Functional Renormalisation Group Equitions for Supersymmetric Field Theories [Elektronische Ressource] / Franziska Synatschke-Czerwonka. Gutachter: Andreas Wipf ; Martin Reuter ; Daniel Litim

Sujets

• Physik

Informations

• Publié par: friedrich-schiller-universitat_jena
• Publié le: 01 janvier 2011
• Nombre de lectures: 51
• Poids de l'ouvrage: 1 Mo

Extrait

Dissertation
zurErlangungdesakademischenGrades
doctorrerumnaturalium(Dr.rer.nat.)
vorgelegtdemRatderPhysikalisch-AstronomischenFakultät
derFriedrich-Schiller-UniversitätJena
vonDipl.-Phys. FranziskaSynatschke-Czerwonka
geborenampv.ov.pxwqinLemgo(NRW)
GroupSupRenoEquationsTheoformalisationrriesunctionalFieldFersymmetricGutachter:
p. Prof. Dr. AndreasWipf,Jena
q. Prof. Dr. MartinReuter,Mainz
r. Dr. habil. DanielLitim,Sussex,UK
TagderDisputation: pp.op.qopp
iContents
p Introduction r
q Functionalrenormalisationgroup v
q.p BasicsofQFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v
q.q ?eRenormalisationGroup . . . . . . . . . . . . . . . . . . . . . . . . . . . . x
q.r Spontaneoussymmetrybreaking . . . . . . . . . . . . . . . . . . . . . . . . . . pp
q.s Derivationoftheowequation . . . . . . . . . . . . . . . . . . . . . . . . . . pq
q.t Propertiesoftheowequation . . . . . . . . . . . . . . . . . . . . . . . . . . pt
q.u Truncations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . pt
q.v Spectrallyadjustedows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . pu
q.w Recoveringperturbationtheory . . . . . . . . . . . . . . . . . . . . . . . . . . pu
r Basicsofsupersymmetry px
r.p Supersymmetryalgebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . px
r.q Superspace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . qo
r.r Spontaneousbreakingofsupersymmetry . . . . . . . . . . . . . . . . . . . . . . qp
r.s Kählerpotential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . qq
s Supersymmetricquantummechanics qr
s.p Descriptionofthemodel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . qs
s.q ?esupersymmetricowequation . . . . . . . . . . . . . . . . . . . . . . . . qt
s.r Localpotentialapproximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . qv
s.s Next-to-leadingorderapproximation . . . . . . . . . . . . . . . . . . . . . . . rq
s.t Beyondnext-to-leadingorder . . . . . . . . . . . . . . . . . . . . . . . . . . . rt
s.u Di erencesbetweentheorieswithandwithoutsupersymmetry . . . . . . . . . ru
s.v LessonstobelearntfromSuSy-QM . . . . . . . . . . . . . . . . . . . . . . . . rw
t etwo-dimensional ¸ =1Wess-Zuminomodel rx
t.p ?eWess-Zuminomodel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . so
t.q ?esupersymmetricowequations . . . . . . . . . . . . . . . . . . . . . . . . . sp
t.r ?elocalpotentialapproximation . . . . . . . . . . . . . . . . . . . . . . . . . . sp
pContents
t.s Fixed-pointanalysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . sr
t.t ?eGaußianWess-Zuminomodel . . . . . . . . . . . . . . . . . . . . . . . . tr
u ethree-dimensional ¸ =1Wess-Zuminomodel tx
u.p ?eWess-Zuminomodel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . uo
u.q ?esupersymmetricowequationsatzerotemperature . . . . . . . . . . . . . . up
u.r Finite-temperatureowequations . . . . . . . . . . . . . . . . . . . . . . . . . ux
v etwo-dimensional ¸ = (2,2)Wess-Zuminomodel vt
v.p Descriptionofthemodel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vt
v.q Supersymmetricowequations . . . . . . . . . . . . . . . . . . . . . . . . . . . vv
v.r ?erenormalisedmass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . wo
v.s Beyondnext-to-leadingorder . . . . . . . . . . . . . . . . . . . . . . . . . . . wu
w Conclusionsandoutlook wx
A eCli ordalgebra xr
B TechnicaldetailsforSuSy-QM xt
B.p Inversionofthepropagator . . . . . . . . . . . . . . . . . . . . . . . . . . . . xt
B.q Flowequationsfromthebosonicandfermionicpart . . . . . . . . . . . . . . . xu
C FlowequationsinMinkowskispace xv
D Flowequationsat nitetemperature xx
E Technicaldetailsforthe¸ = (2,2)Wess-Zuminomodel pop
E.p TwodimensionalEuclidean¸ = (2,2) superspace . . . . . . . . . . . . . . . . . pop
E.q Flowequationforthemomentum-dependentwave-functionrenormalization . por
E.r Determinationoftherenormalizedmass . . . . . . . . . . . . . . . . . . . . . pot
F Diagrammaticdescriptionoftheowequation pov
qp Introduction
Quantumeldtheory[p, q]isanimportantpartofmodernfundamentalresearch. Quantum
electrodynamics(QED),developedinthepxso’sandthestandardmodelofelementaryparticles,
whichwasdevelopedinthepxvo’s,haveproventobeverysuccessful. PredictionsfromQEDhave
beenveri edexperimentallywithveryhighprecisionand,uptonow,thepredictionsfromthe
standardmodelhavebeencon rmedbyallacceleratorexperiments.
Afundamentalconceptofthestandardmodelaresymmetries. ?eyledtotheclassi cation
of the ‘elementary particle zoo’ in the pxuo’s. With the help of symmetries the spectrum of
‘elementary’hadronicparticlescouldbeunderstoodasboundstatesofjustafewbasicbuilding
blocks,thequarks[r,s]. Gaugesymmetriesenforcetheexistenceofgaugebosons,elementary
particlesthatmediatetheforcesinthestandardmodel.
Althoughthestandardmodelhasbeenverysuccessful,therearestillopenquestions. Toname
justafew,thesearethehierarchyproblem,thatthestandardmodelhasnodarkmattercandidate
andthatithasnotbeenuni edwithgravity. ?esearesomeofthereasonswhythestandard
modelisnotconsideredafundamentaltheorybutratheraneectivetheoryofelectroweakand
stronginteractions. Wethusareinneedforatheorybeyondthestandardmodel. Forareviewof
suchtheoriesseee.g. thearticlebyN.Polonsky[t].
With present knowledge supersymmetry, which combines the spacetime symmetry with
a symmetry between bosons and fermions, is a promising candidate for an extension of the
standardmodel. Indeed, it is the onlyknown symmetry thatallowsto combineinternaland
externalsymmetriesinanontrivialway. ?ereforeitisimportanttogaindeeperinsightinto
supersymmetrictheories.
Supersymmetry(SuSy)hasbecomearesearcheldinitselfandisnowanimportantingredient
in most theories that go beyond the standard model. Supersymmetry predicts that for every
elementary particle a superpartner exists. ?ese are particles that have the same quantum
numbersastheparticlesthemselvesexceptforthespin. IfSuSyisunbrokenthesuperpartners
havethesamemassastheoriginalparticles. Sincethesesuperpartnershavenotbeenobserved
yet,supersymmetryhastobebrokeninnature. Ifsupersymmetryisbrokenthesuperpartners
can be much heavier than the particles themselves explaining why theyhave not been found
yetinacceleratorexperimentssofar. Uptonowtherehasbeennoexperimentalevidencefor
supersymmetry. HoweverthehopeisthatitwillbefoundinnewexperimentsdoneattheLHCat
CERN.
rp Introduction
Fortheanalysisofsupersymmetricextensionsofthestandardmodelsimplermodelsarestudied,
e.g. Wess-Zuminomodelsorsupersymmetricsigmamodels[u,v,w]. Wess-Zuminomodelshave
averysimplestructuresincetherearenogaugedegreesoffreedombutonlyYukawainteractions.
Neverthelesstheyexhibitallgenericpropertiesofsupersymmetrictheories. Two-dimensional
sigmamodelsareverysimilartofour-dimensionalgaugetheorieswhichrepresentanessential
partofthestandardmodel. Ofspecialinterestarephasetransitions,especiallytheorderofthe
phasetransitionatcriticalpoints.
However,allthemodelsmentionedaboveareingeneralnotanalyticallysolvableandapproxi-
mationschemesareneeded. Widelyusedapproximationschemessu erfromtheproblemthat
theyeitherbreaksupersymmetryexplicitlyor,iftheypreservesupersymmetry,thepredictions
forphasetransitionsandcriticalexponentsarenotcorrectbecauseuctuationsoflightdegrees
of freedom are not treated properly. For example, the mean eld approximation, which is a
goodapproximationforphasetransitionsinhigherdimensions,breakssupersymmetrydueto
thedi erenttreatmentoffermionsandbosons[x]. ?eloopcalculationcanbeextendedina
supersymmetricway,butitisnotpossibletoobtainresultsonphasetransitions[po,pp].
Non-perturbative results are o?en obtained using lattice calculations where the spacetime
continuum is replaced by a lattice. Although it is a very successful and powerful method,
there are still di culties in formulating supersymmetry on the lattice. One problem is that
Lorentz-symmetryisexplicitlybrokenbythelatticeimplyingbrokensupersymmetryaswell.
However,inrecentyearsalotofprogresshasbeenmadeinrealisingsupersymmetryonthelattice,
seee.g. [pq,pr,ps,pt,pu].
Inordertodeterminetheinuenceofsupersymmetrybreakinginthelatticecalculationonthe
results,manifestlysupersymmetricapproximationschemesareneededandshouldbecompared
tolatticecalculations. Suchanapproachisprovidedbythefunctionalrenormalisationgroup
equations(FRG)[pv,pw]. ?eydealwiththephysicsofscalesandallowtounderstandthephysics
at large scales (small momenta) in terms of fundamental interactions at small scales. ?is is
of particular interest in elementary particle physics where it is desired to gain a macroscopic
descriptionofatomicnucleioutofthesimplelawsthatgovernthefundamentalinteractions.
?e functional renormalisation group equations have been successfully applied to a wide
varietyofphenomena,rangingfromcriticalphenomenaandphasetransitionstoapplicationsin
nitetemperatureeldtheory,QCDandquantumgravity,forreviewssee[px,qo,qp,qq,qr,qs].
For the description of macroscopic behaviour there exist powerful tools such as statistical
descriptionswhereasthemicroscopicphysicsiso?engovernedbysimplelaws. Infact,thereisa
gapbetweenthemicroscopicandmacroscopicdescriptionthathastobebridged. ?efunctional
renormalisation group allows to integrate out uctuations in a systematic way. It acts like a
micr