Predictions for QCD from Supersymmetry

We review the construction of the effective Lagrangians of the Veneziano-Yankielowicz (VY) type for two non-supersymmetric theories containing one Dirac fermion in the two-index antisymmetric or symmetric representation of the gauge group (orientifold theo

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aPREDICTIONSFORQCDFROMSUPERSYMMETRYF.SANNINONORDITA,Blegdamsvej17,CopenhagenØDK-2100,Denmark.Wereviewtheconstructionofthee ectiveLagrangiansoftheVeneziano-Yankielowicz(VY)typefortwonon-supersymmetrictheoriescontainingoneDiracfermioninthetwo-indexantisymmetricorsymmetricrepresentationofthegaugegroup(orientifoldtheories).Sincethesetheoriesareplanarequivalent,atN→∞tosuperYang-Milltheire ectiveLagrangianscoincideswiththebosonicpartoftheVYLagrangian.Wedepartfromthesupersymmetriclimitintwoways.First,weconsider nitebutstilllargevaluesofN.Then1/Ne ectsbreaksupersymme-try.Wesuggestaminimalmodi cationoftheVYLagrangianwhichincorporatesthese1/Ne ects,leadingtoanon-vanishingvacuumenergydensity.Wethenanalyzethespectrumat niteN.ForN=3thetwo-indexantisymmetricrepre-sentation(one avor)isequivalenttoone- avorQCD.Weshowthatinthiscasethescalarquark-antiquarkstateisheavierthanthecorrespondingpseudoscalarstate,“η′”.Second,weaddasmallfermionmassterm.Thefermionmasstermbreakssupersymmetryexplicitly.Thevacuumdegeneracyislifted.Theparitydoubletssplit.Weevaluatethesplitting.Finally,weincludetheθ-angleandstudyitsimplications.1.Introduction

Recentlyithasbeenargued1,2thatnon-supersymmetricYang-Millsthe-orieswithafermioninthetwoindexsymmetricorantisymmetricrepre-sentationarenonperturbativelyequivalenttosupersymmetricYang-Mills(SYM)theoryatlargeN,sothatexactresultsestablishedinSYMtheory(e.g.3,4)shouldholdalsointhese“orientifold”theories.Forexample,theorientifoldtheories,atlargeN,musthaveanexactlycalculablebifermioncondensateandanin nitenumberofdegeneraciesinthespectrumofcolor-singlethadrons.Thephenomenongoesunderthenameofplanarequiv-alence;itdoesnotmean,however,thefullparent-daughtercoincidence.Forinstance,atN→∞thecolor-singletspectrumoftheorientifoldthe-oriesdoesnotincludecompositefermions.Theplanarequivalencerelatescorrespondingbosonicsectorsinthecorrespondingvacuaofthetwothe-ories.Somepredictionsforone- avorQCD(whichistheantisymmetricorientifolddaughteratN=3)weremadealongtheselinesin2,5.Herewe

1

We review the construction of the effective Lagrangians of the Veneziano-Yankielowicz (VY) type for two non-supersymmetric theories containing one Dirac fermion in the two-index antisymmetric or symmetric representation of the gauge group (orientifold theo

February1,200819:2ProceedingsTrimSize:9inx6inMinnesota

2

reviewtheconstructionofthee ectiveLagrangiansfortheorientifold eldtheorieswhichareabletocapturerelevant1/Ncorrections6.Ourstartingpointisthee ectiveLagrangianforsupersymmetricYang-Mills.

Thenameoforientifold eldtheoryisborrowedfromstring-theoryter-minology7.

2.ReviewingSYMe ectiveLagrangian

Thee ectiveLagrangianforsupersymmetricgluodynamicswasfoundbyVenezianoandYankielowicz(VY)8.Intermsofthecompositecolor-singletchiralsuper eldS,

S=3

4α d2θd2θ¯ 1

SS3 d2θ Sln S

2θχ(y)+θ2F(y),whereyµisthe

chiralcoordinate,yµ=xµ iθσµθ¯,and

,√64π2N× λa,αλaα

a,β

wheref.t.standsforfermionterms. Ga+2iDaλaα(4) αβλ 12GaµνG aµν+f.t.

We review the construction of the effective Lagrangians of the Veneziano-Yankielowicz (VY) type for two non-supersymmetric theories containing one Dirac fermion in the two-index antisymmetric or symmetric representation of the gauge group (orientifold theo

February1,200819:2ProceedingsTrimSize:9inx6inMinnesota

3

Thecomplex eld representsthescalarandpseudoscalargluino-ballswhileχistheirfermionicpartner.ItisimportantthattheF eldmustbetreatedasauxiliary.

Thislagrangianisnotcompleteaspointedoutin9.Recentlyanex-tendedVYLagrangianwhichpassesanumberofnontrivialconsistencycheckswhilenaturallyyieldingtheVYe ectivetheoryaugmentedbythemissingtermspointedoutin9hasbeenconstructed10.Suchanextensionrequirestheintroductionofaglueballsuper eld,i.e.achiralsuper eldwithzeroRcharge.EarlierattemptsofgeneralizingtheVYLagrangiancon-tainingglueballdegreesoffreedomarediscussedintheliterature12,13,14,?.Theseextensionsweretriggered,inpart,bylatticesimulationsofSYMspectrum16.SinceinthispaperweareinterestedinthemesonicdegreesoffreedomwewillnotconsiderthegeneralizedversionoftheVY10althoughitisnowstraightforwardtogeneralizedourresultstotheimprovedVYtheory.

Forourpurposesofmostimportancearethescaleandchiralanomalies, µJNµ=g2λaσµ¯λa, µµ= 3N

α( ¯) 2/3 µ ¯ µ 4αN2

9 F+F¯ = 3 N+4GaµνGa,µν

32π2,(7)

µJµ=i4N

16πGaµνGa,µν

2 ,(8)

We review the construction of the effective Lagrangians of the Veneziano-Yankielowicz (VY) type for two non-supersymmetric theories containing one Dirac fermion in the two-index antisymmetric or symmetric representation of the gauge group (orientifold theo

February1,200819:2ProceedingsTrimSize:9inx6inMinnesota

4

where

= 3

1

1+ 1+49N.(10)

Itisjustthiscombinationthatshouldenter Ninthede nitionofthevariable

We review the construction of the effective Lagrangians of the Veneziano-Yankielowicz (VY) type for two non-supersymmetric theories containing one Dirac fermion in the two-index antisymmetric or symmetric representation of the gauge group (orientifold theo

February1,200819:2ProceedingsTrimSize:9inx6inMinnesota

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