Meson loop effects in the Nambu-Jona-Lasionio model

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Meson loop effects in the Nambu-Jona-Lasionio model

M. Oertel, M. Buballa, J. Wambach

To cite this version:

M.Oertel

IPNLyon,43,Bvddu 11novembre1918, 69622VilleurbanneCedex, France

M.Buballa GSIDarmstadt

J.Wambach

Institutf urKernphysik,TUDarmstadt,Schlossgartenstr. 9,64289Darmstadt,Germany (December10, 2001)

We comparetwodierent possibilities to include meson-loopcorrections inthe Nambu{Jona-Lasiniomodel: astrict1=N

c

-expansioninnext-to-leadingorderandanon-perturbativescheme cor-respondingtoaone-meson-loopapproximationtotheeectiveaction. Bothschemesareconsistent withchiralsymmetry,inparticularwiththeGoldstonetheoremandtheGell-Mann{Oakes{Renner relation. Thenumericalpartatzerotemperaturefocusesonthepionandthe-meson. Wealso in-vestigatethetemperaturedependenceofthequarkcondensate. Herewendconsistencywithchiral perturbationtheorytolowestorder. Similarities anddierencesofbothschemesarediscussed.

I. INTRODUCTION

Duringthelast few years oneoftheprincipal goalsin nuclear physicshasbeento explorethephasestructure of QCD.Alongwiththiscomestheinvestigationofhadronpropertiesin thevacuumaswellas inhotordensematter. In principle, all properties of strongly interacting particles should be derived from QCD. However, at least in the low-energy regime, where perturbation theory is notapplicable, this is presently limited to a rather small number of observableswhichcan be studied onthe lattice,while morecomplexprocessescan either be addressedby chiral perturbationtheoryorwithineectivemodel calculationswhichtryto incorporatetherelevantdegreesoffreedom.

Sofarthe best descriptionsof hadronicspectra,decaysand scatteringprocessesareobtainedwithin phenomeno-logicalhadronicmodels. Forinstancethepionelectromagneticformfactorinthetime-likeregioncanbereproduced ratherwellwithin asimplevectordominancemodelwithadressed-mesonwhichisconstructedbycouplingabare -mesonto atwo-pion intermediatestate [1,2]. Models of this typehave been successfully extended to investigate mediummodicationsofvectormesonsandtocalculatedileptonproductionratesinhotanddensehadronicmatter [3].

Inthissituationonemightaskhowthephenomenologicallysuccessfulhadronicmodelsemergefromtheunderlying quarkstructure andthesymmetry propertiesof QCD.Since thisquestioncannot beansweredat present fromrst principles it has to be addressed within quark models. For light hadrons chiral symmetry and its spontaneous breakinginthephysicalvacuumthroughinstantonsplaysthedecisiveroleindescribingthetwo-pointcorrelators[4] with connementbeingmuchless important. Thisfeature iscaptured bythe Nambu{Jona-Lasinio (NJL)model in whichthe four-fermioninteractionscanbeviewedasbeinginducedbyinstantons. Furthermore themodel allowsa studyofthechiralphasetransitionaswellastheexaminationofthein uenceof(partial)chiralsymmetryrestoration onthepropertiesoflighthadrons.

The study of hadrons within the NJL model has of course a longhistory. In fact, mesons of various quantum numbershave already beendiscussed in the original papersby Nambuand Jona-Lasinio[5] and by many authors thereafter (forreviews see [6{8]). Inmost of these works quark masses are calculatedin mean-eld approximation (HartreeorHartree-Fock)whilemesonsareconstructedascorrelatedquark-antiquarkstates(RPA).Thiscorresponds to aleading-orderapproximationin 1=N

c

calculationfortheNJLmodel,althoughthishasbeendonebymanyauthors(seee.g.[7{10]). Inthesecalculationsthe thermodynamicsisentirelydrivenbyunphysicalunconnedquarksevenat lowtemperaturesanddensities,whereas thephysicaldegreesoffreedom,inparticularthepion,aremissing.

This and other reasonshavemotivated severalauthors to go beyond thestandard approximationscheme and to include mesonic uctuations. SincethemostimportantfeatureoftheNJLmodelischiralsymmetry,oneshoulduse an approximation scheme which conserves the symmetryproperties, to ensurethe existence of masslessGoldstone bosons. Within the present article we will discuss two dierent approximation schemes and compare the results: A strict (perturbative) expansionin 1=N

c

upto next-to-leadingorder and anon-perturbativesymmetry conserving approximation scheme which has been derived in Ref. [11] using a one-meson-loop approximation (MLA) to the eectiveactioninabosonizedNJLmodel. Thelatterschemehasbeforebeenpresented,derivedbycompletelyother means,in Ref. [12]. It hasbeenshown[12,11,13]that bothschemesareconsistentwiththeGoldstone theoremand theGell-Mann{Oakes{Rennerrelation,twoofthemostimportantlowenergychiraltheorems.

In vacuum we focus the discussion of our results for the pion and the -meson. The inclusion of meson loop eectsshouldalsoimprovethethermodynamicsofthemodelconsiderably. Arstinsightonthein uenceofmesonic uctuations upon the thermodynamics can be obtained viathe temperature dependence of the quark condensate. The low-temperaturebehaviorin bothschemes is dominated by pionic degreesof freedom which is aconsiderable improvementoncalculationsinHartreeapproximationwherequarksaretheonlydegreesoffreedom. Theresultsare consistentwiththelowest-orderchiralperturbationtheoryresult[13].

Thisarticleisorganizedasfollows. InSec.IIwebeginwithabriefsummaryofthestandardapproximationscheme used in the NJL model to describe quarks and mesons. Afterwards we present the two possibilities to go beyond the standard scheme. The numerical results at zero temperature will be presented in Sec. III. The temperature dependenceofthequarkcondensatewillbestudied inSec.IV. Finally,wewillsummarizein Sec.V.

II. THEMODEL

AsdiscussedintheIntroduction,wewillemploytwodierentschemesfordescribingmesonswithintheNJLmodel includingmesonloops: astrict(perturbative)1=N

c

-expansionandthenon-pertubativeone-mesonloopapproximation (MLA).Inthissectionwegiveabriefoutlineoftheseschemes. DetailsoftheformalismcanbefoundinRefs.[13,14]. WeconsideranNJL-typeLagrangianfortwo avorsandthreecolorswithscalar-isoscalarandpseudoscalar-isovector interaction: L =  (i@= m 0 ) + g s [(  ) 2 +(  i 5 ~  ) 2 ] g v [(   ~  ) 2 +(   5 ~ ) 2 ]: (1) Hereg s andg v

aredimensionfulcouplingconstantsoforder1=N c

. Inthelimitm 0

=0theLagrangianissymmetric underSU(2)

L

SU(2) R

transformations.

Starting point of the standard approximation scheme is to solve the gap equation for the quark propagators in Hartreeapproximation. This gap equation is displayed in Fig.1. The mesons are then described in RPA, i.e., by iteratedquark-antiquarkloops,asillustratedinFig.2. TheHartree+RPAapproximationschemecorrespondstothe leadingorderina1=N

c

-expansion.

= +

FIG.1. ThegapequationforthequarkpropagatorinHartreeapproximation(solid). Dashedlinesdenotepropagatorsofbare quarks.

= +

RPA

FIG.3. Contributionstothemesonpolarizationfunctionsinleading (RPA)andnext-to-leading orderin1=N c

.

The1=N c

-correctiontermstothemesonpolarizationfunctions togetherwiththeleadingorder (RPA)quark anti-quarkloopareshownin Fig.3. Solid linesdenotequarkpropagatorsin Hartreeapproximation,i.e.in leadingorder. ThesediagramscontainmesonpropagatorscalculatedinRPA,asdescribedabove. Toobtainthe\improved"meson propagators,theentirepolarizationfunction,includingthe1=N

c

-correctionterms,isiteratedinthesamemanneras before the RPA polarization function alone. It can be shownanalytically that the pion constructed in this way is againmasslessin thechirallimit[13]. Note thatwedonotuse theimproved mesonpropagatorsforevaluatingthe correctionterms. Such aselfconsistentprocedure,although desirablefrom aphenomenologicalpointof view,spoils the1=N

c

countingforthepolarizationfunctions schemeandleadsto inconsistencieswithchiralsymmetry.

= + +

FIG.4. Extended gapequation(upper part)andcontributions tothemeson polarizationfunctions(lowerpart)intheMLA.

WithintheMLAthegapequationforthequarkpropagatorsismodiedselfconsistentlybyaddingatermcontaining ameson loop. This is illustratedin the upper partof Fig.4. Here thedouble line is constructed in the sameway asamesonpropagatorin RPA(seeFig.2), butstartingfrom quarkswhichemergeassolutionsofthemodiedgap equation itself. The terms contributing to the meson polarization functions are shown in thelowerpart of Fig. 4. Here the solid lines and thedouble lines have thesame meaningas in the upperpart ofthe gure. Toobtainthe mesonpropagatorsthepolarisationfunctionsareiteratedasbefore. Again,itcanbeshownanalyticallythatthepion constructedinthiswayismasslessinthechirallimit[11,?,13].

III. NUMERICAL RESULTS

Anon-renormalizablemodelisincompletewithoutdeninghowtoregularizedivergentloops. Inaddition,itisnot suÆcienttoregularizethedivergentloopsinHartree+RPA,sincenewdivergenciesandhencenewcutoparameters emergeifoneincludes mesonloops

1

. Inourcasewehaveto dealwithtwotypesof dierentdivergentloops: quark and mesonloops. Forexampleletus look at diagram(b) ofFig.3. This diagramcontainsoneintermediatemeson which itself consistsof quarkloops. This intermediatemeson hasto becalculated in arststep. We arethen left with twoloops: one quarkloop withfour verticesand an integration overthefour-momentum of theintermediate meson. It is quite naturalto regularizethe quarkloop in the same way asthe polarization diagrams which enter into theintermediatemeson. Infact,this isnecessaryinorder to preservechiralsymmetry. Wehavechosentouse Pauli-Villarsregularization withacuto 

q

. However,there is nostringentreason, whythe remainingmesonloop should alsoberegularizedinthis way. WethereforefollowRefs.[12,11]andintroduceanindependentmesoncuto

1

M

0

1

2

3

0

2

4

6

8

m

π

2

/ (140 MeV)

2

m

₀

[MeV]

0

1

2

0

2

4

6

8

m

π

2

/ (140 MeV)

2

m

₀

[MeV]

FIG.5. Squaredpionmassasafunctionofthecurrentquarkmassm0 fordierentmeson-loopcutos;(left)1=Nc-expansion scheme: 

M

=0MeV(solid),500MeV(dashed),900MeV(dashed-dotted)and1300MeV(dotted);(right)MLA: M

=0MeV (solid),300MeV(dashed),500MeV(dashed-dotted) and700MeV(dotted). The calculated pointsare explicitlymarked.

Let us rst study the in uence of mesonic uctuations onthe NJL pion propagator and related quantities. We beginwith the leadingorder,which corresponds to ameson cuto 

M = 0. With  q =800 MeV,g 2 q =2.9 and m 0

= 6.1 MeV we obtain a reasonablet for the pion mass, the pion decay constant and the quark condensate: m (0)  =140MeV,f (0)  =93.5MeV andh  i (0) =-2(241.1MeV) 3

. Hereand inthefollowingthesuperscript (0)is used todenote quantitieswhich arecalculatedin Hartree+RPA. Theaboveparameterscorrespondto arelatively smallconstituentquarkmassof260MeV.

Nowweturn onthemesonic uctuationsbytakinganon-zeromesoncuto M

. Fig.5displaysthesquaredpion massasafunctionofthecurrentquarkmassm

0

fordierentvaluesof M

,onthelefthandsideforthe1=N c

-expansion scheme and on the righthand side for the MLA. Obviously allpoints which correspond to the samemeson cuto liealmost onastraightline throughthepoint(m

0 =0;m

2 

=0). Thelatterwas calculatedanalyticallywhereasall other pointsare numericalresults. This demonstratesthe consistencyof ourscheme withchiralsymmetry andthe stabilityofthenumerics.

Ofcourseweshouldnotstaywiththemodelparametersdeterminedinleadingorder,butperformaretofvarious observables includingthe eect of the mesonloops. Here weproceed in twosteps[14,13]: For various xed values of the meson cuto 

M

we rst x the current quarkmass m 0

, the quark-loopcuto  q

, and the scalarcoupling constantg

s

totthepionmass,thepiondecayconstantf 

,andthequarkcondensateh  i. f



iscalculatedfromthe one-piontovacuum axialvectormatrixelement, which basicallycorrespondsto evaluating themesonicpolarization diagramscoupledtoanexternalaxialcurrentandtoapion. Thequarkcondensateisgivenbythetraceofthequark propagator: h  i= i Z d 4 p (2) 4 TrS(p): (2) In the 1=N c

-expansion scheme we have to take into account not only the Hartree contribution but also the two contributionstothequarkselfenergyinnext-to-leadingorderwhichareshownin Fig.6.

Inthesecond stepwetrytodeterminethetworemainingparameters,g v

and M

,byttingthedataforthepion

FIG.6. The1=N c

-correctionterms tothequarkself-energy.

electromagneticform factorin thetime-likeregion. Thisobservableisverywell suitedforthispurpose becauseit is dominatedbythe-mesonwhich, besidesbeingavectorstate,cannotbedescribed reasonablywithoutincluding intermediate pion loops. Inthe 1=N

c

modelthephenomenologicallyimportanttwo-pionintermediatestateconsistsofRPApions,weareforcedtotm  andnotm



totheempiricalpion massifwedesiretogetthecorrectthresholdbehaviorforthe-meson. However, in mostcaseswendthatthedierencebetweenm

 andm

(0) 

isnotverylarge.

;q   ;q    ;q    ;q    ;q    ;q   

FIG.7. Contributionstothepionelectromagneticformfactorinthe1=N c

-expansionscheme. Thepropagatordenotedbythe curlylinecorresponds tothe1=Nc-correctedrho-meson,whilethedoublelinesindicateRPA pionsandsigmas.

Theparameterspaceis furtherrestrictedbytherequirementthattheunphysicalqq-thresholdmustliewellabove thepeakin the-mesonspectralfunction inorder toobtainarealisticdescriptionofthepionelectromagnetic form factor. Thismeans,theconstituentquarkmasshastobelargerthanabout400MeV.

0

10

20

30

40

50

0.2

0.4

0.6

0.8

|F

π

(s)|

2

s [GeV

2

]

0

30

60

90

120

400

600

800

δ

1

1

√ s [MeV]

FIG.8. Pion electromagnetic form factor (left panel) and the -phaseshifts in thevector-isovector channel(right panel) forM =500MeV(dashed),M =600MeV(solid)aswellasM =700MeV(dotted)inthe1=Nc-expansion scheme. The datapointsare takenfromRefs.[17]and[18],respectively.

It was an important result of the analysis in Ref. [14] that such a set of parameters can be found in the 1=N c

-expansionscheme. Ourresultsforthepionelectromagneticformfactorfor

M

=500;600and700MeV,aredisplayed onthelefthandsideofFig.8. Thevalueofg

v

hasbeenchosensuchthatthepositionofthemaximumcoincideswith thedata. It is obviousthat in uence of thetwo-pionintermediate stateis underestimatedfor

M

=500MeVand overestimatedfor

M

=700MeV.Besides, theresultsfor  M

=500Mev suer from thefact that theconstituent quarkmassof396MeVwhichemergesfromthetinthepionsectoristoosmalltokeepthequark-antiquarkthreshold well above the peak in the form factor. This is improvedfor larger valuesof 

M , for 

M

=600 MeV wealready obtaina constituentmass of 446MeVand for 

M

=700MeV we nda massof 550MeV. Wecan concludethat theoverallagreementwiththedata fortheform factoris good for

M

=600MeV.This isconrmed ifwelook at the phase shiftsin the vector-isovectorchannel, which have beencalculated from thedominants-channel -meson exchange. Ourresultsforthesamevaluesofthecutoasbeforeareshownontherighthand sideofFig.8.

Incontrastto the1=N c

-expansionscheme, wedid notsucceedin performingareasonablet fortheMLA. There weencounteredinstabilitiesinthe-mesonpropagatorforvaluesofthemesoncuto

M

It is commonly believed that chiral symmetry, which is spontaneously broken in vacuum, gets restored at high temperatures. At zero densities this is conrmed by lattice results, whereas in wide ranges of densities and tem-peraturesthis notionis basedon model calculationsfor lackof fundamental knowledge. Onlyat lowtemperatures and lowdensitiesmodel independentresultscanbeobtainedbyconsidering agasofpions andnucleons. Theseare thedominantdegreesoffreedomin that rangebecausepions arebyfarthelightesthadrons,while nucleonsarethe lightestparticlescarryingnon-zerobaryonnumber. ThisisinstrongcontrasttostandardNJL-modelcalculationsin Hartree(-Fock)approximation(see e.g. [7{10]), where thethermodynamics is entirelydriven by unconnedquarks, i.e., by unphysical degreesof freedom. Hence, although thefundamental problem of lackof connementcannot be overcome,wecan hopeto improvethe resultsby introducing mesonic degreesof freedom within anapproximation beyond Hartree + RPA, i.e. in the 1=N

c

-expansion scheme or the MLA. Since both schemes do not contain any nucleons, essential ingredients at nonzero densities, wewill restrictour examinations to nonzero temperatures but zerodensity.

We rst compare the low-temperature behaviorof the quark condensatewith model independent considerations from chiral perturbation theory. Tolowest orderin temperaturethechangeof h



iwith temperaturein thechiral limitisgivenby[19] h  i T =h  i  1 T 2 8f 2  +:::  : (3) Hereh 

idenotesthequarkcondensateatzerotemperature. ThetermproportionaltoT 2

arisesfromapure (mass-less) pion gas. In Hartreeapproximationthis behavioris completely failed since excitations of the massivequarks areexponentiallysupressed. Inthe1=N

c

-expansionschemeaswellasintheMLA, however,theT 2

-behaviorcanbe reproduced. The only dierence is that in these schemes the uctuations consist of RPA pions, and thereforethe coeÆcientcorrespondstotheRPAquantitiesinsteadofthefullones. Thegood agreementofourresultswithafree piongasbehavioratlowtemperaturescanalsobeseeninFig.9,wherethequarkcondensateisshownasafunction oftemperature. ForcomparisonwealsoshowthebehaviorinHartreeapproximation.

0

0.2

0.4

0.6

0.8

1

0

50

100

150

200

250

<



ψψ

>

T

/<



ψψ

>

T [MeV]

0

0.2

0.4

0.6

0.8

1

0

50

100 150 200 250

<



ψψ

>

T

/<



ψψ

>

T [MeV]

FIG. 9. Quark condensate as a function of temperature, normalized to the vacuum value in the chiral limit. Left: 1=N

c

-expansion scheme (solid), pion gas (dotted) and Hartree approximation (dashed), right: MLA (solid), pion gas (dot-ted)andHartreeapproximation(dashed).

InbothschemesdeviationsfromthepurepiongasbehaviorbecomevisibleatT > 

100MeV.Thesedeviationsarise from quarkeects which could be tolerated only close to the phase transition. In contrast to the 1=N

c

-expansion schemeanexamination ofthephasetransitionispossiblein theMLA. Thepresentparametersetleadsto acritical temperatureofT

c

=164:5MeV.Hencequarkeectsceasetobenegligibleatatemperatureofabout:6T c

,whereone cannotavoidadmittingthat theyarecompletely unphysical.

In accordance with the ndings in Ref. [20] the phase transition is of rst order whereas it is of second order in Hartreeapproximation. This is already indicated by the jump in the order parameterat T = T

c

four(atT c

massless)bosonicdegreesoffreedomandthatQCDthereforeliesinthesameuniversalityclassastheO(4) model which isknown to exhibit asecond order phasetransition. The sameargumentscanbeapplied tothe NJL model, which hasthesameunderlying symmetryasQCDwithtwomassless avors. However,one oftheobjections onemightraiseagainsttheabovehypothesis isthat atheorywith compositebosonelds notnecessarilybelongsto thesameuniversalityclassastheO(4)model[23]. ThisobjectionisamongothersvalidfortheNJLmodel.

V.SUMMARY

We have investigated quarkand meson properties within the Nambu{Jona-Lasiniomodel, including meson-loop corrections. These have been generated in two dierent ways. The rst method is a systematic expansion of the self-energies in powers of 1=N

c

up to next-to-leading order [12,24,14,13]. In the second scheme, a local correction termto thestandardHartreeself-energyisself-consistentlyincludedin thegapequation [12,11]. Bothschemes,the 1=N

c

-expansion scheme and the MLA, are consistent withchiral symmetry, leadingto massless pions in the chiral limit.

The relative importance of the mesonic uctuations is controlled by aparameter  M

, which cuts o the three-momentaofthemesonloops. Thevalueof

M

,hastobedetermined,togetherwiththeotherparameters,bytting physicalobservables. The-mesonandrelatedquantitiesareverywellsuitedforthispurpose,sincethemesonloops areabsolutelycrucialinordertoincludethedominant!-decaychannel,whiletheHartree+RPAapproximation containsonlyunphysicalqq-decaychannels. Of course,aprioriitis notclearto whatextenttheseunphysical decay modes, which are an unavoidable consequence of the missing connement mechanism in the NJL model, are still presentin theregionofthe-mesonpeak.

For the 1=N c

-expansionscheme weobtain areasonablet of f 

, h 

i and the pion electromagnetic form factor with aconstituent quarkmass of m= 446MeV. This means, theunphysical qq-decay channel opens at 892 MeV, about120MeV abovethemaximumof the-mesonpeak. Unfortunatelywedidnotsucceed to obtainasimilar t within theMLA [13]. Since in thisscheme themeson-loopeects lowertheconstituent quarkmass ascompared to theHartreemass,itismuchmorediÆculttoevadetheproblemofunphysicalqq-decaychannelsinthevicinityofthe -mesonpeak.

Inthelastpartofthisarticlewehaveinvestigatedthetemperaturedependence ofthequarkcondensate. Inboth schemesthelow-temperaturebehaviorisconsistentwithlowest-orderchiralperturbationtheory,i.e.,thetemperature dependence arisingfrom a freepion gas. This isa considerableimprovementoverthemean-eld result,where the temperaturedependenceisentirelydue tothermallyexcitedquarks,i.e.,unphysicaldegreesoffreedom.

Athighertemperatures,however,thermalquarkeectsalsobecomevisibleinthetwoextendedschemes. Weargued thatthiscouldbetolerableonlynearthechiralphaseboundary. Whereastheperturbativetreatmentofthemesonic uctuationswithin the1=N

c

-expansionschemedoes notallowanexamination ofthe chiralphasetransition,this is possibleintheMLA.Forourmodelparametersetwefoundacriticaltemperatureof164.5MeV.Ontheotherhand, quarkeects arevisiblealreadyatatemperatureof 100MeV.Obviouslythisisstilltooearlytoberealistic.

In agreementwith Ref. [20] wefound a rst-orderphase transition in that scheme. This contradicts the general belief that the non-zerotemperature chiral phase transition in a model with two light avors should be of second orderandisprobablyanartifactof theapproximation.

ACKNOWLEDGMENTS

We are indebted to G.J. van Oldenborgh for his assistance in questions related to his program package FF (see http://www.xs4all.nl/gjvo/FF.h tml),whichwasusedinpartsofournumericalcalculations. WealsothankG.Ripka, B.-J.Schaeferand M.Urbanforilluminatingdiscussions. Thisworkwassupportedin partby theBMBFand NSF grantNSF-PHY98-00978.

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