Off-shell pairing correlations from meson-exchange

arXiv:nucl-th/0302040v23Dec2003Off-shellpairingcorrelationsfrommeson-exchangetheoryofnuclearforcesArmenSedrakianInstitutf¨urTheoretischePhysik,Universit¨atT¨ubingen,D-72076T¨ubingen,Germany(Dated:February8,2008)Wedevelopamodelofoff-mass-shellpairingcorrelationsinnuclearsystems,whichisbasedonthemeson-exchangepictureofnuclearinteractions.ThetemporalretardationsinthemodelaregeneratedbytheFockexchangediagrams.Thekernelofthecomplexgapequationforbaryonsisrelatedtothein-mediumspectralfunctionofmesons,whichisevaluatednonperturbativelyintherandomphaseapproximation.Themodelisappliedtothelow-densityneutronmatterinneutronstarcrustsbyseparatingtheinteractionintoalong-rangeone-pion-exchangecomponentandashort-rangecomponentparametrizedintermsofLandauFermiliquidparameters.TheresultingEliashberg-typecouplednonlinearintegralequationsaresolvedbyaniterativeprocedure.Wefindthattheself-energiesextendtooff-shellenergiesoftheorderofseveraltensofMeV.Atlowenergiesthedampingoftheneutronpaircorrelationsduetothecouplingtothepionicmodesissmall,butbecomesincreasinglyimportantastheenergyisincreased.Wediscussanimprovedquasiclassicalapproximationunderwhichthenumericalsolutionsareobtained.I.INTRODUCTIONThelowestorderintheinteractionmean-fieldtreat-mentsofthenuclearpairinginbulknuclearmatterandinfinitenucleiareconceptuallysimpleandhavebeenrea-sonablyconsistentwiththephenomenologyofthesesys-tems.TheHartree-Fock-Bogolyubov(HFB)theorywithzero-orfinite-range(e.g.,theSkyrmeandtheGogny)phenomenologicalforceshasbeenthestandardtoolinthestudiesofthenuclearstructuretodescribethebulkoftheexperimentaldata[1].Atthesametime,thecalcula-tionsofthepairinggapsinbulknuclearmatterbasedontheBardeen-Cooper-Schrieffer(BCS)theorywithinter-actionsmodeledbythebare(realistic)nucleon-nucleonforcesandquasiparticlespectraintheeffectivemassap-proximationarenotinconsistentwiththecoolingsimu-lationsofneutronstarsorthephenomenologyoftheirrotationdynamics.Althoughthereisnodirectevidenceforhigherordercorrelationsinthephenomenologyofnu-clearsystems,thesecannotbediscardedonthebasisofpurelyphenomenologicalarguments.TheeffectiveforcesusedintheHFBcalculationsarefittedtothepairingpropertiesof(some)nuclei,consequentlythecontribu-tionsbeyondmeanfieldarebuilt-inautomatically(e.g.,theeffectiveSkyrmeorGognyforcesincludethecontri-butionsfromtheladderdiagrams).Thevaluesofthepairinggapsdeducedfromthephenomenologyofneu-tronstarsdependsensitivelyonthephysicalinput,suchastheequationofstate,themattercomposition,anditsdynamicalproperties.Providedtheuncertaintiesintheseingredientsthephenomenologicalvaluesofthegapcanbeoffbyanorderofmagnitude.Fromthetheorypointofviewthehigherordercorrelationsmightnotbemanifestduetolargecancellationbetweenthevertexandpropagatorrenormalization.Theon-mass-shellpropertiesofthepairinggapsarewellunderstoodatthemean-fieldlevel.Uponignoringtherenormalizationeffects,theon-mass-shellgapscanbedirectlycomputedfromthenucleon-nucleonscatter-ingphaseshifts.Sincethepairingisalow-temperaturephenomenon,awell-definedFermiseaimpliesthattheon-shellself-energiesofweaklyexcitedquasiparticlescanbeexpandedaroundtheirFermimomentum.Themagni-tudeofthegapisthencontrolledbytherenormalizationofdensityofstatesattheFermisurface.Sincetheratioofthemomentum-dependenteffectivemasstothebaremassislessthanunitythereductionofthedensityofstatescausesareductionofthepairinggap.Theearlyworktoconstructrenormalizationschemesforstronglyinteractingfermionsintroducedwave-functionrenormalizationfactorsinthekernelofthegapequationandmodeledthepairinginteractionintermsofparticle-holeirreduciblevertices[2].Thesewerere-latedtothephenomenologicalLandauFermiliquidpa-rameterswhichcan(inprinciple)befixedbycompari-sonwithexperiments.TheLandauparameterscanbeobtainedfromabinitiomicroscopiccalculationsaswell,whichareneeded,forexample,todescribeisospinasym-metricnuclearsystemsthatarenotdirectlyaccessibleinlaboratories[3,4,5,6].Variousaspectsofthepair-ingintheinfinitenuclearmatterbeyondthemean-field(BCS)approximationwereaddressedduringthepasttwodecadesbyemployingeithermoreadvancedschemestocalculatedtheself-energyandvertexcorrectionsand/ormodern(phase-shiftequivalent)interactions.Thisworkincludesthemicroscopiccalculationsoftheeffectivein-teractionsandLandauparametersusingvariantsoftheBabu-Brown[7]orrelated(e.g.,polarizationpotential)schemesofsummingthepolarizationgraphsinneutronmatter[8,9,10].Thepolarizationeffectswerefoundtosuppresstheon-shellpairinggapsbylargefactors.Ear-lier,similarreductionwasfoundfromthecorrelatedbasisvariationalcalculationsbasedontheJastrowansatzforfermionicwavefunctions[11,12,13].Morerecentworkconcentratedonthecalculationsofthewave-functionrenormalizationsandtheirimplementationsingapequa-tionsemployingself-energiesderivedwithinBruecknertheoryatfirst[14]andsecond[15]orderinthehole-lineexpansion.Whileagainasuppressionwasfoundduetothewavefunctionrenormalizations,supplementingthese2self-energycorrectionswithappropriatevertexcorrec-tionslargelycompensatesforthelattereffect[16].ThefinitetemperatureTmatrixwasutilizedtodeducethepairingcorrel