Table Of ContentLecture Notes in Physics 925
Mikko Laine
Aleksi Vuorinen
Basics of
Thermal
Field Theory
A Tutorial on Perturbative
Computations
Lecture Notes in Physics
Volume 925
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Mikko Laine (cid:129) Aleksi Vuorinen
Basics of
Thermal Field Theory
A Tutorial on Perturbative Computations
123
MikkoLaine AleksiVuorinen
AEC,InstituteforTheoreticalPhysics DepartmentofPhysics
UniversityofBern UniversityofHelsinki
Bern,Switzerland Helsinki,Finland
ISSN0075-8450 ISSN1616-6361 (electronic)
LectureNotesinPhysics
ISBN978-3-319-31932-2 ISBN978-3-319-31933-9 (eBook)
DOI10.1007/978-3-319-31933-9
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Preface
These notes are based on lectures delivered at the Universities of Bielefeld and
Helsinki, between 2004 and 2015, as well as at a number of summer and winter
schools, between 1996 and 2015. The early sections were strongly influenced
by lectures by Keijo Kajantie at the University of Helsinki, in the early 1990s.
Obviously,thelecturesadditionallyoweanenormousgratitudetoexistingtextbooks
andliterature,particularlytheclassicmonographbyJosephKapusta.
There are several good textbooks on finite-temperature field theory, and no
attemptis made here to join that group.Rather, the goalis to offeran elementary
expositionofthebasicsof perturbativethermalfield theory,in an explicit“hands-
on”stylewhichcanhopefullymoreorlessdirectlybetransportedtotheclassroom.
Thepresentationismeanttobeself-containedanddisplayalsointermediatesteps.
The idea is, roughly,thateach numberedsection couldconstitute a single lecture.
Referencingissparse;onmoreadvancedtopics,aswellasonhistoricallyaccurate
references, the reader is advised to consult the textbooks and review articles in
Refs.[1–12].
These notes could not have been put together without the helpful influence
of many people, varying from students with persistent requests for clarification;
colleagues who have used parts of an early version of these notes in their own
lecturesandsharedtheirexperienceswithus;colleagueswhoseinterestinspecific
topics has inspired us to add corresponding material to these notes; alert readers
whohaveinformedusabouttypographicerrorsandsuggestedimprovements;and
collaboratorsfromwhomwehavelearnedpartsofthematerialpresentedhere.Let
us gratefullyacknowledgein particular Gert Aarts, Chris Korthals Altes, Dietrich
Bödeker, Yannis Burnier, Stefano Capitani, Simon Caron-Huot, Jacopo Ghiglieri,
Ioan Ghisoiu, Keijo Kajantie, Aleksi Kurkela, Harvey Meyer, Guy Moore, Paul
Romatschke, Kari Rummukainen, York Schröder, Mikhail Shaposhnikov,Markus
Thoma,andMikkoVepsäläinen.
Bern,Switzerland MikkoLaine
Helsinki,Finland AleksiVuorinen
January2016
v
Notation
Inthermalfieldtheory,bothEuclideanandMinkowskianspacetimesplayarole.
IntheEuclideancase,wewrite
Z
X (cid:2).(cid:2);xi/; x(cid:2)jxj; S D L ; (1)
E E
X
whereiD1;:::;d,
Z Z Z Z Z
ˇ 1
(cid:2) d(cid:2) ; (cid:2) ddx; ˇ (cid:2) ; (2)
X 0 x x T
andd is the space dimensionality.Fourieranalysisis carriedoutin the Matsubara
formalismvia
Z
P
K (cid:2).k ;k/; k(cid:2)jkj; (cid:3).X/D (cid:3)Q.K/eiK(cid:2)X ; (3)
n i
K
where
Z Z Z Z
P X ddk
(cid:2)T ; (cid:2) : (4)
.2(cid:4)/d
K kn k k
Here,knstandsfordiscreteMatsubarafrequencies,whichattimesarealsodePnoted
by ! . In the case of antiperiodicfunctions, the summation is written as T .
n fkng
Thesquaresoffour-vectorsreadK2 Dk2Ck2andX2 D(cid:2)2Cx2,buttheEuclidean
n
scalarproductbetweenK andXisdefinedas
Xd
K(cid:3)X Dk (cid:2) C kxi Dk (cid:2) (cid:4)k(cid:3)x; (5)
n i n
iD1
vii
viii Notation
wherethe vectornotationis reservedforcontravariantMinkowskianvectors:x D
.xi/,kD.ki/.Ifachemicalpotentialisalsopresent,wedenotekQ (cid:2)k Ci(cid:5).
n n
IntheMinkowskiancase,wehave
Z
X (cid:2).t;x/; x(cid:2)jxj; S D L ; (6)
M M
X
R R R
where (cid:2) dx0 .Fourieranalysisproceedsvia
X x
Z
K(cid:2).k0;k/; k(cid:2)jkj; (cid:3).X/D (cid:3)Q.K/eiK(cid:2)X ; (7)
K
R R R
where D dk0 ,andthemetricischosentobeofthe“mostlyminus”form,
K 2(cid:4) k
K(cid:3)X Dk0x0(cid:4)k(cid:3)x: (8)
No special notation is introduced for the case where a Minkowskian four-vector
is on-shell, i.e., when K D .E ;k/; this is to be understood from the context.
k
The argument of a field (cid:3) is taken to indicate whether the configuration space is
EuclideanorMinkowskian.Ifnotspecifiedotherwise,momentumintegrationsare
regulatedby definingthe spatial measure in d D 3(cid:4)2(cid:6) dimensions,whereasthe
spacetimedimensionalityisdenotedbyDD4(cid:4)2(cid:6).AGreekindextakesvaluesin
thesetf0;:::;dgandaLatinoneinf1;:::;dg.
Finally,wenotethatweworkconsistentlyinunitswherethespeedoflightcand
theBoltzmannconstantk havebeensettounity.ThereducedPlanckconstant„also
B
equalsunityinmostplaces,excludingthefirstchapter(onquantummechanics)as
wellassomelaterdiscussionswherewewanttoemphasizethedistinctionbetween
quantumandclassicaldescriptions.
General Outline
Physics Context
From the physics point of view, there are two important contexts in which rela-
tivisticthermalfieldtheoryisbeingwidelyapplied:cosmologyandthetheoretical
descriptionofheavyioncollisionexperiments.
In cosmology, the temperatures considered vary hugely, ranging from T '
1015GeV to T ' 10(cid:3)3eV. Contemporarychallenges in the field include figuring
outexplanationsfortheexistenceofdarkmatter,theobservedantisymmetryinthe
amountsofmatterandantimatter,andtheformationoflarge-scalestructuresfrom
smallinitialdensityperturbations.(Theoriginofinitialdensityperturbationsitself
isgenerallyconsideredtobeanonthermalproblem,associatedwithanearlyperiod
of inflation.) An importantfurther issue is that of equilibration,i.e., details of the
processesthroughwhichtheinflationarystate turnedintoathermalplasmaandin
particularwhatthehighesttemperaturereachedduringthisepochwas.Itisnotable
thatmostofthesetopicsareassumedtobeassociatedwithweakorevensuperweak
interactions, whereas strong interactions (QCD) only play a background role. A
notableexceptiontothisislightelementnucleosynthesis,butthiswell-studiedtopic
isnotinthecenterofourcurrentfocus.
In heavy ion collisions, in contrast, strong interactions do play a major role.
The lifetime of the thermal fireball created in such a collision is (cid:5) 10fm/c, and
the maximal temperature reached is in the range of a few hundred MeV. Weak
interactionsaretooslowtotakeplacewithinthelifetimeofthesystem.Prominent
observablesare the yields of differentparticle species, the quenchingof energetic
jets, and the hydrodynamicpropertiesof the plasmathatcan be deducedfromthe
observedparticle yields. An importantissue is again how fast an initial quantum-
mechanicalstateturnsintoanessentiallyincoherentthermalplasma.
Despitemanydifferencesinthephysicsquestionsposedandinthemicroscopic
forces underlying cosmology and heavy ion collision phenomena, there are also
similarities. Most importantly, gauge interactions (whether weak or strong) are
essential in both contexts.Because of asymptotic freedom,the strong interactions
ix
