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