Supersymmetric calculation of mixed Kähler-gauge and mixed Kähler-Lorentz anomalies

L __

NUCLEARPHYSICS B

ELSEVIER Nuclear PhysicsB 418 (1994) 535—568 ______________

Supersymmetriccalculation of mixedKähler—gaugeand mixed Kähler—Lorentz

anomaliesGabrielLopesCardoso1,Burt A. Ovrut2

TheoryDivision, CERN, CH-1211Geneva23, Switzerland

Received17 August 1993; revised22 November1993;accepted22 November1993

Abstract

We presenta manifestly supersymmetricprocedurefor calculating the contribu-tions from matterloops to the mixed Kähler—gaugeand to the mixed Kähler—Lorentzanomaliesin N = 1, D = 4 supergravity-mattersystems.We show how this proce-dureleadsto the well-known result for the mixed Kähler—gaugeanomaly.For generalsupergravity-mattersystemsthe mixed Kähler—Lorentzanomalyis found to containaterm proportionalto 1?? with a backgroundfield dependentcoefficientas well as termsproportionalto (Cmnpq 2 andto the Gauss—Bonnettopologicaldensity.We briefly com-ment on the relationshipbetweenthe mixed Kähler—Lorentzanomalyandthe modulidependentthresholdcorrectionsto gravitationalcouplingsin ZN orbifolds.

1. Introduction

As is well known [1,2], the tree-level lagrangian describingthe couplingof chiral and antichiral matter supermultiplets~ and ~1~ to N = 1, D = 4supergravityis specified by three fundamentalfunctions,namely the KählerpotentialK, theholomorphicsuperpotentialW andthe holomorphicgaugecou-pling function f. In the conventionalformulation of the theory, the tree-level

Work supportedin part by theDepartmentof EnergyunderContractNo. DOE-ACO2-76-ERO-3071.2 On sabbaticalleavefrom the Departmentof Physics,Universityof Pennsylvania,Philadelphia,PA 19104-6396,USA.

0550-32l3/94/$07.00© 1994 ElsevierScienceB.V. All rights reservedSSDIO55O-3213(93)E0448-9

536 G. LopesCardoso,B.A Ovrut/NuclearPhysicsB 418 (1994)535—568

lagrangian is invariant under combined super-Weyl--Kãhlertransformations[1,21. Theseconsist of superfieldrescalingsof the supervielbein,which leavethe conventionaltorsionconstraintsinvariant, followed by Kähler transforma-tionsof thethreefundamentalfunctionsK, W,f. The chiral matter superfieldsdo not transformunder either the super-Weylor the Kähler transformations.On the other hand, the tree-levellagrangianin the conventionalformulationhas, in general, a non-canonicalgravitationalkinetic energyterm of the formh (A, A)R, whereA representscomponentmatter scalarfields, h is somecal-culable function and 1~is the curvaturescalar. It is also well known [1,21that, by appropriaterescalingof thegraviton andof the gravitinoas well as ofthe auxiliary fields M and ba of minimal supergravity,one can obtain anewlagrangianwith appropriatelyEinsteinnormalizedgravitationalkinetic energy,1~.This rescalingcan eitherbe performedat the componentfield level [1,21or at the superfieldlevel [3—51,the latterhavingthe advantageof beingman-ifestly supersymmetric.It turns out that this rescaledlagrangianis invariantunder an important new symmetry,called Kähler symmetry [1,2,6]. By ex-plicitly addingthe Kähler transformationsto the structuregroup of superspace[3—5],onecangive acompletesuperfielddescriptionof this appropriatelyEin-stein normalizedsupergravity-mattertheory. The new underlyingsuperspaceis calledKählersuperspace.The Kähler invarianceof the Einstein normalizedtree-levellagrangiancan be tracedbackto the super-Weyl—Kahlerinvarianceof the tree-levellagrangianin the conventionalsuperspaceformulationof thetheory [1,2,7]. Hence,a chiral matter superfieldin Kãhler superspacecarriesa zero Kähler charge.Its fermionic componentfield, however,carriesa non-vanishingKãhlerchargeand, consequently,couplesto the componentKählerconnectionaa,which is acompositeconnectionmadeout of componentmatterfields. Other fermions, such as gauginosand the gravitino, also carry a non-vanishingKähler chargeand, hence,also coupleto the Kàhlerconnectionaa.As alwaysin four dimensions,the Kählersymmetryof the Einsteinnormalizedtree-levellagrangianmight get spoiledby one-loopcomponenttriangle graphswith fermions running in the loop and with at least one Kähler connectionstickingout. This is, in fact, what happensin generalsupergravity-matterthe-ories [8—11].On theotherhand,sinceonly fermionscarry Kähler charge,it isimpossibleto constructcomponentgraphshavinglegsof theKähler type stick-ing out andhavingcomponentfields runningin the loop otherthanfermions.This, however,createsa puzzle.Namely, it would appearthat it is then im-possibleto constructa supersymmetricexpressionfor the componentKähleranomalyin Kählersuperspacebecauseof missingcomponentgraphs.

In this paper, we will presenta manifestlysupersymmetricprocedureforcalculatingmixed Kähler—gaugeandmixed Kähler—Lorentzanomalies.Thesearethe onesgeneratedby componentfermionic triangle graphswith only oneKählerconnectionstickingout. Thesemixed Kãhler anomalieshaveimportantphysicalconsequences.Theyhavebeenshownto leadto thresholdcorrectionsto gauge and gravitational couplings [8—15] in string theory as well as toinflationary supergravitymodels[16,171.We will restrictour discussionto the

G. LopesCardoso,B.A Ovrut/NuclearPhysicsB 418 (1994)535—568 537

contributionfrom matterloops to thesemixedanomalies.We believe,though,that the proceduredescribedin this paper can also be extendedto includecontributionsfrom gaugeandgravitationalfields running in the loop.

The origin of the puzzleis not hard to understand.As mentionedabove, achiral mattersuperfieldcarriesvanishingKählerchargew in Kählersuperspace.Thismeansthat thereis no direct couplingc1~~e°~”i of the KählerprepotentialK to quadraticmatter superfields,sincew(~I’)= 0. This is to be contrastedwith the coupling of the gaugeprepotentialV to chargedmatter as ~teJ7~I~.Wereoneto examinethepuregaugeanomalythree-pointfunction, one wouldfind a supersymmetricresult,sinceit could havebeenobtainedby calculatingasuperfieldtrianglegraph with ‘2’ runningin the loop and V-legs stickingout. Itseemsthenclearthat, in order to directlycalculatesupersymmetricexpressionsfor the mixedKähleranomalies,onemustgo to aformalismwhereK couplesto chiral matter as Ite’~. The conventionalsuperspaceformalism is sucha framework. It allows for a superfieldprepotentialformalism describingthecoupling of quantummatter superfields~ to backgroundprepotentialssuchas K. An exampleof such a coupling is precisely~te_”/3~. Sincethereexistsa superfield prepotential formalism, all calculations of anomaloustrianglegraphsin the conventionalformulationcan, in principle,be performedat thesuperfieldlevel, yielding automaticallysupersymmetricresults.

The relevant tree-level symmetry in the conventionalsuperspaceformula-tion is the tree-level super-Weyl—Kahlersymmetry. Hence, of relevance inthe conventionalformulationarethe couplingsof quantummatter superfieldsto Kähler as well as supergravityprepotentials,and one has to look out forthe breakdownof the tree-levelsuper-Weyl—Kählersymmetrydue to the non-vanishingof mixed trianglegraphswith quantummatter superfieldsrunninginthe ioop. We do indeedfind that suchmixed trianglegraphsarenon-vanishing,and,hence,that they contributeto the breakdownof the super-Weyl—Kahlersymmetry.As statedabove,resultsin theconventionalsuperspaceformulationof the theory can be transformedover into Kähler superspaceby particularsuperfieldrescalingsof theunderlyingtorsionconstraintsof conventionalsuper-space[3—5].Upon applyingtheserescalingsto the mixed super-Weyl—Kähleranomaliesin conventionalsuperspace,we arrive at manifestly supersymmet-nc results in Kähler superspacewhich qualify to be called the supersym-metric mixed Kähler—gaugeand the supersymmetricmixed Kähler—Lorentzanomaly,respectively.In this way, we give the first completecalculationfor,and rediscoverthe well-known result [8,10,11] for the supersymmetricmixedKähler—gaugeanomaly.We alsogive the first completecalculationfor thesu-persymmetricmixedKähler—Lorentzanomaly.When putting thegravitationalfields on-shell, the supersymmetricmixed Kähler—Lorentzanomalycalculatedusingour procedurealso reducesto its well-known form given in refs. [9,10].Off-shell, however, it containsan additional term proportional to 7V whichis not part of the Gauss—Bonnettopological density. Two remarksconcern-ing this term needto be made.First of all, this 7V-term, presentin generalsupergravity-mattertheories,arisesfrom graphsconstructedout of verticesof

538 G. LopesCardoso,B.A OvrutINuciearPhysicsB 418 (1994)535—568

the typeEe_K/3~2in the conventionalformulation. Suchverticesare not in-variantunderconformal transformations(thesearedefinedin section6), and,hence,the growing of a term proportionalto R2 not containedin the Gauss—Bonnet combinationis not forbiddenby any symmetryargument.Secondly,the coefficient of this ~ term comesout to be backgroundfield dependent.

We close this paperwith a remark on the relationshipbetweenthe super-symmetric mixed Kãhler—Lorentzanomaly,as computedin this paper, andthe moduli dependentthresholdcorrectionsto gravitationalcouplingsin ZNorbifolds computedin ref. [14]. Thesethresholdcorrectionsturn out to beproportional to the traceanomalycoefficientsof the different fields coupledto gravity. From the field theory analysispresentedin this paperthis can beunderstoodby noticing that thereare two distinct termswhich contributetothe Kähler anomaly.One of them comes with a coefficient proportional tothe traceanomalywhereasthe otheronecomeswith a coefficientproportionalto the chiral anomalyin conventionalsuperspace.It is this latter contributionwhich getsremovedby theGreen—Schwarzmechanism[8—10],yielding thresh-old correctionsproportionalto the traceanomalycoefficients of the variousfields coupledto gravity.

2. UK(l) superspaceandtree-levelsymmetries

In this section,we will briefly review some of the featuresof Kähler super-spacegeometrywhich will berelevantin thesubsequentdiscussion.A completedescriptionof the propertiesof UK (1) superspacecan be found in ref. [5].The structuregroupof Kähler superspaceis takento be SL(2,C) x UK( 1) and,accordingly,oneintroducestwo Lie algebravaluedone-formgaugeconnections

A = dZMcbMBA andA = dzM AM correspondingto theLorentz andUK (1)groups,respectively.In addition,one introducesasupervielbienEM A and theassociatedone-forms EA = dZMEMA~The UK (1) gauge connectionA is acompositegaugeconnectiondefinedby

Aa =

A~= —~K,

Aa~= ~ [Va,~&]K, (2.1)

where the prepotentialK(I~,I~) is the Kähler potential for matter chiralsuperfieldscii,. All mattersuperfieldshavevanishingUK (1) weight,COK (~)=

0. UnderaKähler transformation

K2K(~~,~t) K2K(~~,~it) + F(c11) +F(~1~t), (2.2)

the one-formA transformsas

A ~A + K~2~dImF, (2.3)

G. LopesCardoso, B.A OvrutINuclearPhysicsB 418 (1994)535—568 539

where ImF = (F — ~)/2i and ic2 = 8irM~2. Mp is the Planck mass.Also,

under a Kähler transformationthe supervielbeinone-formsEA canbe shown[5] to transformas

EA EAexp [_~w(E~4)ImF] , (2.4)

where

w(E~)= 1, w(E,~)= —1, w(Ea) = 0. (2.5)

Solving the Bianchi identitiessubject to a set of constraints[18], one findsthatall componentsof the torsion andcurvaturemaybe expressedin termsofa setof superfieldsand their coordinatederivatives:

superfield R Rt Gc,c, W~p7, Xc, ~ ~ (2 6)

UK (1) weight 2 —2 0 1 —1

where

Xc, =V~—~Gc,~= _~(~2_8R)vc,K

r=vrRt +Vc,G~= 4(V2_8Rt)~~K. (2.7)

Xc, is the superfield field strengthof the UK (1) gauge connection. If wefurther assumethat thereis an internal gaugegroup, thenwe mustintroduceyet anotherLie algebravalued one-formgaugeconnectionAab = dZMAMab~Solving the Bianchi identities now introducesa new superfieldfield strength,J4’~,with UK (1) weight w ( J4’~)= 1.

Using these superfields, one can write down the tree-level superfield la-grangianin Kählersuperspace[3—5].It consistsof threeparts,eachspecifiedby a fundamentaland independentfunction. The first part, specifiedentirelyby the Kàhlerpotential, is the supergravity-matterkinetic energyterm givenby

= _3K_2f d4OE[K] , (2.8)

whereE isthesuperdeterminant.Thesecondpart,specifiedby the holomorphic

superpotentialW(cI~),is the potentialenergyterm given by£PE = ~Jd4O~eK~72W(~I)j) + h.c. (2.9)

Finally, the Yang—Mills lagrangianis given by

LYM = ~fd4O~f (~I~i)aj,W~aWc,b+ h.c., (2.10)

wheref (~1)~)ab is theholomorphicgaugecouplingfunction. Thetotal lagrangianpossesses,at the tree-level,threesymmetries.

540 G. LopesCardoso,B.A Ovrut /NuclearPhysicsB 418 (1994)535—568

(1) Kähler invariance. —

UnderKähler transformation,c2K —* ic2K + F + F, it canbe shown [3—5]that

E-+E,

R—~Re_(~~I2. (2.11)

One also,simultaneously,transformsW and Wc, as

W —* e_FW,

Wc, —f e~~’~Wc,. (2.12)

Then

4 = _3~c~fd4OE’ = _31c_2Jd4OE= (2.13)

and

£PE = ~fd4O~e/c21(~/2W/+ h.c.

1 Eel~2~~I2We~’~’~~’2e= ~fd49 Re_(F_F)/2 + h.c. = £PE. (2.14)

Furthermore

~YM = .~.fd4O~f’W’a W~+ h.c.

= ~Jd4OEf~~e~2/ +h.c. = LYM. (2.15)

That is, the complete superfield lagrangian is invariant under the Kählertransformationstc

2K —~ K2K + F + F and (2.12).

(2) Gaugeinvariance.This follows from the fact that both E andR are functionsof the Kähler

potential K which, in turn, is invariant underYang-Mills transformationsofthe chargedsuperfields1k. Thesearegiven by

eV

(2.16)

where

V~A= 0 (2.17)

and V = V(r)T(r) denotesthe Yang—Mills prepotentialvectorsuperfield (theT(r) denotethe hermitiangeneratorsof the Yang—Mills gaugegroup H). The


squareof the Yang—Mills superfieldfield strength,W~Wc,, is by construction

invariantunder Yang—Mills transformations.

(3) Lorentzinvariance.Thisfollows from the fact thatall thesuperfieldcombinationsappearingin Lo

and LYM as well as in £PE arescalarswith respectto Lorentz transformations.Invariances(1)— (3) of the tree-level theory of the supergravity-mattersu-

permultipletsystemcan,of course,alsobe displayedatcomponentlevel. Com-ponentfields aredefinedaccordingto standardnotation [3—5]:A~,x~,.‘F~forchiral multiplets (and similar notationsfor antichiralmultiplets) and2~,Vm,

V for Yang—Mills multiplets. The irreducible minimal supergravitymultipletis realizedby (ema, ~ M, ba). M and ba denotethe auxiliary componentfields of minimal supergravity.The covariantderivative of a generic Weylfermion will, in a theory with invariances(1)—(3), thencontainaconnectionfor eachof thesesymmetries.

(1) The componentconnectionfor gaugingKähler transformationsis givenby the lowestcomponent[3—51of the UK (1) gaugeconnectionsuperfieldAc,~

A 1T~S1V m— V-’~’ 1 — ac,~—

where

am = ~ (OJKVmAJ — ~KVm~~) + ~ig~7(XIamX1’). (2.19)

Here, g~7denotesthe Kähler metric g~7= 8~a-1Kof the matter manifold

parameterizedby A~andA~.UnderKähler transformations

am’ = am + K2~3m ImF. (2.20)

All componentWeyl fermions transformunderKähler transformations(2.2).The componentmatter Weyl fermion,x~= (1/v~)Vc,i~tI,transformsas

= e’~’2tmfl~1, (2.21)

whereasthe gaugino)~transformswith oppositecharge

= e(_u/2tmfl)~ (2.22)

as does the gravitino ~(2) The componentYang—Mills connectionvc,~= vrT(’) is containedin

the Yang—Mills prepotentialV as

~ [V~,~] V~= —2vc,~. (2.23)

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(3) The componentLorentzconnection0~ma‘~ is given by the lowest com-

ponentof the SL(2,C) gaugeconnectionsuperfieldcbmc, ‘~

= Wmc,’6. (2.24)

Therefore,the covariantderivativefor matterfermionsx~reads [3—5]

VmX’ = (ôm + jV~(T~— ~D(r)) Wm+ ~bm?c2am...)X1, (2.25)

where

= ~ b Aut~. (2.26)

The ... standfor theadditionalcouplingto theo-modelChristoffel connection= gYO~JgJ~.Such acouplingwill not be consideredin this paper.

The covariantderivativeof the matterscalarA is given by

DmA’ = amA’ + iv~T~A’. (2.27)

Note the absenceof acouplingof the matterscalarA’ to the Kählerconnectionam given in (2.19). This is in manifestcontrastto the matterfermionx’. Thereasonfor this is that the Kählercharge~OK(c1,) of mattersuperfield i~,is zero,(DK (~,)= 0. Therefore,only the componentfermionscarry Kählerchargeand,hence,only componentfermionsrotate underKähler transformations(2.2).

We will now expandthe supergravity-matterkinetic superfieldlagrangian(2.8) into components.It is, by now, well known that one of the advantagesof the Kählersuperspaceformulation lies in the fact that it immediatelygivesthecorrectlynormalizedkinetic termsfor all the componentfields without anyneedfor rescalingsor complicatedpartial integrationsat the componentfieldlevel. That is, it immediatelygives the correctlynormalizedEinstein—Hilbertactionas well asmaking the componentKãhler structurein the matter sectormanifest.The componentkinetic lagrangianfor the supergravity-mattersystemreads [1,2,6,3—5]

= —~,c2e7~— ~K2 (MV — baba)

— ~ ~ + ~ (Dc,’)g~7a~~’

+ V~’~~D(r)+ ... , (2.28)

wherewe haveonly displayedthe componenttermsrelevantfor this paper.Of importanceto this paperarethe couplingsof mattercurrentsto external

gravitationalfields andto Yang—Mills andKählerconnections.Let us look atthe fermionicmatter currentg,.x~’..From the componentlagrangian(2.28)

it follows that the fermioniccurrentcouplesto the Yang—Mills connectionv,~,

G. LopesCardoso,B.A OvrutINuclearPhysicsB 418 (1994)535—568 543

the Kählerconnectionam, to the gravitationalspin connectionWma as well asto the auxiliary field ba andthe space-timemetric.The bosonicmattercurrentgj7Ai8~AJ,on theotherhand,couplesonly to the Yang—Mills connectionv~,as well as to the space-timemetric gc,~,,.

Let us emphasizeagainhow differently bothcomponentcurrentscouple tothe Kähler connection.The bosoniccomponentcurrent does not couple toam at all! This is in manifestcontrastto the couplingof both currentsto the

Yang—Mills connectionVm. Also note that thereis a couplingof A’4~to the

V-term of the Yang—Mills prepotentialV, throughthe term V(T)D(r), whereasthereis no suchcoupling of A’A~to the highestcomponentof prepotentialK.The reasonfor all of this is, onceagain, obvious. On one hand, the mattersuperfield 1, carries anon-zero Yang—Mills charge,on the otherhandit haszero Kähler charge, ~

0K(~,)= 0. This will be of utmostimportancein thenext section.

Finally, note that we have not expandedthe potential energy (2.9) intocomponentfields, since its componentexpansionwill not be neededin thispaper.Its superfieldform is, however,relevantto ourdiscussion.The additionalmattercouplingscontainedin the Yang—Mills lagrangian(2.10), on the otherhand,areof no relevancefor the subsequentdiscussionofthe Kãhleranomalouscontributions from triangle graphs to the one-particle irreducible effectiveaction.Hence,wewill ignore themthroughoutthis paper.

3. Kähleranomaliesin the Kähler superspaceformalism

Since all of the fermions in the supergravity-mattertheory are chiral, thepossibility exists that all three of the symmetriesdiscussedin the previoussectionare anomalousat the one-looplevel. However, it is well known thatthere are no pure Lorentz anomaliesin four-dimensions.Furthermore,wewill assume,as in the standardelectroweakmodel, that the matter superfieldcontentis so chosenthat thereare no puregaugeanomaliesor mixed gauge—Lorentz anomalies.What about possible one-loop anomaliesin the Kählersymmetry?It hasrecentlybeendemonstrated[8—111thatboth non-vanishingpure Kähler, mixed Kahler—gaugeand Kähler—Lorentzanomalies,in general,exist. We will not discussthe pureKähler anomaliesin this paper, sincetheirphysicalrelevanceis presentlyobscure.The mixedKähleranomalies,however,are known to lead to important phenomenologicaland cosmologicaleffects[8,10,11,16,17,15],and it is thesemixed anomalieswe now analyzein detail.

Webeginby consideringthe mixed Kähler—gaugeanomaly.Wewill, through-out this paper,work to lowestorder in supergravityandin Kählerbackgroundfields. Consideraset of matter chiral superfields,CP~,a subsetof which havenon-vanishinggauge charges,so chosenas to satisfy the condition that theyarefree of puregaugeanomalies.The relevantpart of the supergravity-matterlagrangianis given in (2.28), with the covariantderivativesgiven by (2.25)

544 G. LopesCardoso, B.A Ovrut/NuclearPhysicsB 418(1994)535—568

A

Fig. 1. (a)Fermioniccontributionto themixedKähler—gaugeanomaly.(b) Fermioniccontributionto the mixedKähler—Lorentzanomaly.

and (2.27).The graphin Fig. la givesthe fermioniccontributionto the mixedKähler—gaugeanomaly.This graphcanbe evaluated[19] andthe contributionto the associatedeffectivelagrangianis foundto be

iK2 1

= 2 TrFmnPm~?_&0ap, (3.1)(47r) Eo

whereFm,, is the covariantcurl of Vm, pmn = ~fmnlPF~~ andEo denotestheflat spaced’Alembertian. Implied in (3.1) is a sum over all chargedchiralmatter multiplets ~,. It can readily be seenthat (3.1) is anomalousunderKähler transformations.Using (2.20) we find that

1 2I’ _____ T E’’T’ T’ rmn= 2 (4~)2imi lrrmnr

which, since it doesnot vanish, implies that Kählersymmetry is broken.Now£~by itself is not supersymmetric.Before trying to find the one-loopgraphsthat will supersymmetrize(3.1), it is worth noting that there is a uniquesuperfieldexpressionwhose highest componentcontainsL~.This expressionis given by

= (4~)2f d4OTrWc,2—L~D2K+ h.c. (3.3)

Expanding(3.3) out into componentfields yields

£ = (4)2TrFmnFEÔaP

8(4m)2 TrFmnFm0—(4K~

02~2+ floK~)

(3.4)

where ... refersto termscontainingfermions.K land Kl02~2arethelowestand2_2 .0 0 componentsof superfieldK (~,,c~k,)respectivelygiven by

G. LopesCardoso,B.A Ovrut/Nuclear PhysicsB 418 (1994) 535—568 545

Kl = K(A’,T),

K~02~2= _gjj~~AlVmAi+ g17~1~J+ flo K +..., (3.5)

where ... standfor terms containingfermions. Now the term in (3.4) pro-portional to Tr Fm,,F~flis CP even and would haveto be generatedby aone-loop graph with scalar fields running around the loop. But, it is clearfrom (2.28) that there is no quadraticcoupling of A’ to either KI or Kl02~2.Therefore, thereare no one-loopgraphsthat can generatethe secondterm in(3.4)! It is, apparently,impossibleto constructa supersymmetricexpressionfor themixed Kähler—gaugeanomalyusingone-loopgraphsgeneratedfrom theKählersuperspacelagrangian.Therehavebeensomeattemptsto showthatthemissing graphsarisefrom gaugefield two-point functionsthroughthe regular-ization procedure.However, theseattemptsexplicitly breakgauge invariance

when K(A’, A’) is non-constantandmust, in ouropinion,be discarded.Beforeshowing how to perform a supersymmetriccalculation, we demonstratethatthe same problem exists for the mixed Kähler—Lorentzanomalies.

Again, the relevantpart of the supergravity-matterlagrangianis given in(2.28) with the covariantderivativesgivenby (2.25) and (2.27).The graphinFig. lb gives the fermioniccontributionto the mixed Kähler—Lorentzanomaly.This graph can be evaluated [19] and the contribution to the associatedeffectivelagrangian, stemmingfrom one Weyl fermionrunningin the loop, isfound to be

K2

= —i ~ (3.6)24(4m) t~owhere 7~mnba is the Lie algebra valued curvature tensor and ~mn a b =

if mntP~Rtpab If there are N chiral matter fermions in the theory, then (3.6)would simply be multiplied by N. It is easyto see using (2.20) that (3.6) isnot invariant under Kãhler transformations.Now, L~by itself is not super-symmetric.Beforetrying to find the one-loopgraphsthat will supersymmetrize(3.6), it is worth noting that there is a minimal superfield expression whosehighest component contains £~. This expressionis given by

2 p 1— K d40 W2 ‘ D2K h 3 7— 24(4ir)2 J aP~5

0 + .C.

Expanding(3.7) out into componentfields [20,21] yields

= 24(4~)27~mnbl~aE8ap

+ l92(4~)2Cm~~ Eo (4K~02~2+ EOKI)

(3.8)

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where ... refers to terms containing fermions and Cmnpq denotes the Weyltensor.Now the term in (3.8) proportional to C~~pqCmnP~is CP even andwould have to be generatedby a one-loopgraph with scalar fields runningaroundthe 1oop.But,again, it is clear from (2.28) that there is no quadraticcouplingof A’ to eitherKl or Kl02~2.Therefore,thereare no one-loopgraphsthat cangeneratethe secondterm in (3.8)! Again, it is impossibleto constructasupersymmetricexpressionfor the mixed Kähler—Lorentzanomalyusingone-loopgraphsgeneratedfrom the Kählersuperspacelagrangian.Furthermore,anyattemptto generatethe missing graphs from the graviton two-point functionusingthe regularizationprocedurewill breakLorentz invariance.

As paradoxicalas theseresultsmight seem,it is not too hardto understandtheir origin. In superfields,the gaugeprepotentialcouplesto matterasWere we to examinethe pure gaugeanomalythree-pointfunction, we wouldto lowestorder in V get as a result the supersymmetricexpression

Jd40TrW~2~D2V+ h.c. (3.9)

If the Kähler prepotentialalso coupledto matter as cI~te”~,then we wouldindeed havearrived at the supersymmetricexpressions(3.3) and (3.7) forthe mixed Kähler—gaugeand Kãhler—Lorentzanomalies.However, in Kählersuperspace wK(~i) = 0! It follows that there is no such direct coupling of Kto quadratic matter superfields and, hence, one expects difficulty computinga supersymmetric expression for the anomaly.Two things, then, seemclear.First, it must be possible to directly calculatea supersymmetriesexpressionfor the mixed Kähler anomalies.Secondly, it seems likely that we must go toa formalism where K couples to matter as ~ Weshow how to do thisin the next section.

4. Conventionalsuperspaceandtree-levelsymmetries

As discussedin the introduction, the lagrangiandescribingthe coupling ofD = 4, N = 1 supergravityto matter supermultipletshas, in general,a non-canonicalgravitationalkinetic energyof the form h (A,A )R~.In the previoussectionswe showedthat if we workedwith appropriatelyrescaledfields sothat h (A,A) = 1, thenthe tree-levellagrangianwas explicitly invariantunderKähler symmetry. However, computingthe one-loop supersymmetricmixedKähleranomalieswith this lagrangianis apparentlyimpossible,for the reasonsdiscussedabove. In this section,we go backto the genericsupergravity-matterlagrangianwhere h (A,A) � 1 andgravity is not canonicallynormalized.Wewill showthat, in this case,the Kählerpotentialcouplesto quantummatterinthegenericform ~te”~ andthataccordinglypureKählersymmetryis replacedby a mixedsuper-Weyl—Kählersymmetryat the tree-level.This, as we will see,

G. LopesCardoso, B.A Ovrut/NuclearPhysicsB 418(1994)535—568 547

is thefirst steptowardacoherentcalculationof mixedsupersymmetricKähleranomalies.

In the genericcase,the structuregroup of superspaceis takento be simplySL(2,C) with the associatedone-formgaugeconnection9~’BA = dZMcI5MB ~ Inaddition,oneintroducesthe supervielbeinEMA andthe associatedone-formsEA = dZMEM~ Solving the Bianchi identities subject to a set of constraints[22], one finds that all componentsof the torsion and curvature may beexpressedin termsof asetof superfieldsand their coordinatederivatives:

superfield R,R~ Gaa Wc,py, ~ (4.1)

Since they are obtainedby solving the Bianchi identitieswith respectto avery different set of constraints,the R, Gc,~,and Wc,~~in this section aredifferentthan, andnot to be confusedwith, the field strengthsolutionsof theBianchi identities in Kähler superspace.The relation betweenthem will bediscussedin detail later. If we further assumethat thereis an internal gaugegroup, then we must introduce yet another one-form gauge connection A,, b =

dZMAMa~ Solving the Bianchi identities now introducesa new superfieldstrength,W,,.

Using these superfields, one can write down the tree-level superfield la-grangian in this superspace. Again, it consists of three parts, each specified byafundamentalfunction. The first partis the supergravity-matterkineticenergyterm given by

= —3K2Jd4OEexp(_~1c2K(bi,~jt)). (4.2)

The secondpart is the potentialenergyterm given by

£pE = ~Jd4O~W(~t) +h.c. (4.3)

Finally, the Yang—Mills lagrangianis

1 f~4~E~ \ yz,aaTxy b

L,YM = ~ J U U~J‘~“iJab VV VVc, + ii.C.

Note that E is the superdeterminantin conventionalSL(2,C) superspaceandis not identicalto the superdeterminantin Kählersuperspacediscussedearlier.Now, the total tree-level lagrangianpossesses,by construction,both gaugeandLorentz invariance,as did the Kählersuperspacelagrangian.However, itis clear that under the Kãhler transformationK

2K —* K2K + F + F (4.2),for example, is not invariant. That is, the conventionalsupergravity-matterlagrangian does not possess pure Kähler symmetry. Is there anything thatreplaces it? After a short discussion, we will show that indeed there is, as iswell known [1,2,7].

The conventionaltorsion constraintsare left invariant [23] undera set ofsuper-scalingtransformationscalled super-Weylor Howe—Tuckertransforma-tions. They changethe forms of superspaceas follows:


EMa e~rEMa

EMe2E_~ (EMa + ~EMb (e~)~

EMe2E(EM~~+ ~EMb (~) , (4.5)

whereL’ and~ are superfieldparameterssubjectto the chirality conditions

Va~0. (4.6)

It is not hard to show that under (4.5)

E —* E e2(~~. (4.7)

Chiral superfieldsand K (k,, ~) are invariant under super-Weyl transfor-mations. It is clear that (4.2), for example, and, hence, the conventionalsupergravity-matter lagrangian is not invariant under pure super-Weyl rescal-ings. However, under combined Kähler andsuper-Weyltransformations(4.2)transformsas

4 = _3K_2fd4OE~e_K2~(~I3

= —3K2f d40Ee2~+e2K’/3e_~’+I4’)/3. (4.8)

It is clear that if we take

~=éF(4.9)

then lagrangian (4.2) will be invariant. Similarly, if one demands that under

Kähler transformations,the superpotentialtransformas

W—*e~’~W, (4.10)

then, using the fact that under super-Weyl rescalings

oR = -2(2E-~)R- ~

Wc,4e

3~rWc,, (4.11)

it follows that £pE, (4.3), and LYM, (4.4), will also be invariant under com-binedsuper-Weyl—Kählertransformationswith E given in (4.9).We conclude,thereforethat insteadof possessingpure Kähler symmetry,the conventionalsupergravity-matter tree-level lagrangian exhibits a mixed super-Weyl—Kählerinvariance [1,2,7].

G. LopesCardoso, B.A OvrutINuclearPhysicsB 418 (1994) 535—568 549

Before consideringthe componentfield lagrangian,it is very helpful to firstreexpress£o in (4.2) and £pE in (4.3) in a way that exposes, at the superfieldlevel, theverticesrelevantto the anomalycalculation.As in Kählersuperspace,LYM in (4.4) doesnot contribute to Kähler anomalousone-loopirreduciblegraphsand will, henceforth,be ignored. To exhibit the relevantvertices,weperform a backgroundfield- quantumfield splitting of the chiral matter su-perfields as well as of the supergravitysuperfields in such a way that, allrelevantquantumsuperfieldsareinvariantunder super-Weyl—Kählertransfor-mationsof the backgroundsuperfields.Thisproceedsas follows. First performabackgroundfield-quantumfield splitting of the chiral matter superfieldsas

~I~i= ~‘~ci+ ~/Ji, (4.12)

where ~, is a background superfield and çb1 is the fluctuating quantum su-perfield. Henceforth, for simplicity of notation, we will suppressthe indexi. Recalling that c1 does not transform under super-Weyl—Kählertransfor-mations, and demandingthat P~also not transform, it follows that undersuper-Weyl—Kählertransformations

0q5=0. (4.13)

To proceed,we must discussthe backgroundfield-quantumfield splitting ofthe supergravity superfields. It is well known [24] that the superdeterminantE can be written in termsof a set of supergravityprepotentialsHA andp,where p is a chiral superfield. In terms of these prepotentials E factorsinto

E = E[HA]eI~, (4.14)

whereE is a complicated function of HA whose exact form is irrelevant forthis paper. What is relevant is that under super-Weyl transformationsE isinvariant.Superprepotentialp, on the otherhand, transformas

(4.15)

under super-Weyl transformations,which correctly reproduces(4.7). Recallthat under super-Weyl—Kähler transformations I satisfies (4.9) and,therefore,that

p—÷p+~F. (4.16)

We now performabackgroundfield-quantumfield splitting of p as

PPc+Pq, (4.17)

wherePc is a chiral backgroundsuperfieldand Pq a fluctuatingchiral quantumsuperfield.Using (4.16), anddemandingthat

(4.18)

550 G. LopesCardoso,B.AOvrut/NuclearPhysicsB 418 (1994) 535—568

it follows from (4.17) thatunder super-Weyl—Kählertransformations

~ (4.19)

wherethe subscript~ meansdifferentiationwith respectto thesefields andsubscriptc meansevaluationat the classicalbackgroundsuperfield.Note thatthis transformation mixes quantum gravity superfields with quantum mattersuperfields. Amongst other things, this implies that the relevant part of thenaive path integral measure, [D~][V~t][Vpq][Dp~], is not invariant undersuper-Weyl—Kählertransformations.This can be cured by introducing theappropriate jacobian determinant but there is an easier approach, which wewill now follow. This consistsof defininga new chiral quantumsupergravitysuperfield p~1by

P~= Pq + ~(lnW)~~,~cb+ ~(lnW)~~~2 + .,. (4.20)

Combining (4.10), (4.19) and (4.20), it follows that under super-Weyl—Kählertransformations

Op~= 0 (4.21)

In this new variable, the relevant part of the path integral measure,being[Vq~] [Dçbt] [Dp~] [Dp~] with unit jacobian, is indeed invariant under super-Weyl—Kähler transformations.Henceforth,we will use~ andp~as the fluctu-ating quantumsuperfields.

We will first display the couplings of unchargedquantum matter super-fields q~to backgroundsupergravityandbackgroundmatterfields. ExpandingK (cP, 1 f) to quadraticorderaroundclassicalbackground~ gives

K = K~+ ~ + K~t~~t+ Kt~~t~ + ~ + ~K~t~tc~t2 +...,

(4.22)

where,again, the subscript~ or ~t meansdifferentiationwith respectto thesefields andsubscriptc meansevaluationat the classicalbackgroundsuperfields.A similar expansionholds for the holomorphic superpotentialW. Substitutetheseexpansionsinto lagrangians(4.2) and (4.3) anduse (4.14), (4.17) and(4.20) to express the superdeterminant. Setting the terms linear in ~ to zeroleadsto the following equationof motion for backgroundfield ~

(~2— 8R) [e_/c2~~d/3G4c]= 0, (4.23)

whereGdenotestheKähler invariantcombinationG = K + ln W + ln W.Notethat the equationof motion (4.23) is manifestlycovariantundersuper-Weyl—Kähler transformationsof the backgroundsuperfields.Also note thatwe haveignoredapossiblecontributionof lagrangian (4.4) to the equationof motionfor ~, (4.23). As statedearlier, such matter couplingsdo not lead to super-Weyl—Kähleranomalouscontributionsstemmingfrom irreduciblegraphs.Theresultinglagrangiannow becomes

G. LopesCardoso,B.A Ovrut/Nuclear PhysicsB 418 (1994)535—568 551

£~+ £PE = _3K_2f d4OEce_K2KCI3+ (~fd4e~w~+ h.c.)

+ fd4OEce_1c21Cd/3[(~~tt — çG~G~~)~

+~(~—

+~~ — çG~tG~t)~t2] +..., (4.24)

wherewe havedroppedall termsdependingon p~,sincewe will not computegravitationalradiativecorrectionsin this paper.For the samereasonwe havesimply replacedHA by H~in E. Lagrangian(4.24) displaysthe couplingsofunchargedquantumsuperfieldsq~to supergravityandbackgroundmatterfields.We nowproceedto displaythe couplingsof chargedquantumsuperfields~. Wewill throughoutthe paper assumethat all chiral matter superfieldsthat carrynon-vanishingchargeunder someYang—Mills gaugegroup havea vanishingbackground. Then, the corresponding lagrangian reads

= _3ic_2f d4OE~e_c2~~d/3+ ~ + ... (4.25)

Note that termsproportionalto ~52andto do not appear.Both lagrangians(4.24) and (4.25) are manifestly invariant under super-Weyl—Kählertrans-formations (4.9) of the backgroundsuperfields. Also note that the terms in(4.24) proportionalto ~52and comemultiplied by the super-Weyl—Kählerinvariant, but backgroundfield dependentmetrics ~ — ~K~~ ~ andG~t~t~— ~K2 ~ respectively.

We are now readyto expandlagrangians(4.24) and (4.25) into componentfields usingthe standardtechniques.The relevantpart of the lagrangiangivenin (4.24) reads,to lowestorder in the backgroundfields,

£o/e= ... ±(~AAt — ~GAGAt) exp(—~K2K~I)

[{~AA (7?._ -~--(4K l92~2 + E0K

gm~DmAVnA

-~ (Xaml5mX- (15c,~)amy)}I (GAA—cGAGA) 1 / K2

C —(1~——(4K~2_2

t. (gAAt — cGAGAt) 12 ~, 2 8 0


+ E0KI)) + ~iambm] A2 + h.c.}]

+... (4.26)

wherethe covariantderivativesare

~mA’ = [am — ~i (bm — i2K2am)] A’ (4.27)

and

~ i....j~ 22 ‘~ imX — kUm — Wm— ~K CmJX

andwhere

Cm= am — K2~bm. (4.29)

Note thatbm,am,(Jim andCm areall evaluatedat classicalbackgroundfield val-ues. We omit the subscriptc to preventaconfusingproliferationof notation.The appearanceof the connectionsbm andam in (4.26) and, in particular,theappearanceof theexactcombinationcm requiressomeexplanation.In the con-ventionalsuperspacex’ doesnot transformunderpureKàhlertransformations.However, undersuper-Weyltransformationsx’ transformsas

0~t=_(2~_I)~~~. (4.30)

Similarly, under super-Weyltransformationsit canbe shownthat

Obm = _3iam(r_I)~ . (4.31)

Note thatthis variationof bm doesnot compensatethe variationof x’. It neednot, since the lagrangianis not purely super-Weyl invariant. If we now alsoperform a Kähler transformation, am transforms as in (2.20). However, if wefurther demand that (4.9) hold, we thenfind from (2.20) and (4.31) that

0cm = K_2~am(FF)~, (4.32)

which exactly compensates(4.30), as it must. That is, the exactcombinationCm = am — K2~jbm appearingin the covariantderivative (4.28) acts as aconnectionwhich insuresinvarianceundermixed super-Weyl—Kählertransfor-mations.Also notethatthecombinationbm2iK2am appearingin the covariantderivativeDmA in (4.27) is invariant undermixed super-Weyl—Kählertrans-formations, as it must, since the componentscalarfield A doesnot transformundermixed super-Weyl—Kählertransformations.

The component expansion of lagrangian (4.25) reads, to lowest order in thebackground fields,

G. LopesCardoso, B.A Ovrut/NuclearPhysicsB 418 (1994)535—568 553

£o/e= ~ {~A [~_~ (4K~02~+ E0KI)]

gmm~~Av,,A+ AT(T)AD(T)

~ (x15~Y-(25c,~)amX)}

+... (4.33)

wherethe covariantderivativesare

= (am + jV~T~’~— ‘ (bm — 2iK2am) )A’,

VmX’ = (am — Wm+ jV~rT~’~— 2K2c)x’ (4.34)

Again, v~,bm,am,wm andCm are to be evaluatedatclassicalbackgroundfieldvalues.Note that many terms appearingin (4.26) do not appearin (4.33)sincewe haveset chargedbackgroundfields to zero.

5. Super-Weyl—Kähleranomaliesin conventionalsuperspace:The Yang—Millscase

We are nowpoisedto computemixed super-Weyl—Kähler—gaugeandLorentzanomalies,in exactanalogywith the mixedKãhler—gaugeandLorentzanoma-lies discussedin section 3. A key difference in thesecalculationsis that nowthe anomalousfermionic current couplesto connectionCm rather than to am.We will, in this section,considerthe mixed super-Weyl—Kähler—gaugeanomalyonly. The mixedgravitationalanomalieswill be discussedin the next section.We will work to lowestorder in the supergravity-Kählerbackgroundfields. Asbefore, we considera set of matter chiral superfields,~Ij, the subsetof whichwith non-vanishinggaugechargesso chosenthat they are free of gauge andLorentzanomalies.The relevantpart of the supergravity-matterlagrangianisgiven in (4.33), with the covariantderivativesgiven in (4.34). The graphin Fig. 2a gives the fermionic contributionto the mixed super-Weyl—Kähler—gaugeanomaly.This graph canbe evaluated [19] and the associatedeffectivelagrangianis found to be

£ 2i K2 TF pmnl0pX ~‘n ‘2 r mn Cp.~ ytlr; Eo

In this expression,and in all resultsfor the remainderof this paper, all fieldsareto be evaluatedat their classicalbackgroundvalues.We omit the subscriptcto avoid a confusingproliferationof notation.Implied in (5.1), as everywhereelsein this section,is asumover all chargedmattermultiplets ~,. It is readilyseenthat (5.1) is anomalousundersuper-Weyl—Kählertransformations.Using(4.32), we find that


ci Kj8~~+~UoKcI

umAn

Fig. 2. (a) Fermioniccontribution to the mixed super-Weyl—Khhler-gaugeanomaly. (b) Scalarcontributionto themixedsuper-Weyl—Kähler-gaugeanomaly. (c) Fermionicandscalarcontribu-tions to themixed super-Weyl—Kahler-gaugeanomaly.

OSW_KLX = 1 1 2 ImFTrFm,,Pm~1 (5.2)2 (4ir)

which, sinceit doesnot vanish, implies that the super-Weyl—Kählersymmetryis broken. Now £~by itself is not supersymmetric.Beforetrying to find theone-loopgraphsthat will supersymmetrize(5.1), it is worth noting that thereis auniquesuperfieldexpressionwhosehighestcomponentcontains£~. Thisexpressionis given by

£ = 4(4~)2Jd20 Tr ~ (4Rt — !~~D2K)+ h.c. (5.3)

Expanding(5.3) out into componentfields yields

K2 TF pmn1~p

— 2 r mn Cp3 (4ir) Eo

1 K2 1

12(4)2TrFmnFfl (4K~02~2+ EKI)

l2(4~)2TmnFEO~+...~ (5.4)

where ... refers to termscontainingfermionicandKl02~2andKI are given in(3.5). Can one find graphsthat will producethe secondand third termson


Fig. 3. Thegaugetwo-point function.

the right-handsideof (5.4)? Unlike the casein Kählersuperspace,the answernow is affirmative. To begin with, we considerthe graph shown in Fig. 2bwhere the scalar fields A’ run around the loop. The top vertex originatesthrough the quadraticcoupling of A’ to 4K I Q2~2 + E0K in the componentfield lagrangian (4.33). This is preciselythe coupling that was conspicuouslyabsentin Kähler superspace.Evaluationof this graph leadsto the followingadditional contributionto the effective action:

K2 1

= 12(4ir)2 TrFmn Fm~~(4K~0202 + EK~), (5.5)

precisely the second term on the right-handside of (5.4). The origin andmeaningof the third term in (5.4) is a bit more subtle. The relevantgraphsare shown in Fig. 2c. The top vertices originate through the fermion andscalarkinetic energyterms in (4.33). Evaluationof thesegraphsleadsto thefollowing contributionto the effectiveaction:

= ~3()2 TrFm,,Fm~~~—(~R~). (5.6)

This is exactly the third term on the right-handside of (5.4). Thereare twoissuesthatmustbe addressedas regardsthisterm.Firstof all, onemightwonderwhy we did not considersuchgraphsin our discussionof Kähler superspace.The reasonis that in Kähler superspace,1~doesnot transformunder Kählertransformationsand, hence,thesegraphsare irrelevant for the discussionofthe Kähleranomaly.In the conventionalsuperspaceof this section,however,R.doestransformunder the combinedsuper-Weyl—Kählertransformationsand,therefore, these graphs are relevant. Secondly, we want to point out that thegraphs in Fig. 2c are closely relatedto the gaugetwo-point function and theassociatedtraceanomaly [25—27].The gaugetwo-point function is computedfrom the graphsin Fig. 3. Prior to renormalizationthesegraphsare evaluatedto be

£2 = — (4~)2 ~ (! ln p22) Tr FmnFm° (5.7)

usingdimensionalregularization.The energy—momentumtensoris definedby

2 OSTm,, = ~ ogmn~ (5.8)


Varying the action associatedwith (5.7), one finds that the trace of Tmn ~S

finite in four-dimensionsandgivenby

Ttmm =2(4~)2T~mnF. (5.9)

This result occursbecausethe variationof ~ Tr Fm,, F~flin 4 — 2� dimen-sions is non-vanishingandproportionalto e. This multiplies the I/c poleandyields the finite result in (5.9). Now, this is true for the non-renormalizedthe-ory. However,when (5.7) is renormalized,the pole 1/f 15 removed,and it isclearthat the renormalizedT”’ m = 0 in four-dimensionswhenevaluatedfromthe graph in Fig. 3. Now addgraphs2c to the renormalizedgaugetwo-pointeffectivelagrangian.The result in four-dimensionsis

= 4()2ln[2~02 (l_~—1~)]TrFmnFmn. (5.10)

It follows from (5.8) that the associatedtraceof the energymomentumtensor

Ttmm =

2(4)2TrFmnF (5.11)

which is, up to the sign, the sameas in (5.9). The right-handsideof (5.11) isthe correctexpressionfor the one-loop traceanomaly [27] of gaugeindexedchiral multiplets, eachof which hastwo realscalarsandoneWeyl fermion.Notethat it arisesentirely from varying gmn in the three-pointeffective lagrangian(5.6) and not in the renormalizedtwo-point lagrangian.Hence, £XA is theterm in the renormalizedeffective lagrangianthat gives rise to the one-looptraceanomaly.

We conclude,then, that the one-loopthree-pointgraphsshownin Fig. 2a—cgive rise to the supersymmetriceffective lagrangian(5.4) and, equivalently,tothe superfieldexpressionfor the lagrangian (5.3). Under super-Weyl—Kählertransformations

K2OK = F + F,0R~= —~D2F, (5.12)

it follows that

= 1 2 IdeTr W~W,,F+ h.c. (5.13)2(4,v) j

which is non-vanishing.Hence,expression(5.3) representsthe supersymmet-nc mixed super-Weyl—Kähler—gaugeanomalousterm in the effective action.We point out that expression (5.3) could also have beenobtainedby di-rect three-pointsupergraphcalculationsusingthe gaugeandsupergravitationalprepotentialformalism [24,9]. We closethis sectionby showinghow to trans-form the anomalouslagrangian(5.3), obtainedin theconventionalsuperspace


formulation,overinto aresult in the Kählersuperspaceformulation.Both for-mulationsof supergravity-matterare relatedby particularsuperfieldrescalingsof the underlying torsion constraints[5]. This implies, amongother things,that thesuperfieldsR and Gc,~,of the conventionalsuperspaceformulationarerelatedto the superfieldsR and G,,a of the Kähler superspaceformulationby[5]

R-~R-~2K

Gc,~—k Gc,,~— ~ [D~,~] K, (5.14)

wherewe work to linearizedlevel in K only. It follows from (5.14) that

bm “ bm + 2iK2am. (5.15)

Applying (5.14) to the anomalouslagrangian (5.3) yields

£= (4)2JdOTrWWaER

—— K 2 fde Tr W~’Wc,—D2K+ h.c. (5.16)

8 (4ir) Eo

Notethat lagrangian(5.16) is given in termsof KählersuperspacesuperfieldsW~R~,K~.In Kähler superspacethe superfieldRt transforms,accordingto(2.6),homogeneouslywith Kähler chargeWK = —2 underKähler transforma-tions. It follows that the term proportionalto Rt in (5.16) is invariantunderKähler transformations.Hence, it canbe discardedandthe Kähleranomalyisentirelygiven by the secondterm in (5.16). That is

1 2 1£anom = —— K 2 I d40 Tr W~’Wc,—D2K+ h.c. (5.17)

8 (4ir) j Eo

Comparingthis expressionagainst lagrangian (3.3) shows that both agree!Hence,it follows that aconsistentway of generatingsupersymmetricone-loopresults in the Kähler superspaceformulation is to first perform manifestlysupersymmetriccalculationsin the conventional formulation, and then totransformthem over into the Kähler superspaceformulation by using thesuperfield rescalingsof the torsion constraints given in ref. [5]. We will,in the next section, apply this strategy to the computationof the mixedsupersymmetricKähler—Lorentzanomalyin Kählersuperspace.

Finally, recall that TrT(~’)T(s)= ~ where each H is afactor group of the total gauge group, g~is the gauge coupling parameterassociatedwith H, R sums over variousrepresentationsof multiplets withineachfactorgaugegroup and c~is the quadraticCasimir coefficient of the Rrepresentationof H. It follows that (5.17) canbe written as


£ = —~ (4,t)2 ~g~f d40( ))2.—D2 {~c~K]+ h.c. (5.18)

In this paper, we have,for simplicity, not discussedKähler anomalieswithinternal vector supermultiplet loops. Similarly, we have ignored anomaliesinvolving the sigma-model Christoffel connection [28,11,8,10]. It seems clearthat thesecalculationswill proceedin a mannersimilar to that discussedabove. Assumingthis to be the case,the result (5.18) canbe easilyextendedto give the completesupersymmetricKähler andsigma-modelanomaliesforbothchiral andvectorsuperfieldinternalloops. The final result is [11,8,10]

1 1 \T’ 2 fd4O(WH(rh2 1 D2a I

x [c~K2K + ~c~(2lndetg~ — K2K)] + h.c., (5.19)

wherecV is the quadraticCasimir coefficientof the adjoint representationofH andg~is the Kählermetric with indices restrictedto the R representationof H.

6. Super-Weyl—Kähleranomaliesin conventionalsuperspace:The gravitationalcase

We now turn to the computationof the mixed super-Weyl—Kähler—Lorentzanomaly.We will, again, work to lowest order in supergravity-Kählerback-ground fields. All resultspresentedin this section are for onechiral mattersupermultipletrunning in the loop. The generalizationof theseresultsto A/~chiral quantummattersupermultipletsis obtainedby multiplying all the resultsofthis sectionby .A/~.The relevantpart of the supergravity-matterlagrangianisgiven in (4.26)with the covariantderivativesgivenin (4.27) and (4.28). Thegraph in Fig. 4a gives the fermionic contributionto the mixed super-Weyl—Kähler—Lorentzanomaly.This graphcanbeevaluated[19] andthe associatedeffective lagrangianis found to be

— 1 ~, b~mna1a~c 6 1

X — 3 24 (4ir)2 mna bE p~

Proceedingby analogywith the previouscaseof the mixed gaugeanomaly,we now also evaluatethree-pointgraphswith scalar fields A, A running inthe loop. Unlike the previouscase, however, there are now A2 and A2 aswell as AA vertices to consider.These verticesfall into two categories,the“conformal” verticesAA and the “conformal breaking” verticesA2 and A2The conformalverticesarisefrom the ~t/ term in lagrangian(4.24), which is

G. LopesCardoso,B.A Ovrut/NuclearPhysicsB 418(1994)535—568 559

C1 Kc10202+*UoKcl

A/AAA~rs

a) b)

gm gtpc)

~ g~,,R,Kc~9202+*UoKcI

R, K~I92G2+3.9,,K~I R, Kc~O2G2+*UoKcI

d)

Fig. 4. (a) Fermionic contribution to the mixed super-Weyl—Kähler—Lorentzanomaly. (b)Scalar contribution to the mixed super-Weyl—Kahler—Lorentzanomaly constructedfrom con-formalvertices, only. (c) Fermionicandscalarcontributionsto the mixed super-Weyl—-Lorentzanomalyconstructedfrom conformal vertices, only. (d) Graphscontributing to the mixed su-per-Weyl—Kähler—Lorentzanomalyconstructedfrom non-conformalvertices.

invariantunderthe conformaltransformations.Thesearedefinedby arbitrarysuper-Weyltransformations,as given in (4.7), accompaniedby the superfieldrescalingsof the quantumfields ~ and~t, given by

—+ e2~

~ e~r~t. (6.2)

From (6.2) it follows immediatelythat the q~~2and terms in lagrangian(4.24) are not invariant underthe conformaltransformations.Hence, the A2

560 G. LopesCardoso,B.A Ovrut/NuclearPhysicsB 418(1994)535—568

and A2 vertices in the componentlagrangian (4.26) are of the conformalbreakingtype.

We begin by evaluatingthree-pointgraphsconstructedfrom the conformalverticesonly. Thesegraphsare shownin Figs. 4b and 4c. The graphin Fig. 4bhas scalarsrunning around the loop. We find that its contribution to theeffectivelagrangianis

2K dC ~mnpq d lGA l2.24’(4ir)2 M mnpq — M

~ (4K1 02~2+ E0KI) , (6.3)

whereGB denotesthe Gauss—Bonnetcombination

GB = C~,,pqCm!l~o~— 2fl~mn1~m~+ ~2 (6.4)

andwe havenot computedcoefficient dM as it is irrelevantto the physicswewishto discuss.Notethat the only Lorentz invariantgravitationalcombinationsappearingin (6.3) are the squareof the Weyl tensor,Cm,,pq, and the Gauss—Bonnet combinationGB. Also note that theseare the only quadraticgrav-itational combinationsthat transformhomogeneouslyunder arbitrary Weyl—rescalingsof the space-timemetric gc,,,,. The fact that only thesetwo combi-nationsappearin (6.3) is a consequenceof the conformal propertiesof theAA verticesof the graph shown in Fig. 4b. Similarly, the fermion andscalarloop graphsof Fig. 4c canbe computed,andtheir contributionis foundto be[25,26]

— 1 (~ ~mnpq ‘GB~ 1 6 5X,A — 6 24. (4~)2 ~ mnpq — ~ ) ~j

Once again, only the two gravitational combinationscompatible with theconformal symmetryof the verticesappearin (6.5).

As before in the gaugecase,the contributionof (6.5) to the renormalizedtrace of the energy—momentumtensorcan be evaluatedusing (5.8). We findthat

T~~0~ffm— 1 1 (ç~, (-‘mnpq — ~GB’~ ~66m — 24 (4~)2 ~ mnpq 2 ) ~,

which is the well-known gravitational contribution to the one-loop traceanomalyfrom conformalscalarsand Weyl—fermions [25,29]. Note that (6.6)can be put in the form

Tconfmm = J~ 1 2 (3bcCmnpqCm~~ b,~GB), (6.7)

24 (4ir)

where

G. Lopes Cardoso, B.A Ovrut INuclear Physics B 418 (1994) 535—568 561

b~= ~(Ns + 3NF),

= ~~(Ns + ~NF) (6.8)

and N~( = 2) is the numberof scalarsfields and NF ( = 1) is the numberofWeyl fermions.The subscriptc indicatesthatthesearethe traceanomalycoef-ficients associatedwith achiral supermultiplet.We introducethesecoefficientssoas to “tag” thepart of our resultsthat arisefrom the traceanomaly.Adding(6.1), (6.3) and (6.5) we find that

i 2 1

— — ~1K b ~mn “—~---a~c— 3.24.(4~)2 mna bE ~

+ 12.24. (4ir)2 cm,,pqcmn~~_[6b~7~+ dMK2 (4KI02~2+ E0KI)]

12.24. (4)2GB {2b~+ (dM— l),c2 (4KI

02~2+ LI0KI)]. (6.9)

Using the identities [20]

1 (c~ r’mnpq ‘D b ‘5mn a’\ — w2

~ t,~’—mnpq’.-~ — t”-mna “- b ) — afly 02

GB = ~ + (~2— 8R) (G~c,— 4RtR)] 02 + h.c. (6.10)

it follows that (6.9) is the componentfield expansionof the following superfieldlagrangian

= 3 (4)2 f ~ ((3b~— b~)Rt— ~D2K)

12. (4~)2 fd0 (~2— 8R) (G~,,— 4RtR)~_

x (b~Rt— ~(dM — l)D2K) + h.c. (6.11)

We conclude,then, that the one-loopthree-pointgraphsshownin Figs. 4a—cgive rise to the supersymmetriceffectivelagrangian(6.9), or, equivalently,tothe superfield lagrangian (6.11). Note that the backgroundfield dependentmetric gAAt — ~K2GAGAt, which multiplies the AA-vertices in (4.26), hasdroppedout in the effectivelagrangian (6.9). Hence,the anomalycoefficientin (6.11) is backgroundfield independent.Using (5.12), it follows thatundersuper-Weyl—Kählertransformations

0~conf= 72. (4ir)2 fde [8(1 + 3b~— b~)W~fl~

— (dM — 1 + b~)(~2— 8R) (G~c,— 4R+R)] F + h.c. (6.12)

562 G. LopesCardoso,B.A Ovrut/NuclearPhysicsB 418 (1994) 535—568

whichis non-vanishing.Hence,expression(6.11) representsthe contributiontothe supersymmetricmixed super-Weyl—Kähler—Lorentzanomalyfrom graphsconstructedout of conformalverticesonly.

Let us now turn to the contributionsfrom triangle graphsconstructedoutof the conformal breakingA2 and A2 vertices.The only possible graphsareshownin Fig. 4d. Thesegraphscontributethe following non-local term to theeffective lagrangian

2 2 2nonconf 1 1 I(~)I ~“R, K

~~(4~)2 ~, —-~-‘~~

02~2+Eo

~ (~+ K2 (4KI

02~2+ E0KI))] , (6.13)

whereq is given by

GAA — ~K2GAGA

q 1 2 (6.14)— ~K GAGAt

and is the factormultiplying the A2 vertex, as canbe seenfrom (4.26). Factorq is backgroundfield dependent,but invariantunder Kähler transformationsK2K —p K2K + F + F, since it is given in terms of the Kãhler invariantcombinationG = K + ln W + ln W. Note that it is not q, but the constantvacuum expectationvalue (q) that appearsin (6.13). This occurs,becausewhencalculatingthe anomalouscontributionsfrom the graphsdiscussedin thispaper, one actually first expandsthe backgroundfield dependentmetrics inlagrangians(4.26) and (4.33) aroundconstantvacuumexpectationvalues(A),(A). The (R.— ~K2(4KI

02~2+ E0KI) legs in (6.13) arisethroughthe coupling

to the A2 and A2 vertices,whereasthe 1?- + K2 (4KI

02~2+ E0KI) leg arisesthrough the coupling to the AA vertex. Lagrangian (6.13) is obviously notinvariant under super-Weyl—Kählertransformationsand, hence, anomalous.The componentexpression (6.13) is containedin the following superfieldlagrangian

= 4~q)I2fd4O~2 [(Rt + ~D2K) (R + K2~2K)]

x.~_ (i~j— ~D2K) + h.c. (6.15)

Consequently,the total contributionto the mixedsuper-Weyl—Kähler—Lorentzanomalyis given by the sum of (6.11) and (6.15)

£total = ~ + £I~0u1c0~~. (6.16)

We point out that expression(6.16) could also havebeenobtainedby directthree-pointsupergraphcalculationsusingthe prepotentialformalism [24,9].


Lagrangian(6.16) representsthe mixed super-Weyl—Kähler—Lorentzanomalyin conventionalsuperspace.We would nowlike to transformit overinto Kählersuperspace.Following the manifestly supersymmetricproceduredescribedatthe endof section5, wenow apply thetransformations(5.14)to theanomalouslagrangian(6.16). The result is

£total — 1 1d~O18 ‘3b b’ ‘W2— 12(47r)2J [ “ c c/ afly

—b~(~2— 8R) (G~c,— 4RtR)] .~—Rt

+ f d”O~(RtR).~_Rtb’ W2

24~4~J[~(+ C ~)

— (dM —1 + b~)(~2 — 8R~(G2. — 4RtR~1~±~~D2K12 \. I \ JJ Eo

— I(q)12 (4~)2 fd~O~2(RtR) ~_D2K + h.c. (6.17)

where the lagrangian (6.17) is given in terms of Kähler superspacesuper-fields W~,p

3,,Gc,~,R, K. The first two terms in (6.17) are invariant underKähler transformations, as can be readily seen from (2.6). Hence, they can bediscarded. The Kähler anomaly is entirely given by the last two terms

= —~ (4jr)2 fd4o [~l + 3b~—

(dM 12 b~)(~2— 8R) (G~c,_4RtR)] ~_D2K

I(q)12 (4~)2 Jd~OD2(RtR)~~~D2K+ h.c. (6.18)

Thesetermsare obviously not invariant under Kähler transformations.Com-paring the term proportionalto in (6.18) with the minimal expression(3.7) showsthatboth agree! Lagrangian(6.18) thusqualifies to be called thesupersymmetricKähler—Lorentzanomalyin Kählersuperspace.In componentfields, the anomalouslagrangian(6.18) reads

— — 1K b ñmn a’apa— 24. (4~)2 mna bE P

+ 12.24.(4~)2 [(dM + 3bc)CmnpqCm~~— (dM — 1 + b~)GB]


~ [(~i02~2 + EOKI)]

1 I( )12K2 1+144 (4~)2 R~

2~—(4K~02~2+ E0KI) + ... (6.19)

where ... refers to terms containing fermions that we have not explicitlycomputed.Let us emphasizeagain that the coefficient (q) of the R

2K1-piecein (6.19) is backgroundfield dependant.This is to be contrastedwith thecoefficientsof the C~npqKI - andGBK -piecesin (6.19),which arebackgroundfield independent.

Now, recall that ~anomin (6.18) hasbeenderivedfor asinglechiral multipletrunningaroundthe loop. It is easilyextendedto ~ chiral multiplets by simplymultiplying by A/~.This canbe expressedby everywherereplacingK in (6.18)by >J~,n~Kwhere>~implies summationover all different chiral superfieldsectorsandn~is the numberof chiral superfieldsin eachsector.The anomaliesinvolving the sigma-modelChristoffelconnectioncannow easilybe computed.This is doneby replacing,in theconventionalsuperspacecalculation,>~by >~(K2n

5K — 3lndet ga), whereg, is the Kähler metric restrictedto sectors. Adding theseanomalies,(6.18) generalizesto

— I1d~O2 1 3b b’ w2 (dM —1 + b~)C 24. (4~)2J + ~— c) afly 12

(~2— 8R)(G~c,— 4RTR)] .~_D2( ~K~flsK)

+ 24. (4m)2 JdOWapy~_D(~2lndet~s)

+24.(4~)2fd0[12M8~4’~)]

xJ~~D2(~3lndet~s)

+ 2 .(4m)2fda ~2(RfR)lD2 (~(2lndet~s— K2nSK))

(6.20)

where we haveaddedsubscriptc to ~anom to indicatethat this is the resultassociatedwith chiral supermultiplets.Furthermore, it is also not difficultto extend our resultsto include gaugevector multiplets running aroundtheinternal 1oop. The only changesare that the b and b’ coefficients are nowto be evaluatedfor gaugevector multiplets,and the unknowncoefficient dM,which occurredfor chiral multiplets, here does not appear. Also note that,since the sigma-modelChristoffel connectiondoesnot coupleto gauginos,the

G. Lopes Cardoso, B.A Ovrut/NuclearPhysicsB 418(1994)535—568 565

ln detg termsdo not appear.The anomalouslagrangianassociatedwith vector

supermultipletsrunning aroundthe loop is thengivenby

1 1d~O2~3b b”W2

V — 24• (4,t)2 J 3~ “ V’ afly —

x(~2— 8R)(G~.— 4RtR)] ~-_ D~(~(_K2dimHK))

+h.c. (6.21)

where b~and b~.are the trace anomaly coefficients of (Cmnpq)2 and GBrespectivelydue to a singlevectorsupermultipletgiven by

= ~1~(3NF + l2Ny),

= ~ ~ + 62Nv) , (6.22)

whereNF (= 1) is the numberof Weyl fermions andNv (= 1) is the numberof vectorfields. Again, eachH is a factorgroup of the total gaugegroup anddimH is its dimension. Combining (6.20) and (6.21) we obtain our finalresult for the mixed Kähler—Lorentzand sigma-model—Lorentzanomaliesdueto chiral matter superfieldsandvector gaugesuperfieldsrunning aroundtheloop. It is given by

~anom= 36~(4,r)2 fd4e {w~p~~D2[3bV~(_K2dimHK)

+ (3b~+ dM)(~K2nSK)] + ~fl~~D2 (dM> 3lndet~s)

+SGB~D2[~~( ic2dimHK) — (b~+ dM —1) ~(_K2n5K)]

_SGB.~—D2((dM_ 1) ~3lndet~s)

+ ~2(RfR)~D2 l8~(q)I2~(2lndetg~— K2n~K) + h.c., (6.23)

Eo

whereSGB is the superfieldGauss—Bonnetcombination

SGB = 8J4’~~~+ (~28R)(G~c,4RtR). (6.24)

We reiteratethat b~,b~and b~,b,~.are given in (6.8) and (6.22) respectively,and arise from specific component graphs that also lead to a non-vanishingtrace anomaly.CoefficientdM is a coefficient that we havenot computed.Incomponentfields, the Kähleranomalouspartial lagrangian(6.23) reads

566 G. LopesCardoso,B.A Ovrut/NuclearPhysicsB 418 (1994) 535—568

panom— — 1K ~1V— d~G”D b i5mn a’ç~p— ~i i~i ‘2’ C I mna b p

K2

+ 12.24. (4~)2[(~c(~~M + 3b~)+ dimG 3bv)CmnpqCm”~

— (JtfC(dM — 1 + b~)+ dimGb~,)GB]

~ ([~x~02~2+ EOKI])

~ I( )12,c2 1(4~)2~2(4K~2 + E0KI) + ... , (6.25)

where dim G = ~H dimH is the dimension of the entire gaugegroup G.Note that this expressionhas the correct coefficient multiplying the termproportional to 1ZR. Also note that the completecoefficient of the (Cm,,pq)2

term, including the piece from GB, is also proportionalto JV~— dim G and isthe correctcoefficient.

We close this sectionwith a brief commenton the relationshipbetweenthe mixed supersymmetricKähler—Lorentz anomaly, given by (6.23), andthe moduli dependentthresholdcorrectionsto gravitationalcouplings [14] inZN orbifolds. Thesecorrectionswere computedfor the gravitationalcouplingconstantof C~npq.The coefficient of thesecorrectionswas found to be nonvanishingforZN orbifoldswith N = 2 spacetime sectors and to be proportionalto the traceanomalycoefficientsfor the differentfields coupledto gravity.Thisphenomenacan now be somewhatunderstoodin the field theory frameworkas follows. The coefficientof the C,~,nPqKI term in (6.25) has, as shownabove,its origin in the coefficients of the two distinctive terms (6.3) and (6.5).The coefficient (6.3) is determinedby the chiral anomaly (6.1) computedin the conventionalsuperspaceformulation,whereasthe coefficient in (6.5)is given by the trace anomaly (6.6). By comparingagainstthe results of theexplicit string loop calculationof ref. [14] it thenfollows that, in ZN orbifoldswith N = 2 space-timesectors,only acertainamountof the C~,,pqKI-termisremovedby the Green—Schwarzmechanism[30]. This amountis given by thecoefficient of (6.3), yielding a left-over proportional to (6.6), that is to thetraceanomalycoefficientsfor the different fields coupledto gravity. For ZNorbifolds with no N = 2 spacetime sectors, the entire amountgiven in (6.23)mustbe removedby the Green—Schwarzmechanism.Thiswill be discussedindetail in anotherpublication [31].

7. Conclusion

We have shown how to calculate the matter contributions to the mixedKähler—gaugeandmixedKähler—Lorentzanomaliesin amanifestlysupersym-metricway. Even thoughwehaverestrictedour analysisto matterloops only,


the contributionsfrom Yang—Mills andgravitationalfields runningin the loopcan, in principle, also be computedalong the samelines. All these contri-butions can be obtainedby first performing calculationsin the conventionalformulation of supergravity-mattersystems.In this formulation, there existsa prepotentialsuperfield formalism which enablesone to separatequantumdegreesof freedomfrom backgroundfields in a manifestly supersymmetricway. Calculationsin this superfieldformalism lead, by construction,to man-ifestly supersymmetricresults. Such superfield calculationsof matter loopsshould agreewith the componentfield calculationspresentedabove.All theseresultscan be transformedover to the Kähler superspaceformulationof thetheory. In the latter formulation, the tree-levelgravitationalkinetic energyisappropriatelyEinstein normalized.The Kähler anomalyin Kählersuperspaceis obtainedfrom the onein the conventionalformulationby specificsuperfieldrescalingsof the underlyingtorsion constraints[5]. Since theseredefinitionsare defined at the superfield level, the resulting expressionfor the Kähleranomalyin Kähler superspaceis manifestlysupersymmetric.

We would like to emphasizethe following. First of all, we have shownthat the proceduredescribed in this paper leads to the well known resultfor the mixed Kähler—gaugeanomalyin Kähler superspace[8,10,11], whichis responsiblefor non-holomorphicthresholdcorrectionsto the running ofgauge couplings in string effective field theories [8,10—15].Secondly, themixed Kähler—Lorentzanomalycontains, in general,a term proportionaltothe squareof the curvaturetensor,~ Its coefficient, however, comesout tobe backgroundfield dependent,in contrastto both the gauge caseandto thetermsin the mixedKähler—Lorentzanomalyproportionalto the squareof theWeyl tensor,C,~,,pq,and to the Gauss—Bonnet combination. As shown in refs.

[16,17], a term in the Kähler anomalyproportionalto ~ can drive inflationin the early universeand, hence, it can have very interestingcosmologicalconsequences.Finally, some field theoretical understandingof the modulidependent threshold corrections to gravitational couplings in ZN orbifolds [14]can be gained by comparing our results to the ones obtained in ref. [14] by anexplicit string loop calculation.Thesethresholdcorrections,computedfor thegravitationalcoupling of the C,~npq4erm, turn out to be proportional to thetraceanomalycoefficientsof the different fields coupledto gravity. From thefield theorypointof view this canbeunderstoodby noticingthat therearetwodistinct termswhichcontributeto theKähleranomaly,as describedabove.Oneof them comeswith a coefficient proportional to the traceanomalywhereasthe other one comeswith a coefficient proportionalto the chiral anomalyinconventionalsuperspace.It is this lattercontributionwhich getsremovedby theGreen—Schwarzmechanism [30], yielding thresholdcorrectionsproportionalto the traceanomalycoefficientsof the variousfields coupledto gravity.

568 G. LopesCardoso,B.A Ovrut/NuclearPhysicsB 418(1994)535—568

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Supersymmetric calculation of mixed Kähler-gauge and mixed Kähler-Lorentz anomalies

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