Renormalization group analysis of nuclear force(3)

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

3

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