Finite-Temperature Quantum Electrodynamics: General Theory and Bloch-Nordsieck Estimates of Fermion Damping in a Hot Medium by Y

Finite-Temperature Quantum Electrodynamics: General Theory and Bloch-Nordsieck Estimates of Fermion Damping in a Hot Medium

by Yeuan-Ming Sheu B. Sc., National Taiwan University, 1992 M. Sc., National Taiwan University, 1994 Sc. M., Brown University, 1996

Submitted in partial fulﬁllment of the requirements for the Degree of Doctor of Philosophy in the Department of Physics at Brown University

Providence, Rhode Island May 2008 °c Copyright 2008 by Yeuan-Ming Sheu This dissertation by Yeuan-Ming Sheu is accepted in its present form by the Department of Physics as satisfying the dissertation requirement for the degree of Doctor of Philosophy.

Date Herbert M. Fried, Director

Date Antal Jevicki, Advisor

Recommended to the Graduate Council

Date Gerald Guralnik, Reader Brown University, Department of Physics

Date Chung-I Tan, Reader Brown University, Department of Physics

Approved by the Graduate Council

Date Sheila Bonde Dean of the Graduate School

iii Vita

Yeuan-Ming Sheu was born in a remote mountain village in Tunglo, Miaoli county, Taiwan, Re- public of China, on March 10, 1970. He is the ﬁrst child of an army-oﬃcer-turned-civil-servant father and a fulltime mom. He grew up in the country with his two brothers and two sisters, and was adored (perhaps spoiled a little bit) by his grandparents until leaving home for Hsin-Chu Senior High.

In his high school years, he competed in annual national scientiﬁc exhibitions in both physics and biology, and earned a recommendation to the science task camp for collage admissions. He got admitted to the Department of Physics at the National Taiwan University with a full scholarship from the Education Ministry. Right after his freshman year, he worked on rebuilding instruments and projects in the semiconductor physics lab, and soon fell in love with Physics.

After getting his Bachelor of Science in June, 1992, he continued his graduate study and received his Master of Science in June 1994 under the guidance of Prof. Yuan-Huei Chang to study the impurity properties of semiconductor quantum wells under high magnetic ﬁelds. Mr. Sheu attended Brown University with a fellowship in September 1994, and continued to study condensed matter physics. After a few unproductive years, he took a leave of absence and joined Advanced Power Technologies, Inc. (later merged into BAE Systems) in Washington, DC, as a research physicist in the summer of 2001. After getting his company’s support, he resumed his graduate study under Prof. Antal Jevicki and Prof. Herbert M. Fried, and has been working on problems in Quantum Field Theory since Spring 2003.

Over the years, he has published a number of articles in peer-reviewed journals, and has applied for several patents on inventions of semiconductor and optical devices.

When not pondering the mysteries of nature, he enjoys spending time with his lovely wife, boating, and day dreaming.

iv Acknowledgements

In the long journey of my graduate study, there were ups and downs; it has become an enjoyable experience the past few years. Besides my desk and blackboards, the research leading to this thesis was carried out on airplane tray tables, hotel desks, the Washington, DC metro, and on breakfast tables in C¨oted’Azur. An undertaking such as this could not have been possible without the assistance of countless people.

I would ﬁrst like to thank the faculty in the department of physics at both Brown University and National Taiwan University who have guided me to the wonderful world of Physics, in particular, thanks to Prof. Herbert M. Fried and Prof. Antal Jevicki for their guidance, inspiration, and friendship. Though he is in his 70s, Prof. Fried still works hard to conduct research with his notebooks, blackboards and napkins in various parts of the world. My lively discussion with Prof. Fried inspired me to do Physics more intuitively, and not just via formulation. I would also like to thank several individuals here at Brown and around the world, especially, Dr. Thierry Grandou, at Institut Non-Lin´eaire de Nice of CNRS. Without them, I would not be able to complete this work.

In addition to the support from BAE Systems, colleagues at Advanced Technologies deserve my special thanks, especially Mr. Oved Zucker, Dr. Ramy Shanny, Dr. Michael Grove, Dr. Robert D’Amico, and the many others who have encouraged me to resume my graduate study.

I would also like to thank my parents for unconditional support of my academic pursuits, and my brothers and sisters who have taken care for my aging parents while I am on the opposite side of the globe. Furthermore, I wish to thank my wife, Yu-Jie, who has accompanied me through those tough years with all her love. Finally, I would like to dedicate this thesis to my grandparents and my father who have watched over me in heaven.

v Contents

List of Tables x

List of Figures xi

1 Introduction 1 1.1 Overview ...... 1 1.2 Prior Attempts ...... 2 1.3 Current Work ...... 3 1.4 Thesis Organization ...... 4

2 Basics 5 2.1 The Functional Method in Quantum Field Theory ...... 5 2.1.1 The Functional approach in quantum ﬁeld theory ...... 5 2.1.2 The Fermion Green’s Function and Closed-Loop Functional ...... 7 2.2 Finite-Temperature Quantum Field Theory ...... 7 2.2.1 Statistical Thermodynamics ...... 7 2.2.2 The Functional Approach to Finite-Temperature Field Theory ...... 9 2.2.3 Finite-Temperature Propagators ...... 12 2.2.4 Imaginary-Time (Matsubara) Formalism ...... 13 2.2.5 Real-Time Formalism ...... 15 2.3 Finite-Temperature Green’s Functions ...... 18 2.3.1 QED Finite-Temperature Generating Functional ...... 18 2.3.2 Fully-dressed Finite-Temperature Green’s function ...... 18 2.3.3 Linkage Operator ...... 19 2.3.4 Coupled Thermal Fermion Green’s Function ...... 20 2.3.5 Closed-Fermion-Loop Functional and Thermal Normalization Constant . . . 23 2.4 Proper-Time Representations of Schwinger and Fradkin ...... 23 2.4.1 Schwinger’s Proper-Time Representation ...... 24 2.4.2 Fradkin’s Representation ...... 25 2.4.3 Coupled Green’s Functions in Mixed Space Representation ...... 27

vi 2.5 Mixed Representation of Propagators ...... 28 2.5.1 Free Finite-Temperature Fermion Propagators ...... 28 2.5.2 Free Finite-Temperature Boson Propagators ...... 30 2.5.3 Free Finite-Temperature Photon/Gauge Field Propagators ...... 31 2.5.4 Interpretation of Thermal Parts of Propagators ...... 32 2.6 Relationship Between Formalisms ...... 33 2.7 Damping Rate ...... 34 2.8 Hot Thermal Loops and the Resummation Program ...... 36

3 Finite-Temperature Propagator in a Hot QED Medium 38 3.1 Overview ...... 38 3.2 Dressed Finite-Temperature Fermion Propagator ...... 38 3.3 New Variant of Fradkin Representation ...... 41 3.3.1 Thermal Green’s Functions in a Mixed Representation ...... 41 3.3.2 Free-Field Limit ...... 46 3.3.3 Bloch-Nordsieck Approximation ...... 48 3.4 Dressed Propagator in Mixed Formalisms ...... 49 3.4.1 Approximation for Closed-Fermion-Loop Functional ...... 49 3.4.2 Dressed Finite-Temperature Fermion Propagator ...... 51 3.4.3 Linkage Operations ...... 52 3.4.4 Dropping Spin-related Contributions ...... 53 3.5 Quenched Dressed Finite-Temperature Fermion Propagator ...... 54 3.5.1 Quenched Dressed Finite-Temperature Fermion Propagator ...... 54 3.5.2 Linkage with Real-time Photon Propagators ...... 55 3.5.3 Thermal-Photon assisted Damping ...... 56 3.5.4 Bremsstrahlung Processes as a Damping Mechanism ...... 61 3.5.5 Damping Eﬀects under Quenched Approximation ...... 66 3.6 Non-Quenched Full Finite-Temperature Propagator ...... 71 3.6.1 Thermal Closed-Fermion-Loop and the Photon Polarization Tensor ...... 71 3.6.2 Pair-Productions as a Damping Mechanism ...... 73

4 Discussion and Perspectives 81 4.1 Model Approximation and Damping Mechanisms ...... 81 4.2 Damping Eﬀects ...... 83 4.3 Comparison to Perturbative Theory ...... 85 4.4 Longitudinal and Transverse Disturbance in the Medium ...... 87 4.5 Mass Shift ...... 91 4.6 Impact of Gauge ...... 96 4.7 Non-Zero Chemical Potential ...... 97 4.8 Hot Thermal Loop Approximation Revisited ...... 99

vii 4.9 Possible Extension to QCD ...... 101

5 Conclusions 104

A Units and Metric 107 A.1 Natural Units ...... 107 A.2 Metric ...... 107 A.3 Gordon decomposition ...... 109

B Matsubara Summation 110 B.1 Standard Contour Integral ...... 110 B.2 Saclay Method ...... 111 B.3 Mixed Representation of Finite-Temperature Propagator ...... 111

C Gauge 114 C.1 Gauge Conditions ...... 114 C.2 Photon Propagator and Gauge Parameter ...... 115 C.3 Current Conservation and Gauge Conditions in QED ...... 116 C.4 Gauge Structure of Green’s Function(al) ...... 117 C.5 Gauge Structure of Closed-Fermion-Loop Functional ...... 119

D Reviews of Functional Methods 120 D.1 Functional Diﬀerentiation ...... 120 D.2 Functional Integration ...... 121 D.3 Functional Diﬀerentiation vs Functional Integral ...... 122 D.4 Two Useful Relations ...... 122 D.5 Functional Form of Unity ...... 123 D.6 Linkage Operation ...... 123

E Misc Relations 125 E.1 Useful Relations in Fradkin’s Representation ...... 125 E.2 Representations of Delta- and Heaviside Step- Function ...... 125 E.3 Representation of h Function ...... 126 E.4 Operator Relations ...... 127 E.5 Legendre Function of the Second Kind ...... 127 E.6 Abel’s Trick ...... 128 E.7 Bogoliubov Transformation ...... 128

F Calculations in Full Imaginary-Time Formalism 130

G Calculations in Feynman Gauge 133

viii Bibliography 134

? Parts of this thesis are expected to be published in the May 2008 issue of Phys. Rev. D with H. M. Fried at Brown University and T. Grandou at Institut Non-Lin´eairede Nice Sophia-Antipolis, UMR-CNRS 6618, and can also be found in arXiv e-print server at arXiv:0804.1591v1 [hep-th].

ix List of Tables

A.1 Dimension of physical quantity in natural units. The conversion factor is to be used from energy units to conventional units. (Note1: The conventional electric charge is in Heaviside-Lorentz unit.) ...... 108

x List of Figures

2.1 An example of complex-time path (β > σ > 0)...... 10 2.2 Complex-time contour in ITF...... 14 2.3 Complex-time contour in RTF (β > σ > 0)...... 15

4.1 One-loop representation of Thermal-Photon-Induced Bremsstrahlung through a ther-

mal photon (γth) exchange with the medium (HB)...... 82

4.2 One-loop representation of Ordinary Bremsstrahlung through a virtual photon (γv) exchange with the medium (HB)...... 82

4.3 Two-loop representation of Pair Production from a virtual photon (γv) exchange with the medium (HB)...... 82

xi Chapter 1

Introduction

1.1 Overview

It is of great importance to understand properties of nuclear collision at ultra-relativistic energy. When two heavy nuclei at ultra-relativistic speed collide head-on, it creates a high temperature (and high density) plasma of quarks and gluons. These phenomena are the object of study in the projects of the Relativistic Heavy Ion Collider (RHIC) and the Large Hardron Collider (LHC) [1]. The RHIC and LHC experiments are expected to produce high energy quark-qluon plasmas after two heavy ions collide at ultra-relativistic speeds. While a heavy quark (or fermion) is moving through a plasma, it will interact with particles inside the plasma and the scattering will, in turn, disturb the plasma. The interaction and disturbance of a hot plasma can be probed through the photon emission especially the transverse radiation [2, 3, 4]. Results from RHIC and LHC will provide good tests on various proposed theories and models for the understanding of the nuclear and particle physics. The behavior of a charge particle entering into a high-temperature plasma can be simply described as an ultra-relativistic particle, e.g., electron or quark, incident upon a hot medium, which consists of thermalized electrons, positrons, and photons in Quantum Electrodynamics (QED), or quarks, anti- quarks, and gluons in Quantum Chromodynamics (QCD). A natural question is how the energy and momentum of a quark (or fermion) will change during the scattering, and how the plasma responds to the disturbance induced by the incident quark (fermion). Na¨ıvely, the incident particle will exchange energy and momentum with particles inside the plasma. Depending on the initial energy and the strength of interaction, the incident particle may lose energy and eventually become a part of the medium. Instead of large numbers of individual particles, the plasma can be treated as an ensemble of particles with a thermal distribution, i.e., a heat bath or a hot medium. Therefore, the incident particle interacts with a hot plasma as a high-temperature medium instead of individual relativistic particles, which define the finite temperature field theory. There have been several attempts to

1 2 formulate such energy depletion through ﬁnite temperature perturbation theory [5, 6, 7, 8]. But it has been shown that the ﬁnite temperature theory is intrinsically non-perturbative [9]. In this thesis, a non-perturbative and more physically intuitive method will be presented for the process of energy depletion of the incident fermion and the response of the thermal medium.

1.2 Prior Attempts

Works on the finite-temperature field theory dated back from the era of Matsubara [10] and Schwinger [11]. The relativistic finite-temperature theory was subsequently given by Dolan and Jackiw [12], Weinberg [13], Bernard [14], and others. There are two-type of approaches; the imaginary-time formalism are introduced by Matsubara [10], Kirzhnits and Linde [15], Dolan and Jackiw [12], and Weinberg [13]. The Real-time formalism was started from the time-path formalism by Schwinger [11] and Keldysh [16] on non-equilibrium quantum statistics, and further developed by Umezawa, Matsumoto and Tachiki [17], Ojima [18], Niemi and Semenoff [19, 19], and others. Subsequent applications to finite temperature Quantum Electrodynamics (QED) and Quantum Chromodynamics (QCD) has been done, most notably, by Weldon [20, 21], Cox et al. [22], Donoghue et al. [23], and several others, in terms of thermal average in Born approximations. There are several attempts to estimate the damping rate of an incident particle in a hot plasma [5, 24, 25, 26, 27, 28, 29, 8, 30, 30, 31]. Prior works employed perturbation theory and associated the imaginary part of the pole of the fermion propagator as the lifetime, and the damping rates were estimated by calculating the imaginary part of the fermion self-energy. In additional to the usual ultra-violet (UV) divergence at zero temperature theory, the infrared (IR) divergence also appears in the na¨ıve perturbative method at finite temperature. In the finite-temperature theory, the factor of Bose-Einstein distribution function leads to more severe IR divergence. The IR divergence originates from exchanges of soft photons, and occurs in every order of perturbation; therefore, the problem is inherently non-perturbative [9]. To account for contributions from the thermal fluctuations of the same orders as corresponding tree diagrams, Hot Thermal Loop (HTL) approximations were developed [32, 33, 29]; and Braaten and Pisarski [34, 35], Frenkel and Taylor [36] further developed the Resummation Program (RP) to resolve issues of the gauge dependence for the damping rate. While the RP of HTL has leads to some progress in finite-temperature, the effective HTL approximations only alleviates the severity of IR singularity, but does not completely eliminate the difficulties; instead, the effective theory replaced quadratic-type IR divergence with logarithmic-type [5, 37, 38, 39, 40, 41] at finite temperature. In addition, other problems with HTL and RP also appear in the estimate of the damping rate of a fast moving particle and the production of soft real photons [42, 43] which originated from the lack of static screening of transverse gauge modes and from appearance of collinear singularities when external particles are on-shell or massless [28]. Several attempts to resolve the IR problem of the fermion damping rate failed due to unknown analytic structrue of the full fermion propagator [26, 43]. After the recognition of major contributions from the small momenta, the Bloch-Nordsieck (BN) approximation was employed to cancel the IR 3 divergences first by Weldon [6], and subsequently by Takashiba [44], and by Blaizot and Iancu [30, 45, 46]. The damping rates estimated in the framework of BN approximations and effective HTL propagators appeared to be finite. However, these applications of the BN approximation with the constant momenta or the on-shell momenta, however, are inconsistently used to estimate the lifetime of particles. Therefore, one needs to develop a non-perturbative, IR-divergence free method to approach the problem. In Ref. [47], the first functional approach was employed on a toy model of scalar fields. The current work extends that approach to QED, and points out a possible extension to QCD.

1.3 Current Work

The damping of a fast moving fermion entering into a hot plasma is estimated in terms of the fully-dressed, finite-temperature propagator; this is performed in a functional approach with a new variant of Fradkin presentations. Rather than the conventional momentum space expansion of the proper self-energy part of the inverse fermion propagator, the calculations are carried out in the Matsubara/Martin-Schwinger Imaginary-Time formalism with appropriate modifications. In the current functional approach, various aspects of mass and wave-function renormalization from the specific effects of the medium on the particle can be conveniently separated and discarded. As the energy-momentum of the incident particle is much larger than the temperature scale of the medium, a modified Bloch-Nordsieck approximation is introduced and maintained rigorously in the manner which is consistent to the case when the particle’s momentum is decreasing as it proceeds into the medium. Without requiring the particle to remain continuously on its mass shell, the exchange of virtual and real photons with particles in the medium is viewed as an effective mechanism for the loss of energy and momentum for the incident particle. Three mechanisms of energy depletion are identified and estimated: thermal-photon-assisted Bremsstrahlung, ordinary Bremsstrahlung and pair production. An explicit expression for the time-dependence of the thermalization process is given in terms of a damping exponential operator operating on a non-interacting propagator with respect to the energy of the incident particle. Rather than an exponential decay with an extraneous logarithmical factor appeared in the perturbative approach, the damping of the incident particle is £ ¤ 2 of Gaussian, as exp −Γz0 with a simple function Γ of the soft-momentum cutoff. In contrast to the Hot Thermal Loop approximations, thermalization of the fermion-anti-fermion pairs is not used, because such a description is irrelevant at the instant of pair production. Pre- vious HTL description led unrealistically to the introduction and necessary removal of spurious IR divergence, which the present treatment completely avoids. The result of thermal-photon assisted Bremsstrahlung is of similar order of g2T 2 to that of resummation of Hot Thermal Loops, which prompts the possibility that the two approaches might be equivalent if the later is treated properly in a non-perturbative way. However, the HTL approximation and the associated resummation programs failed to account for the mechanism of pair production. 4

In addition to the damping of the incident particle, the finite-temperature propagator also shows the possibility of short-term growth in the probability factors necessary for a longitudinal and transverse fireball. Furthermore, the probability of building up and shrinking down of such fireball probability can be extracted from the dressed, finite-temperature propagator. Various aspects of the finite-temperature theory, for example, the gauge-invariance of the damping effect and the effective, thermally-induced mass-shift due to the exchange of photon with the medium are also discussed.

1.4 Thesis Organization

In Chapter 2, basic formulations of functional method in Quantum field theory will be first introduced at zero temperature, following the functional linkage approach introduced by Fried with Schwinger/Fradkin representations. Subsequently, the functional method will be extended to the finite-temperature field theory with a new interpretation. Two commonly used formalisms in the finite-temperature theory will also be reviewed along with the correspondent functional method and comparable high-temperature approximations. Full QED theory at finite temperature will be introduced in Chapter 3 along with the detail Bloch-Nordsieck estimates of the fully-dressed fermion propagator with a new variant of Fradkin representation, which leads to three decay mechanisms of an ultra-relativistic moving particle interacting with a hot plasma. Discussions of various aspects of fermion damping in the context of finite temperature QED, comparisons to prior works and a possible extension to QCD are presented in Chapter 4, and a brief summary is given in Chapter 5. Chapter 2

Basics

This chapter provides some background leading to the subject of the thesis. First, quantum field theory is briefly presented with emphasis on functional methods with Schwinger/Fradkin representations. Statistical thermodynamics and finite-temperature field theory will then be discussed along with commonly-used formalisms and a brief review of prior attempts by others. In functional approach to quantum field theory, Fradkin’s representations of Green’s function and closed-fermion- loop functional are popularly employed [48, 49, 50]; then, new variants will also be presented in a form suitable to eikonal models for both conventional and finite-temperature field theory. For consistency of text, the notation will closely follow that of Fried in Refs. [48, 49, 50] using the

Minkowski metric convention and in terms of natural units with c, ~ and kB set to 1.

2.1 The Functional Method in Quantum Field Theory

2.1.1 The Functional approach in quantum ﬁeld theory

The functional approach in quantum ﬁeld theory pioneered by Schwinger and subsequently developed by others provides a better overall view of Physics [48, 49]. In contrast to perturbative approach, it also provides a better treatment for problems which are non-perturbative in nature. The subject of study in this thesis mainly focuses on ﬁnite temperature phenomena in QED and a possible extension to QCD. For convenience, the presentation of the functional formulation will be based on QED. The full QED Lagrangian density can be expressed in the form of

LQED = LDirac + Lphoton + Lint (2.1) 1 = −ψ¯(m + γ · ∂)ψ − F2 + ig ψ¯ γ · A ψ, 4 which is Abelian and invariant under U(1) gauge transformation, and Fµν = ∂µAν − ∂ν Aµ is the ﬁeld strength. To facilitate the functional approach following Schwinger, sources like η(x) andη ¯(y) of spinorial Grassmann variables and jµ(z) of bosonic c-number 4-vector are incorporated into the

5 6

Lagrangian density, L → L + j · A +η ¯ · ψ + ψ¯ · η. (2.2)

Through Schwinger’s Action Principle, the generating functional is given by µ ½ Z ¾¶ £ ¤ ¯ Zc{j, η,¯ η} = h0| exp i j · A +η ¯ · ψ + ψ · η |0i, (2.3) + where (··· )+ denotes time-ordering. The n-point Green’s function can be derived by functional diﬀerentiation with respect to these sources, which are to be set to zero afterwards, e.g.,

(n) G (x1, . . . , xn) = ih0| (A(x1) ··· A(xn)) |0i (2.4) µ ¶ µ + ¶ ¯ ¯ 1 δ 1 δ ¯ = i ··· · Zc{j, η, η¯}¯ . i δj(x1) i δj(xn) η=¯η=j=0 Following either Schwinger’s or Symanzik’s construction, the solution of the QED generating functional is given by ½ Z µ ¶ ¾ ½ Z Z ¾ δ δ δ i hSi Z {j, η,¯ η} = exp ig γ · · exp j · D · j + i η¯ · S · η , (2.5) c δη δj δη¯ 2 c c where Sc and Dc are the free, causal fermion and photon propagators, respectively. The generating functional can be further manipulated with functional diﬀerentiations with the help of the reciprocity relation, as in Fried [48] (cf. Eq. (D.21)):

R R Lc[A] i j·D ·j D(c) i η¯·G [A]·η e Z {j, η,¯ η} = e 2 c · e A · e c , (2.6) c hSi where Z c Aµ(x) = dy Dµν (x − y) · jν (y), (2.7) and the normalization constant hSi is deﬁned by ¯ (c) ¯ DA Lc[A]¯ hSi ≡ h0|S|0i = e · e ¯ , (2.8) A→0 and where the argument of the linkage operator is Z Z (c) i δ µν δ DA = − dx dy · Dc (x − y) · . (2.9) 2 δAµ(x) δAν (y)

The gauge-ﬁeld coupled Green’s functional Gc[A] is

−1 Gc[A] = Sc [1 − igγ · ASc] , (2.10) and the closed-fermion-loop functional Lc[A] is

£ −1 ¤ Lc[A] = Tr ln [1 − ig(γ · A)Sc] = −Tr ln Sc · Gc . (2.11)

Both the subscript ’c’ and superscript ’(c)’ will be used interchangeably throughout this thesis to indicate the causal version of functionals or operators in the zero-temperature theory, in order to 7 distinguish their thermal counterparts with ’th’ or ’(th). Any n-point Green’s function can then be derived by functional differentiation of the generating functional with respect to associated sources, and setting sources to zero afterwards. For example, the fully-dressed, causal propagator of a fermion can be defined as µ ¶ µ ¶ ¯ ¡ ¢ ¯ 0 ¯ 1 δ 1 δ ¯ Sc = ih ψψ +i = i − · hSi Zc{j, η, η¯}¯ . (2.12) i δη¯ i δη η=¯η=j=0 With the aid of Eq. (2.6), and setting all sources to zero, the dressed fermion propagator becomes ¯ · L [A] ¸ (c) e c ¯ 0 DA ¯ Sc = e · Gc[A] ¯ . (2.13) hSi A→0 In an alternative method, the propagator is represented by a functional integral with c-number functions; however, one then needs to worry about specific normalization constants.

2.1.2 The Fermion Green’s Function and Closed-Loop Functional

At zero temperature, the field-coupled Green’s function of a fermion under the influence of gauge fields A(x) satisfies the inhomogeneous differential equation:

(4) (m + γµ[∂µ − igAµ]) Gc(x, y|A) = δ (x − y). (2.14)

Its solution, the causal fermion Green’s function, in formal notation is

−1 −1 Gc[A] = [m + iγ · (∂ − igA)] = (m − iγ · Π) , (2.15) where the Π-operator represents

Πµ = i (∂µ − igAµ) . (2.16) The closed-fermion-loop functional in Eq. (2.11) can be expressed in terms of an integral representation of the logarithm function over the coupling constant g as Z g n o Z g 0 0 −1 0 0 Lc[A] = −i dg Tr (γ · A)Sc [1 − ig (γ · A)Sc] = −i dg Tr {(γ · A)Gc[g A]}, (2.17) 0 0 where the deﬁnition of Gc[A] in Eq. (2.10) is used.

2.2 Finite-Temperature Quantum Field Theory

2.2.1 Statistical Thermodynamics

In statistical thermodynamics, the entropy of a system described by a grand canonical ensemble is

S = −Tr (ˆρG lnρ ˆG), (2.18) where the trace sums over all physical states in the ensemble. The grand canonical ensemble can also be described by the grand partition density operators, " # X −1 ˆ ˆ ρˆG = ZG exp −βH + βµANA , (2.19) A 8 and the grand partition function is given by ( " #) X ZG[β, µ] = Tr exp −βHˆ + βµANÂ . (2.20) A The ensemble average of a physical quantity in a grand canonical ensemble is then defined as ³ ´ hOîG = Tr ρˆGOˆ , (2.21) where angled brackets with a subscript ’G’ stand for the grand canonical ensemble average. The grand canonical ensemble is free to have any number of particles, and particles can have any energy. However, the ensemble is still subjected to constraints of fixed average total particle num- P ˆ ˆ −1 ber A hNAiG and average total energy hHiG, with the inverse temperature β = (kBT ) and

αA = βµA (or the chemical potential µA) as Lagrange multipliers for maximizing the entropy. The formulation of statistical thermodynamics above is non-covariant; a covariant form is needed to extend to relativistic conditions [51, 52, 20]. A thermal distribution of an equilibrium system is deﬁned in the rest frame of the heat bath, which is the intrinsic preferred frame; in turn, the velocity of the heat bath becomes the preferred vector in any other frame. In addition to the temperature

T and chemical potential µA, the four-velocity uµ of the system with u · u = −1 can be used to re-deﬁne these variables in a covariant form as

βµ = βuµ, (2.22) where the metric gµν = diag(−1, +1, +1, +1) is implied. The newly deﬁned 4-vector inverse- temperature βµ is a time-like Lorentz four-vector, and chemical potentials αA’s are Lorentz scalars. 2 In the rest frame of the thermal bath, uµ = (1,~0) and βµ = (β,~0). Instead of q , a particle with 2 2 2 momentum qµ can be characterized by two Lorentz invariants, ω = −u · q and κ with q = κ − ω .

For example, a function of βωq in an integral over ~q can be converted to that of −β · q = −βu · q in an integral over four-vector q, i.e., Z Z d3~q d4q f(βω ) = 2πδ(q2 + m2) θ(q ) 2ω f(−β · q) (2.23) (2π)3 q (2π)4 0 q by inserting Z Z dq dq 0 δ(q − ω ) = 0 2ω δ(q2 − ω2), (2.24) 2π 0 q 2π q 0 q 2 2 2 and a step function θ(q0) to ensure inclusion of only the positive root of ~q +m = q0 [22]. Therefore, the introduction of the Lorentz four-vector inverse-temperature permits a covariant form of thermo- A dynamics. Similarly, the energy-momentum tensor Tµν , entropy ﬂux sµ, and conserved current Jµ of type-A charge particles become

Tµν = ρ uµuν + P (gµν + uµuν ), (2.25)

sµ = suµ, (2.26) A Jµ = nAuµ, (2.27) 9

where ρ, P , s and nA denote the Lorentz-invariant energy density, pressure, entropy density, and number density of type-A particles, respectively [51, 20]. Even though a thermodynamic system has a preferred frame, the inclusion of the four-velocity of the system in the deﬁnition enables a covariant thermodynamic formulation. To simplify the notation, subsequent calculations will be carried out in the rest frame of the medium, i.e., uµ = (1,~0) and βµ = (β,~0).

2.2.2 The Functional Approach to Finite-Temperature Field Theory

The functional methods in the ﬁnite-temperature theory used here are based on the seminal paper of Martin and Schwinger [53], and its modern form can be found in Refs. [48] and [49]. The grand partition function of interest can be rewritten as h i X −β(H−ˆ µNˆ ) −β(H−ˆ µNˆ ) ZG[β, µ] = Tr e = hnA, z0|e |nA, z0i, (2.28) {A} where Hˆ and Nˆ denote the Heisenberg Hamiltionian and number operator, respectively, and the trace (or summation) is over all physical states containing nA particles at time z0 in the system of interest. If we let the inverse-temperature β analytically continue to iτ, the system can be thought to evolve under the eﬀective Hamiltonian H¯ = Hˆ − µNˆ with the probability amplitude

−βH¯ −iτH¯ hA, t2|e |B, t1i → hA, t2|e |B, t1i = hA, t2 + τ|B, t1i. (2.29)

The ’analytically continued’ grand partition function ZG[iτ, µ] is then given by X X −iτH¯ ZG[iτ, µ] = hnA, z0|e |nA, z0i = hnA, z0 + τ|nA, z0i. (2.30) {A} {A}

The form of the grand partition function is similar to the generating function of the zero-temperature P ﬁeld theory, except that the ’time variable’ is now a complex number. Let µ = {A} µAnA be the chemical potential for fermions, and the system can then be described by a Finite Temperature QED Lagrangian density with source terms,

¯ ¯ L¯ = LDirac + Lphoton + Lint + µψψ + j · A +η ¯ · ψ + ψ · η. (2.31)

Following the method of zero-temperature theory, one can deﬁne the Finite-Temperature generating functional as X Zth{j, η,¯ η} = hnA, z0 + τ|nA, z0i. (2.32) {A} In the limit of zero sources, this Finite-Temperature generating functional reduces to the grand partition function,

Zth{0, 0, 0} = ZG[iτ, µ]. (2.33) 10 Im

C1 Re

tf − iσ

C2 ti − iβ

Figure 2.1: An example of complex-time path (β > σ > 0).

Applying Schwinger’s Action Principle on the Finite-Temperature generating functional, one obtains 1 δ X Z {j, η,¯ η} = hn , z + τ|A(z)|n , z i, (2.34) i δj(z) th A 0 A 0 {A} 1 δ X Z {j, η,¯ η} = hn , z + τ|ψ(z)|n , z i, (2.35) i δη¯(z) th A 0 A 0 {A} 1 δ X − Z {j, η,¯ η} = hn , z + τ|ψ¯(z)|n , z i. (2.36) i δη(z) th A 0 A 0 {A}

With the understanding that the ’time variable’ z0 is extended to the complex value, this Finite- Temperature generating functional can now be constructed similarly to the zero-temperature theory as ( µ ½ Z Z ¾¶ ) £ ¤ −βH¯ 3 ¯ Zth{j, η,¯ η} = Tr e exp i dz0 d ~z j · A +η ¯ · ψ + ψ · η (2.37) C C+ µ ½ Z Z ¾¶ X £ ¤ 3 ¯ = hnA, z0 + τ| exp i dz0 d ~z j · A +η ¯ · ψ + ψ · η |nA, z0i, {A} C C+ where the ’time’-integral is along some time-path contour C, which starts from the initial point at z0 = ti and ends at z0 = ti + τ = ti − iβ in the complex z0-plane, e.g., Fig. (2.1). Here the conventional ’time-ordering’, (··· )+, is replaced by the ’contour-ordering’, (··· )C+, along some time 0 00 path from z0 to z0 +τ = z0 −iβ (cf. Ref. [19, 54, 55]). The requirement of −β ≤ Im(z0 −z0 ) ≤ 0 for 0 00 any two points, z0 and z0 , on the contour will ensure the existence and analyticity of thermal Green’s functions to all orders. The choice of contour is almost arbitrary except that the imaginary part of a contour should be decreasing monotonically or constant. Two most common choice of contours lead to the imaginary-time formalism (ITF) [10] and the real-time formalism (RTF) [11, 16, 19, 54]. For the uniqueness of solutions of relevant field equations, the fields (not operators) in the functional integral obey either the periodic or anti-periodic conditions depending on the field statistics, as [55, 56]

Aµ(z0) = Aµ(z0 + τ) = Aµ(z0 − iβ), (2.38) ¯ βµ ¯ βµ ¯ ψ(z0) = −e ψ(z0 + τ) = −e ψ(z0 − iβ), (2.39) βµ βµ ψ(z0) = −e ψ(z0 + τ) = −e ψ(z0 − iβ). (2.40) 11

(0) First, set the chemical potential to zero, µ = 0, and let Zth{j, η,¯ η}|g=0 = Zth {j, η,¯ η} as the interaction is turned off, i.e., g = 0. The generating functional can be derived through the Action Principle with aid of the equations of motion as ½ Z Z ½ ¾¾ 3 1 δ 1 δ 1 δ (0) Zth{j, η,¯ η} = exp i dz0 d ~z Lint , − , · Zth {j, η,¯ η}, (2.41) C i δj i δη i δη¯ where fields in the interacting Lagrangian Lint have been replaced by conjugated field operators similar to the zero-temperature theory, i.e., ½ ¾ 1 δ 1 δ 1 δ L {A, η,¯ η} → L , − , . (2.42) int int i δj i δη i δη¯

For non-zero chemical potential, a similar method can be applied as follows. The chemical potential related term µψψ¯ in Lagrangian contains a equal space-time field product. The associated functional differentiation operators anti-commute, but the field operators do not at equal space- ¯ time. Hence,³ the thermal´ ³ average´ of a field product ψ(z)ψ(z) at equal space-time cannot be näively 1 δ 1 δ replaced by − i δη i δη¯ . To avoid the ambiguity, observing that the equal space-time product ψ¯(z)ψ(z) can be split into a symmetric and an anti-symmetric part as 1 1 ψ¯(z)ψ(z) = {ψ¯(z), ψ(z)} + [ψ¯(z), ψ(z)]. (2.43) 2 2 With the help of the anti-commutation relation for field operators, the symmetric part is just an 1 ¯ 1 (3) ~ 1 ¯ infinite c-number 2 {ψ(z), ψ(z)} = 2 γ0δ (0), and can be identified as 2 h0|(ψ(x)ψ(y))+|0i or as −iSc(0). Here, Sc(0) is ¯ Sc(0) = ih0|(ψ(x)ψ(y))+|0i|x−y→0, (2.44) where |0i represents the zero-fermion states instead of a completely-filled Fermi sea [48]. Hence, the appropriate replacement of an equal-time fermion field product ψ¯(z)ψ(z) is µ ¶ µ ¶ 1 δ 1 δ ψ¯(z)ψ(z) → − − iS (0). (2.45) i δη i δη¯ c

For similar applications to bosons, the anti-symmetric part of an equal space-time field product is replaced by an infinite c-number given in terms of a commutator. After appropriate replacement of equal-time field products for the chemical potential related term, the Finite-Temperature generating functional becomes ½ Z Z µ ¶ µ ¶ ¾ 3 1 δ 1 δ Zth{j, η,¯ η} = exp iµ dz0 d ~z − + µτΩSc(0) · Zth,µ=0{j, η,¯ η}, (2.46) C i δη i δη¯ R R 3 where Ω = d ~z is the volume of the system, τ = C dz0 = −iβ is the complex ”time”, and Zth,µ=0{j, η,¯ η} is the Finite-Temperature generating functional of zero chemical potential. The phase factor exp [+µτΩSc(0)] will be canceled at a later stage [49]. For the convenience of calculation, all chemical-potential related terms will first be omitted from the Finite-Temperature generating functional, and then be inserted back afterwards [49]. Except for the complex time path and the extra chemical potential term, the formalism of the Finite Temperature theory is similar to that of 12 the zero-temperature theory. The free, non-interacting Finite-Temperature generating functional of zero chemical potential is given by

R R (0) µ=0 i i C η¯·Sth ·η 2 C j·Dth·j (0) Zth,µ=0[j, η,¯ η] = e · e ·Z [iτ, 0], (2.47)

(0) (0) where the constant Z [iτ, 0] = Zth,µ=0[0, 0, 0] is the normalization constant of the generating functional without the chemical potential, and is related to the partition function of the non-interacting system. Inserting Eq. (2.47) into (2.46), the Finite-Temperature generating functional becomes

R R (0) µ=0 i i C η¯·Sth·η +Tr ln [1−µSth ] 2 C j·Dth·j (0) Zth {j, η,¯ η} = e · e · e ·Z [iτ, 0], (2.48) where the free thermal propagator Sth with non-zero chemical potential is

h i−1 µ=0 µ=0 Sth = Sth 1 − µSth , (2.49)

µ=0 and satisﬁes the same diﬀerential equation as Sth , except that p0 is replaced by p0 + µ. The determinantal factor can be combined with Z(0)[iτ, 0] as n h io µ=0 (0) (0) exp +Tr ln 1 − µSth ·Z [iτ, 0] = exp [−µτΩSc(0)] ·ZG [iτ, µ], (2.50)

(0) (0) where Z [iτ, µ] = Zth [0, 0, 0] is related to the grand partition function of the non-interacting system with the chemical potential µ, and the ﬁrst phase factor on the right hand side will cancel the previously neglected phase factor exp [+µτΩSc(0)] in Eq. (2.46). Hence, the free Finite-Temperature generating functional becomes

R R (0) i (0) i C η¯·Sth·η 2 C j·Dth·j Zth [j, η,¯ η] = e · e ·ZG [iτ, µ], (2.51) where Sth is deﬁned in Eq.(2.49) with the chemical potential µ. The Finite-Temperature generating functional can then be derived by inserting the free generating functional of Eq. (2.51) into Eq. (2.41). Alternatively, the Finite-Temperature generating functional can be put into a functional integral form as Z · Z ¸ ¯ Zth{j, η,¯ η} = DA Dψ Dψ exp i L¯ . (2.52) C Perturbative approximations can be derived from the expansion of interacting parts of the La- grangian L¯. For applications in QED, the functional method with functional diﬀerentiation will be more convenient and intuitive, compared to the perturbative expansion with functional integral, and will be used in the subsequent calculations.

2.2.3 Finite-Temperature Propagators

While a causal n-point Green’s function in the zero-temperature theory is defined as the expectation value of a n-field operator product over vacuum states, its finite-temperature counterpart is taken 13 as the average over a (grand) canonical ensemble. For example, the Finite-Temperature fermion propagator is defined as h i −βH¯ ¯ Tr e (ψ(x)ψ(y))C+ 0 ¯ S (x − y) = ih(ψ(x)ψ(y))C+iβ = i £ ¤ , (2.53) th Tr e−βH¯ where the ’time-ordering’ of field operators within (··· )C+ is along the contour of the formalism employed. The cyclicity of the trace operator leads to the Kubo-Martin-Schwinger (KMS) condition, e.g., h i ¯ −1 −βH¯ ¯ hψ(x0)ψ(y0)iβ = ZG Tr e ψ(x0)ψ(y0) (2.54) h i −1 −βH¯ +βH¯ ¯ −βH¯ = ZG Tr e e ψ(y0) e ψ(x0) ¯ = hψ(y0 − iβ)ψ(x0)iβ, or 0 0 Sth(x0 − y0) = −Sth(x0 − y0 ± iβ). (2.55) The Finite-Temperature Green’s functions (or propagators) can be derived by functional differentiation of the Finite-Temperature generating functional with respect to conjugated sources. The process is similar to the zero-temperature theory, except the normalization factor changes to

ZG[iτ, µ] instead of hSi. Hence, the Finite-Temperature fermion propagator becomes µ ¶ µ ¶ ¯ ¯ 0 1 δ 1 δ ¯ 1 Sth(x − y) = i − · Zth{j, η, η¯}¯ · , (2.56) i δη¯(x) i δη(y) η=¯η=j=0 ZG[iτ, µ] where the extra minus sign is due to exchange of the fermion ﬁelds and functional diﬀerentiations with respect to Grassmannian variables, and ZG[iτ, µ] becomes the partition function ZG[β, µ] of the interacting system as iτ → β. The full Finite Temperature generating functional in QED with ¯ the interacting part Lint = igψγ · Aψ is R R R i2g − 1 δ γ· 1 δ 1 δ i (0) { C ( i δη )( i δj )( i δη¯ )} i C η¯·Sth·η 2 C j·Dth·j Zth{j, η,¯ η} = e · e · e ·ZG [iτ, µ]. (2.57)

Similar to methods of perturbative expansion with functional integral, the choice of the ’time- path’ contour will determine the formulation of Finite-Temperature propagators or Green’s functions [55, 56].

2.2.4 Imaginary-Time (Matsubara) Formalism

The initial point ti of a time-path contour is arbitrary, but must end at ti − iβ as shown in Fig.

(2.1). One can choose a contour parallel to or directly along the imaginary axis from z0 = 0 to z0 = −iβ = τ as in Fig. (2.2), which leads to the Imaginary-Time Formalism (ITF) [10]; The ’time’-variable is pure imaginary with a range of 0 and τ = −iβ, and any ’time’-integral over the time variable is limited in the same range, i.e., Z Z τ Z τ Z 3 dz → dz = dz0 d ~z. (2.58) C 0 0 14 Im

t i Re

CITF ti − iβ

Figure 2.2: Complex-time contour in ITF.

In momentum space, the ’energy’-component k0 is replaced by a discreet Matsubara frequency ωn [10, 55, 49, 56], which is 2nπ ω = , for bosons, (2.59) n τ or (2n + 1)π ω = , for fermions. (2.60) n τ

The integral over k0 is replaced by an inﬁnite sum over Matsubara frequencies. Most calculations eventually come down to an evaluation of Matsubara sums, but not all summations can easily be accomplished. For solvable cases, Matsubara sums can be performed in several methods [57, 58, 56], which are deferred to Appendix B. After Matsubara summation, the imaginary-time τ can then be analytically continued to −iβ. The imaginary-time form of the ﬁnite-temperature generating functional in QED, Eq. (2.57), becomes

R R R i2g τ − 1 δ γ· 1 δ 1 δ τ i τ (0) { 0 ( i δη )( i δj )( i δη¯ )} i 0 η¯·Sth·η 2 0 j·Dth·j Zth{j, η,¯ η} = e · e · e ·ZG [iτ, µ], (2.61) where the contour integrals are in short for

Z τ Z τ Z τ = dx dy . (2.62) 0 0 0 Then, the derivation of a ﬁnite-temperature quantity in the Imaginary-Time formalism follows that of zero-temperature, except that the ’time’-ordering in a Finite-Temperature fermion propagator is along the path from 0 to τ as  ihψ(x)ψ¯(y)i, if x > y 0 ¯ 0 0 S (x − y) = ih(ψ(x)ψ(y))+i = , (2.63) th  ¯ −ihψ(x)ψ(y)i, if y0 > x0 where 0 ≤ x0, y0 ≤ τ. In momentum space, a thermal n-point Green’s function is similar in form to its zero-temperature counterpart, except that the zero-component of any momentum, e.g., p0 of four-vector p, is replaced by Matsubara frequency ωn, i.e., p = (~p,ωn). For a system of non-interacting fermions, the ﬁnite-temperature generating functional is given by ½ Z τ Z τ ¾ (0) (0) Zth {η,¯ η} = ZG [iτ] · exp i dx dy η¯(x) · Sth(x − y) · η(y) . (2.64) 0 0 15

C ti 1 tf

C3 ti − iσ tf − iσ C2 C4 ti − iβ Figure 2.3: Complex-time contour in RTF (β > σ > 0).

Here Sth(x − y) is the free (non-interacting) ﬁnite-temperature fermion propagator, and its Fourier- transformed expression is given by 1 m − iγ · p S˜ (~p,ω ) = = = (m − iγ · p) · ∆˜ (F)(~p,ω ; m2), (2.65) th n m + iγ · p m2 + p2 th n

2 2 2 ˜ (F) 2 2 2 2 −1 where p = (~p,ωn), p = ~p − ωn and ∆th (~p,ωn; m ) = [m + ~p − ωn − i²] .

For certain problems of interest, one would like to have Green’s function with real energy p0, which can be analytically continued from Matsubara frequency ωn. However, the analytic continuation of discreet frequencies to arbitrary values is not unique in general, and causes some diﬃculties if not chosen properly [55, 56].

2.2.5 Real-Time Formalism

Many problems of interest including n-point Thermal Green’s functions are preferred to have real time arguments. When the ’time’-variable z0 is chosen to be complex with real time and imaginary temperature, it leads to the real-time formalism (RTF) [11, 16, 12, 19, 54, 55, 56]; The ’time-path’ contour starts from an initial point at arbitrary z0 = ti, and must end at z0 = ti −iβ. For analyticity of the thermal expectation value of a physical quantity, the imaginary part along the path must be monotonically decreasing or constant, and one part of the contour must run along the whole real axis. One of standard time-path contours in RTF is shown in Fig. (2.3) with 4 segments. The

first segment C1 of the contour C starts from the initial point at z0 = ti and follows the real axis to z0 = tf . The second segment C3 goes from z0 = tf to z0 = tf − iσ, with 0 ≤ σ ≤ β, along a vertical line. Then, the contour goes from z0 = tf − iσ to z0 = ti − iσ horizontally as C2, and finally follows a vertical line to z0 = ti − iβ as C4. The choice of σ is arbitrary and contours of different σ can be shown to be equivalent by redrawing with the periodic or anti-periodic boundary conditions of the fields. Therefore, no physical quantity will depend on σ [59, 60, 56]. Instead of conventional ’time-ordering’, a n-point Green’s function at finite temperature is defined with ’contour-ordering’ along a contour z0 = z0(ξ) ∈ C, which increases monotonically with the parameter ξ, as 0 ¯ Sth(x − y) = ih(ψ(x)ψ(y))C+iβ, (2.66) where (··· )C+ denotes the ’time-ordering’ is along a given contour C parameterized by ξ. The 16

counterparts of the θ- and δ-functions along the contour z0(ξ) ∈ C are then deﬁned as

0 0 θC(z0 − z0) = θ(ξ − ξ ), (2.67) 0 0 δC(z0 − z0) = δ(ξ − ξ ). (2.68)

For examples, two ﬁeld operators in the contour-ordering are given by

¯ ¯ ¯ (ψ(x)ψ(y))C+ = θC(x0 − y0)ψ(x)ψ(y) + θC(y0 − x0)ψ(y)ψ(x), (2.69) and the functional differentiation along the counter becomes δj(z) = δ (z − z0 ) δ(3)(~z − ~z0). (2.70) δj(z0) C 0 0 Then, Real-Time n-point Green’s functions can be derived by functional differentiations to the C ’contoured’ finite-temperature generating functional Zth{j, η, η¯} with respect to sources, and then setting all sources to zero afterwards. For example, the n-point photon Green’s function Gth(x−y) = ih(A(x1) ··· A(xn))Ci is given by µ ¶ µ ¶ ¯ ¯ (n) 1 δ 1 δ C ¯ 1 Gth = i ··· · Zth{j, η, η¯}¯ · , (2.71) i δj(x1) i δj(xn) η=¯η=j=0 ZG[β, µ] where C = C1 + C2 + C3 + C4 and Z · Z ¸ C ¯ ¯ Zth{j, η,¯ η} = DA Dψ Dψ exp i L . (2.72) C

The second horizontal path C2 from z0 = tf − iσ to z0 = ti − iσ with reverse ’time-ordering’ creates extra degrees of freedom or ’ghost’-fields. Hence, Real-Time Green’s functions are usually expressed in a matrix form. At finite temperature, the perturbative series are based on free Green’s functions derived from the non-interacting finite-temperature generating functional. Assume that the interaction is switched on and off adiabatically, then the initial time ti and final time tf on the real time axis are taken to −∞ and +∞, respectively. For the purpose of evaluating n-point Green’s functions, the non-interacting finite-temperature generating functional can be factorized as [19, 55, 56]

(0) C12 C34 Zth {j, η,¯ η} = N Zth {j, η,¯ η}· Zth {j, η,¯ η}, (2.73) where Cab = Ca ∪ Cb is an union of two contour segments, and N is a normalization constant. The factorization of the generating functional in the speciﬁc contour C may be a bit controversial for evaluation of the partition function [61, 59, 56], but can be justiﬁed by observing

lim Sth(x − y) = 0, (2.74) Re|x0−y0|→∞ or choosing alternative contours similar to Fig. (2.1) as in Ref. [59, 60, 62]. For any real-time Green’s function of interest with external (physical) lines, only real-time segments, C12 = C1 ∪C2, contribute, i.e., the functional diﬀerentiation with respect to sources with real-time arguments. Thus, the factor 17

C34 Zth in the generating functional from C34 is just a multiplicative constant like N for the purpose of evaluation of Green’s functions. For fermions, the eﬀective, non-interacting ﬁnite-temperature generating functional becomes ½ Z Z ¾ (0) (0) 4 4 Zth {j, η,¯ η} = ZG [iτ] · exp i d x d y η¯(x) · Sth(x − y) · η(y) . (2.75) C12 C12

In addition to the physical field (type-1) in the forward time-path C1 along the real axis, the reverse time-path C2 leads to an extra degree of freedom, the ghost field (type-2). Hence, a free fermion propagator is given in a matrix form with both types of fermion fields as [19, 63, 55, 64] Ã ! S˜11(p) S˜12(p) S˜th(p) = (2.76) S˜21(p) S˜22(p) Ã ! ∆˜ (p) − 2πin˜(p)δ(m2 + p2) +²(p )ñ(|p |)eσp0 = (m − iγ · p) F 0 0 , (β−σ)p0 ˜ ∗ 2 2 −²(p0)ñ(|p0|)e ∆F (p) + 2πin˜(p0)δ(m + p )

2 2 −1 where the Feynman propagator ∆˜ F (p) = [m + p − i²] . If σ = β/2 is chosen, the fermion propagator matrix is anti-symmetric, and can be diagonalized by a special unitary Bogoliubov transformation (cf. Appendix E.7) as Ã ! (m − iγ · p) ∆˜ (p) 0 S˜ (p) = U† (p) F U (p), (2.77) th β ˜ ∗ β 0 (m − iγ · p) ∆F (p) where the unitary transformation matrix Ãp p ! 1 − n˜(p0) − n˜(p0) Uβ(p) = p p (2.78) n˜(p0) 1 − n˜(p0) contains all thermal information, andn ˜(p0) is the Fermi-Dirac distribution function: 1 n˜(p0) = . (2.79) eβ|p0| + 1 In the zero-temperature limit, β → ∞, the Bogoliubov transformation matrix becomes an identity matrix and the two types of fields decouple. The generating functional of type-1 fields in such limit leads to the conventional zero-temperature theory. It will be shown in the later sections that a free, non-interacting propagator can be separated into two parts with distinct physical origins. However, double degrees of freedom with extra ghost fields make calculations of interacting systems more complicated and tedious. 18

2.3 Finite-Temperature Green’s Functions

2.3.1 QED Finite-Temperature Generating Functional ³ ´ δ δ In Eq. (2.57), ﬁrst apply the functional operations δη¯ ··· δη in Lint on the free (non-interacting) (0) Finite-Temperature generating functional Zth ,

Zth{j, η,¯ η} (2.80) ( Z · µ ¶ ¸ · µ ¶ ¸) δ −1 δ = exp i η¯ · Sth 1 − g γ · Sth · η + Tr ln 1 − g γ · Sth C δj δj ½ Z ¾ i (0) · exp j · Dth · j ·ZG [iτ, µ]. 2 C Then continue with the δ/δj operation with the aid of the reciprocity relation, Eq. (D.21), as in Eq. (2.6), ½ Z ¾ ½ Z ¾ i i δ δ Z {j, η,¯ η} = exp j · D · j · exp − · D · (2.81) th 2 th 2 δA th δA ½ CZ ¾C (0) exp i η¯ · Gth[A] · η + Lth[A] ·ZG [iτ, µ], C where the functional operations have been changed to gauge ﬁelds A(x) given by Z µν Aµ(x) = dy Dth (x − y) · jν (y). (2.82) C

Both the ﬁeld-coupled, thermal Green’s function Gth[A] and the thermal closed-fermion-loop functional are functionals of gauge ﬁelds, A(x), and are given by

−1 Gth[A] = Sth [1 − g (γ · A) Sth] , (2.83) and

Lth[A] = Tr ln [1 − g (γ · A) Sth], (2.84) respectively. Their forms are similar to those in the zero-temperature theory.

2.3.2 Fully-dressed Finite-Temperature Green’s function

Similar to the zero-temperature theory, the fully-dressed, Finite-Temperature fermion propagator, Eq. (2.56), can be derived with the help of Eq. (2.81) as ¯ · L [A] ¸ (th) e th ¯ 0 DA ¯ Sth = e Gth[A] ¯ , (2.85) Z[iτ] A→0 where Z (th) i δ µν δ DA = − · Dth · , (2.86) 2 C δA δA 19 and Z[iτ], with or without a chemical potential µ, is the thermal normalization constant and is given by ¯ h (th) i¯ DA Lth[A] ¯ (0) Z[iτ] = e · e ¯ ·Z [iτ]. (2.87) A→0 Here Z(0)[iτ] is the normalization constant for the free Finite-Temperature generating functional (0) Zth [j, η,¯ η], and the grand canonical ensemble is implicitly assumed and the subscript G is dropped for convenience. Such form for a fully-dressed Finite-Temperature propagator is generic and can be applied to any formalism of interest.

2.3.3 Linkage Operator

In the Matsubara formalism (ITF), the photon linkage operator in the configuration representation is given by Z τ Z τ (th) i δ δ DA = − dx dy · Dth(x − y) · , (2.88) 2 0 0 δA(x) δA(y) where both time-integrations are limited in the range of (0, τ). In contrast, the linkage operator in RTF is Z Z (th) i δ µν δ DA = − dx dy · Dth (x − y) · , (2.89) 2 C C δAµ(x) δAν (y) where both time-integrations are along a given time-path contour C. A gauge field is local in configuration space, i.e., A|xi = A(x)|xi, but non-local in momentum ~ ~ ~ ~ space, i.e., h~p,n|A|k, li = Añ−l(~p − k) in ITF or h~p,p0|A|k, k0i = A˜(~p − k, p0 − k0) in RTF. The form of a linkage operator in momentum space is quite different. The gauge field in ITF is given by Z 1 X 1 X d3~q A (x) = Aµ(~x) · e−iωlx0 = A˜µ(~q) · ei(~q·~x−ωlx0), (2.90) µ τ l τ (2π)3 l l l and its functional differentiation is Z δ 1 X d3~q δ i(~q·~x−ωlx0) = 3 e · µ . (2.91) δAµ(x) τ (2π) ˜ l δAl (~q) For convenience of notation, the sum-integral can be expressed as Z Z 1 X d3~q dq ≡ , (2.92) τ (2π)3 l and the Matsubara sum can be converted to a counter integral as shown in Appendix B.1. Hence, the field strength Fµν is given by Z 1 X d3~q h i F (x) = ∂ A (x) − ∂ A (x) = q A˜ (q) − q A˜ (q) · eiq·x, (2.93) µν µ ν ν µ τ (2π)3 µ ν ν µ l or in momentum space

F˜ µν (q) = qµA˜ν (q) − qν A˜µ(q). (2.94) 20

In ITF, the linkage operator in the momentum representation becomes Z Z i 1 X d3~k 1 X d3~k0 D(th) = − (2.95) A 2 τ (2π)3 τ (2π)3 l l0 h i δ ~ 3 (3) ~ ~ 0 δ · · D˜ th(k, ωl) · (2π) τδ (k + k )δl,−l0 · ~ ~ 0 δA˜l(k) δA˜l0 (k ) or Z i 1 X d3~k δ δ D(th) = − · D˜ (~k, ω ) · (2.96) A 2 τ (2π)3 ˜ ~ th l ˜ ~ l δAl(k) δA−l(−k) Similarly, the linkage operator in RTF is

Z 4 (th) i d k δ ~ δ D = − · D˜ th(k, k0) · , (2.97) A 2 (2π)4 δA˜(k) δA˜(−k)

~ Here D˜ th(k, k0) is in a matrix form of type-1 and type-2 field, and the gauge field and its functional differential are given by Z d4q A (x) = A˜ (~q, q ) · ei(~q·~x−q0x0), (2.98) µ (2π)4 µ 0 and Z δ d4q δ i(~q·~x−q0x0) = 4 e · , (2.99) δAµ(x) (2π) δA˜µ(q) respectively, and where the matrix indices of type-1 and type-2 are suppressed.

2.3.4 Coupled Thermal Fermion Green’s Function

At zero temperature (T = 0), the Green’s function of a fermion under gauge fields Aµ(x) satisfies the basic differential equation,

(4) (m + γµ[∂µ − igAµ]) Gc(x, y|A) = δ (x − y), (2.100) and its solution, the coupled fermion Green’s function, in operator form is

−1 Gc[A] = (m − iγ · Π) , (2.101) where the Π-operator is

Πµ = i (∂µ − igAµ) . (2.102)

At finite temperature, the details of thermal Green’s functions vary in different formalisms of interest. The formulation of the finite temperature field theory in the imaginary-time (Matsubara) formalism is parallel to that of the T = 0 vacuum theory. Calculations in this thesis will base on the imaginary-time formalism with appropriate modification, along with some comments on the real-time formalism. 21

Matsubara Imaginary-Time Formalism

For thermal Green’s functions, the governing diﬀerential equation in ITF is given by

(4) (m + γµ[∂µ − igAµ]) Gth(x, y|A) = δth (x − y), (2.103) where time coordinates, x0, y0, are pure imaginary, but are treated like real variables and 0 ≤ x0, y0 ≤ τ with τ = −iβ after the Matsubara summation. The δth-function is deﬁned by X (4) 1 δ (x − y) = δ(3)(~x − ~y) · δ (x − y ) = δ(3)(~x − ~y) · e−iωn(x0−y0), (2.104) th th 0 0 τ n and Matsubara frequencies for fermions with n = 0, ±1, ±2,... are

(2n + 1)π ω = . (2.105) n τ

There is no periodicity constraint for a gauge ﬁeld Aµ(z), if it is used to facilitate the linkage.

However, if it represents a thermalized gauge ﬁeld, Aµ(z) is to be periodic in its ’time’ coordinate, i.e.,

Aµ(z0, ~z) = Aµ(z0 + τ, ~z). (2.106)

Thus, the thermal Green’s function would have a similar form of its causal counterpart as

−1 Gth[A] = (m − iγ · Π) . (2.107)

In Matsubara representation, the energy component p0 is replaced by its corresponding Matsubara frequency ωn as

In addition, the thermal Green’s function, Gth(x, y|A), is anti-periodic (for fermions) in its ’time’ coordinate, which leads to the Kubo-Martin-Schwinger condition as

Gth(x0, y0|A) = −Gth(x0 − iβ, y0|A) = −Gth(x0 + τ, y0|A), (2.111) and

Gth(x0, y0|A) = −Gth(x0, y0 − iβ|A) = −Gth(x0, y0 + τ|A). (2.112)

Similarly, the same form of the closed-fermion-loop functional in Eq.(2.17) can be applied for the thermal closed-fermion-loop functional as

Z g 0 0 Lth[A] = −i dg Tr {(γ · A)Gth[g A]}. (2.113) 0 22

Real-Time Formalism

In Real-Time Formalism (RTF), the basic diﬀerential equation is similar to that of ITF,

¡ C ¢ (4) m + γµ[∂µ − igAµ] Gth(x, y|A) = δC (x − y), (2.114) where x0 and y0 are ’complex-time’ variables on a given time-path contour C from ti to ti − iβ, C (4) and the differential operator ∂µ and δ-function δC (x − y) are defined along the contour C [19, 54]. However, its solution, the thermal Green’s function in RTF, is much complicated compared to its Matsubara counterpart. In general, the form of a thermal Green’s functions depends on the choice of the contour, and ITF can be seen as a special case by putting the contour along the imaginary axis from 0 to −iβ. For a free fermion, the finite-temperature propagator is much simpler and can be separated into two parts as [12]

Sth(x, y) = Sc(x, y) + δSth(x, y), (2.115) where the ﬁrst and second term represent contributions from the vacuum and the thermal bath, respectively. Similarly, the coupled thermal Green’s function in RTF may be separated in two parts; one from T = 0 and the other from the thermal contribution, as

Gth(x, y|A) = Gc(x, y|A) + δGth(x, y|A). (2.116)

Due to the involvement of gauge ﬁelds, the relationship between the imaginary-time Green’s function,

G˜ th(~p,ωn|A), and its real-time counterpart G˜ th(~p,p0|A) is not readily transparent. The relationship can be illustrated more clearly through the Fourier (integral) transform as

Z τ Z +iωnz0 3 −i~p·~z G˜ th(~p,ωn|A) = dz0 e d ~ze · Gth(~z, z0|A) (2.117) 0 Z τ Z ∞ dp0 = dz e+iωnz0 e−ip0z0 · G˜ (~p,p |A) 0 2π th 0 Z0 Z −∞ ∞ dp τ 0 −i(p0−ωn)z0 = dz0 e · G˜ th(~p,p0|A). −∞ 2π 0

In general, the integral over z0 is non-trivial with gauge fields Aµ(z). If the gauge field is either constant or periodic in its time argument z0, the integral over z0 can be worked out as Z ∞ dp 1 ³ ´ 0 −i(p0−ωn)τ G˜ th(~p,ωn|A) = e − 1 · G˜ th(~p,p0|A), (2.118) −∞ 2π −i(p0 − ωn) which leads to Z ∞ βp0 dp0 e + 1 G˜ th(~p,ωn|A) = i · G˜ th(~p,p0|A), (2.119) −∞ 2π ωn − p0 as ωnτ = (2n + 1)π and β = iτ. Alternatively, the real-time finite-temperature Green’s function might be derived through a modified Abel-Plana formula from its counterpart in ITF, Z 1 X d3~p G (x, y|A) = G˜ (~p,ω |A) · ei~p·(~x−~y)−iωn(x0−y0), (2.120) th τ (2π)3 th n n 23 where the Matsubara sum may be related to the Abel-Plana formula, and may be separated into two parts; the first part, T = 0, is represented by the causal propagator, Gc(x, y|A), with integration over p0 instead of Matsubara sum. The remaining part, δGth(x, y|A), is temperature-related. However, the detail proof is non-trivial and will be worked out separately; meanwhile, a similar link has been attempted by Fried in Ref. [49].

2.3.5 Closed-Fermion-Loop Functional and Thermal Normalization Con- stant

The thermal closed-fermion-loop functional is given by

Z g 0 0 Lth[A] = −i dg Tr {(γ · A)Gth[g A]} (2.121) 0 Z g 0 0 0 = −i dg Tr {(γ · A)[Gc[g A] + ∆Gth[g A]]} 0 Z g Z g 0 0 0 0 = −i dg Tr {(γ · A) Gc[g A]} − i dg Tr {(γ · A) ∆Gth[g A]} 0 0 ≡ Lc[A] + ∆Lth[A].

The normalization constant Z[iτ] = Zth{0, 0, 0} of the generating functional represents the partition function of the thermal bath, and can be re-written as ¯ (th) ¯ DA Lth[A]¯ (0) Z[iτ] = e · e ¯ ·Z [iτ] (2.122) A→0 ¯ (c) (th) ¯ DA +∆DA Lc[A]+∆Lth[A]¯ (0) → e · e ¯ · Z [β], A→0 in terms of linkage operators and closed-fermion-loop functionals, and where Z(0)[β = iτ] is the partition function of the non-interacting system. Hence, the ratio of normalization constants becomes ¯ (c) (th) ¯ Z[iτ] Z[β] D +∆D L [A]+∆L [A] → = e A A · e c th ¯ (2.123) Z(0)[iτ] Z(0)[β] ¯ A→0 ¯ (th) n³ (c) ´ (c) ³ (c) ´o¯ ∆DA DA Lc[A] D12 DA ∆Lth[A] ¯ = e · e · e · e · e · e ¯ . A→0 The thermal (normalization) constant entangles both causal and thermal parts, which are diﬃcult to be separated. Further approximation will be needed to obtain the Physics of interest.

2.4 Proper-Time Representations of Schwinger and Fradkin

While the causal Green’s function and its thermal counterpart in the imaginary-time formalism (ITF) have similar forms; subsequent calculations will be expressed in the causal case with special notation for the thermal case. The proper time representation was first introduced by Schwinger [65] for the Green’s function and its close-fermion-loop functional in a constant electromagnetic field or plane wave of a single frequency. Fradkin generalized the functional representation of Green’s 24 functions to general external fields [66]. The proper-time representations enable exact representations of Green’s functions for non-perturbative calculations which have been discussed extensively, with modern improvements, by Fried in Refs. [49] and [50]. This brief introduction here will closely follow Fried’s treatment.

2.4.1 Schwinger’s Proper-Time Representation

First rationalize the causal Green’s function in Eq.(2.101) as [49]

h i−1 2 2 Gc[A] = (m + iγ · Π) · m + (γ · Π) , (2.124) and the denominator will becomes

(γ · Π)2 = Π2 + igσ · F (2.125) with the anti-symmetric tensor σµν 1 σ = [γ , γ ] = −σ . (2.126) µν 4 µ ν νµ For the causal Green’s function, the mass m2 → m2 − i² is implicitly associated with an inﬁnites- imal positive parameter ². The ’proper-time’ representation introduced by Schwinger [65] is to parameterize the denominator in an exponential form as Z ∞ n h io 2 2 Gc[A] = (m + iγ · Π) · i ds exp −is m + (γ · Π) . (2.127) 0 With the help of Eq.(2.125), one can further expand (γ · Π)2, Z ∞ −ism2 −is(Π2+igσ·F) Gc[A] = (m + iγ · Π) · i ds e · e , (2.128) 0 where the parameter s is called the ’proper time’, but does not necessarily carry the same dimension of ’time’. Similarly, the closed-fermion-loop functional can be given by Z g ½ Z ∞ ¾ 0 −ism2 −is(γ·Π)2 Lc[A] = −i dg Tr (γ · A)(m + iγ · Π) · i ds e e . (2.129) 0 0 n o 2 Since Tr (γ · A) e−is(γ·Π) = 0, the term proportional to m vanishes and

Z ∞ Z g n o −ism2 0 −is(γ·Π)2 Lc[A] = i ds e dg Tr (γ · A)(γ · Π)e , (2.130) 0 0 or Z ∞ Z g n o 1 −1 −ism2 0 ∂ −is(γ·Π)2 Lc[A] = − ds s e dg 0 Tr e (2.131) 2 0 0 ∂g Z ∞ n o 1 ds 2 2 = − e−ism Tr e−is(γ·Π) − {g = 0} . 2 0 s

The form of Green’s function Gc[A] and closed-fermion-loop functional Lc[A] in Schwinger’s proper-time representation is general and has been applied to problems with constant ﬁelds [48, 49]. The more convenient form of Fradkin representations is also exact and further enables various eikonal approximations. 25

2.4.2 Fradkin’s Representation

In QED, both the causal Green’s function Gc[A] and the closed-fermion-loop functional Lc[A] contain the same factor of n o © ¡ ¢ª U(s) ≡ exp −is (γ · Π)2 = exp −is Π2 + igσ · F , (2.132) that is, Z ∞ −ism2 Gc[A] = (m + iγ · Π) · i ds e U(s), (2.133) 0 and Z ∞ 1 ds −ism2 Lc[A] = − e Tr {U(s)} − {g = 0} . (2.134) 2 0 s Following Fradkin’s approach [66, 49], one can replace U(s) with an ordered exponential U(s, v) of µ ½ Z ¾¶ s £ ¤ U(s, v) = exp −i ds0 Π2 + igσ · F + v(s0) · Π , (2.135) 0 +

0 0 where the ’proper-time’ ordering (··· )+ is with respect to s , and vµ(s ) is an arbitrary vector 0 0 function of s . U(s) is recovered by setting vµ(s ) to zero, i.e., U(s, v)|v→0 = U(s). Further, it is possible to deﬁne U(s, v) in terms of Gaussian-type quadratic functional translation as

½ Z s−² 2 ¾ 0 δ U(s, v) = exp i ds 2 0 · W(s, v)|²→0, (2.136) 0 δvµ(s ) where µ ½ Z s ¾¶ 0 0 W(s, v) = exp −i ds [vµ(s ) · Πµ + igσ · F] . (2.137) 0 +

In the process of introducing Gaussian translation, the non-commutative Πµ-function is replaced by the functional diﬀerentiation with respect to vµ as [67] δ Πµ → i . (2.138) δvµ

In a nutshell, the replacement of U(s) with U(s, v) enables expressing both Gc[A] and Lc[A] in terms of averaging Gaussian ﬂuctuations over v(s0). Again, one can introduce F(s, v) such that

½Z s ¾ 0 0 W(s, v) = exp ds vµ(s ) · ∂µ ·F(s, v) (2.139) 0 with R ³ R ´ − s ds0 v (s0)·∂ −i s ds0 v (s0)·Π +igσ·F F(s, v) = e 0 µ µ · e 0 [ µ µ ] . (2.140) + Then, the Green’s function becomes ¯ Z ∞ ½ Z s 2 ¾ 2 δ ¯ G [A] = (m + iγ · Π) · i ds e−ism exp i ds0 · W(s, v)¯ (2.141) c δv2 (s0) ¯ 0 0 µ vµ→0 Z R ¯ ∞ s 0 δ2 R ¯ 2 i 0 ds 2 0 s 0 0 −ism δvµ(s ) ds vµ(s )·∂µ ¯ = (m + iγ · Π) · i ds e · e · e 0 ·F(s, v)¯ , 0 vµ→0 26 and the leading factor, m + iγ · Π = m + iγ · i(∂ − igA), can be expanded as Z R ¯ ∞ s 0 δ2 R ¯ 2 i 0 ds 2 0 s 0 0 −ism δvµ(s ) ds vµ(s )·∂µ ¯ Gc[A] = i ds e · e · [m − γ · (∂ − igA)] · e 0 ·F(s, v)¯ , 0 vµ→0 (2.142) or in terms of functional diﬀerentiation over v, R · ¸ ¯ Z ∞ s 0 δ2 2 i ds R s 0 0 ¯ −ism 0 δv2 (s0) δ ds v (s )·∂ G [A] = i ds e · e µ · m − γ · · e 0 µ µ ·F(s, v)¯ . (2.143) c δv(s) ¯ 0 vµ→0 To solve F(s, v), diﬀerentiate F(s, v) with respect to s as R R ∂F(s, v) − s ds0 v (s0)·∂ + s ds0 v (s0)·∂ = −ig e 0 µ µ ·{v · A + iσ · F}· e 0 µ µ ·F(s, v). (2.144) ∂s µ µ

Apply h~x,x0| on the left-hand side, and the solution is

h~x,x |F(s, v) (2.145) Ã 0 ( " #)! Z s Z s0 Z s0 = exp −ig ds0 v(s0) · A(x − ds00 v(s00)) + iσ · F(x − ds00 v(s00)) 0 0 0 + ·h~x,x0|

Thus, Z R s 0 0 ds vµ(s )·∂µ h~x,x0|W(s, v)|~y, y0i = dz h~x,x0|e 0 |~z, z0i · h~z, z0|F(s, v)|~y, y0i (2.146)

Z Z s 0 0 = dz hx + ds vµ(s )|zi · hz|F(s, v)|yi 0 R R s 0 0 s 0 0 With hx + 0 ds vµ(s )|zi = δ(x + 0 ds vµ(s ) − z) and hz|yi = δ(z − y), one arrives at Ã ( " #)! Z s Z s0 Z s0 hx|W(s, v)|yi = exp −ig ds0 v(s0) · A(y − v) + iσ · F(y − v) (2.147) 0 0 0 + Z s 0 0 ×δ(x − y + ds vµ(s )), 0 and ½ Z s 2 ¾ 0 δ hx|U(s, v)|yi = exp i ds 2 0 (2.148) 0 δv (s ) Ã ( µ " #)! Z s Z s0 Z s0 × exp −ig ds0 v(s0) · A(y − v) + iσ · F(y − v) 0 0 0 + Z s 0 0 ×δ(x − y + ds vµ(s )). 0 Thus, the causal Green’s function becomes R Z ∞ s 0 δ2 2 i ds −ism 0 δv2 (s0) x hx|Gc[A]|yi = i ds e · e µ · [m − γ · (∂ − igA(x))] (2.149) Ã0 ( " #)! Z s Z s0 Z s0 × exp −ig ds0 v(s0) · A(y − v) + iσ · F(y − v) 0 0 0 Z ¯ + s ¯ 0 0 ¯ ×δ(x − y + ds vµ(s ))¯ . 0 vµ→0 27

x Further replacing Πµ(x) = i[∂µ − igAµ(x)] in the numerator factor with help of Eq.(2.138),

R · ¸ Z ∞ s 0 δ2 2 i ds −ism 0 δv2 (s0) δ hx|G [A]|yi = i ds e · e µ · m − γ · (2.150) c δv(s) Ã0 ( " #)! Z s Z s0 Z s0 × exp −ig ds0 v(s0) · A(y − v) + iσ · F(y − v) 0 0 0 Z ¯ + s ¯ 0 0 ¯ ×δ(x − y + ds vµ(s ))¯ . 0 vµ→0 Similarly, the closed-fermion-loop functional is given by

R Z ∞ Z Z s 0 δ2 2 i ds 1 ds −ism 4 4 0 δv2 (s0) Lc[A] = − e d p d x · e µ (2.151) 2 0 s ( µ h i¶ )¯ R R R 0 R 0 s 0 0 −ig s ds0 v(s0)·A(x− s v)+iσ·F(x− s v) ¯ ds v(s )·p 0 0 0 ¯ ×tr e 0 e − {g = 0} ¯ . + vµ→0

Certainly, other variants of Fradkin’s representations can also be derived from these forms to ﬁt applications of interest. In Chapter 3, a new variant will be introduced and applied to thermal Green’s functions.

2.4.3 Coupled Green’s Functions in Mixed Space Representation

At T = 0, one can apply h~p,p0| ≡ hp| to the causal Green’s function,

R · ¸ Z ∞ s 0 δ2 2 i ds −ism 0 δv2 (s0) δ h~p,p |G [A]|~y, y i = i ds e · e µ · m − γ · (2.152) 0 c 0 δv(s) 0 ¯ R s 0 0 ¯ i ds vµ(s )·pµ ¯ ·e 0 · h~p,p0|F(s, v)|~y, y0i¯ , vµ→0 where £ ¤−1/2 4 −i(~p·~y−p0y0) −2 −ip·y h~p,p0|~y, y0i = (2π) · e = (2π) · e (2.153) is the normalization factor. Hence,

h~p,p0|Gc[A]|~y, y0i (2.154) Z ∞ £ ¤−1/2 2 = (2π)4 · e−i(~p·~y−p0y0) · i ds e−ism 0 R · ¸ s 0 δ2 i ds R s 0 0 0 δv2 (s0) δ i ds v (s )·p ×e µ · m − γ · · e 0 µ µ δv(s) Ã ( " #)! Z Z 0 Z 0 ¯ s s s ¯ 0 0 ¯ × exp −ig ds v(s ) · A(y − v) + iσ · F(y − v) ¯ . 0 0 0 + vµ→0

For the ﬁnite temperature theory, the thermal counterpart in ITF can be derived by replacing p0 with Matsubara frequency ωn. The coupled thermal Green’s function Gth[A] bears the same form of Gc[A]. 28

2.5 Mixed Representation of Propagators

2.5.1 Free Finite-Temperature Fermion Propagators

At T = 0, the free causal fermion propagator Sc is given by

−1 Sc = (m + γ · ∂) (2.155) in the operator form, or

˜ −1 ˜ F Sc(p) = (m + iγ · p) = (m − iγ · p)∆c (p) (2.156) in the momentum representation, where

˜ F 2 ˜ F 1 1 ∆c (~p,p0; m ) = ∆c (ω, p0) = 2 2 2 = 2 2 (2.157) m + ~p − p0 − i² ω − p0 − i² with ω2 = m2 + ~p 2. Similarly, the non-interacting ﬁnite-temperature fermion propagator is given by

−1 F 2 Sth(x − y) = hx| (m − γ · ∂) |yi = (m − γ · ∂x) · ∆th(x − y; m ) (2.158) in the conﬁguration space representation, or £ ¤ ˜ 2 2 −1 ˜ F 2 Sth(~p,ωn) = (m − iγ · p) · m + p = (m − iγ · p) · ∆th(~p,ωn; m ) (2.159) in momentum-Matsubara representation, where Matsubara frequencies for fermions are

(2n + 1)π ω = , (2.160) n τ

F 2 and the function ∆th(x − y; m ) is deﬁned as

F 2 £ 2 2¤−1 ∆th(x − y; m ) = hx| m + (γ · i∂) |yi, (2.161) and its Fourier-transformed expression is

˜ F 2 ˜ F 1 ∆th(~p,ωn; m ) = ∆th(ω, ωn) = 2 2 . (2.162) ω − ωn In the mixed (non-covariant) representation, the free ﬁnite-temperature fermion propagator is given by 1 X S˜ (~p,z ) = e−iωnz0 S˜ (~p,ω ) (2.163) th 0 τ th n n 1 X = e−iωnz0 (m − iγ · p) ∆˜ F (ω, ω ) τ th n n 1 X = e−iωnz0 (m − i~γ · ~p + iγ ω ) ∆˜ F (ω, ω ) τ 0 n th n · n µ ¶¸ 1 ∂ 1 X = m − i~γ · ~p + iγ · − e−iωnz0 ∆˜ F (ω, ω ). 0 i ∂z τ th n 0 n 29

˜ F The summation over n in the last line is the Matsubara-type Fourier transform of ∆th(ω, z0), and can be evaluated as in Appendix B.3 as 1 X ∆˜ F (ω, z ) = e−iωnz0 ∆˜ F (ω, ω ) (2.164) th 0 τ th n n i © ª = [1 − n˜(ω)] e−iωz0 − n˜(ω) e+iωz0 , 2ω wheren ˜(ω) is the Fermi-Dirac distribution function as 1 n˜(ω) = . (2.165) eβω + 1 Thus, · µ ¶¸ 1 ∂ i © ª −iωz0 +iωz0 S˜th(~p,z0) = m − i~γ · ~p + iγ0 · − [1 − n˜(ω)] e − n˜(ω) e . (2.166) i ∂z0 2ω

When working out the diﬀerentiation over z0, i © ª S˜ (~p,z ) = [ m − i~γ · ~p ] [1 − n˜(ω)] e−iωz0 − n˜(ω) e+iωz0 (2.167) th 0 2ω γ0 © ª − [1 − n˜(ω)] e−iωz0 +n ˜(ω) e+iωz0 , 2 where the signs between the two propagating factors are diﬀerent.

For non-zero chemical potential, one can replace the energy-component ωn by ωn +µ, Eq. (2.162) and (2.164) become

˜ F 2 ˜ F 1 ∆th(~p,ωn; m , µ) = ∆th(ω, ωn; µ) = 2 2 (2.168) ω − (ωn + µ) and 1 X ∆˜ F (ω, z ; µ) = e−iωnz0 ∆˜ F (ω, ω ; µ) (2.169) th 0 τ th n n i © ª = [1 − n˜(ω + µ)] e−iωz0 − n˜(ω − µ) e+iωz0 , 2ω respectively. Then, the ﬁnite-temperature fermion propagator becomes

˜ 1 m − i~γ · ~p + iγ0(ωn + µ) Sth(~p,ωn; µ) = = 2 2 2 (2.170) m + i~γ · ~p − iγ0(ωn + µ) m + ~p − (ωn + µ) or

S˜th(~p,z0; µ) (2.171) 1 X = e−iωnz0 S˜ (ω, ω ; µ) τ th n · n µ ¶¸ 1 ∂ i © ª −iωz0 +iωz0 = m − i~γ · ~p + iγ0 · µ − · [1 − n˜(ω + µ)] e − n˜(ω − µ) e , i ∂z0 2ω 30 and the Fourier transform of the ﬁnite-temperature propagator is given by

S˜th(~z, z0; µ) (2.172) Z d3~p = e+i~p·~z S˜ (~p,z ; µ) (2π)3 th 0 Z · µ ¶¸ d3~p 1 ∂ = 3 m − i~γ · ~p + iγ0 · µ − (2π) i ∂z0 i n o × [1 − n˜(ω + µ)] ei(~p·~z−ωz0) − n˜(ω − µ) ei(~p·~z+ωz0) · 2ω µ ¶ µ ¶¸ 1 ∂ 1 ∂ = m − i~γ · + iγ0 · µ − i ∂~z i ∂z0 Z d3~p i n h io × ei(~p·~z−ωz0) − n˜(ω + µ) ei(~p·~z−ωz0) +n ˜(ω − µ) ei(~p·~z+ωz0) . (2π)3 2ω The terms in the second part of the curly brackets are the thermal part of a ﬁnite-temperature th fermion (anti-fermion) propagator, δSµν , and represent the collective eﬀects of fermions inside the thermal bath.

2.5.2 Free Finite-Temperature Boson Propagators

The ﬁnite-temperature boson propagator in the Matsubara representation is

˜ B ~ 2 ˜ B 1 ∆th(k, ωl; m ) = ∆th(ωk, ωl) = 2 2 , (2.173) ωk − ωl 2 ~ 2 2 where ωk = k + m , and the propagator in the mixed representation becomes 1 X ∆˜ B (ω , z ) = e−iωlz0 ∆˜ B (ω , ω ) (2.174) th k 0 τ th k l l i © ª −iωz0 +iωkz0 = [1 + n(ωk)]e + n(ωk)e , 2ωk where the boson Matsubara frequencies are deﬁned as 2lπ ω = , (2.175) l τ and n(ωk) is the Bose-Einstein distribution function: 1 n(ωk) = . (2.176) eβωk − 1 For non-zero chemical potential, the free ﬁnite-temperature boson propagator and its mixed representation become

˜ B ~ 2 ˜ B 1 ∆th(k, ωl; m , µ) = ∆th(ωk, ωl; µ) = 2 2 , (2.177) ωk − (ωl − µ) and 1 X ∆˜ B (ω , z ; µ) = e−iωlz0 ∆˜ B (ω , ω ; µ) (2.178) th k 0 τ th k n l i © ª −iωkz0 +iωkz0 = [1 +n ˜(ωk + µ)] e +n ˜(ωk − µ) e , 2ωk respectively. 31

2.5.3 Free Finite-Temperature Photon/Gauge Field Propagators

The finite-temperature photon or gauge propagator is defined with gauge fields Aµ(x) as

µν Dth (x − y) = ih(Aµ(x)Aν (y))+iβ. (2.179)

Depending on the choice of gauge, the form of a finite-temperature photon propagator is quite different. At finite temperature, the thermal distribution is defined in the rest frame of the medium. Hence, there exists a preferred frame, the rest frame of the medium, and the covariance of formalism is then lost. In general, non-covariant gauges like the Coulomb gauge are more convenient for calculations, and have been commonly employed in the finite-temperature theory. Thus, the Coulomb gauge will be adapted in subsequent calculations. The causal photon propagator at T = 0 in Coulomb gauge is given by [68, 69, 56] · ¸ 1 k k u · k 1 D˜ µν (k) = g − µ ν − (k u + u k ) = Pˆ C , (2.180) c k2 µν k2 + (u · k)2 k2 + (u · k)2 µ ν µ ν k2 µν where uµ is the four-velocity of the medium with u · u = −1, gµν = δµν is the Minkowski metric for

µ, ν = 1, 2, 3, 4 and k4 = ik0, and k k u · k Pˆ C = g − µ ν − (k u + u k ). (2.181) µν µν k2 + (u · k)2 k2 + (u · k)2 µ ν µ ν

In the usual Coulomb gauge, uµ = (1,~0) and the photon propagator becomes

· 2 ¸ ˜ µν 1 1 ˆ T 1 k ˆ T 1 ˆ C Dc (k) = − δµ0δ0ν + Pµν = − δµ0δ0ν + Pµν = Pµν . (2.182) ~k2 k2 k2 ~k2 k2

Here Pˆ T is the spatially transverse projection operator,

ˆ T kikj ˆ T ˆ T ˆ T ˆ T ˆ T Pij = δij − , P00 = P0i = 0, P · P = P , (2.183) ~k 2 which projects vectors onto the plane orthogonal to both kµ and kj as

ˆ T ~ ˆ T P · k = Pij kj = 0, (2.184) ˆ T ˆ T ˆ T P · k = Piν kν = Pij kj = 0, (2.185) and is related to the longitudinal projection operator Pˆ L through the identity:

ˆ L ˆ ˆ T ˆ L ˆ L ˆ L Pµν = Pµν − Pµν , P · P = P , (2.186) where Pˆ µν is the gauge-invariant projection operator on a plane orthogonal to kµ and k k k k Pˆ = δ − µ ν = g − µ ν , Pˆ · Pˆ = Pˆ , (2.187) µν µν k2 µν k2 which will project out any gauge-parameter dependent term and

Pˆ · Pˆ T = Pˆ T , Pˆ · Pˆ L = Pˆ L. (2.188) 32

In the usual Coulomb gauge, ∇·A~ = 0, with gauge parameter suppressed, the photon propagator can also be expressed in a matrix form as [68] Ã ! −1/~k2 0 D˜ µν (k) = , (2.189) c ˆ T ~ 2 2 0 Pij/(k − k0) where µ, ν = 0, 1, 2, 3. A ﬁnite-temperature photon propagator in Matsubara (Imaginary-Time) formalism is given by replacing k0 in the causal propagator with ωn;

µν ~ 1 1 T D˜ (k, ωn) = − δµ0δ0ν + Pˆ . (2.190) th ~ 2 ~ 2 2 µν k k − ωn Similarly, the ﬁnite-temperature photon propagator in the Real-Time formalism can be written as

˜ µν ~ ˜ µν ~ ˜ µν ~ Dth (k, k0) = Dc (k, k0) + δDth (k, k0), (2.191) where the T = 0 part is just the causal propagator, and the T 6= 0 part is h i ˜ µν ~ ˜ µν ~ ˜ µν ~ δDth (k, k0) = f(k0) · Dth (k, ωn = k0 + i²) − Dth (k, ωn = k0 − i²) (2.192) ~ 2 2 T = 2πi ²(k0) f(k0) δ(k − k ) · Pˆ h 0 µν i ~ ~ 2 2 ˆ T = 2πi n(|k|) δ(k − k0) · δµi · Pij · δjν where

²(k0) = θ(k0) − θ(−k0), (2.193) and 1 f(k0) = . (2.194) eβk0 + 1 In QED, it can be shown that there is no contribution from the (µ = 0, ν = 0)-component in ˜ µν ˜ µν ˜ µν Coulomb gauge. Both Dc and Dth are symmetric in their indices, as is the thermal part δDth ; and the quantity in the last square bracket of Eq. (2.192) should be understood symmetrized in its indices µ and ν.

2.5.4 Interpretation of Thermal Parts of Propagators

th The thermal part of a ﬁnite-temperature photon propagator δDµν can be considered as describing the emission and absorption of photons by the heat bath, i.e., the thermal photons as blackbody ~ 2 2 th radiation. The factor of δ(k − k0) in δDµν puts the thermal photons on the mass-shell, so the thermal photons are real instead of virtual. For fermions, one can take the Fourier transform of the non-interacting fermion propagator as Z d3~p S (~z, z ) = S (~p,z ) e+i~p·~z (2.195) th 0 (2π)3 th 0 Z · µ ¶¸ d3~p i ∂ = m − i~γ · ~p + iγ · i (2π)3 2ω 0 ∂z n h 0 io × ei(~p·~z−ωz0) − n˜(ω) ei[~p·~z−ωz0] + ei[~p·~z+ωz0] . 33

The first term in the curely brackets describes the forward propagation for the particle of interest with a phase factor of (~p · ~z − ωz0). The second term with the Fermi-Dirac distribution function is the thermal part of the propagator, which represents the creation and annihilation of particles, or fermion-anti-fermion pairs in the thermal bath. In the non-interacting case, (m + iγ · p) u(p) = 0 for spinor u(p), thenu ¯(p)~γu(p) =u ¯(p)[−i~p/m] u(p). Hence, the thermal part is symmetric under the exchange ~p ↔ −~p with phase factors of (~p · ~z ∓ ωz0). Therefore, the thermal part of a finite- th temperature fermion propagator δSµν can be used to describe the collective effects of particles in the thermal bath [56, 70]. Similar interpretation can also apply to bosons.

2.6 Relationship Between Formalisms

The relation between the Imaginary-Time formalism (ITF) and Real-Time formalisms (RTF) can be found in Refs. [55] and [56]. A ﬁnite-temperature propagator in ITF can be analytically continued to that of RTF [71, 12], and the connection has been extended to n-point thermal Green’s functions [58, 72, 73, 74, 75, 76, 64, 42, 77, 78]. However, the analytical continuation is not unique and depends on the scheme of choice, which creates some confusion [56].

The popular choice of analytical continuation is ωn → q0 ±i², which corresponds to the Feynman and Anti-Feynman (F/F)¯ propagators, respectively, in RTF (cf. Eq. (2.77)). For applications in linear response of ﬁelds, the two formalisms are linked by the analytical continuation of ωn → 0 0 0 0 q0 ± i²q0, where + and − are for Retarded and Advanced propagators, respectively, which leads to the Retarded/Advanced (R/A) basis transformation scheme as in Refs. [64], [77], [76], and [79].

In RTF, a time-path ordered, ﬁnite-temperature propagator ∆˜ th is of a 2 × 2 matrix form, and is related to those in these bases by a special Bogoliubov transformation as [78]

T ∆basis(q) = U(q) · ∆˜ th(q) · U (−q), (2.196) where superscript ’T’ denotes the transpose of a matrix. Propagators in F/F¯ and R/A bases are diagonalized as Ã ! ∆F (q) 0 ∆ ¯ (q) = (2.197) F/F ∗ 0 ∆F (q) and Ã ! ∆R(q) 0 ∆R/A(q) = , (2.198) 0 ∆A(q) respectively, and are more convenient to work on compared to non-diagonalized forms in perturbative theory. In addition, diﬀerent bases correspond to separate sets of Feynman rules in the perturbation approach. Asides from these two bases, there are other transformation schemes like the Klydish- basis, which is not diagonalized, but contains symmetric and retarded/advanced propagators as Ã ! ∆S(q) ∆R(q) ∆Klydish(q) = , (2.199) ∆A(q) 0 34

where ∆S(q) = ∆˜ 11(q) + ∆˜ 22(q) with a common arbitrary function suppressed.

2.7 Damping Rate

At zero temperature, the Dyson-Schwinger equation gives the fully-dressed, causal fermion propagator as h i−1 ˜0 ˜−1 ˜ 1 Sc(p) = Sc (p) − Σ(p) = , (2.200) m0 + iγ · p − Σ˜(p) where m0 is the bare mass, and Σ˜(p) is the fermion self-energy. Replacing p with ω = −iγ · p in the ˜0 ˜ argument of both Sc and Σ, ˜0 −1 ˜ Sc(ω) = m0 − ω − Σ(ω; m0). (2.201) The existence of a pole at the physical mass in the fully-dressed, causal fermion propagator leads to the deﬁnition of the physical mass m in terms of the bare mass m0 and the self-energy Σ˜ at the pole, ω = m, as [48]

m = m0 − Σ˜(m; m0), (2.202) and the residue of the propagator at the pole deﬁnes the wave-function renormalization constant Z2 as ¯ ˜ ¯ ∂Σ(ω; m0)¯ Z−1 = 1 + ¯ . (2.203) 2 ∂ω ¯ ω=m

Both can be easily seen through adding and subtracting Σ˜(m; m0) to Eq. (2.201) as

−1 S˜0 (ω) = m − Σ˜(m; m ) − ω − [Σ˜(ω; m ) − Σ˜(m; m )] (2.204) c 0 " 0 0 # 0 Σ˜(ω; m ) − Σ˜(m; m ) = [m − ω] 1 + 0 0 . ω − m

At ﬁnite temperature, it has been argued that there might exist a pole in the fully-dressed fermion propagator, and the damping rate of a fermion is then given by the imaginary part of the pole [26, 56]. In the framework of the imaginary-time formalism (ITF), the dressed fermion propagator can written as

h i−1 ˜0 ˜−1 ˜ 1 Sth(p) = Sth (p) − Σth(p) = , (2.205) m0 + iγ · p − Σ˜ th(p) where four-vector p denotes (~p,ωn), and Σ˜ th(p) is the associated Euclidean fermion self-energy in

ITF. To locate the complex pole at the lower-half of the complex energy plane, ωn is analytically ˜ ˜ ˜0 continued to p0 +i² with p0 real, and Σth(~p,ωn) → Σth(~p,p0). Then, the pole of Sth(~p,p0) is deﬁned at

det |m0 + iγ · p − Σth(p)| = 0, (2.206) and p0 is complex at ﬁnite temperature, [26]

p0 = E − iΓ. (2.207) 35

In the rest frame of the medium, the self-energy can be written in terms of a combination of matrices

1, γ0, and ~γ · ~p [56, 80], and the fully-dressed fermion propagator becomes

m − iγ · p − Σ˜ + 1 Tr [Σ˜ ] S˜0 (p) = th 2 th , (2.208) th 2 2 ˜ m0 + p − Ξth(p) where the scalar self-energy Ξ˜ th is 1 1 1 Ξ˜ (p) = Tr [(m − iγ · p) Σ˜ ] − Tr [Σ˜ 2 ] + (Tr [Σ˜ ])2. (2.209) th 2 th 4 th 8 th Then, the damping rate can be approximated as [26, 56] ¯ 1 ¯ Γ = −Im[p ] ' − Im{Tr [(m − iγ · p)Σ˜ (p , ~p)]}¯ . (2.210) 0 4E th 0 ¯ p0=E−iΓ

In general, the fermion self-energy Σ˜ th is a complicate function of energy and momentum, which may not be fully analytical in lower-half of the complex energy plane. The fermion self-energy Σ˜ th also contains factors related to renormalization. Therefore, some cautions should be taken with such perturbative approach. To avoid entangling with renormalization-related factors, one can follow the procedure similar to the zero-temperature theory, and let ω = −iγ · p,

˜0 1 ˜0 1 Sth(p) = → Sth(ω) = , (2.211) m0 + iγ · p − Σ˜ th(p) m0 − ω − Σ˜ th(ω; m0) where Σ˜ th(p) = Σ˜ th(ω; m0) is the thermal counterpart of the self-energy Σ˜(p) at T = 0 and µ = 0. Following the renormalization procedure of the zero-temperature theory by adding and subtracting both Σ˜(m; m0) and Σ˜(ω; m0),

˜0 −1 ˜ ˜ ˜ Sth(ω) = m0 − Σ(m; m0) − ω − [Σth(ω; m0) − Σ(m; m0)] (2.212) = m − ω − [Σ˜(ω; m ) − Σ˜(m; m )] − [Σ˜ (ω; m ) − Σ˜(ω; m )] " 0 0 # th 0 0 Σ˜(ω; m ) − Σ˜(m; m ) h i = [m − ω] 1 + 0 0 − Σ˜ (ω; m ) − Σ˜(ω; m ) ω − m th 0 0 n h io −1 ˜ ˜ = Z2 m − ω − Z2 Σth(ω; m0) − Σ(ω; m0) , where the renormalized mass m and wave-function renormalization constant Z2 are defined in Eqs. ˜ 0 ˜ ˜ (2.202) and (2.203), respectively. Let Σth(ω; m) = Z2[Σth(ω; m0) − Σ(ω; m0)], and the wave- function renormalization constant Z2 can be absorbed in the finite-temperature propagator through re-definition of the fields as Z 1 S˜0 (ω) = 2 → . (2.213) th ˜ 0 ˜ 0 m − ω − Σth(ω; m0) m − ω − Σth(ω; m) In such form, effects of renormalization are removed from those of thermal processes. To see the relationship between damping and self-energy, one can take the proper-time representation of the 36

finite-temperature propagator as 1 S˜0 (ω) = (2.214) th ˜ 0 m − ω − Σth(ω) Z ∞ −is m−ω−Σ˜ 0 (ω) = i ds e [ th ] 0 Z ∞ −is m−ω−ReΣ˜ 0 (ω) −s ImΣ˜ 0 (ω) = i ds e [ th ] th 0 Z ∞ ¯ ˜ 0 ∂ ¯ −ImΣth(ω)(i ∂u ) −isu¯ = e i ds e ¯ . 0 ˜ 0 u=m−ω−ReΣth(ω) Thus, the particle damping is related to the imaginary part of the renormalized thermal self-energy, which excludes the effects of renormalization. Alternatively, the fully-dressed, finite-temperature propagator in the proper-time representation can be expressed as 1 S˜0 (p) = (2.215) th ˜ 0 m + iγ · p − Σth(p) h i 1 = m − iγ · p − Σ˜ 0 (p) · th ˜ 0 2 2 [m − Σth(p)] + [γ · p] h i Z ∞ 2 2 2 ˜ 0 ˜ 0 ˜ 0 −is{m +p −δm }−2s[m−ReΣth] ImΣth = m − iγ · p − Σth · i ds e , 0

h i2 h i2 2 ˜ 0 ˜ 0 ˜ 0 where δm = 2m ReΣth− ReΣth + ImΣth is related to the mass-shift. Subsequent calculations will closely follow the same spirit of such approach to estimate the fermion damping in a hot medium.

2.8 Hot Thermal Loops and the Resummation Program

The Hot Thermal Loop (HTL) approximation and the associated Resummation Program (RP) are generally employed in the perturbation theory for the damping rate in a hot plasma or other phenomena [5, 34, 35]. The resummation program is based on the power counting of ~ in the action [81]. At T = 0, the power of ~ is directly related to that of coupling constant g2, and the overall ~ in front of the action can be absorbed into the field definition by re-scaling. At finite temperature, ~ also arises from the boundary condition on β or τ (or the range of integration) as

g2 → ~g2, (2.216) T → T/~. (2.217)

Hence, both the coupling constant g2 and temperature T involve ~ at ﬁnite temperature [82, 33, 32, 20, 21], and need to be kept track of. The scheme divides the momentum of a given line in Feynman diagrams into two scales: When the momentum of a line is on the order of T or larger, it is considered as hard; It is soft if the momentum is on the order of gT . The Hard Thermal Loops denote loop momenta that are hard in a given graph. In general, the contributions from HTLs are approximately g2T 2/~p 2 times the corresponding tree-level amplitude with the external momentum p. It has been shown that contributions from HTLs in a given part of diagram are in the same 37 order of tree diagrams when all ’external’ legs (lines) are soft. When one of the external momenta is hard, the contribution of a given thermal loop is on the order of g2, which is smaller than that of the corresponding tree graph. However, it is the same order as the tree diagram when the external momenta are all soft. The resummed program provides a prescription as to which perturbative terms are to be included. When all ’external’ lines of a given diagrams are soft, the eﬀective (HTL) propagators and vertices will be used. If any ’external’ leg of a given part of a diagram is hard, bare propagators or vertices will be used instead. Such approach will be tedious and complicated when working with higher orders of perturbation. Comparison to functional methods, and a possible link to the Resummed HTL approach, will be presented in Chapter 4. Chapter 3

Finite-Temperature Propagator in a Hot QED Medium

3.1 Overview

When two ultra-relativistic ions collide head-to-head, a plasma is created, which consists of high densities of quarks, gluons, leptons, photons, etc. The subject of interest is to model phenomena of an utltra-relativistic particle interacting with the plasma. It has long been known that the problem is non-perturbative in nature [81]. Earlier works have employed perturbative methods, but a lot of problems and inconsistence have raised doubt on the calculations. The non-perturbative approach of Ref. [47] employed functional method in a φ4 toy model to estimate the damping rate of decay processes in the hot medium, and observed the effect of a ’fireball’ inside the plasma. The goal of this thesis is to extend the previous calculation to the context of Quantum Electro- dynamics (QED), and perhaps pave the way to Quantum Chromodynamics (QCD), in examining processes of an ultra-relativistic, charged particle entering a plasma. Instead of dealing with large numbers of fermions, electrons and positrons in QED, individually, a plasma will be treated as a thermal bath specified by a temperature T . The incident particle, a fermion, at ultra-relativistic speed will interact with photons and fermions in the plasma through the exchange of photons. Fur- thermore, calculations in this thesis will base on the Imaginary-time (Matsubara) formalism with modifications suitable for a simplified version of the Physics.

3.2 Dressed Finite-Temperature Fermion Propagator

Following the previous Chapter, the fully-dressed, Finite-Temperature fermion propagator is given by ¯ · L [A] ¸ (th) e th ¯ 0 DA ¯ Sth = e · Gth[A] ¯ , (3.1) Z[iτ] A→0

38 39 where the partition function Z is written in place of the thermal normalization constant Z[iτ], which is related through the analytical continuation β = iτ after setting all sources to zero; the two will be used interchangeably. The same expression for the fermion propagator at ﬁnite temperature can be applied in any formalism of choice. In the Imaginary-time, Matsubara formalism (ITF), the

ﬁeld-coupled thermal Green’s function Gth(x, y|A) and the thermal closed-fermion-loop functional

Lth[A] are parallel in form to their causal, zero-temperature counterparts. The gauge fields Aµ(x) in Eq. (3.1) are just dummy functions to facilitate the functional differentiations of linkage with (th) the Finite-Temperature linkage operator exp{DA }, and will be set to zero afterwards. Hence, the gauge fields can be treated as dummy fields unless external fields exist. (th) (th) In ITF, the operator DA in the exponent of the Finite-Temperature linkage operator exp{DA } is Z τ Z τ (th) i δ δ DA = − dx dy · Dth(x − y) · , (3.2) 2 0 0 δA(x) δA(y) and its Fourier transformed expression includes a complete Matsubara sum over discrete frequencies of photons in ITF as Z i 1 X d3~k δ δ D(th) = − · D˜ (~k, ω ) · , (3.3) A 2 τ (2π)3 ˜ ~ th l ˜ ~ l δAl(k) δA−l(−k) which can be converted to a complete integral over photon energies k0 as

Z 4 (th) i d k δ ˜ µν δ DA = − 4 · Dth (k) · , (3.4) 2 (2π) δA˜µ(k) δA˜ν (k)

µν where the Finite-Temperature photon propagator Dth (k) is in real-time form àla Dolan and Jackiw (th) [12]. Hence, the Finite-Temperature linkage operator exp{DA } can be expressed in any formalism (th) of choice. Furthermore, the operator DA can be expressed in terms of space integrals as Z Z (th) i δ µν δ DA = − dx dy · Dth (x − y) · , (3.5) 2 δAµ(x) δAν (y) R R where the range of time integrals, dx0 and dy0, are from −∞ to +∞, and the finite temperature µν photon propagator Dth (x − y) is in real-time form. The finite-temperature photon propagator in real-time form can be decomposed into two parts as µν µν µν Dth (x − y) = Dc (x − y) + δDth (x − y), (3.6) where the first term is the causal photon propagator at zero temperature, and the second is the (th) contribution from the heat bath, i.e., the effect of thermal photons. In turn, DA of the Finite- Temperature linkage operator can be split into two parts as

(th) (c) (th) DA = DA + ∆DA , (3.7)

(c) (th) where the causal part, DA , and the thermal part, ∆DA , are given by Z Z (c) i δ µν δ DA = − dx dy · Dc (x − y) · , (3.8) 2 δAµ(x) δAν (y) 40 and Z Z (th) i δ µν δ ∆DA = − dx dy · δDth (x − y) · , (3.9) 2 δAµ(x) δAν (y) respectively. The two separate parts of the Finite-Temperature linkage operator account for eﬀects from diﬀerent physical origins, the thermal and non-thermal contributions.

While thermalized gauge ﬁelds (or thermal photon ﬁelds) Aµ are periodic in ’time’-variables, i.e.,

A(x0) = A(x0 + τ), the ﬁeld-coupled, thermal fermion Green’s function, Gth[A], satisﬁes the Kubo-

Martin-Schwinger conditions with anti-periodic boundary condition, i.e., Gth(x0, ~x; A) = −Gth(x0+

τ, ~x; A). In general, Gth[A] is technically diﬃcult to be expressed in RTF with two distinct parts of diﬀerent physical origins, and will be kept in the Matsubara form for subsequent calculations.

A similar approach will be also applied to the thermal closed-fermion-loop functional, Lth[A]. In contrast, the real-time form of the thermal photon propagator, Dth, in the linkage operator will be used in the calculation. Since the linkage operation consists of pure functional diﬀerentiations, the order of the causal (c) (th) linkage exp{DA } and thermal linkage exp{∆DA } is irrelevant and can be exchanged. If the causal linkage is applied ﬁrst, it leads to ¯ ½ · µ L [A] ¶¸¾ (th) (c) e th ¯ 0 ∆DA DA ¯ Sth = e · e · Gth[A] ¯ . (3.10) Z[β] A→0 (c) The causal linkage operator exp{DA } inside the square brackets represents self-linkage of the fermion of interest, self-linkage of fermions in the medium, and exchange (or cross-linkage) of virtual photons between the fermion of interest and those in the medium, along with various numbers of closed-fermion-loop insertions as

Asides from the exchange between the fermion of interest and those in the medium, self-linkage will also generate the mass and wave-function renormalization factors similar to the zero-temperature (c) D Lc[A] theory, i.e., e A · Gc[A] e . In the ﬁrst approximation, one could treat it as ¯ ½ µ L [A] ¶¾ (th) e th,R ¯ 0 ∆DA ¯ Sth ' e · Gth,R[A] ¯ , (3.12) Z,R[β] A→0 where the mass and wave function renormalization factors are incoporated into both the thermal fermion Green’s functional, Gth,R[A], and the thermal closed-fermion-loop functional, Lth,R[A]. These renormalization factors are irrelevant to the problem of interest, and the renormalization symbol ”,R” for the Green’s function and close-fermion-loop functional will be omitted in subsequent calculations for simplicity of notation.

In the mixed formalism to be used, the real-time coordinates x0 and y0 outside any complete integral are understood to be held between 0 and τ; and after the continuation τ → is performed, x0 and y0 are allowed to be arbitrarily large value. In the mixed representation, the thermal Green’s function becomes ¯ ½ µ L [A] ¶¾ (th) e th ¯ 0 ∆DA ¯ h~p,n|Sth|~y, y0i ' e · h~p,n|Gth[A]|~y, y0i ¯ . (3.13) Z[β] A→0 41

(th) The thermal linkage, exp{∆DA }, represents exchange of thermal photons between the fermion of interest and the thermal bath. With such approach, the eﬀect of the fermion exchanging thermal photons with the medium can be easily distinguished from irrelevant renormalization factors, and the current functional approach can separate mixed contributions. In contrast, the mixture of renormalization and thermal contributions have plagued early calculations with perturbation methods. Even though the introduction of the Resummed Program (RP) along with the Hot Thermal Loop (HTL) approximations can reduce degrees of divergence due to the renormalization, various IR and UV divergence still exist in the perturbation approach [6, 8, 30, 45, 46, 83]; none will be seen here.

3.3 New Variant of Fradkin Representation

There are several variants of Fradkin’s representations which can be found, e.g., in Ref. [50]. For the ﬁnite temperature theory, it is more convenient to work in momentum (or Matsubara) space, in which its formulation is parallel to that of the zero-temperature theory. Fradkin representation of the ﬁeld-coupled, thermal Green’s function will be derived in the mixed Matsubara and imaginary- time representation. With forms parallel to the causal case, the derivation in ITF will be easily converted in the vacuum theory at zero temperature.

3.3.1 Thermal Green’s Functions in a Mixed Representation

Instead of h~p,p0| of Eq. (2.152) in the zero-temperature theory, we can apply the Matsubara representation, h~p,n| ≡ h~p,ωn|, to the ﬁeld-coupled, thermal Green’s function in ITF as · ¸ Z ∞ R 2 2 i s ds0 δ δ h~p,n|G [A]|~y, y i = i ds e−ism · e 0 δv2(s0) · m − γ · (3.14) th 0 δv(s) 0 ¯ R s 0 0 i ds vµ(s )·pµ ¯ · e 0 · h~p,n|F(s, v)|~y, y0i¯ , vµ→0 where F(s, v) is given by

h~x,x |F(s, v) (3.15) Ã 0 ( " #)! Z s Z s0 Z s0 = exp −ig ds0 v(s0) · A(x − ds00 v(s00)) + iσ · F(x − ds00 v(s00)) 0 0 0 + ·h~x,x0|, and the corresponding normalization in the Matsubara representation becomes

£ ¤−1/2 3 −i(~p·~y−ωny0) h~p,n|~y, y0i = (2π) τ · e . (3.16)

First apply the same trick of Fried and Woodward [84] by inserting a functional integral representation of unity, " # Z Z s0 1 = d[u] δ u(s0) − ds00v(s00) , (3.17) 0 42

to simplify the arguments of ﬁelds Aµ and the ﬁeld strength tensor Fµν , inside the v-integral of F(s, v) in Eq. (3.15) as " # Z Z s0 0 00 00 h~x,x0|F(s, v) = d[u] δ u(s ) − ds v(s ) (3.18) 0 µ ½ Z s ¾¶ · exp −ig ds0 [v(s0) · A(x − u(s0)) + iσ · F(x − u(s0))] 0 + ·h~x,x0|.

Then replace the delta functional with its Fourier functional integral representation, " # Z s0 δ u(s0) − ds00v(s00) (3.19) 0 ( " #) Z Z s Z s0 −1 0 0 0 00 00 = CFFI d[Ω] · exp i ds Ω(s ) · u(s ) − ds v(s ) 0 0 Z ½ Z s ¾ ½ Z s Z s ¾ −1 0 0 0 0 0 00 00 = CFFI d[Ω] · exp i ds Ω(s ) · u(s ) · exp −i ds v(s ) · ds Ω(s ) , 0 0 s0 where CFFI is the normalization factor of Fourier functional integral, which is irrelevant to the physics of interest, and will be eventually canceled out at a later stage. For simplicity of notation,

CFFI will be dropped in the subsequent calculation. Thus,

Z Z ½ Z s ¾ 0 0 0 h~x,x0|F(s, v) = d[u] d[Ω] exp i ds Ω(s ) · u(s ) (3.20) 0 ½ Z s ·Z s ¸¾ · exp −i ds0 v(s0) · ds00 Ω(s00) + gA(x − u(s0)) 0 s0 µ ½ Z s ¾¶ · exp +g ds0 σ · F(x − u(s0)) 0 + ·h~x,x0|

≡ F(x|s, v) · h~x,x0|.

From Equation (3.14) and h~p,n|F(s, v)|~y, y0i = h~p,n|~y, y0i · F(y|s, v), the ﬁeld-coupled, Thermal Green’s function(al) becomes

Z ∞ R 2 £ ¤−1/2 2 s 0 δ 4 −i(~p·~y−ωny0) −ism i 0 ds 2 0 h~p,ωn|Gth[A]|~y, y0i = (2π) · e · i ds e · e δv (s ) (3.21) · ¸ 0 R δ i s ds0 v (s0)·p × m − γ · · e 0 µ µ δv(s) Z Z R R R i s ds0 Ω(s0)·u(s0) −i s ds0 v(s0)· s ds00 Ω(s00) × d[u] d[Ω] · e 0 · e 0 s0 ¯ µ ½ Z s ¾¶ ¯ ¯ × exp −ig ds0 [v(s0) · A(y − u(s0)) + iσ · F(y − u(s0))] ¯ . 0 +¯ vµ→0 43

n o δ Begin with the operation m − γ · δv(s) ﬁrst, which leads to

Z ∞ Z Z £ ¤−1/2 2 3 −i(~p·~y−ωny0) −ism h~p,n|Gth[A]|~y, y0i = (2π) τ · e · i ds e d[u] d[Ω] (3.22) µ ½ 0 ¾¶ R Z s i s ds0 Ω(s0)·u(s0) 0 0 ×e 0 · exp g ds σ · F(y − u(s )) 0 + ×{m − iγ · [p − gA(y − u(s))]} R 2 R i s ds0 δ s 0 0 0 δv2(s0) ip· ds v(s ) ×e ½ Z · e 0 ·Z ¸¾¯ s s ¯ 0 0 00 00 0 ¯ × exp −i ds v(s ) · ds Ω(s ) + gA(y − u(s )) ¯ . 0 0 s vµ→0

R 2 R s 0 0 s 0 δ ip· ds v(s ) i ds 2 0 Then, move e 0 across Fradkin’s operator e 0 δv (s ) with aid of Eq.(E.1) in the Ap- pendix E.1, and re-arrange the functional-dependent factors, to obtain

h~p,n|Gth[A]|~y, y0i (3.23) Z ∞ Z £ ¤−1/2 2 2 = (2π)3τ · e−i(~p·~y−ωny0) · i ds e−is(m +p ) d[u] 0 µ ½ Z s ¾¶ ×{m − iγ · [p − gA(y − u(s))]}· exp g ds0σ · F(y − u(s0)) 0 ½ · + ¸¾ Z R Z s Z s i s ds0 Ω(s0)·u(s0) 0 00 00 0 × d[Ω] e 0 · exp 2ip · ds ds Ω(s ) + gA(y − u(s )) 0 s0 R ½ Z ·Z ¸¾¯ s 0 δ2 s s ¯ i 0 ds 2 0 δvµ(s ) 0 0 00 00 0 ¯ ×e · exp −i ds v(s ) · ds Ω(s ) + gA(y − u(s )) ¯ 0 0 s vµ→0

Then, Fradkin’s operation (self-linkage over v) yields

R ½ Z ·Z ¸¾¯ s 0 δ2 s s ¯ i 0 ds 2 0 δvµ(s ) 0 0 00 00 0 ¯ e · exp −i ds v(s ) · ds Ω(s ) + gA(y − u(s )) ¯ (3.24) 0 0 s vµ→0 ½ Z s Z s ¾ = exp −i ds1 ds2 Ω(s1) · h(s1, s2) · Ω(s2) 0 0 ½ Z s ¾ × exp −ig2 ds0 A2(y − u(s0)) ( 0 ) Z s Z s0 × exp −2ig ds0 Ω(s0) · ds00 A(y − u(s00)) , 0 0 where Z s 0 0 0 h(s1, s2) = ds θ(s1 − s )θ(s2 − s ), (3.25) 0 and h(s1, s2) is a real function. Various representations of the h-function can be found in Appendix E.3. Insert Eq. (3.24) into the ﬁeld-coupled, Thermal Green’s function, then re-arrange the Thermal 44

Green function for the functional integral over Ω as

h~p,n|Gth[A]|~y, y0i (3.26) Z ∞ Z £ ¤−1/2 2 2 = (2π)3τ · e−i(~p·~y−ωny0) · i ds e−is(m +p ) d[u] 0 µ ½ Z s ¾¶ ×{m − iγ · [p − gA(y − u(s))]}· exp g ds0σ · F(y − u(s0)) 0 + ½ Z s ¾ ½ Z s ¾ × exp 2igp · ds0 A(y − u(s0)) · exp −ig2 ds0 A2(y − u(s0)) 0 0 Z ½ Z s Z s ¾ × d[Ω] exp −i ds1 ds2 Ω(s1) · h(s1, s2) · Ω(s2) ( 0 " 0 #) Z s Z s0 × exp i ds0 Ω(s0) · u(s0) + 2ps0 − 2g ds00 A(y − u(s00)) . 0 0

The functional integral over Ω can be evaulated by either through Eq. (D.16), or by a change of variables as Z · Z ¸ s 1 s¯ Ω(¯ s0) = Ω(s0) − ds¯ h−1(s0, s¯) u(¯s) + 2ps¯ − 2g ds00 A(y − u(s00)) , (3.27) 0 2 0 which leads to Z ½ Z s Z s ¾ d[Ω]¯ exp −i ds1 ds2 Ω(¯ s1) · h(s1, s2) · Ω(¯ s2) (3.28) 0 0 ½ Z s Z s ¾ × exp +i ds1 ds2 [Ω(s1) − Ω(¯ s1)] · h(s1, s2) · [Ω(s2) − Ω(¯ s2)] ½ 0 0 ¾ 1 © ª = C exp − Tr ln (2h) · exp {ip · u(s)}· exp isp2 FFI 2 ½ Z s ¾ ½ Z s ¾ × exp −2igp · ds0 A(y − u(s0)) · exp ig2 ds0 A2(y − u(s0)) ½ Z 0Z 0¾ i s s × exp ds ds u(s ) · h−1(s , s ) · u(s ) 4 1 2 1 1 2 2 ½ 0 0 ¾ Z s Z s Z s2 −1 0 0 × exp −ig ds1 ds2 u(s1) · h (s1, s2) · ds A(y − u(s )) , 0 0 0 where CFFI is a normalization constant of functional integral, which cancels out the normalization factor dropped at the introduction of the Fourier functional integral of δ-functional in Eq. (3.19). The Thermal Green’s function then becomes Z ∞ £ ¤−1/2 2 1 3 −i(~p·~y−ωny0) −ism − Tr ln (2h) h~p,n|Gth[A]|~y, y0i = (2π) τ · e · i ds e · e 2 (3.29) 0 Z ³ R ´ g s ds0σ·F(y−u(s0)) × d[u]{m − iγ · [p − gA(y − u(s))]}· e 0 + ½ Z Z ¾ i s s × exp {ip · u(s)} × exp ds ds u(s ) · h−1(s , s ) · u(s ) 4 1 2 1 1 2 2 ½ 0 0 ¾ Z s Z s Z s2 −1 0 0 × exp −ig ds1 ds2 u(s1) · h (s1, s2) · ds A(y − u(s )) , 0 0 0 45

− 1 Tr ln (2h) where e 2 has been pulled out of the functional integral over u since h(s1, s2) is not a functional of u(s0).

After inserting the Matsubara representations of the ﬁeld Aµ in Eq. (2.90) and the ﬁeld strength

Fµν in Eq. (2.93) into Eq. (3.29), the Thermal Green’s function becomes

h~p,n|Gth[A]|~y, y0i (3.30) Z ∞ £ 3 ¤−1/2 −i(~p·~y−ω y ) −ism2 − 1 Tr ln (2h) = (2π) τ · e n 0 · i ds e · e 2 0 Z ( " Z #) 1 X d3~q × d[u] m − iγ · p − g A˜(q) · eiq·(y−u(s)) τ (2π)3 Ã ( l )! Z s X Z 3 h i 0 1 d ~q ˜ ˜ iq·(y−u(s0)) × exp g ds σµν · 3 qµAν (q) − qν Aµ(q) · e 0 τ (2π) l + ½ Z s Z s ¾ i −1 × exp {ip · u(s)} × exp ds1 ds2 u(s1) · h (s1, s2) · u(s2) 4 0 0 ( Z Z Z Z ) 3 s s s2 1 X d ~q 0 × exp −ig A˜(q) ds ds u(s ) · h−1(s , s ) · ds0 eiq·(y−u(s )) . τ (2π)3 1 2 1 1 2 l 0 0 0 The exponent of the last exponential factor can be re-written as

Z s Z s Z s2 −1 0 0 −ig ds1 ds2 u(s1) · h (s1, s2) · ds A(y − u(s )) (3.31) 0 Z 0 Z Z 0 Z 3 s s s2 1 X d ~q 0 = −ig ds ds u(s ) · h−1(s , s ) · A˜(q) ds0eiq·(y−u(s )) τ (2π)3 1 2 1 1 2 l 0 0 0 Z 1 X d3~q ≡ −ig f˜(q) · A˜(q), τ (2π)3 l where q = (~q, ωl) and

Z s Z s Z s2 ˜ −1 0 iq·(y−u(s0)) fν (q) = ds1 ds2 uν (s1) · h (s1, s2) ds e . (3.32) 0 0 0

Unless an external ﬁeld exists, the gauge ﬁeld Aµ in the Green’s function Gth[A] is just a dummy function to facilitate the linkage operation. If the linkage operator is represented in the real-time form, the ordinary Fourier representation of Aµ, Eq. (2.98), should be used instead, and

h~p,n|Gth[A]|~y, y0i (3.33) Z ∞ £ 3 ¤−1/2 −i(~p·~y−ω y ) −ism2 − 1 Tr ln (2h) = (2π) τ · e n 0 · i ds e · e 2 Z ½ · Z 0 ¸¾ d4q × d[u] m − iγ · p − g A˜(q) · eiq·(y−u(s)) (2π)4 µ ½ Z s Z 4 h i ¾¶ 0 d q ˜ ˜ iq·(y−u(s0)) × exp g ds σµν · 4 qµAν (q) − qν Aµ(q) · e 0 (2π) ½ Z Z ¾ + i s s × exp {ip · u(s)} × exp ds ds u(s ) · h−1(s , s ) · u(s ) 4 1 2 1 1 2 2 ½ Z Z 0 Z 0 Z ¾ 4 s s s2 d q ˜ −1 0 iq·(y−u(s0)) × exp −ig 4 A(q) ds1 ds2 u(s1) · h (s1, s2) · ds e . (2π) 0 0 0 46

Similarly, the exponent of the last exponential factor is Z Z Z Z 4 s s s2 d q 0 −ig ds ds u(s ) · h−1(s , s ) · A˜(q) ds0eiq·(y−u(s )) (3.34) (2π)4 1 2 1 1 2 Z 0 0 0 d4q ≡ −ig f˜(q) · A˜(q), (2π)4

˜ where f(q) is in the same form of Eq. (3.32) with q = (~q, q0). For the zero-temperature theory, a similar treatment leads to the casual Green’s function

h~p,p0|Gc[A]|~y, y0i (3.35) Z ∞ £ 4¤−1/2 −i(~p·~y−p y ) −ism2 − 1 Tr ln (2h) = (2π) · e 0 0 · i ds e · e 2 0 Z ³ R ´ g s ds0σ·F(y−u(s0)) × d[u]{m − iγ · [p − gA(y − u(s))]}· e 0 + ½ Z Z ¾ i s s × exp {ip · u(s)} × exp ds ds u(s ) · h−1(s , s ) · u(s ) 4 1 2 1 1 2 2 ½ 0 0 ¾ Z s Z s Z s2 −1 0 0 × exp −ig ds1 ds2 u(s1) · h (s1, s2) · ds A(y − u(s )) , 0 0 0 and can be simpliﬁed by utilizing the special form of the inverse h-function in Eq. (E.21) as

h~p,p0|Gc[A]|~y, y0i (3.36) Z ∞ £ 4¤−1/2 −i(~p·~y−p y ) −ism2 − 1 Tr ln (2h) = (2π) · e 0 0 · i ds e · e 2 0 Z ³ R ´ g s ds0σ·F(y−u(s0)) × d[u]{m − iγ · [p − gA(y − u(s))]}· e 0 + ½ Z ¾ ½ Z ¾ i s s × exp {ip · u(s)} × exp ds0 [u0(s0)]2 × exp −ig ds0 u0(s0) · A(y − u(s0)) , 4 0 0 where the details of derivation can be found in Appendix C.4. This form is similar to the original Fradkin representation appearing in Ref. [49]. It will be shown in the later sections that the Bloch-Nordsieck approximation will be very convenient to apply with this speciﬁc variant of Fradkin representation.

3.3.2 Free-Field Limit

If g = 0 or Aµ → 0, the ﬁeld-coupled, Green’s function in Eq. (3.35) reduces to ¯ ¯ ¯ h~p,p0|Gc[A = 0]|~y, y0i¯ (3.37) g=0 Z ∞ £ 4¤−1/2 −i(~p·~y−p y ) −ism2 − 1 Tr ln (2h) = (2π) · e 0 0 · i ds e · e 2 0 Z ½ Z s Z s ¾ ip·u(s) i −1 × d[u][m − iγ · p] · e · exp ds1 ds2 u(s1) · h (s1, s2) · u(s2) , 4 0 0 47 where the functional integral over u is given by

Z R R i s ds s ds u(s )·h−1(s ,s )·u(s ) ip·u(s) d[u] e 4 0 1 0 2 1 1 2 2 · e (3.38)

Z R R R i s ds s ds u(s )· 1 h−1(s ,s )·u(s ) ip· s ds0 u(s0)·δ(s0−s) = d[u] e 2 0 1 0 2 1 2 1 2 2 · e 0

Z R R i s ds s ds 1 h−1(s ,s )[u(s )u(s )+2p·u(s )·2h(s ,s )·δ(s −s)] = d[u] e 2 0 1 0 2 2 1 2 1 2 1 2 1 1 R R − i p2· s ds s ds δ(s −s)·2h(s ,s )·δ(s −s) − 1 Tr ln( 1 h−1) = e 2 0 1 0 2 1 1 2 2 · e 2 2 −isp2 + 1 Tr ln(2h) = e · e 2 .

Alternatively, the delta-function δ(s0 − s) can be replaced by an integral of inverse h-function with respective to s (cf. Eq. (E.22)), and the functional integral over u becomes

Z R R i s ds s ds u(s )·h−1(s ,s )·u(s ) ip·u(s) d[u] e 4 0 1 0 2 1 1 2 2 · e (3.39)

Z R R R i s ds s ds u(s )· 1 h−1(s ,s )·u(s ) ip· s ds0 u(s0)·δ(s0−s) = d[u] e 2 0 1 0 2 1 2 1 2 2 · e 0

Z R R i s ds s ds u(s )· 1 h−1(s ,s )·u(s )+4u(s )· 1 h−1(s ,s )·s p = d[u] e 2 0 1 0 2 [ 1 2 1 2 2 1 2 1 2 2 ]

Z R R R R i s ds s ds [u(s )+2s p]· 1 h−1(s ,s )·[u(s )+2s p] −ip2 s ds s ds s ·h−1(s ,s )·s = d[u] e 2 0 1 0 2 1 1 2 1 2 2 2 e 0 1 0 2 1 1 2 2

− 1 Tr ln( 1 h−1) −isp2 −isp2 + 1 Tr ln(2h) = e 2 2 · e = e · e 2 .

Both make use of the Gaussian functional integral of the form

Z ½ Z s Z s ¾ i 1 −1 + 1 Tr ln (2h) d[w] exp ds1 ds2 w(s1) · h (s1, s2) · w(s2) = e 2 (3.40) 2 0 0 2

Note that the Trace-Log term above cancels with that of the Green’s function. Recall that (γ · p)2 = p2, the Green’s function is reduced to

h~p,p0|Gc[A = 0]|~y, y0i (3.41) Z ∞ £ ¤−1/2 2 2 = (2π)4 · e−i(~p·~y−p0y0) · [m − iγ · p] · i ds e−is(m +p ) 0 £ ¤−1/2 m − iγ · p = (2π)4 · e−i(~p·~y−p0y0) · m2 + (γ · p)2

= h~p,p0|Sc|~y, y0i, or m − iγ · p 1 G˜ (~p,p |A = 0) = = = S˜ (~p,p ) (3.42) c 0 m2 + (γ · p)2 m + iγ · p c 0 in momentum space. When g = 0 or Aµ = 0, the ﬁeld-coupled Green’s function reduces to the free propagator. This shows that the new form of Fradkin’s representation has correct free ﬁeld limit. 48

3.3.3 Bloch-Nordsieck Approximation

To better illustrate the Bloch-Nordseick approximation, let

u(si) = w(si) − 2sip, (3.43) and observe that Z s Z s i −1 ds1 ds2 u(s1) · h (s1, s2) · u(s2) (3.44) 4 0 0 Z s Z s Z s Z s i −1 i −1 = ds1 ds2 w(s1) · h (s1, s2) · w(s2) − ds1 ds2 s1p · h (s1, s2) · w(s2) 4 0 0 2 0 0 Z s Z s Z s Z s i −1 −1 − ds1 ds2 w(s1) · h (s1, s2) · s2p + i ds1 ds2 s1p · h (s1, s2) · s2p 2 0 0 0 0 Z s Z s i −1 2 = ds1 ds2 w(s1) · h (s1, s2) · w(s2) − ip · w(s) + isp , 4 0 0 where Eqs. (E.22) and (E.23) have been used in the last line. Hence,

½ Z s Z s ¾ i −1 exp {ip · u(s)}· exp ds1 ds2 u(s1) · h (s1, s2) · u(s2) (3.45) 4 0 0 ½ Z s Z s ¾ i −1 © 2ª = exp ds1 ds2 w(s1) · h (s1, s2) · w(s2) · exp −isp 4 0 0 and, then, one can replace the functional integral over u(s0) by w(s0) as

h~p,n|Gth[A]|~y, y0i (3.46) Z ∞ £ 3 ¤−1/2 −i(~p·~y−ω y ) −is(m2+p2) − 1 Tr ln (2h) = (2π) τ · e n 0 · i ds e · e 2 0 Z ( " Z #) 1 X d3~q × d[w] m − iγ · p − g A˜(q) eiq·(y−u(s)) τ (2π)3 Ã ( l )! Z s X Z 3 h i 0 1 d ~q ˜ ˜ iq·(y−u(s0)) × exp g ds σµν · 3 qµAν (q) − qν Aµ(q) e 0 τ (2π) l + ½ Z s Z s ¾ i −1 × exp ds1 ds2 w(s1) · h (s1, s2) · w(s2) 4 0 0 ( Z Z Z Z ) 3 s s s2 1 X d ~q 0 × exp −ig A˜(q) · ds ds [w(s ) − 2s p] h−1(s , s ) ds0 eiq·(y−u(s )) τ (2π)3 1 2 1 1 1 2 l 0 0 0

0 where u(s0) = w(s0) − 2s0p in eiq·(y−u(s )) is to be understood. If one ignores the ﬂuctuation over 49 w(s0), i.e., w(s0) − 2s0p → −2s0p for large p,

BN h~p,n|Gth [A]|~y, y0i (3.47) Z ∞ £ ¤−1/2 2 2 = (2π)3τ · e−i(~p·~y−ωny0) · i ds e−is(m +p ) 0 ( " Z #) 1 X d3~q × m − iγ · p − g A˜(q) eiq·(y+2sp) τ (2π)3 Ã ( l )! Z s Z 3 h i 1 X d ~q 0 × exp g ds0σ · q A˜ (q) − q A˜ (q) eiq·(y+2s p) µν τ (2π)3 µ ν ν µ 0 l ( ) + Z s Z 3 h i 1 X d ~q 0 × exp +2ig ds0 p · A˜(q) eiq·(y+2s p) τ (2π)3 0 l where the Trace-Log term is canceled with the functional integral over w as in Eq. (3.40). Similarly, the Bloch-Nordsieck approximation of Eq. (3.33) is

BN h~p,n|Gth [A]|~y, y0i (3.48) Z ∞ £ ¤−1/2 2 2 = (2π)3τ · e−i(~p·~y−ωny0) · i ds e−is(m +p ) ½ · Z 0 ¸¾ d4q × m − iγ · p − g A˜(q) eiq·(y+2sp) (2π)4 µ ½ Z s Z 4 h i ¾¶ 0 d q ˜ ˜ iq·(y+2s0p) × exp g ds σµν · 4 qµAν (q) − qν Aµ(q) e 0 (2π) + ½ Z s Z 4 h i ¾ 0 d q ˜ iq·(y+2s0p) × exp +2ig ds 4 p · A(q) e 0 (2π) Both are the Bloch-Nordsieck approximated Green’s function similar to the usual form used in Ref. [49].

3.4 Dressed Propagator in Mixed Formalisms

The fully-dressed, Finite-Temperature fermion propagator also involves the linkage of both thermal Green’s function and the thermal closed-fermion-loop functional Lth[A]. The new variant of Fradkin’s representations in the previous section gives an exact form of the ﬁeld-coupled, thermal Green’s function, Gth[A]. The proper-time exponent of the thermal Green’s function Gth[A] is linear in the ﬁelds Aµ(x). In contrast, the thermal closed-fermion-loop functional, Lth[A] =

Tr ln [1 − g (γ · A) Sth] as deﬁned in Eq. (2.84), is a non-linear function of ﬁelds Aµ(x); approximations on the close-fermion-loop functional will be needed to simplify further evaluation.

3.4.1 Approximation for Closed-Fermion-Loop Functional

The thermal closed-fermion-loop functional can be written as Z g 0 0 Lth[A] = −i dg Tr {(γ · A)Gth[g A]}. (3.49) 0 50

Up to the order of g2, the closed-fermion-loop functional may be approximated as

Z τ Z τ i µν Lth[A] = dx dy Aµ(x) · Kth (x − y) · Aν (y), (3.50) 2 0 0

µν where Kth is the thermal photon polarization tensor, i.e., the ﬁnite-temperature counterpart of the µν vacuum polarization tensor Kc in the zero-temperature theory, and the range of integration for x0 and y0 is from 0 to τ, or Z Z i L [A] = dq dq0 A˜ (q) · K˜ µν (q, q0) · A˜ (q0), (3.51) th 2 µ th ν

˜ µν 0 where q = (~q, ωl), and the short-hand notation of Eq. (2.92) is used. Note that Kth (q, q ) = 3 0 ˜ µν 0 (3) 0 (2π) τδ(q + q ) Kth (q), and δ(q + q ) = δl,−l0 δ (~q + ~q ), so that the thermal close-fermion-loop functional reduces to Z i 1 X d3~q L [A] = A˜ (q) · K˜ µν (q) · A˜ (−q) (3.52) th 2 τ (2π)3 µ th ν Z l i = dq A˜ (q) · K˜ µν (q) · A˜ (−q). 2 µ th ν

˜ µν 0 ˜ µν 0 4 0 ˜ µν 0 If the real-time form of Kth (q, q ) is used instead, Kth (q, q ) = (2π) δ(q+q ) Kth (q) with δ(q+q ) = δ(4)(q + q0), and Z i d4q L [A] = A˜ (q) · K˜ µν (q) · A˜ (−q) (3.53) th 2 (2π)4 µ th ν Z i = dq A˜ (q) · K˜ µν (q) · A˜ (−q), 2 µ th ν R where the short-hand notation dq in the last line is Z Z d4q dq ≡ . (3.54) (2π)4

Therefore, the short-hand notation should follow the usage of real-time or imaginary-time form of ˜ µν Kth . Since there is no fermion tadpole in QED, the thermal closed-fermion-loop functional in Eq. ˜ µν (3.52) or (3.53) is quite general if the thermal polarization tensor Kth (q) contains all orders of g. In making approximations on the closed-fermion-loop functional, current conservation, usually called gauge invariance, should be observed in the evaluation of the vacuum polarization tensor [48, 85]; similar caution should also be taken at ﬁnite temperatures. 51

3.4.2 Dressed Finite-Temperature Fermion Propagator

With the approximation for the thermal close-fermion-loop functional in Eq. (3.52), the fully- dressed, ﬁnite-temperature fermion propagator becomes

£ ¤−1/2 0 −1 3 −i(~p·~y−ωny0) h~p,n|Sth|~y, y0i = (Z[iτ]) (2π) τ · e (3.55) Z ∞ Z −ism2 − 1 Tr ln (2h) × · i ds e · e 2 d[u] exp {ip · u(s)} 0 ³ R ´ D(th) g s ds0σ·F(y−u(s0)) ×e A ·{m − iγ · [p − gA(y − u(s))]}· e 0 ½ Z Z ¾ + i s s × exp ds ds u(s ) · h−1(s , s ) · u(s ) 4 1 2 1 1 2 2 ½ 0Z 0 ¾ ½ Z ¾ i × exp −ig dq f˜(q) · A˜(q) · exp dq A˜ (q) · K˜ µν (q) · A˜ (−q) , 2 µ th ν R where the short-hand notation of either Eq. (2.92) or (3.54) is used for dq depending on the form (th) of the linkage operator exp{DA }. The spin-related ordered exponential can be re-written as a functional-translation of another ordered-exponential, i.e.,

³ R ´ g s ds0σ·F(y−u(s0)) e 0 (3.56) ½ + ¾ ¯ Z s ³ R ´ δ s 0 0 ¯ 0 0 0 ds σ·χ(s ) ¯ = exp +q ds F(y − u(s )) · 0 · e ¯ , 0 δχµν (s ) + χ→0

0 where χµν (s ) is an anti-symmetric tensor, as is σµν , and will be set to zero after the functional (th) differentiation with respect to χµν is performed. The linkage operator exp{DA } applies only to fields Aµ in the field strength tensor Fµν instead of χµν , and the complication related to the order-exponential can be deferred until after the linkage operation. Rearranging the spin-related part,

0 h~p,n|Sth|~y, y0i (3.57) Z ∞ 1 £ 3 ¤−1/2 −i(~p·~y−ω y ) −ism2 − 1 Tr ln (2h) = (2π) τ · e n 0 · i ds e · e 2 Z[iτ] Z ½ Z 0 Z ¾ i s s × d[u] exp {ip · u(s)}· exp ds ds u(s ) · h−1(s , s ) · u(s ) 4 1 2 1 1 2 2 ½ Z 0 0 ¾ i d4k δ δ × exp − · D˜ th(k) · 2 (2π)4 δA˜(k) δA˜(−k) ½ · Z ¸¾ × m − iγ · p − g dq A˜(q) eiq·(y−u(s)) ½ ¾ Z ³ R ´ s ds0 σ χ (s0) × exp −ig dq A˜(q) · fˆ(q) · e 0 µν µν + ½ Z ¾¯ ¯ i ˜ ˜ ˜ ¯ × exp dq A(q) · Kth(q) · A(q) ¯ , 2 χ=0,A=0 52 where fˆ(q) is deﬁned by

Z s Z s Z s2 ˆ −1 0 iq·(y−u(s0)) fν (q) = ds1 ds2 uν (s1) h (s1, s2) ds e (3.58) 0 0 0 Z s 0 iq·(y−u(s0)) δ +2i ds e qµ 0 . 0 δχµν (s ) 3.4.3 Linkage Operations

(th) With help of Eq. (D.36) in Appendix D, one can move the linkage operator exp {DA } across the numerator factor {m − iγ · [p − gA]} to obtain ½ · Z ¸¾ D(th) iq·(y−u(s)) e A m − iγ · p − g dq A˜(q) · e (3.59) ½ · Z · ¸¸¾ iq·(y−u(s)) δ D(th) = m − iγ · p − g dq e A˜(q) − iD˜ th(q) · e A δA˜(q) ½ · Z ¸¾ iq·(y−u(s)) δ D(th) ⇒ m − iγ · p + ig dq e D˜ th(q) · e A , δA˜(q) where the gauge ﬁeld A˜µ will be set to zero after all linkage operations are carried out, as is performed in the last step. The remaining linkage operation is of Gaussian-type as in Eq. (D.8) in Appendix D, ½ Z Z ¾ D(th) i e A · exp + dq A˜(q) · K˜ (q) · A˜(q) − ig dq A˜(q) · fˆ(q) (3.60) 2 th ½ Z · ¸ ¾ i 1 = exp + dq A˜(q) · K˜ th(q) · · A˜(q) 2 1 − K˜ (q) · D˜ (q) ½ Z th th ¾ 1 × exp −ig dq A˜(q) · · fˆ(q) 1 − K˜ (q) · D˜ (q) ½ Z · th th ¸ ¾ i 2 ˆ 1 ˆ × exp g dq f(q) · D˜ th(q) · · f(q) 2 1 − K˜ (q) · D˜ (q) ½ ¾ th th 1 × exp − Tr ln [1 − K˜ · D˜ ] . 2 th th The dressed, Finite-Temperature fermion propagator under the one closed-fermion-loop approximation becomes

0 h~p,n|Sth|~y, y0i (3.61) Z ∞ 1 £ 3 ¤−1/2 −i(~p·~y−ω y ) −ism2 − 1 Tr ln (2h) = (2π) τ · e n 0 · i ds e · e 2 Z[iτ] Z ½ Z Z0 ¾ i s s × d[u] exp {ip · u(s)}· exp ds ds u(s ) · h−1(s , s ) · u(s ) 4 1 2 1 1 2 2 ½ · Z · 0 0 ¸ ¸¾ 2 1 ˆ ik·(y−u(s)) × m − iγ · p + g dk D˜ th(k) · · f(k) · e 1 − K˜ (k) · D˜ (k) ½ · th th ¸ ¾ ¯ Z ³ R ´ i 1 s 0 0 ¯ 2 ˆ ˜ ˆ ds σµν χµν (s ) ¯ × exp g dk f(k) · Dth(k) · · f(k) · e 0 ¯ 2 1 − K˜ th(k) · D˜ th(k) + ½ ¾ χ=0 1 × exp − Tr ln [1 − K˜ · D˜ ] , 2 th th 53 where the last exponential factor of Trace-Log can be absorbed into the the partition function as

− 1 Tr ln [1−K˜ ·D˜ ] (0) Z[β] = e 2 th th · Z [β], (3.62) which is valid only under the one-loop approximation.

3.4.4 Dropping Spin-related Contributions

In the regime of interest, the momentum of an ultra-relativistic incident fermion is much larger than the temperature of a heat bath (plasma), i.e., p À T . The contribution from spin-related term which introduces powers of k < p is then negligible; and the ﬁeld-coupled, Thermal Green’s function reduces to

h~p,n|Gth[A]|~y, y0i (3.63) Z ∞ £ 3 ¤−1/2 −i(~p·~y−ω y ) −ism2 − 1 Tr ln (2h) = (2π) τ · e n 0 · i ds e · e 2 Z ½ · Z 0 ¸¾ × d[u] m − iγ · p − g dq A˜(q) · eiq·(y−u(s)) ½ Z Z ¾ i s s × exp {ip · u(s)}· exp ds ds u(s ) · h−1(s , s ) · u(s ) 4 1 2 1 1 2 2 ½ 0 0 ¾ Z Z s Z s Z s2 −1 0 iq·(y−u(s0)) × exp −ig dq A˜(q) · ds1 ds2 u(s1) · h (s1, s2) · ds e 0 0 0 Under the quenched approximation, it can be shown that the contribution of spin vanishes in general gauges [86]. Hence, the dressed Finite-Temperature fermion propagator becomes

£ ¤−1/2 0 −1 3 −i(~p·~y−ωny0) h~p,n|Sth|~y, y0i = Z [iτ] · (2π) τ · e (3.64) Z ∞ Z −ism2 − 1 Tr ln (2h) ×i ds e · e 2 d[u] exp {ip · u(s)} 0 ½ Z Z ¾ i s s × exp ds ds u(s ) · h−1(s , s ) · u(s ) 4 1 2 1 1 2 2 ½ 0Z 0 ¾ i d4k δ δ × exp − · D˜ th(k) · 2 (2π)4 δA˜(k) δA˜(−k) ½ · Z ¸¾ × m − iγ · p − g dq A˜(q) · eiq·(y−u(s)) ½ Z ¾ ½ Z ¾¯ ¯ ˜ ˜ i ˜ ˜ ˜ ¯ × exp −ig dq f(q) · A(q) · exp dq A(q) · Kth(q) · A(q) ¯ , 2 A=0 ˆ ˜ where fν (q) reduces to fν (q) of Eq. (3.32), and the linkage operator is expressed explicitly in the real-time form. 54

3.5 Quenched Dressed Finite-Temperature Fermion Propa- gator

The fully-dressed ﬁnite-temperature fermion propagator is ¯ · L [A] ¸ (th) e th ¯ 0 DA ¯ Sth = e · Gth[A] ¯ (3.65) Z[β] A=0 nh i h io¯ (th) (th) (th) 1 D D D Lth[A2] ¯ = e A1 · Gth[A1] · e 12 · e A2 · e ¯ · , A1=A2=0 Z[β]

(th) where the cross-linkage exp [D12 ] operates on A1 to the left and A2 to the right. If one drops the cross-linkage between contributions from Green’s function and closed-fermion-loop functional, h i¯ h i¯ (th) ¯ (th) ¯ 1 0 DA DA Lth[A] Sth ' e · Gth[A] ¯ · e · e ¯ · . (3.66) A=0 A=0 Z[β]

The second part can be related to the deﬁnition of the partition function as ¯ D(th) L [A]¯ (0) Z[β] = e A · e th ¯ · Z [β], (3.67) A→0 and the quenched propagator becomes ¯ (th) ¯ 1 0 DA Sth ' e · Gth[A]¯ · (3.68) A→0 Z(0)[β]

3.5.1 Quenched Dressed Finite-Temperature Fermion Propagator

At ﬁnite temperature, the quenched, dressed fermion propagator without spin-related contributions reduces to

0 h~p,n|Sth|~y, y0iQ (3.69) Z ∞ (0) −1 £ 3 ¤−1/2 −i(~p·~y−ω y ) −is(m2+p2) − 1 Tr ln (2h) = [Z [iτ]] · (2π) τ · e n 0 · i ds e · e 2 0 Z ½ Z s Z s ¾ i −1 × d[w] exp ds1 ds2 w(s1) · h (s1, s2) · w(s2) 4 0 0 ½ · Z 4 Z s ¸¾ d k 0 0 × m − iγ · p + g2 ds0 D˜ (k) · [w0(s0) − 2p] · e−ik·[w(s)−w(s )]+2ik·p(s−s ) (2π)4 th ½ Z 0 i d4k × exp g2 2 (2π)4 Z s Z s ¾ ik·[w(s2)−w(s1)]−2ik·p[s2−s1] 0 0 × ds1 ds2 e · [w (s1) − 2p] · D˜ th(k) · [w (s2) − 2p] , 0 0 where the subscript ’Q’ denotes the quenched-approximated propagator, and the real-time form of the photon propagator D˜ th(k) is used. If p represents the momentum of an ultra-relativistic, incident fermion, the quantum ﬂuctuation over w(s0) becomes negligible, i.e.,

[w(s0) − 2s0p] → −2s0p or [w0(s0) − 2p] → −2p, (3.70) 55 and the quenched, ﬁnite-temperature propagator reduces to

0 BN h~p,n|Sth|~y, y0iQ (3.71) Z ∞ (0) −1 £ 3 ¤−1/2 −i(~p·~y−ω y ) −is(m2+p2) − 1 Tr ln (2h) ' [Z [iτ]] · (2π) τ · e n 0 · i ds e · e 2 0 Z ½ Z s Z s ¾ i −1 × d[w] exp ds1 ds2 w(s1) · h (s1, s2) · w(s2) 4 0 0 ½ Z 4 h i Z s ¾ d k 0 × m − iγ · p + 2ig2 γ · D˜ (k) · p ds0 e2ik·p(s−s ) (2π)4 th ½ Z Z Z 0 ¾ d4k s s h i 2 ˜ 2ik·p[s1−s2] × exp 2ig 4 ds1 ds2 p · Dth(k) · p e , (2π) 0 0 where the superscript ’BN’ denotes the Bloch-Nordsieck approximated propagator. Note that the functional integral over w cancels out the normalization factor, i.e,

Z ½ Z s Z s ¾ i −1 + 1 Tr ln (2h) d[w] exp ds1 ds2 w(s1) · h (s1, s2) · w(s2) = e 2 . (3.72) 4 0 0

Since the ﬁnite-temperature photon propagator Dth(x − y) is symmetric in x − y,

0 BN h~p,n|Sth|~y, y0iQ (3.73) Z ∞ £ ¤−1/2 2 2 ' [Z(0)[iτ]]−1 · (2π)3τ · e−i(~p·~y−ωny0) · i ds e−is(m +p ) 0 ½ Z 4 h i Z s ¾ d k 0 × m − iγ · p + 2ig2 γ · D˜ (k) · p ds0 e2ik·ps (2π)4 th ½ Z Z Z 0 ¾ 4 s s1 h i 2 d k 0 ˜ 2ik·ps0 × exp 4ig 4 ds1 ds p · Dth(k) · p e . (2π) 0 0

An important point here is that the fermion momentum, (~p,ωn), is kept oﬀ mass-shell for the Finite 2 2 2 2 Temperature fermion propagator, i.e., p = ~p − ωn 6= −m until the Matsubara sum over ωn is carried out. If the fermion is set on its mass-shell, the subsequent evaluation will cause an improper sign in the exponential factor, and will be shown in Appendix F.

3.5.2 Linkage with Real-time Photon Propagators

The linkage operator with the real-time form of the ﬁnite-temperature photon propagator is given by Z 4 (th) i d k δ ~ δ D = − · D˜ th(k, k0) · . (3.74) A 2 (2π)4 δA˜(k) δA˜(−k) Here, the photon propagator is gauge-dependent and is given by

˜ µν ˜ µν ˜ µν Dth (k) = Dc (k) + δDth (k), (3.75) where D˜ c(k) and δD˜ th(k) are the causal part and the thermal addition, respectively. Hence, the 56 linkage operator can be decomposed into two parts as

D(th) D(c)+∆D(th) e A = e A A (3.76) ½ Z 4 ¾ i d k δ ~ δ = exp − · D˜ c(k, k0) · 2 (2π)4 δA˜(k) δA˜(−k) ½ Z 4 ¾ i d k δ ~ δ × exp − · δD˜ th(k, k0) · . 2 (2π)4 δA˜(k) δA˜(−k)

D(c) In the zero-temperature theory, the causal linkage operation with e A contributes mostly to the mass and wave-function renormalizations to the fully-dressed fermion propagator. Therefore, the first thing to investigate will be the effect of thermal photons in the heat bath, which is generated ∆D(th) from the linkage operation with e A . The effective dressed, finite-temperature fermion propagator becomes

0 BN h~p,n|Sth|~y, y0iQ (3.77) Z ∞ £ ¤−1/2 2 2 −1 3 −i(~p·~y−ωny0) −is(m +p ) ' [Z(0)[iτ]] · (2π) τ · e · i ds e 0 ½ Z 4 h h i i Z s ¾ d k 0 × m − iγ · p + 2ig2 γ · D˜ (k) + δD˜ (k) · p ds0 e2ik·ps (2π)4 c th ½ Z Z Z ¾0 4 s s1 h i d k 0 × exp 4ig2 ds ds0 p · D˜ (k) · p e2ik·ps (2π)4 1 c ½ Z Z0 Z0 ¾ 4 s s1 h i 2 d k 0 ˜ 2ik·ps0 × exp 4ig 4 ds1 ds p · δDth(k) · p e , (2π) 0 0 where the fermion momentum is p = (~p,ωn), i.e., the fermion is still kept off mass-shell. Since the Matsubara summation have not yet been perform, it is still in the imaginary-time form. Under the quenched approximation, the effect of the thermal photons comes from both the numerator term and the exponential factor with δD˜ th(k). However, the exponential factor will have the more prominent effect, and will be evaluated first.

3.5.3 Thermal-Photon assisted Damping

A physical quantity should not depend on the choice of gauge, but one can choose a gauge for the convenience of calculations. Since the thermal distribution of a heat bath automatically selects a preferred frame, one can use a non-covariant gauge. For the following calculation, the Coulomb gauge will used instead of the Feynman gauge popularly used in the zero-temperature. In the Coulomb gauge, the thermal part of a photon propagator is

˜ th ~ ~ 2 2 ˆ T δDµν (k, k0) = 2πi ²(k0) f(k0) δ(k − k0) · Pµν (3.78) 1 ~ 2 2 ˆ T = 2πi δ(k − k0) · Pµν , eβ|k0| − 1

ˆ T ˆ T ~ 2 ˆ T ˆ T where Pµν is the transverse projection operator with Pij = δij − kikj/k and P00 = P0i = 0. The 57 last exponent in Eq. (3.77) becomes Z Z Z 4 s s1 h i 2 d k 0 ˜ 2ik·ps0 4ig 4 ds1 ds p · δDth(k) · p e (3.79) (2π) 0 0  h i  Z Z Z 4 s s1 2πi δ · Pˆ · δ d k µi ij jν 0 = 4ig2 ds ds0 p · δ(~k 2 − k2) · p  e2ik·ps 4 1 µ β|k | 0 ν (2π) 0 0 e 0 − 1 Z Z Z Z " # 2 s s1 2 2 2 g δ(~k − k ) (~p · ~k) ~ 0 0 = − ds ds0 d3~k dk 0 ~p 2 − e2ik·~ps −2ik0ωns . 3 1 0 β|k | 2π 0 0 e 0 − 1 ~k2 The δ-function can be split into two terms as h i ~ 2 2 1 ~ ~ δ(k − k0) = δ(|k| − k0) + δ(|k| + k0) , (3.80) 2|~k| and then the exponent becomes Z Z Z 4 s s1 h i 2 d k 0 ˜ 2ik·ps0 4ig 4 ds1 ds p · δDth(k) · p e (3.81) (2π) 0 0 Z Z Z " # 2 s s1 ~ 0 ~ 2 g 0 3~ cos(2|k|ωns ) 1 2 (~p · k) 2i~k·~ps0 = − 3 ds1 ds d k ~ ~p − e 2π 0 0 |~k| eβ|k| − 1 ~k2 Z Z Z Z 2 s s1 0 ~ £ ¤ 0 g 2 0 ~ 2 cos(2|k|ωns ) 2 2i|~k||~p| cos θs = − 3 ~p ds1 ds dk dΩ k ~ 1 − cos θ e . 2π 0 0 |~k|(eβ|k| − 1) Changing the dummy variable θ to ξ = cos θ, Z Z Z 4 s s1 h i 2 d k 0 ˜ 2ik·ps0 4ig 4 ds1 ds p · δDth(k) · p e (3.82) (2π) 0 0 Z Z Z Z 2 s s1 ∞ 1 0 ~ £ ¤ 0 g 2 0 ~ 2 cos(2|k|ωns ) 2 2i|~k||~p|ξs = − 2 ~p ds1 ds dk |k| dξ ~ 1 − ξ e , π 0 0 0 −1 |~k|(eβ|k| − 1) the integral over ξ can be reduced to

Z 1 Z 1 £ ¤ ~ 0 £ ¤ dξ 1 − ξ2 e2i|k||~p|ξs = 2 dξ 1 − ξ2 cos(2|~k||~p|ξs0), (3.83) −1 0 where sin(2|~k||~p|ξs0) is odd in ξ and has no contribution to the integral, and Z Z Z 4 s s1 h i 2 d k 0 ˜ 2ik·ps0 4ig 4 ds1 ds p · δDth(k) · p e (3.84) (2π) 0 0 Z Z Z Z 2 s s1 1 ∞ ~ 0 2g 2 0 2 ~ cos(2|k|ωns ) ~ 0 = − 2 ~p ds1 ds dξ (1 − ξ ) dk |k| ~ cos(2|k||~p|ξs ). π 0 0 0 0 eβ|k| − 1 Even though the ξ-integral could be evaluated directly as

Z 1 Z 1 2 2 2 4 2a − 4 dξ ξ cos(aξ) = 2 dξ ξ cos(aξ) = 2 cos a + 3 sin a, (3.85) −1 0 a a subsequent calculations will become messy. Instead, one could defer the evaluation of ξ-integral and re-express those two cosine functions as 1 1 cos(2kω s0) cos(2kpξs0) = cos(2k(pξ + ω )s0) + cos(2k(pξ − ω )s0). (3.86) n 2 n 2 n 58

Setting x = βk, the k-integral then becomes

Z ∞ 0 cos(2kωns ) 0 dk k βk cos(2kpξs ) (3.87) 0 e − 1 Z ∞ 1 k 0 0 = dk βk [cos(2k(pξ + ωn)s ) + cos(2k(pξ − ωn)s )] 2 0 e − 1 Z ∞ · ¸ 1 x pξ + ωn 0 pξ − ωn 0 = 2 dx x cos(2 xs ) + cos(2 xs ) . 2β 0 e − 1 β β Deﬁne pξ ± ω w(±) = n , (3.88) β and then Z Z Z 4 s s1 h i 2 d k 0 ˜ 2ik·ps0 4ig 4 ds1 ds p · δDth(k) · p e (3.89) (2π) 0 0 Z Z Z Z 2 s s1 1 ∞ h i g 2 0 2 x (+) 0 (−) 0 = − 2 2 ~p ds1 ds dξ (1 − ξ ) dx x cos(2w xs ) + cos(2w xs ) π β 0 0 0 0 e − 1 2 Z 1 Z 1 Z 1 2 g 2 2 = −s 2 2 ~p dv v du dξ (1 − ξ ) π β 0 0 0 Z ∞ h i x (+) (−) × dx x cos(2sw xuv) + cos(2sw xuv) , 0 e − 1

0 where s = s1u and s1 = sv. First work out integral over u, then the integral over v as Z Z Z 4 s s1 h i 2 d k 0 ˜ 2ik·ps0 4ig 4 ds1 ds p · δDth(k) · p e (3.90) (2π) 0 0 Z Z Z · ¸ g2 1 1 ∞ 1 sin(2sw(+)xv) sin(2sw(−)xv) = −s ~p 2 dv dξ (1 − ξ2) dx + 2 2 x (+) (−) 2π β 0 0 0 e − 1 w w Z Z · ¸ g2 1 ∞ 1 1 − cos(2sw(+)x) 1 − cos(2sw(−)x) = − ~p 2 dξ (1 − ξ2) dx + . 2 2 x (+) 2 (−) 2 4π β 0 0 x(e − 1) (w ) (w )

The thermal distribution [eβk −1]−1 sets an upper limit in the range of contribution of the k-integral, and in turn, the [ex − 1]−1 factor will limit contributions from the range of x in the integral with x = βk. Another approach is to set x = βk ≤ 1 as momenta of exchanged photons are equal to or less than that of heat bath as the thermal photons originated from the medium. In the s-integral, the 2 2 2 phase factor of [−is(m +~p −ωn)] will have a upper limit smax on s. When s > smax, the integrand (2n+1)π will oscillate dramatically, which will produce no signiﬁcant contribution. For ωn → −iβ and ³ ´2 2 (2n+1)π ωn → − β ,

1 1 1 1 smax ≤ 2 2 2 ≤ 2 2 → 2 2 ≤ 2 , (3.91) m + ~p − ωn ~p − ωn ~p + |ωn| ~p which is valid only for ﬁnite temperature. Then, the arguments of the cosine-functions become

(±) (±) pξ ± ωn p ± ωn x 2sw x ≤ 2smaxw x ≤ 2 2 2 2 x ≤ 2 2 2 x ≤ 2 . (3.92) β(m + ~p − ωn) β(~p + |ωn|) βp 59

In the regime of interest, p À T À m, the 2sw(±)x are small if 1 p x < βp = (3.93) 2 2T and 1 1 1 − cos(2sw(±)x) ' (2sw(±)x)2 − (2sw(±)x)4 + ··· . (3.94) 2! 4! Thus, Z Z Z 4 s s1 h i 2 d k 0 ˜ 2ik·ps0 4ig 4 ds1 ds p · δDth(k) · p e (3.95) (2π) 0 0 Z 1 Z βp · ¸ g2 2 1 1 (2sw(+)x)2 (2sw(−)x)2 ' − ~p 2 dξ (1 − ξ2) dx + 2 2 x (+) 2 (−) 2 4π β 0 0 x(e − 1) 2! (w ) (w ) Z Z βp 2 1 2 2 g 2 2 x = −s 2 2 ~p dξ (1 − ξ ) dx x . π β 0 0 e − 1

βp Observing that 2 À 1,

Z βp Z 2 ∞ 2 2 x x π aT = dx x ≤ dx x = , (3.96) 0 e − 1 0 e − 1 6 and the integral over ξ is trivial as Z 1 2 dξ (1 − ξ2) = . (3.97) 0 3 Thus, Z Z Z 4 s s1 h i 2 d k 0 ˜ 2ik·ps0 4ig 4 ds1 ds p · δDth(k) · p e (3.98) (2π) 0 0 Z βp 2 2 2 2 2 ~p x ' −s 2 g 2 dx x 3π β 0 e − 1 2a2 = −s2 T g2T 2 ~p 2 3π2 1 ' − s2 g2T 2 ~p 2, 9

2 2 where the approximation aT = π /6 is employed. The fully-dressed, ﬁnite-temperature fermion propagator in a heat bath then becomes

0 BN h~p,n|Sth|~y, y0iQ (3.99) £ ¤−1/2 ' (Z(0)[iτ])−1 · (2π)3τ · e−i(~p·~y−ωny0) Z ∞ ½ 2 ¾ 2 2 2 2 2aT 2 2 2 ×i ds exp −is(m + ~p − ωn) − s 2 g T ~p 0 3π ½ Z 4 h h i i Z s ¾ d k 0 × m − iγ · p + 2ig2 γ · D˜ (k) + δD˜ (k) · p ds0 e2ik·ps (2π)4 c th ½ Z Z Z ¾0 4 s s1 h i 2 d k 0 ˜ 2ik·ps0 × exp 4ig 4 ds1 ds p · Dc(k) · p e (2π) 0 0 60

2 2 2aT 2 2 2 Since the exponent {−s 3π2 g T ~p } is negative, the linkage with thermal photons contributes to the damping of the incident fermion in the medium. The modiﬁcation to the dressed fermion propagator from the linkage of thermal photons is proportional to (gT )2, which is similar to that of the perturbative approaches even though the assumptions are diﬀerent. Within the Coulomb 2 gauge, the s -factorh again appearsi in the decay exponent; more details are in Appendix G. The contribution of γ · δD˜ th(k) · p in the numerator of the propagator will contribute to the thermal mass-shift, which is irrelevant to the current problem, and will be absorbed into the physical mass m as

0 BN h~p,n|Sth|~y, y0iQ (3.100) £ ¤−1/2 ' (Z(0)[iτ])−1 · (2π)3τ · e−i(~p·~y−ωny0) Z ∞ ½ 2 ¾ 2 2 2 2 2aT 2 2 2 ×i ds exp −is(m + ~p − ωn) − s 2 g T ~p 0 3π ½ Z 4 h i Z s ¾ d k 0 × m − iγ · p + 2ig2 γ · D˜ (k) · p ds0 e2ik·ps (2π)4 c ½ Z Z Z 0 ¾ 4 s s1 h i 2 d k 0 ˜ 2ik·ps0 × exp 4ig 4 ds1 ds p · Dc(k) · p e . (2π) 0 0

Let ω2 = m2 + ~p 2 and replace s2 by the operator µ ¶ ∂ 2 s2 → i (3.101) ∂ω2

2 2 2aT 2 2 2 in the decay exponent of exp [−s 3π2 g T ~p ]; then the dressed propagator reduces to

˜0 Sth(~p,n) (3.102) Z ∞ ½ 2 ¾ (0) −1 2 2 2 2aT 2 2 2 ' (Z [iτ]) · i ds exp −is(ω − ωn) − s 2 g T ~p 0 3π ½ Z 4 h i Z s ¾ d k 0 × m − iγ · p + 2ig2 γ · D˜ (k) · p ds0 e2ik·ps (2π)4 c ½ Z Z Z 0 ¾ 4 s s1 h i d k 0 × exp 4ig2 ds ds0 p · D˜ (k) · p e2ik·ps (2π)4 1 c ( 0 0) 2 µ ¶2 Z ∞ 2aT 2 2 2 ∂ (0) −1 © 2 2 ª = exp − 2 g T ~p · i 2 · (Z [iτ]) · i ds exp −is(ω − ωn) 3π ∂ω 0 ½ Z 4 h i Z s ¾ d k 0 × m − iγ · p + 2ig2 γ · D˜ (k) · p ds0 e2ik·ps (2π)4 c ½ Z Z Z 0 ¾ 4 s s1 h i 2 d k 0 ˜ 2ik·ps0 × exp 4ig 4 ds1 ds p · Dc(k) · p e . (2π) 0 0 Aside from the leading exponential, if the remaining is considered as a non-interactive, but renormalized, fermion propagator, the leading exponential operator represents the energy decay from the contributions of thermal photons of the thermal bath, and more will be discussed in the later sections. 61

3.5.4 Bremsstrahlung Processes as a Damping Mechanism

In the zero-temperature theory, contributions from the last exponential factor will be absorbed into the mass- and/or wave-function re-normalization under the quenched approximation [48, 49]. However, the (thermal) Green’s function in the ﬁnite temperature theory involves fermions in the medium, and is anti-periodic in its ’time’-coordinate, or as a function of Matsubara frequency, ωn. Continuing to work in the Coulomb gauge, the last exponent of Eq. (3.100) becomes Z Z Z 4 s s1 h i 2 d k 0 ˜ 2ik·ps0 4ig 4 ds1 ds p · Dc(k) · p e (3.103) (2π) 0 0 Z Z Z " Ã ! # 4 s s1 ˆ T 2 d k 0 δµ0δ0ν Pµν 2ik·ps0 = 4ig ds1 ds pµ · − + · pν e (2π)4 ~ 2 ~ 2 2 0 0 k k − k0 − i² Z Z Z ( " # ) 4 s s1 2 ~ 2 2 d k 0 ωn 2 (k · ~p) 1 2ik·ps0 = 4ig ds1 ds − + ~p − · e . (2π)4 ~ 2 ~ 2 ~ 2 2 0 0 k k k − k0 − i² Inside the heat bath, the momentum transfer among fermions in the medium is on the order of its temperature T (or kBT ) in each collision. The incident fermion interacts with fermions in the medium through exchanges of virtual photons, and the momentum transfer during the collision can not exceed to its momentum. Thus, we could set the upper limit to |~p| in the ~k-integral, or add a BN-type limiting factor, exp [−~k2/~p 2], in the photon propagator to set the scale of soft photon contributions as Z Z Z 4 s s1 h i d k 0 4ig2 ds ds0 p · D˜ BN(k) · p e2ik·ps (3.104) (2π)4 1 c 0 0   2 Z Z Z Z 2 (~k·~p) s s1 3~ ∞  2  d k 0 2 2 dk ω ~p − 2 0 2 0 2is ~k·~p−~k /~p 0 n ~k −2is k0 ωn = 4ig ds1 ds e − + e , (2π)3 2π  ~ 2 ~ 2 2  0 0 −∞ k k − k0 − i² where the superscript ’BN’ denotes the photon propagator with the Bloch-Nordsieck limit factor. In the usual Coulomb gauge, ∇ · A~ = 0, where the (0, 0)-component of the photon propagator is zero, i.e., D˜ 00(k) = 0. It is not a surprise that there is no contribution from the ﬁrst term in the curly 0 brackets as s 6= 0 and ωn 6= 0 for fermions, i.e.,

2 Z ∞ 2 ω dk 0 ω n 0 −2is k0 ωn n 0 e = δ(2s ωn) = 0, (3.105) ~k2 −∞ 2π ~k2 and the second term in the bracket involves

Z 0 Z 0 ∞ dk e−2is k0 ωn ∞ dk e−2is k0 ωn 0 = − 0 , (3.106) 2π ~ 2 2 2π (k − k + i²0)(k + k − i²0) −∞ k − k0 − i² −∞ 0 0 0 0 where there are two poles at k0 = k − i² and k0 = −k + i² . Converting this into a contour integral, and employing the residue theorem, one obtains

Z ∞ −2is0k ω h i dk e 0 n i 0 0 0 −2is k ωn +2is k ωn = θ(ωn) e + θ(−ωn) e (3.107) 2π ~ 2 2 2k −∞ k − k0 − i² i = [cos(2s0k ω ) + i²(ω ) sin(2s0k ω )] , 2k n n n 62

where ²(ωn) = θ(ωn) − θ(−ωn) and ωn 6= 0 for fermions. Therefore, Z Z Z 4 s s1 h i 2 d k 0 ˜ BN 2ik·ps0 4ig 4 ds1 ds p · Dc (k) · p e (3.108) (2π) 0 0 Z Z Z " # s s1 3~ ~ 2 2 0 d k 1 2is0~k·~p−~k2/~p 2 2 (k · ~p) = −2g ds1 ds 3 e · ~p − 0 0 (2π) k ~k2 0 0 × [cos(2s k ωn) + i²(ωn) sin(2s k ωn)] .

The exponent is complex and the ~k-integral can be reduced to Z " # 3~ |~k| ~ 2 d k 1 2is0~k·~p− 2 (k · ~p) 0 0 e |~p| · ~p − [cos(2s k ωn) + i²(ωn) sin(2s k ωn)] (3.109) (2π)3 k ~k2 Z ∞ Z 1 1 2 0 0 −~k2/~p 2 2 2is0kpξ = 2 ~p dk k [cos(2s k ωn) + i²(ωn) sin(2s k ωn)] e dξ [1 − ξ ] e , (2π) 0 −1 where ξ = cos θ, k = |~k| and p = |~p|. Similar to the previous estimate in Eq. (3.83), only the symmetric parts contribute to the ξ-integral, which is real. For the damping process, one is only interested in the real part of the exponent, as ½ Z Z Z ¾ 4 s s1 h i 2 d k 0 ˜ BN 2ik·ps0 Re 4ig 4 ds1 ds p · Dc (k) · p e (3.110) (2π) 0 0 Z s Z s1 Z 1 Z ∞ 4 2 2 0 2 0 0 −~k2/~p 2 = − 2 g ~p ds1 ds dξ [1 − ξ ] dk k cos(2s k ωn) cos(2s kp ξ) e (2π) 0 0 0 0 Z 1 Z 1 Z 1 4 2 2 2 2 2 = − 2 s g (~p ) dv · v · du dξ · [1 − ξ ] (2π) 0 0 0 Z ∞ 2 −x2 × dx · x · cos(2sp ωnxuv) · cos(2s~p ξxuv) · e , 0

0 where substitutions of k = xp, s = s1u and s1 = sv have been used. Converting the two cosine functions,

2 cos(2s~p ξxuv) · cos(2sp ωnxuv) (3.111) 1 = [cos(2sp(p ξ + ω )xuv) + cos(2sp(p ξ − ω )xuv)] , 2 n n and carrying out both u- and v-integrals ﬁrst,

Z 1 Z 1 1 − cos(2sp(p ξ ± ωn)x) 1 dv · v · du · cos(2sp(p ξ ± ωn)xuv) = 2 ≤ , (3.112) 0 0 (2sp(p ξ ± ωn)x) 2 where the last inequity utilizes the similar argument of s < smax in previous Section, so that the arguments of the cosine-functions are small, i.e.,

2p(p ξ ± ωn)x 2p(p ξ ± ωn)x 2sp(p ξ ± ωn)x ≤ 2 2 2 < 2 2 ≤ 2x (3.113) m + ~p − ωn ~p − ωn and x < 1/2, i.e., the momentum transfer cannot exceed half of what the incident fermion carries. 63

Then, the real part of the exponent becomes ½ Z Z Z ¾ 4 s s1 h i 2 d k 0 ˜ BN 2ik·ps0 Re 4ig 4 ds1 ds p · Dc (k) · p e (3.114) (2π) 0 0 Z 1 2 2 2 2 2 2 = − 2 s g (~p ) dξ [1 − ξ ] (2π) 0 Z ∞ · ¸ 1 − cos(2sp(p ξ + ωn)x) 1 − cos(2sp(p ξ − ωn)x) −x2 × dx x 2 + 2 e 0 (2sp(p ξ + ωn)x) (2sp(p ξ − ωn)x) Z 1 Z ∞ 2 2 2 2 2 2 −x2 ' − 2 s g (~p ) dξ [1 − ξ ] dx x e . (2π) 0 0 The x-integral is trivial as Z ∞ 2 1 dx x e−x = , (3.115) 0 2 as is the ξ-integral, Eq. (3.97). One arrives at ½ Z Z Z ¾ 4 s s1 h i 2 d k 0 ˜ BN 2ik·ps0 Re 4ig 4 ds1 ds p · Dc (k) · p e (3.116) (2π) 0 0 1 ' − s2g2(~p 2)2, 6π2 and the real part of the exponent has a similar s2-factor to that of the linkage with the thermal photons. Compared to the eﬀect from the thermal photons with a T 2 ~p 2-dependence, the contribution from the Bremsstrahlung process is more prominent, with a (~p 2)2-dependence. The UV divergence usually appearing in the zero-temperature theory has been suppressed by the Bloch-Nordsieck factor exp [−~k2/~p 2]. Instead of the Bloch-Nordsieck limiting factor, if the upper cut-oﬀ of the ~k-integral is set to |~p|, the last exponent in Eq. (3.100) becomes Z Z Z 4 s s1 h i 2 d k 0 ˜ 2ik·ps0 4ig 4 ds1 ds p · Dc(k) · p e (3.117) (2π) 0 0 Z Z Z " Ã ! # 4 s s1 ˆ T 2 d k 0 δµ0δ0ν Pµν 2ik·ps0 ' 4ig ds1 ds pµ · − + · pν e . ~ (2π)4 ~ 2 ~ 2 2 |k|≤|~p| 0 0 k k − k0 − i²

2 ~ 2 Similarly there is no contribution from the ﬁrst term with −ωn/k inside the curly brackets, and the exponent becomes Z Z Z 4 s s1 h i 2 d k 0 ˜ 2ik·ps0 4ig 4 ds1 ds p · Dc(k) · p e (3.118) (2π) 0 0 Z Z Z " # s s1 3~ ~ 2 2 0 d k 1 2 (k · ~p) 2is0~k·~p ' −2g ds1 ds 3 ~p − e 0 0 |~k|≤|~p| (2π) |~k| ~k2 0 0 × [cos(2s k ωn) + i²(ωn) sin(2s k ωn)] . 64

The real part of the exponent becomes ½ Z Z Z ¾ 4 s s1 h i 2 d k 0 ˜ 2ik·ps0 Re 4ig 4 ds1 ds p · Dc(k) · p e (3.119) (2π) 0 0

Z s Z s1 Z 1 Z |~p| 4 2 2 0 2 0 0 = − 2 g ~p ds1 ds dξ [1 − ξ ] dk k cos(2s k ωn) cos(2s kpξ) (2π) 0 0 0 0 Z 1 Z 1 Z 1 4 2 2 2 2 2 = − 2 s g (~p ) dv · v · du dξ · [1 − ξ ] (2π) 0 0 0 Z 1 2 × dx · x · cos(2sp ωnxuv) · cos(2s~p ξxuv), 0

0 where ξ = cos θ, k = xp, s = s1u, and s1 = sv. The u- and v-integrals then yield

Z 1 Z 1 1 − cos(2sp(p ξ ± ωn)x) dv · v · du · cos(2sp(p ξ ± ωn)xuv) = 2 . (3.120) 0 0 (2sp(p ξ ± ωn)x)

± Define a = 2sp(p ξ ± ωn), and the x-integral can be directly evaluated as Z 1 1 − cos(a±x) X∞ (a±)2r−2 dx x = (−1)r+1 = a2 , (3.121) (a±x)2 2r (2r)! B 0 r=1 and when a± ≤ 2, 1 a2 ≤ . (3.122) B 4 Then, one arrives at ½ Z Z Z ¾ 4 s s1 h i 2 d k 0 ˜ 2ik·ps0 Re 4ig 4 ds1 ds p · Dc(k) · p e (3.123) (2π) 0 0 2a2 ' − B s2g2(~p 2)2, 3π2 and the real part of the exponent also contains an s2-factor. Even though a cut-off is used in the ~k-integral as the zero-temperature theory, the cut-off is to set the energy scale of soft-photon exchanges, instead of just avoiding the UV divergence in the conventional theory. If the limiting factor is replaced by exp [−|~k|/|~p|] instead, Eq. (3.115) becomes

Z ∞ dx x e−x = 1, (3.124) 0 which leads to a factor of 2 larger compared to exp [−~k2/~p 2]. If the ~k-integral has a cut-oﬀ |~p| instead of the Bloch-Nordsieck limit factor, the x-integral in Eq. (3.115) becomes Z 1 1 dx x = , (3.125) 0 2 which is the same as Eq. (3.115). 65

To estimate the imaginary part of the last exponent in Eq. (3.100), ½ Z Z Z ¾ 4 s s1 h i 2 d k 0 ˜ 2ik·ps0 Im 4ig 4 ds1 ds p · Dc(k) · p e (3.126) (2π) 0 0

Z s Z s1 Z 1 Z |~p| 4 2 2 0 2 0 0 = − 2 ²(ωn) g ~p ds1 ds dξ [1 − ξ ] dk k cos(2s kp ξ) sin(2s k ωn) (2π) 0 0 0 0 Z 1 Z 1 Z 1 4 2 2 2 2 2 = − 2 ²(ωn) s g (~p ) dv · v · du dξ · [1 − ξ ] (2π) 0 0 0 Z 1 2 × dx x cos(2s~p ξxuv) sin(2sp ωnxuv), 0 which involves a sine-function, and

2 cos(2s~p ξxuv) · sin(2sp ωnxuv) (3.127) 1 = [sin(2sp(p ξ + ω )xuv) − sin(2sp(p ξ − ω )xuv)] . 2 n n Working out the u- and v-integrals,

Z 1 Z 1 1 sin(2sp(p ξ ± ωn)x) dv · v · du · sin(2sp(p ξ ± ωn)xuv) = − 2 , (3.128) 0 0 2sp(p ξ ± ωn)x (2sp(pξ ± ωn)x) which will vanish in the first approximation only if the arguments, 2sp(p ξ ± ωn)x, of the sine ± functions are small. Then, let a = 2sp(p ξ ± ωn), the x-integral in Eq. (3.126) is Z · ¸ 1 1 sin(a±x) 1 X∞ (a±)2r+1 dx x − = − (−1)r , (3.129) a±x (a±x)2 a± (2r + 1) (2r = 1)! 0 r=0 and one can expects that the two summations of a± will cancel each other. Therefore, the imaginary part is negligibly small. The quenched, dressed, finite-temperature fermion propagator with combined effects becomes

˜0 Sth(~p,n) (3.130) Z ∞ ½ 2 ¾ −1 2 2 2 2aT 2 2 2 ' (Z(0)[iτ]) · i ds exp −is(ω − ωn) − s 2 g T ~p 0 3π ½ Z 4 h i Z s ¾ d k 0 × m − iγ · p + 2ig2 γ · D˜ (k) · p ds0 e2ik·ps (2π)4 c ½ Z Z Z 0 ¾ 4 s s1 h i 2 d k 0 ˜ 2ik·ps0 × exp 4ig 4 ds1 ds p · Dc(k) · p e (2π) 0 0 Z ∞ ½ · 2 2 ¸¾ −1 2 2 2 2aT 2 2 2 2aB 2 2 2 ' (Z(0)[iτ]) · i ds exp −is(ω − ωn) − s 2 g T ~p + 2 g (~p ) 0 3π 3π ½ Z 4 h i Z s ¾ d k 0 × m − iγ · p + 2ig2 γ · D˜ (k) · p ds0 e2ik·ps (2π)4 c ½ µ Z Z Z 0 ¶¾ 4 s s1 h i 2 d k 0 ˜ 2ik·ps0 × exp i Im 4g 4 ds1 ds p · Dc(k) · p e , (2π) 0 0

2 2 2 where aT = π /6 and aB = 1/4. While the third term in the numerator can be absorbed into the mass renormalization, the last exponent is pure imaginary and can be treated as the wave-function 66 renormalization constant. The propagator then becomes

˜0 Sth(~p,n) (3.131) Z ∞ −1 ' (Z(0)[iτ]) · i ds {m − iγ · p} ½ 0 · ¸¾ 2a2 2a2 × exp −is(ω2 − ω2 ) − s2 T g2T 2 ~p 2 + B g2(~p 2)2 n 3π2 3π2 ( µ ¶ ) 2 ∂ 2 ' exp − g2 ~p 2 [a2 T 2 + a2 ~p 2] · i 3π2 T B ∂ω2 Z ∞ −1 © 2 2 ª ·(Z(0)[iτ]) · i ds {m − iγ · p} exp −is(ω − ωn) , 0 where the replacement of the s2-factor by an operator is employed in the last line (cf. Eqs. (3.101) and (3.102)). There is no infrared (IR) divergence which plagued the perturbative calculations of the fermion damping. While the perturbative approach required re-summation of large numbers of HTL-approximated, multi-loop Feynman graphs to get rid of infrared divergences [81, 45, 46], the functional method with linkage operations automatically includes all relevant terms of the calculation. Another important point is that the momentum of the fermion is kept oﬀ mass-shell until the Matsubara summation is carried out. Otherwise, the exponent will display the wrong sign as p2 ⇒ −m2, as seen in earlier attempts summarized in Ref. [81].

3.5.5 Damping Eﬀects under Quenched Approximation

Let the contributions inside the numerator factor of the propagator and the last exponential be absorbed into mass and wave-function renormalization by replacing m and g with mR and gR, respectively as in the T = 0 theory, ( µ ¶ ) 2g2 ∂ 2 S˜0 (~p,n) ' exp − ~p 2 [a2 T 2 + a2 ~p 2] · i · S˜ (~p,n; m , g ), (3.132) th 3π2 T B ∂ω2 th R R

2 2 2 ˜ where aT = π /6, aB = 1/4, and Sth is the ’free’, but renormalized, ﬁnite-temperature fermion propagator. In order to make the continuation from imaginary time to real time, one will perform the Matsubara sum over ωn as 1 X h~p,x |S0 |~p,y i = e−iωn(x0−y0) S˜0 (~p,n) (3.133) 0 th 0 τ th n ( µ ¶ ) 1 X 2g2 ∂ 2 ' e−iωn(x0−y0) exp − ~p 2[a2 T 2 + a2 ~p 2] · i · S˜ (~p,n; m ) τ 3π2 T B ∂ω2 th R n ( µ ¶ ) 2g2 ∂ 2 1 X = exp − ~p 2 [a2 T 2 + a2 ~p 2] · i · e−iωn(x0−y0) S˜ (~p,n; m ) 3π2 T B ∂ω2 τ th R n ( µ ¶ ) 2g2 ∂ 2 = exp − ~p 2 [a2 T 2 + a2 ~p 2] · i · S˜ (~p,x − y ), 3π2 T B ∂ω2 th 0 0 67

where the renormalized mass is implicitly implied in the last line, and S˜th(~p,x0 − y0) is the free, ﬁnite-temperature fermion propagator in the mixed representation, · µ ¶¸ ∂ i © ª −iωz0 +iωz0 S˜th(~p,z0) = m − i~γ · ~p + iγ0 · i [1 − n˜(ω)]e − n˜(ω)e , (3.134) ∂z0 2ω andn ˜(ω) is the Fermi-Dirac distribution function (without the chemical potential) as 1 n˜(ω) = . (3.135) eβω + 1

Let z0 = x0 − y0,

0 h~p,x0|Sth|~p,y0i (3.136) ( µ ¶ ) 2g2 ∂ 2 = exp − ~p 2 [a2 T 2 + a2 ~p 2] · i 3π2 T B ∂ω2 · µ ¶¸ ∂ i © ª · m − i~γ · ~p + iγ · i [1 − n˜(ω)]e−iωz0 − n˜(ω)e+iωz0 0 ∂z 2ω 0 ( ) · µ ¶¸ 2 µ ¶2 i ∂ 2g 2 2 2 2 2 ∂ = m − i~γ · ~p + iγ0 · i · exp − 2 ~p [aT T + aB ~p ] · i 2 2 ∂z0 3π ∂ω n √ √ o 1 √ 2 √ 2 ·√ [1 − n˜( ω2)]e−i ω z0 − n˜( ω2)e+i ω z0 , ω2

√ 2 2 2 2g 2 2 2 2 2 2 2 where ω is expressed as ω in the last line. Let b = 3π2 ~p [aT T + aB ~p ] with aT = π /6, 2 2 aB = 1/4, and u = ω ; then, the Gaussian-type translation operator can be re-written as ( µ ¶ ) 2g2 ∂ 2 exp − ~p 2 [a2 T 2 + a2 ~p 2] · i (3.137) 3π2 T B ∂ω2 ( µ ¶ ) ∂ 2 = exp +(ib)2 · i ∂u Z +∞ 1 −v2+2v(ib)(i ∂ ) = √ dv e ∂u π −∞ Z +∞ 1 −v2−2vb ∂ = √ dv e ∂u , π −∞ where the selection of sign, + or −, on the linear term in v in the exponent is arbitrary and will not aﬀect the result of v-integral. The operator exp [−2vb (∂/∂u)] acts as the translation operator with 68 u → u − 2vb; and then

0 h~p,x0|Sth|~p,y0i (3.138) · µ ¶¸ Z +∞ ∂ i −v2−2vb ∂ = m − i~γ · ~p + iγ · i · √ dv e ∂u 0 ∂z 2 π 0 −∞ ¯ 1 n √ √ √ √ o¯ −i uz0 +i uz0 ¯ · √ [1 − n˜( u)]e − n˜( u)e ¯ u u=ω2 · µ ¶¸ Z +∞ i ∂ 2 = √ m − i~γ · ~p + iγ · i · dv e−v 2 π 0 ∂z 0 −∞ ¯ 1 n √ √ √ √ o¯ −i u−2vbz0 +i u−2vbz0 ¯ √ [1 − n˜( u − 2vb)]e − n˜( u − 2vb)e ¯ u − 2vb u=ω2 · µ ¶¸ Z +∞ i ∂ −v2 1 = √ m − i~γ · ~p + iγ0 · i · dv e p 2 2 π ∂z0 −∞ ω 1 − 2vb/ω n √ √ o p 2 p 2 [1 − n˜(ω 1 − 2vb/ω2)]e−i 1−2vb/ω ωz0 − n˜(ω 1 − 2vb/ω2)e+i 1−2vb/ω ωz0 . where u = ω2 and ω > 0 are used in the last line. When only the eﬀect of thermal-photon-assisted Bremsstrahlung is considered, r 2vb 2a2 gT p 2 gT gT = 2 T v < v < v , (3.139) ω2 3π2 ω2 3 ω ω p 2 2 ω ω 2vb gT where 2aT /3π = 1/3 < 1. When v < 2gT and 2gT À 1 as p À T > gT , one has ω2 < v ω < 1 and r µ ¶ √ 2vb vb vb u − 2vb = ω 1 − ' ω 1 − = ω − . (3.140) ω2 ω2 ω On the other hand, when v > ω/2gT , the factor exp [−v2] will limit the contribution to the integral. When both contributions to the Bremsstrahlung eﬀects are considered, r q 2vb 2v 2 = gp a2 T 2 + a2 ~p 2 (3.141) ω2 ω2 3π2 T B r r v 8 gp 2π2 ' T 2 + ~p 2 ω2 3π2 2 3 r 8 ~p 2 ≤ gv 3π2 ω2 ~p 2 < gv ω2 < gv.

Similarly, the integrand with the limited factor of exp [−v2] will contribute only when v < 1/g. Thus, one could replace r µ ¶ 2vb vb vb ω 1 − ' ω 1 − = ω − , (3.142) ω2 ω2 ω 69 and

h~p,x |S0 |~p,y i (3.143) 0 · th 0 µ ¶¸ i ∂ ' √ m − i~γ · ~p + iγ0 · i 2 π ∂z0 Z +∞ ½ ¾ 2 1 vb vb vb vb −v −i(ω− ω )z0 +i(ω− ω )z0 · dv e vb [1 − n˜(ω − )]e − n˜(ω − )e −∞ ω − ω ω · ω µ ¶¸ i ∂ ' √ m − i~γ · ~p + iγ0 · i 2 π ∂z0 Z +∞ n o −v2 1 + β vb −iωz +i vb z + β vb +iωz −i vb z · dv e [1 − n˜(ω)e 2 ω ]e 0 ω 0 − n˜(ω)e 2 ω e 0 ω 0 , −∞ ω where the approximation

− β (ω− vb ) vb 1 e 2 ω 1 + β vb n˜(ω − ) = = ' e 2 ω (3.144) β(ω− vb ) β (ω− vb ) − β (ω− vb ) βω ω e ω + 1 e 2 ω + e 2 ω e + 1 for the Fermi-Dirac distribution function is used in the last line. The two v-integrals can be carried out readily as Z +∞ 2 1 −v2±i vb z − b z2 √ dv e ω 0 = e 4ω2 0 , (3.145) π −∞ and ³ ´ Z +∞ 2 2 2 vb β b2 β 2 b 2 β 1 −v + ( ±iz ) ( ±iz ) − 2 z0 ∓iβz0− √ dv e ω 2 0 = e 4ω2 2 0 = e 4ω 4 . (3.146) π −∞ If a diﬀerent sign is chosen in Eq.(3.137) instead, the results are the same, as

³ ´ Z +∞ 2 2 2 vb β b2 β 2 b 2 β 1 −v − ( ±iz ) ( ±iz ) − 2 z0 ∓iβz0− √ dv e ω 2 0 = e 4ω2 2 0 = e 4ω 4 . (3.147) π −∞ The dressed, Finite-Temperature fermion propagator reduces to

h~p,x |S0 |~p,y i (3.148) · 0 th 0 µ ¶¸ i ∂ ' m − i~γ · ~p + iγ · i 2 0 ∂z ½ 0 ¾ 2 2 h 2 2 i 2 1 −iωz + β b −i[ω−β b ]z +i[ω−β b ]z − b z2 · e 0 − n˜(ω)e 4 4ω2 e 4ω2 0 + e 4ω2 0 e 4ω2 0 . ω

2 2 2aT 2 2 2 If only the exchange of eﬀective thermal photons of the heat bath is considered, b = 3π2 g T ~p , one has b2 a2 T 2~p 2 = T g2 , (3.149) 4ω2 6π2 ω2 70 and

h~p,x |S0 |~p,y i (3.150) ·0 th 0 µ ¶¸ a2 2 i ∂ −g2T 2 T ~p z2 ' m − i~γ · ~p + iγ · i · e 6π2 ω2 0 2ω 0 ∂z ½ 0 · ¸¾ a2 2 a2 2 a2 2 −iωz +g2 T ~p −i(ω−g2T T ~p )z +i(ω−g2T T ~p )z · e 0 − n˜(ω) e 24π2 ω2 e 6π2 ω2 0 + e 6π2 ω2 0 · µ ¶¸ 2 i ∂ − 1 g2T 2 ~p z2 ' m − i~γ · ~p + iγ · i · e 36 ω2 0 2ω 0 ∂z ½ 0 · ¸¾ 2 2 2 −iωz + 1 g2 ~p −i(ω− 1 g2T ~p )z +i(ω− 1 g2T ~p )z · e 0 − n˜(ω) e 144 ω2 e 36 ω2 0 + e 36 ω2 0 .

Similarly, when both the thermal-photon assisted and ordinary Bremsstrahlung processes are in- 2 2g2 2 2 2 2 2 cluded with b = 3π2 ~p [aT T + aB ~p ], b2 g2 a2 (~p 2)2 + a2 T 2~p 2 = B T , (3.151) 4ω2 6π2 ω2 then

h~p,x |S0 |~p,y i (3.152) ·0 th 0 µ ¶¸ 2 a2 (~p 2)2+a2 T 2 ~p 2 i ∂ − g B T z2 ' m − i~γ · ~p + iγ · i · e 6π2 ω2 0 2ω 0 ∂z ½ 0 2 a2 (~p2) 2+a2 T 2~p 2 −iωz + g B T · e 0 − n˜(ω) e 24π2 ω2T 2 · ¸¾ 2 a2 (~p 2)2+a2 T 2~p 2 2 a2 (~p 2)2+a2 T 2 ~p 2 −i(ω− g T B T )z +i(ω− g T B T )z × e 6π2 ω2 0 + e 6π2 ω2 0

This nice result has been obtained from the following assumptions: First, the quenched approximation was used by dropping cross-linkage with the close-fermion-loop functional; Second, the incident fermion is relativistic with momentum much larger than the temperature scale of the medium, i.e., p À T . Subsequently, the Bloch-Nordsieck approximation is used along with the assumption of large fermion momentum. While a limiting factor of the type exp [−a2 ~k2] or a cut-off |~k| ≤ |~p| in the ~k-integral is used in the evaluation of energy-momentum exchange in the ordinary Bremsstrahlung process, it is implicitly applied in the case of the thermal photon with the Fermi-Dirac distribution. The thermal distribution defines a preferred frame, in which the Coulomb gauge is employed to make calculations much simpler, compared to those of covariant gauges. Based on the free, finite- temperature fermion propagator, S˜th in Eq. (3.134), there are two distinct parts, the incoming and thermal parts, given by · µ ¶¸ ∂ i © £ ¤ª −iωz0 −iωz0 +iωz0 S˜th(~p,z0) = m − i~γ · ~p + iγ0 · i e − n˜(ω) e + e . (3.153) ∂z0 2ω £ ¤ The second part with e−iωz0 + e+iωz0 in the square brackets represents the induced disturbance inside the thermal bath. ˜0 In the fully-dressed, finite-temperature propagator,h Sth,i the dampingh factor from the thermal-i 2 2 2 2 2 2 2 2 2 2 2 aT g T ~p 2 g aB (~p ) +aT T ~p 2 photon-assisted Bremsstrahlung alone is exp − 6π2 ω2 z0 , or exp − 6π2 ω2 z0 with 71 both effects, which acts on both the incident fermion and the disturbance in the thermal bath. As the energy is depleting, both decay factors are competing with divergent factor of 1/ω with decreasing ω due to energy and/or momentum loss inside the heat bath. Initially, the effect from 1/ω factor dominates at small z0. As the process proceeds, the exponential decay factor will win out, and the disturbance eventually die down, as in the case of the scalar theory from the early work in Ref. [47]. The decay behavior can also be seen in a simple model with Doppler effects, as shown in Ref. [87].

3.6 Non-Quenched Full Finite-Temperature Propagator

3.6.1 Thermal Closed-Fermion-Loop and the Photon Polarization Tensor

Up to the order of g2, the closed-fermion-loop functional could be approximated as Z Z i L [A] = dx dy A (x) · Kth (x − y) · A (y), (3.154) th 2 µ µν ν or Z i L [A] = dq A˜ (q) · K˜ th (q) · A˜ (q) (3.155) th 2 µ µν ν (th) where K˜ µν is the thermal counterpart of the vacuum photon polarization tensor K˜ µν . Hence, the fully-dressed, ﬁnite-temperature photon propagator is given by

0 −1 Dth = Dth · [1 − Kth · Dth] , (3.156) or ˜ 0 −1 ˜ −1 ˜ Dth(k) = Dth(k) − Kth(k). (3.157)

th Similar to the vacuum photon polarization tensor, Kµν is gauge invariant, and the requirement of the current conservation leads to ˜ th qµ Kµν (q) = 0, (3.158) as the T = 0, vacuum theory (cf. Eq. (C.35) in Appendix C.3). Since the Lorentz covariance is broken due to the preferred frame of the medium, the thermal photon polarization tensor in general has two components as [20, 56] h i ˜ th 2 ˆ T ˜ T 2 ˆ L ˜ L 2 Kµν (q) = −q Pµν Πth(q ) + Pµν Πth(q ) , (3.159)

ˆ T ˆ L where Pµν and Pµν are the transverse and longitudinal projection operators, respectively, defined ˜ T 2 in Appendix C.2 and Sec. 2.5.3. In general, the transverse and longitudinal components, Πth(q ) ˜ L 2 and Πth(q ), are temperature-dependent and different due to broken Lorentz covariance. At zero- ˜ T 2 ˜ L 2 ˜ 2 temperature, the two components are equal, i.e., Πc (q ) = Πc (q ) = Π(q ) as the Lorentz covariance restored [88, 20]. In the zero-temperature theory, the vacuum photon polarization tensor can be written in the form of [48] · ¸ q q K˜ (q) = −q2 Pˆ Π˜ (q2) = −q2 δ − µ ν Π˜ (q2), (3.160) µν µν µν q2 72 where 2 Z 1 ·Z ∞ ¸ ˜ 2 g du −iu[m2+y(1−y)q2] 1 Π(q ) = + 2 dy y(1 − y) e + , (3.161) 2π 0 0 u 3 which is divergent near the mass-shell q2 ∼ 0. One can extract the finite part of Π˜ (q2) by

2 2 Π˜ R(q ) = Π˜ (q ) − Π˜ (0) (3.162)

2 and Π˜ R(0) = 0 by definition. The divergent part of Π˜ R(q ) can be related to the photon’s wave- ˜ (2) −1 function renormalization constant Z3 as Π(0) = (Z3 ) − 1, and Z3 can be absorbed in the re-definition of the physical (renormalized) coupling constant (or charge in QED). The difference Π˜ (q2) − Π˜ (0) can be readily worked out as [48]

2 Z 1 µ 2 2 ¶ ˜ 2 g m + y(1 − y)q ΠR(q ) = − 2 dy y(1 − y) ln 2 , (3.163) 2π 0 m

2 2 and Π˜ R(q ) is complex in general depending on the value of q . Observe that 1 1 1 1 1 y(1 − y) = −[y2 − y + ] + = −(y − )2 + < , (3.164) 4 4 2 4 4 and y is within the range of 0 < y < 1 in the integral, i.e., y(1 − y) > 0, 1 0 < y(1 − y) < . (3.165) 4 Thus, q2 1 m2 + y(1 − y)q2 < m2 + = [4m2 + ~q 2 − q2]. (3.166) 4 4 0 2 2 2 2 m2 When q0 < ~q + 4m < ~q + y(1−y) ,

m2 + y(1 − y)q2 > 0, (3.167)

2 and Π˜ R(q ) and K˜ µν (q) are real. 2 2 m2 2 2 On the other hand, when q0 > ~q + y(1−y) > ~q + (2m) , 1 m2 + y(1 − y)q2 < [4m2 + ~q 2 − q2] < 0. (3.168) 4 0

2 Then Π˜ R(q ) and K˜ µν (q) become complex. Recall that

ln (−1) = (2n + 1)πi, n = 0, ±1, ±2, ··· , (3.169) and the principle value of ln(−1) is πi at n = 0, then ¯ ¯ · ¸ m2 + y(1 − y)q2 ¯m2 + y(1 − y)q2 ¯ m2 m2 = ¯ ¯ θ([~q 2 + ] − q2) − θ(q2 − [~q 2 + ]) , m2 ¯ m2 ¯ y(1 − y) 0 0 y(1 − y) (3.170) and µ ¶ ¯ ¯ m2 + y(1 − y)q2 ¯m2 + y(1 − y)q2 ¯ m2 ln = ln ¯ ¯ + πi θ(q2 − [~q 2 + ]), (3.171) m2 ¯ m2 ¯ 0 y(1 − y) 73 which leads to n o n o 2 2 2 Π˜ R(q ) = Re Π˜ R(q ) + i Im Π˜ R(q ) (3.172) with Z ¯ ¯ n o 2 1 ¯ 2 2 ¯ ˜ 2 g ¯m + y(1 − y)q ¯ Re ΠR(q ) = − 2 dy y(1 − y) ln ¯ 2 ¯, (3.173) 2π 0 m and n o 2 Z 1 2 ˜ 2 g 2 2 m i Im ΠR(q ) = −πi 2 dy y(1 − y) θ(q0 − [~q + ]). (3.174) 2π 0 y(1 − y) 3.6.2 Pair-Productions as a Damping Mechanism

Keeping the closed-fermion-loop functional in the thermal propagator ¯ ½ L [A] ¾ (th) e th ¯ 0 DA ¯ h~p,n|Sth|~y, y0i = e · h~p,n|Gth[A]|~y, y0i · ¯ . (3.175) Z[iτ] A→0 Again, drop the contribution from the spin-related part,

0 h~p,n|Sth|~y, y0i (3.176) Z ∞ 1 £ 3 ¤−1/2 −i(~p·~y−ω y ) −ism2 − 1 T r ln (2h) ' (2π) τ · e n 0 · i ds e · e 2 Z[iτ] Z ½ Z 0 Z ¾ i s s × d[u] exp {ip · u(s)}· exp ds ds u(s ) · h−1(s , s ) · u(s ) 4 1 2 1 1 2 2 ½ Z 0 0 ¾ i d4k δ δ × exp − · D˜ th(k) · 2 (2π)4 δA˜(k) δA˜(−k) ½ · Z ¸¾ × m − iγ · p − g dq A˜(q) · eiq·(y−u(s)) ½ Z ¾ ½ Z ¾¯ ¯ ˜ ˜ i ˜ ˜ ˜ ¯ × exp −ig dq A(q) · f(q) · exp dq A(q) · Kth(q) · A(q) ¯ , 2 A=0 ˜ where fν (q) is deﬁned in Eq. (3.32) as

Z s Z s Z s2 ˜ −1 0 iq·(y−u(s0)) fν (q) = ds1 ds2 uν (s1) h (s1, s2) ds e . (3.177) 0 0 0

Moving the linkage operator across the linear numerator term, and only the exponent linear in A˜µ(q) will contribute after Aµ is set to zero at the end of calculation, yielding

0 h~p,n|Sth|~y, y0i (3.178) Z ∞ 1 £ 3 ¤−1/2 −i(~p·~y−ω y ) −ism2 − 1 Tr ln (2h) ' (2π) τ · e n 0 · i ds e · e 2 Z[iτ] 0 Z ½ Z s Z s ¾ i −1 × d[u] exp {ip · u(s)}· exp ds1 ds2 u(s1) · h (s1, s2) · u(s2) 4 0 0 ½ · Z 4 ¸¾ d q iq·(y−u(s)) δ × m − iγ · p + ig e D˜ th(q) (2π)4 δA˜(q) ½ Z ¾ i d4k δ δ × exp − · D˜ th(k) · 2 (2π)4 δA˜(k) δA˜(−k) ½ Z Z ¾¯ ¯ ˜ ˜ i ˜ ˜ ˜ ¯ × exp −ig dq A(q) · f(q) + dq A(q) · Kth(q) · A(q) ¯ . 2 A=0 74

The linkage over the last line is similar to Eq. (3.60) and is of Gaussian-type, cf. Eq. (D.8) in Appendix D, ½ Z Z ¾ D(th) i e A · exp + dq A˜(q) · K˜ (q) · A˜(q) − ig dq A˜(q) · f˜(q) (3.179) 2 th ½ Z · ¸ ¾ i 1 = exp + dq A˜(q) · K˜ th(q) · · A˜(q) 2 1 − K˜ (q) · D˜ (q) ½ Z th th ¾ 1 × exp −ig dq A˜(q) · · f˜(q) 1 − K˜ (q) · D˜ (q) ½ Z · th th ¸ ¾ i 2 ˜ 1 ˜ × exp g dq f(q) · D˜ th(q) · · f(q) 2 1 − K˜ (q) · D˜ (q) ½ ¾ th th 1 × exp − Tr ln [1 − K˜ · D˜ ] . 2 th th

The determinant factor in the ”Trace-log” form could be absorbed into the deﬁnition of the partition function Z[iτ], and ½ ¾ 1 Z[iτ] = exp − Tr ln [1 − K˜ · D˜ ] · Z(0)[iτ], (3.180) 2 th th where the expression is only valid for the current approximation of Lc[A], and Z[iτ] is of course not the full partition function of the interacting system. Thus,

0 h~p,n|Sth|~y, y0i (3.181) Z ∞ 1 £ 3 ¤−1/2 −i(~p·~y−ω y ) −ism2 − 1 Tr ln (2h) ' (2π) τ · e n 0 · i ds e · e 2 (0) Z [iτ] 0 Z ½ Z s Z s ¾ i −1 × d[u] exp {ip · u(s)}· exp ds1 ds2 u(s1) · h (s1, s2) · u(s2) 4 0 0 ½ · Z 4 · ¶ ¸¾ 2 d k ˜ 1 ˜ ik·(y−u(s)) × m − iγ · p − g 4 Dth(k) · ] · f(−k) · e (2π) 1 − K˜ th(k) · D˜ th(k) ½ Z 4 · ¸ ¾ i 2 d k ˜ ˜ 1 ˜ × exp g 4 f(k) · Dth(k) · · f(k) . 2 (2π) 1 − K˜ th(k) · D˜ th(k) Using the same Bloch-Nordsieck approximation as in the quenched case,

0 h~p,n|Sth|~y, y0i (3.182) Z ∞ 1 £ ¤−1/2 2 2 = (2π)3τ · e−i(~p·~y−ωny0) · i ds e−is(m +p ) (0) Z [iτ] 0 ½ Z 4 · µ ¶ ¸ Z s ¾ 2 d k ˜ 1 0 2ik·ps0 × m − iγ · p + 2ig 4 γ · Dth(k) · · p ds e (2π) 1 − K˜ th(k) · D˜ th(k) 0 ½ Z Z Z · µ ¶ ¸ ¾ 4 s s1 2 d k 0 ˜ 1 2ik·ps0 × exp 4ig 4 ds1 ds p · Dth(k) · · p e , (2π) 0 0 1 − K˜ th(k) · D˜ th(k)

˜ 0 where the quantity in the parentheses is the dressed, ﬁnite-temperature photon propagator Dth(k), i.e., ˜ 0 ˜ 1 Dth(k) = Dth(k) · , (3.183) 1 − K˜ th(k) · D˜ th(k) 75 up to one close-fermion-loop approximation, which is parallel to the T = 0 case. The thermal photon polarization tensor can be decomposed into two parts, the vacuum polarization tensor K˜ c(k) and the thermal contribution δK˜ th(k) from the heat bath. If pair production is considered as a damping process, the fermion-anti-fermion pairs generated in the process are not yet thermalized at the instant of production. For the subject of interest, one can drop δK˜ th(k) and use the zero-temperature vacuum polarization tensor to replace the thermal photon polarization tensor as

K˜ th(k) = K˜ c(k) + δK˜ th(k) ' K˜ c(k). (3.184)

2 In the square brackets of the last exponent in Eq. (3.182), put in K˜ th(k) up to the order of g or α as

˜ 0 ˜ 1 Dth(k) = Dth(k) · (3.185) 1 − K˜ th(k) · D˜ th(k) 1 ' D˜ th(k) · 1 − K˜ c(k) · D˜ th(k) 1 = D˜ th(k) · 2 2 1 + k [Π˜ R(k ) + Π˜ (0)] Pˆ · D˜ th(k) 1 = D˜ th(k) · , 2 1 + Π˜ (0) + Π˜ R(k )

˜ µν 2 ˜ 2 ˆ ˜ 2 ˜ 2 ˜ 2 where Kc (k) = −k Π(k ) Pµν , and ΠR(k ) is the renormalized part of Π(k ) deﬁned by ΠR(k ) ≡ Π˜ (k2) − Π˜ (0). The last line utilizes [68, 89]

0 1 D˜ µν (k) = D˜ µν (k) , (3.186) c c 1 + Π˜ (k2) and 2 k · δD˜ th(k) = 0, (3.187)

2 2 where δD˜ th(k) consists of a δ(k )-factor, which puts photon ﬁelds on the mass-shell, i.e., k = 0, and

K˜ c(k) · D˜ th(k) = K˜ c(k) · D˜ c(k). (3.188)

Observe that

2 0 2 1 g D˜ (k) ' g D˜ th(k) · (3.189) th 2 1 + Π˜ (0) + Π˜ R(k ) 1 1 = g2 D˜ (k) · · th ˜ 2 1 + Π˜ (0) 1 + ΠR(k ) 1+Π˜ (0)  2  Π˜ R(k ) g2 ˜ = D˜ (k) · 1 − 1+Π(0)  . th ˜ 2 1 + Π˜ (0) 1 + ΠR(k ) 1+Π˜ (0) The ﬁrst term is related to what has been calculated in the previous sections except for the renormalization factor of [1 + Π˜ (0)]−1. Recall that Π˜ (0) is the divergent part of Π˜ (k2), it would seem 2 that the second part in the square brackets will vanish with |1 + Π˜ (0)| À |Π˜ R(k )|. However, the 76 coupling constant g so far is not yet renormalized, and the extra factor is related to the renormaliza- −1 tion constant Z3 = [1 + Π˜ (0)] . According to the Ward identity [48], the renormalization constant, 2 Z3, could be absorbed into the deﬁnition of the coupling constant g , i.e., g2 g2 = , (3.190) R 1 + Π˜ (0)

2 2 and the same factor could be similarly absorbed into g inside each Π˜ R(k ) as ˜ 2 2 ΠR(k ; g ) 2 2 → Π˜ R(k ; g ). (3.191) 1 + Π˜ (0) R Thus,

2 0 2 1 g D˜ (k) ' g D˜ th(k) · (3.192) th 1 + Π˜ (0) + Π˜ (k2) " R # ˜ 2 2 ΠR(k ) → g D˜ th(k) · 1 − , R 2 1 + Π˜ R(k ) where the ﬁrst term accounts for both Bremsstrahlung processes: ordinary and thermal-photon assisted, and the second term is the contribution from the pair-production. Notice that even if 2 Π˜ R(k ) is large, the correction from the photon polarization or self-energy is limited to 1, i.e., ¯ ¯ ¯ ˜ 2 ¯ ¯ ΠR(k ) ¯ ¯ ¯ . 1, (3.193) 2 ¯1 + Π˜ R(k )¯ which is an indication that the theory is self-consistent and unitary. Up to order of g4 or to ﬁrst 2 order in Π˜ R(k ), " # ˜ 2 2 0 2 ΠR(k ) g D˜ (k) → g D˜ th(k) · 1 − (3.194) th R 2 1 + Π˜ R(k ) 2 ˜ 2 ˜ ˜ 2 2 ' gR Dth(k) − gR Dth(k) · ΠR(k ; gR).

By deﬁnition, Π˜ R(0) = 0, so 2 δD˜ th(k) · Π˜ R(k ) = 0, (3.195) that is, there is no net contribution from the thermal photons, or the medium does not lose energy balance through pair production internally. Then, h i 2 ˜ 0 2 ˜ ˜ 2 ˜ ˜ 2 gR Dth(k) → gR Dc(k) + δDth(k) − gR Dc(k) · ΠR(k ). (3.196) Hence, the one close-fermion-loop approximated, dressed, ﬁnite-temperature fermion propagator reduces to Z ∞ 1 £ ¤−1/2 2 2 h~p,n|S0 |~y, y i ' (2π)3τ · e−i(~p·~y−ωny0) · i ds e−is(m +p ) (3.197) th 0 (0) Z [iτ] 0 ½ Z 4 h i Z s ¾ d k 0 × m − iγ · p + 2ig2 γ · D˜ (k) · p ds0 e2ik·ps (2π)4 th ½ Z Z Z 0 ¾ 4 s s1 h i d k 0 × exp 4ig2 ds ds0 p · D˜ (k) · p e2ik·ps (2π)4 1 th ½ Z 0Z 0Z ¾ 4 s s1 h i 2 d k 0 ˜ ˜ 2 2ik·ps0 × exp −4ig 4 ds1 ds p · Dc(k) ΠR(k ) · p e , (2π) 0 0 77

2 where all subscripts ’R’ in renormalized gR under one closed-fermion-loop approximation has been dropped for notation simplicity. The exponent with a factor of [p · D˜ th(k) · p] in the third line is 2 what has been calculated in the previous sections. The last exponent with [p · D˜ th(k) Π˜ R(k ) · p] is the contribution from one closed-fermion-loop, i.e., pair production, and Z Z Z 4 s s1 h i d k 0 −4ig2 ds ds0 p · D˜ (k) Π˜ (k2) · p e2ik·ps (3.198) (2π)4 1 c R Z 0 Z 0 Z 4 s s1 h i 2 d k 0 ˜ ˜ 2 2ik·ps0 → −4ig 4 ds1 ds p · Dc(k) ΠR(k ) · p e |~k|≤|~p| (2π) 0 0 Z Z Z " Ã ! # 4 s s1 ˆ T 2 d k 0 δµ0δ0ν Pµν 2ik·ps0 = −4ig ds1 ds pµ · − + · pν e ~ (2π)4 ~ 2 ~ 2 2 |k|≤|~p| 0 0 k k − k0 − i² µ ¶ Z µ ¶ g2 1 m2 + y(1 − y)k2 × − 2 dy y(1 − y) ln 2 2π 0 m Z Z Z Z 4 s s1 1 4 2g 0 d k 2ik·ps0 = +i 2 ds1 ds dy y(1 − y) 4 e π 0 0 0 |~k|≤|~p| (2π) ( " # ) µ ¶ ω2 (~k · ~p)2 1 m2 + y(1 − y)k2 × − n + ~p 2 − · ln . ~ 2 ~ 2 ~ 2 2 m2 k k k − k0 − i²

Here, the upper limit of the ~k-integral is set to |~p| to limit the momentum scale of photons to that of the incident particle. In conﬁguration space, D00(z) = δ(z0)/2π|~z|, which is instantaneous and does not contribute the damping. In the usual Coulomb gauge, there is no (0, 0)-component, 1/~k2, 2 ~ 2 in the photon propagator. The part with the −ωn/k factor will vanish, i.e., there is no longitudinal part in the context of QED as k2 · K˜ = 0, then Z Z Z 4 s s1 h i 2 d k 0 ˜ ˜ 2 2ik·ps0 −4ig 4 ds1 ds p · Dc(k) ΠR(k ) · p e (3.199) (2π) 0 0 Z Z Z Z 4 s s1 1 4 2g d k 0 0 2 0 2is ~k·~p−2is k0 ωn ' +i 2 ~p ds1 ds dy y(1 − y) 4 e π 0 0 0 |~k|≤|~p| (2π) Ã ! 1 − cos2 θ m2 + y(1 − y)(~k2 − k2) × ln 0 ~ 2 2 m2 k − k0 − i² Z Z Z Z Z g4 s s1 1 1 |~p| = +i ~p 2 ds ds0 dy y(1 − y) dξ (1 − ξ2) dk ~k2 cos (2s0|~k||~p|ξ) π4 1 0 0 0 Ã 0 !0 Z 0 +∞ dk e−2is k0 ωn m2 + y(1 − y)(~k2 − k2) × 0 ln 0 , 2π ~ 2 2 m2 −∞ k − k0 − i² where ξ = cos θ = kˆ · pˆ. The logarithm factor from the polarization tensor becomes complex when ζ = [y(1 − y)]−1 ≥ 4 for 0 < y < 1 as Ã ! ¯ ¯ ¯ ¯ m2 + y(1 − y)(~k2 − k2) ¯m2 + y(1 − y)(~k2 − k2)¯ ln 0 = ln ¯ 0 ¯ + πiθ(k2 − (~k2 + ζm2)), (3.200) m2 ¯ m2 ¯ 0 78 and the exponent becomes Z Z Z 4 s s1 h i 2 d k 0 ˜ ˜ 2 2ik·ps0 −4ig 4 ds1 ds p · Dc(k) ΠR(k ) · p e (3.201) (2π) 0 0 Z Z Z Z Z g4 s s1 1 1 |~p| ' +i ~p 2 ds ds0 dy y(1 − y) dξ (1 − ξ2) dk ~k2 cos (2s0|~k||~p|ξ) π4 1 0 0 0 ¯ 0 ¯ 0 Z 0 +∞ −2is k0 ωn ¯ 2 ~ 2 2 ¯ dk0 e ¯m + y(1 − y)(k − k )¯ × ln ¯ 0 ¯ 2π ~ 2 2 ¯ m2 ¯ −∞ k − k0 − i² Z Z Z Z Z 4 s s1 1 1 |~p| g 2 0 2 ~ 2 0 ~ − 3 ~p ds1 ds dy y(1 − y) dξ (1 − ξ ) dk k cos (2s |k||~p|ξ) π 0 0 0 0 0 Z 0 +∞ dk e−2is k0 ωn × 0 θ(k2 − (~k2 + ζm2)). 2π ~ 2 2 0 −∞ k − k0 − i²

The k0-integral in the ﬁrst term with the real part of logarithm can be converted into a contour 0 integral, and the integrand has poles at k0 = ±k ∓ i² . Since ¯ ¯¯ ¯ ¯¯ ¯ ¯ ¯m2 + y(1 − y)(~k2 − k2)¯¯ ¯m2 + 0¯ ln ¯ 0 ¯¯ = ln ¯ ¯ = 0, (3.202) ¯ m2 ¯¯ ¯ m2 ¯ k0=0 the residues of such contour integral are zero, and the ﬁrst term vanishes. Hence, only the second term remains, and Z Z Z 4 s s1 h i 2 d k 0 ˜ ˜ 2 2ik·ps0 −4ig 4 ds1 ds p · Dc(k) ΠR(k ) · p e (3.203) (2π) 0 0 Z Z Z Z 4 s s1 1 1 g 2 0 2 ' − 4 ~p ds1 ds dy y(1 − y) dξ (1 − ξ ) π 0 0 0 0 Z |~p| Z ∞ 0 ~ 2 0 ~ cos (2s k0 ωn) × dk k cos (2s |k||~p|ξ) √ dk0 . ~ 2 2 ~ 2 2 0 k +ζm k − k0 − i² To evaluate the pair production as an energy depletion mechanism, the photon energy is at most up p 2 2 to that of the incident fermion, i.e., k0 ≤ ~p + m ' |~p| as |~p| À m. Thus, the upper limit of the ~ k0-integral can also be set to |~p| as the k-integral, Z Z Z 4 s s1 h i 2 d k 0 ˜ ˜ 2 2ik·ps0 −4ig 4 ds1 ds p · Dc(k) ΠR(k ) · p e (3.204) (2π) 0 0 Z Z Z Z 4 s s1 1 1 g 2 0 2 ' − 4 ~p ds1 ds dy y(1 − y) dξ (1 − ξ ) π 0 0 0 0 Z |~p| Z |~p| 0 ~ 2 0 ~ cos (2s k0 ωn) × dk k cos (2s |k||~p|ξ) √ dk0 . ~ 2 2 ~ 2 2 0 k +ζm k − k0 − i² 0 Similar to steps in the previous section, let k = xp, k0 = zp, s1 = sv and s = s1u with 0 < 0 x, z, u, v < 1. Then, the s1- and s -integrals can be reduced to

Z s Z s1 0 0 ~ 0 ds1 · ds cos (2s |k||~p|ξ) · cos (2s k0 ωn) (3.205) 0 0 Z 1 Z 1 2 2 = s dv · v · du cos(2s~p ξxuv) · cos(2spωnzuv) 0 0 Z Z s2 1 1 = dv · v · du [cos(2sp(pξx + ωnz)uv) + cos(2sp(pξx − ωnz)uv)]. 2 0 0 79

1 Since s < smax = 2 2 2 , m +~p +|ωn|

Z 1 Z 1 1 − cos(2sp(pξx ± ωnz)) 1 dv · v · du · cos(2sp(pξx ± ωnz)uv) = 2 ' . (3.206) 0 0 (2sp(pξx ± ωnz)) 2 Hence, Z Z Z 4 s s1 h i 2 d k 0 ˜ ˜ 2 2ik·ps0 −4ig 4 ds1 ds p · Dc(k) ΠR(k ) · p e (3.207) (2π) 0 0 4 Z 1 Z 1 Z |~p| Z |~p| g 2 2 2 ~ 2 1 ' − 4 s ~p dy y(1 − y) dξ (1 − ξ ) dk k √ dk0 2π ~ 2 2 ~ 2 2 0 0 0 k +ζm k − k0 − i²  q  Z Z µ ¶ g4 1 |~p| p − k ~k2 + ζm2 − k = + s2 ~p 2 dy y(1 − y) dk k ln − ln q  6π4 p + k 0 0 ~k2 + ζm2 + k  q  m2 4 Z 1 Z 1 µ ¶ 2 g 1 − x ~x + ζ ~p 2 − x = + s2 (~p 2)2 dy y(1 − y) dx x ln − ln q , 4 2 6π 0 0 1 + x 2 m ~x + ζ ~p 2 + x where k = xp in the k-integral, · ¸ 1 1 1 1 1 = = − , (3.208) ~ 2 2 (k − k + i²0)(k + k − i²0) 2k k − k + i²0 k + k − i²0 k − k0 − i² 0 0 0 0 and the last line involves the Legendre function of the second kind (cf. Appendix E.5). As ζ = [y(1 − y)]−1 ≥ 4, the integrand of the y-integral has the largest contribution around y = 1/2, i.e., ζ = 4. Since m2 ¿ ~p 2, q m2 2 2 ~x2 + ζ − x m ζm 2 ~p 2 x + ζ 2x~p 2 − x 2~p 2 1 ζm q ' 2 = 2 ' . (3.209) m2 m 2 ζm 2 2 2 x + ζ 2 + x 2x + 2 x 4~p ~x + ζ ~p 2 + x 2x~p 2~p

With help of Z µ ¶ 1 1 − x dx x ln = −1, (3.210) 0 1 + x Z µ ¶ µ ¶ 1 1 ζm2 1 1 ζm2 dx x ln 2 2 = + ln 2 , (3.211) 0 x 4~p 2 2 4~p and Z µ ¶ p x2 a2 p x p dx x ln ( x2 + a2 + x) = + ln ( x2 + a2 + x) − x2 + a2, (3.212) 2 4 4 one arrives at Z Z Z 4 s s1 h i 2 d k 0 ˜ ˜ 2 2ik·ps0 −4ig 4 ds1 ds p · Dc(k) ΠR(k ) · p e (3.213) (2π) 0 0 4 Z 1 · µ 2 ¶¸ g 2 2 2 1 1 ζm ' + 4 s (~p ) dy y(1 − y) −1 + + ln 2 . 6π 0 2 2 4~p Recall that ζ = 1/[y(1 − y)], Z Z 1 1 1 1 5 dy y(1 − y) ln (ζ) = − dy y(1 − y) [ln (y) + ln (1 − y)] = , (3.214) 0 2 2 0 36 80

Z 1 1 dy y(1 − y) = , (3.215) 0 6 and then, it leads to Z Z Z 4 s s1 h i d k 0 −4ig2 ds ds0 p · D˜ (k) Π˜ (k2) · p e2ik·ps (3.216) (2π)4 1 c R 0· µ 0 ¶ ¸ g4 1 ~p 2 2 ' − s2 (~p 2)2 ln − ln 4 − 6π4 12 m2 3 µ ¶ g4 ~p 2 ' − s2 (~p 2)2 ln , 72π4 m2 ¡ ¢ where ln ~p 2/m2 À ln(4) + 2 = 2.05 with ~p 2 À m2. The damping eﬀect from the pair production ¡ 3 ¢ has an extra factor of g2 ln ~p 2/m2 compared to that of the ordinary Bremsstrahlung. Combined all terms from three diﬀerent origins: the thermal-photon-assisted Bremsstrahlung, the ordinary Bremsstrahlung, and the pair production; one then has

S˜0 (~p,n) (3.217) th Z ³ ´−1 ∞ (0) © 2 2 ª ' Z [iτ] · i ds exp −is(ω − ωn) ½ · 0 µ ¶¸¾ 2a2 2a2 2a2 ~p 2 × exp −s2 T g2T 2 ~p 2 + B g2(~p 2)2 + P g4 (~p 2)2 ln 3π2 3π2 3π2 m2 ½ Z 4 h i Z s ¾ d k 0 × m − iγ · p + 2ig2 γ · D˜ 0 (k) · p ds0 e2ik·ps (2π)4 th ½ ½ Z Z Z 0 ¾¾ 4 s s1 h i 2 d k 0 ˜ 2ik·ps0 × exp iIm 4g 4 ds1 ds p · Dc(k) · p e , (2π) 0 0

2 2 2 2 2 where aT = π /6, aB = 1/4, and aP = 1/(48π ) are the coeﬃcients for the thermal-photon- assisted Bremsstrahlung, ordinary Bremsstrahlung, and pair production, respectively. Again, the last exponent, which is imaginary, could be treated as the wave-function renormalization constant. £ ¤2 The decay factor could be moved out of s-integral by converting s2 with i (∂/∂ω2) as

˜0 Sth(~p,n) ' (3.218) ( · µ ¶¸ µ ¶ ) 1 ~p 2 ∂ 2 ³ ´−1 exp − g2 ~p 2 [a2 T 2 + a2 ~p 2] + a2 g2 (~p 2)2 ln · i · Z(0)[iτ] 3π2 T B P m2 ∂ω2 Z ∞ ½ Z 4 h i Z s ¾ © ª 0 2 2 2 d k ˜ 0 2ik·ps ·i ds exp −is(ω − ωn) m − iγ · p + 2ig 4 γ · Dc(k) · p ds e . 0 (2π) 0

Compared to the Bremsstrahlung process, the pair production is on the order of g4 under the one close-fermion-loop approximation. Recall that α = g2/4π = 1/137. When |~p| À m, the energy, ω ∼ |~p|, of the incident fermion is about 100GeV, and the electron mass is about 0.5MeV. The ¡ ¢ extra factor of (g2/4π) ln ~p 2/m2 is less than 1 in QED. Thus, the damping eﬀect due to the pair production in QED is still smaller than the Bremsstrahlung process. Chapter 4

Discussion and Perspectives

4.1 Model Approximation and Damping Mechanisms

In the previous chapter, the decay of an ultra-relativistic fermion in a hot QED plasma is modeled as the scattering of a fast moving fermion within a thermal bath. In the full theory, the scattering process should include both soft- and hard-photon exchange in QED [90, 91, 92]. To include both scales in the calculation of the fully-dressed fermion propagator, the linkage operator can be separated in two parts as (th) (th) (th) DA = DS + DH , (4.1) (S) (H) where subscripts ’S’ and ’H’ denote soft- and hard- parts of linkage with gauge ﬁelds Aµ and Aµ , (S) (H) respectively, and Aµ = Aµ + Aµ . Then, the full fermion propagator becomes " #¯ L [A(S)+A(H)] (th) (th) e th ¯ 0 DS +DH (S) (H) ¯ Sth = e · Gth[A + A ] ¯ . (4.2) Z[iτ] A(S)=A(H)=0

As the momenta of particles in the thermal bath is on the order of T , the energy-momentum scale of the incident particle is assumed to be much larger than that of particles in the medium, i.e., p ω = ~p 2 + m2 À T or |~p| À T , in the regime of interest. The exchange will likely involve only soft photons, real or virtual, such that there is no recoil for the incident fermion. Therefore, only the soft-photon exchange is retained in the calculation while the contribution from the hard-photon exchange is dropped, for which the Bloch-Nordsieck (BN) approximation is ideal. The k-integral of momentum exchange is limited to the temperature T of the thermal bath through the thermal distribution function, or the fermion momentum |~p| as a upper cut-oﬀ. Other forms of the momentum limiting are also possible, e.g., exp [−~k2/~p 2]. While the momentum loss is small through photon emission, the energy loss of the incident particle can be large, e.g., through virtual photon emission, which then decays into a fermion plus anti-fermion. There are three damping mechanisms in this calculation: the thermal-photon-induced and ordinary Bremsstrahlung, and pair production:

81 82

HB γ γth

Figure 4.1: One-loop representation of Thermal-Photon-Induced Bremsstrahlung through a thermal photon (γth) exchange with the medium (HB).

HB γ γv

Figure 4.2: One-loop representation of Ordinary Bremsstrahlung through a virtual photon (γv) exchange with the medium (HB).

HB γv γ

Figure 4.3: Two-loop representation of Pair Production from a virtual photon (γv) exchange with the medium (HB). 83

• Thermal-Photon-induced Bremsstrahlung: The plasma is treated like a thermal bath specified by a temperature T , which absorbs and emits real photons. Hence, the plasma can be considered as a collections of states, and be represented by a direct product of different photon modes with a set of occupation numbers specified by the thermal distribution function [6]. The incident particle scatters through the plasma or the thermal bath, and the interaction changes the energy level of each mode by absorbing and emitting thermal photons. The process is similar to the Compton scattering, and the energy depletion of the incident fermion is by transferring energy into the medium by emitting real photons. The process can be described by Fig. (4.1), and the decay exponent is proportional to g2T 2~p 2.

• Ordinary Bremsstrahlung: The incident particle (fermion) interacts with particles in the thermal bath, and emits and absorbs photons as the incident particle slows down. The starting form of the calculation of the fully-dressed fermion propagator at ﬁnite-temperature is similar to that of the vacuum theory. Contrast to the self virtual photon exchange in the vacuum theory at zero-temperature which has no imaginary part, the ﬁnite-temperature calculation involves virtual photon exchange between the incident particle and the thermal bath as can be represented by Fig. (4.2). The decay exponent is proportional to g2(~p 2)2, which is the dominant damping mechanism compared to the thermal-photon-induced Bremsstrahlung, as the upper limit of the internal momentum integral, k-integral, is set to |~p|.

• Pair Production: In additional to the Bremsstrahlung processes, the scattering of the incident particle with the thermal bath can also induce pair production as pictured in Fig. (4.3). The incident fermion can lose large amounts of energy, but small amounts of momentum through pair production. The decay exponent is proportional to g4(~p 2)2 ln (~p 2/m2), which has an extra factor of g2 ln (~p 2/m2) compared to the ordinary Bremsstrahlung. In the regime of small coupling in QED, the eﬀect of pair-production is smaller than the ordinary Bremsstrahlung.

4.2 Damping Eﬀects

Combining all three processes estimated in the previous chapter, the decay exponent is −b2s2 with µ ¶ 1 1 1 ~p2 b2 = g2 T 2 ~p 2 + g2 (~p 2)2 + g4 (~p 2)2 ln (4.3) 9 6π2 72π4 m2 · µ ¶¸ 2 ~p 2 = g2 ~p 2 a2 T 2 + a2 ~p 2 + a2 g2~p 2 ln , 3π2 T B T m2

2 2 2 2 2 where aT = π /6, aB = 1/4, and aP = 1/(48π ) are the coefficients for the thermal-photon-induced Bremsstrahlung, ordinary Bremsstrahlung, and pair production, respectively. In the fermion propagator, there is the s2-dependent, instead of näive s-dependent, exponential factor; and therefore, the damping is not exponential as appeared in the perturbation calculations. With the replacement, ∂ s → i , (4.4) ∂ω2 84 which follows from the form of Eqs. (3.102) and (3.131), the dressed, finite-temperature fermion propagator becomes

˜0 Sth(~p,n) (4.5) " µ ¶ # ∂ 2 ' exp −b2 · i · S˜ (~p,n) ∂ω2 th ( · µ ¶¸ µ ¶ ) 2 ~p 2 ∂ 2 ' exp − g2 ~p 2 a2 T 2 + a2 ~p 2 + a2 g2~p 2 ln · i · S˜ (~p,n). 3π2 T B T m2 ∂ω2 th

In this form, the complete Matsubara sum can be easily performed, using the mixed representation of the non-interacting fermion propagator S˜th(~p,n) expressed as · µ ¶¸ 1 ∂ i © £ ¤ª −iωz0 −iωz0 +iωz0 S˜th(~p,z0) = m − i~γ · ~p + iγ0 · − e − n˜(ω) e + e . (4.6) i ∂z0 2ω

Following the derivation of Eq. (3.152), the dressed, ﬁnite-temperature fermion propagator with all three damping mechanisms can be written as

h~p,x |S0 |~p,y i (4.7) · 0 th 0 µ ¶¸ i 1 ∂ ' m − i~γ · ~p + iγ · − 2 0 i ∂z ½ 0 ¾ 2 2 h 2 2 i 2 1 −iωz + β b −i[ω−β b ]z +i[ω−β b ]z − b z2 · e 0 − n˜(ω)e 16ω2 e 4ω2 0 + e 4ω2 0 e 4ω2 0 , ω wheren ˜(ω) exp [+β2b2/16ω2] is the relative amplitude of particles in the thermal bath compared to the incident particle, and there is a phase shift (or delay) of β2b2/4ω2 for particles in the medium. In the limit of weak coupling, g2 ¿ 1 and

βb2 ¿ 1, (4.8) 8ω3 then, the relative distribution factor can be approximated as

n˜(ω) · exp [+β2b2/16ω2] ' n˜(ω − βb2/8ω2) (4.9) in comparison to Eq. (3.144). Hence,

h~p,x |S0 |~p,y i (4.10) · 0 th 0 µ ¶¸ i 1 ∂ ' m − i~γ · ~p + iγ0 · − 2 i ∂z0 n h 2 2 io 2 1 −iωz 2 2 −i[ω−β b ]z +i[ω−β b ]z − b z2 · e 0 − n˜(ω − βb /8ω ) e 4ω2 0 + e 4ω2 0 e 4ω2 0 , ω

2 2 2 Compared to the non-interacting propagator, there is an overall factor of exp [−(b /4ω )z0 ], which represents the damping of the fermions. The damping is not a simple exponential as given by the prediction of the perturbative calculations; the incident particle suﬀers much faster damping as it proceeds through the medium. 85

4.3 Comparison to Perturbative Theory

The damping rate was first estimated in the Born approximation with thermal distributions to account for finite-temperature effects, and the interacting rate was given by [20, 45]

Z 3 Z 3 Z 3 1 d ~p1 d ~p2 d ~p3 4 (4) Γ(p) = 3 3 3 (2π) δ (p + p1 − p2 − p3) (4.11) 2ω (2π) 2ω1 (2π) 2ω2 (2π) 2ω3 ˜ 2 × [˜n1(1 − n˜2)(1 − n˜3) + (1 − n˜1)˜n2n˜3] |M| , wheren ˜i is the Fermi-Dirac distribution of the i-th particle, and p1 p2

˜ M(p, p1; p2, p3) = (4.12) p p3 is the scattering matrix element of the incident particle oﬀ particles in the thermal bath. The fermion damping rate can also be evaluated from the imaginary part of the self-energy Σ˜ (p) in the fully-dressed propagator as , where the fermion loop represents fermions in the medium. The interaction rate in Eq. (4.11) is equivalent to twice of the imaginary part of the two-loop self-energy, i.e., Γ(p) = 2ImΣ˜ (p) [20, 37, 40, 45]. Under the perturbative method, there is no contribution from graphs without any fermion loop due to kinematics, e.g., , and the leading contribution of fermion damping rate comes from the two-loop diagram, but is quadratically IR divergent under perturbative approaches. As soft momentum exchange is the dominated process, the damping rate becomes Z Z " µ ¶ # 4 3 Λ +ωk 2 2 g T dk0 2 1 k0 2 Γ(p) ' dωk |D˜ L(ωk, k0)| + 1 − |D˜ T (ωk, k0)| , (4.13) ~ 2 6 0 −ωk (2π) 2 k

~ L T where ωk = |k|, Λ is the upper cutoﬀ for soft momenta, and D˜ (k) = D˜ L(k) Pˆ + D˜ L(k) Pˆ is the ~ 2 2 bare photon propagator with (00)-component D˜ L = −1/k and transverse component D˜ T = 1/k , 86 respectively. The result of the Born approximation gives a quadratic IR divergence in the |~k|-integral as Z g4T 3 Λ dk Γ(p) ' 3 . (4.14) 6 0 k To alleviate IR divergence, the bare photon propagator can be replaced with the resummed ˜ 0 photon propagator, DHTL(k), in Hot-Thermal-Loop (HTL) approximations;

˜ 0−1 ˜ −1 ˜ ˜ −1 2 ˜ 2 DHTL(k) = D (k) − KHTL(k) = D (k) + k ΠHTL(k ), (4.15) and

0 ~ −1 D˜ (k, k0) = , (4.16) L ~ 2 2 ˜ HTL k − k ΠL 0 ~ 1 D˜ (k, k0) = , T 2 2 ˜ HTL k + k ΠT which introduce screening eﬀects of plasma. The damping rate can also be estimated from the imaginary part of fermion self-energy with resummed photon propagator, i.e., Γ(p) = 2 ImΣ˜ (p), and

Z k 1 X d3~k Σ˜ (p) = = −g2 γ · D˜ 0 (k) · γ S˜(p − k), (4.17) τ (2π)3 HTL l p − k

~ ˜ 0 where k = (k, ωl), p = (~p,ωn), and DHTL(k) is the resummed photon propagator represented by a photon line with a black dot. It has been argued that the leading IR contribution mainly comes from the region of k0 ¿ k ¿ T with [45]

g2T 2 k2 Π˜ HTL → (4.18) L 3 2 2 2 ˜ HTL π g T k0 k ΠT → −i 4 3 |~k|

The longitudinal contribution is finite, but the transverse component is Z µ ¶ g2T Λ dk g2T Λ ΓT (p) ' → ln , (4.19) 2π 0 k 2π µsc where µsc is an IR cutoff as the dynamic screening, and is still IR logarithmically divergent due to the exchange of very soft and quasi-static (transverse) photons. The IR divergence of the fermion damping rate has raised doubts on the perturbative approach, while the process itself is intrinsically non-perturbative. Weldon in Ref. [20] has proposed applications of Bloch-Nordsieck approximations to get rid of IR problems; subsequent implementations of such approximations were followed by Takashiba [44], and Blaizot and Iancu [30, 45, 46], which 87 applied a fixed, classical-particle velocity in their calculations. In Refs. [45, 46], the Feynman rules are modified so that the vertex function is replaced by γµ → −ivµ = −i(1,~v) with ~v = ~p/|~p| at zero mass limit, the fermion propagator in Eq. (4.17) is then replaced by the BN-type bare propagator 1 1 S (p − k) = → , (4.20) c ~ m + iγ · (p − k) m + ~v · (~p − k) − (p0 − k0) 0 and the HTL photon propagator DHTL(k) is used for photon lines. The imaginary part of the fermion self-energy still exhibits IR divergence near the mass-shell to all orders of perturbation, which contradicts the assumption of existence of a pole in the dressed fermion propagator (cf. Ref. [26]). To avoid the IR divergence appeared in the momentum space calculations, Blaizot and Iancu converted the formulation into the time domain and showed that there is no IR divergence as the inverse of the time acts as an effective IR cutoff at long time limit [30, 45, 46]. In the massless limit, the resulting fermion damping is not a simple exponential decay, as the retarded fermion propagator is proportional to 0 −it(~v·~p) δSR(t, ~p) ' iθ(t) e exp {−αT t[ln (ωpt) + const]} (4.21) for large time t À 1/gT . Without using the inverse-time as an effective IR cut-off, the methods used by Blaizot and Iancu still suffer the same IR divergence and fail to estimate the initial damping which is physically more interesting.

4.4 Longitudinal and Transverse Disturbance in the Medium

In the medium, the momenta of particles are on the order of T , i.e., |~p| ∼ T . In the calculation of the ordinary Bremsstrahlung and pair production, the upper limit of the ~k-integral is set to the momentum ~p of the incident particle, or the Bloch-Nordsieck limiting factor exp [−~k2/~p 2] is added to evaluate the soft-momentum contribution. To distinguish the energy-momentum of particles in the thermal bath from the contribution from the upper limit, one can temporarily set the upper limit of the ~k-integral to Λ. Then, the real part of exponent for the ordinary Bremsstrahlung (cf. Eq. (3.110)) becomes ½ Z Z Z ¾ 4 s s1 h i 2 d k 0 ˜ BN 2ik·ps0 Re 4ig 4 ds1 ds p · Dc (k) · p e (4.22) (2π) 0 0 Z s Z s1 Z 1 Z ∞ 4 2 2 0 2 0 0 −~k2/Λ2 = − 2 g ~p ds1 ds dξ [1 − ξ ] dk k cos(2s k ωn) cos(2s kp ξ) e (2π) 0 0 0 0 Z 1 Z 1 Z 1 4 2 2 2 2 2 = − 2 s g ~p Λ dv · v · du dξ · [1 − ξ ] (2π) 0 0 0 Z ∞ −x2 × dx · x · cos(2sΛ ωnxuv) · cos(2sΛ|~p|ξxuv) · e , 0 where the Bloch-Nordsieck limiting factor is replaced by exp [−~k2/Λ2] and k = x Λ; and one yields ½ Z Z Z ¾ 4 s s1 h i 2 d k 0 ˜ 2ik·ps0 Re 4ig 4 ds1 ds p · Dc(k) · p e (4.23) (2π) 0 0 2a2 ' − B s2g2~p 2 Λ2. 3π2 88

Similarly, the exponent for the pair production, Eq. (3.207), becomes Z Z Z 4 s s1 h i 2 d k 0 ˜ ˜ 2 2ik·ps0 −4ig 4 ds1 ds p · Dc(k) ΠR(k ) · p e (4.24) (2π) 0 0 4 Z 1 Z 1 Z Λ Z Λ g 2 2 2 ~ 2 1 ' − 4 s ~p dy y(1 − y) dξ (1 − ξ ) dk k √ dk0 2π ~ 2 2 ~ 2 2 0 0 0 k +ζm k − k0 − i²  q  2 4 Z 1 Z 1 µ ¶ 2 m g 1 − x ~x + ζ 2 − x 2 2 2  q Λ  = + 4 s ~p Λ dy y(1 − y) dx x ln − ln , 6π 0 0 1 + x 2 m2 ~x + ζ Λ2 + x where ζ = [y(1 − y)]−1 ≥ 4, which leads to Z Z Z 4 s s1 h i d k 0 −4ig2 ds ds0 p · D˜ (k) Π˜ (k2) · p e2ik·ps (4.25) (2π)4 1 c R 0 µ ¶0 g4 Λ2 ' − s2 ~p 2 Λ2 ln . 72π4 m2

Therefore, the combined decay exponent, Eq. (4.3), becomes · µ ¶¸ 2 Λ2 b2 = g2 ~p 2 a2 T 2 + a2 Λ2 + a2 g2Λ2 ln , (4.26) 3π2 T B P m2

2 2 2 2 2 where aT = π /6, aB = 1/4, and aP = 1/(48π ) as deﬁned in the previous sections. As ω0 with |~p| À T À m, one can approximate ~p 2 ' ω and lead to · µ ¶¸ 2 Λ2 b2 = g2 ω2 a2 T 2 + a2 Λ2 + a2 g2Λ2 ln . (4.27) 3π2 T B P m2

The choice of the upper cutoﬀ Λ is arbitrary as long as it gives a sensible limit of soft momenta, and can be chosen as the momentum of the incident particle, the temperature of the medium, or something in between. Furthermore, · µ ¶¸ b2 g2 Λ2 = a2 T 2 + a2 Λ2 + a2 g2Λ2 ln , (4.28) 2ω2 3π2 T B P m2 which is not a direct function of z0 in the leading approximation. Similar to the non-interacting case in Eq. (2.195), the Fourier transform of the fully-dressed, ﬁnite-temperature fermion propagator is Z d3~p S0 (~z, z ) = S0 (~p,z ) e+i~p·~z (4.29) th 0 (2π)3 th 0 Z · µ ¶¸ 3 2 d ~p i ∂ − b z2 4ω2 0 ' 3 m − i~γ · ~p + iγ0 · i e (2π) 2ω ∂z0 n h 2 2 io i(~p·~z−ωz ) 2 2 i[~p·~z−(ω−β b )z ] i[~p·~z+(ω−β b )z ] × e 0 − n˜(ω − βb /8ω ) e 4ω2 0 + e 4ω2 0 .

The integral is complicated to evaluate explicitly. However, the first term inside the curely brackets shows a phase factor of [i(~p · ~z − ωz0)], which represents the forward propagation of the incident 0 particle. In the second terms with a Fermi-Dirac distribution, the phase factors are [i(~p·~z±ω z0)] with 89 a phase velocity of ω0 = ω−βb2/(4ω2). The phase velocity ω0 is different from the ω00 = ω−βb2/(8ω2) in the distribution function, as the disturbance is not at equilibrium. The disturbance should not follow the equilibrium thermal distribution in the time frame of fast decay. This is the reason why one needs to separate the cut-off from the particle momentum. In the Bloch-Nordsieck or no-recoil approximation with soft photon contributions, ~γ k ~p, so

~γ · ~p ∝ ~p 2. (4.30)

Hence the second term is symmetric under the exchange of ~p and −~p. Rewrite the fully-dressed, ﬁnite-temperature fermion propagator as

h~p,x |S0 |~p,y i (4.31) · 0 th 0 µ ¶¸ 1 ∂ ' m − i~γ · ~p + iγ0 · − i ∂z0 ½ h i ¾ 2 2 h i b2 2 b 2 β b2 b2 i −iωz − z − 2 z0 − −i[ω−β ]z +i[ω−β ]z × e 0 4ω2 0 − n˜(ω)e 4ω 4 e 4ω2 0 + e 4ω2 0 . 2ω The first term represents the incident particle traveling forward, and the second term can be treated as the disturbance inside the hot medium. Both the incident particle and the medium disturbance 2 2 2 have a similar amplitude decay exponent of −(b /4ω ) z0 , but the medium disturbance has a shift of −β b2/(4ω2) in the phase velocity. The initial relative amplitude between the incident particle and the medium disturbance is proportional ton ˜(ω) exp [+β2 b2/(16ω2)] orn ˜(ω − βb2/8ω2). In the assumption that the incident particle is ultra-relativistic, the thermal distribution function will suppress the amplitude of the disturbance in the medium with the same energy-momentum ω ∼ |~p| À T . Under the BN, no-recoil approximation, the incident particle propagates in a straight line, and induces a disturbance in the medium through exchange of energy-momentum. The thermal part is symmetric or back-to-back, and will contain information of both the longitudinal and transverse components. Once the incident proceeds through the medium, one cannot know the status of the incident particle inside the medium without interference. Instead, the presence of the incident particle felt by the medium can only be detected indirectly through real photon emission, or fermion and anti-fermion of pair production escaped from the medium; hence, the finite-temperature propagator can be considered as a measure of the disturbance induced by the incident fermion. The transverse component of the medium disturbance only has contributions from the thermal part of the propagator. One can define transverse part of probability amplitude ∆T as

h 2 i h 2 i b2 2 β Γ 2 β T i − z − i − z − ∆ = − n˜(ω) e 4ω2 0 4 = − n˜(ω) e 2 0 4 , (4.32) 2 ω 2 ω where b2 Γ = . (4.33) 2 ω2 Here, the phase factors, which describe the propagation and are not related to the amplitude, are not included in deﬁnition of ∆T. The square modulus of ∆T becomes

h 2 i 1 −Γ z2− β |∆T|2 = [˜n(ω)]2 e 0 4 . (4.34) 4 ω2 90

The ratio of the square modulus compared to the initial value at z0 = 0 is

· ¸2 T 2 h i2 2 |∆ (z0)| ω0 n˜(ω) β 2 − 4 [Γ−Γ0] −Γz0 T 2 = e e , (4.35) |∆ (0)| ω n˜(ω0) where ω0 = ω(0) and Γ0 are the initial values at z0 = 0. Under the leading approximation, Γ = 2 2 b /2ω is not an explicit function of z0, the ratio reduces to · ¸ T 2 h i2 2 |∆ (z0)| ω0 n˜(ω) 2 −Γz0 T 2 = e . (4.36) |∆ (0)| ω n˜(ω0)

As z0 increases, the particle’s energy decreases, i.e., ω0 ≥ ω(z0),n ˜(ω0) ≤ n˜(ω) and · ¸ hω i2 n˜(ω) 2 0 ≥ 1, ≥ 1. (4.37) ω n˜(ω0)

The probability for the transverse component is controlled by two oppositely trending factors,n ˜(ω)/ω 2 and exp [−Γz0 ]. Initially, the two factors 1/ω andn ˜(ω) dominate compared to the Gaussian-type 2 exponential factor exp [−Γz0 ], the disturbance rises as the energy starts to deplete. Eventually the Gaussian-type damping factor will dominate, and the disturbance decays very fast in the Gaussian- fashion. The energy depletion process is complicated, but the rise and fall of disturbance in the medium can be seen more clearly with the following simple models. If the energy depleting is of the form,

−γz˜ 0 ω = ω0 e , (4.38) whereγ ˜ is some characteristic constant and can be estimated by the simply Doppler model [87]. Then, the ratio becomes

T 2 |∆ (z0)| £ ¤ £ ¤ ' exp 2˜γz + 2βω (1 − e−γz˜ 0 ) − Γz2 ' exp 2(1 + βω )˜γz − Γz2 (4.39) |∆T(0)|2 0 0 0 0 0 0 asγz ˜ 0 is small. The disturbance rises initially until γ˜ z ∼ 2(1 + βω ) . (4.40) 0 0 Γ Afterward, the Gaussian-type decay factor dominates and the transverse disturbance shrinks in the Gaussian way, in contrast to simple exponential decay appeared in the perturbative approach. If the energy depletion follows the same fashion as the propagator,

−Γ0z2 ω = ω0 e 0 , (4.41) than

T 2 h i |∆ (z0)| 0 2 −Γ0z2 2 £ 0 2 2¤ ' exp 2Γ z + 2βω (1 − e 0 ) − Γz ' exp 2(1 + βω )Γ z − Γz (4.42) |∆T(0)|2 0 0 0 0 0 0

Again, the transverse disturbance rises initially, but shrinks after

Γ0 z2 ∼ 2(1 + βω ) . (4.43) 0 0 Γ 91

For the longitudinal component, the amplitude of the forward and backward propagations are proportional to 1 − n˜(ω) exp [(b2/4ω2)(β2/4)] andn ˜(ω) exp [(b2/4ω2)(β2/4)] for a given (ω, ~p), respectively. To evaluate the longitudinal component of disturbance, deﬁne ∆L as · ¸ 2 2 2 2 h i L i b β +iβ b z − b z2 i 00 +iβ Γ z − Γ z2 ∆ = 1 − n˜(ω) e 4ω2 4 e 4ω2 0 e 4ω2 0 = 1 − n˜(ω ) e 2 0 e 2 0 (4.44) f 2ω 2ω for the forward propagating component, and · ¸ 2 2 2 2 h i L −i b β −iβ b z − b z2 −i 00 −iβ Γ z − Γ z2 ∆ = n˜(ω) e 4ω2 4 e 4ω2 0 e 4ω2 0 = n˜(ω ) e 2 0 e 2 0 (4.45) b 2ω 2ω for the backward propagating component, as the fermion proceeds through the medium, and where

b2 β2 β2 ω00 = ω − = Γ . (4.46) 2 ω2 4 4 If one takes the square modulus of both ∆L’s as · µ ¶¸ · µ ¶¸ ¯ ¯ 2 2 L 2 1 00 β Γ −Γz2 1 00 β Γ −Γz2 ¯∆ ¯ = 1 − n˜(ω ) cos z e 0 + n˜(ω ) sin z e 0 (4.47) f 4ω2 2 0 4ω2 2 0 · µ ¶¸ 1 2 00 00 β Γ −Γz2 = 1 +n ˜ (ω ) − 2ñ(ω ) cos z e 0 , 4ω2 2 0 and h i ¯ ¯ 2 β2 L 2 1 2 00 −Γz2 1 2 −Γ z − ¯∆ ¯ = n˜ (ω ) e 0 = [ñ(ω)] e 0 4 , (4.48) b 4ω2 4 ω2 where the backward component is similar to the transverse component; but the forward propagating component is complicated and interwound with both the incident and thermal parts. The magnitude of the longitudinal disturbance for the forward component is dominated by the first term as 1 £ ¤ exp −Γz2 . (4.49) 4ω2 0 Similar to the transverse component, the longitudinal ’fireball’ builds up as ω becomes smaller initially; but subsequently damps away by the Gaussian-type decay factor. Unlike the transverse component, the probability of the forward longitudinal fireball doesn’t include the distribution function; hence, the time scale is different that of transverse fireball. In a nutshell, the phenomenon can be seen as a ”fireball” in the medium induced by the energy depletion of the incident particle, with a probability that first rises and then damps away.

4.5 Mass Shift

In addition to the quantum correction, the physical mass of a fermion also includes a shift due to thermal eﬀect [88, 21, 22, 23]. To see the mass-shift, a fully-dressed, but un-renormalized, ﬁnite- temperature fermion propagator can be written in the form of

˜0 1 Sth(~p,ωn) = , (4.50) m0 + iγ · p − Σ0(~p,ωn) − Σth(~p,ωn) 92

where m0 is the un-renormalized bare mass, Σ0 and Σth are the self-energy at T = 0 and the thermal contribution with a non-vanishing chemical potential µ, respectively. As T → 0 and µ → 0,

˜0 ˜0 µ=0 Sth(~p,ωn) → Sc(~p,p0), Σth → 0, and Σ0 → Σ0 , (4.51) and the theory goes back to the zero-temperature form. It has been argued that the counter terms for the bare mass and wave-function renormalization are independent of temperature and chemical potential [93]. To separate the thermally-induced mass shift from the mass and wave-function renormalization, one can follow the standard procedure of the renormalization at T = 0 and µ = 0

(cf. Ref. [48] or Sec. 2.7); Let ω = −iγ · p = −i(~γ · ~p − γ0ωn), then the inverse of the dressed propagator becomes

0 −1 [S˜ ] = m0 − ω − Σ0(ω; m0) − Σth(ω; m0) (4.52) th h i h i µ=0 µ=0 = m0 − Σ0 (m; m0) − ω − Σ0(ω; m0) − Σ0 (m; m0) − Σth(ω; m0).

The renormalized mass m is deﬁned at zero-temperature with zero chemical potential, i.e., T = 0 ˜0 and µ = 0, and under the condition that Sth has a pole at ω = m with

µ=0 m = m0 − Σ0 (m; m0), (4.53) and the residue of the pole at ω = m is the wave-function renormalization constant as ¯ µ=0 µ=0 ¯ Σ (ω; m0) − Σ (m; m0)¯ Z−1 = 1 + 0 0 ¯ , (4.54) 2 ω − m ¯ ω→m as T → 0 and µ → 0. Then, one can express Σ0(ω; m0) as

µ=0 µ=0 Σ0(ω; m0) = Σ0(ω; m0) − Σ0 (m; m0) + Σ0 (ω; m0) (4.55) µ=0 µ=0 +Σ0 (m; m0) − Σ0 (ω; m0), and the propagator becomes

˜0 Z2 Sth(~p,ωn) = , (4.56) m + iγ · p − Σ0,R − Σth,R where

Σth,R = Z2 Σth, (4.57) and µ=0 Σ0,R = Z2 [Σ0(ω; m0) − Σ0 (ω; m0)], (4.58) which reduces to zero, i.e.,Σ0,R = 0, in the case of vanishing chemical potential. Thus, the renormalized, ﬁnite-temperature fermion propagator becomes

˜0 1 Sth,R(p) = , (4.59) m + iγ · p − Σ0,R − Σth,R where Z2 has been absorbed by redeﬁning the ﬁelds. 93

To see the thermally-induced mass shift, one can express the fully-dressed, ﬁnite-temperature fermion propagator as ˜0 1 Sth,R(p) = 0 , (4.60) m + iγ · p − Σth 0 where Σth = Σ0 +Σth = ReΣ+i ImΣ is the complex self-energy with both temperature dependent and independent components. Following similar procedures in Sec. 2.7, one can ’rationalize’ the denominator as 0 ˜0 m − iγ · p − Σth m − iγ · p − ReΣ − iImΣ Sth(p) = 0 2 2 = 0 2 2 , (4.61) (m − Σth) + (γ · p) (m − Σth) + p which shows that the temperature-dependent part of ReΣ is the thermally-induced mass-shift. From the calculation of the previous chapter, the numerator factor of the fermion propagator in Eq.(3.61) or (3.182) is given by ½ · Z ¸¾ d4k m − iγ · p − g2 D˜ 0 (k) · f˜(−k) · eik·(y−u(s)) (4.62) (2π)4 th ½ Z 4 Z s h i ¾ 2 d k 0 ˜ 0 2ik·ps0 → m − iγ · p + 2ig 4 ds γ · Dth(k) · p e , (2π) 0 where the dressed, ﬁnite-temperature photon propagator is

˜ 0 ˜ 1 Dth(k) = Dth(k) · . (4.63) 1 − K˜ th(k) · D˜ th(k) The temperature dependent part of a non-interacting fermion propagator is proportional to 2πiδ(p2+ 2 m )n ˜(p0), i.e., on the mass-shell (cf. Eq. (2.76)). In one fermion-loop approximation, the thermal photon polarization tensor can be split into two parts as [88, 20]

˜ µν ˜ µν ˜ µν Kth (k) = Kc (k) + δKth (k). (4.64) ˜ µν The leading contribution of the thermal part, δKth (k), comes from the Hot Thermal Loops as [56, 43] h i ˜ ˜ HTL 2 ˆ T ˜ HTL 2 ˆ L ˜ HTL 2 δKµν (k) ' Kµν (k) = −k Pµν ΠT (k ) + Pµν ΠL (k ) , (4.65)

ˆ L ˆ T where Pµν and Pµν are the longitudinal and transverse operator, respectively; and the HTL photon polarization functions are given by [56, 43]

˜ HTL 2 ˜ 2 2 1 ΠL (k ) = 2ΠHTL(k ) − 2msc , (4.66) ~k2 2 ˜ HTL 2 2 1 k0 ˜ 2 ΠT (k ) = msc − ΠHTL(k ), (4.67) k2 ~k2 up to the order of g2T 2, and where Ã ! 2 ~ 2 msc 1 k0 k0 + |k| Π˜ HTL(k ) = − ln (4.68) ~ 2 ~ ~ 2 k |k| k0 − |k| with the screening mass msc as g2T 2 m2 = (4.69) sc 6 94

2 ˜ HTL 2 2 ˜ HTL 2 2 for QED. Since k ΠL (k ) and k ΠT (k ) vanish near k = 0,

˜ HTL ˜ ˜ HTL ˜ KL (k) · δDth(k) = 0, and KT (k) · δDth(k) = 0. (4.70)

Thus, 1 1 D˜ th(k) · ' D˜ c(k) · + δD˜ th(k) (4.71) 1 − K˜ th(k) · D˜ th(k) 1 − K˜ th(k) · D˜ c(k) 1 ' D˜ c(k) · + δD˜ th(k) 1 − K˜ c(k) · D˜ c(k) ˜ 0 ˜ = Dc(k) + δDth(k),

˜ µν where the ﬁrst term only retains Kc (k) in further approximation. The temperature-dependence of the ﬁrst term is at least order g2 smaller compared to that of the second term. Under the leading approximation, one obtains

Z 4 Z s h i 2 d k 0 ˜ 0 2ik·ps0 2ig 2 ds γ · Dth(k) · p e (4.72) (2π) 0 Z 4 Z s h i 2 d k 0 ˜ 0 ˜ 2ik·ps0 ' 2ig 2 ds γ · Dc(k) · p + γ · δDth(k) · p e . (2π) 0

˜ 0 ˜ If the quenched approximation is implied instead, Dth(k) will be replaced by Dth(k). In regular Bloch-Nordsieck approximation, the particle of interest experiences negligible recoil from contributions of soft photons, and the gamma matrix γµ is usually replaced with a four-velocity −ivµ with 2 vµ ∼ pµ/m and v = −1 [49]. Thus, one can assume

(~γ · ~k)(~k · ~p) ' (~γ · ~p) cos2 θ. (4.73) ~k2 Then, the two terms can be evaluated as

Z 4 Z s h i d k 0 2ig2 ds0 γ · D˜ 0 (k) · p e2ik·ps (4.74) (2π)2 c 0 · √ ¸ 1 π ' −s(~γ · ~p) g2~p 2 1 + i ²(ω ) ω , 3π2 4 n n and

Z 4 Z s h i 2 d k 0 ˜ 2ik·ps0 2ig 2 ds γ · δDth(k) · p e (4.75) (2π) 0 1 1 ' −s(~γ · ~p) g2 9π2 β2 1 = −s(~γ · ~p) g2T 2. 9π2 The ﬁrst term is temperature-independent and ﬁnite, but it is the artifact of the Bloch-Nordsieck approximation, which excludes the high frequency components and only retains contributions from soft photons [49]. Following similar procedure for the estimate of damping, one can replace the 95

¡ ¢ s-factor by the operator i ∂/∂ω2 , and yield ½ · Z ¸¾ d4k m − iγ · p − g2 D˜ 0 (k) · f˜(−k) · eik·(y−u(s)) (4.76) (2π)2 th ½ Z 4 Z s h i ¾ d k 0 ' m − iγ · p + 2ig2 ds0 γ · D˜ (k) · p e2ik·ps (2π)2 th ½ · 0 µ √ ¶¸¾ 1 1 π ' m − iγ · p − sg2 (~γ · ~p) T 2 + ~p 2 1 + i ²(ω ) ω 9π2 3π2 4 n n ½ · µ √ ¶¸ µ ¶¾ 1 1 π ∂ → m − iγ · p − g2 (~γ · ~p) T 2 + ~p 2 1 + i ²(ω ) ω i . 9π2 3π2 4 n n ∂ω2 The real, T -dependent term inside the square bracket can be considered as the mass-shift while the imaginary term is related to the total energy of the system. The mass-shift due to the thermal eﬀect, the ﬁrst term, is on the order of α. µ ¶ Z 1 ∂ 1 ∞ © ª δm = −(~γ · ~p) g2T 2 i · · i ds exp −is(ω2 − ω2 ) (4.77) T 2 2 (0) n 9π ∂ω Z [β] 0 1 2 2 1 = +i(~γ · ~p) 2 g T 2 2 2 9π (ω − ωn) 1 2 2 1 → +i(~γ · ~p) 2 g T 2 2 2 , 9π (ω + |ωn| ) where the continuation of τ → −iβ has been taken in the last line. Because ~γ ' −i~v = −i~p/m, the thermal mass-shift becomes 1 g2T 2 ~p 2 δmT ' + 2 2 2 2 (4.78) 9π m (ω + |ωn| ) 1 g2T 2 ~p 2 . + . 9π2 m (ω2)2

Here, the positive size of thermal mass-shift δmT implies that the incident fermion becomes heavier as it slows down. In the limit of p À T , the temperature-dependent mass shift is negligible. If the ˜ F mixed representation ∆th(ω, z0) of Eq. (2.164) is used instead, the thermal mass-shift becomes a complicate function of time as µ ¶ 1 ∂ δm = −(~γ · ~p) g2T 2 i · ∆˜ F (ω, z ) (4.79) T 9π2 ∂ω2 th 0 µ ¶ · ¸ 1 ∂ i © ª = −(~γ · ~p) g2T 2 i · [1 − n˜(ω)] e−iωz0 − n˜(ω) e+iωz0 , 9π2 ∂ω2 2ω where the physics is not readily transparent. Earlier attempts have mostly worked on the case of low temperature limit with one-loop correction to the fermion mass. For example, π T 2 δm = − α ω, for T ¿ m (4.80) T 9 m from Cox et al. [22], and 2παT 2 δm2 = , for T ¿ m (4.81) T 3m 2 2 δmT = παT , for T À m, (4.82) 96 or the phase-space mass shift παT 2 δm = , (4.83) T 3m from Donoghue et al. [23]. Compared to the results from the perturbative theory with one-loop corrections of Cox et al. [22] and Donoghue et al. [23], the mass shift is on the order of g2T 2.

4.6 Impact of Gauge

At zero temperature theory, the photon propagator is gauge dependent as well as the field-coupled fermion Green’s function under influence of gauge fields. At finite temperature, a thermal distribution is defined in the rest frame of the medium, which, in turns, defines an intrinsic preferred frame; a preferred vector is selected in any other frame, which is the velocity of the thermal bath that maintains thermal equilibrium [22]. The presence of plasma modeled as a thermal distribution with a rest frame velocity uµ breaks the Lorentz symmetry [20, 94, 56]. Therefore, the Coulomb gauge, one of non-covariant gauges, is chosen to simplify the calculation. The photon propagator in the general Coulomb gauge for the zero temperature theory is [68]

2 ˜ ζ 1 1 ˆ T k kµkν ˜ c ˜ ζ Dµν (k) = − δµ0δ0ν + Pµν + ζC = Dµν (k) + δDµν (k), (4.84) ~k2 k2 − i² (~k2)2 k2 where ζC is the (Coulomb) gauge parameter. The ﬁnite-temperature photon propagator can be written as ˜ µν . ˜ µν ~ Dth (k) = Dc (k, k0 = ωn). (4.85) After analytical continuation, the ﬁnite-temperature photon propagator is separated into two terms as ˜ µν ˜ µν ~ ˜ µν Dth (k, k0) = Dc (k, k0) + δDth (k) (4.86) with h i ˜ µν ˜ th ~ ˜ th ~ δDth (k) = f(k0) Dµν (k, ωn = k0 + i²) − Dµν (k, ωn = k0 − i²) (4.87) ˜ ζ ˜ ζ Similarly, there is no contribution from the gauge term δDth(k) for the thermal part, i.e., δDth(k) =

δD˜ ζ (k). Therefore, the result of fermion damping from the thermal-photon-induced Bremsstrahlung is gauge-invariant. For the ordinary Bremsstrahlung, the gauge dependent part of the decay exponent is Z Z Z 4 s s1 h i d k 0 4ig2 ds ds0 p · δD˜ (k) · p e2ik·ps (4.88) (2π)4 1 ζ Z 0 Z 0 Z 4 s s1 2 2 d k 0 (p · k) 2ik·ps0 = 4ig ζC 4 ds1 ds e , (2π) 0 0 (~k2)2 Z Z Z 4 s s1 ~ 2 2 d k 0 (~p · k − ωnk0) 2ik·ps0 = 4ig ζC 4 ds1 ds e . (2π) 0 0 (~k2)2

The k0-integral generates delta-function related terms and Z Z · ¸ s s1 2 0 ~ 2 0 ~ ∂ 0 1 ∂ 0 2i~k·~ps0 ds1 ds (~p · k) δ(2ωns ) − 2i(~p · k) 0 δ(2ωns ) + 02 δ(2ωns ) e = 0 (4.89) 0 0 ∂s 4 ∂s 97

as ωn 6= 0 for fermions. Therefore, the decay exponent for the ordinary Bremsstrahlung is also independent of gauge. For pair production, the decay exponent contains

2 k kµkν 2 2 δD˜ ζ · K˜ c = −ζC k Π˜ (k )Pˆ νσ = 0, (4.90) (~k2)2 k2 where kν Pˆ νσ = 0; and the the damping exponent from pair production is again gauge-independent. On the other hand, the gauge-dependent, ﬁnite-temperature photon propagator can be written as ˜ ζ ˜ ˜ ˜ ˜ ˜ Dth(k) = Dth(k) + δDζ (k) = Dc(k) + δDth(k) + δDζ (k), (4.91)

The fully-dressed, ﬁnite-temperature photon propagator under the assumption of K˜ th(k) ' K˜ c(k) becomes h i ˜ 0ζ ˜ ˜ 1 Dth(k) ' Dth(k) + δDζ (k) · h i (4.92) 1 − K˜ c(k) · D˜ th(k) + δD˜ ζ (k) h i 1 = D˜ th(k) + δD˜ ζ (k) · 1 − K˜ c(k) · D˜ c(k) 1 = D˜ c(k) · + δD˜ th(k) + δD˜ ζ (k), 2 1 + Π˜ c(k ) where δD˜ ζ (k)·Pˆ = 0 and δD˜ th(k)·K˜ c(k) = 0 are used in the derivation to the last line as kµ Pˆ µν = 0 2 and δ(k ) K˜ c(k) = 0, respectively. Thus, the dressed, ﬁnite-temperature photon propagator is gauge- dependent under the current approximation. However, the decay exponents estimated in the previous chapter are gauge invariant, where the gauge term δD˜ ζ (k) vanishes in the decay exponent as shown in Eqs.(4.88) and (4.89).

4.7 Non-Zero Chemical Potential

For the case of non-vanishing chemical potential, the calculations above can be modiﬁed by simply changing the energy component of 4-momentum of fermions to [88, 56]

p0 → p0 + µ (4.93) in the real-time formalism, or

ωn → ωn + µ (4.94) in the imaginary-time formalism.

Let P ≡ (~p,ωn +µ), the Bloch-Nordsieck-approximated, ﬁnite-temperature Green’s function, Eq. 98

(3.48), becomes

BN h~p,n|Gth [A]|~y, y0i (4.95) Z ∞ £ ¤−1/2 2 2 2 = (2π)3τ · e−i[~p·~y−(ωn+µ)y0] · i ds e−is [m +~p −(ωn+µ) ] ½ · Z 0 ¸¾ d4q × m − iγ · p − g A˜(q) eiq·(y+2sP ) (2π)4 µ ½ Z s Z 4 h i ¾¶ 0 d q ˜ ˜ iq·(y+2s0P ) × exp g ds σµν · 4 qµAν (q) − qν Aµ(q) e 0 (2π) + ½ Z s Z 4 h i ¾ 0 d q ˜ iq·(y+2s0P ) × exp +2ig ds 4 p · A(q) e . 0 (2π) Similarly, the quenched ﬁnite-temperature fermion propagator, Eq. (3.73), becomes

0 BN h~p,n|Sth|~y, y0iQ (4.96) Z ∞ £ ¤−1/2 2 2 2 ' [Z(0)[iτ]]−1 · (2π)3τ · e−i[~p·~y−(ωn+µ)y0] · i ds e−is [m +~p −(ωn+µ) ] 0 ½ Z 4 h i Z s ¾ d k 0 × m − iγ · P + 2ig2 γ · D˜ (k) · P ds0 e2ik·P s (2π)4 th ½ Z Z Z 0 ¾ 4 s s1 h i 2 d k 0 ˜ 2ik·P s0 × exp 4ig 4 ds1 ds P · Dth(k) · P e , (2π) 0 0 and the closed-fermion-loop approximated quenched ﬁnite-temperature fermion propagator, Eq. (3.182), becomes

0 h~p,n|Sth|~y, y0i (4.97) Z ∞ 1 £ ¤−1/2 2 2 2 = (2π)3τ · e−i[~p·~y−(ωn+µ)y0] · i ds e−is [m +~p −(ωn+µ) ] (0) Z [iτ] 0 ½ Z 4 h i Z s ¾ d k 0 × m − iγ · P + 2ig2 γ · D˜ 0 (k) · P ds0 e2ik·P s (2π)4 th ½ Z Z Z 0 ¾ 4 s s1 h i 2 d k 0 ˜ 0 2ik·P s0 × exp 4ig 4 ds1 ds P · Dth(k) · P e . (2π) 0 0 There is also the modiﬁcation in the denominator of the propagator including the thermal mass-shift, and the decay exponents, i.e., the real part of Z Z Z 4 s s1 h i 2 d k 0 ˜ 0 2ik·P s0 4ig 4 ds1 ds P · Dth(k) · P e , (4.98) (2π) 0 0 can be separated into three damping mechanisms; " # 2 ~ 2 (ωn + µ) 2 (k · ~p) 1 P · D˜ c(k) · P = − + ~p − , (4.99) ~ 2 ~ 2 ~ 2 2 k k k − k0 − i² for ordinary Bremsstrahlung, " # ~ 2 2 ~ 2 δ(k − k0) 2 (k · ~p) P · δD˜ th(k) · P = ~p − (4.100) eβ|k0| − 1 ~k2 99 for thermal photon-assisted Bremsstrahlung, and " # µ ¶ h i ~ 2 2 2 2 (k · ~p) 1 m + y(1 − y)k P · D˜ c(k)Π˜ c(k) · P = ~p − ln , (4.101) ~ 2 ~ 2 2 m2 k k − k0 − i² for pair production, under the ordinary Coulomb gauge, and D˜ th(k) = D˜ c(k) + δD˜ th(k). Only the term from the ordinary Bremsstrahlung contains the chemical potential, but vanishes after k0- integration. Other chemical potential dependent terms are in the phase factors exp [2ik · P s0], and the contributions are within cosine or sine functions of w(±), where pξ ± ω pξ ± (ω + µ) w(±) = n → n , (4.102) β β which are oscillatory factors, and vanish in the approximation used. Hence, the decay exponents do not change with the chemical potential added.

Instead of S˜th(~p,z0; µ = 0), the non-interacting ﬁnite-temperature fermion propagator is replaced by Eq. (2.171) as · µ ¶¸ 1 ∂ S˜th(~p,z0; µ) = m − i~γ · ~p + iγ0 · µ − (4.103) i ∂z0 i © ª × [1 − n˜(ω + µ)] e−iωz0 − n˜(ω − µ) e+iωz0 , 2ω which contains asymmetric thermal-terms, and

h~p,x |S0 |~p,y i (4.104) · 0 th 0 µ ¶¸ i 1 ∂ ' m − i~γ · ~p + iγ · µ − 2 0 i ∂z ½ 0 ¾ 2 2 h 2 2 i 2 1 −iωz + β b −i[ω−β b ]z +i[ω−β b ]z − b z2 × e 0 − e 16ω2 n˜(ω + µ) e 4ω2 0 +n ˜(ω − µ) e 4ω2 0 e 4ω2 0 , ω where b2 is the same as that of vanishing chemical potential.

4.8 Hot Thermal Loop Approximation Revisited

When the momenta of all ’external’ legs of a graph are soft, corrections from certain subsets of graphs, such as Hot Thermal Loops (HTL), to the bare graph are of the same order of the coupling constant g as the ’bare’ amplitude at finite temperature. Under the Hot Thermal Loop approximations, the internal loop momenta are hard compared to the external momenta, and the contributions from Hot Thermal Loops are similar to the tree graphs in the same order; and the perturbative approaches resort to the resummation of Hot Thermal Loops to account for finite-temperature effects. It has been shown that the HTL contributions to photon amplitudes are only in the two-point function [5, 36]. The effective HTL Lagrangian can be deduced from the effective, or resummed, photon propagator as [95, 96] * + ˜ ˜ 1 2 kαkβ δLHTL = m Fµα Fαµ, (4.105) 4 γ −(k˜ · ∂)2 100

˜ ~ ˆ where kµ = |k| (i, k), the angled bracket h· · · i is the average over all possible directions of the 2 2 2 loop momenta k, i.e., the angular integral over Ωkˆ, and the thermal photon mass is mγ = g T . Alternatively, the eﬀective HTL Lagrangian addition can be written as [45, 46] Z Z 1 4 4 δLHTL = d x d y A(x) · KHTL(x − y) · A(y), (4.106) 2 C C

HTL where Kµν is the effective HTL photon polarization tensor. The addition δLHTL is then appended into the effective photon Lagrangian as [46] Z Z 1 2 1 4 4 LHTL = Lphoton + δLHTL = − F + d x d y A(x) · KHTL(x − y) · A(y), (4.107) 4 2 C C which generates the resummed HTL photon propagators. To make a comparison to the resummed HTL approach, the fully-dressed, finite-temperature fermion propagator can also be given in terms of functional integral instead of functional method with linkage operator used in Eq. (2.85) as Z ½ Z ¾ i 0 −1 Lth[A] ˜ −1 Sth = ZG [iτ, µ] DA Gth[A] e exp − A · Dth · A . (4.108) 2 C Under the HTL approximation, the fermion propagator becomes [44, 30, 45, 46] Z ½ Z Z ¾ 0 −1 i ˜ −1 i ˜ Sth = ZG [iτ, µ] DA Gth[A] exp − A · Dth · A + A · KHTL · A , (4.109) 2 C 2 C where the closed-fermion-loop functional Lth[A] is replaced by the effective HTL Lagrangian iδLHTL. With help of the Schwinger-Dyson equation, one can define the re-summed photon propagator as [20] ˜ −1 ˜ −1 ˜ DHTL = Dth − KHTL, (4.110) which leads to Z ½ Z ¾ 0 −1 i −1 Sth = ZG [iτ, µ] DA Gth[A] exp − A · DHTL · A , (4.111) 2 C in the form of functional integral, or ½ Z ¾ ¯ ¯ 0 −1 i δ δ ¯ Sth = Z(0) [iτ, µ] exp − · DHTL · · Gth[A]¯ (4.112) 2 C δA δA A→0 in terms of linkage operator with the resummed photon propagator DHTL, where the normalization constant of the functional integral is absorbed into Z(0). The finite-temperature fermion propagator under the resummed HTL approximations is similar in form of the quenched approximated propagator in Eq. (3.68), and leads to the set of modified Feynman rules with photon lines replaced by resummed photon propagators. Interestingly enough, the result from the resummed program is of the same g2T 2-dependence, if the calculation is performed properly without IR divergence, as that of the thermal-photon assisted Bremsstrahlung. ˜ HTL Separate the HTL photon polarizationh tensor into longitudinali and transverse parts as Kµν = ˜ HTL L ˜ HTL T 2 ˜ HTL L ˜ HTL T KL Pµν + KT Pµν = −k ΠL Pµν + ΠT Pµν , and the resummed photon propagator in 101 the usual Coulomb gague becomes [56]

δ δ PT D˜ HTL = − µ0 0ν + µν , (4.113) µν ~ 2 ˜ HTL 2 2 ˜ HTL 2 k + KCL (k ) k + KCT (k ) ˜ HTL 2 ˜ HTL 2 where KCL (k ) and KCT (k ) are the (00)-component and transverse part of the HTL approximated photon polarization tensor in Coulomb gauge. µ ¶ ~k 2 iω K˜ HTL(k2) = K˜ HTL(k2) = −2m2Q , (4.114) CL k2 L 1 k µ ¶ "Ã µ ¶ ! µ ¶ µ ¶# iω iω 2 iω iω K˜ HTL(k2) = K˜ HTL(k2) = m2 1 − + Q + , CT T k k 0 k k where Q0 and Q1 are Legendre Function of the second kind (cf. Appendix E.5). In the previous chapter, the closed-fermion-loop functional is approximated by a gauge-invariant one-fermion-loop functional, which leads to pair production as a damping mechanism. In contrast, the HTL approximation along with the resummation program only leads to the Bremsstrahlung processes, but fails to include pair production. When the HTL approximation is employed in the perturbation theory, there are at least two problems:

• The resummed HTL approximation cannot produce the static screening of transverse gauge ˜ HTL mode, i.e., KT → 0 as k0 → 0, such that the calculation of the damping rate of a fast moving particle in the hot medium is still plagued with an IR divergence.

• The resummed HTL approximated calculation cannot account for the production of soft, real photons, so the collinear singularity exists in the massless on-shell limit.

In addition, it has been argued that one needs to sum an infinite class of multi-loop Feynman graphs with effective HTL photon propagators to get rid of IR divergence [45]. If the functional method with correct linkages is used instead, the IR divergence will not appear in the proper Bloch- Nordsieck approximation. In essence, the resummed HTL approximation could be equivalent to the quenched approximation used in the calculation of the Bremsstrahlung-related damping in the previous chapter, were the HTL appropriately redefined to remove its problem noted directly above.

4.9 Possible Extension to QCD

The QCD Lagrangian can be expressed as 1 L = L + L + L = −ψ¯ [m + γ · (∂ − igA)] ψ − G2, (4.115) quark qloun int 4 a a a a where Aµ = Aµλ are gauge ﬁelds, Gµν = Gµν λ is the ﬁeld strength,

a a a b c Gµν = ∂µAν − ∂ν Aµ + gfabcAµAν , (4.116) and λa are the color matrices in the fundamental representations of SU(N);

a b c a a b [λ , λ ] = 2ifabcλ , tr [λ ] = 0, tr [λ λ ] = 2δab, (4.117) 102

a where fabc is the structure constant and deﬁnes the adjoint representation (¯τ )bc of SU(N). The ﬁeld strength becomes

Gµν = ∂µAν − ∂ν Aµ − ig[Aµ,Aν ]. (4.118)

Since the last term of the field strength Gµν is quadratic in the gauge field Aµ, the gluon sector of the QCD Lagrangian contains terms both cubic and quartic in Aµ, which involves closed-gluon- loops in the generating functional. To alleviate the difficulty of calculations with the higher-than quadratic couplings of gauge fields, one can replace it with effective field strengths as [97, 98] · Z ¸ Z · Z Z ¸ i exp − G2 = N d[σ] exp i σ2 + i σ · G , (4.119) 4 R a where σµν is the effective field strength, N is a normalization constant, and σ · G is only quadratic in A. Alternatively, the gluon term in the QCD Lagrangian can be re-written as [98, 49]

G2 = F2 + (G2 − F2) = F2 + (G − F) · (G + F), (4.120) where Fµν = ∂µAν − ∂ν Aµ are the QED-type ﬁeld strengths. Similarly, · Z ¸ i exp − (G − F) · (G + F) (4.121) 4 Z Z · Z Z Z ¸ i i = N 0 d[φ] d[ϕ] exp i φ · ϕ + φ · (G − F) + ϕ · (G + F) , 2 2

a a where φµν and ϕµν are antisymmetric, auxiliary ﬁelds. Instead of cubic and quartic in gauge ﬁleds, the exponent in this representation is only linear or quadratic in Aµ as

G + F = 2F + gA · τ¯ · A, G − F = gA · τ¯ · A. (4.122)

(0) Thus, the gluon sector of the QCD Lagrangian can be separated into two parts as Lqloun = Lqloun + 0 Lgluon, where 1 1 L(0) = − F2, and L0 = − (G2 − F2), (4.123) qloun 4 gluon 4 (0) 0 and the QCD Lagrangian can be rewritten as the ’modiﬁed’ QED form as LQCD = L +Lint, where 1 L(0) = L + L(0) = −ψ¯ [m + γ · ∂] ψ − F2 (4.124) quark qloun 4 is the ’free’ Lagrangian, which is similar to the non-interacting QED lagrangian, and 1 L0 = L + L0 = +ig ψ¯ γ · A ψ − (G2 − F2) (4.125) int int gluon 4 is the ’modiﬁed’ interacting part. Then, the generating functional becomes ½ Z · ¸¾ 1 δ 1 δ −1 δ Z {j, η,¯ η} = hSi−1 exp i L0 , , · Z(0){j, η,¯ η}, (4.126) c int i δj i δη¯ i δη c

a a b b c c where jµ = jµλ , ην = ην λ , andη ¯σ =η ¯σλ are gluon and quark sources, respectively. Following similar derivation of QED, the non-interacting generating functional is given by ½ Z Z ¾ i Z(0){j, η,¯ η} = exp j · D · j + i η¯ · S · η . (4.127) c 2 c c 103

0 Since L does not involve η orη ¯, one can ﬁrst perform the Gaussian translation operation in Lint gluonR over exp [i η¯ · Sc · η] as

hSi Z {j, η,¯ η} (4.128) ½c Z · ¸¾ ½ Z µ ¶ ¾ ½ Z Z ¾ 1 δ δ δ δ i = exp i L0 · exp ig γ · · exp j · D · j + i η¯ · S · η gluon i δj δη δj δη¯ 2 c c ½ Z · ¸¾ ½ Z · ¸ · ¸¾ ½ Z ¾ 1 δ 1 δ 1 δ i = exp i L0 · exp i η¯ · G · η + L · exp j · D · j , gluon i δj c i δj i δj 2 c

−1 1 δ where Gc[A] = Sc · [1 − g (γ · A) Sc] and L[A] = Tr ln [1 − g (γ · A) Sc] with A replaced by i δj , which are similar to QED. Use the reciprocity relation in Eq. (D.21), the generating functional becomes ½ Z ¾ ½ Z ¾ i i δ δ hSi Z {j, η,¯ η} = exp j · D · j · exp − · D · (4.129) c 2 c 2 δA c δA ½ Z ¾ ½ Z ¾ 0 × exp i η¯ · Gc[A] · η + L[A] · exp i Lgluon[A] , R 4 where Aµ(x) = d y Dc(x − y) · j(y) as before. R 0 The formalism in QCD is similar to that in QED except the extra factor of exp [i Lgluon] in the 0 generating functional. The diﬃculty of cubic and quartic dependence of gauge ﬁelds in Lgluon can be avoided by utilizing the functional representation of Eq. (4.121) as

hSi Z {j, η,¯ η} (4.130) Zc Z ½ Z ¾ ½ Z ¾ ½ Z ¾ i i δ δ = N 0 d[φ] d[ϕ] exp i φ · ϕ · exp j · D · j · exp − · D · 2 c 2 δA c δA ½ Z ¾ ½ Z Z ¾ i i × exp i η¯ · G [A] · η + L[A] · exp φ · (G − F) + ϕ · (G + F) . c 2 2

Any n-point Green’s function can then be derived from the generating functional as that of QED, and extend to the ﬁnite-temperature theory by following similar procedures for QED. The ﬁeld-coupled

Green’s function Gc[A] and the closed-quark-loop functional L[A] can be constructed similarly to a those in QED. However, gauge fields Aµ in QCD are non-Abelian, any ordinary exponential involving gauge fields in the representation of both Gc[A] and L[A] becomes an ordered-exponential, which is complicated to evaluate even at zero-temperature. At zero temperature, quasi-Abelian approximations have been proposed [99, 100, 101, 102, 103, 50] for both Gc[A] and L[A]. In addition, the a color degrees of freedom of non-Abelian gauge fields Aµ in SU(N) complicate any approximation of the closed-loop functional L[A] through color current. Once proper approximations are made at zero temperature, the extension to finite-temperature should be readily applicable. Chapter 5

Conclusions

The finite-temperature quantum field theory is introduced in functional formalism in the spirit of Schwinger, Fradkin, and Fried [48, 49, 50], with a special focus on QED. The main subject of interest is to investigate the phenomenon of a fast moving fermion entering into a hot QED plasma, with the incident fermion exchanging virtual and real photons with particles in a thermal bath. The information of the particle and the thermal medium (heat bath) is encoded in the finite-temperature propagators, which are treated in the Matsubara, imaginary-time formalism, using functional methods. The dressed, finite-temperature fermion propagator is calculated by a functional method linking the particle’s Green’s function with the close-fermion-loop functional as ¯ · L [A] ¸ (th) e th ¯ 0 DA ¯ Sth = e · Gth[A] ¯ . (5.1) Z[iτ] A→0 A new variant of Fradkin representation of the Green function is introduced, to set the stage for the application of an eikonal approximation. As the thermal distribution of the medium defines a preferred frame, the Coulomb gauge, a non-covariant gauge, is used in the calculation, which turns out to be most convenient when summing over all Matsubara frequencies. The thermal photon propagator in the linkage operator is separated into T = 0 and T 6= 0 parts, such that the calculation of physical processes is separated from that of the conventional renormalization constants, so that all integrals of internal loop momenta are finite, although complicated. In addition, the finite- temperature fermion propagator is estimated under a proper Bloch-Nordsieck approximation without fixing the incident particle momentum to be constant as the previous publication in Ref. [47]. A leading approximation is made to evaluate effects of the energy-momentum depletion. Under quenched approximation, the calculation with the thermal part of photon propagators leads to the damping process through thermal-photon-assisted Bremsstrahlung; and the energy depletion through ordinary Bremsstrahlung is estimated with linkage of the causal part of photon propagator. The closed-fermion-loop functional is approximated with one-fermion-loop in the non-quenched calculation; the results from the quenched approximation are recovered through redef- inition (or renormalization) of the coupling constant; and the one-fermion-loop approximation also leads to energy depletion through pair production. The contribution of fermion spin is negligibly 104 105 small and not included in the estimate as the internal loop-momentum is much smaller than that of incident particle. The fully-dressed, finite-temperature fermion propagator sums over all Matsubara frequencies instead of focusing on the static mode with n = 0, as have previous attempts using perturbative methods in Refs. [6, 45, 46]. The finite-temperature fermion propagator is presented as a Gaussian- translation, damping operator operating on a non-interacting propagator with respect to the energy square ω2 as " µ ¶ # ∂ 2 S˜0 (~p,z ) ' exp −2 ω2 Γ i · S˜ (~p,z ), (5.2) th 0 ∂ω2 th 0 and the essence of damping is separated from the non-interacting propagator. Estimates of three mechanisms of energy-momentum depletion are presented: thermal-photon- assisted Bremsstrahlung, ordinary bremsstrahlung, and pair production. In contrast to an exponential decay (with an extraneous logarithmical factor in the exponent) from perturbative approach, £ ¤ 2 the damping of incident particle is of the Gaussian-type, as exp −Γz0 with · µ ¶¸ 2 Λ2 2ω2Γ = g2 ~p 2 a2 T 2 + a2 Λ2 + a2 g2Λ2 ln , (5.3) 3π2 T B P m2 where Λ is the soft-momentum cutoff appropriate for the Bloch-Nordsieck approximation used in the estimate with Λ ≤ |~p|, and

π2 1 1 a2 = , a2 = , and a2 = , (5.4) T 6 B 4 P 48π2 are the numerical coefficients for the processes of thermal-photon-assisted Bremsstrahlung, ordinary Bremsstrahlung, and pair production, respectively. The energy depletion through pair- production is of higher ordered compared to that of Bremsstrahlung processes, with an extra factor ¡ ¢ of g2 ln Λ2/m2 which is small in weak coupling. Even if the coupling constant were large instead of its QED value, the correction from the photon polarization function to the fermion self-energy is limited to 1 by a unitary denominator, which suppresses the contribution of such pair production. In contrast to thermalization of particles in Hot Thermal Loops, fermion-anti-fermion pairs are not thermalized at the instant of pair production relevant to the energy depletion of the incident particle. IR divergence, which appeared in the perturbative computation and is removed only after resummation of an infinite class of multi-loop HTL graphs, is completely avoided in the present treatment. The result of thermal-photon assisted Bremsstrahlung is of similar order of g2T 2 to that of the resummed Hot Thermal Loops, which prompts the possibility that the two approaches might be equivalent if the later is treated properly in a non-perturbative way. However, the resummation of Hot Thermal Loops failed to include pair production as a damping mechanism. In addition to damping of incident particle, the possibility of short-term growth in probability factors necessary for a longitudinal and transverse fireball can also be extracted from the finite- temperature propagator. Furthermore, the probability of building up and shrinking down of such fireball probability can be extracted from the dressed, finite-temperature propagator. 106

The eﬀective, temperature-induced mass-shift of incident particle is also discussed, which is small when ω À T with small coupling constant g. The eﬀect of gauge and chemical potential is also discussed; there is no gauge-dependence in damping factors of three energy-depletion mechanisms, and nor is chemical potential in the leading approximation. A possible extension to QCD is also pointed out, using a parallelism to QED, which can be drawn in the context of certain functional application. Appendix A

Units and Metric

A.1 Natural Units

Natural units are frequently employed to simplify the notation in the calculation of Quantum Field Theory. Except few noted places for explicit dimensional expression, the concise convention will be used throughout this thesis. In natural units, both length and time are set to be in terms of the mass unit with

c = 1, ~ = 1, kB = 1 (A.1)

Furthermore, the unit of mass is expressed in terms of energy in MeV or GeV, i.e.,

m → mc2 (A.2)

Other quantities are also in units of energy. For temperature, the Boltzmann constant is also set to 1 as

kB = 1 (A.3)

Thus, all quantities used here are in units of energy. Imaginary time τ runs from 0 → ~/T with ~ = 1. The conversion factors from energy units to conventional units are listed in Table (A.1) [104].

A.2 Metric

Throughout the thesis, the Minkowski metric is adapted as the following: Four vector:

xµ = (~x,x4) = (~x,ix0), µ = 1, 2, 3, 4 (A.4)

Scalar product

x · y ≡ x1y1 + x2y2 + x3y3 + x4y4 = ~x · ~y − x0y0 (A.5)

107 108

Table A.1: Dimension of physical quantity in natural units. The conversion factor is to be used from energy units to conventional units. (Note1: The conventional electric charge is in Heaviside-Lorentz unit.)

Quantity Dimension Conversion factor Form Mass [M] 1/c2 mc2 Length [M]−1 ~c x/~c Time [M]−1 ~ t/~ Energy [M] 1 E Momentum [M] 1/c pc Force [M]2 1/~c F ~c 0 Action [M] √~ S/√~ Electric charge1 [M]0 ~c e/ ~c Temperature [M] 1/kB kBT

Causal structure:

2 2 2 x is time-like : x = ~x − x0 < 0 (A.6) 2 2 2 x is space-like : x = ~x − x0 > 0 2 2 2 x is light-like : x = ~x − x0 = 0

For any four-vector, pµ, to be ’physical’, i.e., time-like,

2 2 p0 > ~p (A.7) d’Alembertian operator: 2 ~ 2 2 ∂ = ∇ − ∂0 (A.8) Metric tensor

gµν = diag(+1, +1, +1, +1) = δµν µ, ν = 1, 2, 3, 4 (A.9) or equivalently

gµν = diag(−1, +1, +1, +1), µ, ν = 0, 1, 2, 3 (A.10)

Fourier transform (form is independent of metric) Z Z 4 ip0·x0−i~p·~x 4 −ip·x f(~p,p0) = d x e f(~x,x0) = d x e f(x) (A.11)

Gamma matrices:

{γµ, γν } = 2δµν (A.12) for µ, ν = 1, 2, 3, 4, and † γµ = γµ (A.13)

γ5 matrix:

γ5 = γ1γ2γ3γ4; (A.14) 109

{γ5, γµ} = 0, µ = 1, 2, 3, 4, (A.15) and † 2 γ5 = γ5, γ5 = 1; (A.16)

1 σ = [γ , γ ] = −iγ γ , µ 6= ν; (A.17) µν 2i µ ν µ ν Gamma matrices in Pauli representation: Ã ! Ã ! Ã ! 0 −iσi 1 0 0 1 γi = , γ4 = , γ5 = − (A.18) iσi 0 0 −1 1 0 where the Pauli matrices are Ã ! Ã ! Ã ! 0 1 0 −i 1 0 σ1 = , σ2 = , σ3 = . (A.19) 1 0 i 0 0 −1

A.3 Gordon decomposition

† Dirac equations for u(p) andu ¯(p) = u γ4:

(m + iγ · p) u(p) = 0, (A.20) u¯(p)(m − iγ · p) = 0, (A.21) where the spin index is suppressed. Gordon decomposition of current [85]: · ¸ (p + p0) σ (p − p0) u¯(p0) γ u(p) =u ¯(p0) −i µ + µν ν u(p) (A.22) µ 2m 2m Appendix B

Matsubara Summation

There are several techniques to evaluate Matsubara sums. At ﬁrst, the inﬁnite frequency sums can be carried out by conventional contour integral [55, 105, 56]. The summation over discrete frequencies (or energies) is represented by a contour integral with an appropriate integrand.

B.1 Standard Contour Integral

After introduction of a proper seed function as integrand, the Matsubara summation over any function f(z0) which is analytic in the neighbourhood of the imaginary axis, and the product 2nπ f(z0) exp [−β|z0|] vanishes suﬃciently fast at inﬁnity. For bosons, k0 = τ Z Z 1 X dz +i∞ dz f(k ) = 0 n(z )[f(z ) + f(−z )] + 0 f(z ) (B.1) τ 0 2π 0 0 0 2π 0 n C+ −i∞

(2n+1)π For fermions, p0 = τ + µ Z Z 1 X dz +i∞ dz f(p ) = − 0 [˜n (z )f(z ) +n ˜ (z )f(−z )] + 0 f(z ) (B.2) τ 0 2π + 0 0 − 0 0 2π 0 n C+ −i∞ where the Bose-Einstein and Fermi-Dirac distribution functions are 1 n(z0) = (B.3) eβz0 − 1

1 n˜±(z0) = (B.4) eβ(z0∓µ) + 1

The contour C+ circumscribes clockwise all singularities of the functions f(±z0) in the right half plane without any pole of distribution functions n(z0) andn ˜±(z0) at the Matsubara frequencies [55].

110 111

B.2 Saclay Method

Alternatively, the Saclay method [57, 106, 107, 34, 56] uses a Fourier integral representation of the propagator as a function of τ instead of k0 = ωn: Z τ iωnτ˜ ∆(ωn) = dτ˜ e ∆(˜τ) (B.5) 0 and its inverse 1 X ∆(τ) = e−iωnτ ∆(ω ). (B.6) τ n n In terms, it will result in a delta-function of τ for eliminating τ-integrals with the following identities.

For |x0 − y0| < τ, the closure of Matsubara functions is X hx0|nihn|y0i = δ(x0 − y0), (B.7) n

√1 −iωnx0 where hx0|ni = τ e . Alternatively, 1 X e−iωn(x0−y0) = δ(x − y ). (B.8) τ 0 0 n The delta-function in Matsubara representation is given by

Z 3 1 X d k ~ δ(4)(x − y) = eik·(~x−~y)−iωn(x0−y0), (B.9) τ (2π)3 n or Z τ (2π)3τδ(4)(p − q) = dx ei(~p−~q)·~x−i(ωn−ωl)x0 . (B.10) 0

B.3 Mixed Representation of Finite-Temperature Propaga- tor

The third method of performing Matsubara summation is to employ the Feynman or proper-time representation of denominators and the closure of Matsubara functions. The Fourier transform from a function of Matsubara frequencies, ωn, to that of ’time’, τ, is replacing the conventional integral with inﬁnite sum over Matsubara frequencies. The mixed presentation of propagators used in the Saclay method above is one of the examples. The thermal propagator (or the Matsubara propagator) in the Imaginary-Time Formalism (ITF) is given by ˜ ~ 2 ˜ (B) 1 ∆th(k, ωn; m ) = ∆th (ω, ωn) = 2 2 (B.11) ω − ωn for bosons with 2nπ ω = , ω2 = m2 + ~k2 (B.12) n τ or £ ¤ ˜ 2 2 −1 ˜ (F ) Sth(~p,ωn) = (m − iγ · p) · m + p = (m − iγ · p) · ∆th (ω, ωn) (B.13) 112 for fermions with (2n + 1)π ω = , ω2 = m2 + ~p 2. (B.14) n τ

The superscripts, (B) and (F ), of ∆˜ th(ω, ωn) denote the Boson and Fermion Matsubara frequencies

ωn to be used, respectively, and will be dropped for simplicity of notation. The propagator ∆˜ th(ω, ωn) can be converted to ∆˜ th(ω, z0) through the Matsubara-type Fourier transform as

1 X+∞ ∆˜ (ω, τ) = e−iωnz0 ∆˜ (ω, ω ), (B.15) th τ th n n=−∞ then, the ’imaginary-time variable’ τ in ∆˜ th(ω, τ) can be analytically continued to −iβ once Matsub- ara summation is performed. Due to diﬀerent statistics, bosonic and fermionic boundary conditions will have diﬀerent signs. While the summation procedure in the bosonic case has been presented in most of literature, the Matsubara-type Fourier transform of a fermion propagator will be shown explicitly here.

1 X+∞ ∆˜ (ω, τ) = e−iωnz0 ∆˜ (ω, ω ) (B.16) th τ th n n=−∞ +∞ 1 X e−iωnz0 = τ ω2 − ω2 − i² n=−∞ n · ¸ 1 X+∞ 1 1 1 = e−iωnz0 + . τ 2ω ω − ω − i² ω + ω − i² n=−∞ n n

Use the proper time representation for the two terms in square brackets, Z i X+∞ ∞ h i ∆˜ (ω, τ) = ds e−iωnz0 e−is(ω−ωn) + e−is(ω+ωn) (B.17) th 2ωτ n=−∞ 0 Z i ∞ 1 X+∞ h i = ds e−isω e−iωn(z0−s) + e−iωn(z0+s) . 2ω τ 0 n=−∞

To utilize the closure of Matsubara functions, Eq. (B.8), one need to change the form of z0 ± s such that their magnitudes are less than τ. |z0| < τ by deﬁnition, and let s = θ + lτ with 0 < θ < τ, then the s-integral converts to Z i X∞ τ 1 X+∞ h i ∆˜ (ω, τ) = dθ e−iω(θ+lτ) e−iωn(z0−θ−lτ) + e−iωn(z0+θ+lτ) (B.18) th 2ω τ l=0 0 n=−∞ Z i X∞ τ 1 X+∞ h i = dθ e−iω(θ+lτ) (−1)le−iωn(z0−θ) + (−1)l+1e−iωn(z0+θ−τ) , 2ω τ l=0 0 n=−∞ where e−iωn(±lτ) = e∓i(2n+1)πl = (e∓iπ)l = (−1)l. (B.19) 113

Notice that there is extra (−1) factor in the second term in square brackets to ensure |z0 +θ−τ| < τ. Z i X∞ τ 1 X+∞ h i ∆˜ (ω, τ) = dθ (−1)l e−iω(θ+lτ) e−iωn(z0−θ) − e−iωn(z0+θ−τ) (B.20) th 2ω τ l=0 0 n=−∞ Z i X∞ τ = dθ (−1)l e−iω(θ+lτ) [δ(z − θ) − δ(z + θ − τ)] 2ω 0 0 l=0 0 i X∞ h i = (−1)l e−iωlτ e−iωz0 − e−iω(τ−z0) . 2ω l=0 Since X∞ X∞ 1 1 (−1)l e−iωlτ = e−i(ωτ+π)l = = , (B.21) 1 − e−i(ωτ+π) 1 + e−iωτ l=0 l=0 and the Matsubara sum has been carried out, one can convert τ → −iβ and i 1 h i ∆˜ (ω, τ) = e−iωz0 − e−iω(τ−z0) (B.22) th 2ω 1 + e−iωτ i 1 £ ¤ = e−iωz0 − e−βωe+iωz0 2ω 1 + e−βω i eβω £ ¤ = e−iωz0 − e−βωe+iωz0 2ω eβω + 1 · ¸ i 1 1 = e−iωz0 − e−iωz0 − e+iωz0 . 2ω eβω + 1 eβω + 1 Thus, the ﬁnite-temperature fermion propagator in the mixed representation becomes

(F ) i £ ¡ ¢¤ ∆˜ (ω, τ = −iβ) = e−iωz0 − n˜(ω) e−iωz0 + e+iωz0 , (B.23) th 2ω where the Fermi-Dirac distribution functionn ˜(ω) is given by 1 n˜(ω) = . (B.24) eβω + 1 In contrast, the bosonic Matsubara frequency ensures

e−iωn(±lτ) = e∓i2nπl = (+1)l = 1, (B.25) and the boson thermal propagator in the mixed representation is

(B) i £ ¡ ¢¤ ∆˜ (ω, τ = −iβ) = e−iωz0 + n(ω) e−iωz0 + e+iωz0 , (B.26) th 2ω where the Bose-Einstein distribution function n(ω) is 1 n(ω) = . (B.27) eβω − 1 Alternatively, the mixed representation could also be derived from the contour integral through a contour with poles on the imaginary axis of q0 as Z −iq0z0 dq0 1 e ∆˜ th(ω, τ = −iβ) = , (B.28) −βq0 2 2 2πi e ∓ 1 ω − ωn − i² where the upper (or lower) sign is for bosons (or fermions). Appendix C

Gauge

Let Aµ be a gauge ﬁeld, either Abelian for photons in QED or Non-Abelian for gluons in QCD. The ﬁeld strength is

Fµν = ∂µAν − ∂ν Aµ − ig[Aµ,Aν ] (C.1) where [Aµ,Aν ] = 0 for Abelian gauge ﬁelds and [Aµ,Aν ] = −igFµν for non-Abelian gauge ﬁelds. The Maxwell equation (in QED) is µν ∂µF = 0 (C.2)

The QED gauge ﬁeld Aµ has 4 components, but only two are physical. The ﬁeld strength Fµν is physical and invariant while Aµ is under gauge transformation with gauge parameter ω(x),

0 Aµ → A µ = Aµ + ∂µω(x) (C.3)

C.1 Gauge Conditions

Linear gauge for any 4-vector fµ from a ﬁxed vector and derivatives or a matrix in a group space,

µ fµA = 0 (C.4)

For various fµ [108],

µ ∂µA = 0, Lorenz gauge (C.5)

∇ · A~ = ∂jAj = 0, Coulomb or radiation gauge (C.6) µ 2 nµA = 0, (n = 0), light-cone gauge (C.7)

A0 = 0, Hamiltionian or temporal axial gauge (C.8)

A3 = 0, axial gauge (C.9) µ xµA = 0, Fock-Schwinger gauge (C.10)

xjAj = 0, Poincar´e gauge (C.11)

114 115

Note that Coulomb (or Radiation) gauge sometimes also imposes A0 = 0 such that Aµ has only 2 free-components.

C.2 Photon Propagator and Gauge Parameter

The form of a photon propagator D˜ c(k) (or D˜ th(k) at ﬁnite temperature) is gauge-dependent. In covariant gauge, · ¸ k k 1 D˜ (k) = g − ζ µ ν · , (C.12) µν µν k2 − i² k2 − i² where ζ is the (covariant) gauge parameter; ζ = 0 for Feynman gauge, ζ = 1 for Landau gauge, and ζ = −2 for Yennie gauge (or Fried-Yennie gauge) [109, 110]. Note that the metric convention here is gµν = (−1, +1, +1, +1, ) so that 2 ~ 2 2 k = k − k0. (C.13) In conﬁguration space [86], " £ ¤ # ς δµν 1 − z z D˜ (z) = 4π2 2 + ζ µ ν . (C.14) µν z2 + i² (z2 + i²)2

For the metric convention of gµν = (+1, −1, −1, −1, ), · ¸ k k 1 D˜ (k) = − g − ζ µ ν · . (C.15) µν µν k2 − i² k2 − i²

In Coulomb gauge, ∂iAi = 0, the photon propagator becomes

2 ˜ 1 ˆ T 1 k kµkν Dµν (k) = −δµ0δ0ν + Pµν + ζC , (C.16) ~k2 k2 − i² (~k2)2 k2

ˆ T where ζC is the Coulomb gauge parameter, Pµν is the transverse projection operator

ˆ T kikj ˆ ˆ ˆ T ˆ T Pij = δij − = δij − kikj, P00 = P0i = 0. (C.17) ~k2

ˆ L The longitudinal projection operator Pµν is deﬁned as

ˆ L ˆ ˆ T Pµν = Pµν − Pµν , (C.18) with k k Pˆ = g − µ ν . (C.19) µν µν k2

In the strict (usual) Coulomb gauge, the parameter is set to zero, ζC = 0. 116

C.3 Current Conservation and Gauge Conditions in QED

The Fermion current operator in the QED interaction Lagrangian L0 is deﬁned as

¯ jµ(x) = ig ψ(x)γµψ(x) (C.20)

The vacuum expectation value of a current density operator vanishes in the limit of zero external

ﬁeld Aν , i.e., hjµiAν =0 = 0. Since total charge in vacuum is zero, the induced current in vacuum by an external ﬁeld is conserved for any Aν , i.e.,

∂µhjµiAν = 0. (C.21)

The current operator is deﬁned with fermion operators in same space-time point. To avoid the singularity of ﬁelds at the same space-time, the current operator can be re-written as X ¡ ¢ X £ ¤ αβ ¯ 0 αβ ¯ 0 jµ(x) = ig lim γµ ψβ(x )ψα(x) = ig lim γµ ψβ(x ), ψα(x) , (C.22) x0−x→0 + x0−x→0 α,β α,β

0 2 2 where the points x and x are relatively space-like to maintain causality, i.e., ~x − x0 > 0, and the commutator of fermion operators is equivalent to the normal-ordered form for free fermions. Through the deﬁnition of Green’s function, the vacuum expectation value of induced current becomes

0 hjµ(x)igA = − lim tr [γµGc(x, x |gA)], (C.23) x0−x→0 and the closed-fermion-loop functional is then given by

Z g Z 0 4 Lc[A] = i dg d x Aµ hjµ(x)ig0A (C.24) 0 with δLc[A] = ighjµ(x)igA. (C.25) δAµ(x) 0 0 To avoid the singularity of Gc(x, x |gA) at the limit of x → x, the deﬁnition of the current operator hjµ(x)igA can be modiﬁed to ensure [111, 112, 48]

y δ ∂ν hjµ(x)igA = 0 (C.26) δAν (y) for any y, which is equivalent to the current conservation of Eq. (C.21) under the translation invariance. This imply y δ δ ∂ν Lc[A] = 0, (C.27) δAν (y) δAµ(x) which also ensure that both the closed-fermion-loop functional Lc[A] and current operator are gauge invariant as shown below.

For any gauge transformation of the form Aµ → Aµ + ∂µΛ,

Gc(x, y|A) → Gc(x, y|A + ∂Λ) = Gc(x, y|A) · exp [ig (Λ(x) − Λ(y))]. (C.28) 117

The gauge-independent version of Green’s function Gc[A] is · Z x ¸ GF (x, y|A) = Gc(x, y|A) · exp −ig dξν Aν (ξ) (C.29) y £R ¤ Since Gc[A] can be represented by exp i f · A as an expansion of functional diﬀerentiations. By one integration-by-parts,

δ y δ GF [A + ∂Λ] = −∂µ GF [A + ∂Λ] (C.30) δΛ(y) δAµ(y) for any Λ(y). By deﬁnition, GF [A + ∂Λ] is independent of Λ, which implies that

y δ 0 ∂ν GF (x, x |gA) = 0. (C.31) δAν (y) Thus, · · Z x ¸¸ hjµ(x)iF = − lim tr γµGF (x, y|gA) · exp −iq dξν Aν (ξ) , (C.32) y→x y which satisﬁes the current conservation Eq. (C.26) for any y, and the gauge-dependent factor will vanish as y = x to ensure that the current operator is gauge invariant. In the perturbation theory, the closed-fermion-loop functional can be expressed in Taylor series of vacuum polarization tensors, Z Z i L[A] = d4x d4y A (x) K(2)(x − y) A (y) (C.33) 2 µ µν ν Z Z Z Z i + d4x d4y d4z d4w A (x) A (y) K(4) (x, y, z, w) A (z) A (w) + ··· , 4 µ ν µνλσ λ σ and Eq. (C.27) leads to (2l) ∂µ Kµν··· = 0. (C.34) The momentum space representation is

(2l) kµ K˜ µν··· = 0, (C.35)

(2l) which implies that the vacuum polarization tensors Kµν··· are transverse.

C.4 Gauge Structure of Green’s Function(al)

−1 Note that h (s1, s2) can be represented in terms of δ(s1, s2) as in Eq. (E.21), ½ Z Z ¾ i s s exp ds ds u(s ) · h−1(s , s ) · u(s ) (C.36) 4 1 2 1 1 2 2 ½ 0 Z 0 Z · ¸ ¾ i s s ∂ ∂ = exp ds1 ds2 u(s1) · δ(s1 − s2) · u(s2) 4 ∂s1 ∂s2 ½ Z0 0 Z · ¸ ¾ i s ∂ s ∂ = exp ds1 u(s1) · ds2 δ(s1 − s2) · u(s2) 4 ∂s1 ∂s2 ½ Z0 Z 0 ¾ i s s = exp ds ds u0(s ) · δ(s , s ) · u0(s ) 4 1 2 1 1 2 2 ½ Z0 0 ¾ i s = exp ds0 u02(s0) , 4 0 118 where there are discarded surface terms u2(s)δ(0)+u2(0)δ(0)−2u(s)·u0(s)+2u(0)·u0(0) by imposing

0 < s1(or s2) < s. Similarly,

Z s Z s Z s2 −1 0 0 ds1 ds2 u(s1) · h (s1, s2) · ds A(y − u(s )) (C.37) 0 0 0 Z s 0 0 ∂ 0 = ds u(s ) · 0 A(y − u(s )) 0 ∂s Z s = ds0 u0(s0) · A(y − u(s0)). 0

R s−² Note that 0 < s1, s2 < s and take lim²→0 0+² ds1 ···, or

Z s Z s Z s2 −1 0 0 ds1 ds2 u(s1) · h (s1, s2) · ds A(y − u(s )) (C.38) 0 0 0 Z s 0 0 ∂ 0 = ds u(s ) · 0 A(y − u(s )) 0 ∂s Z s 0 0 0 0 0 = ds uµ(s ) · ∂ν Aµ(y − u(s )) · uν (s ). 0 The Green’s function can be simpliﬁed as

Z ∞ x −ism2 − 1 Tr ln (2h) h~x,x0|Gc[A]|~y, y0i = [m − γ · (∂ − igA(x))] · i ds e · e 2 (C.39) 0 Z µ ½ Z s ¾¶ × d[u] exp g ds0σ · F(y − u(s0)) 0 ½ Z ¾ ½ Z + ¾ i s s × exp ds0 [u0(s0)]2 · exp −ig ds0 u0(s0) · A(y − u(s0)) 4 0 0 ×δ(x − y + u(s)). h i x δ δ Note γ·A(x) in [m − γ · (∂ − igA(x))] can be replaced by iγ· δu0(s) , which is similar to m − γ · δv(s) before gauge transformation. Under gauge transformation, Aµ → Aµ + ∂µΛ, there is an extra term linear in A like Z s 0 0 0 0 ds uµ(s ) · ∂µΛ(y − u(s )) (C.40) 0 Z s 0 ∂ 0 = − ds 0 Λ(y − u(s )) 0 ∂s = − [Λ(y − u(s)) − Λ(y − u(0))] .

Upon imposing the condition in u(s) = y − x from δ(x − y + u(s)), and the boundary condition u(0) = 0, Green’s function becomes

−ig[Λ(x)−Λ(y)] h~x,x0|Gc[A + ∂Λ]|~y, y0i = e · h~x,x0|Gc[A]|~y, y0i. (C.41) 119

C.5 Gauge Structure of Closed-Fermion-Loop Functional

Start from the form of Eq. (2.151) and follow the similar derivation for Green’s function, Fradkin’s representation of the closed-fermion-loop functional can be written as

Z ∞ 1 ds −ism2 − 1 Tr ln (2h) Lc[A] = − e · e 2 (C.42) 2 0 s Z Z µ ½ Z s ¾¶ × d4x d[u] δ(4)(u(s)) · tr exp g ds0σ · F(x − u(s0)) 0 ½ Z Z ¾ + i s s × exp ds ds u(s ) · h−1(s , s ) · u(s ) 4 1 2 1 1 2 2 ½ 0 0 ¾ Z s Z s Z s2 −1 0 0 × exp −ig ds1 ds2 u(s1) · h (s1, s2) · ds A(x − u(s )) 0 0 0 − {g = 0} ,

(4) where the delta functional δ (u(s)) enforces that uµ(s) = 0, and the boundary condition uµ(0) = 0 by deﬁnition. The seemly gauge-dependent factor in the last exponential could be rewritten under any gauge transformation, Aµ → Aµ + ∂µΛ as

Z s 0 0 0 0 ds uµ(s ) · Aµ(x − u(s )) (C.43) 0 Z s Z s 0 0 0 0 0 0 0 0 → ds uµ(s ) · Aµ(x − u(s )) + ds uµ(s ) · ∂µΛ(x − u(s )), 0 0 where Z s 0 0 0 0 ds uµ(s ) · ∂µΛ(x − u(s )) (C.44) 0 Z s 0 ∂ 0 = − ds 0 Λ(x − u(s )) 0 ∂s = − [Λ(x − u(s)) − Λ(x − u(0))] = − [Λ(x) − Λ(x)] = 0

Thus, the closed-fermion-loop functional is gauge-independent. Appendix D

Reviews of Functional Methods

Functional methods and notations used throughout this thesis are adapted from Ref. [49] and summarized in following sections.

D.1 Functional Diﬀerentiation

Linear translation: ·Z ¸ δ exp f · F [j] = F [j + f] (D.1) δj Functional in terms of operator representation: ¯ · ¸ · Z ¸¯ 1 δ ¯ F [j] = F · exp i gj ¯ (D.2) i δg ¯ g→0 Quadratic (Gaussian) translation operator or linkage operator: · Z ¸ i δ δ eD = exp − A (D.3) 2 δj δj

Quadratic (Gaussian) translation: · Z ¸ · Z ¸ · Z Z ¸ i δ δ i exp − A · exp i jg = exp gAg + i jg (D.4) 2 δj δj 2 Functional diﬀerential on a product of two functionals: · Z ¸ i δ δ exp − A · F [j] · F [j] (D.5) 2 δj δj 1 2 · Z ¸ · Z ¸¯ ¯ 1 δ 1 δ i δ δ ¯ = F1[ ] · F1[ ] · exp − A · exp i j(g1 + g2) ¯ i δg1 i δg2 2 δj δj g→0 · Z Z Z ¸ · Z ¸¯ ¯ 1 δ 1 δ i i ¯ = F1[ ] · F1[ ] · exp g1Ag1 + g2Ag2 + i g1Ag2 · exp i j(g1 + g2) ¯ i δg1 i δg2 2 2 g→0 h R i h R i i δ δ δ δ or with D = exp − A and D12 = exp −i A , 2 δj δj δj1 δj2 120 121

£¡ ¢ ¡ ¢¤¯ D D12 D1 D2 e · F1[j]F2[j] = e e F1[j1] e F2[j2] ¯ (D.6) j1=j2=j Quadratic (Gaussian) translation operator on the Gaussian functional: · Z ¸ · Z ¸ · Z ¸ i δ δ i i 1 exp − A · exp jBj = exp jB(1 − AB)−1j − Tr ln(1 − AB) (D.7) 2 δj δj 2 2 2

Quadratic (Gaussian) translation operator on the general Gaussian functional: · Z ¸ · Z Z ¸ i δ δ i exp − A · exp jBj + i fj (D.8) 2 δj δj 2 · Z Z Z ¸ i i 1 = exp jB(1 − AB)−1j + i j(1 − BA)−1f + fA(1 − BA)−1f − Tr ln(1 − AB) 2 2 2 Charged Bosons: · Z ¸ · Z ¸ δ δ exp −i A · exp i j∗Bj (D.9) δj δj∗ · Z ¸ = exp i j∗B(1 − AB)−1j − Tr ln(1 − AB)

Fermions: · Z ¸ · Z ¸ δ δ exp −i Aαβ · exp i η¯µBµν ην (D.10) δηα δη¯β · Z ¸ = exp i η¯B(1 + AB)−1η + Tr ln(1 + AB)

Proper time representation:

Z ∞ Z ∞ (A − i²)−1 = i ds e−is(A−i²) → i ds e−isA (D.11) 0 0

D.2 Functional Integration

Measure of functional integration (FI): u(x) → u(xi) = ui

Z YN Z +∞ d[u] = lim dui (D.12) N→∞ i=1 −∞ Functional representation of δ-functional: Z · Z ¸ δ[j − f] = η−1 d[u] exp i u(j − f) , (D.13) or

YN Z +∞ −1 δ[j − f] = lim η dui exp [i∆ui(ji − fi)] N→∞ i=1 −∞ µ ¶N YN −1 2π = lim η δ(ji − fi), (D.14) N→∞ ∆ i=1 122 where η is the normalization constant and can be chosen as (2π/∆)N , and ∆ is the 4-volume of space-time cells. Functional Fourier transform (FFT): Z ½ Z ¾ F [j] = η−1 d[u] F˜[u] · exp i ju (D.15)

Functional Integration of a Gaussian: Z ½ Z ¾ i I[A] = d[u] exp ± u · A · u (D.16) 2 ½ ¾ 1 = C· exp − Tr ln A , 2 and Z ½ Z Z ¾ i I[j; A] = d[u] exp ± u · A · u + i j · u (D.17) 2 ½ Z ¾ i 1 = C· exp ∓ j · A−1 · j − Tr ln A , 2 2 where C is a divergent normalization constant and ∆ is the 4-volume of space-time cells µ ¶ 2πi N/2 C = lim = η. (D.18) N→∞ ∆

D.3 Functional Diﬀerentiation vs Functional Integral

Functional linkage operation can be recast into a Gaussian-weighted functional integral: ¯ Z R 2 R i s ds0 δ ¯ s 0 2 0 0 δv(s0)2 ¯ (i/4) ds v (s ) e ·F[v]¯ = N d[v]e 0 ·F[v], (D.19) v→0 where Z · Z s ¸ N −1 = d[v] exp (i/4) dsv2(s0) . (D.20) 0

D.4 Two Useful Relations

Reciprocity Relation: for any functional F [A] of polynomials in A, · Z ¸ · Z ¸ · Z ¸ 1 δ i i i δ δ F [ ] · exp j · D · j = exp j · D · j · exp − · D · · F [A], (D.21) i δj 2 2 2 δA δA where Z A(x) = dy D(x, y) · j(j). (D.22)

Gauge Formula: for arbitrary Λ(z),

δ z δ F [A + ∂Λ] = −∂µ F [A + ∂Λ]. (D.23) δΛ(z) δAµ(z) 123

D.5 Functional Form of Unity

Z 1 = d[u]δ[u − j], (D.24) or for any functional Z F [j] = d[u]F [u] · δ[u − j]. (D.25)

D.6 Linkage Operation

Z Z i δ δ DA = − dx dy · Dµν (x − y) · (D.26) 2 δAµ(x) δAν (y) Z Z δ δ D = −i dx dy · D (x − y) · (D.27) 12 (1) µν (2) δAµ (x) δAν (y) h i h i DA D12 D (1) (1) D (2) (2) e · Aα(z)F[A] = e e A Aα (z) e A F[A ] (D.28)

· ¸ 1 eDA A (z) = 1 + D + D ·D + ··· A (z) = A (z) (D.29) α A 2! A A α α

· ¸ 1 eD12 A(1)(z) = 1 + D + D ·D + ··· A(1)(z) (D.30) α 12 2! 12 12 α = [1 + D ] A(1)(z) 12 αZ δ = A(1)(z) − i dy D (z − y) · α αν (2) δAν (y) · Z ¸ δ £ ¤ DA DA e · Aα(z)F[A] = Aα(z) − i dy Dαν (z − y) · e F[A] (D.31) δAν (y) In momentum and Matsubara space, the linkage operator is given by

Z 4 Z 4 0 i d k d k δ 0 δ DA = − 4 4 · Dµν (k − k ) · 0 (D.32) 2 (2π) (2π) δAµ(k) δAν (k ) Z i d4k δ δ = − 4 · Dµν (k) · , 2 (2π) δAµ(k) δAν (−k)

0 4 0 where D˜ µν (k − k ) = (2π) D˜ µν (k) δ(k + k ), and the integral for the Matsubara representation is Z Z d4k 1 X d3~k → (D.33) (2π)4 τ (2π)3

˜ th 0 3 ˜ th ~ ~0 with Dµν (k − k ) = (2π) τ Dµν (k) δ(k + k ) δl,−l0 . · Z ¸ d4k δ DA ˜ ˜ ˜ ˜ DA ˜ e Aα(q)F[A] = Aα(q) − i 4 Dαν (q − k) · e F[A] (D.34) (2π) δA˜ν (k) 124

" Z # 1 X d3~k δ eDA A˜ (q)F[A˜] = A˜ (q) − i D˜ th (q − k) · eDA F[A˜], (D.35) α α τ (2π)3 αν ˜ l δAν (k) 4 where D˜ µν (q − k) = (2π) D˜ µν (q) δ(q + k). · ¸ δ DA DA e A˜α(q)F[A˜] = A˜α(q) − iD˜ αν (q) · e F[A˜], (D.36) δA˜ν (−q)

˜ th 3 ˜ th ~ or Dµν (q − k) = (2π) τ Dµν (q) δ(q + k) and δ(q + k) = δ(~q + k) δl0,−l, · ¸ δ DA ˜ ˜ ˜ ˜ th DA ˜ e Aα(q)F[A] = Aα(q) − iDαν (q) · e F[A]. (D.37) δA˜ν (−q) Appendix E

Misc Relations

E.1 Useful Relations in Fradkin’s Representation

Z s 2 Z s 0 δ 0 0 exp {i ds 2 0 }· exp {ip · ds v(s )}F[v] (E.1) 0 δv (s ) 0 Z s Z s 2 2 0 0 0 δ = exp {−isp }· exp {ip · ds v(s )}· exp {i ds 2 0 }F[v − 2p] 0 0 δv (s )

Z δ s {m − γ · }· exp {ip · ds0v(s0)}F[v] (E.2) δv(s) Z 0 · ¸ s δ = exp {ip · ds0v(s0)}·{m − iγ · p − i }F[v] 0 δv(s)

E.2 Representations of Delta- and Heaviside Step- Function

For 0 < s1, s2 < s, δ-function in the Fourier series µ ¶ µ ¶ 2X0 Nπs Nπs δ(s − s ) = sin 1 · sin 2 , (E.3) 1 2 s 2s 2s N µ ¶ µ ¶ 2X0 Nπs Nπs δ(s − s ) = cos 1 · cos 2 , (E.4) 1 2 s 2s 2s N where the prime on the sum indicates that N runs over only odd integers as

∞ X0 X = . (E.5) N N=1,3,5,··· For ² > 0, Lorentzian representation:

Z +∞ 1 ² 1 ikx−²|k| δ(x) = lim 2 2 = lim dk e (E.6) ²→0 π x + ² ²→0 2π −∞ 125 126

Z e−|x|/² 1 +∞ eikx δ(x) = lim = lim dk 2 2 (E.7) ²→0 2² ²→0 2π −∞ 1 + ² k Gaussian representation or limit of normal distribution:

2 1 − x δ(x) = lim √ e ²2 (E.8) ²→0 ² π Z 1 +∞ δ(x) = dk e−ikx (E.9) 2π −∞ Diﬀraction peak representation: Z 1 ³x´ 1 +1/² δ(x) = lim sin = lim dk cos(kx) (E.10) ²→0 πx ² ²→0 2π −1/²

² ³x´ δ(x) = lim sin2 (E.11) ²→0 πx2 ² Derivative of the sigmoid (or Fermi-Dirac) function 1 1 δ(x) = lim ∂x = − lim ∂x (E.12) ²→0 1 + e−x/² ²→0 1 + ex/² Limit of a rectangular function Z µ ¶ 1 ³x´ 1 +∞ ²k δ(x) = lim rect = lim dk sinc e−ikx (E.13) ²→0 ² ² ²→0 2π −∞ 2π Limit of the Airy function 1 ³x´ δ(x) = lim Ai (E.14) ²→0 ² ² Limit of a Bessel function µ ¶ 1 x + 1 δ(x) = lim J1/² (E.15) ²→0 ² ² Representation of Heaviside step function or θ-function, Z +∞ dk e−ik0t θ(t) = lim i 0 (E.16) + ²→0 −∞ 2π k0 + i²

E.3 Representation of h Function

Deﬁnition: Z s 0 0 0 1 h(s1, s2) = ds θ(s1 − s )θ(s2 − s ) = [(s1 + s2) + |s1 − s2|] , (E.17) 0 2 and the h-function is symmetric in its parameters as Z s 0 0 0 h(s1, s2) = ds θ(s1 − s )θ(s2 − s ) = h(s2, s1). (E.18) 0 The h-function can be expended in series of sine and cosine functions as µ ¶ µ ¶ 8s X0 1 Nπs Nπs h(s , s ) = cos 1 · cos 2 (E.19) 1 2 π2 N 2 2s 2s N 127 and its inverse is given by µ ¶ µ ¶ π2 X0 Nπs Nπs h−1(s , s ) = N 2 sin 1 · sin 2 (E.20) 1 2 2s3 2s 2s N Compare with the second representation of the delta function, Equation (E.3),

−1 ∂ ∂ h (s1, s2) = δ(s1 − s2) (E.21) ∂s1 ∂s2

Z s −1 dsi si · h (si, sj) = δ(s − sj) (E.22) 0

Z s Z s −1 ds1 ds2 s1 · h (s1, s2) · s2 = s (E.23) 0 0

Z s Z s 0 0 0 −1 0 0 0 ds1 ds2 θ(s1 − s1) h (s1, s2) θ(s2 − s2) = δ(s1 − s2) (E.24) 0 0

Z s 0 0 −1 0 0 ∂ ds1 θ(s1 − s1) h (s1, s2) = − δ(s1 − s2) (E.25) 0 ∂s2

E.4 Operator Relations

If [A, [A, B]] = [B, [A, B]] = 0,

eA+B = eAeBe−[A,B]/2, (E.26)

λ2 eλBAe−λB = A + λ[B,A] + [B, [B,A]] + ··· (E.27) 2!

E.5 Legendre Function of the Second Kind

The Legendre diﬀerential equation is of second-order d2Q dQ (1 − x2) − 2x + l(l + 1)Q = 0 (E.28) dx2 dx The equation has regular singular points at −1, 1, and ∞. It has two linearly independent solutions,

Pl(x) and Ql. While Pl(x), called the Legendre function of the ﬁrst kind, is regular at ﬁnite point, the Legendre function of the second kind, Ql(x), is singular at ±1. First few of Ql(x) are µ ¶ 1 1 + x Q (x) = ln (E.29) 0 2 1 − x µ ¶ x 1 + x Q (x) = ln − 1 (E.30) 1 2 1 − x µ ¶ 3x2 − 1 1 + x 3x Q (x) = ln − (E.31) 2 4 1 − x 2 µ ¶ 5x3 − 3x 1 + x 5x2 2 Q (x) = ln − + . (E.32) 3 4 1 − x 2 3 128

First few of Pl(x) are

P0(x) = 1 (E.33)

P1(x) = x (E.34) 1 P (x) = (3x2 − 1) (E.35) 2 2 1 P (x) = (5x3 − 3x) (E.36) 3 2 1 P (x) = (35x4 − 30x2 + 3) (E.37) 4 8 1 P (x) = (63x5 − 70x3 + 15x). (E.38) 5 8 Both functions satisfy the same recurrence relation:

(l + 1)Pl+1(x) − (2l + 1)xPl + lPl−1(x) = 0. (E.39)

E.6 Abel’s Trick

If F (s1, s2) = F (s2, s1), [113]

Z s Z s Z s Z s1 ds1 ds2 F (s1, s2) = 2 ds1 ds2 F (s1, s2) (E.40) 0 0 0 0

E.7 Bogoliubov Transformation

The Bogoliubov transformation is commonly used to diagonalize Hamiltonians of a quadratic form, and can be easily carried out in terms of creation and annihilation operators [114]. It is a unitary transformation which transforms a unitary representation of a canonical commutation or anti- commutation relation algebra into another unitary representation isomorphically. If a set of creation and annihilation operatorsa ˆ(k) anda ˆ† of bosons obey the canonical commutation relation,

[ˆa(k), aˆ†(k)] = 1, (E.41) then the correspondent operators, ˆb(k) and ˆb†, of a Bogoliubov transformation also have to satisfy the canonical commutation relation, i.e.,

[ˆb(k), ˆb†(k)] = 1 (E.42)

Let the transformation be

ˆb(k) = uaˆ(k) + vaˆ†(k), (E.43) ˆb†(k) = u∗aˆ†(k) + v∗aˆ(k), (E.44) which yield [ˆb(k), ˆb†(k)] = (|u|2 − |v|2) [ˆa(k), aˆ†(k)], (E.45) 129

To ensure the transformation is canonical, the commutation relation of ˆb(k) and ˆb† leads to

|u|2 − |v|2 = 1, (E.46) for bosons. Similarly, |u|2 + |v|2 = 1, (E.47) for fermions with anti-commutation relations. The canonical commutation or anti-commutation relations impose restrictions on u and v, but the ratio of u and v is still arbitrary and can be chosen to simplify the Hamiltonian or any function of creation and annihilation operators. To diagonalize the real-time propagator matrix, one can use the temperature-dependent Bogoli- ubov transformation; e.g., [56] p p ˆ † b(k) = 1 + n(k0)â(k) + n(k0)â (k), (E.48) p p ˆ† † b (k) = 1 + n(k0)â (k) + n(k0)â(k), (E.49) p ~ 2 2 for bosons, where k0 = k + m . Appendix F

Calculations in Full Imaginary-Time Formalism

If both fermion and photon propagators are represented in the full imaginary-time formalism, the dressed, ﬁnite-temperature fermion propagator under the Bloch-Nordsieck approximation can be written as

0 BN h~p,n|Sth|~y, y0iQuenched (F.1) Z ∞ £ ¤−1/2 2 2 −1 3 −i(~p·~y−ωny0) −is(m +p ) = Z(0)[iτ] · (2π) τ · e · i ds e ( 0 ) Z 3 h i Z s X d ~k 0 × m − iγ · p + 2ig2 γ · D˜ BN (k) · p ds0 e2ik·p(s−s ) (2π)3τ th l 0 ( Z Z Z ) 3 h i s s1 X d ~k 0 × exp 4ig2 p · D˜ BN (k) · p ds ds0 e2is k·p . (2π)3τ th 1 l 0 0 First use the Feynman gauge in photon propagator, ¯ ~2 δµν ~ 2 ¯ δµν − k BN ~ −iαk ~p2 D˜ (k, ωl) = · e ¯ = · e , (F.2) th 2 2 ¯ 2 2 ~k − ω − i² α=− i ~k − ω − i² l ~p2 l where the Bloch-Nordsieck limiting factor exp [−iα~k2] is included with α−1 = i~p 2. After dropping the renormalization factor, the exponent becomes Z Z Z 3 h i s s1 X d ~k 0 4ig2 p · D˜ BN (k) · p ds ds0 e2is k·p (F.3) (2π)3τ th 1 l 0 0 Z Z Z 2 2 s s1 ∞ g p 02 2 2 1 X −2is0ω ω +( 1 − 1 )ω2 0 −s ~p q l n q2 ~p 2 l = −i 3 ds1 ds dq e e , 2 τ π 0 0 |~p| l 2 2 1 1 where the leading sign is minus. While q > ~p , the factor q2 − ~p 2 < 0 in the exponent, and it is relatively small, i.e. ¯µ ¶ ¯ ¯ 1 1 ¯ ¯ − ω2¯ < s02~p 2q2, (F.4) ¯ q2 ~p 2 l ¯ 130 131

2 2 2 and the contribution from exp [(1/q − 1/~p )ωl ] can be dropped. It leads to Z Z Z 3 h i s s1 X d ~k 0 4ig2 p · D˜ BN (k) · p ds ds0 e2ik·ps (F.5) (2π)3τ th 1 l 0 0 Z Z Z 2 2 s s1 ∞ g p 02 2 2 1 X 0 0 −s ~p q −2is ωlωn ' −i 3 ds1 ds dq e e . 2 τ π 0 0 |~p| l

The Matsubara sum over l is similar to the previous work of Candelpergher, Fried and Grandou [47] as

µ 0 ¶ 1 X 0 X l e−2is ωnωl = Θ δ(l0τ − 2s0ω ), (F.6) τ n n l l0 and the exponent becomes Z Z Z X d3~k h i s s 4ig2 p · D˜ BN (k) · p ds ds e2ik·p(s1−s2) (F.7) (2π)3τ th 1 2 l 0 0 µ ¶ µ ¶ µ ¶ ³ ´ 2 2 0 0 0 Z ∞ 0 2 g p X l l τ l τ − l τ ~p 2q2 2ωn ' −i 3 Θ s − Θ s − dq e . 2 ωn n 2ωn 2ωn 2π l0 |~p| Similar evaluation can apply to the linear or denominator factor in the dressed, ﬁnite-temperature propagator as

Z 3 h i Z s X d ~k 0 2ig2 γ · D˜ BN (k) · p ds0 e2ik·p(s−s ) (F.8) (2π)3τ th l 0 3 Z s Z ∞−i|α| ¯ ³ ´− X 02 ¯ 2 π 2 2 0 −2is0ω ω 00 00− 3 i s ~p 2+i(t0+i|α|)ω2 l n 2 t00 l ¯ = − 3 g (γ · p) ds e dt t e ¯ (2π) τ i 0 0−i|α| |α|= 1 l ~p 2 √ µ ¶ Z ³ ´2 X 0 1 l0 2 2 2 π 2 |~p| l − (~p ) λ = g (γ · p) Θ dλ e 4πnT 2 . 2(2π)3 nT n l0 0 Thus, the dressed, ﬁnite-temperature fermion propagator becomes

0 BN h~p,n|Sth|~y, y0iQuenched (F.9) Z ∞ £ ¤−1/2 2 2 −1 3 −i(~p·~y−ωny0) −is(m +p ) ' Z(0)[iτ] · (2π) τ · e · i ds e ( 0 ) √ µ ¶ Z ³ ´2 X 0 1 l0 2 2 2 π 2 |~p| l − (~p ) λ × m − iγ · p + g (γ · p) Θ dλ e 4πnT 2 2(2π)3 nT n l0 0 ( µ ¶ µ ¶ ³ ´ ) 2 2 Z 1 0 0 0 2 g p X l l − l (~p 2)2λ2 × exp + |~p| dλ Θ s + e 4πnT 2 . 4π5/2 nT n 4πnT 2 0 l0

Notice that p2 is kept oﬀ the mass-shell until calculation is fully performed; and when the mass-shell 132 condition, p2 = −m2, is set,

0 BN h~p,n|Sth|~y, y0iQuenched (F.10) Z ∞ £ ¤−1/2 2 2 −1 3 −i(~p·~y−ωny0) −is(m +p ) ' Z(0)[iτ] · (2π) τ · e · i ds e ( 0 ) √ µ ¶ Z ³ ´2 X 0 1 l0 2 2 2 π 2 |~p| l − (~p ) λ × m − iγ · p + g (γ · p) Θ dλ e 4πnT 2 2(2π)3 nT n l0 0 ( µ ¶ µ ¶ ³ ´ ) 2 Z 1 n 0 0 0 2 mg |~p| X l l − l (~p 2)2λ2 × exp − dλ Θ s + e 4πnT 2 , 4π5/2 nT n 4πnT 2 0 l0=−n which the sum over l0 is hard to evaluate unless resorting to Ramanujan Constants Approximation. Meanwhile, the result contains both thermal and non-thermal contribution, which is diﬃcult to be separated. Appendix G

Calculations in Feynman Gauge

In Feynman gauge, the thermal part of photon propagator is

˜ µν δµν ~ 2 2 δDth (k) = 2πi δ(k − k0). (G.1) eβ|k0| − 1 The last exponent in Eq. (3.77) becomes Z Z Z 4 s s1 h i d k 0 4ig2 ds ds0 p · δD˜ (k) · p e2ik·ps (G.2) (2π)4 1 th Z 0Z 0Z · ¸ 4 s s1 d k 2πi 0 = 4ig2p2 ds ds0 δ(~k2 − k2) e2ik·ps 4 1 β|k | 0 (2π) 0 0 e 0 − 1 Z Z Z Z 2 s s1 2 2 g δ(~k − k ) ~ 0 0 = − (~p2 − ω2 ) ds ds0 d3~k dk 0 e2ik·~ps −2ik0ωns . 3 n 1 0 β|k | 2π 0 0 e 0 − 1 Applying similar approximations used in Coulomb gauge, Z Z Z 4 s s1 h i 2 d k 0 ˜ 2ik·ps0 4ig 4 ds1 ds p · δDth(k) · p e (G.3) (2π) 0 0 2 2 2 Z 1 g ~p − ωn 2sp x ' −s 2 dx x 2π βp β 0 e − 1 2 2 2 Z 1 2 g ~p − ωn x = −s 2 2 dx x π β 0 e − 1 c = − s2g2T 2 (~p2 − ω2 ). π2 n

The exponent still contains Matsubara frequency ωn, and the Matsubara sum needs to be evaluated similar to Appendix F, which is much complicate compared to that of Coulomb gauge.

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