NUCLEARNuclearPhysicsB407 (1993)373—384 P H Y S I CS BNorth-Holland

TheS, U andAp parametersin theZaragozaproposalfor latticechiralgaugefermions

JoséLuis Alonsoa,PhilippeBoucaudb,JoséLuis Cortésa,FelipeLesmesa andElenaRivasC

a DepartamentodeFIsica Teórica, Universidadde Zaragoza,50009Zaragoza,Spainb LaboratoiredePhysiqueThéoriqueet HautesEnergies, UniversitéParis XI, 91405 Orsay Cedex,

FranceDepartmentof Physics, WashingtonUniversity,St. Louis, MO 63130-4899,USA

Received7 December1992(Revised 22 April 1993)

Acceptedfor publication 11 June 1993

UsingtheZaragozaproposalfor latticechiralgaugefermions,the 5, U andApparametershavebeencalculatedatone loop. It is shownthat the continuumvaluesfor thesequantitiescan be reproducedwithout requiring explicit fine tuning of counterterms.Furthermore,fermion fields doubling is not necessary.To the best of our knowledge, the Zaragozaproposalis theonly schemewhich hastheseproperties.

1. Introduction

A recentpublicationby DuganandRandall [1] exhibitedcalculationsof theone-loopperturbativecontributionto the S parameterin severalproposalsforimplementinglatticechiral fermions.In this paperwewould like to report theresultsfor 5, U, andz~pobtainedin the Zaragozaproposal[2—4].Thismodelis alsoa candidatefor the descriptionof latticechiral fermions,with one majordifferentiatingcharacteristicwith respectto thosestudiedin ref. [1]: doublerfermions arefree andmasslessinsteadof coupledbut very massive.This is tobe contrastedto the situationthat ariseswhenWilson termsareused.

In euclideanspacethe parameters5, U andz\p aredefinedby

aS 4e2[H~3(O) —H~Q(O)I,

aU 4e2[11

1’1(O)—TI~3(O)],

e2

2 2 [1111(O)—H33(O)]. (1)

s(Here (Wa,~u(p)W6,~,Lp)) ö,wllab(P

2) + (p,~p~—terms);a, b 1,2,3, Q andH’(O) = dH/dp2atp2= 0.)Theseparametersareof greatimportancebecausethey describeradiativecorrectionsto the electroweakinteractionobservablescomingfrom the Higgs sector[51.0550-32l3/93/$06.00 © 1993—ElsevierSciencePublishersB.V. All rights reserved

374 J.L. Alonso et a!. /5, U, andApin Zaragozaproposal

The nice thing about the 5, U, andAp parametersis that they are finitequantitiesthat canbe calculatedin the “target” (unregularized)theoryweareaiming to construct,i.e. a chiral gaugetheory,in particularthe StandardModel.The targetcontinuumlagrangianis

~target — ptarget—

+WLY~9p— 14)WL + WRY~(8,i— iA~)wR

+‘/‘L~YWR + WRY~tWL. (2)Here, togetherwith the fermiondependentterms,onehasthe lagrangian£~et

of thebosonicsectorof theStandardModel.For instance,consideradoubletwith massesm

1, m2, whereAm mi —m21<<

m1, m2 m. Using the target lagrangianone obtainsthe following one-loopvaluesfor S, U andAp

5target = 1

6irut~get— 2 (Am)

2— l5m m2

~~tar~et = 2 2 (Am)2. (3)l2ms m~

However,foragivenlatticeregularizationofthistargetchiraltheory (if indeedanysuchnon-perturbativeregularizationexists), thereis no guaranteethat theresultsgiven by (3) will be reproduced.Thereforetheseconditionsmust beimposed(as a necessarythoughnot sufficient) constrainton the latticetheoryin orderto correctly reproducethe “targettheory” in the continuumlimit.

In theirpaperDuganandRandallinvestigatedthe valueof theS parameterin(1) atoneloopin perturbationtheoryforseverallatticechiral fermionproposals.They discussedthe mirror fermions [6], the Smit—Swift model [7], and themodelof Borelli et al. [81. They concludedthatonly the last of theseproposalscorrectly reproducesthe continuumresult for S without need of explicit finetuning of counterterms.

Here, we will see that in the Zaragozaproposalno explicit fine tuning ofcountertermsis requiredin order to reproducethe continuumoneloop valuesfor the5, U andAp parameters.Thisresultpairsup ourmodelwith theproposalof Borelli etal. studiedin ref. [11. However, in that case,they were forced todoublethe numberof variablesin the regularizationprocess.

Among the modelswith Wilsonterms [6—9]two differentsituationsmayoc-cur: either fine tuning of countertermsis requiredto recoverthe continuumvaluesof S, U and Ap or not. When tuning of countertermsis necessary,twodifferentsituationscanbeencountered.For instance,therearecaseswherethesecountertermsaredeterminedby BRST invarianceat the continuumlimit (asin ref. [9]). In othercasesthe tuning of countertermscannotbefixed by gauge

J.L. Alonsoet a!. / S,U, andLxp in Zaragozaproposal 375

invariance(as in ref. [7]). For yet other models [6,8], no specific tuning ofcountertermsis requiredto get the one-loop targetvalues for 5, U andAp. Inthesecases,thecontinuumvaluesarealwaysachievedat the expenseof addingextravariables.(Whetheror not the regularizationgives the targetvaluesde-pendson how manyfermionsarecoupled).

TheZaragozaproposaldoesnot useWilson terms.The fermiondoublersaredecoupledin sucha way thatonehasthe target valuesfor 5, U andAp in per-turbationtheorywithoutfine tuningof countertermsandwith minimalfermionfield content.Explicit one-loopcalculationswill be exhibited.

2. The model

The Zaragozaproposal [2,4] belongsto a categoryof models whereinthechiral gaugesymmetryis explicitly brokenby the regularization.Thereforead-ditional countertermsmustbeincludedin orderto restoregaugeinvarianceinthecontinuumlimit.

The lagrangianfor the StandardModel is, in euclideanspace,

£ = £u~+ 4’ + £Wj,fl + £ghost + £fix + £Count, (4)

where

= —-~~Tr[U,~(x) + h.c.]

+ ~t(x)~(x) — 1)2]

(5)

standsfor the usuallatticetermscorrespondingto gaugefieldsandbosonfields.Here~ (x) is the StandardModel Higgs field in a2 x 2 matrixnotation.

The latticeactionfor free fermionsis

4, = (±)i~(x)y,~W(x±/~). (6)

The ±signsarecorrelatedandthe hermitiany matricessatisfy {~1j, yv} =

2c~uv.

TheDirac fermion ij~i is in thefundamentalrepresentationof SUL(2) 0 SUR(2)

WL (~), WR (~). (7)

The sum ~ runsoverall threefamilies of leptonsand 3N~families of quarks.It is importantto mentionthat in the interactionsbetweenthe fermionsand

thegaugeandbosonfields, describedby theterm ~ wewishto arrangethings

376 IL. Alonsoet a!. / S.U, andAp in Zaragozaproposal

in sucha way that only the physicalfermionis coupled.This is accomplishedby usingin £~flt only the quasi-localfield component

~i(x) = ~ çu(y). (8)yEh(x)

Here h (x) is the elemental hypercube that starts from “x” in the positive direc-tion,

= ~ (9)

+~(~iL(x)i�(x)yw~iR(x) +WR(X)ytj4)~(X)WL(X)),

in which

[~çt’(x) = [U~(x)—1]~(x+~), (10)

I~yi(x)=

with

U~(x)=U~(x)PL+U~(x)PR, (11)

U~(x) = exp(—i4(x)), U,~(x) = exp(—i4(x)). (12)

The gauge fields associated with isospin J+~and hypercharge B,L areintroducedas 4x = ~ (13)

~(x) = ~ ~(x)( 0

Herethecrt are the Pauli matrices, Y denotes the hypercharge of the differentfermionsandthe Yukawacouplings are

Y~= (J’~i 0 )~ (14)

Yyi2

Finally, Leount includes all the counterterms needed to recover BRST invari-ancein the continuumlimit. The quantum theory is defined through the pathintegral

= Z_if[dU][d~][dw][dwt][dc][de](...)exp [_~~],(15)

where [dc] [dë] stands for the measure of the ghost fields and Z is defined sothat (1) = 1.

IL. Alonsoet a!. / S,U, andAp in Zaragozaproposal 377

To understand why we must choose ~ to describe the fermion interactions in(9), wego to momentumspace.There~i is givenby

= F(q)yi’(q), F(q) =

f(q,~) = cos(q~/2), q E (—it, it]4. (16)

Note that ~ is the original field ~‘ modulatedby a form factor f which isresponsiblefor the decouplingof the doublersat treelevel. This is achievedbyforcing the form factor to vanishfor momentacorrespondingto the doublers(notethat f (0) = 1 andf (it) = 0). In particular (16) correspondsto themostlocal possiblechoice.

In summary,thisregularizationis hermitian,invariantunderdiscretetransla-tionsandrotations,andno redundantvariablesarerequired.The lagrangianisinvariantunderglobalchiral transformations,but thefull chiralgaugesymmetryis lost. To recoverBRST invariancein thecontinuumlimit, suitablecountert-ermsmustbe added.The lagrangianis constructedsuchway thatdoublersarenot coupled,sono Wilson termsare introduced.The lagrangianis also invari-antunder a shift transformationof thefermionfields. This shift-symmetryalsoassures,if acontinuumlimit exists,the non-perturbativedecouplingof thedou-blers. (Seeref. [4] for moredetails.)

3. Calculation of S, U and Ap

Before starting the calculation, we consider what can be learned by judicioususe of symmetries.

Thetargettheory~target as given in (2) is invariantundertheglobal transfor-mation

WL(x) —÷ gL~L(x),

WR(x) —~ gR~R(x),

UL(x) —p gLUL(x)g~,

UR(x) —~ gRUR(x)g~, with UL,R = exp(—iAL,R)

~(x) —i gLç5(x)g~, (17)

where g~= g h YL for g e SU(2), h eU(1) and

hYiR 0gR= ( 0 h)’2R). (18)

(Note that this transformation, when made local, is the gauge transformationSUL (2) 0 Uy (1) under which ~target is invariant by construction.)

BecauseS, U andAp arenot invariantunder (17) anycontinuumregulariza-tionof ~target respecting(17) will giveS = U = Ap = 0in the symmetricphase.

378 IL. Alonsoet a!. / S,U, andAp in Zaragozaproposal

Therefore, the S, U and Ap parameters measure the effects of the spontaneoussymmetrybreaking (SSB) SUL(2) ® Uy(1) —~ UQ(l).

In the continuum,the only effect of SSB ((q~)= v) relevant for the one-loop calculationsat handis the massterm in the fermionpropagatorgeneratedthrough the Yukawa coupling y, m = yv.

For any lattice regularization of the Standard Model, possible contributionsto 5, U and Ap parameters will arise from either explicit breaking of the globalsymmetry (17) or through SSB.

For the Zaragoza proposal, the global SUL(2) 0 Uy (1) symmetry is unbrokenin the regularized model. Since the doublers were not coupled, and the SSBcomes through a Yukawa coupling as in the continuum, we expect to get thetarget results for 5, U and Ap without tuning of counterterms.

Explicit calculations of the one-loop contributionsto 5, U and Ap have beenmadeandtheyconfirm the previousexpectations.

Introduce (a beingthe latticespacing)

2H~~(ap) 42

H~(ap) —~—[H,~(ap)—H,~(ap)]. (19)

In termsof thesequantities, one has

11 82HS (ap) 1

~ aaii~aaiis ~0 caPPj~= ~

Ii 32HU(ap) )

aO 12 ~ 9ap~8app ~iPci~PflJ’= (Up2 + U’p~)ô~~+ K”p~p~,

—a2 2 lim1J~(0)= Apô~,,+ K (20)4s m~a—~O

Here 5, U and Ap are those defined in (1) and 5’, U’ are extra terms thatmayappearbecauseH~,and H,~,for finite a, have non-covariant expressions.The KS, KU and ~ are not of interest. They appear because in (19) we havenot yet extractedtheö,~componentsfrom andH,~.

5, S’, Ap, U, and U’ are rathercomplicated integrals. Explicit expressionsare given in the appendix. Wecan rigorously prove (see the appendix for somedetails) that the calculation of those integrals leads in fact to the target results

IL. Alonso et a!. / 5,U, andAp in Zaragozaproposal 379

i.e.

S = —,

6it

- 2(Am)2lSir m2Ap = 2 2 (Am)2, (21)

l2irs m~

and5’ = U’ = 0.Thederivationof the previousresultfollows the samestepsusedin ref. [2] to

show the decouplingof the doublersin a chiral YukawamodelusingZaragozafermions. The calculationof 5, U andAp involvesa combinationof vacuumpolarizationcontributionswhich arefree of ultravioletdivergences.

One first decomposesthe momentumintegrals into two parts, an exteriorintegral over momentaaq~> �, and the remaining interior integralovera ballcenteredatthe origin in momentumspacewith radius~ suchthat� << 1.

One can show that the integral over the outer region, which includes all thedetails of the regularization, does not give any contribution in the continuumlimit. The reason is simple, since in this region the momenta of integration areof order 0(1/a), all the masses involved in the calculation can be neglected.Thereforethosequantities,vanishingin the masslesscase,suchasS,U andAp,will not receiveanycontributionfrom the exteriorintegrals.

As for the remaininginner region (laqi < � << 1) integral, onecan approx-imateall the verticesandpropagatorsby those of the continuum [note thatF(q = 0) = 1]. Thus the target resultsare reproduced.SeeappendixA fordetails.

4. Comparisonwith othermodelsandfinal remarks

Otherlatticeregularizationresultscanbe explainedeasily in termsof the waytheybreakthe global SUL(2) ® Uy (1) symmetry.

For instance,in the Smit—Swift model [7], the global symmetryis unbroken.Howeverthe Wilson—Yukawaterm introducesan additionaleffect of SSB (notproportionalto y) with respectto the continuum,due to the regularization.Therefore,oneshouldexpectsomenon-conventionalresultsfor thisregulariza-tion.

Indeedthisis whathappens,aswasfirst reportedin ref. [1] for theS parame-ter. All threeparametershave,atoneloop,r-dependentexpressionsS(r), U(r),Ap (r), wherethe r —~ 0 limit is 16 timesthe targetcontribution.

Theseresultsreflect the fact that the doublerscontributeto thosequantitiesthroughthe Wilson—Yukawaterm.In the r —÷ 0 limit, this term disappearsbutatthe priceof having16 fermionsequivalentto eachother.For r ~ 0 the result

380 IL. A!onsoet aL / S,U, andL~pin Zaragozaproposal

is, in general,a complicatedfunctionof r.The introductionof all possible terms of dimensionsix (as the Wilson—

Yukawaterm)compatiblewith the SUL(2)0Uy (1) symmetryandafine tuningof its coefficientswould be necessaryin orderto obtainthe targetresults.It isimportantto notethat the coefficientof thesetermswill not be determinedbygaugeinvariance. (Notethat, in someparticularcasesthe resultsfor the Smit—Swift model may agreewith the target results. For instance,when Am = 0,U(r) = Ap(r) = 0 as in (3), this is justan accidentconsequenceof a custodialsymmetry.Also particularvaluesof r couldaccidentallygive thetargetresults.)

Thereareways of building up Wilson termsthat do not give any SSB con-tribution dueto the regularization.Examplesof this are the mirror fermionsmodel [6], andthe model of Borelli et al. [8]. In both casesthis is achievedatthe expense of doubling the numberof fermionfieldsandextendingthe globaltransformation (17) to include the extra fields.

Furthermore,asreportedin ref. [1], in themirror fermionmodel(becausethemirror fermionsarecoupled)the mirror fermionscontributeto thosequantitiesasmuchas thephysicalfermionsdo. Onthe contrary,for themodelofBorelli etal. (sincethe extrafermionsarenot coupled),the continuumresult is achieved.

A subsidiarymodel of the model of Borelli et al. hasappeared,in whichredundantvariablesarenot used[9]. It breaksthe global symmetry (17) so itbreaksgaugeinvariance.Thereforeexplicit tuning of countertermsto recoverBRST in the continuumlimit will be required. This tuning of countertermsleads,atone1oop,to anon-trivialcancellationof contributionscomingfrom thecountertermsandfrom theWilsontermof theactionwhichbreaksexplicitly theglobal SUL(2) 0 Uy (1) symmetry.This cancellationof contributionsis crucialto gettingthe targetone-loopvaluesfor 5, U andAp.

In our model, sinceno Wilson termsare required, no extraSSB termsareintroduced.Furthermore,the global SUL(2)0 Uy(l) is preserved.Thuswe getthe sameresultas in the modelof Borelli et al. but without needof redundantfermion fields. This is a big economyif one were to implementa numericalsimulation.

Of course, many more aspects have to beinvestigatedin any modelof chiralfermions (includingthosepreviouslymentioned)in orderto concludewhetheror not they are good candidates to describethe targettheory.One hasfirst toextend the perturbative analysis beyond one loop, to see if the target values of5, U andAp areobtained.Onehasalsoto test thevalidity of theformulationatthenon-perturbativelevel. In fact, wenow knowthatthe Smit—Swiftmodel [7]andotherrelatedmodels [10], basedon Wilson—Yukawaterms,do not succeedin describingchiral fermions [11,12] and the search for a fully satisfactorylatticizationis still open [13].

So far, the Zaragozaproposalis still a viable candidatefor describingchiralfermions on the lattice. Unfortunately,we have not yet provedconclusively

IL. Alonso et a!. / S.U, andApin Zaragozaproposal 381

that it is in all regards satisfactory. In the meantime, the result presented herecanbe takenas a technicalimprovement:if the Zaragozaproposalprovestobe successfulin describingchiral fermions,we have shownthat some of thepossiblecountertermsthatmight otherwisehavebeenrequiredat oneloopwillnot actuallyappear.

The studyof the non-perturbativebehaviorof the Ap parameteris of greatinterestwith respectto probing the effectsof heavy fermionsin theorieswithspontaneoussymmetrybreaking[4]. In this regard,the implicationsof the re-sultspresentedin thisletterareinteresting:sincethe one-loopperturbativecal-culationsof 5, U andAp do not requirea fine tuning of counterterms,thereissomehopethat in the Zaragozaproposalafull non-perturbativecalculationoftheseparametersmight be tractable[14].

Oneof us (E.R.) wishesto thankthe WashingtonUniversityDepartmentofPhysicsfor its hospitality, in particularM. Golterman,D. PetcherandM. Visserfor acritical readingofthismanuscript.Theauthorswantto thankD. EspriuandB. Grinsteinfor very interestingdiscussions.This work is partially supportedby the CICYT (proyectoAEN 90-0030).E.R. is supportedby a FormacióndelPersonalInvestigadorfellowship from the Spanishgovernment.

Appendix A

Usingthe notation

F(q) = flcos(q~/2), qe (—m,ir]4, f = fd4q/(21t)4, (A.!)

sp(q) = sin(q4, cp(q) = cos(qp), s2(q) = ~sin2(qp), (A.2)

D(q) = s2(q) + a2m2F4(q) for in = yv in the brokenphase. (A.3)

The expressionsfor (19) are (m1 m2 m, and mi — m21 = Am)

.FS(q,ap,m)

H,~(ap)= o,~j~ +O(Am), (A.4)

H~(ap)= fa2F ~ + O(Am3), (A.5)

where

c2(q + ap/2)J~(q,ap,m) = —l6mm2F4(q)F4(q+ ap) “ D(q) (A.6)

FU(qaprnAm) = +32itm2(Am)2F6(q)F6(q+ aP)D2~)~(A.7)

382 IL. Alonsoet a!. / S.U, andL~pin Zaragozaproposal

and

g~~(q,ap) = c~(q+ap/2)c~(q+ap/2)

x ~

(A.8)

Usingall thesepreviousexpressions,(20) canbe written as

Sp2 + S’p,~ = (Sj~ap + 5~20~)p0p/~, (A 9)

nfl

K”p~p~+ (Up2 + U’p~)ö,~,,= >(U~

0p + U,~np)papfl, (A.l0)nfl

Ap = 2 2 (A.ll)4ms m~

in which

— U [a2 1 (~s F~

0(q)Dp(q)+

p,nfl — a~Jq2D(q)~ ~,0~(q) D(q)

2 [a2Y~~po(2D,~(q)D,p(q)5p,afl = ~ Jq 2 D2(q) ~ D(q) — D,afl (q)

U~np= ~ ~ ~ + F~(q)c~~,~(q)+

.F~’(q)Dp(q)+F~(q)D,n(q)D(q)

U2 F [a4FUIpo (2c~,,n(q)D,p(q)+ cpv,p(q)D,n(q)~w,nfl = a~Jq 2 D

2 (q) ~Gpv,afl(q) — D(q)

~ D,0(q)D,p(q) D+ D(q) D(q) - ,ap(q)

I1~’= iimfa232itm2(Am)2Fi2(q) (2s1(q)-s(q))c1(q) (A.12)

with the notation

00(q) = ôO(q,ap) , O~(q)= 020(q,ap) . (A.13)8ap

0Oapp ~

The KU term in (A. 10) is not of interest.It appearsbecausewe havenot yetextracted the ô,~componentfrom H,~,in (A.5).

The calculationof integralsH~,~ and U,j’~npproceedsas follows: eachintegral can be split into two integrals,one interior to andone exterior to an�-ball= {q; I~I= �} with � >> a.

IL. A!onsoet a!. /5, U, andL~pin Zaragozaproposal 383

UsingLebesgue’stheoremof dominated convergence we canshowthat in theexterior integralsthe limit a —~ 0 andthe integrationcanbe exchanged.As aconsequenceof thiswecaneasilyprovethat all of the exteriorintegralsgivezerocontribution.

To apply thistheoremwehaveto placea Lebesgueintegrableupperboundonthe modulusof the variousintegrandsin (A. 12). In thisregularizationwecanuse the bound

/

F2(q) ‘cos4(�/2)’~ I . for q~~ �, (A.14)D(q) ~ 2sin� )

to successfullyget upperboundsfor the integrandsin (A. 12).As an example,for H~’

~l2~ ~(2s~(q)—s2(q))c~(q) (cos4(�/2)\ 4.F 1n21 \\4 . ~or q ~

ti-.’ ~q,, .~.sin� (A 15)

Heretheright-handsideis aLebesgueintegrablefunction.A quitesimilaranaly-sis sufficesfortheotherintegralsin (A. 12). Thuswehaverigorouslyprovedthatwhencalculatingexteriorcontributionsto (A. 12), the limit a —~ 0 canbe takenprior to do the integration.Suchlimit tells us that all the exteriorcontributionsto (A.l2) vanish.

The vanishing of the exterior integrals is a consequence of fact that whencalculatingtheexteriorcontributions,becausethemomentaareoforder0(1/a),we can neglectall the Yukawa couplings.Therefore, the exterior integralsdonot seeany SSB contributionandsincethereis not explicit breakingin theregularization,no contributionto 5, U andAp shouldbe expectedfrom them.

Overthe innerregion,the limit a —~ 0 of the integralsis easyto do. Thingshavebeenarrangedsuchthat thereis no contributionfrom integralsS)~afland~ inside the �-ball.All the otherintegralsin (A.12) canbe calculatedbypicking up the first term in a Taylorexpansionaroundq 0.

Finally, the resultsare

5p,nfl = 0, Up’vnp = 0, (A.16)

and introducingU~~nflApv,ap + U~fl~5/41,,

c’2 1‘-~p,nfl =

2 2 (Am)2Unfl = önflf~ m2

= _~i~(Am)2, (A.17)3m

so that (21) is accomplished.

384 IL. Alonsoet aL / S,U, and/~pin Zaragozaproposal

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