ThomasPapenbrock

and

EFTformanybodysystems Madrid1.16.2014ResearchpartlyfundedbytheUSDepartmentofEnergy

Effec%vetheoryfordeformednuclei

TP,Nucl.Phys.A852,36(2011);arXiv:1011.5026JialinZhangandTP,Phys.Rev.C87,034323(2013);arXiv:1302.3775

TPandH.Weidenmüller,arXiv.1307.1181ToñoCoelloandTP,inpreparaYon

2

SeparaYonofscalesinnuclearphysics

Fig.:Bertsch,Dean,Nazarewicz,SciDACreview(2007)

ChiraleffecYvefieldtheory

EnergyorRe

soluYo

n

EffecYvetheoryofdeformednuclei

EffecYvetheoriesprovideuswithmodelindependence

DeformaYonofatomicnuclei

Modelsrule!• BohrHamiltonian• GeneralcollecYvemodel• InteracYngbosonmodel

Rotors: E(4+)/E(2+) = 10/3 Vibrators: E(4+)/E(2+) = 2

EffecYvetheory!

ModelsfornuclearrotaYons

1.  BohrHamiltonian(1952),Bohr&Modelson’scollecYvemodel(1953).•  fivequadrupoledegreesoffreedommodelsurfacedeformaYons

threeEuleranglesandtwoshapeparameters2.  GeneralcollecYvemodel“Frankfurtmodel”[Gneuss,Mosel&Greiner

(1969);Hess,Maruhn,Greiner(1981)]3.  Arima&Iachello’sInteracYngBosonModel(1976)

•  sanddbosonaredegreesoffreedom;algebraicapproachwithsymmetriesimposedonHamiltonianinlimiYngcases

• TheexisYngmodelsdescribemanyaspectsofvastsetsofdataquitewell.

• TheyaredifficulttogeneralizeinatractablewayduetothedifficulYesincouplingofangularmomentaandthecomputaYonofmatrixelements.

• RenewedinterestincomputaYonallytractablemodels[M.A.Caprio,Phys.Rev.C68,054303(2003);D.J.Rowe,Nucl.Phys.A735,372(2004)]

ApproachwithinaneffecYvefieldtheorypossible.

ξ

Ω

SeparaYonofscale:ξ<<Ω

Spectrumofadeformednucleus

g.s.‐band

γ‐band

“β”‐band

γγ‐band

“Complete”spectrumof168Er[Davidsonet al.,J.Phys.G7,455(1981)]

Keyfeatures:rotaYonalbandsontopof“vibraYonal”bandheadsTypicaldeformedeven‐evennuclei:lowestvibraYonalstatehasJπ=0+,2+

Similarlycomplete:spectrumof162DyA.Aprahamianet al.,Nucl.Phys.A764,42(2006)

EFTforfinitesystemswith“emergentsymmetrybreaking”*ConsiderenergyscalesinanatomicnucleuswithmassnumberA

Radius

MomentofinerYa

 RotaYonalenergy~

TypicalwavenumberofNGmode

Thus,forA>>1thereisaregimewhererotaYonalmoYonisacorrecYontoNGmodes

*Yannouleas&Landman,Rep.Progr.Phys.70,2067(2007)

Relatedwork:CorrecYonstoparYYonfuncYonsofinfinitesystemsfromfinitesimulaYons.[Horsch&vonderLinden,Z.Phys.B72,181(1988);H.Leutwyler,Phys.Led.B189,197(1987);Gasser&Leutwyler,Nucl.Phys.B307,763(1988);Hasenfratz&Niedermayer,Z.Phys.B92,91(1993),hep‐lat/9212022.]

CosetSO(3)/SO(2)ininfinitesystems:[Leutwyler,Phys.Rev.D49,3033(1994);Roman&Soto,Int.J.Mod.Phys.B13,755(1999);Hofmann,Phys.Rev.B60,388(1999);Bär,Imboden&Wiese,Nucl.Phys.B686,347(2004).]

Keystep:singleoutthezeromode[Leutwyler,Phys.Led.B189,197(1987)]

Cosetspaceformoleculesanddeformednuclei:

ParameterizaYonintermsofYme‐dependentangles(α,β)andNambu‐Goldstonefields(x,y).

ProperYesofNGfields:

UponquanYzaYon,angles(α,β)restoresphericalsymmetry(goodangularmomentum)while(x,y)becomeintrinsicquanYzedexcitaYonmodes.

BehaviorunderrotaYonsBehaviorofU=guunderrotaYonsr,with

 (α,β)transformnonlinearly(likeanglesonthesphere). (x,y)areintrinsicvariablesandtransformlinearly(withacomplicatedangle)

Steps:• idenYfyinvariants• powercounYng:rotaYons<<vibraYons<<breakupscale(100keV<<1MeV<<3MeV)• Expansionoffields(x,y)innormalmodesandcanonicalquanYzaYon• QuantumNumbers:TotalangularmomentumJ,intrinsicprojecYonK• Keydifferencebetweennucleiandmolecules:Nucleiarepairedintheirgroundstate.Thisexcludeslow‐energyexcitaYonswithposiYveparityandoddK• Result:RotaYonalbandsontopofvibraYonal(spinK)bandheads

TPandH.Weidenmüller,arXiv.1307.1181(v2soon),Phys.Rev.C(2014)

9

ConstrucYonofasimplereffecYvetheoryfordeformednuclei

StaywithinQM,considereffecYvetheoryforangles(α,β),andaddphysicsofintrinsicquadrupoledegreesoffreedomwithSSB.

Quadrupolefield• canbeviewedasoneofthenormalmodesoftheintrinsicfields(x,y)• takenfromphenomenologicalapproachbylookingatlow‐energystates

1.IdenYfyrelevantdegreesoffreedom

Quadrupoledegreesoffreedomdescribespinsandparityoflow‐energyspectra

2.IdenYfyrelevantsymmetriesandsymmetrybreaking

Symmetry:RotaYonalinvarianceVerylowenergyexcitaYons(“zeromodes”)indicateemergentsymmetrybreaking

ξ

Ω

SeparaYonofscale:ξ<<Ω

NonlinearrealizaYonoftherotaYonalsymmetry:QuanYYeswithdefinitesymmetryproperYes

1.  ExandEytransformasthexandy‐componentsofavectorunderrotaYons.(ThesearethevelocitycomponentsofaparYcleonthesphere)

2.  The“covariantderivaYve”Dttransformsasthez‐componentofavectorunderrotaYons.(ThisisthecovariantderivaYveonthesphere)

AnyLagrangianconsisYngofcombinaYonsofEx,Ey,andDt(acYngonotherfields)thatisformallyinvariantunderSO(2)(i.e.axiallysymmetric)isindeedinvariantunderSO(3).

Weinberg(1967);Coleman,Wess,Zumino(1969);Callan,Coleman,Wess&Zumino(1969).Pedagogicalreviews:S.Weinberg,TheQuantumTheoryofFields,Vol.II,chap.19;C.P.Burgess,PhysicsReports330(2000)193;T.Brauner,Symmetry2,2010;arXiv:1001.5212.

Physicsofzeromodes

RotaYonalbandsarequanYzedNambu‐Goldstonemodes.Low‐energyconstantC0ismomentofinerYaandfittodata.

Lagrangian

Hamiltonian

QuanYzaYon

Spectrum

Nucleiwithnonzeroground‐statespins:WZtermsAfiniteground‐statespinbreaksYmereversalinvariance. ConsidertermsthatarefirstorderintheYmederivaYve NosuchtermsareinvariantunderrotaYons.BUT:AcYonremainsessenYallyinvariantunderoneparYcularcombinaYon(correspondstomagneYcmonopoleinsidesphere)

Lagrangian

Hamiltonian

EigenvaluesandeigenfuncYons(IdenYfyqwithground‐statespin!)

(WignerDfuncYons)

Chandrasekharan,Jiang,Pepe,&Wiese,Phys.Rev.D78,077901(2008)

PowercounYngandbeyondleadingorder

Es%mates(naïvedimensionalanalysis)HLO~ξ

LagrangianatNLO

Powercoun%ngMainidea:higher‐odertermduetoneglectedcouplingsbetweenNGmodesandvibraYons.[C2/C0]=energy–2

Spectrum:AI(I+1)+B(I(I+1))2foreven‐evennuclei.(Bohr&Modelson)Ingeneral:

173Yb:RelaYveerrorinLOandNLO

Smallparameter:ξ/Ω≈79/350≈1/4.5

BeyondNGmodes:couplingtovibraYons

HigherenergeYcdegreesoffreedomneedtobeincluded.

Quadrupolefieldexhibitsspontaneoussymmetrybreaking.

 5DoF–2NG=3DoF

ξ

Ω

SeparaYonofscale:ξ<<Ω

CouplingstovibraYons:powercounYngLowenergyscaleξHighenergyscaleΩ>>ξ

Dimensionalanalysis

PotenYalexpandedaroundminiumum

PowercounYng:largeamplitudesφ0≈vrestorerotaYonalsymmetrybreakdownofEFT

Leadingorder~O(Ω)Lagrangianatleadingorder:

Spectrum

LeadingorderLagrangianyieldsthebandheadsasharmonicvibraYons

LagrangianconsistsofEx,Ey,Dtφ0,Dtϕ2,φ0,ϕ2,andneedstobeformallyinvariantunderSO(2)(axialsymmetry)only.

βvibraYon γvibraYon

Next‐to‐leadingorder~O(ξ)

Lagrangian

Hamiltonian(kineYcenergy)

Spectrum:harmonicvibraYons&rotaYonalbandoneveryvibraYonalbandhead

(CorrecYons~ξofbandheadsduetoanharmoniciYesinthepotenYalneglected.)

Innext‐to‐leadingorder,theresultsoftherotaYonal‐vibraYonalmodelarereproduced.

MulY‐phononexcitaYonsW.F.Davidsonet al.,J.Phys.G7,455(1981)

TheoryatNLOpredictsvibraYonalspectrumwithrotaYonalbandsontop(andconstantmomentofinerYas/rotaYonalconstants)

Data• VibraYonalbandheadsclearlynotharmonic• MomentsofinerYaexhibitsmaller,band‐head‐dependentvariaYons• Manybandheads(neg.parity)notaccountedfor(octupole,hexadecupole,…)

gsband

γband

βband

γγband

MulY‐phononbandsindeformednucleiMulY‐phononγbandsunambiguously*observedin166,68Er,232Th

168Er[H.G.Börneret al.,Phys.Rev.Led.66,691(1991);M.Oshimaet al.,Phys.Rev.C52,3492(1995);T.Härtleinet al.,Eur.Phys.J.A2,253(1998).]

166Er[C.Fahlanderet al.,Phys.Led.B388,475(1996);P.E.Garredet al.,Phys.Rev.Led.78,4545(1997).]

232Th[W.Kortenet al.,Phys.Led.B317,19(1993);A.MarYnet al.,Phys.Rev.C62,067302(2000)].

EnergiesofrotaYonalbands

• NonlineartermsinthevibraYonsonly(anharmoniciYes;sufficientparameters)• CouplingsbetweenrotaYonalandvibraYonaldegreesoffreedom

Methods:Fukuda’sinversionmethodforperturbaYveLegendretransformaYons

Result:rotaYonalconstantsarelinearlyintheexcitedvibraYonalquanta

PredicYonsingoodagreementfor168Er,fairagreementfor166Er,andcorrecttrendin232Th

Thetheory’saccountforsmallvariaYonsinthemomentofinerYaovercomesasmallerbutlong‐standingproblemoftheIBMandcollecYvemodels.Originofsuccess:couplingbetweenkineYctermsofrotorandvibraYons.

RotaYonalconstantsoftwo‐phononγγbandsfromNNLOcorrecYons

ElectromagneYctransiYonsindeformednuclei

KeyfeaturesofgeometriccollecYvemodel

✓ RotaYonalbandsontopofvibraYonalbandheads✓ Strongin‐bandE2transiYons✓ Weakerinter‐bandE2transiYons

✗Inter‐bandtransiYonsarefactors2‐10toostrong[Garred,J.Phys.G27(2001)R1;Rowe&Wood“FundamentalsofNuclearModels”(2010)]

CollecYvemodelconfrontsdata

25

x=B(E2,2g.s.0g.s.)

y=B(E2,200g.s.)

z=B(E2,2γ0g.s.)

Rowe &Wood, Fundamentals of nuclear models, World Scientific (2010)

++

++

++

2+

• Theproblemlieswiththeabsolutestrengthsofinter‐bandtransiYons.• RaYosofinter‐bandtransiYonsok• In‐bandtransiYonsok

Lagrangianexpansioninpowersofξ/Ω

Ω(ξ/Ω)3/2

ξ

Ω

+ … Gaugingyields

ElectromagneYctransiYons

27

• AddelectromagneYctermstotheLagrangian

• TheinteracYonHamiltoniandefinesthetransiYonoperatorsandtransiYonprobabiliYes

• ALegendretransformaYonyieldthecorrespondingHamiltonian

GaugedHamiltonian

28

RicherelectromagneYcstructurethancollecYvemodels:“Radial”charge,and“angular”charge.

Resultsfor168Er

0.0391

0.568

Adiaba%cBohrmodel 5.92

aBaglin,Nucl.DataSheets111,1807(2010)bLehmannetal.,Phys.Rev.C57,569(1998)cValueemployedtoadjustlow‐energyconstant

EffecYvetheory:

• GaugingofLagrangianyieldsEMcurrentsconsistentwithHamiltonian

• PowercounYngalsoforEMcouplings

• Richerstructurethangeometricmodel;moreparametersinasystemaYcexpansion

[ToñoCoelloandTP,inpreparaYon]

B(E2,if)ine2b2

Summary

EffecYvetheoryfordeformednuclei

• EFTdevelopedforfinitesystem(singleoutthesymmetry‐restoringcosetmodes)

• rotaYonsarelarge‐amplitudezero‐modes

• vibraYonalstatesarequanYzedNambu‐Goldstonemodes

• exploitsseparaYonofenergybetweenrotaYonsandvibraYons• essenYallyreproducesspectraofphenomenologicalmodelsatNLO

• allowsforsmallvariaYonsinrotaYonalconstantsofbandheads

• predicYonsforrotaYonalconstantsofmulY‐phononγvibraYonsinreasonableagreementwithdata

• electromagneYccouplingviagaugingpromising:consistentandricherstructurethancollecYvemodels;improvedstrengthsofinter‐bandtransiYons