Quantum Superfield Supersymmetry

160 Pages·2021·2.504 MB·English

by  Albert Petrov

Fundamental Theories of Physics 202
Albert Petrov
Quantum
Superfield
Supersymmetry
Fundamental Theories of Physics
Volume 202
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Albert Petrov
Quantum Superfield
Supersymmetry
AlbertPetrov
DepartamentodeFisica
UniversidadeFederaldaParaiba
JoãoPessoa,Brazil
ISSN0168-1222 ISSN2365-6425 (electronic)
FundamentalTheoriesofPhysics
ISBN978-3-030-68135-7 ISBN978-3-030-68136-4 (eBook)
https://doi.org/10.1007/978-3-030-68136-4
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Preface
This review represents itself as a collection of lecture notes on superfield
supersymmetrybasedonlecturesgivenatInstitutodeFísica,UniversidadedeSão
Paulo(SãoPaulo),InstitutodeFísica,UniversidadeFederaldoRioGrandedoSul
(PortoAlegre),andDepartamentodeFísica,UniversidadeFederaldaParaiba(João
Pessoa). The book includes many examples of explicit quantum calculations
performed with use of the superfield formalism and is intended for students at
levelsofundergraduateandgraduatestudiesinquantumfieldtheoryandtheoryof
elementaryparticlesandforresearchersworkinginrelatedsubjects.
AuthorisgratefultoC.A.S.Almeida,E.A.Asano,L.C.T.Brito,I.L.Buchbinder,
M.Cvetic,A.F.Ferrari,F.S.Gama,H.O.Girotti,M.Gomes,S.M.Kuzenko,A.
C.Lehum,R.V.Maluf,J.R.Nascimento,P.Porfirio,A.A.Ribeiro,V.O.Rivelles,
A.J.daSilva,andE.O.Silvaforfruitfulcollaborationandinterestingdiscussions.
TheworkhasbeenpartiallysupportedbyCNPq.
JoãoPessoa,Brazil AlbertPetrov
v
Contents
1 Introduction ................................................... 1
2 EffectiveActionandLoopExpansion:GeneralFormulation ....... 5
3 SuperfieldDescriptionofThree-DimensionalSupersymmetric
Theories ....................................................... 11
3.1 DefinitionsandConventions ................................ 11
3.2 FieldTheoryModels ....................................... 17
3.3 Non-AbelianGaugeModels ................................ 23
3.4 QuantumDescriptionfortheSuperfieldModels ............... 25
3.5 Effective Action of the Three-Dimensional Superfield
TheoriesandtheProper-TimeMethod ........................ 32
3.6 Supersymmetry in Three-Dimensional Space-Time
andNoncommutativity ..................................... 40
3.7 On the Noncommutativity in the Fermionic Sector
oftheSuperspace ......................................... 44
3.8 Conclusion ............................................... 46
4 Four-DimensionalSuperfieldSupersymmetry ..................... 49
4.1 GeneralPropertiesoftheFour-DimensionalSuperspace ........ 49
4.2 FieldTheoryModelsintheFour-DimensionalSuperspace ...... 57
4.2.1 ChiralSuperfieldModels ........................... 58
4.2.2 AbelianGaugeSuperfieldModel ..................... 60
4.2.3 Non-AbelianGaugeTheories ........................ 62
4.3 GeneratingFunctionalandGreenFunctionsforSuperfields ...... 63
4.4 FeynmanSupergraphs ...................................... 69
4.5 SuperficialDegreeofDivergence.Renormalization ............. 75
4.6 Effective Action in Superfield Theories. Superfield
Proper-TimeTechnique .................................... 81
4.7 ProblemofSuperfieldEffectivePotential ..................... 85
4.8 The Wess-Zumino Model and a Problem of the Chiral
EffectivePotential ......................................... 89
4.9 GeneralChiralSuperfieldModel ............................ 104
vii
viii Contents
4.10 TheHigher-DerivativeChiralSuperfieldModels ............... 110
4.11 SupergaugeTheories ....................................... 121
4.11.1 GeneralDescriptionofSupergaugeTheories ........... 122
4.11.2 BackgroundFieldMethod ........................... 129
4.11.3 Proper-TimeMethodfortheSupergaugeTheories ...... 134
4.12 Conclusions .............................................. 140
5 SupersymmetryBreaking ....................................... 143
5.1 ExplicitSupersymmetryBreaking ........................... 143
5.2 SpontaneousSupersymmetryBreaking ....................... 145
6 Conclusions .................................................... 147
Bibliography ...................................................... 151
Index ............................................................. 157
Chapter 1
Introduction
Weprove,onceandforall,thatpeoplewhodon’tuse
superspacearereallyoutofit.
“Stuperspace”
Theideaofsupersymmetryisnowconsideredasoneofthebasicconceptsoftheoret-
icalhighenergyphysics(seee.g.[1]).Thesupersymmetry,beingafundamentalsym-
metryallowingtorelatebosonsandfermions,providespossibilitiestoconstructthe-
orieswithmuchbetterrenormalizationpropertiessincesomebosonicandfermionic
divergent contributions in supersymmetric theories cancel each other. Moreover,
thereareessentiallyfinitefour-dimensionalsupersymmetrytheorieswithouthigher
derivatives,e.g.N =4super-Yang-Mills(SYM)theory(thedetaileddiscussionof
thefinitenessofthistheoryispresentedin[2]),and,probably,N =8supergravity
[3].Inthethree-dimensionalcase,N =6andN =8supersymmetricChern-Simons
theoriesareknowntobefinite[4].Duetothisessentialimprovementofrenormal-
ization properties, there is a common expectation that the expected unified theory
ofallfundamentalinteractionsmustbesupersymmetric(seee.g.[5]andreferences
therein).
TheconceptofsupersymmetrywasintroducedinknownpapersbyVolkovand
Akulov[6]andGolfandandLichtman[7]inearly70s.Itgotafurtheradvance in
[8, 9](thehistoryofarisinganddevelopmentoftheconceptofthesupersymmetry,
includingbiographicalaspects,canbefoundinthebook[10]).Theessentialbreak-
through in supersymmetric field theory was achieved with introducing the idea of
a superfield [11] (see also [12, 13]). The reason for it consists in the fact that the
superfieldformulation,first,allowstomaintainmanifestsupersymmetriccovariance
at all steps of calculations, with all calculations are performed in a very compact
manner(indeed,anysupergraphcorrespondstoasetoftheFeynmandiagramsfor
componentfields),second,automaticallytakesintoaccountthefamous“miraculous
cancellations” of ultraviolet divergences responsible for an essential improvement
oftherenormalizationbehaviorofsupersymmetricfieldtheories[14].Moreover,in
thecontextofthenoncommutativefieldtheoryitturnsouttobethatthesupersym-
©TheAuthor(s),underexclusivelicensetoSpringerNatureSwitzerlandAG2021 1
A.Petrov,QuantumSuperfieldSupersymmetry,FundamentalTheoriesofPhysics202,
https://doi.org/10.1007/978-3-030-68136-4_1
2 1 Introduction
metry allows for the cancellation not only of ultraviolet divergences, as usual, but
alsoofdangerousinfrareddivergencesarisingduetotheUV/IRmixingmechanism
(seee.g.[15]).Further,theconceptoftheextendedsupersymmetryhasbeenintro-
duced, and various versions of an extended supersymmetric formalism have been
elaborated. Within this lecture course, nevertheless, we concentrate on the N =1
superfieldformalismwhichisknownasoneofthemostuniversaltoolsforstudying
thesupersymmetricfieldtheories.
Now, let us briefly describe the main steps in development of the superfield
methodology.Thefirstexampleofthesuccessfulapplicationofthesuperfieldcon-
ceptwasthemodelproposedbyWessandZuminointheirseminalpaper[11]where
thesimplestsuperfieldmodel,furthercalledtheWess-Zuminomodel,hasbeenfor-
mulated.Further,in[9]theyintroducedthesuperfieldgaugemodel,thatis,theSYM
theory. Intensive studies of different issues related quantum aspects of these theo-
ries began to be carried out. One of the most important results is the finiteness of
N =4 SYM theory that was proved in [16] (for discussions of finiteness of the
supersymmetricgaugetheoriesseealso[17]),whichimpliedinastrongestinterest
to the supergauge theories. Then, the superfield supergravity was formulated [18].
In1984,thefirstconsistentmethodologypossessinganexplicitextended(N =2)
supersymmetryhasbeendeveloped,thatis,theharmonicsuperspace[19,20].Atthe
sameyear,thesuperfieldapproachhasbeensuccessfullyappliedtothesuperstring
theory[21],whichemphasizedtheimportanceofthismethodologywithinthestring
context.Abitearlier,thesuperfieldformulationhasbeendevelopedforthesupersym-
metrictheoriesinathree-dimensionalspace-time[22,23],whichfurthermanifested
itself as a very convenient laboratory for studies of different issues related to the
supersymmetry. It should be noted that actually, the interest to three-dimensional
field theories, besides of their simplicity and better renormalization properties, is
drivenbystudiesofgraphene.Inourcontext,itworthmentioningthatthesuperfield
supersymmetricmodelforgraphenehasbeenformulatedaswell[24].
A new epoch for studies of the superfield theories began in 1991 when, in the
papers [25, 26], the chiral quantum contributions to the effective action were dis-
cussedforthefirsttime.Thesepaperscalledanattentiontothesuperfieldmethod-
ologyforevaluatingtheeffectivepotentialwhosedevelopmenthasbeencarriedout
in a series of works initiated by the paper [27], where this methodology has been
successfully applied to the Wess-Zumino model. Further, the success of the paper
[28] strongly increased the interest to various issues, both perturbative and non-
perturbative ones, related to supergauge theories, especially those ones possessing
theextendedsupersymmetry.
Amongthemostsuccessfulapplicationsofthesuperfieldmethodology,alsoitsuse
forthenoncommutativesupersymmetricfieldtheoriesdeservestobementioned.The
famouspaper[29]establishedthefactthat,sincethespace-timenoncommutativity
does not affect the anticommuting (Grassmannian) coordinates of superfields, the
superfieldmethodologycanbeusedasaverypowerfulinstrumenttodealwiththe
famous problem of the UV/IR mixing [30] known for generating the new infrared
divergences which are able to break the perturbative expansion. It was shown first
in [29] that the supersymmetric extension of field theories, implying in improving