Table Of ContentLecture Notes in Physics
Urko Reinosa
Perturbative Aspects
of the Deconfi nement
Transition
Beyond the Faddeev-Popov Paradigm
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Urko Reinosa
Perturbative Aspects
of the Deconfinement
Transition
Beyond the Faddeev-Popov Paradigm
UrkoReinosa
CenterforTheoreticalPhysics
ÉcolePolytechnique
Palaiseau,France
ISSN0075-8450 ISSN1616-6361 (electronic)
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To myparents,
Preface
Inthecaseofnon-abeliangaugetheories,thestandardFaddeev-Popovgauge-fixing
procedurein the Landau gauge is known to be incomplete due to the presence of
gauge-equivalent gluon field configurations that fulfill the gauge condition, also
known as Gribov copies. A widespread belief is that the proper analysis of the
lowenergypropertiesofnon-abeliantheoriesinthisgaugerequires,therefore,the
extensionofthegauge-fixingprocedure,beyondtheFaddeev-Popovrecipe.
ThismanuscriptreviewsvariousapplicationsoftheCurci-Ferrarimodel,aphe-
nomenologicalproposalforsuchanextension,basedonthedecouplingproperties
ofLandaugaugecorrelatorsascomputedonthelattice.Inparticular,weinvestigate
the predictionsof the modelconcerningthe confinement/deconfinementtransition
of strongly interacting matter at finite temperature, first in the case of pure Yang-
Mills theory for various gauge groups, and then, in a formal regime of quantum
chromodynamicswhereallquarksareconsideredheavy.Weshowthatmostquali-
tativeaspectsandalsomanyquantitativefeaturesofthedeconfinementtransitionin
these theoriescan be accountedforwithin the Curci-Ferrarimodel,with only one
additional parameter, adjusted from comparison to lattice simulations. Moreover,
these features emerge in a systematic and controlled perturbative expansion, as
opposedtotheill-definedperturbativeexpansionwithintheFaddeev-Popovmodel
intheinfrared.
TheapplicationsoftheCurci-Ferrarimodelatfinitetemperatureand/ordensity
requireonetoconsiderabackgroundextensionoftheLandaugauge,knownasthe
Landau-DeWitt gauge. Therefore, besides the above-mentioned applications, the
manuscript is intended as a thorough but pedagogical introduction to these tech-
niquesat finite temperatureand/ordensity, includingthe rationale for considering
self-consistent backgrounds and the associated background-field effective action,
theroleoftheWeylchambersindiscussingthevariousphysicalsymmetriesofthe
problem, and the complications that emerge due to the sign problem in the case
ofa realquarkchemicalpotential.Italso discusses a novelimplementationofthe
background-fieldmethodatfinitetemperaturewhichdoesnotsufferfromsomeof
thelimitationsofthestandardapproachandshould,therefore,providemorerobust
results.
vii
viii Preface
Thematerialcoveredintheselecturenotes,whichderivesfrommyhabilitation
thesis work defended in June 2019, is intended for any graduate student or
confirmed researcher that wishes to deepen their understanding of the confine-
ment/deconfinementtransitionfromananalyticalperspectiveinYang-Millstheory
or related theories. Basic knowledge of these theories at finite temperature is
required, although the text is designed in a self-contained manner, with most
concepts and tools introducedwhen needed. In particular, the Cartan-Weyl bases,
so useful in the context of background-field applications at finite temperature,
are thoroughly reviewed in a dedicated Appendix. The same goes for the various
techniquesof computationof Matsubarasums at finite temperature.At the end of
eachchapter,aseriesofexercisesisproposedinsupportofthematerialinthemain
text.Exercicesmarkedwithone(cid:2)aredirectapplicationsofthematerialinthemain
text,thosewithtwo(cid:2)saremoderatelydemanding,whilethosewiththree(cid:2)srequire
adeeperinvestment.
I would like to thank Duifje van Egmond for checking the main text and the
problems. Also, the new version of the final sections of Chap.3 derives from
enlighteningdiscussionsthatIhadwithherduringourworkincommon.Mymost
sincegratitudegoesalsotoLisaScaloneforsmoothandefficientinteractionsduring
thewholesubmissionprocess.
Palaiseau,France UrkoReinosa
May2022
Acknowledgements
The time that I have spent writing this manuscript strangely reminds me that of
the imaginarytime formalism with periodic boundaryconditionsused throughout
this work: it does not flow in the usual direction, but rather orthogonalto it, and
itis characterizedbyperpetual,almostperiodic,repetitionsuntilthe finalresultis
achieved.NowthatIamfreedfromtheconstraintsofthisunusualtime,Iwouldlike
to expressmy most sincere gratitudeto all the peoplethat supportedme on many
differentlevels,prior,during,andafterthewritingprocess.Thismanuscriptwould
clearlynotbethesamewithoutthem.
Firstofall, mywarmestthankstothemembersofthejury,MaximChernodub,
FrançoisGelis,AntalJakovac,Jean-LoïcKneur,DominiqueMouhanna,andSamuel
Wallon, who accepted the invitation to read and review the manuscript and to
participatein the oral defense.Thanksalso to FrédéricFleuret, Régine Perzynski,
MarcoPicco,andNathalieSuircofortheirmanyadvisesandforacceptingtomove
thedeadlinebyafewmonths,inordertoallowmetocopewithsomerealtime,real
lifeimperatives.ThanksfinallytoJuanaIsabelMallmannformakingthedefensein
AlanTuring’sbuildingpossibleandincrediblysmooth.
I want to express my deepest gratitude to the collaborators with whom this
adventure started: Marcela Peláez, Julien Serreau, Matthieu Tissier, and Nicolás
Wschebor. I consider myself very lucky to have crossed paths with these top-
notch researchers, and I am very proud of the work that we have accomplished
together. Thanks also to their respective laboratories (and by this I do not mean
the walls but the people within them) where a substantial part of the present
workwasdeveloped:LaboratoiredePhysiqueThéoriquedelaMatièreCondensée
(UniversitéParisSorbonne),LaboratoireAstroParticuleetCosmologie(Université
Paris-Diderot),andInstitutodeFísica (Universidaddela Républica,Montevideo).
Of course, I have also benefited from interactions with many other collaborators,
whichhavecontributedtoshapingmyknowledgeinonewayoranotherandwhose
insightcertainly contributesto the flavor (or I should say the color) of this thesis.
Inparticular,aspecialthankyougoestoJean-PaulBlaizot, EdmondIancu,David
Dudal,JanM.Pawlowski,andZsoltSzép.
ix
x Acknowledgements
Thisthesiswaswrittenpartlyatthe“CentredePhysiqueThéorique”(CPHT)on
EcolePolytechniquecampus(eventhoughmanyhourswerealsospentatthecentral
library,amidst the noisy workersthatwere renewingit). I would like to thankthe
wholeCPHTmembersfortheoutstandingworkingatmospherethattheycontribute
tocreate,muchabovetheaverageIwouldsay.Inparticular,IwanttothankFlorence
Auger,Fadila Debbou,and Malika Lang for their promptnessin dealing with any
kindofadministrativeissue.Thanksalsotoourdirector,Jean-RenéChazottes,and
toEcolePolytechnique,foracceptingtopaymyregistrationattheUniversity.
A specialthankyougoesofcoursetopastandpresentmembersoftheParticle
PhysicsgroupatCPHT,TranTruong,GeorgesGrunberg,TriNangPham,Bernard
Pire, Claude Roiesnel, StéphaneMunier, Cyrille Marquet, and Cédric Lorcé (plus
postdoctoralfellowsaswellasmaster’sandPhDstudents)fortheenjoyableworking
atmosphereandforacceptingmeasamemberofthegroupdespitethecrazytaste
for gluonic mass operators. Hopefully, this thesis will succeed in conveying the
message that this is not that crazy after all. An even more special thank you goes
toBernardPireandStéphaneMunierwhohaveinvestedaconsiderableamountof
energyformanyyearsnowintostrengtheningthevisibilityofourgroup/laboratory.
Iowethemmuchsincemyhiring12yearsago.Theyhavealwaysbeenavailableto
listentomyqueriesandtoprovidevaluablepiecesofadvice.
Thank you to all the friendsand colleaguesthat, every now and then, inquired
abouttheprogressofthethesis,inparticularthose,LaetitiaandMathieu,Xoanaand
Daniel,wholenttheirhousessothatIcouldfindafewhoursoffocusonthewriting
ofthemanuscript.
Finally, my most special thank you to my dear wife and kids: Rosa, Aitor,
Guillem, and Rosa (did I mention periodic boundary conditions?). Your patience
hasbeeninfinite.Mine,certainlytheinverse.
