finite-temperature quantum electrodynamics: by yeuan …
TRANSCRIPT
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.
DateHerbert M. Fried, Director
DateAntal Jevicki, Advisor
Recommended to the Graduate Council
DateGerald Guralnik, Reader
Brown University, Department of Physics
DateChung-I Tan, Reader
Brown University, Department of Physics
Approved by the Graduate Council
DateSheila 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 Cote d’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-Lineaire 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-Lineaire de 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). Naıvely, 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 ap-
plications 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 naıve 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 ap-
proximations 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
of Gaussian, as exp[−Γz2
0
]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 prop-
erly 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 trans-
verse 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 damp-
ing 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 in-
troduced 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 inter-
acting 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 repre-
sentations. 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)
= −ψ(m + γ · ∂)ψ − 14F2 + ig ψ γ ·Aψ,
which is Abelian and invariant under U(1) gauge transformation, and Fµν = ∂µAν − ∂νAµ is the
field strength. To facilitate the functional approach following Schwinger, sources like η(x) and η(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
Zcj, η, η = 〈0|(
exp
i
∫ [j ·A + η · ψ + ψ · η])
+
|0〉, (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.,
G(n)(x1, . . . , xn) = i〈0| (A(x1) · · ·A(xn))+ |0〉 (2.4)
= i
(1i
δ
δj(x1)
)· · ·
(1i
δ
δj(xn)
)· Zcj, η, η
∣∣∣∣η=η=j=0
.
Following either Schwinger’s or Symanzik’s construction, the solution of the QED generating func-
tional is given by
〈S〉Zcj, η, η = exp
ig
∫δ
δη
(γ · δ
δj
)δ
δη
· exp
i
2
∫j ·Dc · j + i
∫η · Sc · η
, (2.5)
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)):
Zcj, η, η = ei2
∫j·Dc·j · eD
(c)A · ei
∫η·Gc[A]·η eLc[A]
〈S〉 , (2.6)
where
Aµ(x) =∫
dy Dcµν(x− y) · jν(y), (2.7)
and the normalization constant 〈S〉 is defined by
〈S〉 ≡ 〈0|S|0〉 = eD(c)A · eLc[A]
∣∣∣∣A→0
, (2.8)
and where the argument of the linkage operator is
D(c)A = − i
2
∫dx
∫dy
δ
δAµ(x)·Dµν
c (x− y) · δ
δAν(y). (2.9)
The gauge-field coupled Green’s functional Gc[A] is
Gc[A] = Sc [1− igγ ·ASc]−1
, (2.10)
and the closed-fermion-loop functional Lc[A] is
Lc[A] = Tr ln [1− ig(γ ·A)Sc] = −Tr ln[S−1
c ·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
S′c = i〈(ψψ)+〉 = i
(1i
δ
δη
)(−1
i
δ
δη
)· 〈S〉Zcj, η, η
∣∣∣∣η=η=j=0
. (2.12)
With the aid of Eq. (2.6), and setting all sources to zero, the dressed fermion propagator becomes
S′c = eD(c)A ·
[Gc[A]
eLc[A]
〈S〉]∣∣∣∣
A→0
. (2.13)
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:
(m + γµ[∂µ − igAµ])Gc(x, y|A) = δ(4)(x− y). (2.14)
Its solution, the causal fermion Green’s function, in formal notation is
Gc[A] = [m + iγ · (∂ − igA)]−1 = (m− iγ ·Π)−1, (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 repre-
sentation of the logarithm function over the coupling constant g as
Lc[A] = −i
∫ g
0
dg′Tr
(γ ·A)Sc [1− ig′(γ ·A)Sc]−1
= −i
∫ g
0
dg′Tr (γ ·A)Gc[g′A], (2.17)
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 (ρG ln ρG), (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,
ρG = Z−1G exp
[−βH+
∑
A
βµANA
], (2.19)
8
and the grand partition function is given by
ZG[β, µ] = Tr
exp
[−βH+
∑
A
βµANA
]. (2.20)
The ensemble average of a physical quantity in a grand canonical ensemble is then defined as
〈O〉G = Tr(ρGO
), (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 en-
ergy. However, the ensemble is still subjected to constraints of fixed average total particle num-
ber∑
A 〈NA〉G and average total energy 〈H〉G, with the inverse temperature β = (kBT )−1 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.
In the rest frame of the thermal bath, uµ = (1,~0) and βµ = (β,~0). Instead of q2, a particle with
momentum qµ can be characterized by two Lorentz invariants, ω = −u · q and κ with q2 = κ2 − ω2.
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.,∫
d3~q
(2π)3f(βωq) =
∫d4q
(2π)42πδ(q2 + m2) θ(q0) 2ωq f(−β · q) (2.23)
by inserting ∫dq0
2πδ(q0 − ωq) =
∫dq0
2π2ωq δ(q2
0 − ω2q ), (2.24)
and a step function θ(q0) to ensure inclusion of only the positive root of ~q 2+m2 = q20 [22]. Therefore,
the introduction of the Lorentz four-vector inverse-temperature permits a covariant form of thermo-
dynamics. Similarly, the energy-momentum tensor Tµν , entropy flux sµ, and conserved current JAµ
of type-A charge particles become
Tµν = ρ uµuν + P (gµν + uµuν), (2.25)
sµ = suµ, (2.26)
JAµ = 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
ZG[β, µ] = Tr[e−β(H−µN )
]=
∑
A〈nA, z0|e−β(H−µN )|nA, z0〉, (2.28)
where H and N 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 = H − µN with the probability amplitude
〈A, t2|e−βH|B, t1〉 → 〈A, t2|e−iτH|B, t1〉 = 〈A, t2 + τ |B, t1〉. (2.29)
The ’analytically continued’ grand partition function ZG[iτ, µ] is then given by
ZG[iτ, µ] =∑
A〈nA, z0|e−iτH|nA, z0〉 =
∑
A〈nA, z0 + τ |nA, z0〉. (2.30)
The form of the grand partition function is similar to the generating function of the zero-temperature
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
Zthj, η, η =∑
A〈nA, z0 + τ |nA, z0〉. (2.32)
In the limit of zero sources, this Finite-Temperature generating functional reduces to the grand
partition function,
Zth0, 0, 0 = ZG[iτ, µ]. (2.33)
10
C1
C2ti − iβ
ti
tf − iσ
Re
Im
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
1i
δ
δj(z)Zthj, η, η =
∑
A〈nA, z0 + τ |A(z)|nA, z0〉, (2.34)
1i
δ
δη(z)Zthj, η, η =
∑
A〈nA, z0 + τ |ψ(z)|nA, z0〉, (2.35)
−1i
δ
δη(z)Zthj, η, η =
∑
A〈nA, z0 + τ |ψ(z)|nA, z0〉. (2.36)
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
Zthj, η, η = Tr
e−βH
(exp
i
∫
Cdz0
∫d3~z
[j ·A + η · ψ + ψ · η])
C+
(2.37)
=∑
A〈nA, z0 + τ |
(exp
i
∫
Cdz0
∫d3~z
[j ·A + η · ψ + ψ · η])
C+|nA, z0〉,
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
path from z0 to z0 +τ = z0− iβ (cf. Ref. [19, 54, 55]). The requirement of −β ≤ Im(z′0−z′′0 ) ≤ 0 for
any two points, z′0 and z′′0 , 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 func-
tional 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)
ψ(z0) = −eβµψ(z0 + τ) = −eβµψ(z0 − iβ), (2.39)
ψ(z0) = −eβµψ(z0 + τ) = −eβµψ(z0 − iβ). (2.40)
11
First, set the chemical potential to zero, µ = 0, and let Zthj, η, η|g=0 = Z(0)th 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
Zthj, η, η = exp
i
∫
Cdz0
∫d3~z Lint
1i
δ
δj,−1
i
δ
δη,1i
δ
δη
· Z(0)
th j, η, η, (2.41)
where fields in the interacting Lagrangian Lint have been replaced by conjugated field operators
similar to the zero-temperature theory, i.e.,
LintA, η, η → Lint
1i
δ
δj,−1
i
δ
δη,1i
δ
δη
. (2.42)
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, the thermal average of a field product ψ(z)ψ(z) at equal space-time cannot be naively
replaced by(− 1
iδδη
) (1i
δδη
). 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
ψ(z)ψ(z) =12ψ(z), ψ(z)+
12[ψ(z), ψ(z)]. (2.43)
With the help of the anti-commutation relation for field operators, the symmetric part is just an
infinite c-number 12ψ(z), ψ(z) = 1
2γ0δ(3)(~0), and can be identified as 1
2 〈0|(ψ(x)ψ(y))+|0〉 or as
−iSc(0). Here, Sc(0) is
Sc(0) = i〈0|(ψ(x)ψ(y))+|0〉|x−y→0, (2.44)
where |0〉 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
ψ(z)ψ(z) →(−1
i
δ
δη
)(1i
δ
δη
)− iSc(0). (2.45)
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
Zthj, η, η = exp
iµ
∫
Cdz0
∫d3~z
(−1
i
δ
δη
)(1i
δ
δη
)+ µτΩSc(0)
· Zth,µ=0j, η, η, (2.46)
where Ω =∫
d3~z is the volume of the system, τ =∫C dz0 = −iβ is the complex ”time”, and
Zth,µ=0j, 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
Z(0)th,µ=0[j, η, η] = ei
∫C η·Sµ=0
th ·η · e i2
∫C j·Dth·j · Z(0)[iτ, 0], (2.47)
where the constant Z(0)[iτ, 0] = Z(0)th,µ=0[0, 0, 0] is the normalization constant of the generating func-
tional 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
Z(0)th j, η, η = ei
∫C η·Sth·η · e+Tr ln [1−µSµ=0
th ] · e i2
∫C j·Dth·j · Z(0)[iτ, 0], (2.48)
where the free thermal propagator Sth with non-zero chemical potential is
Sth = Sµ=0th
[1− µSµ=0
th
]−1
, (2.49)
and satisfies the same differential equation as Sµ=0th , except that p0 is replaced by p0 + µ. The
determinantal factor can be combined with Z(0)[iτ, 0] as
exp
+Tr ln[1− µSµ=0
th
]· Z(0)[iτ, 0] = exp [−µτΩSc(0)] · Z(0)
G [iτ, µ], (2.50)
where Z(0)[iτ, µ] = Z(0)th [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
Z(0)th [j, η, η] = ei
∫C η·Sth·η · e i
2
∫C j·Dth·j · Z(0)
G [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
Zthj, η, η =∫DADψDψ exp
[i
∫
C
L]. (2.52)
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
S′th(x− y) = i〈(ψ(x)ψ(y))C+〉β = iTr
[e−βH (ψ(x)ψ(y))C+
]
Tr[e−βH] , (2.53)
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.,
〈ψ(x0)ψ(y0)〉β = Z−1G Tr
[e−βH ψ(x0)ψ(y0)
](2.54)
= Z−1G Tr
[e−βH e+βH ψ(y0) e−βH ψ(x0)
]
= 〈ψ(y0 − iβ)ψ(x0)〉β ,
or
S′th(x0 − y0) = −S′th(x0 − y0 ± iβ). (2.55)
The Finite-Temperature Green’s functions (or propagators) can be derived by functional dif-
ferentiation 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 〈S〉. Hence, the Finite-Temperature fermion propagator becomes
S′th(x− y) = i
(1i
δ
δη(x)
)(−1
i
δ
δη(y)
)· Zthj, η, η
∣∣∣∣η=η=j=0
· 1ZG[iτ, µ]
, (2.56)
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
Zthj, η, η = ei2g∫C (− 1
iδ
δη )(γ· 1i δδj )( 1
iδ
δη ) · ei∫C η·Sth·η · e i
2
∫C j·Dth·j · Z(0)
G [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.,∫
Cdz →
∫ τ
0
dz =∫ τ
0
dz0
∫d3~z. (2.58)
14
CITF
Re
Im
ti
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
ωn =2nπ
τ, for bosons, (2.59)
or
ωn =(2n + 1)π
τ, for fermions. (2.60)
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
Zthj, η, η = ei2g∫ τ0 (− 1
iδ
δη )(γ· 1i δδj )( 1
iδ
δη ) · ei∫ τ0 η·Sth·η · e i
2
∫ τ0 j·Dth·j · Z(0)
G [iτ, µ], (2.61)
where the contour integrals are in short for∫ τ
0
=∫ τ
0
dx
∫ τ
0
dy . (2.62)
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
S′th(x− y) = i〈(ψ(x)ψ(y))+〉 =
i〈ψ(x)ψ(y)〉, if x0 > y0
−i〈ψ(x)ψ(y)〉, if y0 > x0
, (2.63)
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
Z(0)th η, η = Z(0)
G [iτ ] · exp
i
∫ τ
0
dx
∫ τ
0
dy η(x) · Sth(x− y) · η(y)
. (2.64)
15
C4C2
C1
C3
ti − iβ
ti − iσ tf − iσ
ti tf
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
Sth(~p, ωn) =1
m + iγ · p =m− iγ · pm2 + p2
= (m− iγ · p) · ∆(F)th (~p, ωn; m2), (2.65)
where p = (~p, ωn), p2 = ~p 2 − ω2n and ∆(F)
th (~p, ωn; m2) = [m2 + ~p 2 − ω2n − iε]−1.
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
S′th(x− y) = i〈(ψ(x)ψ(y))C+〉β , (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
θC(z0 − z′0) = θ(ξ − ξ′), (2.67)
δC(z0 − z′0) = 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)δj(z′)
= δC(z0 − z′0) δ(3)(~z − ~z′). (2.70)
Then, Real-Time n-point Green’s functions can be derived by functional differentiations to the
’contoured’ finite-temperature generating functional ZCthj, 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) =
i〈(A(x1) · · ·A(xn))C〉 is given by
G(n)th = i
(1i
δ
δj(x1)
)· · ·
(1i
δ
δj(xn)
)· ZCthj, η, η
∣∣∣∣η=η=j=0
· 1ZG[β, µ]
, (2.71)
where C = C1 + C2 + C3 + C4 and
ZCthj, η, η =∫DADψDψ exp
[i
∫
CL
]. (2.72)
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]
Z(0)th j, η, η = N ZC12th j, η, η · ZC34th 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
limRe|x0−y0|→∞
Sth(x− y) = 0, (2.74)
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
ZC34th 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
Z(0)th j, η, η = Z(0)
G [iτ ] · exp
i
∫
C12d4x
∫
C12d4y η(x) · Sth(x− y) · η(y)
. (2.75)
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]
Sth(p) =
(S11(p) S12(p)
S21(p) S22(p)
)(2.76)
= (m− iγ · p)
(∆F (p)− 2πin(p)δ(m2 + p2) +ε(p0)n(|p0|)eσp0
−ε(p0)n(|p0|)e(β−σ)p0 ∆∗F (p) + 2πin(p0)δ(m2 + p2)
),
where the Feynman propagator ∆F (p) = [m2 + p2 − iε]−1. 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
Sth(p) = U†β(p)
((m− iγ · p) ∆F (p) 0
0 (m− iγ · p) ∆∗F (p)
)Uβ(p), (2.77)
where the unitary transformation matrix
Uβ(p) =
(√1− n(p0) −
√n(p0)√
n(p0)√
1− n(p0)
)(2.78)
contains all thermal information, and n(p0) is the Fermi-Dirac distribution function:
n(p0) =1
eβ|p0| + 1. (2.79)
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
In Eq. (2.57), first apply the functional operations(
δδη · · · δ
δη
)in Lint on the free (non-interacting)
Finite-Temperature generating functional Z(0)th ,
Zthj, η, η (2.80)
= exp
i
∫
Cη · Sth
[1− g
(γ · δ
δj
)Sth
]−1
· η + Tr ln[1− g
(γ · δ
δj
)Sth
]
· exp
i
2
∫
Cj ·Dth · j
· Z(0)
G [iτ, µ].
Then continue with the δ/δj operation with the aid of the reciprocity relation, Eq. (D.21), as in
Eq. (2.6),
Zthj, η, η = exp
i
2
∫
Cj ·Dth · j
· exp
− i
2
∫
C
δ
δA·Dth · δ
δA
(2.81)
exp
i
∫
Cη ·Gth[A] · η + Lth[A]
· Z(0)
G [iτ, µ],
where the functional operations have been changed to gauge fields A(x) given by
Aµ(x) =∫
Cdy Dµν
th (x− y) · jν(y). (2.82)
Both the field-coupled, thermal Green’s function Gth[A] and the thermal closed-fermion-loop func-
tional are functionals of gauge fields, A(x), and are given by
Gth[A] = Sth [1− g (γ ·A)Sth]−1, (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
S′th = eD(th)A
[Gth[A]
eLth[A]
Z[iτ ]
]∣∣∣∣A→0
, (2.85)
where
D(th)A = − i
2
∫
C
δ
δA·Dµν
th ·δ
δA, (2.86)
19
and Z[iτ ], with or without a chemical potential µ, is the thermal normalization constant and is
given by
Z[iτ ] =[eD
(th)A · eLth[A]
]∣∣∣∣A→0
· Z(0)[iτ ]. (2.87)
Here Z(0)[iτ ] is the normalization constant for the free Finite-Temperature generating functional
Z(0)th [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
D(th)A = − i
2
∫ τ
0
dx
∫ τ
0
dyδ
δA(x)·Dth(x− y) · δ
δA(y), (2.88)
where both time-integrations are limited in the range of (0, τ). In contrast, the linkage operator in
RTF is
D(th)A = − i
2
∫
Cdx
∫
Cdy
δ
δAµ(x)·Dµν
th (x− y) · δ
δAν(y), (2.89)
where both time-integrations are along a given time-path contour C.A gauge field is local in configuration space, i.e., A|x〉 = A(x)|x〉, but non-local in momentum
space, i.e., 〈~p, n|A|~k, l〉 = An−l(~p−~k) in ITF or 〈~p, p0|A|~k, k0〉 = 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
Aµ(x) =1τ
∑
l
Aµl (~x) · e−iωlx0 =
1τ
∑
l
∫d3~q
(2π)3Aµ
l (~q) · ei(~q·~x−ωlx0), (2.90)
and its functional differentiation is
δ
δAµ(x)=
1τ
∑
l
∫d3~q
(2π)3ei(~q·~x−ωlx0) · δ
δAµl (~q)
. (2.91)
For convenience of notation, the sum-integral can be expressed as∫
dq ≡ 1τ
∑
l
∫d3~q
(2π)3, (2.92)
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
Fµν(x) = ∂µAν(x)− ∂νAµ(x) =1τ
∑
l
∫d3~q
(2π)3[qµAν(q)− qνAµ(q)
]· eiq·x, (2.93)
or in momentum space
Fµν(q) = qµAν(q)− qνAµ(q). (2.94)
20
In ITF, the linkage operator in the momentum representation becomes
D(th)A = − i
21τ
∑
l
∫d3~k
(2π)31τ
∑
l′
∫d3~k′
(2π)3(2.95)
· δ
δAl(~k)·[Dth(~k, ωl) · (2π)3τδ(3)(~k + ~k′)δl,−l′
]· δ
δAl′(~k′)
or
D(th)A = − i
21τ
∑
l
∫d3~k
(2π)3δ
δAl(~k)· Dth(~k, ωl) · δ
δA−l(−~k)(2.96)
Similarly, the linkage operator in RTF is
D(th)A = − i
2
∫d4k
(2π)4δ
δA(k)· Dth(~k, k0) · δ
δA(−k), (2.97)
Here Dth(~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
Aµ(x) =∫
d4q
(2π)4Aµ(~q, q0) · ei(~q·~x−q0x0), (2.98)
andδ
δAµ(x)=
∫d4q
(2π)4ei(~q·~x−q0x0) · δ
δAµ(q), (2.99)
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,
(m + γµ[∂µ − igAµ])Gc(x, y|A) = δ(4)(x− y), (2.100)
and its solution, the coupled fermion Green’s function, in operator form is
Gc[A] = (m− iγ ·Π)−1, (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 in-
terest. 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
(m + γµ[∂µ − igAµ])Gth(x, y|A) = δ(4)th (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
δ(4)th (x− y) = δ(3)(~x− ~y) · δth(x0 − y0) = δ(3)(~x− ~y) · 1
τ
∑n
e−iωn(x0−y0), (2.104)
and Matsubara frequencies for fermions with n = 0,±1,±2, . . . are
ωn =(2n + 1)π
τ. (2.105)
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
Gth[A] = (m− iγ ·Π)−1. (2.107)
In Matsubara representation, the energy component p0 is replaced by its corresponding Matsubara
frequency ωn as
Gc(~p, p0|A) → Gth(~p, ωn|A), (2.108)
where
Gc(x, y|A) =∫
dp0
2π
∫d3~p
(2π)3Gc(~p, p0|A) · ei~p·(~x−~y)−ip0(x0−y0), (2.109)
and
Gth(x, y|A) =1τ
∑n
∫d3~p
(2π)3Gth(~p, ωn|A) · ei~p·(~x−~y)−iωn(x0−y0). (2.110)
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
Lth[A] = −i
∫ g
0
dg′Tr (γ ·A)Gth[g′A]. (2.113)
22
Real-Time Formalism
In Real-Time Formalism (RTF), the basic differential equation is similar to that of ITF,
(m + γµ[∂Cµ − igAµ]
)Gth(x, y|A) = δ
(4)C (x− y), (2.114)
where x0 and y0 are ’complex-time’ variables on a given time-path contour C from ti to ti − iβ,
and the differential operator ∂Cµ and δ-function δ(4)C (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,
Gth(~p, ωn|A), and its real-time counterpart Gth(~p, p0|A) is not readily transparent. The relationship
can be illustrated more clearly through the Fourier (integral) transform as
Gth(~p, ωn|A) =∫ τ
0
dz0 e+iωnz0
∫d3~z e−i~p·~z ·Gth(~z, z0|A) (2.117)
=∫ τ
0
dz0 e+iωnz0
∫ ∞
−∞
dp0
2πe−ip0z0 · Gth(~p, p0|A)
=∫ ∞
−∞
dp0
2π
∫ τ
0
dz0 e−i(p0−ωn)z0 · Gth(~p, p0|A).
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
Gth(~p, ωn|A) =∫ ∞
−∞
dp0
2π
1−i(p0 − ωn)
(e−i(p0−ωn)τ − 1
)· Gth(~p, p0|A), (2.118)
which leads to
Gth(~p, ωn|A) = i
∫ ∞
−∞
dp0
2π
eβp0 + 1ωn − p0
· Gth(~p, p0|A), (2.119)
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,
Gth(x, y|A) =1τ
∑n
∫d3~p
(2π)3Gth(~p, ωn|A) · ei~p·(~x−~y)−iωn(x0−y0), (2.120)
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
Lth[A] = −i
∫ g
0
dg′Tr (γ ·A)Gth[g′A] (2.121)
= −i
∫ g
0
dg′Tr (γ ·A) [Gc[g′A] + ∆Gth[g′A]]
= −i
∫ g
0
dg′Tr (γ ·A)Gc[g′A] − i
∫ g
0
dg′Tr (γ ·A)∆Gth[g′A]≡ Lc[A] + ∆Lth[A].
The normalization constant Z[iτ ] = Zth0, 0, 0 of the generating functional represents the partition
function of the thermal bath, and can be re-written as
Z[iτ ] = eD(th)A · eLth[A]
∣∣∣∣A→0
· Z(0)[iτ ] (2.122)
→ eD(c)A +∆D
(th)A · eLc[A]+∆Lth[A]
∣∣∣∣A→0
· Z(0)[β],
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
Z[iτ ]Z(0)[iτ ]
→ Z[β]Z(0)[β]
= eD(c)A +∆D
(th)A · eLc[A]+∆Lth[A]
∣∣∣∣A→0
(2.123)
= e∆D(th)A ·
(eD
(c)A · eLc[A]
)· eD
(c)12 ·
(eD
(c)A · e∆Lth[A]
)∣∣∣∣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]
Gc[A] = (m + iγ ·Π) ·[m2 + (γ ·Π)2
]−1
, (2.124)
and the denominator will becomes
(γ ·Π)2 = Π2 + igσ · F (2.125)
with the anti-symmetric tensor σµν
σµν =14
[γµ, γν ] = −σνµ. (2.126)
For the causal Green’s function, the mass m2 → m2 − iε is implicitly associated with an infinites-
imal positive parameter ε. The ’proper-time’ representation introduced by Schwinger [65] is to
parameterize the denominator in an exponential form as
Gc[A] = (m + iγ ·Π) · i∫ ∞
0
ds exp−is
[m2 + (γ ·Π)2
]. (2.127)
With the help of Eq.(2.125), one can further expand (γ ·Π)2,
Gc[A] = (m + iγ ·Π) · i∫ ∞
0
ds e−ism2 · e−is(Π2+igσ·F), (2.128)
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
Lc[A] = −i
∫ g
0
dg′Tr
(γ ·A) (m + iγ ·Π) · i∫ ∞
0
ds e−ism2e−is(γ·Π)2
. (2.129)
Since Tr
(γ ·A) e−is(γ·Π)2
= 0, the term proportional to m vanishes and
Lc[A] = i
∫ ∞
0
ds e−ism2∫ g
0
dg′Tr
(γ ·A)(γ ·Π)e−is(γ·Π)2
, (2.130)
or
Lc[A] = −12
∫ ∞
0
ds s−1 e−ism2∫ g
0
dg′∂
∂g′Tr
e−is(γ·Π)2
(2.131)
= −12
∫ ∞
0
ds
se−ism2
Tr
e−is(γ·Π)2− g = 0 .
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
U(s) ≡ exp−is (γ ·Π)2
= exp
−is(Π2 + igσ · F)
, (2.132)
that is,
Gc[A] = (m + iγ ·Π) · i∫ ∞
0
ds e−ism2U(s), (2.133)
and
Lc[A] = −12
∫ ∞
0
ds
se−ism2
Tr U(s) − g = 0 . (2.134)
Following Fradkin’s approach [66, 49], one can replace U(s) with an ordered exponential U(s, v) of
U(s, v) =(
exp−i
∫ s
0
ds′[Π2 + igσ · F + v(s′) ·Π])
+
, (2.135)
where the ’proper-time’ ordering (· · · )+ is with respect to s′, and vµ(s′) is an arbitrary vector
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
U(s, v) = exp
i
∫ s−ε
0
ds′δ2
δv2µ(s′)
·W(s, v)|ε→0, (2.136)
where
W(s, v) =(
exp−i
∫ s
0
ds′ [vµ(s′) ·Πµ + igσ · F])
+
. (2.137)
In the process of introducing Gaussian translation, the non-commutative Πµ-function is replaced by
the functional differentiation with respect to vµ as [67]
Πµ → iδ
δvµ. (2.138)
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(s′). Again, one can introduce F(s, v) such that
W(s, v) = exp∫ s
0
ds′ vµ(s′) · ∂µ
· F(s, v) (2.139)
with
F(s, v) = e−∫ s0 ds′ vµ(s′)·∂µ ·
(e−i
∫ s0 ds′[vµ(s′)·Πµ+igσ·F]
)+
. (2.140)
Then, the Green’s function becomes
Gc[A] = (m + iγ ·Π) · i∫ ∞
0
ds e−ism2exp
i
∫ s
0
ds′δ2
δv2µ(s′)
·W(s, v)
∣∣∣∣vµ→0
(2.141)
= (m + iγ ·Π) · i∫ ∞
0
ds e−ism2 · ei∫ s0 ds′ δ2
δv2µ(s′) · e
∫ s0 ds′ vµ(s′)·∂µ · F(s, v)
∣∣∣∣vµ→0
,
26
and the leading factor, m + iγ ·Π = m + iγ · i(∂ − igA), can be expanded as
Gc[A] = i
∫ ∞
0
ds e−ism2 · ei∫ s0 ds′ δ2
δv2µ(s′) · [m− γ · (∂ − igA)] · e
∫ s0 ds′ vµ(s′)·∂µ · F(s, v)
∣∣∣∣vµ→0
,
(2.142)
or in terms of functional differentiation over v,
Gc[A] = i
∫ ∞
0
ds e−ism2 · ei∫ s0 ds′ δ2
δv2µ(s′) ·
[m− γ · δ
δv(s)
]· e
∫ s0 ds′ vµ(s′)·∂µ · F(s, v)
∣∣∣∣vµ→0
. (2.143)
To solve F(s, v), differentiate F(s, v) with respect to s as
∂F(s, v)∂s
= −ig e−∫ s0 ds′ vµ(s′)·∂µ · vµ ·Aµ + iσ · F · e+
∫ s0 ds′ vµ(s′)·∂µ · F(s, v). (2.144)
Apply 〈~x, x0| on the left-hand side, and the solution is
〈~x, x0|F(s, v) (2.145)
=
(exp
−ig
∫ s
0
ds′[v(s′) ·A(x−
∫ s′
0
ds′′ v(s′′)) + iσ · F(x−∫ s′
0
ds′′ v(s′′))
])
+
·〈~x, x0|
Thus,
〈~x, x0|W(s, v)|~y, y0〉 =∫
dz 〈~x, x0|e∫ s0 ds′ vµ(s′)·∂µ |~z, z0〉 · 〈~z, z0|F(s, v)|~y, y0〉 (2.146)
=∫
dz 〈x +∫ s
0
ds′ vµ(s′)|z〉 · 〈z|F(s, v)|y〉
With 〈x +∫ s
0ds′ vµ(s′)|z〉 = δ(x +
∫ s
0ds′ vµ(s′)− z) and 〈z|y〉 = δ(z − y), one arrives at
〈x|W(s, v)|y〉 =
(exp
−ig
∫ s
0
ds′[v(s′) ·A(y −
∫ s′
0
v) + iσ · F(y −∫ s′
0
v)
])
+
(2.147)
×δ(x− y +∫ s
0
ds′ vµ(s′)),
and
〈x|U(s, v)|y〉 = exp
i
∫ s
0
ds′δ2
δv2µ(s′)
(2.148)
×(
exp
−ig
∫ s
0
ds′[v(s′) ·A(y −
∫ s′
0
v) + iσ · F(y −∫ s′
0
v)
])
+
×δ(x− y +∫ s
0
ds′ vµ(s′)).
Thus, the causal Green’s function becomes
〈x|Gc[A]|y〉 = i
∫ ∞
0
ds e−ism2 · ei∫ s0 ds′ δ2
δv2µ(s′) · [m− γ · (∂x − igA(x))] (2.149)
×(
exp
−ig
∫ s
0
ds′[v(s′) ·A(y −
∫ s′
0
v) + iσ · F(y −∫ s′
0
v)
])
+
×δ(x− y +∫ s
0
ds′ vµ(s′))∣∣∣∣vµ→0
.
27
Further replacing Πµ(x) = i[∂xµ − igAµ(x)] in the numerator factor with help of Eq.(2.138),
〈x|Gc[A]|y〉 = i
∫ ∞
0
ds e−ism2 · ei∫ s0 ds′ δ2
δv2µ(s′) ·
[m− γ · δ
δv(s)
](2.150)
×(
exp
−ig
∫ s
0
ds′[v(s′) ·A(y −
∫ s′
0
v) + iσ · F(y −∫ s′
0
v)
])
+
×δ(x− y +∫ s
0
ds′ vµ(s′))∣∣∣∣vµ→0
.
Similarly, the closed-fermion-loop functional is given by
Lc[A] = −12
∫ ∞
0
ds
se−ism2
∫d4p
∫d4x · ei
∫ s0 ds′ δ2
δv2µ(s′) (2.151)
×tr
e∫ s0 ds′ v(s′)·p
(e−ig
∫ s0 ds′
[v(s′)·A(x−∫ s′
0 v)+iσ·F(x−∫ s′0 v)
])
+
− 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 〈~p, p0| ≡ 〈p| to the causal Green’s function,
〈~p, p0|Gc[A]|~y, y0〉 = i
∫ ∞
0
ds e−ism2 · ei∫ s0 ds′ δ2
δv2µ(s′) ·
[m− γ · δ
δv(s)
](2.152)
·ei∫ s0 ds′ vµ(s′)·pµ · 〈~p, p0|F(s, v)|~y, y0〉
∣∣∣∣vµ→0
,
where
〈~p, p0|~y, y0〉 =[(2π)4
]−1/2 · e−i(~p·~y−p0y0) = (2π)−2 · e−ip·y (2.153)
is the normalization factor. Hence,
〈~p, p0|Gc[A]|~y, y0〉 (2.154)
=[(2π)4
]−1/2 · e−i(~p·~y−p0y0) · i∫ ∞
0
ds e−ism2
×ei∫ s0 ds′ δ2
δv2µ(s′) ·
[m− γ · δ
δv(s)
]· ei
∫ s0 ds′ vµ(s′)·pµ
×(
exp
−ig
∫ s
0
ds′[v(s′) ·A(y −
∫ s′
0
v) + iσ · F(y −∫ s′
0
v)
])
+
∣∣∣∣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
Sc = (m + γ · ∂)−1 (2.155)
in the operator form, or
Sc(p) = (m + iγ · p)−1 = (m− iγ · p)∆Fc (p) (2.156)
in the momentum representation, where
∆Fc (~p, p0; m2) = ∆F
c (ω, p0) =1
m2 + ~p 2 − p20 − iε
=1
ω2 − p20 − iε
(2.157)
with ω2 = m2 +~p 2. Similarly, the non-interacting finite-temperature fermion propagator is given by
Sth(x− y) = 〈x| (m− γ · ∂)−1 |y〉 = (m− γ · ∂x) ·∆Fth(x− y; m2) (2.158)
in the configuration space representation, or
Sth(~p, ωn) = (m− iγ · p) · [m2 + p2]−1
= (m− iγ · p) · ∆Fth(~p, ωn;m2) (2.159)
in momentum-Matsubara representation, where Matsubara frequencies for fermions are
ωn =(2n + 1)π
τ, (2.160)
and the function ∆Fth(x− y; m2) is defined as
∆Fth(x− y;m2) = 〈x| [m2 + (γ · i∂)2
]−1 |y〉, (2.161)
and its Fourier-transformed expression is
∆Fth(~p, ωn;m2) = ∆F
th(ω, ωn) =1
ω2 − ω2n
. (2.162)
In the mixed (non-covariant) representation, the free finite-temperature fermion propagator is given
by
Sth(~p, z0) =1τ
∑n
e−iωnz0 Sth(~p, ωn) (2.163)
=1τ
∑n
e−iωnz0 (m− iγ · p) ∆Fth(ω, ωn)
=1τ
∑n
e−iωnz0 (m− i~γ · ~p + iγ0ωn) ∆Fth(ω, ωn)
=[m− i~γ · ~p + iγ0 ·
(−1
i
∂
∂z0
)]1τ
∑n
e−iωnz0 ∆Fth(ω, ωn).
29
The summation over n in the last line is the Matsubara-type Fourier transform of ∆Fth(ω, z0), and
can be evaluated as in Appendix B.3 as
∆Fth(ω, z0) =
1τ
∑n
e−iωnz0 ∆Fth(ω, ωn) (2.164)
=i
2ω
[1− n(ω)] e−iωz0 − n(ω) e+iωz0
,
where n(ω) is the Fermi-Dirac distribution function as
n(ω) =1
eβω + 1. (2.165)
Thus,
Sth(~p, z0) =[m− i~γ · ~p + iγ0 ·
(−1
i
∂
∂z0
)]i
2ω
[1− n(ω)] e−iωz0 − n(ω) e+iωz0
. (2.166)
When working out the differentiation over z0,
Sth(~p, z0) =i
2ω[ m− i~γ · ~p ]
[1− n(ω)] e−iωz0 − n(ω) e+iωz0
(2.167)
−γ0
2[1− n(ω)] e−iωz0 + n(ω) e+iωz0
,
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
∆Fth(~p, ωn;m2, µ) = ∆F
th(ω, ωn; µ) =1
ω2 − (ωn + µ)2(2.168)
and
∆Fth(ω, z0;µ) =
1τ
∑n
e−iωnz0 ∆Fth(ω, ωn;µ) (2.169)
=i
2ω
[1− n(ω + µ)] e−iωz0 − n(ω − µ) e+iωz0
,
respectively. Then, the finite-temperature fermion propagator becomes
Sth(~p, ωn;µ) =1
m + i~γ · ~p− iγ0(ωn + µ)=
m− i~γ · ~p + iγ0(ωn + µ)m2 + ~p 2 − (ωn + µ)2
(2.170)
or
Sth(~p, z0;µ) (2.171)
=1τ
∑n
e−iωnz0 Sth(ω, ωn; µ)
=[m− i~γ · ~p + iγ0 ·
(µ− 1
i
∂
∂z0
)]· i
2ω
[1− n(ω + µ)] e−iωz0 − n(ω − µ) e+iωz0
,
30
and the Fourier transform of the finite-temperature propagator is given by
Sth(~z, z0; µ) (2.172)
=∫
d3~p
(2π)3e+i~p·~z Sth(~p, z0; µ)
=∫
d3~p
(2π)3
[m− i~γ · ~p + iγ0 ·
(µ− 1
i
∂
∂z0
)]
× i
2ω
[1− n(ω + µ)] ei(~p·~z−ωz0) − n(ω − µ) ei(~p·~z+ωz0)
=[m− i~γ ·
(1i
∂
∂~z
)+ iγ0 ·
(µ− 1
i
∂
∂z0
)]
×∫
d3~p
(2π)3i
2ω
ei(~p·~z−ωz0) −
[n(ω + µ) ei(~p·~z−ωz0) + n(ω − µ) ei(~p·~z+ωz0)
].
The terms in the second part of the curly brackets are the thermal part of a finite-temperature
fermion (anti-fermion) propagator, δSthµν , 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
∆Bth(~k, ωl; m2) = ∆B
th(ωk, ωl) =1
ω2k − ω2
l
, (2.173)
where ω 2k = ~k 2 + m2, and the propagator in the mixed representation becomes
∆Bth(ωk, z0) =
1τ
∑
l
e−iωlz0 ∆Bth(ωk, ωl) (2.174)
=i
2ωk
[1 + n(ωk)]e−iωz0 + n(ωk)e+iωkz0
,
where the boson Matsubara frequencies are defined as
ωl =2lπ
τ, (2.175)
and n(ωk) is the Bose-Einstein distribution function:
n(ωk) =1
eβωk − 1. (2.176)
For non-zero chemical potential, the free finite-temperature boson propagator and its mixed
representation become
∆Bth(~k, ωl; m2, µ) = ∆B
th(ωk, ωl; µ) =1
ω2k − (ωl − µ)2
, (2.177)
and
∆Bth(ωk, z0; µ) =
1τ
∑
l
e−iωlz0 ∆Bth(ωk, ωn; µ) (2.178)
=i
2ωk
[1 + n(ωk + µ)] e−iωkz0 + n(ωk − µ) e+iωkz0
,
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
Dµνth (x− y) = i〈(Aµ(x)Aν(y))+〉β . (2.179)
Depending on the choice of gauge, the form of a finite-temperature photon propagator is quite dif-
ferent. 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 cal-
culations, 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]
Dµνc (k) =
1k2
[gµν − kµkν
k2 + (u · k)2− u · k
k2 + (u · k)2(kµuν + uµkν)
]=
1k2
PCµν , (2.180)
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
PCµν = gµν − kµkν
k2 + (u · k)2− u · k
k2 + (u · k)2(kµuν + uµkν). (2.181)
In the usual Coulomb gauge, uµ = (1,~0) and the photon propagator becomes
Dµνc (k) = − 1
~k2δµ0δ0ν +
1k2
PTµν =
1k2
[−k2
~k2δµ0δ0ν + PT
µν
]=
1k2
PCµν . (2.182)
Here PT is the spatially transverse projection operator,
PTij = δij − kikj
~k 2, PT
00 = PT0i = 0, PT · PT = PT , (2.183)
which projects vectors onto the plane orthogonal to both kµ and kj as
PT · ~k = PTij kj = 0, (2.184)
PT · k = PTiν kν = PT
ij kj = 0, (2.185)
and is related to the longitudinal projection operator PL through the identity:
PLµν = Pµν − PT
µν , PL · PL = PL, (2.186)
where Pµν is the gauge-invariant projection operator on a plane orthogonal to kµ and
Pµν = δµν − kµkν
k2= gµν − kµkν
k2, P · P = P, (2.187)
which will project out any gauge-parameter dependent term and
P · PT = PT , P · PL = PL. (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]
Dµνc (k) =
(−1/~k2 0
0 PTij/(~k2 − k2
0)
), (2.189)
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;
Dµνth (~k, ωn) = − 1
~k2δµ0δ0ν +
1~k2 − ω2
n
PTµν . (2.190)
Similarly, the finite-temperature photon propagator in the Real-Time formalism can be written as
Dµνth (~k, k0) = Dµν
c (~k, k0) + δDµνth (~k, k0), (2.191)
where the T = 0 part is just the causal propagator, and the T 6= 0 part is
δDµνth (~k, k0) = f(k0) ·
[Dµν
th (~k, ωn = k0 + iε)− Dµνth (~k, ωn = k0 − iε)
](2.192)
= 2πi ε(k0) f(k0) δ(~k 2 − k20) · PT
µν
= 2πi n(|~k|) δ(~k 2 − k20) ·
[δµi · PT
ij · δjν
]
where
ε(k0) = θ(k0)− θ(−k0), (2.193)
and
f(k0) =1
eβk0 + 1. (2.194)
In QED, it can be shown that there is no contribution from the (µ = 0, ν = 0)-component in
Coulomb gauge. Both Dµνc and Dµν
th are symmetric in their indices, as is the thermal part δDµνth ;
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
The thermal part of a finite-temperature photon propagator δDthµν can be considered as describing
the emission and absorption of photons by the heat bath, i.e., the thermal photons as blackbody
radiation. The factor of δ(~k2 − k20) in δDth
µν 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
Sth(~z, z0) =∫
d3~p
(2π)3Sth(~p, z0) e+i~p·~z (2.195)
=∫
d3~p
(2π)3i
2ω
[m− i~γ · ~p + iγ0 ·
(i
∂
∂z0
)]
×
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), then u(p)~γ u(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-
temperature fermion propagator δSthµν 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± iε, 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 →q0 ± iεq0, 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]
∆basis(q) = U(q) · ∆th(q) · UT(−q), (2.196)
where superscript ’T’ denotes the transpose of a matrix. Propagators in F/F and R/A bases are
diagonalized as
∆F/F(q) =
(∆F (q) 0
0 ∆∗F (q)
)(2.197)
and
∆R/A(q) =
(∆R(q) 0
0 ∆A(q)
), (2.198)
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
∆Klydish(q) =
(∆S(q) ∆R(q)
∆A(q) 0
), (2.199)
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 propa-
gator as
S′c(p) =[S−1
c (p)− Σ(p)]−1
=1
m0 + iγ · p− Σ(p), (2.200)
where m0 is the bare mass, and Σ(p) is the fermion self-energy. Replacing p with ω = −iγ · p in the
argument of both S′c and Σ,
S′c(ω)−1
= 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
Z−12 = 1 +
∂Σ(ω; m0)∂ω
∣∣∣∣∣ω=m
. (2.203)
Both can be easily seen through adding and subtracting Σ(m; m0) to Eq. (2.201) as
S′c(ω)−1
= m0 − Σ(m; m0)− ω − [Σ(ω; m0)− Σ(m; m0)] (2.204)
= [m− ω]
[1 +
Σ(ω; m0)− Σ(m;m0)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
S′th(p) =[S−1
th (p)− Σth(p)]−1
=1
m0 + iγ · p− Σth(p), (2.205)
where four-vector p denotes (~p, ωn), and Σth(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
continued to p0 +iε with p0 real, and Σth(~p, ωn) → Σth(~p, p0). Then, the pole of S′th(~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
S′th(p) =m− iγ · p− Σth + 1
2Tr [Σth]
m20 + p2 − Ξth(p)
, (2.208)
where the scalar self-energy Ξth is
Ξth(p) =12
Tr [(m− iγ · p) Σth]− 14
Tr [Σ2th] +
18
(Tr [Σth])2. (2.209)
Then, the damping rate can be approximated as [26, 56]
Γ = −Im[p0] ' − 14E
ImTr [(m− iγ · p)Σth(p0, ~p)]∣∣∣∣p0=E−iΓ
. (2.210)
In general, the fermion self-energy Σth 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 Σth
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,
S′th(p) =1
m0 + iγ · p− Σth(p)→ S′th(ω) =
1m0 − ω − Σth(ω; m0)
, (2.211)
where Σth(p) = Σth(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),
S′th(ω)−1
= m0 − Σ(m;m0)− ω − [Σth(ω; m0)− Σ(m;m0)] (2.212)
= m− ω − [Σ(ω;m0)− Σ(m; m0)]− [Σth(ω; m0)− Σ(ω; m0)]
= [m− ω]
[1 +
Σ(ω; m0)− Σ(m;m0)ω −m
]−
[Σth(ω; m0)− Σ(ω; m0)
]
= Z−12
m− ω − Z2
[Σth(ω; m0)− Σ(ω; m0)
],
where the renormalized mass m and wave-function renormalization constant Z2 are defined in Eqs.
(2.202) and (2.203), respectively. Let Σ′th(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
S′th(ω) =Z2
m− ω − Σ′th(ω; m0)
→ 1m− ω − Σ′
th(ω; m). (2.213)
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
S′th(ω) =1
m− ω − Σ′th(ω)
(2.214)
= i
∫ ∞
0
ds e−is[m−ω−Σ′th(ω)]
= i
∫ ∞
0
ds e−is[m−ω−ReΣ′th(ω)]−s ImΣ′th(ω)
= e−ImΣ′th(ω)(i ∂∂u ) i
∫ ∞
0
ds e−isu
∣∣∣∣u=m−ω−ReΣ′th(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
S′th(p) =1
m + iγ · p− Σ′th(p)
(2.215)
=[m− iγ · p− Σ′
th(p)]· 1[m− Σ′
th(p)]2 + [γ · p]2
=[m− iγ · p− Σ′
th
]· i
∫ ∞
0
ds e−ism2+p2−δm2−2s[m−ReΣ′th] ImΣ′th ,
where δm2 = 2mReΣ′th−
[ReΣ′
th
]2
+[ImΣ′
th
]2
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
S′th = eD(th)A ·
[Gth[A]
eLth[A]
Z[iτ ]
]∣∣∣∣A→0
, (3.1)
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
the Finite-Temperature linkage operator expD(th)A , and will be set to zero afterwards. Hence, the
gauge fields can be treated as dummy fields unless external fields exist.
In ITF, the operator D(th)A in the exponent of the Finite-Temperature linkage operator expD(th)
A is
D(th)A = − i
2
∫ τ
0
dx
∫ τ
0
dyδ
δA(x)·Dth(x− y) · δ
δA(y), (3.2)
and its Fourier transformed expression includes a complete Matsubara sum over discrete frequencies
of photons in ITF as
D(th)A = − i
21τ
∑
l
∫d3~k
(2π)3δ
δAl(~k)· Dth(~k, ωl) · δ
δA−l(−~k), (3.3)
which can be converted to a complete integral over photon energies k0 as
D(th)A = − i
2
∫d4k
(2π)4δ
δAµ(k)· Dµν
th (k) · δ
δAν(k), (3.4)
where the Finite-Temperature photon propagator Dµνth (k) is in real-time form a la Dolan and Jackiw
[12]. Hence, the Finite-Temperature linkage operator expD(th)A can be expressed in any formalism
of choice. Furthermore, the operator D(th)A can be expressed in terms of space integrals as
D(th)A = − i
2
∫dx
∫dy
δ
δAµ(x)·Dµν
th (x− y) · δ
δAν(y), (3.5)
where the range of time integrals,∫
dx0 and∫
dy0, are from −∞ to +∞, and the finite temperature
photon propagator Dµνth (x− y) is in real-time form.
The finite-temperature photon propagator in real-time form can be decomposed into two parts
as
Dµνth (x− y) = Dµν
c (x− y) + δDµνth (x− y), (3.6)
where the first term is the causal photon propagator at zero temperature, and the second is the
contribution from the heat bath, i.e., the effect of thermal photons. In turn, D(th)A of the Finite-
Temperature linkage operator can be split into two parts as
D(th)A = D
(c)A + ∆D
(th)A , (3.7)
where the causal part, D(c)A , and the thermal part, ∆D
(th)A , are given by
D(c)A = − i
2
∫dx
∫dy
δ
δAµ(x)·Dµν
c (x− y) · δ
δAν(y), (3.8)
40
and
∆D(th)A = − i
2
∫dx
∫dy
δ
δAµ(x)· δDµν
th (x− y) · δ
δAν(y), (3.9)
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
linkage expD(c)A and thermal linkage exp∆D
(th)A is irrelevant and can be exchanged. If the causal
linkage is applied first, it leads to
S′th =
e∆D(th)A ·
[eD
(c)A ·
(Gth[A]
eLth[A]
Z[β]
)]∣∣∣∣A→0
. (3.10)
The causal linkage operator expD(c)A 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
eD(c)A ·
[(Gc[A] + δGth[A])
(eLc[A]+∆Lth[A]
)]. (3.11)
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
theory, i.e., eD(c)A ·Gc[A] eLc[A]. In the first approximation, one could treat it as
S′th '
e∆D(th)A ·
(Gth,R[A]
eLth,R[A]
Z,R[β]
)∣∣∣∣A→0
, (3.12)
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
〈~p, n|S′th|~y, y0〉 '
e∆D(th)A ·
(〈~p, n|Gth[A]|~y, y0〉 eLth[A]
Z[β]
)∣∣∣∣A→0
. (3.13)
41
The thermal linkage, exp∆D(th)A , 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 renor-
malization 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 〈~p, p0| of Eq. (2.152) in the zero-temperature theory, we can apply the Matsubara repre-
sentation, 〈~p, n| ≡ 〈~p, ωn|, to the field-coupled, thermal Green’s function in ITF as
〈~p, n|Gth[A]|~y, y0〉 = i
∫ ∞
0
ds e−ism2 · ei∫ s0 ds′ δ2
δv2(s′) ·[m− γ · δ
δv(s)
](3.14)
· ei∫ s0 ds′ vµ(s′)·pµ · 〈~p, n|F(s, v)|~y, y0〉
∣∣∣vµ→0
,
where F(s, v) is given by
〈~x, x0|F(s, v) (3.15)
=
(exp
−ig
∫ s
0
ds′[v(s′) ·A(x−
∫ s′
0
ds′′ v(s′′)) + iσ · F(x−∫ s′
0
ds′′ v(s′′))
])
+
·〈~x, x0|,
and the corresponding normalization in the Matsubara representation becomes
〈~p, n|~y, y0〉 =[(2π)3τ
]−1/2 · e−i(~p·~y−ωny0). (3.16)
First apply the same trick of Fried and Woodward [84] by inserting a functional integral represen-
tation of unity,
1 =∫
d[u] δ
[u(s′)−
∫ s′
0
ds′′v(s′′)
], (3.17)
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
〈~x, x0|F(s, v) =∫
d[u] δ
[u(s′)−
∫ s′
0
ds′′v(s′′)
](3.18)
·(
exp−ig
∫ s
0
ds′ [v(s′) ·A(x− u(s′)) + iσ · F(x− u(s′))])
+
·〈~x, x0|.
Then replace the delta functional with its Fourier functional integral representation,
δ
[u(s′)−
∫ s′
0
ds′′v(s′′)
](3.19)
= C−1FFI
∫d[Ω] · exp
i
∫ s
0
ds′Ω(s′) ·[u(s′)−
∫ s′
0
ds′′v(s′′)
]
= C−1FFI
∫d[Ω] · exp
i
∫ s
0
ds′ Ω(s′) · u(s′)· exp
−i
∫ s
0
ds′ v(s′) ·∫ s
s′ds′′ Ω(s′′)
,
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,
〈~x, x0|F(s, v) =∫
d[u]∫
d[Ω] exp
i
∫ s
0
ds′Ω(s′) · u(s′)
(3.20)
· exp−i
∫ s
0
ds′ v(s′) ·[∫ s
s′ds′′ Ω(s′′) + gA(x− u(s′))
]
·(
exp
+g
∫ s
0
ds′ σ · F(x− u(s′)))
+
·〈~x, x0|≡ F(x|s, v) · 〈~x, x0|.
From Equation (3.14) and 〈~p, n|F(s, v)|~y, y0〉 = 〈~p, n|~y, y0〉 · F(y|s, v), the field-coupled, Thermal
Green’s function(al) becomes
〈~p, ωn|Gth[A]|~y, y0〉 =[(2π)4
]−1/2 · e−i(~p·~y−ωny0) · i∫ ∞
0
ds e−ism2 · ei∫ s0 ds′ δ2
δv2(s′) (3.21)
×[m− γ · δ
δv(s)
]· ei
∫ s0 ds′ vµ(s′)·pµ
×∫
d[u]∫
d[Ω] · ei∫ s0 ds′ Ω(s′)·u(s′) · e−i
∫ s0 ds′ v(s′)·∫ s
s′ ds′′ Ω(s′′)
×(
exp−ig
∫ s
0
ds′ [v(s′) ·A(y − u(s′)) + iσ · F(y − u(s′))])
+
∣∣∣∣∣vµ→0
.
43
Begin with the operation
m− γ · δδv(s)
first, which leads to
〈~p, n|Gth[A]|~y, y0〉 =[(2π)3τ
]−1/2 · e−i(~p·~y−ωny0) · i∫ ∞
0
ds e−ism2∫
d[u]∫
d[Ω] (3.22)
×ei∫ s0 ds′ Ω(s′)·u(s′) ·
(exp
g
∫ s
0
ds′σ · F(y − u(s′)))
+
×m− iγ · [p− gA(y − u(s))]×e
i∫ s0 ds′ δ2
δv2(s′) · eip·∫ s0 ds′ v(s′)
× exp−i
∫ s
0
ds′ v(s′) ·[∫ s
s′ds′′ Ω(s′′) + gA(y − u(s′))
]∣∣∣∣vµ→0
.
Then, move eip·∫ s0 ds′ v(s′) across Fradkin’s operator e
i∫ s0 ds′ δ2
δv2(s′) with aid of Eq.(E.1) in the Ap-
pendix E.1, and re-arrange the functional-dependent factors, to obtain
〈~p, n|Gth[A]|~y, y0〉 (3.23)
=[(2π)3τ
]−1/2 · e−i(~p·~y−ωny0) · i∫ ∞
0
ds e−is(m2+p2)
∫d[u]
×m− iγ · [p− gA(y − u(s))] ·(
exp
g
∫ s
0
ds′σ · F(y − u(s′)))
+
×∫
d[Ω] ei∫ s0 ds′ Ω(s′)·u(s′) · exp
2ip ·
∫ s
0
ds′[∫ s
s′ds′′ Ω(s′′) + gA(y − u(s′))
]
×ei∫ s0 ds′ δ2
δv2µ(s′) · exp
−i
∫ s
0
ds′ v(s′) ·[∫ s
s′ds′′ Ω(s′′) + gA(y − u(s′))
]∣∣∣∣vµ→0
Then, Fradkin’s operation (self-linkage over v) yields
ei∫ s0 ds′ δ2
δv2µ(s′) · exp
−i
∫ s
0
ds′ v(s′) ·[∫ s
s′ds′′ Ω(s′′) + gA(y − u(s′))
]∣∣∣∣vµ→0
(3.24)
= exp−i
∫ s
0
ds1
∫ s
0
ds2 Ω(s1) · h(s1, s2) · Ω(s2)
× exp−ig2
∫ s
0
ds′A2(y − u(s′))
× exp
−2ig
∫ s
0
ds′ Ω(s′) ·∫ s′
0
ds′′A(y − u(s′′))
,
where
h(s1, s2) =∫ s
0
ds′ θ(s1 − s′)θ(s2 − s′), (3.25)
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
〈~p, n|Gth[A]|~y, y0〉 (3.26)
=[(2π)3τ
]−1/2 · e−i(~p·~y−ωny0) · i∫ ∞
0
ds e−is(m2+p2)
∫d[u]
×m− iγ · [p− gA(y − u(s))] ·(
exp
g
∫ s
0
ds′σ · F(y − u(s′)))
+
× exp
2igp ·∫ s
0
ds′A(y − u(s′))· exp
−ig2
∫ s
0
ds′A2(y − u(s′))
×∫
d[Ω] exp−i
∫ s
0
ds1
∫ s
0
ds2 Ω(s1) · h(s1, s2) · Ω(s2)
× exp
i
∫ s
0
ds′ Ω(s′) ·[u(s′) + 2ps′ − 2g
∫ s′
0
ds′′A(y − u(s′′))
].
The functional integral over Ω can be evaulated by either through Eq. (D.16), or by a change of
variables as
Ω(s′) = Ω(s′)−∫ s
0
ds12
h−1(s′, s)[u(s) + 2ps− 2g
∫ s
0
ds′′A(y − u(s′′))], (3.27)
which leads to∫
d[Ω] exp−i
∫ s
0
ds1
∫ s
0
ds2 Ω(s1) · h(s1, s2) · Ω(s2)
(3.28)
× exp
+i
∫ s
0
ds1
∫ s
0
ds2 [Ω(s1)− Ω(s1)] · h(s1, s2) · [Ω(s2)− Ω(s2)]
= CFFI exp−1
2Tr ln (2h)
· exp ip · u(s) · exp
isp2
× exp−2igp ·
∫ s
0
ds′A(y − u(s′))· exp
ig2
∫ s
0
ds′A2(y − u(s′))
× exp
i
4
∫ s
0
ds1
∫ s
0
ds2 u(s1) · h−1(s1, s2) · u(s2)
× exp−ig
∫ s
0
ds1
∫ s
0
ds2 u(s1) · h−1(s1, s2) ·∫ s2
0
ds′A(y − u(s′))
,
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
〈~p, n|Gth[A]|~y, y0〉 =[(2π)3τ
]−1/2 · e−i(~p·~y−ωny0) · i∫ ∞
0
ds e−ism2 · e− 12Tr ln (2h) (3.29)
×∫
d[u]m− iγ · [p− gA(y − u(s))] ·(eg
∫ s0 ds′σ·F(y−u(s′))
)+
× exp ip · u(s) × exp
i
4
∫ s
0
ds1
∫ s
0
ds2 u(s1) · h−1(s1, s2) · u(s2)
× exp−ig
∫ s
0
ds1
∫ s
0
ds2 u(s1) · h−1(s1, s2) ·∫ s2
0
ds′A(y − u(s′))
,
45
where e−12Tr ln (2h) has been pulled out of the functional integral over u since h(s1, s2) is not a
functional of u(s′).
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
〈~p, n|Gth[A]|~y, y0〉 (3.30)
=[(2π)3τ
]−1/2 · e−i(~p·~y−ωny0) · i∫ ∞
0
ds e−ism2 · e− 12Tr ln (2h)
×∫
d[u]
m− iγ ·
[p− g
1τ
∑
l
∫d3~q
(2π)3A(q) · eiq·(y−u(s))
]
×(
exp
g
∫ s
0
ds′σµν · 1τ
∑
l
∫d3~q
(2π)3[qµAν(q)− qνAµ(q)
]· eiq·(y−u(s′))
)
+
× exp ip · u(s) × exp
i
4
∫ s
0
ds1
∫ s
0
ds2 u(s1) · h−1(s1, s2) · u(s2)
× exp
−ig
1τ
∑
l
∫d3~q
(2π)3A(q)
∫ s
0
ds1
∫ s
0
ds2 u(s1) · h−1(s1, s2) ·∫ s2
0
ds′ eiq·(y−u(s′))
.
The exponent of the last exponential factor can be re-written as
−ig
∫ s
0
ds1
∫ s
0
ds2 u(s1) · h−1(s1, s2) ·∫ s2
0
ds′A(y − u(s′)) (3.31)
= −ig1τ
∑
l
∫d3~q
(2π)3
∫ s
0
ds1
∫ s
0
ds2 u(s1) · h−1(s1, s2) · A(q)∫ s2
0
ds′eiq·(y−u(s′))
≡ −ig1τ
∑
l
∫d3~q
(2π)3f(q) · A(q),
where q = (~q, ωl) and
fν(q) =∫ s
0
ds1
∫ s
0
ds2 uν(s1) · h−1(s1, s2)∫ s2
0
ds′eiq·(y−u(s′)). (3.32)
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
〈~p, n|Gth[A]|~y, y0〉 (3.33)
=[(2π)3τ
]−1/2 · e−i(~p·~y−ωny0) · i∫ ∞
0
ds e−ism2 · e− 12Tr ln (2h)
×∫
d[u]
m− iγ ·[p− g
∫d4q
(2π)4A(q) · eiq·(y−u(s))
]
×(
exp
g
∫ s
0
ds′σµν ·∫
d4q
(2π)4[qµAν(q)− qνAµ(q)
]· eiq·(y−u(s′))
)
+
× exp ip · u(s) × exp
i
4
∫ s
0
ds1
∫ s
0
ds2 u(s1) · h−1(s1, s2) · u(s2)
× exp−ig
∫d4q
(2π)4A(q)
∫ s
0
ds1
∫ s
0
ds2 u(s1) · h−1(s1, s2) ·∫ s2
0
ds′ eiq·(y−u(s′))
.
46
Similarly, the exponent of the last exponential factor is
−ig
∫d4q
(2π)4
∫ s
0
ds1
∫ s
0
ds2 u(s1) · h−1(s1, s2) · A(q)∫ s2
0
ds′eiq·(y−u(s′)) (3.34)
≡ −ig
∫d4q
(2π)4f(q) · A(q),
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
〈~p, p0|Gc[A]|~y, y0〉 (3.35)
=[(2π)4
]−1/2 · e−i(~p·~y−p0y0) · i∫ ∞
0
ds e−ism2 · e− 12Tr ln (2h)
×∫
d[u]m− iγ · [p− gA(y − u(s))] ·(eg
∫ s0 ds′σ·F(y−u(s′))
)+
× exp ip · u(s) × exp
i
4
∫ s
0
ds1
∫ s
0
ds2 u(s1) · h−1(s1, s2) · u(s2)
× exp−ig
∫ s
0
ds1
∫ s
0
ds2 u(s1) · h−1(s1, s2) ·∫ s2
0
ds′A(y − u(s′))
,
and can be simplified by utilizing the special form of the inverse h-function in Eq. (E.21) as
〈~p, p0|Gc[A]|~y, y0〉 (3.36)
=[(2π)4
]−1/2 · e−i(~p·~y−p0y0) · i∫ ∞
0
ds e−ism2 · e− 12Tr ln (2h)
×∫
d[u]m− iγ · [p− gA(y − u(s))] ·(eg
∫ s0 ds′σ·F(y−u(s′))
)+
× exp ip · u(s) × exp
i
4
∫ s
0
ds′ [u′(s′)]2× exp
−ig
∫ s
0
ds′ u′(s′) ·A(y − u(s′))
,
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
〈~p, p0|Gc[A = 0]|~y, y0〉∣∣∣∣g=0
(3.37)
=[(2π)4
]−1/2 · e−i(~p·~y−p0y0) · i∫ ∞
0
ds e−ism2 · e− 12Tr ln (2h)
×∫
d[u][m− iγ · p] · eip·u(s) · exp
i
4
∫ s
0
ds1
∫ s
0
ds2 u(s1) · h−1(s1, s2) · u(s2)
,
47
where the functional integral over u is given by∫
d[u] ei4
∫ s0 ds1
∫ s0 ds2 u(s1)·h−1(s1,s2)·u(s2) · eip·u(s) (3.38)
=∫
d[u] ei2
∫ s0 ds1
∫ s0 ds2 u(s1)· 12 h−1(s1,s2)·u(s2) · eip·∫ s
0 ds′ u(s′)·δ(s′−s)
=∫
d[u] ei2
∫ s0 ds1
∫ s0 ds2
12 h−1(s1,s2)[u(s1)u(s2)+2p·u(s1)·2h(s2,s1)·δ(s1−s)]
= e−i2 p2·∫ s
0 ds1∫ s0 ds2 δ(s1−s)·2h(s1,s2)·δ(s2−s) · e− 1
2Tr ln( 12 h−1)
= e−isp2 · e+ 12Tr ln(2h).
Alternatively, the delta-function δ(s′ − 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∫
d[u] ei4
∫ s0 ds1
∫ s0 ds2 u(s1)·h−1(s1,s2)·u(s2) · eip·u(s) (3.39)
=∫
d[u] ei2
∫ s0 ds1
∫ s0 ds2 u(s1)· 12 h−1(s1,s2)·u(s2) · eip·∫ s
0 ds′ u(s′)·δ(s′−s)
=∫
d[u] ei2
∫ s0 ds1
∫ s0 ds2 [u(s1)· 12 h−1(s1,s2)·u(s2)+4u(s1)· 12 h−1(s1,s2)·s2p]
=∫
d[u] ei2
∫ s0 ds1
∫ s0 ds2 [u(s1)+2s1p]· 12 h−1(s1,s2)·[u(s2)+2s2p] e−ip2 ∫ s
0 ds1∫ s0 ds2 s1·h−1(s1,s2)·s2
= e−12Tr ln( 1
2 h−1) · e−isp2= e−isp2 · e+ 1
2Tr ln(2h).
Both make use of the Gaussian functional integral of the form∫
d[w] exp
i
2
∫ s
0
ds1
∫ s
0
ds2 w(s1) · 12h−1(s1, s2) · w(s2)
= e+ 1
2Tr ln (2h) (3.40)
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
〈~p, p0|Gc[A = 0]|~y, y0〉 (3.41)
=[(2π)4
]−1/2 · e−i(~p·~y−p0y0) · [m− iγ · p] · i∫ ∞
0
ds e−is(m2+p2)
=[(2π)4
]−1/2 · e−i(~p·~y−p0y0) · m− iγ · pm2 + (γ · p)2
= 〈~p, p0|Sc|~y, y0〉,
or
Gc(~p, p0|A = 0) =m− iγ · p
m2 + (γ · p)2=
1m + iγ · p = Sc(~p, p0) (3.42)
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
i
4
∫ s
0
ds1
∫ s
0
ds2 u(s1) · h−1(s1, s2) · u(s2) (3.44)
=i
4
∫ s
0
ds1
∫ s
0
ds2 w(s1) · h−1(s1, s2) · w(s2)− i
2
∫ s
0
ds1
∫ s
0
ds2 s1p · h−1(s1, s2) · w(s2)
− i
2
∫ s
0
ds1
∫ s
0
ds2 w(s1) · h−1(s1, s2) · s2p + i
∫ s
0
ds1
∫ s
0
ds2 s1p · h−1(s1, s2) · s2p
=i
4
∫ s
0
ds1
∫ s
0
ds2 w(s1) · h−1(s1, s2) · w(s2)− ip · w(s) + isp2,
where Eqs. (E.22) and (E.23) have been used in the last line. Hence,
exp ip · u(s) · exp
i
4
∫ s
0
ds1
∫ s
0
ds2 u(s1) · h−1(s1, s2) · u(s2)
(3.45)
= exp
i
4
∫ s
0
ds1
∫ s
0
ds2 w(s1) · h−1(s1, s2) · w(s2)· exp
−isp2
and, then, one can replace the functional integral over u(s′) by w(s′) as
〈~p, n|Gth[A]|~y, y0〉 (3.46)
=[(2π)3τ
]−1/2 · e−i(~p·~y−ωny0) · i∫ ∞
0
ds e−is(m2+p2) · e− 12Tr ln (2h)
×∫
d[w]
m− iγ ·
[p− g
1τ
∑
l
∫d3~q
(2π)3A(q) eiq·(y−u(s))
]
×(
exp
g
∫ s
0
ds′σµν · 1τ
∑
l
∫d3~q
(2π)3[qµAν(q)− qνAµ(q)
]eiq·(y−u(s′))
)
+
× exp
i
4
∫ s
0
ds1
∫ s
0
ds2 w(s1) · h−1(s1, s2) · w(s2)
× exp
−ig
1τ
∑
l
∫d3~q
(2π)3A(q) ·
∫ s
0
ds1
∫ s
0
ds2 [w(s1)− 2s1p] h−1(s1, s2)∫ s2
0
ds′ eiq·(y−u(s′))
where u(s′) = w(s′) − 2s′p in eiq·(y−u(s′)) is to be understood. If one ignores the fluctuation over
49
w(s′), i.e., w(s′)− 2s′p → −2s′p for large p,
〈~p, n|GBNth [A]|~y, y0〉 (3.47)
=[(2π)3τ
]−1/2 · e−i(~p·~y−ωny0) · i∫ ∞
0
ds e−is(m2+p2)
×
m− iγ ·[p− g
1τ
∑
l
∫d3~q
(2π)3A(q) eiq·(y+2sp)
]
×(
exp
g
∫ s
0
ds′σµν · 1τ
∑
l
∫d3~q
(2π)3[qµAν(q)− qνAµ(q)
]eiq·(y+2s′p)
)
+
× exp
+2ig
∫ s
0
ds′1τ
∑
l
∫d3~q
(2π)3[p · A(q)
]eiq·(y+2s′p)
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
〈~p, n|GBNth [A]|~y, y0〉 (3.48)
=[(2π)3τ
]−1/2 · e−i(~p·~y−ωny0) · i∫ ∞
0
ds e−is(m2+p2)
×
m− iγ ·[p− g
∫d4q
(2π)4A(q) eiq·(y+2sp)
]
×(
exp
g
∫ s
0
ds′σµν ·∫
d4q
(2π)4[qµAν(q)− qνAµ(q)
]eiq·(y+2s′p)
)
+
× exp
+2ig∫ s
0
ds′∫
d4q
(2π)4[p · A(q)
]eiq·(y+2s′p)
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 ther-
mal 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, ther-
mal 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); approxima-
tions 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
Lth[A] = −i
∫ g
0
dg′Tr (γ ·A)Gth[g′A]. (3.49)
50
Up to the order of g2, the closed-fermion-loop functional may be approximated as
Lth[A] =i
2
∫ τ
0
dx
∫ τ
0
dy Aµ(x) ·Kµνth (x− y) ·Aν(y), (3.50)
where Kµνth is the thermal photon polarization tensor, i.e., the finite-temperature counterpart of the
vacuum polarization tensor Kµνc in the zero-temperature theory, and the range of integration for x0
and y0 is from 0 to τ , or
Lth[A] =i
2
∫dq
∫dq′ Aµ(q) · Kµν
th (q, q′) · Aν(q′), (3.51)
where q = (~q, ωl), and the short-hand notation of Eq. (2.92) is used. Note that Kµνth (q, q′) =
(2π)3τδ(q + q′) Kµνth (q), and δ(q + q′) = δl,−l′ δ
(3)(~q + ~q′), so that the thermal close-fermion-loop
functional reduces to
Lth[A] =i
21τ
∑
l
∫d3~q
(2π)3Aµ(q) · Kµν
th (q) · Aν(−q) (3.52)
=i
2
∫dq Aµ(q) · Kµν
th (q) · Aν(−q).
If the real-time form of Kµνth (q, q′) is used instead, Kµν
th (q, q′) = (2π)4δ(q+q′) Kµνth (q) with δ(q+q′) =
δ(4)(q + q′), and
Lth[A] =i
2
∫d4q
(2π)4Aµ(q) · Kµν
th (q) · Aν(−q) (3.53)
=i
2
∫dq Aµ(q) · Kµν
th (q) · Aν(−q),
where the short-hand notation∫
dq in the last line is
∫dq ≡
∫d4q
(2π)4. (3.54)
Therefore, the short-hand notation should follow the usage of real-time or imaginary-time form of
Kµνth .
Since there is no fermion tadpole in QED, the thermal closed-fermion-loop functional in Eq.
(3.52) or (3.53) is quite general if the thermal polarization tensor Kµνth (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
〈~p, n|S′th|~y, y0〉 = (Z[iτ ])−1[(2π)3τ
]−1/2 · e−i(~p·~y−ωny0) (3.55)
× · i∫ ∞
0
ds e−ism2 · e− 12Tr ln (2h)
∫d[u] exp ip · u(s)
×eD(th)A · m− iγ · [p− gA(y − u(s))] ·
(eg
∫ s0 ds′σ·F(y−u(s′))
)+
× exp
i
4
∫ s
0
ds1
∫ s
0
ds2 u(s1) · h−1(s1, s2) · u(s2)
× exp−ig
∫dq f(q) · A(q)
· exp
i
2
∫dq Aµ(q) · Kµν
th (q) · Aν(−q)
,
where the short-hand notation of either Eq. (2.92) or (3.54) is used for∫
dq depending on the form
of the linkage operator expD(th)A . The spin-related ordered exponential can be re-written as a
functional-translation of another ordered-exponential, i.e.,(eg
∫ s0 ds′σ·F(y−u(s′))
)+
(3.56)
= exp
+q
∫ s
0
ds′ F(y − u(s′)) · δ
δχµν(s′)
·(e∫ s0 ds′σ·χ(s′)
)+
∣∣∣∣χ→0
,
where χµν(s′) is an anti-symmetric tensor, as is σµν , and will be set to zero after the functional
differentiation with respect to χµν is performed. The linkage operator expD(th)A 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,
〈~p, n|S′th|~y, y0〉 (3.57)
=1
Z[iτ ][(2π)3τ
]−1/2 · e−i(~p·~y−ωny0) · i∫ ∞
0
ds e−ism2 · e− 12Tr ln (2h)
×∫
d[u] exp ip · u(s) · exp
i
4
∫ s
0
ds1
∫ s
0
ds2 u(s1) · h−1(s1, s2) · u(s2)
× exp− i
2
∫d4k
(2π)4δ
δA(k)· Dth(k) · δ
δA(−k)
×
m− iγ ·[p− g
∫dq A(q) eiq·(y−u(s))
]
× exp−ig
∫dq A(q) · f(q)
·(e∫ s0 ds′ σµνχµν(s′)
)+
× exp
i
2
∫dq A(q) · Kth(q) · A(q)
∣∣∣∣χ=0, A=0
,
52
where f(q) is defined by
fν(q) =∫ s
0
ds1
∫ s
0
ds2 uν(s1)h−1(s1, s2)∫ s2
0
ds′ eiq·(y−u(s′)) (3.58)
+2i
∫ s
0
ds′ eiq·(y−u(s′)) qµδ
δχµν(s′).
3.4.3 Linkage Operations
With help of Eq. (D.36) in Appendix D, one can move the linkage operator exp D(th)A across the
numerator factor m− iγ · [p− gA] to obtain
eD(th)A
m− iγ ·
[p− g
∫dq A(q) · eiq·(y−u(s))
](3.59)
=
m− iγ ·[p− g
∫dq eiq·(y−u(s))
[A(q)− iDth(q)
δ
δA(q)
]]· eD
(th)A
⇒
m− iγ ·[p + ig
∫dq eiq·(y−u(s)) Dth(q)
δ
δA(q)
]· eD
(th)A ,
where the gauge field Aµ 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,
eD(th)A · exp
+
i
2
∫dq A(q) · Kth(q) · A(q)− ig
∫dq A(q) · f(q)
(3.60)
= exp
+i
2
∫dq A(q) ·
[Kth(q) · 1
1− Kth(q) · Dth(q)
]· A(q)
× exp−ig
∫dq A(q) · 1
1− Kth(q) · Dth(q)· f(q)
× exp
i
2g2
∫dq f(q) ·
[Dth(q) · 1
1− Kth(q) · Dth(q)
]· f(q)
× exp−1
2Tr ln [1− Kth · Dth]
.
The dressed, Finite-Temperature fermion propagator under the one closed-fermion-loop approxima-
tion becomes
〈~p, n|S′th|~y, y0〉 (3.61)
=1
Z[iτ ][(2π)3τ
]−1/2 · e−i(~p·~y−ωny0) · i∫ ∞
0
ds e−ism2 · e− 12Tr ln (2h)
×∫
d[u] exp ip · u(s) · exp
i
4
∫ s
0
ds1
∫ s
0
ds2 u(s1) · h−1(s1, s2) · u(s2)
×
m− iγ ·[p + g2
∫dk
[Dth(k) · 1
1− Kth(k) · Dth(k)
]· f(k) · eik·(y−u(s))
]
× exp
i
2g2
∫dk f(k) ·
[Dth(k) · 1
1− Kth(k) · Dth(k)
]· f(k)
·(e∫ s0 ds′ σµνχµν(s′)
)+
∣∣∣∣χ=0
× exp−1
2Tr ln [1− Kth · Dth]
,
53
where the last exponential factor of Trace-Log can be absorbed into the the partition function as
Z[β] = e−12 Tr ln [1−Kth·Dth] · Z(0)[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
〈~p, n|Gth[A]|~y, y0〉 (3.63)
=[(2π)3τ
]−1/2 · e−i(~p·~y−ωny0) · i∫ ∞
0
ds e−ism2 · e− 12Tr ln (2h)
×∫
d[u]
m− iγ ·[p− g
∫dq A(q) · eiq·(y−u(s))
]
× exp ip · u(s) · exp
i
4
∫ s
0
ds1
∫ s
0
ds2 u(s1) · h−1(s1, s2) · u(s2)
× exp−ig
∫dq A(q) ·
∫ s
0
ds1
∫ s
0
ds2 u(s1) · h−1(s1, s2) ·∫ s2
0
ds′ eiq·(y−u(s′))
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
〈~p, n|S′th|~y, y0〉 = Z−1[iτ ] · [(2π)3τ]−1/2 · e−i(~p·~y−ωny0) (3.64)
×i
∫ ∞
0
ds e−ism2 · e− 12Tr ln (2h)
∫d[u] exp ip · u(s)
× exp
i
4
∫ s
0
ds1
∫ s
0
ds2 u(s1) · h−1(s1, s2) · u(s2)
× exp− i
2
∫d4k
(2π)4δ
δA(k)· Dth(k) · δ
δA(−k)
×
m− iγ ·[p− g
∫dq A(q) · eiq·(y−u(s))
]
× exp−ig
∫dq f(q) · A(q)
· exp
i
2
∫dq A(q) · Kth(q) · A(q)
∣∣∣∣A=0
,
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
S′th = eD(th)A ·
[Gth[A]
eLth[A]
Z[β]
]∣∣∣∣A=0
(3.65)
=[
eD(th)A1 ·Gth[A1]
]· eD
(th)12 ·
[eD
(th)A2 · eLth[A2]
]∣∣∣A1=A2=0
· 1Z[β]
,
where the cross-linkage exp [D(th)12 ] 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,
S′th '[eD
(th)A ·Gth[A]
]∣∣∣A=0
·[eD
(th)A · eLth[A]
]∣∣∣A=0
· 1Z[β]
. (3.66)
The second part can be related to the definition of the partition function as
Z[β] = eD(th)A · eLth[A]
∣∣∣A→0
· Z(0)[β], (3.67)
and the quenched propagator becomes
S′th ' eD(th)A ·Gth[A]
∣∣∣A→0
· 1Z(0)[β]
(3.68)
3.5.1 Quenched Dressed Finite-Temperature Fermion Propagator
At finite temperature, the quenched, dressed fermion propagator without spin-related contributions
reduces to
〈~p, n|S′th|~y, y0〉Q (3.69)
= [Z(0)[iτ ]]−1 · [(2π)3τ]−1/2 · e−i(~p·~y−ωny0) · i
∫ ∞
0
ds e−is(m2+p2) · e− 12Tr ln (2h)
×∫
d[w] exp
i
4
∫ s
0
ds1
∫ s
0
ds2 w(s1) · h−1(s1, s2) · w(s2)
×
m− iγ ·[p + g2
∫d4k
(2π)4
∫ s
0
ds′ Dth(k) · [w′(s′)− 2p] · e−ik·[w(s)−w(s′)]+2ik·p(s−s′)]
× exp
i
2g2
∫d4k
(2π)4
×∫ s
0
ds1
∫ s
0
ds2 eik·[w(s2)−w(s1)]−2ik·p[s2−s1] · [w′(s1)− 2p] · Dth(k) · [w′(s2)− 2p]
,
where the subscript ’Q’ denotes the quenched-approximated propagator, and the real-time form of
the photon propagator Dth(k) is used. If p represents the momentum of an ultra-relativistic, incident
fermion, the quantum fluctuation over w(s′) becomes negligible, i.e.,
[w(s′)− 2s′p] → −2s′p or [w′(s′)− 2p] → −2p, (3.70)
55
and the quenched, finite-temperature propagator reduces to
〈~p, n|S′th|~y, y0〉BNQ (3.71)
' [Z(0)[iτ ]]−1 · [(2π)3τ]−1/2 · e−i(~p·~y−ωny0) · i
∫ ∞
0
ds e−is(m2+p2) · e− 12Tr ln (2h)
×∫
d[w] exp
i
4
∫ s
0
ds1
∫ s
0
ds2 w(s1) · h−1(s1, s2) · w(s2)
×
m− iγ · p + 2ig2
∫d4k
(2π)4[γ · Dth(k) · p
] ∫ s
0
ds′ e2ik·p(s−s′)
× exp
2ig2
∫d4k
(2π)4
∫ s
0
ds1
∫ s
0
ds2
[p · Dth(k) · p
]e2ik·p[s1−s2]
,
where the superscript ’BN’ denotes the Bloch-Nordsieck approximated propagator. Note that the
functional integral over w cancels out the normalization factor, i.e,∫
d[w] exp
i
4
∫ s
0
ds1
∫ s
0
ds2 w(s1) · h−1(s1, s2) · w(s2)
= e+ 12Tr ln (2h). (3.72)
Since the finite-temperature photon propagator Dth(x− y) is symmetric in x− y,
〈~p, n|S′th|~y, y0〉BNQ (3.73)
' [Z(0)[iτ ]]−1 · [(2π)3τ]−1/2 · e−i(~p·~y−ωny0) · i
∫ ∞
0
ds e−is(m2+p2)
×
m− iγ · p + 2ig2
∫d4k
(2π)4[γ · Dth(k) · p
] ∫ s
0
ds′ e2ik·ps′
× exp
4ig2
∫d4k
(2π)4
∫ s
0
ds1
∫ s1
0
ds′[p · Dth(k) · p
]e2ik·ps′
.
An important point here is that the fermion momentum, (~p, ωn), is kept off mass-shell for the Finite
Temperature fermion propagator, i.e., p2 = ~p2 − ω2n 6= −m2 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
D(th)A = − i
2
∫d4k
(2π)4δ
δA(k)· Dth(~k, k0) · δ
δA(−k). (3.74)
Here, the photon propagator is gauge-dependent and is given by
Dµνth (k) = Dµν
c (k) + δDµνth (k), (3.75)
where Dc(k) and δDth(k) are the causal part and the thermal addition, respectively. Hence, the
56
linkage operator can be decomposed into two parts as
eD(th)A = eD
(c)A +∆D
(th)A (3.76)
= exp− i
2
∫d4k
(2π)4δ
δA(k)· Dc(~k, k0) · δ
δA(−k)
× exp− i
2
∫d4k
(2π)4δ
δA(k)· δDth(~k, k0) · δ
δA(−k)
.
In the zero-temperature theory, the causal linkage operation with eD(c)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
from the linkage operation with e∆D(th)A . The effective dressed, finite-temperature fermion propagator
becomes
〈~p, n|S′th|~y, y0〉BNQ (3.77)
' [Z(0)[iτ ]]−1 · [(2π)3τ]−1/2 · e−i(~p·~y−ωny0) · i
∫ ∞
0
ds e−is(m2+p2)
×
m− iγ · p + 2ig2
∫d4k
(2π)4[γ ·
[Dc(k) + δDth(k)
]· p
] ∫ s
0
ds′ e2ik·ps′
× exp
4ig2
∫d4k
(2π)4
∫ s
0
ds1
∫ s1
0
ds′[p · Dc(k) · p
]e2ik·ps′
× exp
4ig2
∫d4k
(2π)4
∫ s
0
ds1
∫ s1
0
ds′[p · δDth(k) · p
]e2ik·ps′
,
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 δDth(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
δDthµν(~k, k0) = 2πi ε(k0) f(k0) δ(~k 2 − k2
0) · PTµν (3.78)
= 2πi1
eβ|k0| − 1δ(~k 2 − k2
0) · PTµν ,
where PTµν is the transverse projection operator with PT
ij = δij − kikj/~k2 and PT
00 = PT0i = 0. The
57
last exponent in Eq. (3.77) becomes
4ig2
∫d4k
(2π)4
∫ s
0
ds1
∫ s1
0
ds′[p · δDth(k) · p
]e2ik·ps′ (3.79)
= 4ig2
∫d4k
(2π)4
∫ s
0
ds1
∫ s1
0
ds′
pµ ·
2πi[δµi · Pij · δjν
]
eβ|k0| − 1δ(~k 2 − k2
0) · pν
e2ik·ps′
= − g2
2π3
∫ s
0
ds1
∫ s1
0
ds′∫
d3~k
∫dk0
δ(~k 2 − k20)
eβ|k0| − 1
[~p 2 − (~p · ~k)2
~k2
]e2i~k·~ps′−2ik0ωns′ .
The δ-function can be split into two terms as
δ(~k 2 − k20) =
1
2|~k|[δ(|~k| − k0) + δ(|~k|+ k0)
], (3.80)
and then the exponent becomes
4ig2
∫d4k
(2π)4
∫ s
0
ds1
∫ s1
0
ds′[p · δDth(k) · p
]e2ik·ps′ (3.81)
= − g2
2π3
∫ s
0
ds1
∫ s1
0
ds′∫
d3~kcos(2|~k|ωns′)
|~k|1
eβ|~k| − 1
[~p 2 − (~p · ~k)2
~k2
]e2i~k·~ps′
= − g2
2π3~p 2
∫ s
0
ds1
∫ s1
0
ds′∫
dk
∫dΩ~k 2 cos(2|~k|ωns′)
|~k|(eβ|~k| − 1)
[1− cos2 θ
]e2i|~k||~p| cos θs′ .
Changing the dummy variable θ to ξ = cos θ,
4ig2
∫d4k
(2π)4
∫ s
0
ds1
∫ s1
0
ds′[p · δDth(k) · p
]e2ik·ps′ (3.82)
= − g2
π2~p 2
∫ s
0
ds1
∫ s1
0
ds′∫ ∞
0
dk |~k|2∫ 1
−1
dξcos(2|~k|ωns′)
|~k|(eβ|~k| − 1)
[1− ξ2
]e2i|~k||~p|ξs′ ,
the integral over ξ can be reduced to∫ 1
−1
dξ[1− ξ2
]e2i|~k||~p|ξs′ = 2
∫ 1
0
dξ[1− ξ2
]cos(2|~k||~p|ξs′), (3.83)
where sin(2|~k||~p|ξs′) is odd in ξ and has no contribution to the integral, and
4ig2
∫d4k
(2π)4
∫ s
0
ds1
∫ s1
0
ds′[p · δDth(k) · p
]e2ik·ps′ (3.84)
= −2g2
π2~p 2
∫ s
0
ds1
∫ s1
0
ds′∫ 1
0
dξ (1− ξ2)∫ ∞
0
dk |~k| cos(2|~k|ωns′)
eβ|~k| − 1cos(2|~k||~p|ξs′).
Even though the ξ-integral could be evaluated directly as∫ 1
−1
dξ ξ2 cos(aξ) = 2∫ 1
0
dξ ξ2 cos(aξ) =4a2
cos a +2a2 − 4
a3sin a, (3.85)
subsequent calculations will become messy. Instead, one could defer the evaluation of ξ-integral and
re-express those two cosine functions as
cos(2kωns′) cos(2kpξs′) =12
cos(2k(pξ + ωn)s′) +12
cos(2k(pξ − ωn)s′). (3.86)
58
Setting x = βk, the k-integral then becomes∫ ∞
0
dk kcos(2kωns′)
eβk − 1cos(2kpξs′) (3.87)
=12
∫ ∞
0
dkk
eβk − 1[cos(2k(pξ + ωn)s′) + cos(2k(pξ − ωn)s′)]
=1
2β2
∫ ∞
0
dxx
ex − 1
[cos(2
pξ + ωn
βxs′) + cos(2
pξ − ωn
βxs′)
].
Define
w(±) =pξ ± ωn
β, (3.88)
and then
4ig2
∫d4k
(2π)4
∫ s
0
ds1
∫ s1
0
ds′[p · δDth(k) · p
]e2ik·ps′ (3.89)
= − g2
π2β2~p 2
∫ s
0
ds1
∫ s1
0
ds′∫ 1
0
dξ (1− ξ2)∫ ∞
0
dxx
ex − 1
[cos(2w(+)xs′) + cos(2w(−)xs′)
]
= −s2 g2
π2β2~p 2
∫ 1
0
dv v
∫ 1
0
du
∫ 1
0
dξ (1− ξ2)
×∫ ∞
0
dxx
ex − 1
[cos(2sw(+)xuv) + cos(2sw(−)xuv)
],
where s′ = s1u and s1 = sv. First work out integral over u, then the integral over v as
4ig2
∫d4k
(2π)4
∫ s
0
ds1
∫ s1
0
ds′[p · δDth(k) · p
]e2ik·ps′ (3.90)
= −sg2
2π2β2~p 2
∫ 1
0
dv
∫ 1
0
dξ (1− ξ2)∫ ∞
0
dx1
ex − 1
[sin(2sw(+)xv)
w(+)+
sin(2sw(−)xv)w(−)
]
= − g2
4π2β2~p 2
∫ 1
0
dξ (1− ξ2)∫ ∞
0
dx1
x(ex − 1)
[1− cos(2sw(+)x)
(w(+))2+
1− cos(2sw(−)x)(w(−))2
].
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
phase factor of [−is(m2 +~p 2−ω2n)] will have a upper limit smax on s. When s > smax, the integrand
will oscillate dramatically, which will produce no significant contribution. For ωn → (2n+1)π−iβ and
ω2n → −
((2n+1)π
β
)2
,
smax ≤ 1m2 + ~p 2 − ω2
n
≤ 1~p 2 − ω2
n
→ 1~p 2 + |ω2
n|≤ 1
~p 2, (3.91)
which is valid only for finite temperature. Then, the arguments of the cosine-functions become
2sw(±)x ≤ 2smaxw(±)x ≤ 2pξ ± ωn
β(m2 + ~p 2 − ω2n)
x ≤ 2p± ωn
β(~p 2 + |ω2n|)
x ≤ 2x
βp. (3.92)
59
In the regime of interest, p À T À m, the 2sw(±)x are small if
x <12βp =
p
2T(3.93)
and
1− cos(2sw(±)x) ' 12!
(2sw(±)x)2 − 14!
(2sw(±)x)4 + · · · . (3.94)
Thus,
4ig2
∫d4k
(2π)4
∫ s
0
ds1
∫ s1
0
ds′[p · δDth(k) · p
]e2ik·ps′ (3.95)
' − g2
4π2β2~p 2
∫ 1
0
dξ (1− ξ2)∫ βp
2
0
dx1
x(ex − 1)12!
[(2sw(+)x)2
(w(+))2+
(2sw(−)x)2
(w(−))2
]
= −s2 g2
π2β2~p 2
∫ 1
0
dξ (1− ξ2)∫ βp
2
0
dxx
ex − 1.
Observing that βp2 À 1,
a2T =
∫ βp2
0
dxx
ex − 1≤
∫ ∞
0
dxx
ex − 1=
π2
6, (3.96)
and the integral over ξ is trivial as ∫ 1
0
dξ (1− ξ2) =23. (3.97)
Thus,
4ig2
∫d4k
(2π)4
∫ s
0
ds1
∫ s1
0
ds′[p · δDth(k) · p
]e2ik·ps′ (3.98)
' −s2 23π2
g2 ~p 2
β2
∫ βp2
0
dxx
ex − 1
= −s2 2a2T
3π2g2T 2 ~p 2
' −19
s2 g2T 2 ~p 2,
where the approximation a2T = π2/6 is employed. The fully-dressed, finite-temperature fermion
propagator in a heat bath then becomes
〈~p, n|S′th|~y, y0〉BNQ (3.99)
' (Z(0)[iτ ])−1 · [(2π)3τ]−1/2 · e−i(~p·~y−ωny0)
×i
∫ ∞
0
ds exp−is(m2 + ~p 2 − ω2
n)− s2 2a2T
3π2g2T 2 ~p 2
×
m− iγ · p + 2ig2
∫d4k
(2π)4[γ ·
[Dc(k) + δDth(k)
]· p
] ∫ s
0
ds′ e2ik·ps′
× exp
4ig2
∫d4k
(2π)4
∫ s
0
ds1
∫ s1
0
ds′[p · Dc(k) · p
]e2ik·ps′
60
Since the exponent −s2 2a2T
3π2 g2T 2 ~p 2 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
gauge, the s2-factor again appears in the decay exponent; more details are in Appendix G. The
contribution of[γ · δDth(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
〈~p, n|S′th|~y, y0〉BNQ (3.100)
' (Z(0)[iτ ])−1 · [(2π)3τ]−1/2 · e−i(~p·~y−ωny0)
×i
∫ ∞
0
ds exp−is(m2 + ~p 2 − ω2
n)− s2 2a2T
3π2g2T 2 ~p 2
×
m− iγ · p + 2ig2
∫d4k
(2π)4[γ · Dc(k) · p
] ∫ s
0
ds′ e2ik·ps′
× exp
4ig2
∫d4k
(2π)4
∫ s
0
ds1
∫ s1
0
ds′[p · Dc(k) · p
]e2ik·ps′
.
Let ω2 = m2 + ~p 2 and replace s2 by the operator
s2 →(
i∂
∂ω2
)2
(3.101)
in the decay exponent of exp [−s2 2a2T
3π2 g2T 2 ~p 2]; then the dressed propagator reduces to
S′th(~p, n) (3.102)
' (Z(0)[iτ ])−1 · i∫ ∞
0
ds exp−is(ω2 − ω2
n)− s2 2a2T
3π2g2T 2 ~p 2
×
m− iγ · p + 2ig2
∫d4k
(2π)4[γ · Dc(k) · p
] ∫ s
0
ds′ e2ik·ps′
× exp
4ig2
∫d4k
(2π)4
∫ s
0
ds1
∫ s1
0
ds′[p · Dc(k) · p
]e2ik·ps′
= exp
−2a2
T
3π2g2T 2 ~p 2 ·
(i
∂
∂ω2
)2· (Z(0)[iτ ])−1 · i
∫ ∞
0
ds exp−is(ω2 − ω2
n)
×
m− iγ · p + 2ig2
∫d4k
(2π)4[γ · Dc(k) · p
] ∫ s
0
ds′ e2ik·ps′
× exp
4ig2
∫d4k
(2π)4
∫ s
0
ds1
∫ s1
0
ds′[p · Dc(k) · p
]e2ik·ps′
.
Aside from the leading exponential, if the remaining is considered as a non-interactive, but renor-
malized, 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
4ig2
∫d4k
(2π)4
∫ s
0
ds1
∫ s1
0
ds′[p · Dc(k) · p
]e2ik·ps′ (3.103)
= 4ig2
∫d4k
(2π)4
∫ s
0
ds1
∫ s1
0
ds′[pµ ·
(−δµ0δ0ν
~k2+
PTµν
~k2 − k20 − iε
)· pν
]e2ik·ps′
= 4ig2
∫d4k
(2π)4
∫ s
0
ds1
∫ s1
0
ds′−ω2
n
~k2+
[~p 2 − (~k · ~p)2
~k2
]· 1~k2 − k2
0 − iε
e2ik·ps′ .
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
4ig2
∫d4k
(2π)4
∫ s
0
ds1
∫ s1
0
ds′[p · DBN
c (k) · p]
e2ik·ps′ (3.104)
= 4ig2
∫ s
0
ds1
∫ s1
0
ds′∫
d3~k
(2π)3e2is′~k·~p−~k2/~p2
∫ ∞
−∞
dk0
2π
−
ω2n
~k2+
~p 2 − (~k·~p)2
~k2
~k2 − k20 − iε
e−2is′k0 ωn ,
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., D00(k) = 0. It is not a surprise that there is no contribution from the first term in the curly
brackets as s′ 6= 0 and ωn 6= 0 for fermions, i.e.,
ω2n
~k2
∫ ∞
−∞
dk0
2πe−2is′k0 ωn =
ω2n
~k2δ(2s′ωn) = 0, (3.105)
and the second term in the bracket involves∫ ∞
−∞
dk0
2π
e−2is′k0 ωn
~k2 − k20 − iε
= −∫ ∞
−∞
dk0
2π
e−2is′k0 ωn
(k0 − k + iε′)(k0 + k − iε′), (3.106)
where there are two poles at k0 = k− iε′ and k0 = −k + iε′. Converting this into a contour integral,
and employing the residue theorem, one obtains∫ ∞
−∞
dk0
2π
e−2is′k0 ωn
~k2 − k20 − iε
=i
2k
[θ(ωn) e−2is′k ωn + θ(−ωn) e+2is′k ωn
](3.107)
=i
2k[cos(2s′k ωn) + iε(ωn) sin(2s′k ωn)] ,
62
where ε(ωn) = θ(ωn)− θ(−ωn) and ωn 6= 0 for fermions. Therefore,
4ig2
∫d4k
(2π)4
∫ s
0
ds1
∫ s1
0
ds′[p · DBN
c (k) · p]
e2ik·ps′ (3.108)
= −2g2
∫ s
0
ds1
∫ s1
0
ds′∫
d3~k
(2π)31k
e2is′~k·~p−~k2/~p 2 ·[~p 2 − (~k · ~p)2
~k2
]
× [cos(2s′k ωn) + iε(ωn) sin(2s′k ωn)] .
The exponent is complex and the ~k-integral can be reduced to
∫d3~k
(2π)31k
e2is′~k·~p− |~k||~p| ·[~p 2 − (~k · ~p)2
~k2
][cos(2s′k ωn) + iε(ωn) sin(2s′k ωn)] (3.109)
=1
(2π)2~p 2
∫ ∞
0
dk k [cos(2s′k ωn) + iε(ωn) sin(2s′k ωn)] e−~k2/~p 2
∫ 1
−1
dξ [1− ξ2] e2is′kpξ,
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
Re
4ig2
∫d4k
(2π)4
∫ s
0
ds1
∫ s1
0
ds′[p · DBN
c (k) · p]
e2ik·ps′
(3.110)
= − 4(2π)2
g2~p 2
∫ s
0
ds1
∫ s1
0
ds′∫ 1
0
dξ [1− ξ2]∫ ∞
0
dk k cos(2s′k ωn) cos(2s′kp ξ) e−~k2/~p 2
= − 4(2π)2
s2g2(~p 2)2∫ 1
0
dv · v ·∫ 1
0
du
∫ 1
0
dξ · [1− ξ2]
×∫ ∞
0
dx · x · cos(2spωnxuv) · cos(2s~p 2ξxuv) · e−x2,
where substitutions of k = xp, s′ = s1u and s1 = sv have been used. Converting the two cosine
functions,
cos(2s~p 2ξxuv) · cos(2spωnxuv) (3.111)
=12
[cos(2sp(p ξ + ωn)xuv) + cos(2sp(p ξ − ωn)xuv)] ,
and carrying out both u- and v-integrals first,∫ 1
0
dv · v ·∫ 1
0
du · cos(2sp(p ξ ± ωn)xuv) =1− cos(2sp(p ξ ± ωn)x)
(2sp(p ξ ± ωn)x)2≤ 1
2, (3.112)
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.,
2sp(p ξ ± ωn)x ≤ 2p(p ξ ± ωn)xm2 + ~p 2 − ω2
n
<2p(p ξ ± ωn)x
~p 2 − ω2n
≤ 2x (3.113)
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
Re
4ig2
∫d4k
(2π)4
∫ s
0
ds1
∫ s1
0
ds′[p · DBN
c (k) · p]
e2ik·ps′
(3.114)
= − 2(2π)2
s2g2(~p 2)2∫ 1
0
dξ [1− ξ2]
×∫ ∞
0
dxx
[1− cos(2sp(p ξ + ωn)x)
(2sp(p ξ + ωn)x)2+
1− cos(2sp(p ξ − ωn)x)(2sp(p ξ − ωn)x)2
]e−x2
' − 2(2π)2
s2g2(~p 2)2∫ 1
0
dξ [1− ξ2]∫ ∞
0
dxx e−x2.
The x-integral is trivial as ∫ ∞
0
dx x e−x2=
12, (3.115)
as is the ξ-integral, Eq. (3.97). One arrives at
Re
4ig2
∫d4k
(2π)4
∫ s
0
ds1
∫ s1
0
ds′[p · DBN
c (k) · p]
e2ik·ps′
(3.116)
' − 16π2
s2g2(~p 2)2,
and the real part of the exponent has a similar s2-factor to that of the linkage with the thermal pho-
tons. 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
4ig2
∫d4k
(2π)4
∫ s
0
ds1
∫ s1
0
ds′[p · Dc(k) · p
]e2ik·ps′ (3.117)
' 4ig2
∫
|~k|≤|~p|
d4k
(2π)4
∫ s
0
ds1
∫ s1
0
ds′[pµ ·
(−δµ0δ0ν
~k2+
PTµν
~k2 − k20 − iε
)· pν
]e2ik·ps′ .
Similarly there is no contribution from the first term with −ω2n/~k2 inside the curly brackets, and
the exponent becomes
4ig2
∫d4k
(2π)4
∫ s
0
ds1
∫ s1
0
ds′[p · Dc(k) · p
]e2ik·ps′ (3.118)
' −2g2
∫ s
0
ds1
∫ s1
0
ds′∫
|~k|≤|~p|
d3~k
(2π)31
|~k|
[~p 2 − (~k · ~p)2
~k2
]e2is′~k·~p
× [cos(2s′k ωn) + iε(ωn) sin(2s′k ωn)] .
64
The real part of the exponent becomes
Re
4ig2
∫d4k
(2π)4
∫ s
0
ds1
∫ s1
0
ds′[p · Dc(k) · p
]e2ik·ps′
(3.119)
= − 4(2π)2
g2~p 2
∫ s
0
ds1
∫ s1
0
ds′∫ 1
0
dξ [1− ξ2]∫ |~p|
0
dk k cos(2s′k ωn) cos(2s′kpξ)
= − 4(2π)2
s2g2(~p 2)2∫ 1
0
dv · v ·∫ 1
0
du
∫ 1
0
dξ · [1− ξ2]
×∫ 1
0
dx · x · cos(2sp ωnxuv) · cos(2s~p 2ξxuv),
where ξ = cos θ, k = xp, s′ = s1u, and s1 = sv. The u- and v-integrals then yield∫ 1
0
dv · v ·∫ 1
0
du · cos(2sp(p ξ ± ωn)xuv) =1− cos(2sp(p ξ ± ωn)x)
(2sp(p ξ ± ωn)x)2. (3.120)
Define a± = 2sp(p ξ ± ωn), and the x-integral can be directly evaluated as
∫ 1
0
dxx1− cos(a±x)
(a±x)2=
∞∑r=1
(−1)r+1 (a±)2r−2
2r (2r)!= a2
B , (3.121)
and when a± ≤ 2,
a2B ≤ 1
4. (3.122)
Then, one arrives at
Re
4ig2
∫d4k
(2π)4
∫ s
0
ds1
∫ s1
0
ds′[p · Dc(k) · p
]e2ik·ps′
(3.123)
' −2a2B
3π2s2g2(~p 2)2,
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∫ ∞
0
dxx e−x = 1, (3.124)
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
∫ 1
0
dxx =12, (3.125)
which is the same as Eq. (3.115).
65
To estimate the imaginary part of the last exponent in Eq. (3.100),
Im
4ig2
∫d4k
(2π)4
∫ s
0
ds1
∫ s1
0
ds′[p · Dc(k) · p
]e2ik·ps′
(3.126)
= − 4(2π)2
ε(ωn) g2 ~p 2
∫ s
0
ds1
∫ s1
0
ds′∫ 1
0
dξ [1− ξ2]∫ |~p|
0
dk k cos(2s′kp ξ) sin(2s′k ωn)
= − 4(2π)2
ε(ωn) s2g2(~p 2)2∫ 1
0
dv · v ·∫ 1
0
du
∫ 1
0
dξ · [1− ξ2]
×∫ 1
0
dxx cos(2s~p 2ξxuv) sin(2spωnxuv),
which involves a sine-function, and
cos(2s~p 2ξxuv) · sin(2spωnxuv) (3.127)
=12
[sin(2sp(p ξ + ωn)xuv)− sin(2sp(p ξ − ωn)xuv)] .
Working out the u- and v-integrals,∫ 1
0
dv · v ·∫ 1
0
du · sin(2sp(p ξ ± ωn)xuv) =1
2sp(p ξ ± ωn)x− sin(2sp(p ξ ± ωn)x)
(2sp(pξ ± ωn)x)2, (3.128)
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
∫ 1
0
dx x
[1
a±x− sin(a±x)
(a±x)2
]=
1a±
−∞∑
r=0
(−1)r (a±)2r+1
(2r + 1) (2r = 1)!, (3.129)
and one can expects that the two summations of a± will cancel each other. Therefore, the imag-
inary part is negligibly small. The quenched, dressed, finite-temperature fermion propagator with
combined effects becomes
S′th(~p, n) (3.130)
' (Z(0)[iτ ])−1 · i∫ ∞
0
ds exp−is(ω2 − ω2
n)− s2 2a2T
3π2g2T 2 ~p 2
×
m− iγ · p + 2ig2
∫d4k
(2π)4[γ · Dc(k) · p
] ∫ s
0
ds′ e2ik·ps′
× exp
4ig2
∫d4k
(2π)4
∫ s
0
ds1
∫ s1
0
ds′[p · Dc(k) · p
]e2ik·ps′
' (Z(0)[iτ ])−1 · i∫ ∞
0
ds exp−is(ω2 − ω2
n)− s2
[2a2
T
3π2g2T 2 ~p 2 +
2a2B
3π2g2(~p 2)2
]
×
m− iγ · p + 2ig2
∫d4k
(2π)4[γ · Dc(k) · p
] ∫ s
0
ds′ e2ik·ps′
× exp
i Im(
4g2
∫d4k
(2π)4
∫ s
0
ds1
∫ s1
0
ds′[p · Dc(k) · p
]e2ik·ps′
),
where a2T = π2/6 and a2
B = 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
S′th(~p, n) (3.131)
' (Z(0)[iτ ])−1 · i∫ ∞
0
ds m− iγ · p
× exp−is(ω2 − ω2
n)− s2
[2a2
T
3π2g2T 2 ~p 2 +
2a2B
3π2g2(~p 2)2
]
' exp
− 2
3π2g2 ~p 2 [a2
T T 2 + a2B~p 2] ·
(i
∂
∂ω2
)2
·(Z(0)[iτ ])−1 · i∫ ∞
0
ds m− iγ · p exp−is(ω2 − ω2
n),
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 calcu-
lation. 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,
S′th(~p, n) ' exp
− 2g2
3π2~p 2 [a2
T T 2 + a2B~p 2] ·
(i
∂
∂ω2
)2· Sth(~p, n;mR, gR), (3.132)
where a2T = π2/6, a2
B = 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
〈~p, x0|S′th|~p, y0〉 =1τ
∑n
e−iωn(x0−y0) S′th(~p, n) (3.133)
' 1τ
∑n
e−iωn(x0−y0) exp
− 2g2
3π2~p 2[a2
T T 2 + a2B~p 2] ·
(i
∂
∂ω2
)2· Sth(~p, n;mR)
= exp
− 2g2
3π2~p 2 [a2
T T 2 + a2B~p 2] ·
(i
∂
∂ω2
)2· 1τ
∑n
e−iωn(x0−y0) Sth(~p, n;mR)
= exp
− 2g2
3π2~p 2 [a2
T T 2 + a2B~p 2] ·
(i
∂
∂ω2
)2· Sth(~p, x0 − y0),
67
where the renormalized mass is implicitly implied in the last line, and Sth(~p, x0 − y0) is the free,
finite-temperature fermion propagator in the mixed representation,
Sth(~p, z0) =[m− i~γ · ~p + iγ0 ·
(i
∂
∂z0
)]i
2ω
[1− n(ω)]e−iωz0 − n(ω)e+iωz0
, (3.134)
and n(ω) is the Fermi-Dirac distribution function (without the chemical potential) as
n(ω) =1
eβω + 1. (3.135)
Let z0 = x0 − y0,
〈~p, x0|S′th|~p, y0〉 (3.136)
= exp
− 2g2
3π2~p 2 [a2
T T 2 + a2B~p 2] ·
(i
∂
∂ω2
)2
·[m− i~γ · ~p + iγ0 ·
(i
∂
∂z0
)]i
2ω
[1− n(ω)]e−iωz0 − n(ω)e+iωz0
=i
2
[m− i~γ · ~p + iγ0 ·
(i
∂
∂z0
)]· exp
− 2g2
3π2~p 2 [a2
T T 2 + a2B~p 2] ·
(i
∂
∂ω2
)2
· 1√ω2
[1− n(
√ω2)]e−i
√ω2z0 − n(
√ω2)e+i
√ω2z0
,
where ω is expressed as√
ω2 in the last line. Let b2 = 2g2
3π2 ~p 2 [a2T T 2 + a2
B~p 2] with a2T = π2/6,
a2B = 1/4, and u = ω2; then, the Gaussian-type translation operator can be re-written as
exp
− 2g2
3π2~p 2 [a2
T T 2 + a2B~p 2] ·
(i
∂
∂ω2
)2
(3.137)
= exp
+(ib)2 ·
(i
∂
∂u
)2
=1√π
∫ +∞
−∞dv e−v2+2v(ib)(i ∂
∂u )
=1√π
∫ +∞
−∞dv e−v2−2vb ∂
∂u ,
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
〈~p, x0|S′th|~p, y0〉 (3.138)
=[m− i~γ · ~p + iγ0 ·
(i
∂
∂z0
)]· i
2√
π
∫ +∞
−∞dv e−v2−2vb ∂
∂u
· 1√u
[1− n(
√u)]e−i
√uz0 − n(
√u)e+i
√uz0
∣∣∣∣u=ω2
=i
2√
π
[m− i~γ · ~p + iγ0 ·
(i
∂
∂z0
)]·∫ +∞
−∞dv e−v2
1√u− 2vb
[1− n(
√u− 2vb)]e−i
√u−2vbz0 − n(
√u− 2vb)e+i
√u−2vbz0
∣∣∣∣u=ω2
=i
2√
π
[m− i~γ · ~p + iγ0 ·
(i
∂
∂z0
)]·∫ +∞
−∞dv e−v2 1
ω√
1− 2vb/ω2
[1− n(ω
√1− 2vb/ω2)]e−i
√1−2vb/ω2 ωz0 − n(ω
√1− 2vb/ω2)e+i
√1−2vb/ω2 ωz0
.
where u = ω2 and ω > 0 are used in the last line.
When only the effect of thermal-photon-assisted Bremsstrahlung is considered,
2vb
ω2= 2
√2a2
T
3π2v
gTp
ω2<
23vgT
ω< v
gT
ω, (3.139)
where√
2a2T /3π2 = 1/3 < 1. When v < ω
2gT and ω2gT À 1 as p À T > gT , one has 2vb
ω2 < v gTω < 1
and√
u− 2vb = ω
√1− 2vb
ω2' ω
(1− vb
ω2
)= ω − vb
ω. (3.140)
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,
2vb
ω2=
2v
ω2
√2
3π2gp
√a2
T T 2 + a2B ~p 2 (3.141)
' v
ω2
√8
3π2
gp
2
√2π2
3T 2 + ~p 2
≤√
83π2
gv~p 2
ω2
< gv~p 2
ω2
< gv.
Similarly, the integrand with the limited factor of exp [−v2] will contribute only when v < 1/g.
Thus, one could replace
ω
√1− 2vb
ω2' ω
(1− vb
ω2
)= ω − vb
ω, (3.142)
69
and
〈~p, x0|S′th|~p, y0〉 (3.143)
' i
2√
π
[m− i~γ · ~p + iγ0 ·
(i
∂
∂z0
)]
·∫ +∞
−∞dv e−v2 1
ω − vbω
[1− n(ω − vb
ω)]e−i(ω− vb
ω )z0 − n(ω − vb
ω)e+i(ω− vb
ω )z0
' i
2√
π
[m− i~γ · ~p + iγ0 ·
(i
∂
∂z0
)]
·∫ +∞
−∞dv e−v2 1
ω
[1− n(ω)e+ β
2vbω ]e−iωz0+i vb
ω z0 − n(ω)e+ β2
vbω e+iωz0−i vb
ω z0
,
where the approximation
n(ω − vb
ω) =
1
eβ(ω− vbω ) + 1
=e−
β2 (ω− vb
ω )
eβ2 (ω− vb
ω ) + e−β2 (ω− vb
ω )' 1
eβω + 1e+ β
2vbω (3.144)
for the Fermi-Dirac distribution function is used in the last line. The two v-integrals can be carried
out readily as
1√π
∫ +∞
−∞dv e−v2±i vb
ω z0 = e−b2
4ω2 z20 , (3.145)
and
1√π
∫ +∞
−∞dv e−v2+ vb
ω ( β2±iz0) = e
b2
4ω2 ( β2±iz0)
2= e
− b2
4ω2
(z20∓iβz0− β2
4
). (3.146)
If a different sign is chosen in Eq.(3.137) instead, the results are the same, as
1√π
∫ +∞
−∞dv e−v2− vb
ω ( β2±iz0) = e
b2
4ω2 ( β2±iz0)
2= e
− b2
4ω2
(z20∓iβz0− β2
4
). (3.147)
The dressed, Finite-Temperature fermion propagator reduces to
〈~p, x0|S′th|~p, y0〉 (3.148)
' i
2
[m− i~γ · ~p + iγ0 ·
(i
∂
∂z0
)]
· 1ω
e−iωz0 − n(ω)e+ β2
4b2
4ω2
[e−i[ω−β b2
4ω2 ]z0 + e+i[ω−β b2
4ω2 ]z0]
e−b2
4ω2 z20 .
If only the exchange of effective thermal photons of the heat bath is considered, b2 = 2a2T
3π2 g2T 2~p 2,
one hasb2
4ω2=
a2T
6π2g2 T 2~p 2
ω2, (3.149)
70
and
〈~p, x0|S′th|~p, y0〉 (3.150)
' i
2ω
[m− i~γ · ~p + iγ0 ·
(i
∂
∂z0
)]· e−g2T 2 a2
T6π2
~p 2
ω2 z20
·
e−iωz0 − n(ω) e+g2 a2T
24π2~p 2
ω2
[e−i(ω−g2T
a2T
6π2~p 2
ω2 )z0 + e+i(ω−g2Ta2
T6π2
~p 2
ω2 )z0
]
' i
2ω
[m− i~γ · ~p + iγ0 ·
(i
∂
∂z0
)]· e− 1
36 g2T 2 ~p 2
ω2 z20
·
e−iωz0 − n(ω) e+ 1144 g2 ~p 2
ω2
[e−i(ω− 1
36 g2T ~p 2
ω2 )z0 + e+i(ω− 136 g2T ~p 2
ω2 )z0
].
Similarly, when both the thermal-photon assisted and ordinary Bremsstrahlung processes are in-
cluded with b2 = 2g2
3π2 ~p2 [a2T T 2 + a2
B~p 2],
b2
4ω2=
g2
6π2
a2B(~p 2)2 + a2
T T 2~p 2
ω2, (3.151)
then
〈~p, x0|S′th|~p, y0〉 (3.152)
' i
2ω
[m− i~γ · ~p + iγ0 ·
(i
∂
∂z0
)]· e− g2
6π2a2
B(~p 2)2+a2T T2~p 2
ω2 z20
·
e−iωz0 − n(ω) e+ g2
24π2a2
B(~p2) 2+a2T T2~p 2
ω2T2
×[e−i(ω− g2T
6π2a2
B(~p 2)2+a2T T2~p 2
ω2 )z0 + e+i(ω− g2T
6π2a2
B(~p 2)2+a2T T2~p 2
ω2 )z0
]
This nice result has been obtained from the following assumptions: First, the quenched approxima-
tion 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, Sth in Eq. (3.134), there are two distinct parts, the incoming and
thermal parts, given by
Sth(~p, z0) =[m− i~γ · ~p + iγ0 ·
(i
∂
∂z0
)]i
2ω
e−iωz0 − n(ω)
[e−iωz0 + e+iωz0
]. (3.153)
The second part with[e−iωz0 + e+iωz0
]in the square brackets represents the induced disturbance
inside the thermal bath.
In the fully-dressed, finite-temperature propagator, S′th, the damping factor from the thermal-
photon-assisted Bremsstrahlung alone is exp[− a2
T
6π2g2T 2~p 2
ω2 z20
], or exp
[− g2
6π2a2
B(~p 2)2+a2T T 2~p 2
ω2 z20
]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
Lth[A] =i
2
∫dx
∫dy Aµ(x) ·Kth
µν(x− y) ·Aν(y), (3.154)
or
Lth[A] =i
2
∫dq Aµ(q) · Kth
µν(q) · Aν(q) (3.155)
where K(th)µν is the thermal counterpart of the vacuum photon polarization tensor Kµν . Hence, the
fully-dressed, finite-temperature photon propagator is given by
D′th = Dth · [1−Kth ·Dth]−1, (3.156)
or
D′th(k)
−1= Dth(k)
−1 − Kth(k). (3.157)
Similar to the vacuum photon polarization tensor, Kthµν is gauge invariant, and the requirement of
the current conservation leads to
qµ Kthµν(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]
Kthµν(q) = −q2
[PT
µν ΠTth(q2) + PL
µν ΠLth(q2)
], (3.159)
where PTµν and PL
µν are the transverse and longitudinal projection operators, respectively, defined
in Appendix C.2 and Sec. 2.5.3. In general, the transverse and longitudinal components, ΠTth(q2)
and ΠLth(q2), are temperature-dependent and different due to broken Lorentz covariance. At zero-
temperature, the two components are equal, i.e., ΠTc (q2) = ΠL
c (q2) = Π(q2) as the Lorentz covari-
ance restored [88, 20].
In the zero-temperature theory, the vacuum photon polarization tensor can be written in the
form of [48]
Kµν(q) = −q2 Pµν Π(q2) = −q2
[δµν − qµqν
q2
]Π(q2), (3.160)
72
where
Π(q2) = +g2
2π2
∫ 1
0
dy y(1− y)[∫ ∞
0
du
ue−iu[m2+y(1−y)q2] +
13
], (3.161)
which is divergent near the mass-shell q2 ∼ 0. One can extract the finite part of Π(q2) by
ΠR(q2) = Π(q2)− Π(0) (3.162)
and ΠR(0) = 0 by definition. The divergent part of ΠR(q2) can be related to the photon’s wave-
function renormalization constant Z3 as Π(0) = (Z(2)3 )−1 − 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]
ΠR(q2) = − g2
2π2
∫ 1
0
dy y(1− y) ln(
m2 + y(1− y)q2
m2
), (3.163)
and ΠR(q2) is complex in general depending on the value of q2. Observe that
y(1− y) = −[y2 − y +14] +
14
= −(y − 12)2 +
14
<14, (3.164)
and y is within the range of 0 < y < 1 in the integral, i.e., y(1− y) > 0,
0 < y(1− y) <14. (3.165)
Thus,
m2 + y(1− y)q2 < m2 +q2
4=
14
[4m2 + ~q 2 − q20 ]. (3.166)
When q20 < ~q 2 + 4m2 < ~q 2 + m2
y(1−y) ,
m2 + y(1− y)q2 > 0, (3.167)
and ΠR(q2) and Kµν(q) are real.
On the other hand, when q20 > ~q 2 + m2
y(1−y) > ~q 2 + (2m)2,
m2 + y(1− y)q2 <14
[4m2 + ~q 2 − q20 ] < 0. (3.168)
Then ΠR(q2) and Kµν(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=
∣∣∣∣m2 + y(1− y)q2
m2
∣∣∣∣[θ([~q 2 +
m2
y(1− y)]− q2
0)− θ(q20 − [~q 2 +
m2
y(1− y)])]
,
(3.170)
and
ln(
m2 + y(1− y)q2
m2
)= ln
∣∣∣∣m2 + y(1− y)q2
m2
∣∣∣∣ + πi θ(q20 − [~q 2 +
m2
y(1− y)]), (3.171)
73
which leads to
ΠR(q2) = ReΠR(q2)
+ i Im
ΠR(q2)
(3.172)
with
ReΠR(q2)
= − g2
2π2
∫ 1
0
dy y(1− y) ln∣∣∣∣m2 + y(1− y)q2
m2
∣∣∣∣, (3.173)
and
i ImΠR(q2)
= −πi
g2
2π2
∫ 1
0
dy y(1− y) θ(q20 − [~q 2 +
m2
y(1− y)]). (3.174)
3.6.2 Pair-Productions as a Damping Mechanism
Keeping the closed-fermion-loop functional in the thermal propagator
〈~p, n|S′th|~y, y0〉 = eD(th)A ·
〈~p, n|Gth[A]|~y, y0〉 · eLth[A]
Z[iτ ]
∣∣∣∣A→0
. (3.175)
Again, drop the contribution from the spin-related part,
〈~p, n|S′th|~y, y0〉 (3.176)
' 1Z[iτ ]
[(2π)3τ
]−1/2 · e−i(~p·~y−ωny0) · i∫ ∞
0
ds e−ism2 · e− 12 Tr ln (2h)
×∫
d[u] exp ip · u(s) · exp
i
4
∫ s
0
ds1
∫ s
0
ds2 u(s1) · h−1(s1, s2) · u(s2)
× exp− i
2
∫d4k
(2π)4δ
δA(k)· Dth(k) · δ
δA(−k)
×
m− iγ ·[p− g
∫dq A(q) · eiq·(y−u(s))
]
× exp−ig
∫dq A(q) · f(q)
· exp
i
2
∫dq A(q) · Kth(q) · A(q)
∣∣∣∣A=0
,
where fν(q) is defined in Eq. (3.32) as
fν(q) =∫ s
0
ds1
∫ s
0
ds2 uν(s1)h−1(s1, s2)∫ s2
0
ds′ eiq·(y−u(s′)). (3.177)
Moving the linkage operator across the linear numerator term, and only the exponent linear in Aµ(q)
will contribute after Aµ is set to zero at the end of calculation, yielding
〈~p, n|S′th|~y, y0〉 (3.178)
' 1Z[iτ ]
[(2π)3τ
]−1/2 · e−i(~p·~y−ωny0) · i∫ ∞
0
ds e−ism2 · e− 12Tr ln (2h)
×∫
d[u] exp ip · u(s) · exp
i
4
∫ s
0
ds1
∫ s
0
ds2 u(s1) · h−1(s1, s2) · u(s2)
×
m− iγ ·[p + ig
∫d4q
(2π)4eiq·(y−u(s)) Dth(q)
δ
δA(q)
]
× exp− i
2
∫d4k
(2π)4δ
δA(k)· Dth(k) · δ
δA(−k)
× exp−ig
∫dq A(q) · f(q) +
i
2
∫dq A(q) · Kth(q) · A(q)
∣∣∣∣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,
eD(th)A · exp
+
i
2
∫dq A(q) · Kth(q) · A(q)− ig
∫dq A(q) · f(q)
(3.179)
= exp
+i
2
∫dq A(q) ·
[Kth(q) · 1
1− Kth(q) · Dth(q)
]· A(q)
× exp−ig
∫dq A(q) · 1
1− Kth(q) · Dth(q)· f(q)
× exp
i
2g2
∫dq f(q) ·
[Dth(q) · 1
1− Kth(q) · Dth(q)
]· f(q)
× exp−1
2Tr ln [1− Kth · Dth]
.
The determinant factor in the ”Trace-log” form could be absorbed into the definition of the partition
function Z[iτ ], and
Z[iτ ] = exp−1
2Tr ln [1− Kth · Dth]
· Z(0)[iτ ], (3.180)
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,
〈~p, n|S′th|~y, y0〉 (3.181)
' 1Z(0)[iτ ]
[(2π)3τ
]−1/2 · e−i(~p·~y−ωny0) · i∫ ∞
0
ds e−ism2 · e− 12Tr ln (2h)
×∫
d[u] exp ip · u(s) · exp
i
4
∫ s
0
ds1
∫ s
0
ds2 u(s1) · h−1(s1, s2) · u(s2)
×
m− iγ ·[p− g2
∫d4k
(2π)4
[Dth(k) · 1
1− Kth(k) · Dth(k)
)] · f(−k) · eik·(y−u(s))
]
× exp
i
2g2
∫d4k
(2π)4f(k) ·
[Dth(k) · 1
1− Kth(k) · Dth(k)
]· f(k)
.
Using the same Bloch-Nordsieck approximation as in the quenched case,
〈~p, n|S′th|~y, y0〉 (3.182)
=1
Z(0)[iτ ][(2π)3τ
]−1/2 · e−i(~p·~y−ωny0) · i∫ ∞
0
ds e−is(m2+p2)
×
m− iγ · p + 2ig2
∫d4k
(2π)4
[γ ·
(Dth(k) · 1
1− Kth(k) · Dth(k)
)· p
] ∫ s
0
ds′ e2ik·ps′
× exp
4ig2
∫d4k
(2π)4
∫ s
0
ds1
∫ s1
0
ds′[p ·
(Dth(k) · 1
1− Kth(k) · Dth(k)
)· p
]e2ik·ps′
,
where the quantity in the parentheses is the dressed, finite-temperature photon propagator D′th(k),
i.e.,
D′th(k) = Dth(k) · 1
1− Kth(k) · Dth(k), (3.183)
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 polariza-
tion tensor Kc(k) and the thermal contribution δKth(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 δKth(k) and use
the zero-temperature vacuum polarization tensor to replace the thermal photon polarization tensor
as
Kth(k) = Kc(k) + δKth(k) ' Kc(k). (3.184)
In the square brackets of the last exponent in Eq. (3.182), put in Kth(k) up to the order of g2 or α
as
D′th(k) = Dth(k) · 1
1− Kth(k) · Dth(k)(3.185)
' Dth(k) · 11− Kc(k) · Dth(k)
= Dth(k) · 11 + k2 [ΠR(k2) + Π(0)] P · Dth(k)
= Dth(k) · 11 + Π(0) + ΠR(k2)
,
where Kµνc (k) = −k2 Π(k2) Pµν , and ΠR(k2) is the renormalized part of Π(k2) defined by ΠR(k2) ≡
Π(k2)− Π(0). The last line utilizes [68, 89]
D′µνc (k) = Dµν
c (k)1
1 + Π(k2), (3.186)
and
k2 · δDth(k) = 0, (3.187)
where δDth(k) consists of a δ(k2)-factor, which puts photon fields on the mass-shell, i.e., k2 = 0,
and
Kc(k) · Dth(k) = Kc(k) · Dc(k). (3.188)
Observe that
g2 D′th(k) ' g2 Dth(k) · 1
1 + Π(0) + ΠR(k2)(3.189)
= g2 Dth(k) · 11 + Π(0)
· 1
1 + ΠR(k2)
1+Π(0)
=g2
1 + Π(0)Dth(k) ·
1−
ΠR(k2)
1+Π(0)
1 + ΠR(k2)
1+Π(0)
.
The first term is related to what has been calculated in the previous sections except for the renor-
malization factor of [1 + Π(0)]−1. Recall that Π(0) is the divergent part of Π(k2), it would seem
that the second part in the square brackets will vanish with |1 + Π(0)| À |ΠR(k2)|. However, the
76
coupling constant g so far is not yet renormalized, and the extra factor is related to the renormaliza-
tion constant Z3 = [1+ Π(0)]−1. According to the Ward identity [48], the renormalization constant,
Z3, could be absorbed into the definition of the coupling constant g2, i.e.,
g2R =
g2
1 + Π(0), (3.190)
and the same factor could be similarly absorbed into g2 inside each ΠR(k2) as
ΠR(k2; g2)1 + Π(0)
→ ΠR(k2; g2R). (3.191)
Thus,
g2 D′th(k) ' g2 Dth(k) · 1
1 + Π(0) + ΠR(k2)(3.192)
→ g2R Dth(k) ·
[1− ΠR(k2)
1 + ΠR(k2)
],
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
ΠR(k2) is large, the correction from the photon polarization or self-energy is limited to 1, i.e.,∣∣∣∣∣
ΠR(k2)1 + ΠR(k2)
∣∣∣∣∣ . 1, (3.193)
which is an indication that the theory is self-consistent and unitary. Up to order of g4 or to first
order in ΠR(k2),
g2 D′th(k) → g2
R Dth(k) ·[1− ΠR(k2)
1 + ΠR(k2)
](3.194)
' g2R Dth(k)− g2
R Dth(k) · ΠR(k2; g2R).
By definition, ΠR(0) = 0, so
δDth(k) · ΠR(k2) = 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,
g2R D′
th(k) → g2R
[Dc(k) + δDth(k)
]− g2
R Dc(k) · ΠR(k2). (3.196)
Hence, the one close-fermion-loop approximated, dressed, finite-temperature fermion propagator
reduces to
〈~p, n|S′th|~y, y0〉 ' 1Z(0)[iτ ]
[(2π)3τ
]−1/2 · e−i(~p·~y−ωny0) · i∫ ∞
0
ds e−is(m2+p2) (3.197)
×
m− iγ · p + 2ig2
∫d4k
(2π)4[γ · Dth(k) · p
] ∫ s
0
ds′ e2ik·ps′
× exp
4ig2
∫d4k
(2π)4
∫ s
0
ds1
∫ s1
0
ds′[p · Dth(k) · p
]e2ik·ps′
× exp−4ig2
∫d4k
(2π)4
∫ s
0
ds1
∫ s1
0
ds′[p · Dc(k) ΠR(k2) · p
]e2ik·ps′
,
77
where all subscripts ’R’ in renormalized g2R under one closed-fermion-loop approximation has been
dropped for notation simplicity. The exponent with a factor of [p · Dth(k) · p] in the third line is
what has been calculated in the previous sections. The last exponent with [p · Dth(k) ΠR(k2) · p] is
the contribution from one closed-fermion-loop, i.e., pair production, and
−4ig2
∫d4k
(2π)4
∫ s
0
ds1
∫ s1
0
ds′[p · Dc(k) ΠR(k2) · p
]e2ik·ps′ (3.198)
→ −4ig2
∫
|~k|≤|~p|
d4k
(2π)4
∫ s
0
ds1
∫ s1
0
ds′[p · Dc(k) ΠR(k2) · p
]e2ik·ps′
= −4ig2
∫
|~k|≤|~p|
d4k
(2π)4
∫ s
0
ds1
∫ s1
0
ds′[pµ ·
(−δµ0δ0ν
~k2+
PTµν
~k2 − k20 − iε
)· pν
]e2ik·ps′
×(− g2
2π2
) ∫ 1
0
dy y(1− y) ln(
m2 + y(1− y)k2
m2
)
= +i2g4
π2
∫ s
0
ds1
∫ s1
0
ds′∫ 1
0
dy y(1− y)∫
|~k|≤|~p|
d4k
(2π)4e2ik·ps′
×−ω2
n
~k2+
[~p 2 − (~k · ~p)2
~k2
]· 1~k2 − k2
0 − iε
ln
(m2 + y(1− y)k2
m2
).
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,
in the photon propagator. The part with the −ω2n/~k2 factor will vanish, i.e., there is no longitudinal
part in the context of QED as k2 · K = 0, then
−4ig2
∫d4k
(2π)4
∫ s
0
ds1
∫ s1
0
ds′[p · Dc(k) ΠR(k2) · p
]e2ik·ps′ (3.199)
' +i2g4
π2~p 2
∫ s
0
ds1
∫ s1
0
ds′∫ 1
0
dy y(1− y)∫
|~k|≤|~p|
d4k
(2π)4e2is′~k·~p−2is′k0 ωn
× 1− cos2 θ
~k2 − k20 − iε
ln
(m2 + y(1− y)(~k2 − k2
0)m2
)
= +ig4
π4~p 2
∫ s
0
ds1
∫ s1
0
ds′∫ 1
0
dy y(1− y)∫ 1
0
dξ (1− ξ2)∫ |~p|
0
dk~k2 cos (2s′|~k||~p|ξ)
×∫ +∞
−∞
dk0
2π
e−2is′k0 ωn
~k2 − k20 − iε
ln
(m2 + y(1− y)(~k2 − k2
0)m2
),
where ξ = cos θ = k · p. The logarithm factor from the polarization tensor becomes complex when
ζ = [y(1− y)]−1 ≥ 4 for 0 < y < 1 as
ln
(m2 + y(1− y)(~k2 − k2
0)m2
)= ln
∣∣∣∣∣m2 + y(1− y)(~k2 − k2
0)m2
∣∣∣∣∣ + πiθ(k20 − (~k2 + ζm2)), (3.200)
78
and the exponent becomes
−4ig2
∫d4k
(2π)4
∫ s
0
ds1
∫ s1
0
ds′[p · Dc(k) ΠR(k2) · p
]e2ik·ps′ (3.201)
' +ig4
π4~p 2
∫ s
0
ds1
∫ s1
0
ds′∫ 1
0
dy y(1− y)∫ 1
0
dξ (1− ξ2)∫ |~p|
0
dk~k2 cos (2s′|~k||~p|ξ)
×∫ +∞
−∞
dk0
2π
e−2is′k0 ωn
~k2 − k20 − iε
ln
∣∣∣∣∣m2 + y(1− y)(~k2 − k2
0)m2
∣∣∣∣∣
− g4
π3~p 2
∫ s
0
ds1
∫ s1
0
ds′∫ 1
0
dy y(1− y)∫ 1
0
dξ (1− ξ2)∫ |~p|
0
dk~k2 cos (2s′|~k||~p|ξ)
×∫ +∞
−∞
dk0
2π
e−2is′k0 ωn
~k2 − k20 − iε
θ(k20 − (~k2 + ζm2)).
The k0-integral in the first term with the real part of logarithm can be converted into a contour
integral, and the integrand has poles at k0 = ±k ∓ iε′. Since
ln
∣∣∣∣∣m2 + y(1− y)(~k2 − k2
0)m2
∣∣∣∣∣
∣∣∣∣∣k0=0
= ln∣∣∣∣m2 + 0
m2
∣∣∣∣ = 0, (3.202)
the residues of such contour integral are zero, and the first term vanishes. Hence, only the second
term remains, and
−4ig2
∫d4k
(2π)4
∫ s
0
ds1
∫ s1
0
ds′[p · Dc(k) ΠR(k2) · p
]e2ik·ps′ (3.203)
' − g4
π4~p 2
∫ s
0
ds1
∫ s1
0
ds′∫ 1
0
dy y(1− y)∫ 1
0
dξ (1− ξ2)
×∫ |~p|
0
dk~k2 cos (2s′|~k||~p|ξ)∫ ∞√
~k2+ζm2dk0
cos (2s′k0 ωn)~k2 − k2
0 − iε.
To evaluate the pair production as an energy depletion mechanism, the photon energy is at most up
to that of the incident fermion, i.e., k0 ≤√
~p 2 + m2 ' |~p| as |~p| À m. Thus, the upper limit of the
k0-integral can also be set to |~p| as the ~k-integral,
−4ig2
∫d4k
(2π)4
∫ s
0
ds1
∫ s1
0
ds′[p · Dc(k) ΠR(k2) · p
]e2ik·ps′ (3.204)
' − g4
π4~p 2
∫ s
0
ds1
∫ s1
0
ds′∫ 1
0
dy y(1− y)∫ 1
0
dξ (1− ξ2)
×∫ |~p|
0
dk~k2 cos (2s′|~k||~p|ξ)∫ |~p|√
~k2+ζm2dk0
cos (2s′k0 ωn)~k2 − k2
0 − iε.
Similar to steps in the previous section, let k = xp, k0 = zp, s1 = sv and s′ = s1u with 0 <
x, z, u, v < 1. Then, the s1- and s′-integrals can be reduced to∫ s
0
ds1 ·∫ s1
0
ds′ cos (2s′|~k||~p|ξ) · cos (2s′k0 ωn) (3.205)
= s2
∫ 1
0
dv · v ·∫ 1
0
du cos(2s~p2ξxuv) · cos(2spωnzuv)
=s2
2
∫ 1
0
dv · v ·∫ 1
0
du [cos(2sp(pξx + ωnz)uv) + cos(2sp(pξx− ωnz)uv)].
79
Since s < smax = 1m2+~p2+|ωn|2 ,
∫ 1
0
dv · v ·∫ 1
0
du · cos(2sp(pξx± ωnz)uv) =1− cos(2sp(pξx± ωnz))
(2sp(pξx± ωnz))2' 1
2. (3.206)
Hence,
−4ig2
∫d4k
(2π)4
∫ s
0
ds1
∫ s1
0
ds′[p · Dc(k) ΠR(k2) · p
]e2ik·ps′ (3.207)
' − g4
2π4s2 ~p 2
∫ 1
0
dy y(1− y)∫ 1
0
dξ (1− ξ2)∫ |~p|
0
dk~k2
∫ |~p|√
~k2+ζm2dk0
1~k2 − k2
0 − iε
= +g4
6π4s2 ~p 2
∫ 1
0
dy y(1− y)∫ |~p|
0
dk k
ln
(p− k
p + k
)− ln
√~k2 + ζm2 − k√~k2 + ζm2 + k
= +g4
6π4s2 (~p 2)2
∫ 1
0
dy y(1− y)∫ 1
0
dxx
ln
(1− x
1 + x
)− ln
√~x2 + ζ m2
~p 2 − x√
~x2 + ζ m2
~p 2 + x
,
where k = xp in the k-integral,
1~k2 − k2
0 − iε=
1(k0 − k + iε′)(k0 + k − iε′)
=12k
[1
k0 − k + iε′− 1
k0 + k − iε′
], (3.208)
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,√
~x2 + ζ m2
~p 2 − x√
~x2 + ζ m2
~p 2 + x'
x + ζ m2
2x~p 2 − x
x + ζ m2
2x~p 2 + x=
ζm2
2~p 2
2x2 + ζm2
2~p 2
' 1x2
ζm2
4~p 2. (3.209)
With help of ∫ 1
0
dx x ln(
1− x
1 + x
)= −1, (3.210)
∫ 1
0
dxx ln(
1x2
ζm2
4~p 2
)=
12
+12
ln(
ζm2
4~p 2
), (3.211)
and ∫dxx ln (
√x2 + a2 + x) =
(x2
2+
a2
4
)ln (
√x2 + a2 + x)− x
4
√x2 + a2, (3.212)
one arrives at
−4ig2
∫d4k
(2π)4
∫ s
0
ds1
∫ s1
0
ds′[p · Dc(k) ΠR(k2) · p
]e2ik·ps′ (3.213)
' +g4
6π4s2 (~p 2)2
∫ 1
0
dy y(1− y)[−1 +
12
+12
ln(
ζm2
4~p 2
)].
Recall that ζ = 1/[y(1− y)],∫ 1
0
dy y(1− y)12
ln (ζ) = −12
∫ 1
0
dy y(1− y) [ln (y) + ln (1− y)] =536
, (3.214)
80
∫ 1
0
dy y(1− y) =16, (3.215)
and then, it leads to
−4ig2
∫d4k
(2π)4
∫ s
0
ds1
∫ s1
0
ds′[p · Dc(k) ΠR(k2) · p
]e2ik·ps′ (3.216)
' − g4
6π4s2 (~p 2)2
112
[ln
(~p 2
m2
)− ln 4− 2
3
]
' − g4
72π4s2 (~p 2)2 ln
(~p 2
m2
),
where ln(~p 2/m2
) À ln(4) + 23 = 2.05 with ~p 2 À m2. The damping effect from the pair production
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′th(~p, n) (3.217)
'(Z(0)[iτ ]
)−1
· i∫ ∞
0
ds exp−is(ω2 − ω2
n)
× exp−s2
[2a2
T
3π2g2T 2 ~p 2 +
2a2B
3π2g2(~p 2)2 +
2a2P
3π2g4 (~p 2)2 ln
(~p 2
m2
)]
×
m− iγ · p + 2ig2
∫d4k
(2π)4[γ · D′
th(k) · p] ∫ s
0
ds′ e2ik·ps′
× exp
iIm
4g2
∫d4k
(2π)4
∫ s
0
ds1
∫ s1
0
ds′[p · Dc(k) · p
]e2ik·ps′
,
where a2T = π2/6, a2
B = 1/4, and a2P = 1/(48π2) 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.
The decay factor could be moved out of s-integral by converting s2 with[i (∂/∂ω2)
]2 as
S′th(~p, n) ' (3.218)
exp
− 1
3π2g2
[~p 2 [a2
T T 2 + a2B ~p 2] + a2
P g2 (~p 2)2 ln(
~p 2
m2
)]·(
i∂
∂ω2
)2·(Z(0)[iτ ]
)−1
·i∫ ∞
0
ds exp−is(ω2 − ω2
n)
m− iγ · p + 2ig2
∫d4k
(2π)4[γ · Dc(k) · p
] ∫ s
0
ds′ e2ik·ps′
.
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 scatter-
ing 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
D(th)A = D
(th)S + D
(th)H , (4.1)
where subscripts ’S’ and ’H’ denote soft- and hard- parts of linkage with gauge fields A(S)µ and A
(H)µ ,
respectively, and Aµ = A(S)µ + A
(H)µ . Then, the full fermion propagator becomes
S′th = eD(th)S +D
(th)H ·
[Gth[A(S) + A(H)]
eLth[A(S)+A(H)]
Z[iτ ]
]∣∣∣∣A(S)=A(H)=0
. (4.2)
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.,
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 ordi-
nary Bremsstrahlung, and pair production:
81
82
γth γHB
Figure 4.1: One-loop representation of Thermal-Photon-Induced Bremsstrahlung through a thermalphoton (γth) exchange with the medium (HB).
γvγHB
Figure 4.2: One-loop representation of Ordinary Bremsstrahlung through a virtual photon (γv)exchange with the medium (HB).
γv γHB
Figure 4.3: Two-loop representation of Pair Production from a virtual photon (γv) exchange withthe 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 consid-
ered 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 inci-
dent 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 ther-
mal 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
b2 =19g2 T 2 ~p 2 +
16π2
g2 (~p 2)2 +1
72π4g4 (~p 2)2 ln
(~p2
m2
)(4.3)
=2
3π2g2 ~p 2
[a2
T T 2 + a2B ~p 2 + a2
T g2~p 2 ln(
~p 2
m2
)],
where a2T = π2/6, a2
B = 1/4, and a2P = 1/(48π2) are the coefficients for the thermal-photon-induced
Bremsstrahlung, ordinary Bremsstrahlung, and pair production, respectively. In the fermion prop-
agator, there is the s2-dependent, instead of naive s-dependent, exponential factor; and therefore,
the damping is not exponential as appeared in the perturbation calculations.
With the replacement,
s → i∂
∂ω2, (4.4)
84
which follows from the form of Eqs. (3.102) and (3.131), the dressed, finite-temperature fermion
propagator becomes
S′th(~p, n) (4.5)
' exp
[−b2 ·
(i
∂
∂ω2
)2]· Sth(~p, n)
' exp
− 2
3π2g2 ~p 2
[a2
T T 2 + a2B ~p 2 + a2
T g2~p 2 ln(
~p 2
m2
)]·(
i∂
∂ω2
)2· Sth(~p, n).
In this form, the complete Matsubara sum can be easily performed, using the mixed representation
of the non-interacting fermion propagator Sth(~p, n) expressed as
Sth(~p, z0) =[m− i~γ · ~p + iγ0 ·
(−1
i
∂
∂z0
)]i
2ω
e−iωz0 − n(ω)
[e−iωz0 + e+iωz0
]. (4.6)
Following the derivation of Eq. (3.152), the dressed, finite-temperature fermion propagator with all
three damping mechanisms can be written as
〈~p, x0|S′th|~p, y0〉 (4.7)
' i
2
[m− i~γ · ~p + iγ0 ·
(−1
i
∂
∂z0
)]
· 1ω
e−iωz0 − n(ω)e+ β2b2
16ω2
[e−i[ω−β b2
4ω2 ]z0 + e+i[ω−β b2
4ω2 ]z0]
e−b2
4ω2 z20 ,
where n(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
8ω3¿ 1, (4.8)
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,
〈~p, x0|S′th|~p, y0〉 (4.10)
' i
2
[m− i~γ · ~p + iγ0 ·
(−1
i
∂
∂z0
)]
· 1ω
e−iωz0 − n(ω − βb2/8ω2)
[e−i[ω−β b2
4ω2 ]z0 + e+i[ω−β b2
4ω2 ]z0]
e−b2
4ω2 z20 ,
Compared to the non-interacting propagator, there is an overall factor of exp [−(b2/4ω2)z20 ],
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]
Γ(p) =12ω
∫d3~p1
(2π)32ω1
∫d3~p2
(2π)32ω2
∫d3~p3
(2π)32ω3(2π)4 δ(4)(p + p1 − p2 − p3) (4.11)
× [n1(1− n2)(1− n3) + (1− n1)n2n3] |M|2,
where ni is the Fermi-Dirac distribution of the i-th particle, and
M(p, p1; p2, p3) = p p3
p1 p2
(4.12)
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
Γ(p) ' g4T 3
6
∫ Λ
0
dωk
∫ +ωk
−ωk
dk0
(2π)
[|DL(ωk, k0)|2 +
12
(1− k2
0
~k2
)2
|DT (ωk, k0)|2], (4.13)
where ωk = |~k|, Λ is the upper cutoff for soft momenta, and D(k) = DL(k) PL + DL(k) PT is the
bare photon propagator with (00)-component DL = −1/~k2 and transverse component DT = 1/k2,
86
respectively. The result of the Born approximation gives a quadratic IR divergence in the |~k|-integral
as
Γ(p) ' g4T 3
6
∫ Λ
0
dk
k3. (4.14)
To alleviate IR divergence, the bare photon propagator can be replaced with the resummed
photon propagator, D′HTL(k), in Hot-Thermal-Loop (HTL) approximations;
D′−1HTL(k) = D−1(k)− KHTL(k) = D−1(k) + k2 ΠHTL(k2), (4.15)
and
D′L(~k, k0) =
−1~k2 − k2 ΠHTL
L
, (4.16)
D′T (~k, k0) =
1k2 + k2 ΠHTL
T
,
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
Σ(p) =p− k
k= −g2 1
τ
∑
l
∫d3~k
(2π)3γ · D′
HTL(k) · γ S(p− k), (4.17)
where k = (~k, ωl), p = (~p, ωn), and D′HTL(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]
k2 ΠHTLL → g2T 2
3(4.18)
k2 ΠHTLT → −i
π
4g2T 2
3k0
|~k|The longitudinal contribution is finite, but the transverse component is
ΓT (p) ' g2T
2π
∫ Λ
0
dk
k→ g2T
2πln
(Λ
µsc
), (4.19)
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 appli-
cations 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
Sc(p− k) =1
m + iγ · (p− k)→ 1
m + ~v · (~p− ~k)− (p0 − k0), (4.20)
and the HTL photon propagator D′HTL(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
δS′R(t, ~p) ' iθ(t) e−it(~v·~p) exp −αTt[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
Re
4ig2
∫d4k
(2π)4
∫ s
0
ds1
∫ s1
0
ds′[p · DBN
c (k) · p]
e2ik·ps′
(4.22)
= − 4(2π)2
g2~p 2
∫ s
0
ds1
∫ s1
0
ds′∫ 1
0
dξ [1− ξ2]∫ ∞
0
dk k cos(2s′k ωn) cos(2s′kp ξ) e−~k2/Λ2
= − 4(2π)2
s2g2~p 2 Λ2
∫ 1
0
dv · v ·∫ 1
0
du
∫ 1
0
dξ · [1− ξ2]
×∫ ∞
0
dx · x · cos(2sΛ ωnxuv) · cos(2sΛ|~p|ξxuv) · e−x2,
where the Bloch-Nordsieck limiting factor is replaced by exp [−~k2/Λ2] and k = xΛ; and one yields
Re
4ig2
∫d4k
(2π)4
∫ s
0
ds1
∫ s1
0
ds′[p · Dc(k) · p
]e2ik·ps′
(4.23)
' −2a2B
3π2s2g2~p 2 Λ2.
88
Similarly, the exponent for the pair production, Eq. (3.207), becomes
−4ig2
∫d4k
(2π)4
∫ s
0
ds1
∫ s1
0
ds′[p · Dc(k) ΠR(k2) · p
]e2ik·ps′ (4.24)
' − g4
2π4s2 ~p 2
∫ 1
0
dy y(1− y)∫ 1
0
dξ (1− ξ2)∫ Λ
0
dk~k2
∫ Λ
√~k2+ζm2
dk01
~k2 − k20 − iε
= +g4
6π4s2 ~p 2 Λ2
∫ 1
0
dy y(1− y)∫ 1
0
dxx
ln
(1− x
1 + x
)− ln
√~x2 + ζ m2
Λ2 − x√
~x2 + ζ m2
Λ2 + x
,
where ζ = [y(1− y)]−1 ≥ 4, which leads to
−4ig2
∫d4k
(2π)4
∫ s
0
ds1
∫ s1
0
ds′[p · Dc(k) ΠR(k2) · p
]e2ik·ps′ (4.25)
' − g4
72π4s2 ~p 2 Λ2 ln
(Λ2
m2
).
Therefore, the combined decay exponent, Eq. (4.3), becomes
b2 =2
3π2g2 ~p 2
[a2
T T 2 + a2B Λ2 + a2
P g2Λ2 ln(
Λ2
m2
)], (4.26)
where a2T = π2/6, a2
B = 1/4, and a2P = 1/(48π2) as defined in the previous sections. As ω0 with
|~p| À T À m, one can approximate ~p 2 ' ω and lead to
b2 =2
3π2g2 ω2
[a2
T T 2 + a2B Λ2 + a2
P g2Λ2 ln(
Λ2
m2
)]. (4.27)
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,
b2
2ω2=
g2
3π2
[a2
T T 2 + a2B Λ2 + a2
P g2Λ2 ln(
Λ2
m2
)], (4.28)
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
S′th(~z, z0) =∫
d3~p
(2π)3S′th(~p, z0) e+i~p·~z (4.29)
'∫
d3~p
(2π)3i
2ω
[m− i~γ · ~p + iγ0 ·
(i
∂
∂z0
)]e−
b2
4ω2 z20
×
ei(~p·~z−ωz0) − n(ω − βb2/8ω2)[ei[~p·~z−(ω−β b2
4ω2 )z0] + ei[~p·~z+(ω−β b2
4ω2 )z0]]
.
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
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 ω′ = ω−βb2/(4ω2). The phase velocity ω′ is different from the ω′′ = 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 ‖ ~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
〈~p, x0|S′th|~p, y0〉 (4.31)
'[m− i~γ · ~p + iγ0 ·
(−1
i
∂
∂z0
)]
× i
2ω
e−iωz0− b2
4ω2 z20 − n(ω)e−
b2
4ω2
[z20− β2
4
] [e−i[ω−β b2
4ω2 ]z0 + e+i[ω−β b2
4ω2 ]z0]
.
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
have a similar amplitude decay exponent of −(b2/4ω2) z20 , 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 to n(ω) exp [+β2 b2/(16ω2)] or n(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 parti-
cle 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
∆T = − i
2 ωn(ω) e
− b2
4ω2
[z20− β2
4
]= − i
2 ωn(ω) e
−Γ2
[z20− β2
4
], (4.32)
where
Γ =b2
2 ω2. (4.33)
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
|∆T|2 =1
4 ω2[n(ω)]2 e
−Γ[z20− β2
4
]. (4.34)
90
The ratio of the square modulus compared to the initial value at z0 = 0 is
|∆T(z0)|2|∆T(0)|2 =
[ω0
ω
]2[
n(ω)n(ω0)
]2
e−β2
4 [Γ−Γ0] e−Γz20 , (4.35)
where ω0 = ω(0) and Γ0 are the initial values at z0 = 0. Under the leading approximation, Γ =
b2/2ω2 is not an explicit function of z0, the ratio reduces to
|∆T(z0)|2|∆T(0)|2 =
[ω0
ω
]2[
n(ω)n(ω0)
]2
e−Γz20 . (4.36)
As z0 increases, the particle’s energy decreases, i.e., ω0 ≥ ω(z0), n(ω0) ≤ n(ω) and
[ω0
ω
]2
≥ 1,
[n(ω)n(ω0)
]2
≥ 1. (4.37)
The probability for the transverse component is controlled by two oppositely trending factors, n(ω)/ω
and exp [−Γz20 ]. Initially, the two factors 1/ω and n(ω) dominate compared to the Gaussian-type
exponential factor exp [−Γz20 ], 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,
ω = ω0 e−γz0 , (4.38)
where γ is some characteristic constant and can be estimated by the simply Doppler model [87].
Then, the ratio becomes
|∆T(z0)|2|∆T(0)|2 ' exp
[2γz0 + 2βω0(1− e−γz0)− Γz2
0
] ' exp[2(1 + βω0)γz0 − Γz2
0
](4.39)
as γz0 is small. The disturbance rises initially until
z0 ∼ 2(1 + βω0)γ
Γ. (4.40)
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,
ω = ω0 e−Γ′z20 , (4.41)
than
|∆T(z0)|2|∆T(0)|2 ' exp
[2Γ′z2
0 + 2βω0(1− e−Γ′z20 )− Γz2
0
]' exp
[2(1 + βω0)Γ′z2
0 − Γz20
](4.42)
Again, the transverse disturbance rises initially, but shrinks after
z20 ∼ 2(1 + βω0)
Γ′
Γ. (4.43)
91
For the longitudinal component, the amplitude of the forward and backward propagations are
proportional to 1− n(ω) exp [(b2/4ω2) (β2/4)] and n(ω) exp [(b2/4ω2) (β2/4)] for a given (ω, ~p), re-
spectively. To evaluate the longitudinal component of disturbance, define ∆L as
∆Lf =
i
2ω
[1− n(ω) e
b2
4ω2β2
4 e+iβ b2
4ω2 z0
]e−
b2
4ω2 z20 =
i
2ω
[1− n(ω′′) e+iβ Γ
2 z0
]e−
Γ2 z2
0 (4.44)
for the forward propagating component, and
∆Lb =
−i
2ω
[n(ω) e
b2
4ω2β2
4 e−iβ b2
4ω2 z0
]e−
b2
4ω2 z20 =
−i
2ω
[n(ω′′) e−iβ Γ
2 z0
]e−
Γ2 z2
0 (4.45)
for the backward propagating component, as the fermion proceeds through the medium, and where
ω′′ = ω − b2
2 ω2
β2
4= Γ
β2
4. (4.46)
If one takes the square modulus of both ∆L’s as
∣∣∆Lf
∣∣2 =1
4ω2
[1− n(ω′′) cos
(β Γ2
z0
)]2
e−Γz20 +
14ω2
[n(ω′′) sin
(β Γ2
z0
)]2
e−Γz20 (4.47)
=1
4ω2
[1 + n2(ω′′)− 2n(ω′′) cos
(β Γ2
z0
)]e−Γz2
0 ,
and ∣∣∆Lb
∣∣2 =1
4ω2n2(ω′′) e−Γz2
0 =1
4 ω2[n(ω)]2 e
−Γ[z20− β2
4
], (4.48)
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
14ω2
exp[−Γz2
0
]. (4.49)
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 func-
tion; 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
S′th(~p, ωn) =1
m0 + iγ · p−Σ0(~p, ωn)−Σth(~p, ωn), (4.50)
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,
S′th(~p, ωn) → S′c(~p, p0), Σth → 0, and Σ0 → Σµ=00 , (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
[S′th]−1 = m0 − ω −Σ0(ω; m0)−Σth(ω;m0) (4.52)
=[m0 −Σµ=0
0 (m;m0)]− ω −
[Σ0(ω; m0)−Σµ=0
0 (m; m0)]−Σth(ω; m0).
The renormalized mass m is defined at zero-temperature with zero chemical potential, i.e., T = 0
and µ = 0, and under the condition that S′th has a pole at ω = m with
m = m0 −Σµ=00 (m; m0), (4.53)
and the residue of the pole at ω = m is the wave-function renormalization constant as
Z−12 = 1 +
Σµ=00 (ω; m0)−Σµ=0
0 (m; m0)ω −m
∣∣∣∣∣ω→m
, (4.54)
as T → 0 and µ → 0. Then, one can express Σ0(ω; m0) as
Σ0(ω;m0) = Σ0(ω;m0)−Σµ=00 (m;m0) + Σµ=0
0 (ω; m0) (4.55)
+Σµ=00 (m;m0)−Σµ=0
0 (ω; m0),
and the propagator becomes
S′th(~p, ωn) =Z2
m + iγ · p−Σ0,R −Σth,R, (4.56)
where
Σth,R = Z2 Σth, (4.57)
and
Σ0,R = Z2 [Σ0(ω; m0)−Σµ=00 (ω;m0)], (4.58)
which reduces to zero, i.e., Σ0,R = 0, in the case of vanishing chemical potential. Thus, the renor-
malized, finite-temperature fermion propagator becomes
S′th,R(p) =1
m + iγ · p−Σ0,R −Σth,R, (4.59)
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
S′th,R(p) =1
m + iγ · p−Σ′th
, (4.60)
where Σ′th = 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
S′th(p) =m− iγ · p−Σ′
th
(m−Σ′th)2 + (γ · p)2
=m− iγ · p−ReΣ− iImΣ
(m−Σ′th)2 + p2
, (4.61)
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
m− iγ ·[p− g2
∫d4k
(2π)4D′
th(k) · f(−k) · eik·(y−u(s))
](4.62)
→
m− iγ · p + 2ig2
∫d4k
(2π)4
∫ s
0
ds′[γ · D′
th(k) · p]
e2ik·ps′
,
where the dressed, finite-temperature photon propagator is
D′th(k) = Dth(k) · 1
1− Kth(k) · Dth(k). (4.63)
The temperature dependent part of a non-interacting fermion propagator is proportional to 2πiδ(p2+
m2) 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]
Kµνth (k) = Kµν
c (k) + δKµνth (k). (4.64)
The leading contribution of the thermal part, δKµνth (k), comes from the Hot Thermal Loops as
[56, 43]
δKµν(k) ' KHTLµν (k) = −k2
[PT
µν ΠHTLT (k2) + PL
µν ΠHTLL (k2)
], (4.65)
where PLµν and PT
µν are the longitudinal and transverse operator, respectively; and the HTL photon
polarization functions are given by [56, 43]
ΠHTLL (k2) = 2ΠHTL(k2)− 2m2
sc
1~k2
, (4.66)
ΠHTLT (k2) = m2
sc
1k2
k20
~k2− ΠHTL(k2), (4.67)
up to the order of g2T 2, and where
ΠHTL(k2) = −m2sc
21~k2
k0
|~k|ln
(k0 + |~k|k0 − |~k|
)(4.68)
with the screening mass msc as
m2sc =
g2T 2
6(4.69)
94
for QED. Since k2 ΠHTLL (k2) and k2 ΠHTL
T (k2) vanish near k2 = 0,
KHTLL (k) · δDth(k) = 0, and KHTL
T (k) · δDth(k) = 0. (4.70)
Thus,
Dth(k) · 11− Kth(k) · Dth(k)
' Dc(k) · 11− Kth(k) · Dc(k)
+ δDth(k) (4.71)
' Dc(k) · 11− Kc(k) · Dc(k)
+ δDth(k)
= D′c(k) + δDth(k),
where the first term only retains Kµνc (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
2ig2
∫d4k
(2π)2
∫ s
0
ds′[γ · D′
th(k) · p]
e2ik·ps′ (4.72)
' 2ig2
∫d4k
(2π)2
∫ s
0
ds′[γ · D′
c(k) · p + γ · δDth(k) · p]
e2ik·ps′ .
If the quenched approximation is implied instead, D′th(k) will be replaced by Dth(k). In regular
Bloch-Nordsieck approximation, the particle of interest experiences negligible recoil from contribu-
tions of soft photons, and the gamma matrix γµ is usually replaced with a four-velocity −ivµ with
vµ ∼ pµ/m and v2 = −1 [49]. Thus, one can assume
(~γ · ~k)(~k · ~p)~k2
' (~γ · ~p) cos2 θ. (4.73)
Then, the two terms can be evaluated as
2ig2
∫d4k
(2π)2
∫ s
0
ds′[γ · D′
c(k) · p]
e2ik·ps′ (4.74)
' −s(~γ · ~p)1
3π2g2~p 2
[1 + i
√π
4ε(ωn)ωn
],
and
2ig2
∫d4k
(2π)2
∫ s
0
ds′[γ · δDth(k) · p
]e2ik·ps′ (4.75)
' −s(~γ · ~p)1
9π2g2 1
β2
= −s(~γ · ~p)1
9π2g2T 2.
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
m− iγ ·
[p− g2
∫d4k
(2π)2D′
th(k) · f(−k) · eik·(y−u(s))
](4.76)
'
m− iγ · p + 2ig2
∫d4k
(2π)2
∫ s
0
ds′[γ · Dth(k) · p
]e2ik·ps′
'
m− iγ · p− sg2 (~γ · ~p)[
19π2
T 2 +1
3π2~p 2
(1 + i
√π
4ε(ωn)ωn
)]
→
m− iγ · p− g2 (~γ · ~p)[
19π2
T 2 +1
3π2~p 2
(1 + i
√π
4ε(ωn)ωn
)] (i
∂
∂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.
δmT = −(~γ · ~p)1
9π2g2T 2
(i
∂
∂ω2
)· 1Z(0)[β]
· i∫ ∞
0
ds exp−is(ω2 − ω2
n)
(4.77)
= +i(~γ · ~p)1
9π2g2T 2 1
(ω2 − ω2n)2
→ +i(~γ · ~p)1
9π2g2T 2 1
(ω2 + |ωn|2)2 ,
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
δmT ' +1
9π2
g2T 2
m
~p 2
(ω2 + |ωn|2)2 (4.78)
. +1
9π2
g2T 2
m
~p 2
(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
mixed representation ∆Fth(ω, z0) of Eq. (2.164) is used instead, the thermal mass-shift becomes a
complicate function of time as
δmT = −(~γ · ~p)1
9π2g2T 2
(i
∂
∂ω2
)· ∆F
th(ω, z0) (4.79)
= −(~γ · ~p)1
9π2g2T 2
(i
∂
∂ω2
)·[
i
2ω
[1− n(ω)] e−iωz0 − n(ω) e+iωz0
],
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,
δmT = −π
9α
T 2
mω, for T ¿ m (4.80)
from Cox et al. [22], and
δm2T =
2παT 2
3m, for T ¿ m (4.81)
δm2T = παT 2, for T À m, (4.82)
96
or the phase-space mass shift
δmT =παT 2
3m, (4.83)
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 distri-
bution 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]
Dζµν(k) = − 1
~k2δµ0δ0ν +
1k2 − iε
PTµν + ζC
k2
(~k2)2kµkν
k2= Dc
µν(k) + δDζµν(k), (4.84)
where ζC is the (Coulomb) gauge parameter. The finite-temperature photon propagator can be
written as
Dµνth (k) .= Dµν
c (~k, k0 = ωn). (4.85)
After analytical continuation, the finite-temperature photon propagator is separated into two terms
as
Dµνth (k, k0) = Dµν
c (~k, k0) + δDµνth (k) (4.86)
with
δDµνth (k) = f(k0)
[Dth
µν(~k, ωn = k0 + iε)− Dthµν(~k, ωn = k0 − iε)
](4.87)
Similarly, there is no contribution from the gauge term δDζth(k) for the thermal part, i.e., δDζ
th(k) =
δDζ(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
4ig2
∫d4k
(2π)4
∫ s
0
ds1
∫ s1
0
ds′[p · δDζ(k) · p
]e2ik·ps′ (4.88)
= 4ig2 ζC
∫d4k
(2π)4
∫ s
0
ds1
∫ s1
0
ds′(p · k)2
(~k2)2e2ik·ps′ ,
= 4ig2 ζC
∫d4k
(2π)4
∫ s
0
ds1
∫ s1
0
ds′(~p · ~k − ωnk0)2
(~k2)2e2ik·ps′ .
The k0-integral generates delta-function related terms and∫ s
0
ds1
∫ s1
0
ds′[(~p · ~k)2 δ(2ωns′)− 2i(~p · ~k)
∂
∂s′δ(2ωns′) +
14
∂2
∂s′2δ(2ωns′)
]e2i~k·~ps′ = 0 (4.89)
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
δDζ · Kc = −ζCk2
(~k2)2kµkν
k2k2Π(k2)Pνσ = 0, (4.90)
where kν Pνσ = 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
Dζth(k) = Dth(k) + δDζ(k) = Dc(k) + δDth(k) + δDζ(k), (4.91)
The fully-dressed, finite-temperature photon propagator under the assumption of Kth(k) ' Kc(k)
becomes
D′ζth(k) '
[Dth(k) + δDζ(k)
]· 1
1− Kc(k) ·[Dth(k) + δDζ(k)
] (4.92)
=[Dth(k) + δDζ(k)
]· 11− Kc(k) · Dc(k)
= Dc(k) · 11 + Πc(k2)
+ δDth(k) + δDζ(k),
where δDζ(k)·P = 0 and δDth(k)·Kc(k) = 0 are used in the derivation to the last line as kµ Pµν = 0
and δ(k2) Kc(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 δDζ(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
〈~p, n|GBNth [A]|~y, y0〉 (4.95)
=[(2π)3τ
]−1/2 · e−i[~p·~y−(ωn+µ)y0] · i∫ ∞
0
ds e−is [m2+~p 2−(ωn+µ)2]
×
m− iγ ·[p− g
∫d4q
(2π)4A(q) eiq·(y+2sP )
]
×(
exp
g
∫ s
0
ds′σµν ·∫
d4q
(2π)4[qµAν(q)− qνAµ(q)
]eiq·(y+2s′P )
)
+
× exp
+2ig∫ s
0
ds′∫
d4q
(2π)4[p · A(q)
]eiq·(y+2s′P )
.
Similarly, the quenched finite-temperature fermion propagator, Eq. (3.73), becomes
〈~p, n|S′th|~y, y0〉BNQ (4.96)
' [Z(0)[iτ ]]−1 · [(2π)3τ]−1/2 · e−i[~p·~y−(ωn+µ)y0] · i
∫ ∞
0
ds e−is [m2+~p 2−(ωn+µ)2]
×
m− iγ · P + 2ig2
∫d4k
(2π)4[γ · Dth(k) · P
] ∫ s
0
ds′ e2ik·Ps′
× exp
4ig2
∫d4k
(2π)4
∫ s
0
ds1
∫ s1
0
ds′[P · Dth(k) · P
]e2ik·Ps′
,
and the closed-fermion-loop approximated quenched finite-temperature fermion propagator, Eq.
(3.182), becomes
〈~p, n|S′th|~y, y0〉 (4.97)
=1
Z(0)[iτ ][(2π)3τ
]−1/2 · e−i[~p·~y−(ωn+µ)y0] · i∫ ∞
0
ds e−is [m2+~p 2−(ωn+µ)2]
×
m− iγ · P + 2ig2
∫d4k
(2π)4[γ · D′
th(k) · P] ∫ s
0
ds′ e2ik·Ps′
× exp
4ig2
∫d4k
(2π)4
∫ s
0
ds1
∫ s1
0
ds′[P · D′
th(k) · P]
e2ik·Ps′
.
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
4ig2
∫d4k
(2π)4
∫ s
0
ds1
∫ s1
0
ds′[P · D′
th(k) · P]
e2ik·Ps′ , (4.98)
can be separated into three damping mechanisms;
P · Dc(k) · P = − (ωn + µ)2
~k2+
[~p 2 − (~k · ~p)2
~k2
]1
~k2 − k20 − iε
, (4.99)
for ordinary Bremsstrahlung,
P · δDth(k) · P =δ(~k 2 − k2
0)eβ|k0| − 1
[~p 2 − (~k · ~p)2
~k2
](4.100)
99
for thermal photon-assisted Bremsstrahlung, and
P ·[Dc(k)Πc(k)
]· P =
[~p 2 − (~k · ~p)2
~k2
]1
~k2 − k20 − iε
ln(
m2 + y(1− y)k2
m2
), (4.101)
for pair production, under the ordinary Coulomb gauge, and Dth(k) = Dc(k) + δDth(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 · Ps′], and the con-
tributions are within cosine or sine functions of w(±), where
w(±) =pξ ± ωn
β→ pξ ± (ωn + µ)
β, (4.102)
which are oscillatory factors, and vanish in the approximation used. Hence, the decay exponents do
not change with the chemical potential added.
Instead of Sth(~p, z0; µ = 0), the non-interacting finite-temperature fermion propagator is replaced
by Eq. (2.171) as
Sth(~p, z0; µ) =[m− i~γ · ~p + iγ0 ·
(µ− 1
i
∂
∂z0
)](4.103)
× i
2ω
[1− n(ω + µ)] e−iωz0 − n(ω − µ) e+iωz0
,
which contains asymmetric thermal-terms, and
〈~p, x0|S′th|~p, y0〉 (4.104)
' i
2
[m− i~γ · ~p + iγ0 ·
(µ− 1
i
∂
∂z0
)]
× 1ω
e−iωz0 − e+ β2b2
16ω2
[n(ω + µ) e−i[ω−β b2
4ω2 ]z0 + n(ω − µ) e+i[ω−β b2
4ω2 ]z0]
e−b2
4ω2 z20 ,
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]
δLHTL =14m2
γ Fµα
⟨kαkβ
−(k · ∂)2
⟩Fαµ, (4.105)
100
where kµ = |~k| (i, k), the angled bracket 〈· · · 〉 is the average over all possible directions of the
loop momenta k, i.e., the angular integral over Ωk, and the thermal photon mass is m2γ = g2T 2.
Alternatively, the effective HTL Lagrangian addition can be written as [45, 46]
δLHTL =12
∫
Cd4x
∫
Cd4y A(x) ·KHTL(x− y) ·A(y), (4.106)
where KHTLµν is the effective HTL photon polarization tensor. The addition δLHTL is then appended
into the effective photon Lagrangian as [46]
LHTL = Lphoton + δLHTL = −14F2 +
12
∫
Cd4x
∫
Cd4y A(x) ·KHTL(x− y) ·A(y), (4.107)
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
S′th = Z−1G [iτ, µ]
∫DAGth[A] eLth[A] exp
− i
2
∫
CA · D−1
th ·A
. (4.108)
Under the HTL approximation, the fermion propagator becomes [44, 30, 45, 46]
S′th = Z−1G [iτ, µ]
∫DAGth[A] exp
− i
2
∫
CA · D−1
th ·A +i
2
∫
CA · KHTL ·A
, (4.109)
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]
D−1HTL = D−1
th − KHTL, (4.110)
which leads to
S′th = Z−1G [iτ, µ]
∫DAGth[A] exp
− i
2
∫
CA ·D−1
HTL ·A
, (4.111)
in the form of functional integral, or
S′th = Z−1(0) [iτ, µ] exp
− i
2
∫
C
δ
δA·DHTL · δ
δA
·Gth[A]
∣∣∣∣A→0
(4.112)
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 propa-
gator 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.
Separate the HTL photon polarization tensor into longitudinal and transverse parts as KHTLµν =
KHTLL PL
µν + KHTLT PT
µν = −k2[ΠHTL
L PLµν + ΠHTL
T PTµν
], and the resummed photon propagator in
101
the usual Coulomb gague becomes [56]
DHTLµν = − δµ0 δ0ν
~k2 + KHTLCL (k2)
+PT
µν
k2 + KHTLCT (k2)
, (4.113)
where KHTLCL (k2) and KHTL
CT (k2) are the (00)-component and transverse part of the HTL approxi-
mated photon polarization tensor in Coulomb gauge.
KHTLCL (k2) =
~k 2
k2KHTL
L (k2) = −2m2Q1
(iω
k
), (4.114)
KHTLCT (k2) = KHTL
T (k2) = m2
(iω
k
) [(1−
(iω
k
)2)
+ Q0
(iω
k
)+
(iω
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
mode, i.e., KHTLT → 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
L = Lquark + Lqloun + Lint = −ψ [m + γ · (∂ − igA)] ψ − 14G2, (4.115)
where Aµ = Aaµλa are gauge fields, Gµν = Ga
µνλa is the field strength,
Gaµν = ∂µAa
ν − ∂νAaµ + gfabcA
bµAc
ν , (4.116)
and λa are the color matrices in the fundamental representations of SU(N);
[λa, λb] = 2ifabcλc, tr [λa] = 0, tr [λaλb] = 2δab, (4.117)
102
where fabc is the structure constant and defines the adjoint representation (τa)bc of SU(N). The field
strength becomes
Gµν = ∂µAν − ∂νAµ − 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]
exp[− i
4
∫G2
]= N
∫d[σ] exp
[i
∫σ2 + i
∫σ ·G
], (4.119)
where σaµν 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 − ∂νAµ are the QED-type field strengths. Similarly,
exp[− i
4
∫(G− F) · (G + F)
](4.121)
= N ′∫
d[φ]∫
d[ϕ] exp[i
∫φ · ϕ +
i
2
∫φ · (G− F) +
i
2
∫ϕ · (G + F)
],
where φaµν and ϕa
µν 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)
Thus, the gluon sector of the QCD Lagrangian can be separated into two parts as Lqloun = L(0)qloun +
L′gluon, where
L(0)qloun = −1
4F2, and L′gluon = −1
4(G2 − F2), (4.123)
and the QCD Lagrangian can be rewritten as the ’modified’ QED form as LQCD = L(0)+L′int, where
L(0) = Lquark + L(0)qloun = −ψ [m + γ · ∂] ψ − 1
4F2 (4.124)
is the ’free’ Lagrangian, which is similar to the non-interacting QED lagrangian, and
L′int = Lint + L′gluon = +ig ψ γ ·Aψ − 14(G2 − F2) (4.125)
is the ’modified’ interacting part. Then, the generating functional becomes
Zcj, η, η = 〈S〉−1 exp
i
∫L′int
[1i
δ
δj,1i
δ
δη,−1i
δ
δη
]· Z(0)
c j, η, η, (4.126)
where jµ = jaµλa, ην = ηb
νλb, and ησ = ηcσλc are gluon and quark sources, respectively. Following
similar derivation of QED, the non-interacting generating functional is given by
Z(0)c j, η, η = exp
i
2
∫j ·Dc · j + i
∫η · Sc · η
. (4.127)
103
Since L′gluon does not involve η or η, one can first perform the Gaussian translation operation in Lint
over exp [i∫
η · Sc · η] as
〈S〉Zcj, η, η (4.128)
= exp
i
∫L′gluon
[1i
δ
δj
]· exp
ig
∫δ
δη
(γ · δ
δj
)δ
δη
· exp
i
2
∫j ·Dc · j + i
∫η · Sc · η
= exp
i
∫L′gluon
[1i
δ
δj
]· exp
i
∫η ·Gc
[1i
δ
δj
]· η + L
[1i
δ
δj
]· exp
i
2
∫j ·Dc · j
,
where Gc[A] = Sc · [1− g (γ ·A)Sc]−1 and L[A] = Tr ln [1− g (γ ·A)Sc] with A replaced by 1
iδδj ,
which are similar to QED. Use the reciprocity relation in Eq. (D.21), the generating functional
becomes
〈S〉Zcj, η, η = exp
i
2
∫j ·Dc · j
· exp
− i
2
∫δ
δA·Dc · δ
δA
(4.129)
× exp
i
∫η ·Gc[A] · η + L[A]
· exp
i
∫L′gluon[A]
,
where Aµ(x) =∫
d4y Dc(x− y) · j(y) as before.
The formalism in QCD is similar to that in QED except the extra factor of exp [i∫ L′gluon] in the
generating functional. The difficulty of cubic and quartic dependence of gauge fields in L′gluon can
be avoided by utilizing the functional representation of Eq. (4.121) as
〈S〉Zcj, η, η (4.130)
= N ′∫
d[φ]∫
d[ϕ] exp
i
∫φ · ϕ
· exp
i
2
∫j ·Dc · j
· exp
− i
2
∫δ
δA·Dc · δ
δA
× exp
i
∫η ·Gc[A] · η + L[A]
· exp
i
2
∫φ · (G− F) +
i
2
∫ϕ · (G + F)
.
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
those in QED. However, gauge fields Aaµ 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 approxi-
mations have been proposed [99, 100, 101, 102, 103, 50] for both Gc[A] and L[A]. In addition, the
color degrees of freedom of non-Abelian gauge fields Aaµ 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
S′th = eD(th)A ·
[Gth[A]
eLth[A]
Z[iτ ]
]∣∣∣∣A→0
. (5.1)
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 pho-
ton 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
S′th(~p, z0) ' exp
[−2 ω2 Γ
(i
∂
∂ω2
)2]· Sth(~p, z0), (5.2)
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 exponen-
tial decay (with an extraneous logarithmical factor in the exponent) from perturbative approach,
the damping of incident particle is of the Gaussian-type, as exp[−Γz2
0
]with
2ω2Γ =2
3π2g2 ~p 2
[a2
T T 2 + a2B Λ2 + a2
P g2Λ2 ln(
Λ2
m2
)], (5.3)
where Λ is the soft-momentum cutoff appropriate for the Bloch-Nordsieck approximation used in
the estimate with Λ ≤ |~p|, and
a2T =
π2
6, a2
B =14, and a2
P =1
48π2, (5.4)
are the numerical coefficients for the processes of thermal-photon-assisted Bremsstrahlung, or-
dinary 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 fromenergy units to conventional units. (Note1: The conventional electric charge is in Heaviside-Lorentzunit.)
Quantity Dimension Conversion factor FormMass [M ] 1/c2 mc2
Length [M ]−1 ~c x/~cTime [M ]−1 ~ t/~Energy [M ] 1 EMomentum [M ] 1/c pcForce [M ]2 1/~c F~cAction [M ]0 ~ S/~Electric charge1 [M ]0
√~c e/
√~c
Temperature [M ] 1/kB kBT
Causal structure:
x is time-like : x2 = ~x 2 − x20 < 0 (A.6)
x is space-like : x2 = ~x 2 − x20 > 0
x is light-like : x2 = ~x 2 − x20 = 0
For any four-vector, pµ, to be ’physical’, i.e., time-like,
p20 > ~p 2 (A.7)
d’Alembertian operator:
∂2 = ~∇2 − ∂20 (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)
f(~p, p0) =∫
d4x eip0·x0−i~p·~x f(~x, x0) =∫
d4x e−ip·x f(x) (A.11)
Gamma matrices:
γµ, γν = 2δµν (A.12)
for µ, ν = 1, 2, 3, 4, and
ㆵ = 㵠(A.13)
γ5 matrix:
γ5 = γ1γ2γ3γ4; (A.14)
109
γ5, γµ = 0, µ = 1, 2, 3, 4, (A.15)
and
γ†5 = γ5, γ25 = 1; (A.16)
σµν =12i
[γµ, γν ] = −iγµγν , µ 6= ν; (A.17)
Gamma matrices in Pauli representation:
γi =
(0 −iσi
iσi 0
), γ4 =
(1 0
0 −1
), γ5 = −
(0 1
1 0
)(A.18)
where the Pauli matrices are
σ1 =
(0 1
1 0
), σ2 =
(0 −i
i 0
), σ3 =
(1 0
0 −1
). (A.19)
A.3 Gordon decomposition
Dirac equations for u(p) and u(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]:
u(p′) γµ u(p) = u(p′)[−i
(p + p′)µ
2m+
σµν (p− p′)ν
2m
]u(p) (A.22)
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
f(z0) exp [−β|z0|] vanishes sufficiently fast at infinity. For bosons, k0 = 2nπτ
1τ
∑n
f(k0) =∫
C+
dz0
2πn(z0) [f(z0) + f(−z0)] +
∫ +i∞
−i∞
dz0
2πf(z0) (B.1)
For fermions, p0 = (2n+1)πτ + µ
1τ
∑n
f(p0) = −∫
C+
dz0
2π[n+(z0)f(z0) + n−(z0)f(−z0)] +
∫ +i∞
−i∞
dz0
2πf(z0) (B.2)
where the Bose-Einstein and Fermi-Dirac distribution functions are
n(z0) =1
eβz0 − 1(B.3)
n±(z0) =1
eβ(z0∓µ) + 1(B.4)
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) and n±(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:
∆(ωn) =∫ τ
0
dτ eiωnτ ∆(τ) (B.5)
and its inverse
∆(τ) =1τ
∑n
e−iωnτ ∆(ωn). (B.6)
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∑
n
〈x0|n〉〈n|y0〉 = δ(x0 − y0), (B.7)
where 〈x0|n〉 = 1√τe−iωnx0 . Alternatively,
1τ
∑n
e−iωn(x0−y0) = δ(x0 − y0). (B.8)
The delta-function in Matsubara representation is given by
δ(4)(x− y) =1τ
∑n
∫d3k
(2π)3ei~k·(~x−~y)−iωn(x0−y0), (B.9)
or
(2π)3τδ(4)(p− q) =∫ τ
0
dx ei(~p−~q)·~x−i(ωn−ωl)x0 . (B.10)
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
∆th(~k, ωn;m2) = ∆(B)th (ω, ωn) =
1ω2 − ω2
n
(B.11)
for bosons with
ωn =2nπ
τ, ω2 = m2 + ~k2 (B.12)
or
Sth(~p, ωn) = (m− iγ · p) · [m2 + p2]−1
= (m− iγ · p) · ∆(F )th (ω, ωn) (B.13)
112
for fermions with
ωn =(2n + 1)π
τ, ω2 = m2 + ~p 2. (B.14)
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
∆th(ω, τ) =1τ
+∞∑n=−∞
e−iωnz0 ∆th(ω, ωn), (B.15)
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.
∆th(ω, τ) =1τ
+∞∑n=−∞
e−iωnz0 ∆th(ω, ωn) (B.16)
=1τ
+∞∑n=−∞
e−iωnz0
ω2 − ω2n − iε
=1τ
+∞∑n=−∞
e−iωnz012ω
[1
ω − ωn − iε+
1ω + ωn − iε
].
Use the proper time representation for the two terms in square brackets,
∆th(ω, τ) =i
2ωτ
+∞∑n=−∞
∫ ∞
0
ds e−iωnz0
[e−is(ω−ωn) + e−is(ω+ωn)
](B.17)
=i
2ω
∫ ∞
0
ds e−isω 1τ
+∞∑n=−∞
[e−iωn(z0−s) + e−iωn(z0+s)
].
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
∆th(ω, τ) =i
2ω
∞∑
l=0
∫ τ
0
dθ e−iω(θ+lτ) 1τ
+∞∑n=−∞
[e−iωn(z0−θ−lτ) + e−iωn(z0+θ+lτ)
](B.18)
=i
2ω
∞∑
l=0
∫ τ
0
dθ e−iω(θ+lτ) 1τ
+∞∑n=−∞
[(−1)le−iωn(z0−θ) + (−1)l+1e−iωn(z0+θ−τ)
],
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 .
∆th(ω, τ) =i
2ω
∞∑
l=0
∫ τ
0
dθ (−1)l e−iω(θ+lτ) 1τ
+∞∑n=−∞
[e−iωn(z0−θ) − e−iωn(z0+θ−τ)
](B.20)
=i
2ω
∞∑
l=0
∫ τ
0
dθ (−1)l e−iω(θ+lτ) [δ(z0 − θ)− δ(z0 + θ − τ)]
=i
2ω
∞∑
l=0
(−1)l e−iωlτ[e−iωz0 − e−iω(τ−z0)
].
Since ∞∑
l=0
(−1)l e−iωlτ =∞∑
l=0
e−i(ωτ+π)l =1
1− e−i(ωτ+π)=
11 + e−iωτ
, (B.21)
and the Matsubara sum has been carried out, one can convert τ → −iβ and
∆th(ω, τ) =i
2ω
11 + e−iωτ
[e−iωz0 − e−iω(τ−z0)
](B.22)
=i
2ω
11 + e−βω
[e−iωz0 − e−βωe+iωz0
]
=i
2ω
eβω
eβω + 1[e−iωz0 − e−βωe+iωz0
]
=i
2ω
[e−iωz0 − 1
eβω + 1e−iωz0 − 1
eβω + 1e+iωz0
].
Thus, the finite-temperature fermion propagator in the mixed representation becomes
∆(F )th (ω, τ = −iβ) =
i
2ω
[e−iωz0 − n(ω)
(e−iωz0 + e+iωz0
)], (B.23)
where the Fermi-Dirac distribution function n(ω) is given by
n(ω) =1
eβω + 1. (B.24)
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)th (ω, τ = −iβ) =
i
2ω
[e−iωz0 + n(ω)
(e−iωz0 + e+iωz0
)], (B.26)
where the Bose-Einstein distribution function n(ω) is
n(ω) =1
eβω − 1. (B.27)
Alternatively, the mixed representation could also be derived from the contour integral through a
contour with poles on the imaginary axis of q0 as
∆th(ω, τ = −iβ) =∫
dq0
2πi
1e−βq0 ∓ 1
e−iq0z0
ω2 − ω2n − iε
, (B.28)
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 − ∂νAµ − 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
∂µFµν = 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),
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,
fµAµ = 0 (C.4)
For various fµ [108],
∂µAµ = 0, Lorenz gauge (C.5)
∇ · ~A = ∂jAj = 0, Coulomb or radiation gauge (C.6)
nµAµ = 0, (n2 = 0), light-cone gauge (C.7)
A0 = 0, Hamiltionian or temporal axial gauge (C.8)
A3 = 0, axial gauge (C.9)
xµAµ = 0, Fock-Schwinger gauge (C.10)
xjAj = 0, Poincare 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 Dc(k) (or Dth(k) at finite temperature) is gauge-dependent. In
covariant gauge,
Dµν(k) =[gµν − ζ
kµkν
k2 − iε
]· 1k2 − iε
, (C.12)
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
k2 = ~k2 − k20. (C.13)
In configuration space [86],
Dµν(z) = 4π2
[δµν
[1− ς
2
]
z2 + iε+ ζ
zµzν
(z2 + iε)2
]. (C.14)
For the metric convention of gµν = (+1,−1,−1,−1, ),
Dµν(k) = −[gµν − ζ
kµkν
k2 − iε
]· 1k2 − iε
. (C.15)
In Coulomb gauge, ∂iAi = 0, the photon propagator becomes
Dµν(k) = −δµ0δ0ν1~k2
+ PTµν
1k2 − iε
+ ζCk2
(~k2)2kµkν
k2, (C.16)
where ζC is the Coulomb gauge parameter, PTµν is the transverse projection operator
PTij = δij − kikj
~k2= δij − kikj , PT
00 = PT0i = 0. (C.17)
The longitudinal projection operator PLµν is defined as
PLµν = Pµν − PT
µν , (C.18)
with
Pµν = gµν − kµkν
k2. (C.19)
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 L′ 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., 〈jµ〉Aν=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.,
∂µ〈jµ〉Aν= 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
jµ(x) = ig limx′−x→0
∑
α,β
γαβµ
(ψβ(x′)ψα(x)
)+
= ig limx′−x→0
∑
α,β
γαβµ
[ψβ(x′), ψα(x)
], (C.22)
where the points x′ and x are relatively space-like to maintain causality, i.e., ~x2 − x20 > 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
〈jµ(x)〉gA = − limx′−x→0
tr [γµGc(x, x′|gA)], (C.23)
and the closed-fermion-loop functional is then given by
Lc[A] = i
∫ g
0
dg′∫
d4xAµ 〈jµ(x)〉g′A (C.24)
withδLc[A]δAµ(x)
= ig〈jµ(x)〉gA. (C.25)
To avoid the singularity of Gc(x, x′|gA) at the limit of x′ → x, the definition of the current operator
〈jµ(x)〉gA can be modified to ensure [111, 112, 48]
∂yν
δ
δAν(y)〈jµ(x)〉gA = 0 (C.26)
for any y, which is equivalent to the current conservation of Eq. (C.21) under the translation
invariance. This imply
∂yν
δ
δAν(y)δ
δAµ(x)Lc[A] = 0, (C.27)
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
GF (x, y|A) = Gc(x, y|A) · exp[−ig
∫ x
y
dξν Aν(ξ)]
(C.29)
Since Gc[A] can be represented by exp i[∫
f ·A]as an expansion of functional differentiations. By
one integration-by-parts,
δ
δΛ(y)GF [A + ∂Λ] = −∂y
µ
δ
δAµ(y)GF [A + ∂Λ] (C.30)
for any Λ(y). By definition, GF [A + ∂Λ] is independent of Λ, which implies that
∂yν
δ
δAν(y)GF (x, x′|gA) = 0. (C.31)
Thus,
〈jµ(x)〉F = − limy→x
tr[γµGF (x, y|gA) · exp
[−iq
∫ x
y
dξν Aν(ξ)]]
, (C.32)
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,
L[A] =i
2
∫d4x
∫d4y Aµ(x)K(2)
µν (x− y) Aν(y) (C.33)
+i
4
∫d4x
∫d4y
∫d4z
∫d4w Aµ(x)Aν(y)K(4)
µνλσ(x, y, z, w) Aλ(z) Aσ(w) + · · · ,
and Eq. (C.27) leads to
∂µ K(2l)µν··· = 0. (C.34)
The momentum space representation is
kµ K(2l)µν··· = 0, (C.35)
which implies that the vacuum polarization tensors K(2l)µν··· are transverse.
C.4 Gauge Structure of Green’s Function(al)
Note that h−1(s1, s2) can be represented in terms of δ(s1, s2) as in Eq. (E.21),
exp
i
4
∫ s
0
ds1
∫ s
0
ds2 u(s1) · h−1(s1, s2) · u(s2)
(C.36)
= exp
i
4
∫ s
0
ds1
∫ s
0
ds2 u(s1) ·[
∂
∂s1
∂
∂s2δ(s1 − s2)
]· u(s2)
= exp
i
4
∫ s
0
ds1 u(s1) · ∂
∂s1
∫ s
0
ds2
[∂
∂s2δ(s1 − s2)
]· u(s2)
= exp
i
4
∫ s
0
ds1
∫ s
0
ds2 u′(s1) · δ(s1, s2) · u′(s2)
= exp
i
4
∫ s
0
ds′ u′2(s′)
,
118
where there are discarded surface terms u2(s)δ(0)+u2(0)δ(0)−2u(s)·u′(s)+2u(0)·u′(0) by imposing
0 < s1(or s2) < s. Similarly,∫ s
0
ds1
∫ s
0
ds2 u(s1) · h−1(s1, s2) ·∫ s2
0
ds′A(y − u(s′)) (C.37)
=∫ s
0
ds′ u(s′) · ∂
∂s′A(y − u(s′))
=∫ s
0
ds′ u′(s′) ·A(y − u(s′)).
Note that 0 < s1, s2 < s and take limε→0
∫ s−ε
0+εds1 · · ·, or
∫ s
0
ds1
∫ s
0
ds2 u(s1) · h−1(s1, s2) ·∫ s2
0
ds′A(y − u(s′)) (C.38)
=∫ s
0
ds′ u(s′) · ∂
∂s′A(y − u(s′))
=∫ s
0
ds′ uµ(s′) · ∂νAµ(y − u(s′)) · u′ν(s′).
The Green’s function can be simplified as
〈~x, x0|Gc[A]|~y, y0〉 = [m− γ · (∂x − igA(x))] · i∫ ∞
0
ds e−ism2 · e− 12Tr ln (2h) (C.39)
×∫
d[u](
exp
g
∫ s
0
ds′σ · F(y − u(s′)))
+
× exp
i
4
∫ s
0
ds′ [u′(s′)]2· exp
−ig
∫ s
0
ds′ u′(s′) ·A(y − u(s′))
×δ(x− y + u(s)).
Note γ·A(x) in [m− γ · (∂x − igA(x))] can be replaced by iγ· δδu′(s) , which is similar to
[m− γ · δ
δv(s)
]
before gauge transformation. Under gauge transformation, Aµ → Aµ + ∂µΛ, there is an extra term
linear in A like∫ s
0
ds′ u′µ(s′) · ∂µΛ(y − u(s′)) (C.40)
= −∫ s
0
ds′∂
∂s′Λ(y − u(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
〈~x, x0|Gc[A + ∂Λ]|~y, y0〉 = e−ig[Λ(x)−Λ(y)] · 〈~x, x0|Gc[A]|~y, y0〉. (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
Lc[A] = −12
∫ ∞
0
ds
se−ism2 · e− 1
2Tr ln (2h) (C.42)
×∫
d4x
∫d[u] δ(4)(u(s)) · tr
(exp
g
∫ s
0
ds′σ · F(x− u(s′)))
+
× exp
i
4
∫ s
0
ds1
∫ s
0
ds2 u(s1) · h−1(s1, s2) · u(s2)
× exp−ig
∫ s
0
ds1
∫ s
0
ds2 u(s1) · h−1(s1, s2) ·∫ s2
0
ds′A(x− u(s′))
−g = 0 ,
where the delta functional δ(4)(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∫ s
0
ds′ u′µ(s′) ·Aµ(x− u(s′)) (C.43)
→∫ s
0
ds′ u′µ(s′) ·Aµ(x− u(s′)) +∫ s
0
ds′ u′µ(s′) · ∂µΛ(x− u(s′)),
where∫ s
0
ds′ u′µ(s′) · ∂µΛ(x− u(s′)) (C.44)
= −∫ s
0
ds′∂
∂s′Λ(x− u(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:
exp[∫
fδ
δj
]· F [j] = F [j + f ] (D.1)
Functional in terms of operator representation:
F [j] = F
[1i
δ
δg
]· exp
[i
∫gj
]∣∣∣∣∣g→0
(D.2)
Quadratic (Gaussian) translation operator or linkage operator:
eD = exp[− i
2
∫δ
δjA
δ
δj
](D.3)
Quadratic (Gaussian) translation:
exp[− i
2
∫δ
δjA
δ
δj
]· exp
[i
∫jg
]= exp
[i
2
∫gAg + i
∫jg
](D.4)
Functional differential on a product of two functionals:
exp[− i
2
∫δ
δjA
δ
δj
]· F1[j] · F2[j] (D.5)
= F1[1i
δ
δg1] · F1[
1i
δ
δg2] · exp
[− i
2
∫δ
δjA
δ
δj
]· exp
[i
∫j(g1 + g2)
]∣∣∣∣g→0
= F1[1i
δ
δg1] · F1[
1i
δ
δg2] · exp
[i
2
∫g1Ag1 +
i
2
∫g2Ag2 + i
∫g1Ag2
]· exp
[i
∫j(g1 + g2)
]∣∣∣∣g→0
or with D = exp[− i
2
∫δδj A
δδj
]and D12 = exp
[−i
∫δ
δj1A δ
δj2
],
120
121
eD · F1[j]F2[j] = eD12[(
eD1F1[j1]) (
eD2F2[j2])]∣∣
j1=j2=j(D.6)
Quadratic (Gaussian) translation operator on the Gaussian functional:
exp[− i
2
∫δ
δjA
δ
δj
]· exp
[i
2
∫jBj
]= exp
[i
2
∫jB(1−AB)−1j − 1
2Tr ln(1−AB)
](D.7)
Quadratic (Gaussian) translation operator on the general Gaussian functional:
exp[− i
2
∫δ
δjA
δ
δj
]· exp
[i
2
∫jBj + i
∫fj
](D.8)
= exp[
i
2
∫jB(1−AB)−1j + i
∫j(1−BA)−1f +
i
2
∫fA(1−BA)−1f − 1
2Tr ln(1−AB)
]
Charged Bosons:
exp[−i
∫δ
δjA
δ
δj∗
]· exp
[i
∫j∗Bj
](D.9)
= exp[i
∫j∗B(1−AB)−1j −Tr ln(1−AB)
]
Fermions:
exp[−i
∫δ
δηαAαβ
δ
δηβ
]· exp
[i
∫ηµBµνην
](D.10)
= exp[i
∫ηB(1 + AB)−1η + Tr ln(1 + AB)
]
Proper time representation:
(A− iε)−1 = i
∫ ∞
0
ds e−is(A−iε) → i
∫ ∞
0
ds e−isA (D.11)
D.2 Functional Integration
Measure of functional integration (FI): u(x) → u(xi) = ui
∫d[u] = lim
N→∞
N∏
i=1
∫ +∞
−∞dui (D.12)
Functional representation of δ-functional:
δ[j − f ] = η−1
∫d[u] exp
[i
∫u(j − f)
], (D.13)
or
δ[j − f ] = limN→∞
η−1N∏
i=1
∫ +∞
−∞dui exp [i∆ui(ji − fi)]
= limN→∞
η−1
(2π
∆
)N N∏
i=1
δ(ji − fi), (D.14)
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):
F [j] = η−1
∫d[u] F [u] · exp
i
∫ju
(D.15)
Functional Integration of a Gaussian:
I[A] =∫
d[u] exp± i
2
∫u ·A · u
(D.16)
= C · exp−1
2Tr lnA
,
and
I[j; A] =∫
d[u] exp± i
2
∫u ·A · u + i
∫j · u
(D.17)
= C · exp∓ i
2
∫j ·A−1 · j − 1
2Tr lnA
,
where C is a divergent normalization constant and ∆ is the 4-volume of space-time cells
C = limN→∞
(2πi
∆
)N/2
= η. (D.18)
D.3 Functional Differentiation vs Functional Integral
Functional linkage operation can be recast into a Gaussian-weighted functional integral:
ei∫ s0 ds′ δ2
δv(s′)2 · F [v]∣∣∣∣v→0
= N∫
d[v]e(i/4)∫ s0 ds′v2(s′) · F [v], (D.19)
where
N−1 =∫
d[v] exp[(i/4)
∫ s
0
dsv2(s′)]. (D.20)
D.4 Two Useful Relations
Reciprocity Relation: for any functional F [A] of polynomials in A,
F [1i
δ
δj] · exp
[i
2
∫j ·D · j
]= exp
[i
2
∫j ·D · j
]· exp
[− i
2
∫δ
δA·D · δ
δA
]· F [A], (D.21)
where
A(x) =∫
dy D(x, y) · j(j). (D.22)
Gauge Formula: for arbitrary Λ(z),
δ
δΛ(z)F [A + ∂Λ] = −∂z
µ
δ
δAµ(z)F [A + ∂Λ]. (D.23)
123
D.5 Functional Form of Unity
1 =∫
d[u]δ[u− j], (D.24)
or for any functional
F [j] =∫
d[u]F [u] · δ[u− j]. (D.25)
D.6 Linkage Operation
DA = − i
2
∫dx
∫dy
δ
δAµ(x)·Dµν(x− y) · δ
δAν(y)(D.26)
D12 = −i
∫dx
∫dy
δ
δA(1)µ (x)
·Dµν(x− y) · δ
δA(2)ν (y)
(D.27)
eDA ·Aα(z)F [A] = eD12
[eDA(1) A(1)
α (z)] [
eDA(2)F [A(2)]]
(D.28)
eDAAα(z) =[1 +DA +
12!DA · DA + · · ·
]Aα(z) = Aα(z) (D.29)
eD12A(1)α (z) =
[1 +D12 +
12!D12 · D12 + · · ·
]A(1)
α (z) (D.30)
= [1 +D12] A(1)α (z)
= A(1)α (z)− i
∫dy Dαν(z − y) · δ
δA(2)ν (y)
eDA ·Aα(z)F [A] =[Aα(z)− i
∫dy Dαν(z − y) · δ
δAν(y)
] [eDAF [A]
](D.31)
In momentum and Matsubara space, the linkage operator is given by
DA = − i
2
∫d4k
(2π)4
∫d4k′
(2π)4δ
δAµ(k)·Dµν(k − k′) · δ
δAν(k′)(D.32)
= − i
2
∫d4k
(2π)4δ
δAµ(k)·Dµν(k) · δ
δAν(−k),
where Dµν(k − k′) = (2π)4 Dµν(k) δ(k + k′), and the integral for the Matsubara representation is
∫d4k
(2π)4→ 1
τ
∑ ∫d3~k
(2π)3(D.33)
with Dthµν(k − k′) = (2π)3τ Dth
µν(k) δ(~k + ~k′) δl,−l′ .
eDAAα(q)F [A] =[Aα(q)− i
∫d4k
(2π)4Dαν(q − k)
δ
δAν(k)
]· eDAF [A] (D.34)
124
eDAAα(q)F [A] =
[Aα(q)− i
1τ
∑
l
∫d3~k
(2π)3Dth
αν(q − k)δ
δAν(k)
]· eDAF [A], (D.35)
where Dµν(q − k) = (2π)4 Dµν(q) δ(q + k).
eDAAα(q)F [A] =[Aα(q)− iDαν(q)
δ
δAν(−q)
]· eDAF [A], (D.36)
or Dthµν(q − k) = (2π)3τ Dth
µν(q) δ(q + k) and δ(q + k) = δ(~q + ~k) δl′,−l,
eDAAα(q)F [A] =[Aα(q)− iDth
αν(q)δ
δAν(−q)
]· eDAF [A]. (D.37)
Appendix E
Misc Relations
E.1 Useful Relations in Fradkin’s Representation
exp i∫ s
0
ds′δ2
δv2(s′) · exp ip ·
∫ s
0
ds′v(s′)F [v] (E.1)
= exp −isp2 · exp ip ·∫ s
0
ds′v(s′) · exp i∫ s
0
ds′δ2
δv2(s′)F [v − 2p]
m− γ · δ
δv(s) · exp ip ·
∫ s
0
ds′v(s′)F [v] (E.2)
= exp ip ·∫ s
0
ds′v(s′) · m− iγ ·[p− i
δ
δv(s)
]F [v]
E.2 Representations of Delta- and Heaviside Step- Function
For 0 < s1, s2 < s, δ-function in the Fourier series
δ(s1 − s2) =2s
∑
N
′sin
(Nπs1
2s
)· sin
(Nπs2
2s
), (E.3)
δ(s1 − s2) =2s
∑
N
′cos
(Nπs1
2s
)· cos
(Nπs2
2s
), (E.4)
where the prime on the sum indicates that N runs over only odd integers as
∑
N
′=
∞∑
N=1,3,5,···. (E.5)
For ε > 0, Lorentzian representation:
δ(x) = limε→0
1π
ε
x2 + ε2= lim
ε→0
12π
∫ +∞
−∞dk eikx−ε|k| (E.6)
125
126
δ(x) = limε→0
e−|x|/ε
2ε= lim
ε→0
12π
∫ +∞
−∞dk
eikx
1 + ε2k2(E.7)
Gaussian representation or limit of normal distribution:
δ(x) = limε→0
1ε√
πe−
x2
ε2 (E.8)
δ(x) =12π
∫ +∞
−∞dk e−ikx (E.9)
Diffraction peak representation:
δ(x) = limε→0
1πx
sin(x
ε
)= lim
ε→0
12π
∫ +1/ε
−1/ε
dk cos(kx) (E.10)
δ(x) = limε→0
ε
πx2sin2
(x
ε
)(E.11)
Derivative of the sigmoid (or Fermi-Dirac) function
δ(x) = limε→0
∂x1
1 + e−x/ε= − lim
ε→0∂x
11 + ex/ε
(E.12)
Limit of a rectangular function
δ(x) = limε→0
1ε
rect(x
ε
)= lim
ε→0
12π
∫ +∞
−∞dk sinc
(εk
2π
)e−ikx (E.13)
Limit of the Airy function
δ(x) = limε→0
1ε
Ai
(x
ε
)(E.14)
Limit of a Bessel function
δ(x) = limε→0
1ε
J1/ε
(x + 1
ε
)(E.15)
Representation of Heaviside step function or θ-function,
θ(t) = limε→0+
i
∫ +∞
−∞
dk0
2π
e−ik0t
k0 + iε(E.16)
E.3 Representation of h Function
Definition:
h(s1, s2) =∫ s
0
ds′ θ(s1 − s′)θ(s2 − s′) =12
[(s1 + s2) + |s1 − s2|] , (E.17)
and the h-function is symmetric in its parameters as
h(s1, s2) =∫ s
0
ds′ θ(s1 − s′)θ(s2 − s′) = h(s2, s1). (E.18)
The h-function can be expended in series of sine and cosine functions as
h(s1, s2) =8s
π2
∑
N
′ 1N2
cos(
Nπs1
2s
)· cos
(Nπs2
2s
)(E.19)
127
and its inverse is given by
h−1(s1, s2) =π2
2s3
∑
N
′N2 sin
(Nπs1
2s
)· sin
(Nπs2
2s
)(E.20)
Compare with the second representation of the delta function, Equation (E.3),
h−1(s1, s2) =∂
∂s1
∂
∂s2δ(s1 − s2) (E.21)
∫ s
0
dsi si · h−1(si, sj) = δ(s− sj) (E.22)
∫ s
0
ds1
∫ s
0
ds2 s1 · h−1(s1, s2) · s2 = s (E.23)
∫ s
0
ds′1
∫ s
0
ds′2 θ(s′1 − s1)h−1(s′1, s′2) θ(s′2 − s2) = δ(s1 − s2) (E.24)
∫ s
0
ds′1 θ(s′1 − s1) h−1(s′1, s′2) = − ∂
∂s2δ(s1 − s2) (E.25)
E.4 Operator Relations
If [A, [A,B]] = [B, [A,B]] = 0,
eA+B = eAeBe−[A,B]/2, (E.26)
eλBAe−λB = A + λ[B,A] +λ2
2![B, [B, A]] + · · · (E.27)
E.5 Legendre Function of the Second Kind
The Legendre differential equation is of second-order
(1− x2)d2Q
dx2− 2x
dQ
dx+ l(l + 1)Q = 0 (E.28)
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
Q0(x) =12
ln(
1 + x
1− x
)(E.29)
Q1(x) =x
2ln
(1 + x
1− x
)− 1 (E.30)
Q2(x) =3x2 − 1
4ln
(1 + x
1− x
)− 3x
2(E.31)
Q3(x) =5x3 − 3x
4ln
(1 + x
1− x
)− 5x2
2+
23. (E.32)
128
First few of Pl(x) are
P0(x) = 1 (E.33)
P1(x) = x (E.34)
P2(x) =12
(3x2 − 1) (E.35)
P3(x) =12
(5x3 − 3x) (E.36)
P4(x) =18
(35x4 − 30x2 + 3) (E.37)
P5(x) =18
(63x5 − 70x3 + 15x). (E.38)
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]∫ s
0
ds1
∫ s
0
ds2 F (s1, s2) = 2∫ s
0
ds1
∫ s1
0
ds2 F (s1, s2) (E.40)
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 uni-
tary 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 operators a(k) and a† of bosons obey the canonical commutation relation,
[a(k), a†(k)] = 1, (E.41)
then the correspondent operators, b(k) and b†, of a Bogoliubov transformation also have to satisfy
the canonical commutation relation, i.e.,
[b(k), b†(k)] = 1 (E.42)
Let the transformation be
b(k) = ua(k) + va†(k), (E.43)
b†(k) = u∗a†(k) + v∗a(k), (E.44)
which yield
[b(k), b†(k)] = (|u|2 − |v|2) [a(k), a†(k)], (E.45)
129
To ensure the transformation is canonical, the commutation relation of b(k) and b† 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]
b(k) =√
1 + n(k0)a(k) +√
n(k0)a†(k), (E.48)
b†(k) =√
1 + n(k0)a†(k) +√
n(k0)a(k), (E.49)
for bosons, where k0 =√
~k2 + m2.
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
〈~p, n|S′th|~y, y0〉BNQuenched (F.1)
= Z−1(0)[iτ ] · [(2π)3τ
]−1/2 · e−i(~p·~y−ωny0) · i∫ ∞
0
ds e−is(m2+p2)
×
m− iγ · p + 2ig2∑
l
∫d3~k
(2π)3τ
[γ · DBN
th (k) · p] ∫ s
0
ds′ e2ik·p(s−s′)
× exp
4ig2
∑
l
∫d3~k
(2π)3τ
[p · DBN
th (k) · p] ∫ s
0
ds1
∫ s1
0
ds′ e2is′k·p
.
First use the Feynman gauge in photon propagator,
DBNth (~k, ωl) =
δµν
~k2 − ω2l − iε
· e−iα~k2∣∣∣∣α=− i
~p2
=δµν
~k2 − ω2l − iε
· e−~k2
~p2 , (F.2)
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
4ig2∑
l
∫d3~k
(2π)3τ
[p · DBN
th (k) · p] ∫ s
0
ds1
∫ s1
0
ds′ e2is′k·p (F.3)
= −ig2 p2
π32
∫ s
0
ds1
∫ s1
0
ds′∫ ∞
|~p|dq e−s′2~p 2q2 1
τ
∑
l
e−2is′ωlωn+( 1
q2− 1~p 2 )ω2
l ,
where the leading sign is minus. While q2 > ~p 2, the factor 1q2 − 1
~p 2 < 0 in the exponent, and it is
relatively small, i.e. ∣∣∣∣(
1q2− 1
~p 2
)ω2
l
∣∣∣∣ < s′2~p 2q2, (F.4)
130
131
and the contribution from exp [(1/q2 − 1/~p 2)ω2l ] can be dropped. It leads to
4ig2∑
l
∫d3~k
(2π)3τ
[p · DBN
th (k) · p] ∫ s
0
ds1
∫ s1
0
ds′ e2ik·ps′ (F.5)
' −ig2 p2
π32
∫ s
0
ds1
∫ s1
0
ds′∫ ∞
|~p|dq e−s′2~p 2q2 1
τ
∑
l
e−2is′ωlωn .
The Matsubara sum over l is similar to the previous work of Candelpergher, Fried and Grandou [47]
as
1τ
∑
l
e−2is′ωnωl =∑
l′Θ
(l′
n
)δ(l′τ − 2s′ωn), (F.6)
and the exponent becomes
4ig2∑
l
∫d3~k
(2π)3τ
[p · DBN
th (k) · p] ∫ s
0
ds1
∫ s
0
ds2 e2ik·p(s1−s2) (F.7)
' −ig2
2π32
p2
ωn
∑
l′Θ
(l′
n
) (s− l′τ
2ωn
)Θ
(s− l′τ
2ωn
) ∫ ∞
|~p|dq e
−(
l′τ2ωn
)2~p 2q2
.
Similar evaluation can apply to the linear or denominator factor in the dressed, finite-temperature
propagator as
2ig2∑
l
∫d3~k
(2π)3τ
[γ · DBN
th (k) · p] ∫ s
0
ds′ e2ik·p(s−s′) (F.8)
= − 2(2π)3τ
(π
i
)− 32
g2 (γ · p)∫ s
0
ds′∑
l
e−2is′ωlωn
∫ ∞−i|α|
0−i|α|dt′′ t′′−
32 ei s′2
t′′ ~p 2+i(t′+i|α|)ω2l
∣∣∣∣|α|= 1
~p 2
=√
π
2(2π)3g2 (γ · p)
|~p|nT
∑
l′Θ
(l′
n
) ∫ 1
0
dλ e−
(l′
4πnT2
)2(~p 2)2λ2
.
Thus, the dressed, finite-temperature fermion propagator becomes
〈~p, n|S′th|~y, y0〉BNQuenched (F.9)
' Z−1(0)[iτ ] · [(2π)3τ
]−1/2 · e−i(~p·~y−ωny0) · i∫ ∞
0
ds e−is(m2+p2)
×
m− iγ · p +√
π
2(2π)3g2 (γ · p)
|~p|nT
∑
l′Θ
(l′
n
) ∫ 1
0
dλ e−
(l′
4πnT2
)2(~p 2)2λ2
× exp
+
g2
4π5/2
p2
nT|~p|
∫ 1
0
dλ∑
l′Θ
(l′
n
)(s +
l′
4πnT 2
)e−
(l′
4πnT2
)2(~p 2)2λ2
.
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,
〈~p, n|S′th|~y, y0〉BNQuenched (F.10)
' Z−1(0)[iτ ] · [(2π)3τ
]−1/2 · e−i(~p·~y−ωny0) · i∫ ∞
0
ds e−is(m2+p2)
×
m− iγ · p +√
π
2(2π)3g2 (γ · p)
|~p|nT
∑
l′Θ
(l′
n
) ∫ 1
0
dλ e−
(l′
4πnT2
)2(~p 2)2λ2
× exp
− mg2
4π5/2
|~p|nT
∫ 1
0
dλ
n∑
l′=−n
Θ(
l′
n
) (s +
l′
4πnT 2
)e−
(l′
4πnT2
)2(~p 2)2λ2
,
which the sum over l′ 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
δDµνth (k) = 2πi
δµν
eβ|k0| − 1δ(~k2 − k2
0). (G.1)
The last exponent in Eq. (3.77) becomes
4ig2
∫d4k
(2π)4
∫ s
0
ds1
∫ s1
0
ds′[p · δDth(k) · p
]e2ik·ps′ (G.2)
= 4ig2p2
∫d4k
(2π)4
∫ s
0
ds1
∫ s1
0
ds′[
2πi
eβ|k0| − 1δ(~k2 − k2
0)]
e2ik·ps′
= − g2
2π3(~p2 − ω2
n)∫ s
0
ds1
∫ s1
0
ds′∫
d3~k
∫dk0
δ(~k2 − k20)
eβ|k0| − 1e2i~k·~ps′−2ik0ωns′ .
Applying similar approximations used in Coulomb gauge,
4ig2
∫d4k
(2π)4
∫ s
0
ds1
∫ s1
0
ds′[p · δDth(k) · p
]e2ik·ps′ (G.3)
' −sg2
2π2
~p2 − ω2n
βp
2sp
β
∫ 1
0
dxx
ex − 1
= −s2 g2
π2
~p2 − ω2n
β2
∫ 1
0
dxx
ex − 1
= − c
π2s2g2T 2 (~p2 − ω2
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.
133
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