Basics of thermal field theory - a tutorial on perturbative computations

1.6 MB·

by  Mikko Laine

December 2016
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Basics of Thermal Field Theory
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A Tutorial on Perturbative Computations 1
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h Mikko Lainea and Aleksi Vuorinenb
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p aAEC, Institute for Theoretical Physics, University of Bern,
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h Sidlerstrasse 5, CH-3012 Bern, Switzerland
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bDepartment of Physics, University of Helsinki,
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v P.O. Box 64, FI-00014 University of Helsinki, Finland
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Abstract
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1 These lecture notes, suitable for a two-semesterintroductory course or self-study, offer an elemen-
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taryandself-containedexpositionofthebasictoolsandconceptsthatareencounteredinpractical
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1 computations in perturbative thermal field theory. Selected applications to heavy ion collision
: physics and cosmology are outlined in the last chapter.
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1A corresponding ebook has been published as Springer Lecture Notes in Physics 925 (2016) and is available
throughhttp://dx.doi.org/10.1007/978-3-319-31933-9.

Contents
Foreword . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i
Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
General outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
1 Quantum mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Path integral representation of the partition function . . . . . . . . . . . . . 1
1.2 Evaluation of the path integral for the harmonic oscillator . . . . . . . . . . 6
2 Free scalar fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.1 Path integral for the partition function. . . . . . . . . . . . . . . . . . . . . 13
2.2 Evaluation of thermal sums and their low-temperature limit . . . . . . . . . 16
2.3 High-temperature expansion. . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3 Interacting scalar fields. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.1 Principles of the weak-coupling expansion . . . . . . . . . . . . . . . . . . . 30
3.2 Problems of the naive weak-coupling expansion . . . . . . . . . . . . . . . . 38
3.3 Proper free energy density to (λ): ultraviolet renormalization . . . . . . . 40
O
3.4 Proper free energy density to (λ32): infrared resummation . . . . . . . . . 44
O
4 Fermions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.1 Path integral for the partition function of a fermionic oscillator . . . . . . . 49
4.2 The Dirac field at finite temperature . . . . . . . . . . . . . . . . . . . . . . 53
5 Gauge fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
5.1 Path integral for the partition function. . . . . . . . . . . . . . . . . . . . . 61
5.2 Weak-coupling expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
5.3 Thermal gluon mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
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5.4 Free energy density to (g3) . . . . . . . . . . . . . . . . . . . . . . . . . . 79
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6 Low-energy effective field theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
6.1 The infrared problem of thermal field theory . . . . . . . . . . . . . . . . . 84
6.2 Dimensionally reduced effective field theory for hot QCD . . . . . . . . . . 90
7 Finite density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
7.1 Complex scalar field and effective potential . . . . . . . . . . . . . . . . . . 98
7.2 Dirac fermion with a finite chemical potential . . . . . . . . . . . . . . . . . 104
8 Real-time observables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
8.1 Different Green’s functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
8.2 From a Euclidean correlator to a spectral function . . . . . . . . . . . . . . 123
8.3 Real-time formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
8.4 Hard Thermal Loops . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
9 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
9.1 Thermal phase transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
9.2 Bubble nucleation rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
9.3 Particle production rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
9.4 Embedding rates in cosmology . . . . . . . . . . . . . . . . . . . . . . . . . 176
9.5 Evolution of a long-wavelengthfield in a thermal environment. . . . . . . . 185
9.6 Linear response theory and transport coefficients . . . . . . . . . . . . . . . 190
9.7 Equilibration rates / damping coefficients . . . . . . . . . . . . . . . . . . . 199
9.8 Resonances in medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
Appendix: Extended Standard Model in Euclidean spacetime . . . . . . . . . . . . . . . 215
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
3
Foreword
These notes are based on lectures delivered at the Universities of Bielefeld and Helsinki, between
2004and2015,aswellasatanumberofsummerandwinterschools,between1996and2015. The
earlysections were stronglyinfluenced by lectures by Keijo Kajantieat the University of Helsinki,
inthe early1990s. Obviously,the lecturesadditionallyoweanenormousgratitudetoexisting text
books and literature, particularly the classic monographby Joseph Kapusta.
There are several good text books on finite-temperature field theory, and no attempt is made
here to join that group. Rather, the goal is to offer an elementary exposition of the basics of
perturbative thermal field theory, in an explicit “hands-on” style which can hopefully more or
less directly be transported to the classroom. The presentation is meant to be self-contained and
display also intermediate steps. The idea is, roughly, that each numbered section could constitute
asinglelecture. Referencingis sparse;onmoreadvancedtopics,aswellasonhistoricallyaccurate
references, the reader is advised to consult the text books and review articles in refs. [0.1]–[0.13].
Thesenotescouldnothavebeenputtogetherwithoutthehelpfulinfluenceofmanypeople,vary-
ing from students with persistent requests for clarification; colleagues who have used parts of an
early version of these notes in their own lectures and shared their experiences with us; colleagues
whose interest in specific topics has inspired us to add corresponding material to these notes;
alert readers who have informed us about typographic errors and suggested improvements; and
collaborators from whom we have learned parts of the material presented here. Let us gratefully
acknowledgeinparticularGertAarts,ChrisKorthalsAltes,DietrichB¨odeker,YannisBurnier,Ste-
fanoCapitani,SimonCaron-Huot,JacopoGhiglieri,IoanGhisoiu,KeijoKajantie,AleksiKurkela,
Harvey Meyer, Guy Moore, Paul Romatschke, Kari Rummukainen, York Schr¨oder, Mikhail Sha-
poshnikov, Markus Thoma, and Mikko Vepsa¨l¨ainen.
Mikko Laine and Aleksi Vuorinen
i
Notation
In thermal field theory, both Euclidean and Minkowskian spacetimes play a role.
In the Euclidean case, we write
X (τ,xi), x x , S = L , (0.1)
≡ ≡| | E E
ZX
where i=1,...,d,
β 1
dτ , ddx, β , (0.2)
≡ ≡ ≡ T
ZX Z0 Zx Zx Z
and d is the space dimensionality. Fourier analysis is carried out in the Matsubara formalism via
K (k ,k ), k k , φ(X)= φ˜(K)eiKX , (0.3)
n i ·
≡ ≡| |
ZK
P
where
ddk
T , . (0.4)
≡ ≡ (2π)d
PZK Xkn Zk Zk Z
Here, k stands for discrete Matsubarafrequencies, which at times are also denoted by ω . In the
n n
case of antiperiodic functions, the summation is written as T . The squares of four-vectors
read K2 = k2 +k2 and X2 = τ2 +x2, but the Euclidean scal{akrn}product between K and X is
n P
defined as
d
K X =k τ + k xi =k τ k x, (0.5)
n i n
· − ·
i=1
X
where the vector notation is reserved for contravariant Minkowskian vectors: x = (xi), k = (ki).
If a chemical potential is also present, we denote k˜ k +iµ.
n n
≡
In the Minkowskian case, we have
(t,x), x x , = , (0.6)
X ≡ ≡| | SM LM
ZX
where dx0 . Fourier analysis proceeds via
≡ x
X
R R R (k0,k), k k , φ( )= φ˜( )eiK·X , (0.7)
K≡ ≡| | X K
ZK
where = dk0 , and the metric is chosen to be of the “mostly minus” form,
2π k
K
R R R =k0x0 k x. (0.8)
K·X − ·
No special notation is introduced for the case where a Minkowskian four-vector is on-shell, i.e.
when =(E ,k); this is to be understood from the context.
k
K
The argument of a field φ is taken to indicate whether the configuration space is Euclidean or
Minkowskian. If not specified otherwise, momentum integrations are regulated by defining the
spatial measure in d = 3 2ǫ dimensions, whereas the spacetime dimensionality is denoted by
−
D =4 2ǫ. A Greek index takes values in the set 0,...,d , and a Latin one in 1,...,d .
− { } { }
Finally,wenotethatweworkconsistentlyinunitswherethespeedoflightcandtheBoltzmann
constant k have been set to unity. The reduced Planck constant ~ also equals unity in most
B
places,excludingthe firstchapter(onquantummechanics)aswellassomelaterdiscussionswhere
we want to emphasize the distinction between quantum and classical descriptions.
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General outline
Physics context
Fromthephysicspointofview,therearetwoimportantcontextsinwhichrelativisticthermalfield
theory is being widely applied: cosmology and the theoretical description of heavy ion collision
experiments.
In cosmology, the temperatures considered vary hugely, ranging from T 1015 GeV to T
≃ ≃
10 3 eV. Contemporary challenges in the field include figuring out explanations for the existence
−
of dark matter, the observed antisymmetry in the amounts of matter and antimatter, and the
formation of large-scale structures from small initial density perturbations. (The origin of initial
density perturbations itself is generally considered to be a non-thermal problem, associated with
an early period of inflation.) An important further issue is that of equilibration, i.e. details of the
processesthroughwhichtheinflationarystateturnedintoathermalplasma,andinparticularwhat
the highesttemperature reachedduring this epoch was. It is notable that most ofthese topics are
assumed to be associated with weak or even superweak interactions, whereas strong interactions
(QCD) only play a background role. A notable exception to this is light element nucleosynthesis,
but this well-studied topic is not in the center of our current focus.
In heavy ion collisions, in contrast, strong interactions do play a major role. The lifetime of the
thermalfireballcreatedinsuchacollisionis 10fm/candthe maximaltemperaturereachedisin
∼
the range of a few hundred MeV. Weak interactions are too slow to take place within the lifetime
of the system. Prominent observables are the yields of different particle species, the quenching
of energetic jets, and the hydrodynamic properties of the plasma that can be deduced from the
observedparticleyields. Animportantissue isagainhowfastaninitial quantum-mechanicalstate
turns into an essentially incoherent thermal plasma.
Despite many differences in the physics questions posed and in the microscopic forces underly-
ing cosmology and heavy ion collision phenomena, there are also similarities. Most importantly,
gaugeinteractions(whether weak or strong)areessentialin both contexts. Because ofasymptotic
freedom, the strong interactions of QCD also become “weak” at sufficiently high temperatures. It
isforthisreasonthatmanytechniques,suchastheresummationsthatareneededfordevelopinga
formally consistent weak-coupling expansion, can be applied in both contexts. The topics covered
in the present notes have been chosen with both fields of application in mind.
Organization of these notes
The notes start with the definition and computation of basic “static” thermodynamic quantities,
such as the partition function and free energy density, in various settings. Considered are in turn
quantum mechanics (sec. 1), free and interacting scalar field theories (secs. 2 and 3, respectively),
fermionic systems (sec. 4), and gauge fields (sec. 5). The main points of these sections include
the introduction of the so-called imaginary-time formalism; the functioning of renormalization at
finite temperature; and the issue of infrared problems that complicates almost every computation
in relativistic thermal field theory. The last of these issues leads us to introduce the concept of
effectivefieldtheories(sec.6),afterwhichweconsiderthe changescausedby the introductionofa
finite density or chemicalpotential(sec.7). After these topics, we moveonto a new setof observ-
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ables,so-calledreal-timequantities,whichplayanessentialroleinmanymodernphenomenological
applicationsofthermalfieldtheory(sec.8). Inthe finalchapterofthe book,a numberofconcrete
applications of the techniques introduced are discussed in some detail (sec. 9).
We note that secs. 1–7 are presented on an elementary and self-contained level and require no
backgroundknowledgebeyondstatisticalphysics,quantum mechanics,andrudiments ofquantum
fieldtheory. Theycouldconstitutethecontentsofaone-semesterbasicintroductiontoperturbative
thermalfieldtheory. Insec.8,thelevelincreasesgradually,andpartsofthediscussioninsec.9are
already close to the research level, requiring more background knowledge. Conceivably the topics
of secs. 8 and 9 could be covered in an advanced course on perturbative thermal field theory, or
in a graduate student seminar. In addition the whole book is suitable for self-study, and is then
advised to be read in the order in which the material has been presented.
Recommended literature
Apedagogicalpresentationofthermalfieldtheory,concentratingmostlyonEuclideanobservables
andtheimaginary-timeformalism,canbefoundinref.[0.1]. Thecurrentnotesborrowsignificantly
from this classic treatise.
Inthermalfieldtheory,thecommunityissomewhatdividedbetweenthosewhofindtheimaginary-
time formalism more practicable, and those who prefer to use the so-called real-time formalism
from the beginning. Particularly for the latter community, the standard reference is ref. [0.2],
which also contains an introduction to particle production rate computations.
A modern textbook, partly an update of ref. [0.1] but including also a full account of real-time
observables, as well as reviews on many recent developments, is provided by ref. [0.3].
Lecture notes on transportcoefficients, infraredresummations, and non-equilibriumphenomena
suchasthermalization,canbefoundinref.[0.4]. Reviewswithvaryingfociareofferedbyrefs.[0.5]–
[0.12].
Finally, an extensive review of current efforts to approach a non-perturbative understanding of
real-time thermal field theory has been presented in ref. [0.13].
iv
Literature
[0.1] J.I. Kapusta, Finite-temperature Field Theory (Cambridge University Press, Cambridge,
1989).
[0.2] M. Le Bellac, Thermal Field Theory (Cambridge University Press, Cambridge, 2000).
[0.3] J.I. Kapusta and C. Gale, Finite-Temperature Field Theory: Principles and Applications
(Cambridge University Press, Cambridge, 2006).
[0.4] P. Arnold, Quark-Gluon Plasmas and Thermalization, Int. J. Mod. Phys. E 16 (2007) 2555
[0708.0812].
[0.5] V.A. Rubakov and M.E. Shaposhnikov, Electroweak Baryon Number Non-Conservation in
the Early Universe and in High-Energy Collisions, Usp. Fiz. Nauk 166 (1996) 493 [Phys.
Usp. 39 (1996) 461] [hep-ph/9603208].
[0.6] L.S.BrownandR.F.Sawyer,Nuclear reaction rates in a plasma, Rev.Mod.Phys.69(1997)
411 [astro-ph/9610256].
[0.7] J.P. Blaizot and E. Iancu, The Quark-Gluon Plasma: Collective Dynamics and Hard Ther-
mal Loops, Phys. Rept. 359 (2002) 355 [hep-ph/0101103].
[0.8] D.H. Rischke, The Quark-Gluon Plasma in Equilibrium, Prog. Part. Nucl. Phys. 52 (2004)
197 [nucl-th/0305030].
[0.9] U. Kraemmer and A. Rebhan, Advances in perturbative thermal field theory, Rept. Prog.
Phys. 67 (2004) 351 [hep-ph/0310337].
[0.10] S. Davidson, E. Nardi and Y. Nir, Leptogenesis, Phys. Rept. 466 (2008) 105 [0802.2962].
[0.11] D.E.Morrisseyand M.J.Ramsey-Musolf,Electroweak baryogenesis, New J. Phys.14(2012)
125003[1206.2942].
[0.12] J. Ghiglieri and D. Teaney, Parton energy loss and momentum broadening at NLO in high
temperature QCD plasmas, Int. J. Mod. Phys. E 24 (2015) 1530013 [1502.03730].
[0.13] H.B. Meyer, Transport Properties of the Quark-Gluon Plasma: A Lattice QCD Perspective,
Eur. Phys. J. A 47 (2011) 86 [1104.3708].
v
1. Quantum mechanics
Abstract: After recallingsomebasicconceptsofstatisticalphysicsandquantummechanics,the
partition function of a harmonic oscillator is defined and evaluated in the standard canonical for-
malism. Animaginary-timepathintegralrepresentationissubsequentlydevelopedforthepartition
function, the path integral is evaluated in momentum space, and the earlier result is reproduced
upon a careful treatment of the zero-mode contribution. Finally, the concept of 2-point functions
(propagators)is introduced, and some of their key properties are derived in imaginary time.
Keywords: Partition function, Euclidean path integral, imaginary-time formalism, Matsubara
modes, 2-point function.
1.1. Path integral representation of the partition function
Basic structure
The properties of a quantum-mechanical system are defined by its Hamiltonian, which for non-
relativistic spin-0 particles in one dimension takes the form
pˆ2
Hˆ = +V(xˆ), (1.1)
2m
where m is the particle mass. The dynamics of the states ψ is governed by the Schro¨dinger
| i
equation,
∂
i~ ψ =Hˆ ψ , (1.2)
∂t| i | i
whichcanformally be solvedin terms of a time-evolution operator Uˆ(t;t ). This operatorsatisfies
0
the relation
ψ(t) =Uˆ(t;t )ψ(t ) , (1.3)
0 0
| i | i
and for a time-independent Hamiltonian takes the explicit form
Uˆ(t;t0)=e−~iHˆ(t−t0) . (1.4)
It is useful to note that in the classical limit, the system of eq. (1.1) can be described by the
Lagrangian
1
= = mx˙2 V(x), (1.5)
L LM 2 −
which is related to the classical version of the Hamiltonian via a simple Legendre transform:
∂ p2
p LM , H =x˙p = +V(x). (1.6)
≡ ∂x˙ −LM 2m
Returning to the quantum-mechanicalsetting, various bases can be chosenfor the state vectors.
The so-called x -basis satisfies the relations
| i
xxˆx =x xx =xδ(x x), xpˆx = i~∂ xx = i~∂ δ(x x), (1.7)
′ ′ ′ ′ x ′ x ′
h | | i h | i − h | | i − h | i − −
whereas in the energy basis we simply have
Hˆ n =E n . (1.8)
n
| i | i
1