Renormalization group analysis of nuclear force

We study a relation between nuclear forces based on phenomenological approach (V_ph) and nuclear effective field theory (V_EFT) from a viewpoint of renormalization group. We find the relation between these two types of nuclear force using Wilsonian renorma

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arXiv:nucl-th/0610034v1 10 Oct 2006RenormalizationgroupanalysisofnuclearforceS.X.NakamuraaaTheoryGroup,TRIUMF,4004WesbrookMall,Vancouver,BCV6T2A3,CanadaWestudyarelationbetweennuclearforcesbasedonphenomenologicalapproach(Vph)andnucleare ective eldtheory(VEFT)fromaviewpointofrenormalizationgroup.We ndtherelationbetweenthesetwotypesofnuclearforceusingWilsonianrenormalizationgroupequation.ConsideringthefactthatVEFTisde nedinacertainsmallmodelspace,weshowthatasimplecontactinteractionaccuratelysimulatesshort-distancephysics,andthatVEFTisfreefromdependenceonmodellingthedetailsoftheshort-distancephysics.Basedonourresult,wediscusssomefeaturesofnucleare ective eldtheory.1.IntroductionNuclearforceisoneoftheoldestprobleminnuclearphysics.However,itstheoreticaldescriptionisstillanunsettledissue.Here,we rstexplainatraditional,phenomenolog-icalapproachtothenuclearforce,andafairlynewapproachbasedonnucleare ective eldtheory(NEFT)[Aftertheexplanation,weformulateascenarioforarelationbetweennuclearforcesbasedonthetwoapproaches.Themainsubjectinthisreportistodiscusstherelationfromaviewpointoftherenormalizationgroup(RG).Inthephenomenologicalnuclearforce(Vph),thelong-rangepartisdescribedbythe

one-pion-exchangepotential(OPEP).Theshort-rangepartispoorlyknown,andthereforeaphenomenologicalmodelisused;heavy-mesonexchangesorpurelyphenomenologicalparameterization.Parametersinvolvedinthemodelare xedsoastoreproducelow-energyNNscatteringdataandthedeuteronbindingenergy.Inthisway,severalhigh-precisionNNpotentialshavebeenconstructed,suchastheCD-BonnandtheNijmegenpotentials.Althoughthesemodelsarephenomenologicallyverysuccessful,therearesomeproblemsasfollows.At rst,eachmodeldescribestheshort-rangepartdi erently,andislargelymodel-dependent.Secondly,thereisnosystematicwaytoconstructtheshort-rangemechanism.Thirdly,thereislittleconnectiontotheunderlyingtheory,QCD.Incontrast,itisclaimedthatanuclearforcebasedonNEFT(VEFT)isfreefromtheproblemsinherentinVph.TheclaimisbasedonthederivationprocedureofVEFT,whichisbrie ygivenasfollows.Onestartswithane ectivechiralLagrangian,takingcareofthespontaneouslybrokenchiralsymmetryofQCD.TheLagrangianconsistsofe ectivedegreesoffreedomforasysteminquestion,andisthemostgeneralaslongasassumedsymmetriesaresatis ed.Oneidenti esasetofirreduciblediagramsfromtheLagrangianwithVEFT.Theimportanceofanirreduciblediagramisassignedbyacountingrule.However,onemay ndquestionsaboutVEFT:(Q1)InVEFT,short-distancephysicsisdescribedbycontactinteractions,whichismuchsimplercomparedtotheshort-range

We study a relation between nuclear forces based on phenomenological approach (V_ph) and nuclear effective field theory (V_EFT) from a viewpoint of renormalization group. We find the relation between these two types of nuclear force using Wilsonian renorma

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mechanismemployedinVph.Isthecontactinteractionreallyappropriatetodescribetheshort-distancephysics?(Q2)IsVEFTstilloneofmanyphase-equivalentpotentials,asVphis?(Q3)IsthereanyrelationbetweenmodelindependentVEFTandmodeldependentVph?Infact,theanswerto(Q3)naturallyleadsustotheanswersto(Q1)and(Q2).Thus,weformulateascenariofortherelationbetweenVEFTandVph.

Itisnotedthatthemodelspace(thestatespaceforthenucleon)ofVEFTisconsiderablysmallerthanthatofVph.Withthispointinmind,wecanformulatethefollowingscenariofortherelationbetweenVEFTandVph.WemayconstructmanyVphwhichreproduceNNscatteringdataandthedeuteronbindingenergy.Theyaredi erentindescribingtheshort-distancephysics,andthereforemodeldependent.StartingwithsuchVph,wereducetheirmodelspacebyintegratingoutthehighmomentumstatesofthenucleon.Reducingthemodelspacecorrespondstoviewingthesysteminacoarse-grainedmanner.Asthemodelspaceisreduced,informationaboutdetailsoftheshort-distancephysicsisgraduallylost.Eventually,weobtainalow-momentume ectiveinteraction(VM)de nedinareducedmodelspace.VMdoesnothavethemodeldependencewhichVphhave.AparameterizationofVMconstitutesVEFT.ThisisthescenariofortherelationbetweenVEFTandVph.VEFTobtainedinthiswayis,byconstruction,doesnothaveadependenceonmodellingtheshort-distancephysics.Theshort-rangepartofVMisexpectedtobeaccuratelysimulatedbysimplecontactinteractionsbecausethedetailedinformationhasbeenintegratedout.Therefore,ifweshowthatthisscenarioisrealized,thenwecananswerallofthequestionsraisedinthepreviousparagraph.

Thepurposeofthisreportistocon rmthescenario,therebyproposingtherelationbetweenVEFTandVph[2].Furthermore,keepingtherelationinmind,wecanunderstandfeaturesofNEFTmoredeeply,whichwewilldiscusslater.

2.Wilsonianrenormalizationgroupequation

Inordertoexaminethescenario,statedintheprevioussection,fortherelationbetweenVEFTandVph,anappropriatemodel-spacereductionmethodisnecessary.Ine ective ledtheory,weuseane ectiveLagrangianinwhichhigh-energydegreesoffreedomhavebeenintegratedoutusingapathintegral:

Z= DΨHDΨLei d4xLH= DΨLei d4xLL,(1)whereΨH(ΨL)ishigh-(low-)energydegreesoffreedom,andLH(LL)consistsofΨHandΨL(onlyΨL).Ane ective(morefundamental)LagrangianisLL(LH).Themodel-spacereduction,integratingouthigh-momentumstatesofthenucleon,alsoshouldbedonewiththepathintegral.Althoughitisdi cultingeneral,incasetheLagrangianiscomposedbyonlythenucleon eldandtwonucleonsinteractthroughcontactinteractions,wecanperformthepathintegral.Itisnotedthatwerestrictourselvestotwo-nucleonsystemthroughoutthisreport.Inthecenterofmassframe,performingthepathintegralisequivalenttosolvingthefollowingWilsonianrenormalizationgroup(WRG)equation: V(α)(k′,k;p,Λ)

2π2V(α)(k,Λ;p,Λ)′Λ2

We study a relation between nuclear forces based on phenomenological approach (V_ph) and nuclear effective field theory (V_EFT) from a viewpoint of renormalization group. We find the relation between these two types of nuclear force using Wilsonian renorma

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whereV(α)isNNpotentialforapartialwaveα,andMthenucleonmass.InV(α)(k′,k;p,Λ),k(k′)denotestheo -shellrelativemomentumofthetwonucleonsbefore(after)theinter-action,pdenotestheon-shellmomentum,andΛisthecuto valuespecifyingthemodelspace.ThisequationcontrolstheevolutionofV(α)withrespecttoachangeofΛ.Thisequationwas rstlyderivedbyBirseetal.inadi erentmanner[3].3.Result

StartingwithseveralphenomenologicalNNpotentials,wereducetheirmodelspacesusingtheWRGequation,andexaminetheevolution.Then,wesimulatetheobtainede ectivemodel-spaceinteraction(VM)withVEFTbyadjustingparametersinit.Ifthesimulationisaccurate,thentherelationbetweenVEFTandVphthroughRGisrealized.Inthesimulation,forsimplicity,weincludeonlytheOPEPinVEFTasamechanismex-plicitlyincludingthepion.ThisisnotfullyconsistentwithWeinberg’scounting.Becauseweconsidercontactinteractionswithzero,two,andfourderivatives,weshouldincludemoreirreduciblegraphs,suchasatwo-pionexchangepotential(TPEP).However,weemployarathersmallmodelspacewherethedetailsoftheTPEPplayessentiallynorole;theTPEPisaccuratelysimulatedbycontactinteractions.Therefore,oursimpli cationdoesnotdeterioratetheaccuracyofthesimulation.

InFig.1,shownarediagonalcomponentsof 1S0|V|1S0 inthemomentumspace.ThereistheclearmodeldependenceamongVph,whichalmostdisappearsafterthemodel-spacereductiondowntoΛ=200MeV.TheobtainedVMisaccuratelysimulatedbytheOPEPandafewcontactinteractions,whichwedonotshowhere.InFig.2,diagonalcomponentsof 3D1|VM|3S1 (Λ=200MeV)obtainedfromtheCD-BonnpotentialissimulatedusingtheOPEPandoneortwocontactinteractions.Weseethatthesimulationisveryaccurate.VEFTobtainedinthiswayalsoaccuratelysimulatestheo -diagonalcomponentsofVM.

4.Discussionandsummary

WehaveseenintheprevioussectionthatthescenariofortherelationbetweenVEFTandVphisrealized.Althoughwehaveshowntherelationemployingarathersmallmodelspace,iftheTPEPandhigherorderOPEPareproperlyincludedinVEFT,thentherelationshouldholdtrueforalargermodelspaceaswell;probablysafeuptoΛ～400MeV.InacertainlargemodelspacewheretheverydetailsoftheTPEPisconsiderable,i.e.,aphenomenologicalmodelcannotmimictheTPEPanymore,therelationmaynotholdtrue.Wehavebeenconcernedwithonlythestrongnucleon-nucleoninteraction.However,thiskindofrelationprobablyexistsalsoinothernuclearoperators,suchaselectroweakcurrentsandpionproductionoperators.

Now,basedonourresult,wediscussinthefollowingsomefeaturesofNEFTwhichthetraditionalapproachdoesnothave;thediscussionisnotalwaysrestrictedtothenu-clearforce.OneisthatnuclearoperatorsbasedonNEFTareconstructedperturbatively.Therefore,onecansystematicallyimprovetheaccuracy,andcandiscusstheoreticaluncer-tainty.Wenotethat,foraconvergentperturbation,itisnecessarytode neanoperatorinasuitablysmallmodelspace.WealsonotethattheconvergenceoftheNEFT-basedperturbativeexpansionisnotalwaysgoodforsomeprocesses.

AnotheristhatNEFT-basedoperatorsaremodelindependent,whichhasbeenoften

We study a relation between nuclear forces based on phenomenological approach (V_ph) and nuclear effective field theory (V_EFT) from a viewpoint of renormalization group. We find the relation between these two types of nuclear force using Wilsonian renorma

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00V(k,k) (MeV 2) × 10 4V(k,k) (MeV 2) × 10 5

0 50 100

k (MeV) 150 200-1-0.5-2-1-3-1.5-4 0 50 100k (MeV) 150 200

Figure1.NN-potential 1S0|V|1S0 ;the

upper,theReid93,theNijmegenIandthe

CD-Bonnpotentials,respectively.The

solidanddash-dottedcurvesareVM(Λ=

200MeV),thesolidoneresultsfromthe

NijmegenIandtheCD-Bonnpotentials,

whilethedash-dottedfromtheReid93.Figure2.NN-potential 3D1|V|3S1 .ThesolidcurveisVM(Λ=200MeV)result-ingfromtheCD-Bonnpotential.Thedashed(dotted)curveisasimulationofVMusingtheOPEPplusone(two)contactinteraction(s).Thedash-dottedcurveistheOPEP.

claimed.However,theclaimisbasedontheprocedureofderivingthenuclearoperators,whichisnotaquantitativeargument.Inthiswork,wequantitativelyshowedthatVEFTdoesnothavethedependenceonmodellingthedetailsofshort-distancephysics.Thisisaconsequenceofviewingthesystemroughly,therebyignoringthedetailedstructureoftheshort-distancephysics.Itisnoted,therefore,adescriptionofasystembasedonNEFTisnotalwaysbetterthanthosebasedonphenomenologicalmodels.Inmanycases,theygiveessentiallythesamelow-energyobservablesbecausetheyareequivalentthroughtheRG.NEFTmayhavesuperiorityincaseswheredetailedinformationaboutmulti-pionexchangemechanismplayanimportantrole.

Finally,withNEFT,onecandescribeasystemmoree cientlyandsimply,comparedtothephenomenologicalmodels.Thee ciencyisduetotheperturbativecalculation,andthesimplicityisduetotheignoranceofthedetailsoftheshort-distancephysics.Tosummarize,weconsideredthenuclearforcefromtheviewpointoftheRG,andshowedthatthereexiststheequivalencerelationbetweenVEFTandVph.Wesimultane-ouslyshowedthatVEFTdoesnothavethemodeldependenceduetothedetailsoftheshort-distancephysics,andthatthesimplecontactinteractionsinVEFTaccuratelyde-scribetheshort-distancephysicsinacertainsmallmodelspace.Basedontheresult,wediscussedthenewfeaturesinNEFT.

REFERENCES

1.S.Weinberg,Phys.Lett.B251(1990)288.

2.S.X.Nakamura,Prog.Theor.Phys.114(2005)77.

3.M.C.Birse,J.A.McGovernandK.G.Richardson,Phys.Lett.B464,(1999)169.

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