(m ) is not analytic in mˆ 2 around mˆ 2 = 0. Putting this aside for a second, the zeroth order term in this expansion is trivial to compute

JT(0) =

T4 2π 2

∫0

∞

dx x 2 log (1 − e−x ) = −

π 2T 4 , 90

(4.88)

which is nothing but the free energy density (negative of the pressure) for black body radiation with one massless degree of freedom, aka the Stefan-Boltzmann limit. The correction at order mˆ 2 can also be evaluated by naively Taylor expanding; however, the integral coefficient of the mˆ 4 term is power-divergent at small x = k /T . So we need a more clever technique. A general method was invented by Dolan and Jackiw with the result being [6]

⎧ π2 m m m ˆ2 ˆ3 ˆ4 JT(m) = T 4⎨ − + − − 24 12π 2(4π )2 ⎩ 90 ⎡ ⎫ ⎛ me ˆ γ⎞ 3⎤ m ˆ 6ζ(3) ⎟ − × ⎢log ⎜ + O(m ˆ 8) ⎬ , ⎥+ 4 ⎝ 4π ⎠ 4 ⎦ ⎣ ⎭ 3(4π )

4-15

(4.89)

Relativistic Quantum Field Theory, Volume 3

where mˆ ≡ mˆ 2 and ζ (x ) is the Riemann zeta function. The non-analyticity of the result is evidenced by the third term which, when viewed as an expansion in mˆ 2 has a branch cut. We will now try to derive this result by using the form given in equation (4.66) instead. We will evaluate the momentum integral ∫ first and then do the Matsubara k sum. Of course, equation (4.66) contains some constant terms that are not listed; however, we know the mass independent term J (0, T ) = JT(0) already since this is simply the high-temperature Stefan-Boltzmann limit. As a result, we can focus our attention on I (m, T ) and then integrate using

J (m , T ) = J (0, T ) +

∫0

m

dm′ m′I (m′ , T ).

(4.90)

Focusing on I (m, T ) it turns out to be essential to separate out the zero Matsubara mode from the non-zero Matsubara modes using

∫∑k f (ωn, k) = ∫∑′k f (ωn, k) + T ∫k

f (0, k),

(4.91)

where, as before, the primed sum-integral is the sum-integral with the zero-mode removed. Let us focus first on the second term, which we will denote as I (n=0) and use the function Φ introduced previously in equation (4.74)

I (n=0) = T

∫k

1 mT = T Φ(m , 3 − 2ε , 1) = − + O(ε ). 2 k +m 4π 2

(4.92)

Integrating to obtain J (n=0), we obtain

J (n=0) = −

m3T + O(ε ), 12π

(4.93)

which, as we can see, is precisely the non-analytic term in the result quoted above. So, we see that the non-analyticity is associated with the infrared physics of the Matsubara zero mode. To prove this conclusively, we need to include the contributions from the non-zero Matsubara modes4. Turning to the contributions from the non-zero Matsubara, one finds that these contributions can be Taylor-expanded since the integrals encountered are of the form

∫k 4

(m 2 ) n 2 (ωn + k 2 )

n +1

,

ωn ≠ 0,

(4.94)

This is not an artifact of dimensional regularization. If cutoff regularization were used, the result is Λ

limΛ→∞I (n=0) = T [ 2π 2 −

m 4π

2

m + O( Λ )]. The first term is canceled by similar terms coming from the non-zero

modes since the temperature-dependent part is manifestly finite. This leaves behind the second term. In dimensional regularization, such power law divergences simply do not appear in the first place.

4-16

Relativistic Quantum Field Theory, Volume 3

which are all finite in the infrared (small k). There may still be ultraviolet divergences but these will be taken care of systematically by the regularization. Explicitly, one obtains

I ′(m , T ) = T ∑

∫k

n′ ∞

ωn2

1 + k 2 + m2

∞

= 2T ∑ n=1

∫k

m 2ℓ ∑( −1) [(2 nT )2 k 2 ] ℓ+1 . π + ℓ=0

(4.95)

ℓ

Performing the k integration using equation (4.78), one obtains

2Tμ¯ 2ε I ′(m , T ) = (4π )d /2

∞

∞

∑ ∑( −1) ℓ m2ℓ n=1 ℓ=0

1 Γ(ℓ + 1 − d /2) . (2πnT )2ℓ+2−d Γ(ℓ + 1)

(4.96)

Reordering the sums and recognizing that the sum over n gives a Riemann zeta function ∞

ζ (s ) ≡

1 , ns n=1

∑

(4.97)

we obtain

I ′(m , T ) =

2Tμ¯ 2ε d /2 (4π ) (2πT )2−d

⎡ −m 2 ⎤ ℓ Γ(ℓ + 1 − d /2) ζ(2ℓ + 2 − d ). (4.98) Γ(ℓ + 1) ⎣ ⎦ ℓ=0 ∞

∑⎢ (2πT )2 ⎥

Note that this is a series in even powers of m. Computing this term by term, combining it with the zero-mode contribution, and then integrating to obtain J one obtains

⎧ π2 ˆ2 ˆ3 ˆ4 m m m + − − J (m , T ) = T 4⎨ − 24 12π 2(4π )2 ⎩ 90 ⎫ ⎡1 ⎛ μe γ ⎞⎤ m ˆ 6ζ(3) ⎟ + ×⎢ + log ⎜ + O(m ˆ 8 )⎬ . ⎥ 4 ⎝ 4πT ⎠⎦ ⎣ 2ε 3(4π ) ⎭

(4.99)

Subtracting the zero temperature contribution J0(m ) given by equation (4.80) gives the expansion of the purely thermal contribution JT(m ) quoted in equation (4.89) above. Exercise 4.7. Verify equation (4.99).

4.4 The need for resummation In this section, we consider thermal field theories at high temperatures, which means temperatures much higher than all zero-temperature masses or any mass scales 4-17

Relativistic Quantum Field Theory, Volume 3

generated at zero temperature. It has been known for many years that naive perturbation theory, or the loop expansion, breaks down at high temperature due to infrared divergences. Diagrams which are nominally of higher order in the coupling constant contribute to leading order in g. A consistent perturbative expansion requires the resummation of an infinite subset of diagrams from all orders of perturbation theory. We discuss these issues first in the context of scalar field theory. In what follows in this chapter, we will work in Euclidean space, with Euclidean four-vectors indicated with capital letters, e.g. P, and the length of the space-like (vector) part indicated with a lower case letter, e.g. p. We start our discussion by considering the simplest interacting thermal field theory, namely that of a single massless scalar field with a ϕ4 -interaction. In this case, the Euclidean Lagrangian density is

L=

1 g2 4 (∂μϕ)2 + ϕ. 2 4!

(4.100)

Perturbative calculations at zero temperature proceed by dividing the Lagrangian density into a free part and an interacting part according to

Lfree = Lint =

1 (∂μϕ)2 , 2

(4.101)

g2 4 ϕ. 4!

(4.102)

Radiative corrections are then calculated in a loop expansion, which is equivalent to a power series in g2. We shall see that the naive perturbative expansion breaks down at finite temperature and the weak-coupling expansion becomes an expansion in g rather than g2. To demonstrate this, we will calculate the self-energy by evaluating the relevant diagrams. The Feynman diagrams that contribute to the self-energy up to two loops are

one loop:

two loop:

.

(4.103)

The one-loop diagram is independent of the external momentum and the resulting integral expression is

1 1 Π(1) = g 2 ∑p 2 , P 2 g2 2 = T , 24 ≡ m2 ,

∫

4-18

(4.104)

Relativistic Quantum Field Theory, Volume 3

where the superscript indicates the number of loops. The sum-integral above has an ultraviolet power-law divergence, which is zero in dimensional regularization. We are then left with the finite result (4.104), which shows that thermal fluctuations generate a mass for the scalar field of order gT. This thermal mass is analogous to the Debye mass, which is well known from non-relativistic QED plasmas. We next focus on the two-loop diagrams and first consider the double-bubble diagram (b) in the sequence of bubble diagrams:

=

+ (a)

+ (b)

+ ···

(4.105)

(c)

This diagram is also independent of the external momentum and results in the following sum-integral:

1 1 1 Π(2b) = − g 4 ∑pq 2 4 . 4 P Q

∫

(4.106)

This integral is infrared divergent. The problem stems from the loop with two propagators. In order to isolate the source of the divergence, we look at the contribution from the zeroth Matsubara mode to the Q integration

1 1 − g 4 ∑p 2 T 4 P

∫

∫q

1 , q4

(4.107)

which is quadratically infrared divergent due to the three-momentum integral over q. This infrared divergence indicates that naive perturbation theory breaks down at finite temperature. However, in practice this infrared divergence is screened by the thermally generated mass and we must somehow take this into account. The thermal mass can be incorporated by using an effective propagator

Δ(ωn, p ) =

1 , P + m2 2

(4.108)

with m ∼ gT ≪ T . If the momentum of the propagator is of order T or hard, clearly the thermal mass is a perturbation and can be omitted. However, if the momentum of the propagator is of order gT or soft, the thermal mass is as large as the bare inverse propagator and cannot be omitted. The mass term in the propagator (4.108) provides an infrared cutoff of order gT. The contribution from (4.107) would then be

1 1 − g 4 ∑p 2 T 4 P

∫

∫q

1 1 4⎛ T 2 ⎞⎛ T ⎞ ⎟ + O(g 4mT ) . = − g ⎜ ⎟⎜ (q 2 + m 2 ) 2 4 ⎝ 12 ⎠⎝ 8πm ⎠

(4.109)

Since m ∼ gT , this shows that the double-bubble contributes at order g 3T 2 to the self-energy and not at order g 4T 2 as one might have expected. Similarly, one can show that the diagrams with any number of bubbles like equation (4.105) (c) are all of order g3. Clearly, naive perturbation theory breaks down since the order-g3

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Relativistic Quantum Field Theory, Volume 3

correction to the thermal mass receives contributions from all loop orders. On the other hand, the three-loop diagram

,

(4.110)

is of order g 4T 2 and thus subleading. Therefore, we only need to resum a subset of all possible Feynman graphs in order to obtain a consistent expansion in g. If we use the effective propagator to recalculate the one-loop self-energy, we obtain

1 1 Π(1)(P ) = g 2 ∑p 2 2 P + m2 ⎤ 1 ⎡ 1 1 2 ′ O ( ) = g 2⎢T + ∑ + m ⎥ p 2 2 ⎦ 2 ⎣ P2 p p + m ⎤ g2 ⎡ g 6 = T 2⎢1 − + O(g 2 )⎥ , 4π 24 ⎣ ⎦

∫

∫

∫

(4.111)

where here, once again, the prime on the sum-integral indicates that we have excluded the n = 0 mode from the sum over the Matsubara frequencies. The order g3 corresponds to the summation of the bubble diagrams in equation (4.105), which can be verified by expanding the effective propagator (4.108) around m = 0. Thus, by taking the thermal mass into account, one is resumming an infinite set of diagrams from all orders of perturbation theory. The self-energy (4.104) is the first example of a hard thermal loop (HTL). Hard thermal loops are loop corrections that are g 2T 2 /P2 times the corresponding treelevel amplitude, where P is a momentum that characterizes the external lines. From this definition, we see that, whenever P is hard, the loop correction is suppressed by g2 and is thus a perturbative correction. However, for soft P, the hard thermal loop is O(1) and is therefore as important as the tree-level contribution to the amplitude. These loop corrections are called ‘hard’ because the relevant integrals are dominated by momenta of order T. Also, note that the hard thermal loop in the two-point function is finite since it is exclusively due to thermal fluctuations. Quantum fluctuations do not enter. Both properties are shared by all hard thermal loops [7]. What about higher-order n-point functions in scalar thermal field theory? One can show that, within scalar theory, the one-loop correction to the four-point function for high temperature behaves as

Γ (4) ∝ g 4 log (T / p ),

(4.112)

where p is the external momentum. Thus, the loop correction to the four-point function increases logarithmically with temperature. It is therefore always down by one power of g, and it suffices to use a bare vertex. More generally, it can be shown that the only hard thermal loop in scalar field theory is the tadpole diagram and 4-20

Relativistic Quantum Field Theory, Volume 3

resummation is taken care of by including the thermal mass in the propagator. In gauge theories, the situation is much more complicated as we shall see in the next chapter. Exercise 4.8. Verify equation (4.104).

4.5 Perturbative expansion of thermodynamics for a scalar field theory The simplest way of dealing with the infrared divergences in scalar field theory is to reorganize perturbation theory in such a way that it incorporates the effects of the thermally generated mass m into the free part of the Lagrangian. One possibility is to divide the Lagrangian (4.100) into free and interacting parts according to

Lfree = Lint =

1 1 (∂μϕ)2 + m 2ϕ 2 , 2 2

(4.113)

g2 4 1 2 2 ϕ − mϕ . 24 2

(4.114)

Both terms in equation (4.114) are treated as interaction terms of the same order, namely g2. However, the resummation implied by the above is rather cumbersome when it comes to calculating Green’s function with zero external energy. Static Green’s functions can always be calculated directly in imaginary time without having to analytically continue them back to real time. This implies that we can use Euclidean propagators with discrete energies when analyzing infrared divergences, which greatly simplifies the treatment. In particular, since only propagators with zero Matsubara frequency have no infrared cutoff of order T, only for these modes is the thermal mass of order gT relevant as an IR cutoff. Thus, another possibility is to add and subtract a mass term only for the zero-frequency mode. This approach has the advantage that we do not need to expand the sum-integrals in powers of m2 /T 2 in order to obtain the contribution from a given term in powers of g2. We will follow this path in the remainder of this section and write

Lfree = Lint =

1 1 (∂μϕ)2 + m 2ϕ 2δ p0 ,0, 2 2

(4.115)

g2 4 1 2 2 ϕ − m ϕ δ p0 ,0. 24 2

(4.116)

The free propagator then takes the form

Δ(ωn, p ) =

1 − δ p0 ,0 P

2

+

4-21

δ p0 ,0 2

p + m2

.

(4.117)

Relativistic Quantum Field Theory, Volume 3

This way of resumming is referred to as static resummation. It is important to point out that this simplified resummation scheme can only be used to calculate static quantities such as the pressure or screening masses. Calculation of dynamical quantities requires the full Braaten-Pisarski resummation program [7]. The problem is that the calculation of correlation functions with zero external frequencies cannot unambiguously be analytically continued to real time. We next consider the calculation of the free energy through order g5 in scalar field theory. This involves the evaluation of vacuum graphs up to three-loop order. The diagrams are those shown in figure 4.1. 4.5.1 One loop The one-loop contribution to the free energy is

F0 =

1 T 2

∫p

log (p 2 + m 2 ) +

1 ∑′p log P 2. 2

∫

(4.118)

Evaluating the necessary integrals/sum-integrals, the result for this diagram in the limit ε → 0 is

F0 = −

π 2 4 Tm3 . T − 90 12π

(4.119)

Exercise 4.9. Show that equation (4.119) follows from equation (4.118). [Hint: d log P2 /dm2 = 1/P2 .]

4.5.2 Two loops The two-loop contribution to the free-energy is given by

F1 = F1a + F1b,

(4.120)

with

1 ⎛ F1a = g 2⎜T 8 ⎝

∫p

2 1 1 ⎞ ′ + ∑p 2 ⎟ , p2 + m2 P ⎠

∫

(4.121)

Figure 4.1. Diagrams which contribute up to three-loop order in scalar perturbation theory. A boldfaced × indicates an insertion of m2.

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Relativistic Quantum Field Theory, Volume 3

1 F1b = − m 2T 2

∫p

1 . p + m2

(4.122)

2

The result for these diagrams in the limit ε → 0 is

⎡ ⎛ ⎪5 π 2T 4 ⎧ ζ′( −1) ⎞⎤⎛⎜ μ ⎞⎟4ε ⎨ ⎢ 1 4 4 α + ε + ⎜ ⎟⎥ 90 ⎪ ζ( −1) ⎠⎦⎝ 4πT ⎠ ⎝ ⎩4 ⎣

F1a =

⎛ ⎪ 5 6 3/2⎡ ζ′( −1) ⎞⎤⎛⎜ μ ⎞⎟2ε ⎛⎜ μ ⎞⎟2ε 15 2⎫ , + − α ⎢1 + ε⎜4 + 2 α⎬ ⎟⎥ ⎪ 2 ⎭ 2 ζ( −1) ⎠⎦⎝ 4πT ⎠ ⎝ 2m ⎠ ⎝ ⎣ F1b =

π 2T 4 5 6 3/2 α , 90 2

(4.123)

(4.124)

where we have kept all terms that contribute through order ε because they are needed for the counterterm diagrams in the three-loop free energy and α ≡ g 2(μ)/16π 2 . Exercise 4.10. Show that equations (4.123) and (4.124) follow from equations (4.121) and (4.122).

4.5.3 Three loops The three-loop contribution is given by

F2 = F2a + F2b + F2c + F2d +

F1a Δ1g 2, g2

(4.125)

where the expressions for the diagrams are

F2a = −

1 4⎛ g ⎜T 16 ⎝

∫p

2 1 1 ⎞⎛ ′p ⎟ ⎜T + ∑ p2 + m2 P2 ⎠ ⎝

∫

∫p

1 1 ⎞ ′p ⎟ , + ∑ (p 2 + m 2 ) 2 P4 ⎠

∫

1 4 1 1 1 1 g ∑′pqr 2 2 2 P Q R (P + Q + R ) 2 48 1 4 3 1 1 1 1 − gT 2 2 2 2 2 2 2 2 48 pqr p + m q + m r + m (p + q + r) + m + O(g 6),

F2b = −

∫

∫

F2c =

(4.126)

1 2 2⎛ g m ⎜T 4 ⎝

∫p

1 1 ⎞⎛ + ∑′p 2 ⎟⎜T 2 (p + m ) P ⎠⎝ 2

∫

4-23

∫p

⎞ 1 ⎟, 2 2 (p + m ) ⎠ 2

(4.127)

(4.128)

Relativistic Quantum Field Theory, Volume 3

1 F2d = − m 4T 4

∫p

1 . (p + m 2 ) 2 2

(4.129)

The result for these diagrams in the limit ε → 0 is

F2a =

π 2T 4 ⎧ 5 6 3/2 5 2⎡ 1 ζ′( −1) ⎤⎛⎜ μ ⎞⎟6ε ⎨− α − α ⎢ + 2γE + 4 + 4 ⎥ ζ( −1) ⎦⎝ 4πT ⎠ 90 ⎩ 8 8 ⎣ε ζ′( −1) ⎤⎛⎜ μ ⎞⎟2ε ⎛⎜ μ ⎞⎟6ε ⎫ 5 6 5/2⎡ 1 ⎬, + α ⎢ + 2γE + 4 + ⎥ ζ( −1) ⎦⎝ 4πT ⎠ ⎝ 2m ⎠ ⎭ ⎣ε 4

F2b =

π 2T 4 ⎧ 5 2⎡ 1 ζ′( −1) ζ′( −3) 91 ⎤⎛⎜ μ ⎞⎟6ε ⎨− α ⎢ + 8 + − ⎥ ζ( −1) ζ( −3) 90 ⎩ 4 ⎣ ε 15 ⎦⎝ 4πT ⎠ ⎤⎛ μ ⎞6ε ⎫ 5 6 5/2⎡ 1 ⎟ ⎬, + α ⎢ + 8 − 4 log 2⎥⎜ ⎣ε ⎦⎝ 2m ⎠ ⎭ 2 F2c =

π 2T 4 ⎡ 5 6 3/2 15 2⎤ ⎢ α − α ⎥, 90 ⎣ 4 2 ⎦

F2d = −

π 2T 4 5 6 3/2 α . 90 8

(4.130)

(4.131)

(4.132)

(4.133)

4.5.4 Pressure through g5 Combining the one-, two-, and three-loop contributions given by equations (4.118), (4.120), and (4.125), respectively, gives the free energy through order g5

F =−

5 5 6 3/2 15 ⎛ 59 π 2T 4 ⎡ μ ⎜log ⎢1 − α + − − 3 log 2 + γ α + 4 3 4⎝ 2πT 15 90 ⎣

5 2 μ ζ′( −1) ζ′( −3) ⎞ 2 15 6 ⎛ ⎜log − log α + −2 ⎟α − 6 2 ⎝ 2πT 3 ζ( −1) ζ( −3) ⎠ ⎤ 1 2 ζ′( −1) ⎞ 5/2 2 5 − log 2 + log 3 + γ − ⎟α + O(α 3 log α )⎥ . 3 3 ζ( −1) ⎠ 3 3 ⎦ +4

(4.134)

The pressure through order g5 was first calculated using resummation by Parwani and Singh [8] and later by Braaten and Nieto [9] using effective field theory. This calculation was recently extended to four-loop order by Andersen and Kyllingstad [10, 11].

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Relativistic Quantum Field Theory, Volume 3

The renormalization group equation for the coupling g2 is

μ

dα = 3α 2 . dμ

(4.135)

Using equation (4.135), one can verify that the free energy (4.134) is RG-invariant up to corrections of order g 6 log g . In figure 4.2, we show the successive perturbative approximations to the pressure as a function of g (2πT ). The bands are obtained by varying the renormalization scale μ by a factor of two around the central value μ = 2πT . The lack of convergence of the weak-coupling expansion is evident from this figure. The band obtained by varying μ by a factor of two is not necessarily a good measure of the error, but it is certainly a lower bound on the theoretical error. Another indicator of the theoretical error is the deviation between successive approximations. We can infer from figure 4.2 that the error grows rapidly when g (2πT ) exceeds 1.5.

4.6 Screened perturbation theory In this section, we discuss one possibility to reorganize perturbation theory, which is screened perturbation theory (SPT), which was introduced by Karsch, Patkós, and Petreczky [12]. It can be made more systematic by using a framework called ‘optimized perturbation theory’ that Chiku and Hatsuda have applied to a spontaneously broken scalar field theory [13]. In this approach, one introduces a single variational parameter m, which has a simple interpretation as a thermal mass. The advantage of SPT is that it is relatively easy to apply and that higher-order corrections can be calculated so that one can study convergence properties.

Figure 4.2. Weak-coupling expansion for the pressure to orders g2, g3, g4, and g5 normalized to that of an ideal gas as a function of g (2πT ).

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Relativistic Quantum Field Theory, Volume 3

In SPT, a mass term is added and subtracted to the scalar Lagrangian with the added bit kept as part of the free Lagrangian and the subtracted bit associated with the interactions. This can be accomplished by introducing a parameter δ in the spirit of the linear delta expansion [14, 15]. The resulting Euclidean Lagrangian is

L=

1 g 2δ 4 (∂μϕ)2 + (1 − δ )m 2ϕ 2 + ϕ, 2 24

(4.136)

where we have also introduced a factor of δ in the ϕ4 term. In the limit δ → 1, the SPT Lagrangian reduces to the standard massless scalar Lagrangian. The Lagrangian (4.136) is then expanded into free and interacting parts with all interaction terms proportional to δ

Lfree =

1 1 (∂μϕ)2 + m 2ϕ 2 2 2

⎛ g2 ⎞ 1 Lint = δ⎜ ϕ4 − m 2ϕ 2⎟ . ⎝ 24 ⎠ 2

(4.137)

(4.138)

The loop expansion then simply becomes an expansion in δ. Note also that the counterterms that are necessary to renormalize the theory also have expansions in powers of δ. After expansion of all of the contributions to the appropriate order in δ, we set δ = 1. If we were able to calculate the result to all orders in δ, the result would independent of the mass parameter m; however, at any finite order in the expansion there will be a residual dependence on m. We will discuss three possible ways to fix the parameter m including a variational prescription. The diagrams that contribute to the free energy up to three-loop order in SPT are the same as shown in figure 4.1. These were first evaluated in [16]. 4.6.1 One-loop contribution The one-loop free energy is

F0 = F0a + Δ0E 0.

(4.139)

Computing the sum-integrals and counterterm Δ0E 0, we obtain the final result for the one-loop free energy

1 1 (4π )2 F0 = − (2L + 3)m 4 − J0T 4, 2 8

(4.140)

where Jn are functions of m/T

4e γε π

Jn(βm) = Γ

(

5 2

−n−ε

)

β 4 − 2nm 2ε

∫0

∞

dk

evaluated at ε = 0 and L = log(μ2 /m2 ).

4-26

1 k 4 −2n −2ε 2 2 1/2 β (k 2+m 2 )1/2 (k + m ) e −1

(4.141)

Relativistic Quantum Field Theory, Volume 3

Exercise 4.11. Verify equation (4.140). 4.6.2 Two-loop contribution The contribution to the free energy of order δ is

F1 = F1a + F1b + Δ1E 0 +

∂F0a Δ1m 2 , ∂m 2

(4.142)

where Δ1E 0 and Δ1m2 are the order-δ vacuum and mass counterterms, respectively. Evaluating the sum-integrals and counterterms, the two-loop contribution to the free energy becomes

1 1 [(L + 1)m 2 − J1T 2 ]m 2 + α[(L + 1)m 2 − J1T 2 ]2 . 8 2

(4π )2 F1 =

(4.143)

Exercise 4.12. Verify equation (4.143).

4.6.3 Three-loop contribution The contribution to the free energy of order δ 2 is

F2 = F2a + F2b + F2c + F2d + Δ2E 0 1 ∂ 2F0a ∂F 2 + 0a2 Δ2m 2 + (Δ1m 2 ) 2 (∂m 2 )2 ∂m ⎛ ∂F ∂F1b ⎞ F1a ⎟Δ1m 2 + Δ1g 2, + + ⎜ 1a 2 ⎝ ∂m ∂m 2 ⎠ g2

(4.144)

where we have included all of the order-δ 2 counterterms. Evaluating the integrals and counterterms, the three-loop contribution to the free energy becomes

1 α (L + J2 )m14 − (L + J2 )[(L + 1)m 2 − J1T 2 ]m 2 4 4 ⎡ ⎛ 41 1 2 − L − 23 α ⎢⎜5L3 + 17L2 + 2 48 ⎣⎝ ⎞ 23 2 − π − ψ ″(1) + C 0 + 3(L + 1)2 J2 ⎟m 4 ⎠ 12

(4π )2 F2 = −

− (12L2 + 28L − 12 − π 2 − 4C1 + 6(L + 1)J2 )J1m 2T 2 ⎤ + (3(3L + 4)J12 + 3J12J2 + 6K2 + 4K3)T 4 ⎥ , ⎦

4-27

(4.145)

Relativistic Quantum Field Theory, Volume 3

where C0 = 39.429, C1 = −9.8424, and K2 and K3 are functions of m/T given in [17]. 4.6.4 Pressure to three loops The contributions to the free energy of zeroth, first, and second order in δ are given in (4.140), (4.143), and (4.145), respectively. Adding them, we obtain the successive approximations to the free energy in screened perturbation theory. Using the fact that P = −F , we obtain the one-loop approximation to the pressure:

(4π )2 P0 =

1 [4J0T 4 + (2L + 3)m 4 ]. 8

(4.146)

The two-loop approximation is obtained by including (4.143)

1 (4π )2 P0+1 = [4J0T 4 + 4J1m 2T 2 − (2L + 1)m 4 ] 8 1 − α[J1T 2 − (L + 1)m 2 ]2 . 8

(4.147)

The three-loop approximation is obtained by including (4.145) to obtain

(4π )2 P0+1+2 =

1 [4J0T 4 + 4J1m 2T 2 + 2J2m 4 − m 4 ] 8 1 − α[J1T 2 − (L + 1)m 2 ][J1T 2 + 2J2m 2 + (L − 1)m 2 ] 8 1 2⎡ + α ⎢ 3J2(J1T 2 − (L + 1)m 2 )2 48 ⎣

(4.148)

+ (3(3L + 4)J12 + 6K2 + 4K3)T 4 − (12L2 + 28L − 12 − π 2 − 4C1)J1m 2T 2 ⎛ ⎞ ⎤ 23 2 41 L − 23 − + ⎜5L3 + 17L2 + π − ψ ″(1) + C 0⎟m 4⎥ . ⎝ ⎠ ⎦ 12 2 The only remaining task is to determine the mass parameter m which we will discuss next. 4.6.5 Mass prescription The mass parameter m in screened perturbation theory is completely arbitrary. In order to complete a calculation using SPT, we need a prescription for the mass parameter m = m*(T ). The prescription of Karsch, Patkós, and Petreczky for m*(T ) is the solution to the one-loop gap equation [12]:

m*2 =

⎡ ⎛ ⎞ ⎤ μ 1 α(μ*)⎢J1(m*/ T )T 2 − ⎜2 log * + 1⎟m*2⎥ , ⎝ ⎠ ⎦ 2 m* ⎣

4-28

(4.149)

Relativistic Quantum Field Theory, Volume 3

where the function J1 is defined in (4.141). Their choice for the scale was μ* = T . In the weak-coupling limit, the solution to (4.149) is m* = g (μ*)T / 24 . There are many possibilities for generalizing (4.149) to higher orders in g. In this section, we will only consider one. 4.6.6 The tadpole mass prescription The tadpole mass is generalization of equation (4.149) to higher loops. The tadpole mass is defined by

mt2 = g 2〈ϕ 2〉.

(4.150)

Note that the tadpole mass is well defined at all orders in scalar field theory, but the generalization to gauge theories is problematic. The natural replacement of 〈ϕ2〉 would be 〈AμAμ〉, which is a gauge-variant quantity. In the next chapter, we will discuss gauge theories and we will have to find a suitable gauge-invariant generalization of this mass prescription. 4.6.7 Three-loop SPT Pressure In figure 4.3, we show the one-, two-, and three-loop SPT-improved approximations to the pressure using the tadpole gap equation (4.150). The bands are obtained by varying μ by a factor of two around the central values μ = 2πT . The one-loop bands in figure 4.3 lie below the other bands; however, the two- and three-loop bands both lie within the g5 band of the naive weak-coupling expansion in figure 4.2. The one-, two-, and three-loop approximations to the pressure are perturbatively correct up to

Figure 4.3. One-, two-, and three-loop SPT-improved pressure as a function of g (2πT ). Bands correspond to a variation of the renormalization scale πT < μ < 4πT . Figure taken from [16].

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Relativistic Quantum Field Theory, Volume 3

Figure 4.4. Two-, three-, and four-loop SPT-improved pressure as a function of g (2πT ). Figure taken from [10].

order g1, g3, and g5, respectively; however, we see a dramatic improvement in the apparent convergence compared to the weak-coupling expansion. Improved convergence properties were also found for the entropy and the Debye mass [16, 18]. This has recently been extended to four-loop order [10]. The result is shown in figure 4.4. These results demonstrate the effectiveness of SPT in providing stable and apparently converging predictions for the thermodynamic functions of a massless scalar field theory. An essential ingredient of this approach is using the solution to a gap equation as the prescription for the mass parameter m. In the following section, we will see how this idea can be extended to gauge theories.

References [1] Wang X N 2016 Quark-Gluon Plasma 5 (Singapore: World Scientific) [2] Kapusta J I and Gale C 2011 Finite-temperature field theory: principles and applications Cambridge Monographs on Mathematical Physics (Cambridge: Cambridge University Press) [3] Le Bellac M 1996 Thermal Field Theory Cambridge Monographs on Mathematical Physics (Cambridge: Cambridge University Press) [4] Andersen J O and Strickland M 2005 Ann. Phys. 317 281–353 [5] Laine M and Vuorinen A 2016 Lect. Notes Phys. 925 1–281 [6] Dolan L and Jackiw R 1974 Phys. Rev. D 9 3320–41 [7] Braaten E and Pisarski R D 1990 Nucl. Phys. B 337 569–634 [8] Parwani R and Singh H 1995 Phys. Rev. D 51 4518–24 [9] Braaten E and Nieto A 1995 Phys. Rev. D 51 6990–7006 [10] Andersen J O and Kyllingstad L 2008 Phys. Rev. D 78 076008 [11] Andersen J O, Kyllingstad L and Leganger L E 2009 JHEP 08 066 [12] Karsch F, Patkos A and Petreczky P 1997 Phys. Lett. B 401 69–73 [13] Chiku S and Hatsuda T 1998 Phys. Rev. D 58 076001

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[14] [15] [16] [17] [18]

Okopinska A 1987 Phys. Rev. D 35 1835–47 Duncan A and Moshe M 1988 Phys. Lett. B 215 352–8 Andersen J O, Braaten E and Strickland M 2001 Phys. Rev. D 63 105008 Andersen J O, Braaten E and Strickland M 2000 Phys. Rev. D 62 045004 Andersen J and Strickland M 2001 Phys. Rev. D 64 105012

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IOP Concise Physics

Relativistic Quantum Field Theory, Volume 3 Applications of quantum field theory Michael Strickland

Chapter 5 Hard-thermal-loops for QED and QCD

In the previous chapter, we demonstrated the need for resummation in a hot scalar theory. For scalar theories, resummation simply amounts to including the thermal mass in the propagator and since the running coupling depends logarithmically on the temperature, corrections to the bare vertex are always down by powers of g2. In gauge theories, the situation is more complicated. The equivalent HTL self-energies are no longer local, but depend in a non-trivial way on the external momentum. In addition, it is also necessary to use effective vertices that depend on the external momentum. It turns out that all hard thermal loops are gauge-fixing independent. This was shown explicitly in covariant gauges, Coulomb gauges, and axial gauges. They also satisfy tree-level-like Ward identities. Furthermore, a gauge invariant effective Lagrangian exists, found independently by Braaten and Pisarski [1] and by Taylor and Wong [2], which generates all of the hard thermal loop n-point functions. From a renormalization group point of view, this is an effective Lagrangian for the soft scale gT that is obtained by integrating out the hard scale T. First, we review the basics of hard-thermal-loops applied to finite temperature gauge theory, specifically QED and QCD. For a recent review, which details the derivation of the hard-loop vertex functions both in and out of equilibrium using a variety of different approaches, we refer the reader to [3]. After this brief review, we then present state-of-the-art results for resummed three-loop QCD thermodynamics at finite temperature and quark chemical potentials.

5.1 Photon polarization tensor We next discuss in some detail the hard thermal loop for the polarization tensor Πμν . For simplicity, we discuss QED. In the high-temperature limit, this suffices because one finds that in QCD the gluon polarization tensor (and fermion self-energy) are the same up to scalings by Casimirs. The Feynman diagram for the one-loop selfenergy

doi:10.1088/2053-2571/ab3a99ch5

5-1

ª Morgan & Claypool Publishers 2019

Relativistic Quantum Field Theory, Volume 3

which results in the following sum-integral

⎡ K γμ( P + K )γν ⎤ ⎥, Πμν(ωn, p ) = e 2 ∑{k}Tr ⎢ ⎣ K 2 (P + K ) 2 ⎦

∫

(5.2)

where Tr denotes the trace over Dirac indices. After taking the trace, the self-energy becomes

K μK ν 1 − 4δμνe 2 ∑{k} 2 K (P + K ) 2 K PμK ν + PνK μ 1 , + 2δμνP 2e 2 ∑{k} 2 + 4e 2 ∑ { k } 2 2 K (P + K ) 2 K (P + K )

∫

Πμν(ωn, p ) = 8e 2 ∑{k}

∫

2

∫

(5.3)

∫

where we have assumed, for now, that d = 3. We first consider the spatial components of Πμν(ωn , p ). The sum over Matsubara frequencies can be written as a contour integral in the complex energy plane. After analytic continuation, we obtain

Πij (ω, p ) = e 2

∫

∫

dω0 2πi

⎡ 2 2 2δij ⎢ 4kikj − (ω − p )δij + 2(pi kj + kipj ) − ⎢ 2 ⎡ ⎤ 2 2 2 2 k k − ω 02 ⎣ k − ω 0 ⎣(p + k) − (ω + ω0) ⎦

(

tanh

)

(

)

⎤ ⎥ ⎥ ⎦

(5.4)

βω0 . 2

In order to extract the hard thermal loop, we notice that terms that contain one or more power of the external momentum can be omitted since the external momentum p is assumed to be soft. The self-energy then becomes

Πij (ω, p ) = 2e 2

∫

∫

dω 0 2πi

⎡ 2kikj δij ⎢ − ⎢ 2 ⎡ ⎤ 2 2 2 2 k k − ω 02 ⎣ k − ω 0 ⎣(p + k) − (ω + ω0) ⎦ βω0 tanh . 2

(

)

(

5-2

)

⎤ ⎥ ⎥ ⎦

(5.5)

Relativistic Quantum Field Theory, Volume 3

After integrating over the energy ω0 , we obtain

Πij (ω, p ) = − 2e 2δij

∫k

1 (1 − 2nF (k )) + 2e 2 k

ki kj k ∣ p + k∣

∫k

⎧ × ⎨ (1 − nF (k ) − nF (∣p + k∣)) ⎩ ⎡ ⎤ 1 1 + ⎢ ⎥ ⎣ k + ∣p + k∣ + ω k + ∣p + k∣ − ω ⎦

(5.6)

⎡ ⎤⎫ 1 1 − [nF (k ) − nF (∣p + k∣)] ⎢ + ⎥⎬ , ⎣ ∣p + k∣ − k + ω ∣p + k∣ − k − ω ⎦⎭ where nF (x ) = 1/(exp(βx ) + 1) is the Fermi–Dirac distribution function. The zerotemperature part of equation (5.6) is logarithmically divergent in the ultraviolet. This term depends on the external momentum and is canceled by standard zerotemperature wavefunction renormalization. We next consider the terms that depend on temperature. In the case that the loop momentum is soft, the Fermi–Dirac distribution functions can be approximated by a constant. The contribution from the integral over the magnitude of k is then of order g3 and subleading. When the loop momentum is hard, one can expand the terms in the integrand in powers of the external momentum. We can then make the following approximations:

nF (∣p + k∣) ≈ nF (k ) +

dnF (k ) ˆ p · k, dk

(5.7)

1 1 ≈ , ∣p + k∣ + k ± ω 2k

(5.8)

1 1 ≈ , ∣p + k∣ − k ± ω p · kˆ ± ω

(5.9)

where kˆ = k/k is a unit vector. Thus the angular integration decouples from the integral over the magnitude k. This implies

Πij (ω, p) =

4e 2 π2

=

m D2

∞

∫0 ∫

dk k nF (k )

∫

dΩ ω kˆikˆj , 4π ω − p · kˆ

dΩ ω kˆikˆj , 4π ω − p · kˆ

where we have defined m D2 = e 2T 2 /3.

5-3

(5.10)

Relativistic Quantum Field Theory, Volume 3

The other components of the self-energy tensor Πμν(ω, p) are derived in the same manner. One finds

⎛ Π 00(ω, p) = m D2⎜ ⎝

∫

Π 0j (ω, p) = m D2

⎞ dΩ ω − 1⎟ , 4π ω − p · kˆ ⎠

(5.11)

d Ω ωkˆj . 4π ω − p · kˆ

(5.12)

∫

Exercise 5.1. Verify equations (5.11) and (5.12).

5.1.1 Generalization to d-dimensions In d dimensions, we can compactly write the self-energy tensor as

Πμν(p ) = m D2[T μν(p , −p ) − n μn ν ],

(5.13)

where n specifies the thermal rest frame is canonically given by n = (1, 0), we have defined

∫

m D2 = −4(d − 1)e 2 ∑{k}

1 , K2

(5.14)

and the tensor T μν(p, q ), which is defined only for momenta that satisfy p + q = 0, is

T μν(p , −p ) =

y μy ν

p·n p·y

.

(5.15)

yˆ

The angular brackets indicate averaging over the spatial directions of the light-like vector y = (1, yˆ ). The sum-integral above is

∫∑{p} P12

=−

T2 + O(ε ). 24

Exercise 5.2. Derive equation (5.15).

5-4

(5.16)

Relativistic Quantum Field Theory, Volume 3

5.1.2 The HTL polarization tensor The tensor T μν is symmetric in μ and ν and the self-energy satisfies the Ward identity:

pμ Πμν( p ) = 0.

(5.17)

Because of this Ward identity and the rotational symmetry around the pˆ -axis, one can express the self-energy in terms of two independent functions, ΠT(ω, p) and ΠL(ω, p):

Πμν(ω, p) = ΠL(ω, p)

(ω 2 − p 2 )gμν − pμ pν p2

⎡ ⎤ ω2 − p2 + ⎢ΠT(ω, p) − ΠL(ω, p)⎥gμi δij − pˆi pˆj gjν, 2 p ⎣ ⎦

(

(5.18)

)

where the functions ΠT(ω, p) and ΠL(ω, p) are

ΠT(ω, p ) =

1 (δij − pˆi pˆj )Πij (ω, p ), 2

ΠL(ω, p ) = −Π 00(ω, p ).

(5.19) (5.20)

In three dimensions, the self-energies ΠT(ω, p) and ΠL(ω, p) reduce to

ΠT(ω, p ) =

m D2 ω 2 ⎡ p2 − ω2 ω + p⎤ 1 log + ⎢ ⎥, 2 2 p ⎣ 2ωp ω − p⎦

⎡ ω ω + p⎤ log ΠL(ω, p ) = m D2⎢1 − ⎥, 2p ω − p⎦ ⎣

(5.21)

(5.22)

where here p = ∣p∣. Exercise 5.3. Derive equations (5.21) and (5.22). 5.1.3 Generalization to QCD The hard thermal loop in the photon propagator was first calculated by Silin more than 40 years ago [4]. The hard thermal loop in the gluon self-energy was first calculated by Klimov and Weldon. For QCD, we have to calculate the hard loops in all polarization graphs

5-5

Relativistic Quantum Field Theory, Volume 3

.

The final result of combining all of these graphs has the same form as in QED, but where the Debye mass mD is replaced by

⎡ 1 1 ⎤ m D2 = g 2⎢(d − 1)2 CA ∑k 2 − 2(d − 1)Nf ∑{k} 2 ⎥ , ⎣ K K ⎦

∫

∫

(5.24)

where CA = Nc is the number of colors and Nf is the number of flavors and

∫∑p P12 ∫∑{p} P12

=

T2 + O(ε ), 12

=−

T2 + O(ε ), 24

(5.25)

(5.26)

with the second sum-integral being the fermionic sum-integral over odd Matsubara frequencies. When d = 3, the QCD gluon Debye mass becomes

m D2 =

1⎛ 1 ⎞ ⎜CA + Nf ⎟g 2T 2. 3⎝ 2 ⎠

Exercise 5.4. Derive equation (5.24).

5.2 Fermionic self-energy The electron self-energy

is given by

5-6

(5.27)

Relativistic Quantum Field Theory, Volume 3

Σ(P ) = m f2γμT μ(p ),

(5.29)

where

T μ(p ) =

yμ p·y

,

(5.30)

yˆ

and mf

∫

m 2f = −3e 2 ∑{k}

1 . K2

(5.31)

In QCD, the quark mass is given by

∫

m 2q = −3CFg 2 ∑{k}

1 . K2

(5.32)

Exercise 5.5. Derive equation (5.29).

5.3 Collective modes Let us consider the quark–gluon plasma in a homogeneous and stationary state in which case there are no net local color charges and no net currents. The state is perturbed by either a random fluctuation or an external field. As a result, local charges or currents appear that generate chromoelectric and chromomagnetic fields. The fields in turn interact with colored partons, which contribute to the local charges and currents. If the wavelength of the perturbation exceeds the typical inter-particle spacing, the plasma undergoes a collective motion involving many partons that are present within the interaction range. The collective motions caused by imbalanced charges or currents are classically called the plasma oscillations or plasma waves. Quantum mechanically we deal with quasiparticle collective excitations of the plasma. The spectrum of collective excitations is a fundamental characteristic of any many-body system. It carries a great deal of information on the thermodynamic and transport properties of an equilibrium system, it also controls to a large extent the temporal evolution of a non-equilibrium one. In the quark–gluon plasma, there are collective modes corresponding to plasma particles, that is quarks and (transverse) gluons, there are also collective excitations being genuine many-body phenomena like longitudinal gluon modes (longitudinal plasmons), plasminos, and phonons, which are density fluctuations. We discuss here the longitudinal and transverse gluon, as well as the quark collective modes.

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Relativistic Quantum Field Theory, Volume 3

5.3.1 Gluon modes The dispersion relations for gluonic collective modes naturally appear when one considers the equation of motion of the Fourier-transformed gauge potential Aμ(k ) which represents the chromodynamic mean field. The equation of motion of Aμ(k ) in the absence of external sources is

[k 2g μν − k μk ν − Πμν(k )]Aν (k ) = 0,

(5.33)

where k ≡ (ω, k) and Πμν(k ) is the retarded polarization tensor that contains dynamical information about the system. Specifically, it describes how the plasma responds to chromodynamic perturbations. Solutions of the homogeneous equation (5.33) exist if

det [k 2g μν − k μk ν − Πμν(k )] = 0.

(5.34)

Equation (5.34) is the general dispersion equation to be solved by ω(k) which gives the dispersion relation for the collective modes. The gauge potential Aμ from equation (5.33) can be understood as a classical field or expectation value of the gauge field operator. This is justified due to the long wavelengths of collective excitations. However, we obtain the same dispersion equation (5.34) when, instead of the classical field Aμ, one starts with the equation of motion of a retarded propagator of the gauge field. Then, the dispersion relations correspond to poles of the propagator. We recall that, in general, Πμν(k ) is transversal, that is kμ Πμν(k ) = 0, which guarantees its gauge independence and current conservation. Due to the transversality, not all components of Πμν(k ) are independent, and consequently the dispersion equation (5.34), which involves a determinant of a 4 × 4 matrix, can be simplified to the determinant of a 3 × 3 matrix. For this purpose, one can introduce the chromodielectric tensor εij (ω, k) which is related to the polarization tensor by

ε ij (k ) = δ ij +

1 ij Π (k ). ω2

(5.35)

Using the chromodielectric tensor, we write down the dispersion equation as

det [Ξ(ω , k)] = 0,

(5.36)

where the matrix Ξ(ω, k), which equals the inverse gluon propagator in the temporal axial gauge, is defined as

Ξij (ω, k) ≡ k2δ ij − k ik j − ω 2ε ij (ω, k).

(5.37)

The relationship between equations (5.34) and (5.36) is most easily seen in the radiation gauge when A0 = 0 and k · A(k ) = 0. Then, E = iω A and equation (5.33)

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Relativistic Quantum Field Theory, Volume 3

is immediately transformed into an equation of motion for E(k ) which further provides the dispersion equation (5.36). Solutions to the dispersion equations ω(k), which are, in general, complex-valued, represent plasma modes. If the imaginary part of a mode’s frequency Iω is negative, the mode is damped—its amplitude exponentially decays in time as e Iωt . The mode is over-damped when additionally Rω(k) = 0. When Iω = 0, we have a stable mode with a constant amplitude. Finally, if Iω > 0, the mode’s amplitude exponentially grows in time, i.e. there is an instability. When the electric field of a mode is parallel to its wave vector k , the mode is called longitudinal. A mode is called transverse when the electric field is transverse to the wave vector. The Maxwell equations show that the longitudinal modes, also known as electric or electrostatic modes, are associated with electric charge oscillations. The transverse modes, which are also known as magnetic, correspond to oscillations of electric current. 5.3.2 Quark modes Since the quark field ψ (k ) obeys the equation

[ k − Σ(k )]ψ (k ) = 0,

(5.38)

where Σ(k ) is the retarded quark self-energy, the dispersion equation of quark modes is

det [ k − Σ(k )] = 0.

(5.39)

The quark self-energy has the following spinor structure

Σ(k ) = γ μΣμ(k ),

(5.40)

with

Σ μ(k ) = g 2

CF 4

∫

d 3p f˜ (p) pμ , (2π )3 Ep k · p + i 0+

(5.41)

where

f˜ (p) = 2( fq (p) + f¯q (p)) + 4fg (p).

(5.42)

Substituting the expression (5.40) into equation (5.39) and computing the determinant, one obtains

[(k μ − Σ μ(k ))(kμ − Σμ(k ))]2 = 0. This can be used to determine the fermionic dispersion relation.

5-9

(5.43)

Relativistic Quantum Field Theory, Volume 3

5.3.3 Collective modes in an isotropic QGP The above discussions are general. We would now like to specialize to the spectrum of collective modes in the case of an isotropic QGP which, in particular, includes the case of an equilibrium plasma. 5.3.4 Gluon modes If the plasma is isotropic, the dielectric tensor can be expressed as

⎛ k ik j ⎞ k ik j ε ij (ω, k) = εT(ω, k)⎜δ ij − 2 ⎟ + ε L(ω, k) 2 , ⎝ k ⎠ k

(5.44)

where the longitudinal (εL(ω, k)) and transverse (εT(ω, k)) components equal

ε L ( ω , k) =

k ik j ij ε (ω, k), k2

εT(ω, k) =

1 ii ( ε ( ω , k) − ε L ( ω , k) ) . 2

(5.45)

Then, the dispersion equation (5.36) splits into two equations

ω 2εT(ω, k) − k2 = 0.

ε L(ω, k) = 0,

(5.46)

We note that in vacuum, where εL = εT = 1, equations (5.46) give two transverse modes ω = ±∣k∣ and no longitudinal one. For further analysis, one needs explicit expressions for εL(ω, k) and εT(ω, k). Using

ε ij (k ) = δ ij +

g2 2ω

∫

vi d 3p k · v ⎞ kj k kv j ⎤ ∂f (p) ⎡⎛⎜ ⎟δ + ⎥ , (5.47) ⎢ 1− 3 k ⎣⎝ + (2π ) ω − k · v + i 0 ∂p ω ⎠ ω ⎦

the longitudinal and transverse chromodielectric functions are found to be

ε L ( ω , k) = 1 +

g2 2ω k2

∫

∂f (p) d 3p (k · v)2 , 3 + (2π ) ω − k · v + i 0 ∂E p

(5.48)

εT(ω, k) = 1 +

g2 4ω k2

∫

∂f (p) d 3p (k × v)2 , 3 + (2π ) ω − k · v + i 0 ∂E p

(5.49)

where we have used the fact that in an isotropic plasma the distribution function f (p) depends on a particle’s momentum only through E p and thus

∂f (p) ∂f (p) =v . ∂E p ∂p

(5.50)

One computes εL(ω, k) and εT(ω, k) in spherical coordinates with the z-axis chosen to lie along the vector k . As a result, the integral over the azimuthal angle is trivial while the integrals over E p = ∣p∣ and factorize when plasma particles are massless and ∣v∣ = 1. In this way, one finds

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Relativistic Quantum Field Theory, Volume 3

ε L ( ω , k) = 1 +

2 m D2 ⎛ ω ω + ∣k∣ ⎞ π 2 2 m Dω ⎟ ⎜ 1 − ln + i Θ ( k − ω ) , ω − ∣k∣ ⎠ k2 ⎝ 2∣ k ∣ 2 ∣k∣3

εT(ω, k) = 1 −

m D2 ⎛ ω 2 − k2 ω + ∣k∣ ⎞ ⎟ ⎜ − 1 ln ω − ∣k∣ ⎠ 2k 2 ⎝ 2ω∣k∣

m 2 (ω 2 − k 2 ) π + i Θ(k2 − ω 2 ) D , ω∣k∣3 4

(5.51)

(5.52)

where mD is the Debye mass given as

m D2 ≡ −

g2 4π 2

∞

∫0

dE pEp2

df (p) g2 = dE p 2π 2

∞

∫0

dE pE pf (p) = g 2

∫

d 3p f (p) . (5.53) (2π )3 E p

As can be seen from equations (5.48) and (5.49), the denominators of the integrands vanish for ω = k · v , giving rise to IεL and IεT through the relation

⎡1 ⎤ 1 = P⎢ ⎥ ∓ iπδ(x ), + ⎣x ⎦ x ± i0

(5.54)

where P denotes the principal value of the integral. As a consequence, the imaginary contributions to εL(ω, k) and εT(ω, k) appear when ω 2 < k2 , as seen in equations (5.51) and (5.52). The dielectric functions (5.51) and (5.52) of an ultrarelativistic plasma were derived for the first time by Silin in 1960 [4]. Substituting the dielectric functions (5.51) and (5.52) into equation (5.46), one obtains explicit dispersion equations that must be solved numerically. However, they can easily be solved in the long wavelength limit ω ≫ ∣k∣. Then, the logarithms in formulas (5.51) and (5.52) can be expanded in powers of ∣k∣/ω and

ε L ( ω , k) = 1 −

⎛ k 4 ⎞⎤ m D2 ⎡ 3k2 ⎢1 + + O⎜ 4 ⎟⎥ , 2 2 ⎝ ω ⎠⎦ 5ω 3ω ⎣

(5.55)

εT(ω, k) = 1 −

⎛ k 4 ⎞⎤ m D2 ⎡ k2 ⎢ 1 + + O ⎜ 4 ⎟⎥ . ⎝ ω ⎠⎦ 3ω 2 ⎣ 5ω 2

(5.56)

With the approximate expressions of εL and εT , one easily obtains the following dispersion relations for long-wavelength longitudinal and transverse modes:

ω L2(k) = m g2 +

⎛ k4 ⎞ 3 2 k + O⎜⎜ 2 ⎟⎟ , 5 ⎝ mg ⎠

(5.57)

ω T2(k) = m g2 +

⎛ k4 ⎞ 6 2 k + O⎜⎜ 2 ⎟⎟ , 5 ⎝ mg ⎠

(5.58)

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Relativistic Quantum Field Theory, Volume 3

where mg ≡ mD / 3 is the plasmon mass. Classically, this is the lowest frequency of plasma oscillations, which is the same for transverse and longitudinal modes. The dispersion equations (5.46) with the dielectric functions (5.51) and (5.52) can also be solved analytically in the short wavelength limit (k2 ≫ m D2) and the dispersion relations are

ω T2(k) ≈

1 2 m D + k2 , 2

⎡ ⎛ ⎞⎤ k2 ω L2(k) ≈ k2⎢1 + 4 exp⎜ −2 2 − 2⎟⎥ . ⎢⎣ ⎝ mD ⎠⎥⎦

(5.59)

(5.60)

Numerical solutions of the dispersion equations (5.46) are shown in figure 5.1. The transverse and longitudinal modes are both above the light cone ω = ±∣k∣, even so the longitudinal mode approaches the light cone as k2 → ∞, in agreement with formula (5.60). Since the dispersion curves stay above the light cone, the phase velocity of the plasma waves exceeds the speed of light. Consequently, there is no Landau damping that occurs when the velocity of a plasma particle equals the phase velocity of the wave. 5.3.5 Landau damping For space-like momentum the gluon polarization function is complex-valued. This is related to the phenomenon of Landau damping. We would now like to understand this in the context of classical QED plasma. To do this, let us consider a plane wave of the electric field with the wave vector along the z axis. For a charged particle, which moves along the axis z with a velocity v = pz /E p equal to the phase velocity of

Figure 5.1. Gluon collective modes (plasmons) in an isotropic plasma. The solid line represents the transverse mode and the dashed line the longitudinal one. The grey double-dashed line denotes the light cone ω = ∣k∣.

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Relativistic Quantum Field Theory, Volume 3

Figure 5.2. The Landau-damping mechanism for energy transfer between particles and fields.

the wave vϕ = ω /k , the electric field does not oscillate but it is constant. The particle is then either accelerated or decelerated depending on the field′s phase. For an electron with v = vϕ the probability to be accelerated and to be decelerated are equal to each other, as the time intervals spent by the particle in the acceleration zone and in the deceleration zone are equal to each other. Let us now consider electrons with velocities somewhat smaller than the phase velocity of the wave. Such particles spend more time in the acceleration zone than in the deceleration zone, and the net result is that the particles with v < vϕ are accelerated. Consequently, the energy is transferred from the electric field to the particles. The particles with v > vϕ spend more time in the deceleration zone than in the acceleration zone, and thus they are effectively decelerated—the energy is transferred from the particles to the field. If the momentum distribution is such that there are more electrons in a system with v < vϕ than with v > vϕ , the wave loses energy, which is gained by the particles, as shown in figure 5.2. This is the mechanism of collisionless Landau damping of plasma oscillations. 5.3.6 Quark modes As mentioned previously, when the plasma is isotropic, the distribution function f˜ (p), which enters the quark self-energy (5.41), depends on the particle′s momentum only through E p . Computing the self-energy (5.41) in spherical coordinates with the z-axis chosen along the vector k , the integral over the azimuthal angle is trivial while the integrals over E p = ∣p∣ and θ factorize when plasma particles are massless and ∣v∣ = 1. As a result, one finds

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Relativistic Quantum Field Theory, Volume 3

Σ0(k ) =

Σ⃗(k ) =

m q2 k2

⎤ − iπ Θ(k2 − ω 2 )⎥ , ⎦

m q2 ⎡ ω + ∣k∣ ⎢ln 2∣k∣ ⎣ ω − ∣k∣

k−

m q2ω ⎡ ⎢ln 2∣k∣3 ⎣

ω + ∣k∣ ω − ∣k∣

⎤ − iπ Θ(k2 − ω 2 )⎥ k , ⎦

(5.61)

(5.62)

where

m q2 ≡

g2 CF 8π 2

∞

∫0

dE pE p f˜ (p).

(5.63)

For the case of an equilibrium plasma, the self-energies (5.61) and (5.62) were found originally by Weldon [5] and verified independently by Blaizot and Ollitrault [6]. With the self-energies (5.61) and (5.62), the dispersion equation (5.43), which can be rewritten as

(

2 (ω − Σ0(k )) − k − Σ⃗(k )

)

2

= 0,

(5.64)

cannot be solved analytically. However, in the long wavelength limit (ω 2 ≫ k2 ) the dispersion equation (5.64) simplifies to 2 ⎛ m q2 ⎞ 2 ⎜⎜1 − ⎟ ω − k2 = 0, 2 ⎟ ω ⎝ ⎠

(5.65)

which is solved by

ω±2(k) ≈ m q2 +

1 2 k ± m q ∣k∣ . 2

(5.66)

Figure 5.3. Quark collective modes in an isotropic plasma. The solid black line represents the normal mode and the red dashed line the plasmino. The grey double-dashed line denotes the light cone ω = ∣k∣.

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Relativistic Quantum Field Theory, Volume 3

Numerical solutions of dispersion equation (5.64) with the self-energies (5.61) and (5.62) are shown for a broad range of momenta in figure 5.3. As seen, ω+(k) monotonically grows as a function of ∣k∣. In the case of the ω− mode, its frequency initially decreases with ∣k∣ and there is a shallow minimum at ∣k∣ ≈ mq and further ω−(k) monotonically grows. The modes ω±(k) have opposite helicity over chirality ratio. The ω+(k) mode corresponds to the positive energy fermion, the ω−(k), which is sometimes called a plasmino, is a specific medium effect. Needless to say, both modes are stable. As discussed above, the resummed high-temperature gluon propagator can be compactly expressed in terms of the so-called hard thermal loop (HTL) gluon propagator. There is also an HTL resummed quark propagator and resummed vertices, which are necessary in order to maintain gauge invariance. As it turns out these can be collected into a compact HTL effective action, which is manifestly gauge invariant. Before presenting this effective action, we would first like to present explicit expressions for the low order n-point functions. We will present the Feynman rules in Minkowski space and then give some simple rules that can be used to obtain the Euclidean imaginary-time expressions. Additionally, for generality, we will present expressions that are valid in d + 1 dimensions since these are necessary when performing calculations in dimensional regularization. Exercise 5.6. Derive equations (5.61) and (5.62).

5.4 Hard-thermal-loop effective action The story with hard-thermal loops (HTL) does not end at the two-point functions. Since HTLs are formulated in gauge theories, there are constraints (Ward– Takahashi and Slavov–Taylor identities) that must be obeyed by all n-point functions. If there is modification of the two-point functions (quark and gluon propagators), then there must be corresponding modifications of the quark–gluon and gluon–gluon vertex functions that ensure gauge invariance. In this section, I summarize the ingredients that are necessary to construct a gauge-invariant effective HTL action from which all resummed vertex functions can be computed [1]. 5.4.1 Minkowski-space HTL gluon propagator Based on the results presented in the previous section, one finds that the HTL inverse gluon propagator in a general covariant gauge can be expressed as −1 Δ−1(p ) μν = Δ∞ (p ) μν −

1 μ ν p p , ξ

(5.67)

where ξ is the gauge-fixing parameter and −1 Δ∞ (p ) μν ≡ −p 2 g μν + p μ p ν − Πμν(p ),

5-15

(5.68)

Relativistic Quantum Field Theory, Volume 3

with Πμν being the HTL resummed gluon polarization tensor. The HTL gluon polarization tensor reads in turn

Πμν(p ) = m D2[T μν(p , −p ) − n μn ν ],

(5.69)

where n μ is again the heat-bath four-velocity in the local rest frame. The tensor T μν(p, q ), which is defined only for momenta that satisfy p + q = 0, is

T μν(p , −p ) =

y μy ν

p·n p·y

,

(5.70)

yˆ

where the angular brackets indicate averaging over the spatial directions of the lightlike vector y = (1, yˆ ). The tensor T μν is symmetric in μ and ν and satisfies the identity

pμ T μν( p , −p ) = ( p · n ) n ν .

(5.71)

The polarization tensor Πμν is therefore also symmetric in μ and ν and satisfies the Ward–Takahashi identity pμ Πμν(p ) = 0 as well as gμν Πμν(p ) = −m D2. Just as its full theory counterpart, the HTL gluon polarization tensor can be expressed in terms of two scalar functions, the transverse and longitudinal polarization functions ΠT and ΠL

ΠT(p ) =

1 (δ ij − pˆi pˆ j )Πij (p ), d−1

ΠL(p ) = −Π 00(p ),

(5.72) (5.73)

where pˆ is the unit vector in the direction of p . In terms of these functions, the polarization tensor reads

Πμν(p ) = −ΠT(p )T pμν −

1 ΠL(p )L pμν, n p2

(5.74)

where the transverse and longitudinal projectors Tp and Lp are

T pμν = g μν −

L pμν

n pμn pν p μ pν − , p2 n p2

=

n pμn pν n p2

.

(5.75)

(5.76)

The four-vector n pμ is n pμ = n μ − (n · p ) p μ /p2 and satisfies p · np = 0 and n p2 = 1 − (n · p )2 /p2 . In the local rest frame of the heat bath, one has n p2 = −p 2 /p2 . Note that the identity pμ Πμν( p ) = 0 reduces to (d − 1)ΠT(p ) + ΠL(p )/n p2 = m D2 which implies that there is only one independent polarization function.

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Relativistic Quantum Field Theory, Volume 3

We can express both gluon polarization functions in terms of the function T defined in equation (5.70):

ΠT(p ) =

m D2 ⎡⎣T 00(p , −p ) − 1 + n p2⎤⎦ , (d − 1)n p2

ΠL(p ) = m D2[1 − T 00(p , −p )].

00

(5.77) (5.78)

For consistency of higher order radiative corrections, it is essential to take the angular average in the definition of T μν( p, −p ) (5.70) in d = 3 − 2ε dimensions and analytically continue to d = 3 only after all poles in ε have been canceled. Expressing the angular average as an integral over the cosine of an angle, the expression for the 00 component of the tensor becomes

T 00(p , −p ) =

w(ε ) 2

1

∫− 1

dc (1 − c 2 )−ε

p0 , p0 − ∣p∣c

(5.79)

where the weight function reads w(ε ) = Γ( 32 − ε )/(Γ( 32 )Γ(1 − ε )). The integral in equation (5.79) must be defined so that it remains analytic as p0 → ∞. It then has a branch cut running from p0 = −∣p∣ to p0 = +∣p∣, and if we take the limit ε → 0, the result reduces to

T 00(p , −p ) =

p0 p + ∣ p∣ log 0 , 2∣p∣ p0 − ∣p∣

(5.80)

i.e. the function appearing in the usual d = 3 HTL polarization functions. Working in d = 3 and in the rest frame of the heat bath, we hereby obtain

ΠT(p ) =

2 p0 p0 − p 2 p0 + ∣p∣ ⎤ m D2 ⎡ p2 ⎢1 + 2 ⎥, log − 2 ⎢⎣ 2∣p∣ p 2 p0 − ∣p∣ ⎥⎦ p0 − p 2

⎡ p p + ∣ p∣ ⎤ ⎥, ΠL(p ) = m D2⎢1 − 0 log 0 2∣p∣ p0 − ∣p∣ ⎦ ⎣

(5.81)

(5.82)

which in the static limit ( p0 → 0) produce lim p0 →0ΠT = 0 and lim p0 →0ΠL = m D2. As discussed in the previous subsection, the vanishing of the static limit of the transverse polarization function means that chromomagnetic fields are not screened, while the finiteness of the static ΠL corresponds to the Debye screening of the chromoelectric interaction. Returning to the HTL gluon propagator, equation (5.68) can also be written as −1 Δ∞ (p ) μν = −

1 1 T pμν + 2 L pμν, ΔT(p ) n p ΔL(p )

where ΔT and ΔL are the transverse and longitudinal propagators:

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(5.83)

Relativistic Quantum Field Theory, Volume 3

1 , p − ΠT(p )

ΔT(p ) =

ΔL(p ) =

(5.84)

2

−n p2p 2

1 . + ΠL(p )

(5.85)

Note that for d = 3 and in the heat bath rest frame the second relation reduces to Δ−L1 = p 2 + ΠL and, furthermore, in the static limit this becomes lim p0 →0Δ−L1 = p 2 + m D2. Finally, we mention that the general covariant gauge HTL gluon propagator can be obtained by inverting equation (5.67) to obtain

Δμν(p ) = −ΔT(p )T pμν + ΔL(p )n pμn pν − ξ

p μ pν . (p 2 ) 2

(5.86)

5.4.2 Minkowski-space HTL quark propagator One can also extract the HTL resummed quark propagator using a similar procedure as we outlined for the gluon propagator. The result reads

1 , p − Σ( p )

S (p ) =

(5.87)

where the quark self-energy is given by

Σ(p ) = m q2 T (p ).

(5.88)

Here, we have defined

yμ p·y

T μ(p ) =

,

(5.89)

yˆ

and, for d = 3, one furthermore obtains

m q2 =

CF 2 2 gT . 8

(5.90)

Expressing the angular average as an integral over the cosine of an angle, the expression for T μ(p ) reads

T μ(p ) =

w(ε ) 2

1

∫− 1

dc (1 − c 2 )−ε

yμ . p0 − ∣p∣c

(5.91)

As before, the integral in equation (5.91) must be defined so that it is analytic as p0 → ∞. It then has a branch cut running from p0 = −∣p∣ to p0 = +∣p∣. For d = 3 and in the heat bath rest frame, the fermion self-energy reduces to

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Relativistic Quantum Field Theory, Volume 3

Σ( p ) =

m q2 2∣p∣

γ0 log

⎛ m q2 p0 + ∣p∣ p p + ∣p∣ ⎞ + γ ⃗ · pˆ ⎜1 − 0 log 0 ⎟. p0 − ∣p∣ 2∣p∣ p0 − ∣p∣ ⎠ ∣p∣ ⎝

(5.92)

5.4.3 Three-gluon vertex The three-gluon vertex for gluons with outgoing four-momenta p, q, and r, Lorentz indices μ, ν, and λ, and color indices a, b, and c is μνλ i Γ abc (p , q , r ) = −gfabc Γμνλ(p , q , r ),

(5.93)

where the three-gluon vertex tensor is

Γμνλ(p , q , r ) = g μν(p − q ) λ + g νλ(q − r ) μ + g λμ(r − p )ν − m D2T μνλ(p , q , r ). (5.94) The tensor T μνλ in the HTL correction term is defined only for p + q + r = 0:

⎛ p·n r·n ⎞ T μνλ(p , q , r ) = − y μy ν y λ ⎜ − ⎟ . ⎝ p · y q · y r · y q · y⎠

(5.95)

This tensor is totally symmetric in its three indices and traceless in any pair of indices: gμν T μνλ = 0. It is odd (even) under odd (even) permutations of the momenta p, q, and r, and it satisfies the identity

qμT μνλ(p , q , r ) = T νλ(p + q , r ) − T νλ(p , r + q ).

(5.96)

The three-gluon vertex tensor therefore also obeys the Ward–Takahashi identity −1 −1 pμ Γμνλ(p , q , r ) = Δ∞ (q )νλ − Δ∞ (r )νλ .

(5.97)

5.4.4 Four-gluon vertex The four-gluon vertex for gluons with outgoing momenta p, q, r, and s, Lorentz indices μ, ν, λ, and σ, and color indices a, b, c, and d reads

{

μνλσ i Γ abcd (p , q , r , s ) = − ig 2 fabx fxcd (g μλg νσ − g μσg νλ )

+ 2m D2tr[T a(T bT cT d + T d T cT b)] T

μνλσ

(p , q , r , s )

(5.98)

}

+ 2 cyclic permutations, where the cyclic permutations are of (q, ν, b ), (r, λ , c ), and (s, σ , d ). The matrices T a are the generators of the fundamental representation of the SU(3) group with the standard normalization tr(T aT b ) = 12 δ ab. The tensor T μνλσ in the HTL correction term is defined only for p + q + r + s = 0:

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Relativistic Quantum Field Theory, Volume 3

T μνλσ(p , q , r , s ) =

⎛ p·n y μy ν y λ y σ ⎜ ⎝ p · y q · y (q + r ) · y

(5.99)

(p + q ) · n (p + q + r ) · n ⎞ + + ⎟ . q · y r · y (r + s ) · y r · y s · y (s + p ) · y ⎠

This tensor is totally symmetric in its four indices and traceless in any pair of indices: gμν T μνλσ = 0. It is even under cyclic or anti-cyclic permutations of the momenta p, q, r, and s, and satisfies the identity

qμT μνλσ(p , q , r , s ) = T νλσ(p + q , r , s ) − T νλσ(p , r + q , s ).

(5.100)

When the color indices are traced in pairs, the four-gluon vertex becomes much simpler μνλσ δ abδ cd i Γ abcd (p , q , r , s ) = −ig 2Nc(Nc2 − 1)Γμν, λσ(p , q , r , s ),

(5.101)

where the color-traced four-gluon vertex tensor is

Γμν, λσ(p , q , r , s ) = 2g μνg λσ − g μλg νσ − g μσg νλ − m D2T μνλσ(p , s , q , r ).

(5.102)

The tensor (5.102) is symmetric under the interchange of μ and ν, under the interchange of λ and σ, and under the interchange of (μ, ν ) and (λ , σ ). It is also symmetric under the interchange of p and q, under the interchange of r and s, and under the interchange of (p, q ) and (r, s ). Finally, it satisfies the Ward–Takahashi identity

pμ Γμν, λσ(p , q , r , s ) = Γ νλσ(q , r + p , s ) − Γ νλσ(q , r , s + p ).

(5.103)

5.4.5 Quark-gluon three-vertex The HTL resummed quark–gluon vertex with outgoing gluon momentum p, incoming quark momentum q, and outgoing quark momentum r, Lorentz index μ and color index a reads

(

)

μ

Γ aμ(p , q , r ) = gta γ μ − m q2T˜ (p , q , r ) .

(5.104)

The tensor in the HTL correction term is only defined for p − q + r = 0 and is given by μ T˜ (p , q , r ) =

⎛ ⎞ y y μ⎜ ⎟ ⎝q · y r · y ⎠

.

(5.105)

yˆ

This tensor is even under the permutation of q and r. It satisfies the identity μ μ μ pμ T˜ (p , q , r ) = T˜ (q ) − T˜ (r ),

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(5.106)

Relativistic Quantum Field Theory, Volume 3

and the quark–gluon vertex therefore satisfies the Ward–Takahashi identity

pμ Γμ(p , q , r ) = S −1(q ) − S −1(r ).

(5.107)

5.4.6 Quark-gluon four-vertex We define the quark–gluon four-point vertex with outgoing gluon momenta p and q, incoming fermion momentum r, and outgoing fermion momentum s. Generally, this vertex has both adjoint and fundamental indices; however, for our presentation we will only need the quark–gluon four-point vertex traced over the adjoint color indices, μν μν δ abΓ abij (p , q , r , s ) = − g 2m q2CFδij T˜ (p , q , r , s )

(5.108)

≡ g 2CFδij Γμν . The tensor T˜

μν

is only defined for p + q − r + s = 0

μν T˜ (p , q , r , s ) =

⎛ 1 1 ⎞ y/ . + y μy ν ⎜ ⎟ s · y ⎠ [(r − p ) · y ] [(s + p ) · y ] ⎝r · y

(5.109)

It is is traceless and symmetric in μ and ν and satisfies the Ward–Takahashi identity

pμ Γμν(p , q , r , s ) = Γ ν(q , r − p , s ) − Γ ν(q , r , s + p ).

(5.110)

5.4.7 Hard thermal loop effective Lagrangian The HTL effective Lagrangian can be written compactly as [1]

L = LQCD + LHTL ,

(5.111)

where LQCD is the usual vacuum QCD Lagrangian. The HTL contribution to the effective Lagrangian can be written as

⎛ 1 yαy β LHTL = − m D2 Tr ⎜⎜Gμα 2 (y · D ) 2 ⎝

⎞ yμ G μβ⎟⎟ + im q2ψγ ¯ μ y·D ⎠ y

ψ,

(5.112)

y

where G μν is the gluon field strength tensor, D is the covariant derivative, y μ = (1, yˆ ) is a light-like vector, and, once again, 〈⋯〉 is the verage over all possible directions of yˆ . The HTL effective action is gauge invariant and can generate all HTL n-point functions [1], which satisfy the necessary Ward–Takahashi identities by construction. This includes all of the n-point functions we have listed thus far. For example,

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Relativistic Quantum Field Theory, Volume 3

when the HTL contribution to the effective Lagrangian is expanded in powers of the quark and gluon fields, there will be a term of the form

∫y ∫z ψ¯ (x) Γμ(x, y, z) ψ (y ) Aμ(z), where Γμ(x , y, z ) is the quark–gluon vertex function. To obtain this vertex function, we only need to expand the HTL effective Langrangian to leading order in the gluon field strength

y y·D

ψ¯ Aψ ) (x ) = im q2 ψ¯ (x ) L(HTL

=

im q2

ψ¯ (x )γ

μ

ψ (x ) y

y y·∂

⎛ i g y · A(x ) ⎞n ∑⎜ y · ∂ ⎟ ⎝ ⎠ n=0

(5.113)

∞

ψ (x ). y

After a Fourier transformation, the O(g 3) contribution gives

Γ aμ(p , q , r ) = igt a(2π )4δ (4)(p + q + r ) Γμ(p , q , r ),

(5.114)

with

⎛ ⎞ y/ Γμ(p , q , r ) = −m q2 y μ⎜ ⎟ ⎝q · y r · y ⎠

,

(5.115)

yˆ

where q and r are the incoming and outgoing quark momenta and p is the outgoing gluon momentum. This corresponds precisely to the HTL correction to the bare QCD vertex presented earlier. 5.4.8 Euclidean space HTL effective Lagrangian and vertex functions The HTL effective Lagrangian and vertex functions listed above were specified for Minkowski space. As mentioned earlier, in the imaginary-time formalism, one has discrete imaginary energies, i.e. the Matsubara sums p0 = i 2πnT . Continuing to use a capital letter for Euclidean momenta, e.g. P = (P0, p), the inner product of two Euclidean vectors reads P · Q = P0Q0 + p · q , while the vector that specifies the thermal rest frame remains n = (1, 0). The Feynman rules for Minkowski space given in the prior subsections can then be easily adapted to Euclidean space. The Euclidean tensor corresponding to a given Feynman rule is obtained from the corresponding Minkowski tensor with all indices raised by replacing each Minkowski energy p0 by iP0 and multiplying for every 0 index by −i . This prescription transforms p = (p0 , p) into P = (P0, p), g μν into −δ μν , and p · q into −P · Q . The effect on the HTL tensors defined in equations (5.15), (5.95),

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Relativistic Quantum Field Theory, Volume 3

and (5.99) is equivalent to substituting p · n → −P · N , where N = ( −i , 0); p · y → −P · Y , where Y = ( −i , yˆ ); and y μ → Y μ.

5.5 Hard-thermal-loop resummed thermodynamics In this section, I review the hard-thermal-loop resummed calculations of QCD thermodynamics to three-loop order. The calculations have been presented at finite temperature in [7, 8] and finite temperature/quark chemical potentials in [9, 10]. The calculation makes use of the so-called hard-thermal-loop perturbation theory (HTLpt) reorganization of finite tempeature QCD [11–14]. This method was designed to address the poor convergence of finite-temperature field theory for the free energy. It relies on a loop expansion around a non-trivial HTL background, which is the appropriate expansion point in the high-temperature limit. We start with the QCD Lagrangian density in Minkowski space, which can be written as

LQCD = −

1 Tr[GμνG μν ] + iψγ ¯ μDμψ + Lgh + Lgf + ΔLQCD , 2

(5.116)

where the field strength is G μν = ∂ μAν − ∂ νAμ − ig[Aμ , Aν ] and the covariant derivative is D μ = ∂ μ − igAμ. The term ΔLQCD contains the renormalization counterterms necessary to cancel ultraviolet divergences in perturbative calculations. The ghost term Lgh depends on the form of the gauge-fixing term Lgf . As mentioned above, HTLpt is a reorganization of in-medium perturbation theory for QCD. The HTLpt Lagrangian density can be written as

L = (LQCD + LHTL ) g→

δg

+ ΔLHTL ,

(5.117)

where the HTL improvement term is [1, 13] LHTL = (1 − δ )im q2ψγ ¯ μ

yμ y·D

ψ− yˆ

⎛ y α yβ 1 (1 − δ )m D2 Tr ⎜⎜Gμα 2 (y · D ) 2 ⎝

⎞ G μβ⎟⎟ , (5.118) ⎠ yˆ

and y μ = (1, yˆ ) is a light-like four-vector with yˆ being a three-dimensional unit vector. The two parameters mD and mq can be identified with the Debye screening mass and the thermal quark mass, respectively, and account for screening effects. HTLpt is defined by treating δ as a formal expansion parameter. By adding the HTL improvement term (5.118) to the QCD Lagrangian (5.116), HTLpt systematically shifts the perturbative expansion from being around an ideal gas of massless particles to being around a gas of massive quasiparticles, which are the appropriate physical degrees of freedom at high temperature and/or chemical potential. The HTLpt Lagrangian (5.117) reduces to the QCD Lagrangian (5.116) if we set δ = 1. Physical observables are calculated in HTLpt by expanding in powers of δ,

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Relativistic Quantum Field Theory, Volume 3

truncating at some specified order, and then setting δ = 1. This defines a reorganization of the perturbative series in which the effects of m D2 and mq2 terms in (5.118) are included to leading order but then systematically subtracted out at higher orders in perturbation theory by the δm D2 and δmq2 terms in (5.118). To obtain leading order (LO), next-to-leading order (NLO), and next-to-next-leading order (NNLO) results, one expands to orders δ 0, δ1, δ 2 , respectively. Note that HTLpt is gauge-invariant order-by-order in the δ expansion and, consequently, the results obtained are independent of the gauge-fixing parameter ξ. If the expansion in δ could be calculated to all orders, the final result would not depend on mD and mq when we set δ = 1. However, any truncation of the expansion in δ produces results that depend on mD and mq . As a consequence, a prescription is required to determine mD and mq as a function of T, μ and αs . Several prescriptions had been discussed in [8] at zero chemical potential. The HTLpt expansion generates additional ultraviolet divergences. In QCD perturbation theory, renormalizability constrains the ultraviolet divergences to have a form that can be canceled by the counterterm Lagrangian ΔLQCD. We will demonstrate that the renormalization of HTLpt can be implemented by including a counterterm Lagrangian ΔLHTL in the interaction terms in (5.118). There is no all-order proof that the HTL perturbation expansion is renormalizable, so the general structure of the ultraviolet divergences is unknown. However, as shown previously in [8, 13–15], it is possible to renormalize the NNLO HTLpt thermodynamic potential using only a vacuum counterterm, a Debye mass counterterm, a fermion mass counterterm, and a coupling constant counterterm. The necessary counterterms for renormalization of the NNLO thermodynamic potential are [8]

dA (1 − δ )2 m D4 , 128π 2ε

(5.119)

11cA − 4sF αsδ(1 − δ )m D2 , 12πε

(5.120)

3 dA αsδ(1 − δ )m q2 , 8πε cA

(5.121)

ΔE 0 = Δm D2 =

Δm q2 =

δΔαs = −

11cA − 4sF 2 2 αs δ , 12πε

(5.122)

where, with the standard normalization, the QCD Casimir numbers are cA = Nc , dA = Nc2 − 1, sF = Nf /2, dF = NcNf , and s2F = CFsf with CF = (Nc2 − 1)/2Nc . Note that the coupling constant counterterm (5.122) is consistent with one-loop running of αs . In addition to the δ expansion, it is also necessary to make a Taylor expansion in the mass parameters scaled by the temperature, mD /T and mq /T , in order to obtain analytically tractable sum-integrals. An added benefit of this procedure is that the

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Relativistic Quantum Field Theory, Volume 3

Figure 5.4. QCD diagrams contributing through NLO in HTLpt. Figure adapted from [8].

final result obtained at NNLO is completely analytic. In order to truncate the series in mD /T and mq /T , one treats these quantities as being O(g ) at leading order, keeping all terms that naively contribute to the thermodynamic potential through O(g 5). In practice, such an truncated expansion works well [16, 17] and the radius of convergence of the scaled mass expansion seems to be quite large, giving us confidence in this approximate treatment of the necessary sum-integrals. 5.5.1 Contributions to the HTLpt thermodynamic potential through NNLO The diagrams needed for the computation of the HTLpt thermodynamic potential through NNLO can be found in figure 5.4 and 5.5 with a key given in figure 5.6. In [8] the authors computed the NNLO thermodynamic potential at zero chemical potential. Here, I present the NNLO calculation at finite chemical potential. For this purpose, one needs to only consider diagrams that contain at least one quark propagator; however, for completeness we also list the purely gluonic contributions below. In the results, we will express thermodynamic quantities in terms of two ˆ = Λ/(2πT ). dimensionless variables: mˆ D = mD /(2πT ), μˆ = μ /(2πT ), and Λ The complete NNLO HTLpt thermodynamic potential can be expressed in terms of these diagrams as

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Relativistic Quantum Field Theory, Volume 3

Figure 5.5. QCD diagrams contributing to NNLO in HTLpt. Figure adapted from [8].

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Relativistic Quantum Field Theory, Volume 3

Figure 5.6. Key explaining various quantities appearing in figure 5.4 and 5.5. Figure adapted from [8].

ΩNNLO

=

dA⎡⎣ F 1ga + F 1gb + F 2gd + F 3gm⎤⎦ + dF ⎡⎣ F 1fb + F 2fd + F 3fi⎤⎦ + dAcA⎡⎣F 2ga + F 2gb + F 2gc + F 3gh + F 3gi + F 3gj + F 3gk + F 3gl ⎤⎦ + dAsF ⎡⎣F 2fa + F 2fb + F 3fd + F 3fe + F 3ff + F 3fg + F 3fk + F 3fl ⎤⎦ + dAcA2⎡⎣F 3ga + F 3gb + F 3gc + F 3gd + F 3ge + F 3gf + F 3gg⎤⎦ + dAs2F ⎡⎣ F 3fa + F 3fb⎤⎦ ⎡ 1 ⎤ + dAcAsF ⎢ − F 3fa + F 3fm + F 3fn + F 3fo⎥ + dAsF2⎡⎣F 3fc + F 3fj⎤⎦ ⎣ 2 ⎦ ∂ ∂ + Δ0E 0 + Δ1E 0 + Δ2E 0 + Δ1m D2 Ω + Δ1m q2 ΩLO (5.123) 2 LO ∂m D ∂m q2 + Δ2m D2

∂ ∂ ∂ ΩNLO Ω + Δ2m q2 Ω + Δ1m D2 2 LO 2 LO ∂m D2 ∂m D ∂m q

+ Δ1m q2

∂ ΩNLO ∂m q2

+

⎤ ⎤ 1 ⎡ ∂2 1 ⎡ ∂2 2 2 ⎢ ⎥(Δ1m q2 )2 m Ω Δ + Ω ⎢ ⎥ ( ) LO 1 LO D 2 2 2 2 ⎥⎦ 2 ⎣ (∂m D) 2 ⎢⎣ (∂m q ) ⎦

f ⎡c F g ⎤ A 2a +2b +2c + sF F 2a +2b ⎥Δ1αs , + dA⎢ ⎢⎣ ⎥⎦ αs

where the necessary counterterms at any order in δ can be calculated using equations (5.119)–(5.122). The expressions for the one- and two-loop diagrams above can be found in [13, 14]. The expressions for the three-loop bosonic diagrams F 3ga –F 3gm are presented in section 3 of [18], and the three-loop diagrams with fermions F 3fa –F 3fi can be found 5-27

Relativistic Quantum Field Theory, Volume 3

in section 3 of [19]. The three-loop diagrams specific to QCD, i.e. the non-Abelian diagrams involving quarks, are given by F 3fm =

1 ∑ 6 {pqr} (5.124) Tr [Γ α(R − P , R , P )S (P )Γ β(P − Q , P , Q )S (Q )Γ γ(Q − R , Q , R )S (R )]

∫

× Γ μνδ(P − R , Q − P, R − Q )Δαμ(P − R )Δβν (Q − P )Δγδ (R − Q ), βμ ¯ gμν(P )Δνα(P )Π ¯ αβ F 3fn = −∑p Π f (P )Δ (P ),

(5.125)

1 ¯ gμν(P ), F 3fo = − g 2 ∑p{q} Tr [Γ αβ(P, −P, Q , Q )S (Q )]Δαμ(P )Δβν(P )Π 2

(5.126)

∫

∫

where

¯ gμν(P ) = 1 g 2 ∑ Γμν, αβ(P, −P, Q , −Q )Δαβ (Q ) Π q 2 1 + g 2 ∑q Γμαβ(P, Q , −P − Q )Δαβ (Q )Γ νγδ 2 × (P, Q , −P − Q )Δγδ ( −P − Q ) Q μ(P + Q )ν , + g 2 ∑q 2 Q (P + Q ) 2

∫

∫

(5.127)

∫

∫

2 μ ν ¯ μν Π f (P ) = − g ∑{q}Tr [Γ (P , Q , Q − P )S (Q )Γ (P , Q , Q − P )S (Q − P )] .

(5.128)

¯ μν(P ) is the one-loop gluon self-energy with HTL-resummed propagators Thus, Π and vertices as in [8]:

¯ μν(P ) = cAΠ ¯ gμν(P ) + sF Π ¯ μν Π f (P ).

(5.129)

5.5.2 NNLO HTLpt thermodynamic potential One can evaluate the sum-integrals necessary analytically by expanding in the ratios mD /T and mq /T . For details concerning this expansion and intermediate results, we refer the reader to the appendices [10]. We consider first the case that all quarks have the same chemical potential μf = μ = μB /3 where f is a flavor index and μf ∈ {μu , μd , μs , ⋯ , μ Nf }. After presenting the steps needed for this case, we present the general result with separate chemical potentials for each quark flavor. 5.5.3 NNLO result for equal chemical potentials The final result for the NNLO HTLpt thermodynamic potential in the case that all quarks have the same chemical potential is

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Relativistic Quantum Field Theory, Volume 3

s α ⎡5 Ω NNLO 7 dF ⎛⎜ 120 2 240 4 ⎞⎟ μˆ + μˆ − F s ⎢ (1 + 12μˆ 2 )(5 + 12μˆ 2 ) 1+ = ⎠ π ⎣8 4 dA ⎝ 7 7 Ω0 ⎞ ⎛ ˆ 15 Λ 15 − (1 + 12μˆ 2 )mˆ D − ⎜2ln − 1 − ℵ(z )⎟mˆ D3 2 ⎝ 2 2 ⎠ ⎛ α ⎞2 +90mˆ q2mˆ D⎤⎦ + s2F ⎜ s ⎟ ⎝π⎠ ⎡ 15 ⎧ ζ′(−1) ×⎢ ⎨35 − 32(1 − 12μˆ 2 ) + 472μˆ 2 + 1328μˆ 4 64 ζ (−1) ⎩ ⎣ +64(−36iμˆℵ(2, z ) + 6(1 + 8μˆ 2 )ℵ(1, z ) + 3iμˆ (1 + 4μˆ 2 )ℵ(0, z ))} ⎤ ⎛ s α ⎞2 45 mˆ D(1 + 12μˆ 2 )⎥ + ⎜ F s ⎟ ⎦ ⎝ π ⎠ 2 ⎡ mˆ q2 5 2 2 2 ×⎢ + + + 1 12 30 1 12 μ μ ˆ ˆ ( ) ( ) ⎢⎣ 4mˆ D mˆ D ˆ 25 ⎧⎛ 72 2 144 4 ⎟⎞ Λ + ⎨⎜ 1 + μˆ + μˆ ln + ⎠ 2 12 ⎩⎝ 5 5 −

3 1 2 4 2 2 (1 + 168μˆ + 2064μˆ ) + (1 + 12μˆ ) γE 5 20 72 8 34 ζ′(−3) 2 ζ ′(− 1) − − (1 + 12μˆ ) − 25 ζ (−3) 5 5 ζ (−1) [8ℵ(3, z ) + 3ℵ(3, 2z ) − 12μˆ 2 ℵ(1, 2z )

(5.130)

+12iμˆ (ℵ(2, z ) + ℵ(2, 2z )) − iμˆ (1 + 12μˆ 2 ) ℵ(0, z ) − 2(1 + 8μˆ 2 )ℵ(1, z )]} ⎤ ⎞ ⎛ Λ ˆ 15 ⎧ − ⎨(1 + 12μˆ 2 )⎜2ln − 1 − ℵ(z )⎟}mˆ D⎥ ⎥⎦ 2 2 ⎩ ⎠ ⎝ ⎪

⎪

⎛ c α ⎞⎛ s α ⎞⎡ 15 235 1 + 12μˆ 2 ) − +⎜ A s ⎟⎜ F s ⎟⎢ ⎝ 3π ⎠⎝ π ⎠⎣ 2mˆ D ( 16 ⎧⎛ ˆ 792 2 1584 4 ⎞ Λ × ⎨⎜ 1 + μˆ + μˆ ⎟ln ⎠ 2 47 47 ⎩⎝ 319 ⎛ 2040 2 38640 4 ⎞ 144 2 ⎜1 + μˆ + μˆ ⎟ (1 + 12μˆ )lnmˆ D + ⎠ 940 ⎝ 319 319 47 24γE 2 − (1 + 12μ ) 47 44 ⎛ 156 2 ⎞ ζ′(−1) 268 ζ′(−3) 72 − ⎜1 + − − μˆ ⎟ [4iμˆℵ(0, z ) ⎝ ⎠ 47 11 235 ζ (−3) 47 ζ (−1) −

+(5 − 92μˆ 2 )ℵ(1, z ) + 144iμˆℵ(2, z ) + 52ℵ(3, z )]} + 90

mˆ q2

mˆ D ⎧ ˆ 9 ⎛ 132 2 ⎞ 315 ⎛ 132 2 ⎞ Λ 11 2 ⎜1 + ⎨⎜ 1 + + μˆ ⎟ln + μˆ ⎟ (1 + 12μˆ )γE + ⎠ 2 ⎠ 14 ⎝ 9 4 ⎩⎝ 7 7 ⎤ Ω YM 2 + ℵ(z )}mˆ D⎥ + NNLO ⎦ 7 Ω0

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Relativistic Quantum Field Theory, Volume 3

where

Ω YM NNLO

is

the

Ω 0 = −dAπ 2T 4 /45, and

NNLO

Ω3loop,YM Ω0

pure-glue

thermodynamic

potential

[18],

is the pure Yang Mills three-loop thermodynamic

potential [8]. Above, ℵ(z ) = Ψ(z ) + Ψ(z *) with z = 1/2 − iμˆ and Ψ being the digamma function. The pure-glue result is [8] Ω YM NNLO Ω0

= 1−

cα 15 3 mˆ D + A s 4 3π

⎡ ⎛ Λ ⎞ ⎤ ˆ ⎢ − 15 + 45 mˆ D − 135 mˆ D2 − 495 ⎜⎜ln g + 5 + γE ⎟⎟mˆ D3⎥ ⎢⎣ 4 2 2 4 ⎝ 2 22 ⎠ ⎥⎦ +

ˆ ⎛ cAαs ⎞2 ⎡ 45 165 ⎛ Λ 72 84 6 ⎜ ⎟ ⎜⎜ln g − − − γE ln mˆ D − ⎢ ⎝ 3π ⎠ ⎣ 4mˆ D 8 ⎝ 2 11 55 11 −

+

(5.131)

74 ζ′( − 1) 19 ζ′( − 3) ⎞ ⎟⎟ + 11 ζ ( − 1) 11 ζ ( − 3) ⎠

⎤ ˆ π2 ⎞ ⎥ 1485 ⎛ Λ 79 ⎜⎜ln g − ⎟⎟mˆ D . + γE + ln 2 − 4 ⎝ 2 44 11 ⎠ ⎥⎦

The result contained in equation (5.130) was first presented in [9]. Note that the full thermodynamic potential (5.130) reduces to thermodynamic potential of [8] in the limit μ → 0. In addition, the above thermodynamic potential produces the correct O(g 5) perturbative result when expanded in a strict power series in g [20, 21]1. 5.5.4 NNLO result—general case It is relatively straightforward to generalize the previously obtained result (5.130) to the case that each quark has a separate chemical potential μf . The final result is

There is a mismatch in one term proportional to s2F α s2 compared to result published in [20, 21]. We found that the second term proportional to s2F α s2 is 32(1 − 12μˆ 2 )ζ′(−1)/ζ (−1), whereas in [20, 21] it was listed as 32(1 − 4μˆ 2 )ζ′(−1)/ζ (−1). The author of [20, 21] has agreed that this was a typo in his article. 1

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ΩNNLO 7 dF 1 = Ω0 4 dA Nf

⎛

∑⎜⎝1 + f

s α 1 120 2 240 4 ⎞ μˆ + μˆ ⎟ − F s π Nf 7 f 7 f⎠

⎡5

∑⎢⎣ 8 (1 + 12μˆf2 )(5 + 12μˆf2 ) f

⎤ ⎞ ˆ Λ 15 15 ⎛ ⎜2ln − 1 − ℵ(zf )⎟mˆ D3 + 90mˆ q2mˆ D⎥ − 1 + 12μˆf2 mˆ D − ⎥⎦ 2 2 ⎝ 2 ⎠

(

+

)

s2F ⎛ αs ⎞2 ⎡ 15 ⎧ ζ ′(−1) ⎜ ⎟ ∑⎢ ⎨35 − 32 1 − 12μˆf2 + 472μˆf2 + 1328μˆf4 ζ (−1) Nf ⎝ π ⎠ f ⎣ 64 ⎩

(

)

(

+64 −36iμˆf ℵ(2, zf ) + 6(1 + 8μˆf2 )ℵ(1, zf )

)

45 +3iμˆf (1 + 4μˆf2 )ℵ(0, zf ) − mˆ D 1 + 12μˆf2 2 ⎡ 2 mˆ q2 ⎛s α ⎞ 1 5 ⎢ 4 +⎜ F s ⎟ + 96 1 + 12μˆf2 ∑ ⎝ π ⎠ Nf ⎢ ˆ 16 3 m D ⎣ f

⎤

}

(

)⎥⎦

(

)

(1 + 12μˆ )(5 + 12μˆ )ln Λ2 2 f

ˆ

2 f

ζ ′(−1) 1 64 ζ ′(−3) 32 − + 4γE + 8(7 + 12γE )μˆf2 + 112μf4 − (1 + 12μˆf2 ) ζ (−1) 3 15 ζ (−3) 3 ⎤ 2 −96 8ℵ(3, zf ) + 12iμˆf ℵ(2, zf ) − 2(1 + 2μˆf )ℵ(1, zf ) − iμˆf ℵ(0, zf ) ⎦⎥ +

{

}

⎡ 5 ⎛s α ⎞ 1 + ⎜ F s ⎟ 2 ∑⎢ 1 + 12μˆf2 1 + 12μˆg2 + 90 2(1 + γE )μˆf2 μˆg2 ⎝ π ⎠ N ˆ 4 m ⎣ D f f ,g 2

(

)(

)

{

(5.132)

{

− ℵ(3, zf + zg ) + ℵ(3, zf + zg*) +4iμˆf ⎡⎣ℵ(2, zf + zg ) + ℵ(2, zf + zg*)⎤⎦ − 4μˆg2 ℵ(1, zf ) −(μˆf + μˆg )2ℵ(1, zf + zg ) − (μˆf − μˆg )2ℵ(1, zf + zg*) − 4iμˆf μˆg2 ℵ(0, zf )

}}

⎞ ⎤ ⎛ ˆ Λ 15 − 1 + 12μˆf2 ⎜2ln − 1 − ℵ(zg )⎟mˆ D⎥ 2 2 ⎠ ⎥⎦ ⎝ ˆ ⎛ c α ⎞⎛ s α ⎞ ⎡ 15 235 ⎧⎛ 792 2 1584 4 ⎞ Λ ⎨⎜1 + +⎜ A s ⎟⎜ F s ⎟ ∑⎢ μˆf + μˆf ⎟ln 1 + 12μˆf2 − ⎝ 3π ⎠⎝ πNf ⎠ ⎢⎣ 2mˆ D ⎠ 2 16 ⎩⎝ 47 47 f

(

)

(

)

24γE 144 319 ⎛ 2040 2 38640 4 ⎞ ⎜1 + μˆ + μˆ ⎟ − 1 + 12μˆf2 lnmˆ D + 1 + 12μˆf2 47 940 ⎝ 319 f 319 f ⎠ 47 44 ⎛ 156 2 ⎞ ζ ′(−1) 268 ζ ′(−3) 72 ⎡ 2 − − ⎜1 + μˆ ⎟ − ⎢4iμˆ ℵ(0, zf ) + 5 − 92μˆf ℵ 47 ⎝ 11 f ⎠ ζ (−1) 235 ζ (−3) 47 ⎣ f

(

−

)

(

(

)

)

(1, zf ) ˆ mˆ q2 ⎤⎫ 315 ⎧⎛ 132 2 ⎞ Λ ⎨⎜1 + μˆf ⎟ln + 144iμˆf ℵ(2, zf ) + 52ℵ(3, zf )⎥⎬ + 90 + ⎠ 2 ⎦⎭ 4 ⎩⎝ 7 mˆ D YM ⎫ ⎤ Ω 11 9 ⎛ 132 2 ⎞ 2 ⎜1 + μˆ ⎟ + ℵ(zf )⎬mˆ D⎥ + NNLO , + 1 + 12μˆf2 γE + ⎭ ⎦ Ω0 7 14 ⎝ 9 f⎠ 7 ⎪ ⎪

(

)

where the sums over f and g include all quark flavors, zf = 1/2 − iμˆf , and Ω YM NNLO is the pure-glue contribution as before.

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Relativistic Quantum Field Theory, Volume 3

5.5.5 Mass prescription As discussed in [8], the two-loop perturbative electric gluon mass, first introduced by Braaten and Nieto in [22, 23], is the most suitable for three-loop HTLpt calculations. Originally, the two-loop perturbative mass was calculated in [22, 23] for zero chemical potential; however, Vuorinen has generalized it to finite chemical potential. The resulting expression for m D2 is [20, 21] ⎧ ⎡ ˆg⎞ Λ c 2α ⎛ αs ⎪ 1 2 ⎢ sF 1 + 12μˆ 2 ⎟⎟ + ˆ D = ⎨cA + A s ⎜⎜5 + 22γE + 22 ln m ∑ f ⎢ π 3π ⎪ 12 2 N f ⎝ ⎠ f ⎣ ⎩

(

cAsF αs ⎛ 2 2 ⎜ 9 + 132μˆ f + 22 1 + 12μˆ f γE 12π ⎝ ⎞ ˆ Λ + 2 7 + 132μˆ f2 ln + 4ℵ(zf )⎟ 2 ⎠

(

+

)

(

+

(

)

)

)

(5.133)

⎛ ⎞ ˆ sF2αs Λ 1 + 12μˆ f2 ⎜1 − 2 ln + ℵ(zf )⎟ 3π 2 ⎝ ⎠ ⎤⎫ ⎪ 3 s2F αs 1 + 12μˆ f2 ⎥⎬ . − ⎪ ⎥⎦⎭ 2 π

(

)

(

)

The effect of the in-medium quark mass parameter mq in thermodynamic functions is small and following [8] we take mq = 0 which is the three-loop variational solution. The maximal effect on the susceptibilities comparing the perturbative quark mass, mˆ q2 = αs(1 + 4μˆ 2 )/6π , with the variational solution, mq = 0, is approximately 0.2% at T = 200MeV. At higher temperatures, the effect is much smaller, e.g. 0.02% at T = 1 GeV. 5.5.6 Thermodynamic functions and susceptibilities With equation (5.132) as the final result for thermodynamic potential, one can compute the pressure, energy density, and entropy density as

P = − ΩNNLO(T , Λ , μ), ∂P ∂P − P, +μ E =T ∂μ ∂T ∂P . S= ∂T

(5.134)

From these, one can compute the trace anomaly I = E − 3P . For a conformal system with no running coupling, the trace anomaly vanishes, but for QCD it is nonzero due to the running coupling. In figure 5.7 and 5.8, we present the QCD pressure (left) and trace anomaly (right) as a function of temperature in the range of temperatures, which are relevant for studies of the quark–gluon plasma at LHC. Figure 5.7 shows the two quantities at 5-32

Relativistic Quantum Field Theory, Volume 3

Figure 5.7. The QCD pressure and trace anomaly at μB = 0 . In both panels, we compare the perturbative results with lattice data from the Wuppertal–Budapest (WB) collaboration [24]. Figure from [25].

Figure 5.8. The QCD pressure and trace anomaly at μB = 400 MeV . In both panels, we compare the perturbative results with lattice data from the Wuppertal–Budapest (WB) collaboration [24]. Figure from [25].

zero baryochemical potential and figure 5.8 shows the same at finite baryochemical potential, μB = 400 MeV . The theory curves shown correspond (1) the three-loop HTLpt results obtained from equation (5.132) shown as a solid black line and (2) the electric QCD (EQCD) effective field theory result truncated at order g5 [25]. In both cases, one-loop running of the strong coupling constant was used and we show bands that correspond to varying the gluon and quark renormalization scales by a factor of two around their central values, which were taken to be Λ 0g = 2πT and Λ q0 = 2π T 2 + μ2 /π 2 . This band reflects the residual renormalization scale dependence which would, in principle, be reduced if one goes to higher orders in the HTLpt framework. In both figure 5.7 and 5.8, data from the Wuppertal–Budapest collaboration is shown. From these figures, we see that HTLpt with the central scale choice does a good job at reproducing the temperature dependence of the pressure and trace-anomaly at both zero and finite baryochemical potential. We note that for the lattice data at finite μB a Taylor expansion around μB = 0 was used to obtain the results shown. To wrap up this discussion, we note that both resummed

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Relativistic Quantum Field Theory, Volume 3

Figure 5.9. Comparison of the Nf = 2 + 1, μB = 0 (left) and μB = 400 MeV (right) NNLO HTLpt speed of sound squared with lattice data. The μB = 0 lattice data is from [24] and the μB = 400MeV lattice data is from [26]. For the HTLpt results, a one-loop running coupling constant was used.

perturbative calculations give a quite good description of the trace anomaly, with the central-RG scale HTLpt result being marginally better at reproducing the pressure than the central-RG scale EQCD calculation. Another quantity that is phenomenologically interesting is the speed of sound. The speed of sound squared is defined as

cs2 =

∂P . ∂E

(5.135)

In figure 5.9, we show comparisons of the three-loop HTLpt result with lattice data from the Wuppertal–Budapest collaboration. As before the black line is the HTLpt result using the central value of the renormalization scales and the blue band shows the effect of varying these scales by a factor of two around these central values. The left panel shows the case of zero baryochemical potential and the right panel shows the case of finite baryochemical potential. In both cases, we see good agreement between theory and lattice simulation for T > 250 MeV. 5.5.7 Quark number susceptibilities Now we turn to the calculation of various susceptibilities. Using the full thermodynamic potential as a function of chemical potential(s) and temperature, we can compute the quark number susceptibilities. In general, one can introduce a separate chemical potential for each quark flavor, giving a Nf -dimensional vector μ ⃗ ≡ (μu , μd , … , μ Nf ). By taking derivatives of the pressure with respect to chemical potentials in this set, we obtain the quark number susceptibilities2

2 We have specified that the derivatives should be evaluated at μ⃗ = 0 . In general, one could define the susceptibilities at μ ⃗ = μ0⃗ .

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Relativistic Quantum Field Theory, Volume 3

χijk ⋯ (T ) ≡

∂ i +j +k+⋯ P(T , μ ⃗ ) ∂μui ∂μdj ∂μsk ⋯

.

(5.136)

μ⃗ = 0

In figure 5.10, we present a comparison of the three-loop HTLpt and EQCD predictions for the fourth-order up-quark susceptibility (left) and the fourth-order mixed ‘uudd’ susceptibility. In both cases, the perturbative calculations do a surprisingly good job at reproducing the lattice measurements taken from the Wuppertal–Budapest (WB) and Brookhaven–Bielefeld (BNLB) collaborations, with the HTLpt results appearing to be marginally better. Importantly we note that for the case of the fourth-order up-quark susceptibility the RG bands are quite small, meaning that HTLpt and EQCD are making quite constrained predictions. The highly constrained HTLpt predictions, in particular, seem to describe the lattice measurements down to temperatures on the order of 250 MeV. This is surprising since at these temperatures the strong coupling constant gs ∼ 2. 5.5.8 Baryon number susceptibilities Next, we consider the baryon number susceptibilities. The n th -order baryon number susceptibility is defined as

χBn (T ) ≡

∂ nP ∂μBn

.

(5.137)

μB=0

For a three-flavor system consisting of (u, d , s ), the baryon number susceptibilities can be related to the quark number susceptibilities [30]

χ2B =

1 ⎡ uu dd ss ud ds us ⎤ ⎢⎣χ2 + χ2 + χ2 + 2χ2 + 2χ2 + 2χ2 ⎥⎦ , 9

(5.138)

and

Figure 5.10. Left: the fourth-order diagonal light QNS. Right: the fourth-order off-diagonal light QNS. In both panels, we compare with lattice data from the Wuppertal–Budapest (WB) [27, 28] and BNLB collaborations [29]. Figure from [25].

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Figure 5.11. (Left) The second-order light quark (and baryon) number susceptibilities. (Right) The fourthorder baryon number susceptibility. In both panels, we compare with lattice data from the Wuppertal– Budapest (WB) [27, 28] and BNLB collaborations [29]. Figure from [25].

χ4B =

1 ⎡ uuuu + χ4dddd + χ4ssss + 4χ4uuud + 4χ4uuus + 4χ4dddu + 4χ4ddds + 4χ4sssu ⎣χ 81 4 (5.139) sssd uudd ddss uuss uuds ddus ssud ⎤ + 4χ4 + 6χ4 + 6χ4 + 6χ4 + 12χ4 + 12χ4 + 12χ4 ⎦ .

If one treats all quarks as having the same chemical potential μu = μd = μs = 1 μ = 3 μB, equations (5.138) and (5.139) reduce to χ2B = χ2uu and χ4B = χ4uuuu . This allows us to straightforwardly compute the baryon number susceptibility by computing derivatives of (5.130) with respect to μ. We present the HTLpt and EQCD results for the scaled second-order baryon number susceptibility in the left panel of figure 5.11 and the fourth-order baryon number susceptibility in the right panel of figure 5.11. Once again at high temperatures (T ≳ 250MeV ), we see excellent agreement between resummed perturbation theory and the lattice measurements from both the WB and BNLB collaborations. The results for the quark number and baryon number susceptibilities strongly suggest that the quark sector of the QGP, in particular, can be understood in terms of resummed perturbative approach, which has an underlying quasiparticle picture at its core.

Appendix A. Useful integrals Here, I collect some useful integrals and sum-integrals. For details of all integrals and sum-integrals necessary for the three-loop HTLpt calculation, I refer the reader to the appendices of [8, 10, 13, 14]. Gaussian integrals The most basic one-dimensional integral is ∞

⎛ ⎞1/2

∫−∞ dx e−ax +bx = e b /4a⎝ πa ⎠ 2

2

5-36

⎜

⎟

.

(5.140)

Relativistic Quantum Field Theory, Volume 3

Sum-integrals 2

∫∑p log P 2 = − π45 T 4 + O(ε), ∫∑p P12

∫

⎤ ⎛ ⎛ μ ⎞2ε 1 ⎡ ζ′( −1) ⎞ ⎟ ⎢1 + ⎜2 + 2 = T 2⎜ ⎟ε + O(ε 2 )⎥ , ⎝ 4πT ⎠ 12 ⎣ ζ( −1) ⎠ ⎝ ⎦

∑p

⎤ 1 1 ⎛⎜ μ ⎞⎟2ε ⎡ 1 = + 2γ + O ( ε ) ⎥ , ⎢ 2 2 2⎝ ⎦ (P ) (4π ) 4πT ⎠ ⎣ ε

∫∑{p} P12

=−

T2 + O(ε ). 24

(5.141)

(5.142)

(5.143)

(5.144)

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] [25]

Braaten E and Pisarski R D 1992 Phys. Rev. D 45 R1827 Taylor J C and Wong S M H 1990 Nucl. Phys. B 346 115–28 Mrowczynski S, Schenke B and Strickland M 2017 Phys. Rep. 682 1–97 Silin V P 1960 Sov. Phys. JETP 11 1136–40 [Zh. Eksp. Teor. Fiz. 38, 1577 (1960)] Weldon H A 1982 Phys. Rev. D 26 2789 Blaizot J P and Ollitrault J Y 1993 Phys. Rev. D 48 1390–408 Andersen J O, Leganger L E, Strickland M and Su N 2011 Phys. Lett. B 696 468–72 Andersen J O, Leganger L E, Strickland M and Su N 2011 JHEP 08 053 Haque N, Andersen J O, Mustafa M G, Strickland M and Su N 2014 Phys. Rev. D 89 061701 Haque N, Bandyopadhyay A, Andersen J O, Mustafa M G, Strickland M and Su N 2014 JHEP 05 027 Andersen J O, Braaten E and Strickland M 1999 Phys. Rev. Lett. 83 2139–42 Andersen J O, Braaten E and Strickland M 2000 Phys. Rev. D 61 074016 Andersen J O, Braaten E, Petitgirard E and Strickland M 2002 Phys. Rev. D 66 085016 Andersen J O, Petitgirard E and Strickland M 2004 Phys. Rev. D 70 045001 Haque N, Mustafa M G and Strickland M 2013 Phys. Rev. D 87 105007 Mogliacci S, Andersen J O, Strickland M, Su N and Vuorinen A 2013 JHEP 12 055 Andersen J and Strickland M 2001 Phys. Rev. D 64 105012 Andersen J O, Strickland M and Su N 2010 JHEP 08 113 Andersen J O, Strickland M and Su N 2009 Phys. Rev. D 80 085015 Vuorinen A 2003 Phys. Rev. D 67 074032 Vuorinen A 2003 Phys. Rev. D 68 054017 Braaten E and Nieto A 1995 Phys. Rev. D 51 6990–7006 Braaten E and Nieto A 1996 Phys. Rev. D 53 3421–37 Borsanyi S, Endrodi G, Fodor Z, Jakovac A, Katz S D, Krieg S, Ratti C and Szabo K K 2010 JHEP 11 077 Ghiglieri J, Kurkela A, Strickland M and Vuorinen A 2019 in press

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[26] Borsanyi S, Endrodi G, Fodor Z, Katz S D, Krieg S, Ratti C and Szabo K K 2012 JHEP 08 053 [27] Borsanyi S 2013 Nucl. Phys. A904-A905 270c–7c [28] Borsanyi S, Fodor Z, Katz S D, Krieg S, Ratti C and Szabo K K 2013 Phys. Rev. Lett. 111 062005 [29] Bellwied R, Borsanyi S, Fodor Z, Katz S D, Pasztor A, Ratti C and Szabo K K 2015 Phys. Rev. D 92 114505 [30] Petreczky P 2012 J. Phys. G39 093002

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