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Nuclearinterac-onsandnuclei

ThomasPapenbrock

Aimoftheselectures:

Giveoverviewofnuclearforcesfromeffec-vefieldtheory,resultsoffirstprinciplescomputa-ons,andaneffec-vetheoryforheavydeformednuclei

30thEcoleJoliotCurie:Physicsatthefemtometerscale

September12‐172011,LaColleSurLoup/France

and

Content

1.  Lecture:Overview,interac-onsfromchiraleffec-vefieldtheory,andresultsfromnuclearstructurecomputa-ons

2.  Lecture:Restoflecture1+effec-vetheoryfordeformednuclei

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Readingsugges-ons

Moreisdifferent,P.W.Anderson,Science177,393(1972)

Elementaryfeaturesofnuclearstructure,B.R.MoTelson,in:H.Nifenecker,J.P.Blaizot,G.Bertsch,W.Weise,F.David(Eds.),Trends in Nuclear Physics, 100 Years Later,North‐Holland,Amsterdam,1998

Chiraleffec-vefieldtheoryandnuclearforces• Machleidt,Nuclear Forces from chiral effec<ve field theory,arXiv:0704.0807• EpelbaumHammer&Meißner,Modern theory of nuclear forces, Rev.Mod.Phys.81,1773(2009);arXiv:0811.1338

Low‐momentuminterac-onsandsimilaritytransforms• Bogner,Furnstahl&Schwenk,From low‐momentum interac<ons to nuclear structure, Prog.Part.Nucl.Phys.65,94(2010);arXiv:0912.3688

Effec-vetheoryfordeformednuclei• HaruoUiandGyoTakeda,A Class of Simple Hamiltonians with Degenerate Ground State. II A Model of Nuclear Rota<on,Prog.Theor.Phys.70,176(1983)• Papenbrock, Effec<ve theory for deformed nuclei,Nucl.Phys.A852,36(2011),arXiv:1011.5026

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Scienceques-ons

Basicscienceques-onsinthephysicsofnuclei:

1.  Whatbindsnucleonsintostablenucleiandrareisotopes?

2.  Whatarethelimitsofnuclearbinding?

3.  Whatistheoriginofsimplemodesincomplexsystems?

Basicscienceques-onsinnuclearastrophysics:

4.  Howaretheelementsfromirontouraniummade?

5.  Whatisthefateofmassivestars?

6.  Whatisthemassoftheneutrino,andwhatisitsnature(Dirac/Majoranafermion)?

Understandingofrareisotopescentraltoaddressingfouroftheseques-ons(1,2,4,5).

Rareisotopesalsorelevantformedicalapplica-ons(imagingandtherapy),andenergyproduc-on

Quantumchromodynamics–theoryofthestronginterac-on

Mostimpressiveprogress(lecturebyJ.P.Lansberg)But:first‐principlecomputa-onofnucleifromQCDares-llfaraway…Worse:LookingattheQCDLagrangian,itisnotobviouswhatthelow‐energyQCDphysicsis.NeitherthespontaneousbreakingofchiralsymmetrynortheemergenceofselmoundnucleiisobviousorpredictedfromQCD.(TheQEDLagrangianalsodoesnottellusaboutemergingphenomenasuchassuperconduc-vityorcrystals.)Weneedanotherapproach!

Dürretal.,Science322,1224(2008)

HadronmassesfromlapceQCD

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Energyscalesandrelevantdegreesoffreedom

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

Chiraleffec-vefieldtheory

Energy

Effec-vetheoryofdeformednuclei

Effec-vefieldtheoryQ:Howcanweeconomicallysolveaphysicalproblem(byemployingappropriatedegrees

offreedom)?

A:ExploitaseparaQonofscales.

Examples:

1.  Mul-poleexpansionfortheelectromagne-cfield.

Q:Why/whendoesitwork?A:Distancefromchargedistribu-on>>extensionofchargedistribu-on

2. QuantumchemistryemploystheCoulombpoten-alandnotQED

Q:Whydoesitwork?

3. Nucleiaredescribedintermsofprotonsandneutronsandnotviaquarksandgluons

Q:Whydoesitwork?

A:e+e‐pairproduc-onthreshold(~1MeV)>>chemicalbonds(~eV)

A:Excita-onofnucleon(~300MeV)>>excita-onenergiesofnuclei(~1MeV)

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Construc-onofnuclearpoten-alsviachiralEFTWeinberg,vanKolck,Epelbaum,Machleidt,…

1.  Iden-fytherelevantdegreesoffreedomfortheresolu-onscaleofatomicnuclei:nucleonsandpions.

2.  Iden-fytherelevantsymmetriesoflow‐energyQCDandinves-gateifandhowtheyarebroken:spontaneouslybrokenchiralsymmetry

3.  ConstructthemostgeneralLagrangianconsistentwiththosesymmetriesandthesymmetrybreaking.

4.  DesignanorganizaQonalschemethatcandis-nguishbetweenmoreandlessimportantcontribu-ons:alow‐momentumexpansion:powercounQng

5.  Guidedbytheexpansion,calculateFeynmandiagramstothedesiredaccuracyfortheproblemunderconsidera-on.

Reviews:

BedaqueandvanKolck,Ann.Rev.Nucl.Part.Sci.52(2002)339,nucl‐th/0205058.

Machleidt,arxiv:0704.0807.

Epelbaum,Hammer,Meißner,Rev.Mod.Phys.81,1773(2009);arXiv:0811.1338.

1.Iden-fyrelevantdegreesoffreedom

140

300

700

780

940

80

0

Energy(MeV)

Nucleon

OmegamesonRhomeson

Nucleonresonance(Delta)

Pion

Cutoffinchiraleffec-vefieldtheory

Fermienergy

Cutoffinchiraleffec-vefieldtheorywithDeltas

–Separa-onofscale!

Cutoffinpion‐lesseffec-vefieldtheory

2.Iden-fica-onofrelevantsymmetries

1.  SU(3)colorsymmetryfromQCD(Nucleonsandpionsarecolorsinglets)

2.  Chiralsymmetry:Levandright‐handedmasslessuanddquarksdonotmix:SU(2)LxSU(2)Rsymmetry.Expectlev‐rightparitydoubletsinnature.

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Explicitbreakingofchiralsymmetry:uanddquarkshaveasmallmass.Smallcorrec-onstoabovepicturearise.

Reason:Spontaneousbreakingofchiralsymmetry(Moreisdifferent!)

•  SU(2)LxSU(2)RsymmetryspontaneouslybrokentoSU(2)V

•  PionsaretheNambu‐Goldstonebosonsofspontaneouslybrokenchiralsymmetry

•  Low‐energypionLagrangiancompletelydetermined

But:Therearenoparitydoubletsobservedinnature!

Intermezzo:Spontaneoussymmetrybreaking

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Thefundamentallawsofphysicsareinvariantunderrota-ons.Howcannon‐sphericalthings(e.g.watermolecules,pencils,

humans)exist?

1.  Non‐sphericalthingshaveagroundstatewithnonzerospinandfixedspinprojec-on.

2.  Thenon‐sphericalthingsarenotintheirgroundstate

3.  Macroscopicthings,evenintheirgroundstates,donotneedtobeeigenstatesofangularmomentum.

Thefundamentallawsofphysicsareinvariantunderrota-ons.Howcannon‐sphericalthings(e.g.watermolecules,pencils,

humans)exist?

1.  Non‐sphericalthingshaveagroundstatewithnonzerospinandfixedspinprojec-on.

2.  Thenon‐sphericalthingsarenotintheirgroundstate

3.  Macroscopicthings,evenintheirgroundstates,donotneedtobeeigenstatesofangularmomentum.

SpontaneoussymmetrybreakingDef.:Spontaneoussymmetrybreakinghappenswhenanarbitrarilysmallexternalperturba-onyieldsanon‐symmetricgroundstate.

Q1:Whatistheexcita-onenergyforhorizontaltransla-onalmo-onofamacroscopicobjectinthisroom?

A1:E=ħ2/(2mL2)≈10‐70Joule(Objectofmassm=1kginaboxofsizeL=10m)

Q2:Whatistheexcita-onenergyforrota-onalmo-onofamacroscopicobjectinthisroom?

A2:E=ħ2J(J+1)/(2mR2)≈10‐64Joule(Objectofmassm=1kgandradiusR=10cm)

Consequence:Superposi-onsofprac-callydegenerateplanewavestatesofthecenter‐of‐masswillyieldalocalizedgroundstatebecauseofspontaneoussymmetrybreaking.

Consequence:Superposi-onsofprac-callydegeneratestateswithdifferentangularmomentumJwillyieldadeformedgroundstatebecauseofspontaneoussymmetrybreaking.

Strictlyspeaking,thiscanonlyhappeninmacroscopicsystemsthathaveagaplessexcita-onspectrum.

Nambu‐Goldstonemodesarelow‐lyingexcita-onsinthepresenceofspontaneoussymmetrybreaking

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16Picture:www‐llb.cea.fr

Spontaneoussymmetrybreakingofrota-onalsymmetry:ferromagnet

• AxiallysymmetricgroundstatebreaksSO(3)rota-onalsymmetry.• Nambu‐Goldstonemodesgeneratelocal(i.e.posi-onand-medependent)rota-onsofthespins.• exp(‐iψx(x,y,z,t)Jx‐iψy(x,y,z,t)Jy)withNambu‐Goldstonefields(ψx,ψy)andangularmomentumoperators(Jx,Jy)• Spinwaves(ormagnons)arelow‐energyexcita-onswithlongwavelength

Spontaneousbreakingoftransla-onalsymmetry:crystal

17Source:wikipedia(FlorianMarquard)

• Crystallapcebreakstransla-onalsymmetry.• Nambu‐Goldstonemodesgeneratelocal(i.e.posi-onand-medependent)transla-onsofthelapcepoints(ions).• exp(‐iψx(x,y,z,t)Px‐iψy(x,y,z,t)Py‐iψz(x,y,z,t)Pz)withNambu‐Goldstonefields(ψx,ψy,ψz)andmomentumoperators(Px,Py,Pz)• Phononsarelow‐energyexcita-ons(wavelengthλmuchlargerthanlapcespacing)

DerivaQve(low‐momentum)expansionindicatedbysuperscripts

Pion‐pionLagrangian:UisSU(2)matrixparameterizedbythreepionfields

Leadingorderpion‐nucleonLagrangian

Leadingordernucleon‐nucleonLagrangian(encodesunknownshort‐rangedphysics)

3.ConstructmostgeneralLagrangianconsistentwithsymmetries;organiza-onalschemepowercoun-ng

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Effec-vefieldtheory:chiralpoten-alatorderN3LO

Nucleons:fulllinesPions:dashedlines

Features:1.  Systema-cexpansionof

nucleonpoten-al;smallparameter(Q/Λ)

2.  Low‐energyconstantsfromfittodata

3.  HierarchyofforcesNN>>NNN>>NNNN

[fromMachleidtarXiv:0704.0807]

Chiralnucleon‐nucleonpoten-alatleadingorder

One‐pionexchangepoten-al(p,p’areini-alandfinalrela-vemomenta)

Leadingordercontactterm(encodeunknownshort‐rangephysics)

Higher‐ordercontacttermsalsoserveascountertermsthatrenormalizeloopintegrals.

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Whycontactterms?

1.  Onlycontacttermscanmodelreallyshortrangephysics.

2.  Anyshort‐rangeterms(e.g.deltafunc-ons,Gaussians…)withrangesmaller1/Λwoulddothejob,butcontactsarecomputa-onallyveryconvenient.

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Whycontactterms?

1.  Onlycontacttermscanmodelreallyshortrangephysics.

2.  Anyshort‐rangeterms(e.g.deltafunc-ons,Gaussians…)withrangesmaller1/Λwoulddothejob,butcontactsarecomputa-onallyveryconvenient.

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HowdoesthemomentumcutoffΛentertheEFT?

1.  Theconstruc-onofthechiralpoten-alinvolvessolvingtheLippmann‐Schwingerequa-on.Λisthecutoffinthisequa-on.

2.  Theloopintegralsthatappearbeyondleadingorderneedtoberegularized.Onewayofregulariza-onisbyimposingacutoffoftheorderofΛ.

3.  Asaresult,thelow‐energyconstantsdependimplicitlyontheregulariza-onschemeandthecutoff.

4.  Thereare(infinitely)manydifferentchiralpoten-als!Differencesofpoten-alsthatemploydifferentvaluesforthecutoffmustbeofhigherorder.

5.  Regulariza-onschemes,andformofpoten-alsthatencodeshort‐rangedphysics(contactpoten-alorpoten-alswithaveryshortrange)areatthepoten-albuilder’sdiscre-on.Thismakestheapproachmodelindependent.

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FromBogneretal,arXiv:0912.3688

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Higherorder(morework)higheraccuracy

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Three‐nucleonforces–Why?•  Nucleonsarenotpointpar-cles(i.e.notelementary).

•  Weneglectedsomeinternaldegreesoffreedom(e.g.Δ‐resonance,“polariza-oneffects”,…),andunconstrainedhigh‐momentummodes.

ExamplefromcelesQalmechanics:Earth‐Moonsystem:pointmassesandmodifiedtwo‐bodyinterac-on

OtherQdaleffectscannotbeincludedinthetwo‐bodyinteracQon!Three‐bodyforceunavoidableforpointmasses.

Renormaliza-ongrouptransforma-on:Removalof“s-ff”degreesoffreedomatexpenseofaddi-onalforces.

Three‐bodyforcescont’d

(fromBogner,Furnstahl,Schwenk,arXiv:0912.3688)

Leadingthree‐nucleonforce1.  Long‐rangedtwo‐pionterm(Fujita&Miyazawa…)2.  Intermediate‐rangedone‐pointerm3.  Short‐rangedthree‐nucleoncontact

Theques-onisnot:Dothree‐bodyforcesenterthedescrip-on?The(only)quesQonis:Howlargearethree‐bodyforces? 26

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Non‐uniquenessofthree‐nucleonforces

A.Nogga,S.K.Bogner,andA.Schwenk,Phys.Rev.C70(2004)061002

AscutoffΛ isvaried,mo-onalong“Tjonline”.

Addi-onofΛ‐dependentthree‐nucleonforceyields(almost)agreementwithexperiment.Q:What’smissing?

A:Thecompletedescrip-onof4Hewouldrequirefour‐nucleonforces!

Ques-on:Yourfavoritephysicsfriendcomestoyouandsuggeststodeterminetheeffectsofthethree‐bodyforceonthestructureofyourfavoritenucleus.Youreply

1.  Let’sdothis.ThiswillputusonthefasttracktoStockholm.

2.  Thisisdifficulttodisentangle.Butitcanbedoneinathree‐bodysystemsuchas3H.

3.  Whichinterac-onareyoulookingat?

4.  Answers2&3arecorrect.

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Ques-on:Yourfavoritephysicsfriendcomestoyouandsuggeststodeterminetheeffectsofthethree‐bodyforceonthestructureofyourfavoritenucleus.Youreply

1.  Let’sdothis.ThiswillputusonthefasttracktoStockholm.

2.  Thisisdifficulttodisentangle.Butitcanbedoneinathree‐bodysystemsuchas3H.

3.  Whichinterac-onareyoulookingat?

4.  Answers2&3arecorrect.

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Thesizeandformofthree‐bodyforcesdependsonthecutoff,andthechosenrenormaliza-onscheme.Differentschemes(“implementa-onsoftheEFTatordern”)yieldpredic-onsthatexpectedtoagreewithintheerrores-mate(Q/Λ)n+1.Onlythesumofinterac-onscanbeprobed.

What’stheroleofthree‐nucleonforces?

Contribu-onstobindingof4He.[Hagen,TP,Dean,Schwenk,Nogga,Wloch,Piecuch,PRC76,034302(2007)]

Errores-mateduetoCCSDapproxima-on

NNonly

0‐body3NF

1‐body3NF

2‐body3NF

residual3NF

Monopoleshivsfrom3NFasdensity‐dependentNNforce.[A.Zuker,PRL90,42502(2003)]

22Na

Is28Oaboundnucleus?ExperimentalsituaQon

•  “Last”stableoxygenisotope24O

•  25Ounstable(Hoffmanetal2008)

•  26,28Onotseeninexperiments

•  31Fexists(addingonprotonshivsdriplineby6neutrons!?)

Shellmodel(sdshell)withmonopolecorrec-onsbasedonthree‐nucleonforcepredicts2ndOaslaststableisotopeofoxygen.[Otsuka,Suzuki,Holt,Schwenk,Akaishi,PRL(2010),arXiv:0908.2607]

Intermission

•  Systema-cconstruc-onofnuclearforceswithin(chiral)effec-vefieldtheory

•  Thereisarecipetofollow

•  Highlights:powercoun-ng,hierarchyofNN>>NNN>>NNNNforces

•  Approachismodelindependent•  Resul-ngpoten-aldependsonregulariza-onschemeandcutoff

•  Thereare(infinitely)manygoodwaystoimplementthis

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Highlights:first‐principlecomputa-onsofnuclei

Pieper and Wiringa, Ann.Rev.Nucl.Part.Sci. 51 (2001) 53

QuaglioniandNavrá-l,PRL101,092501(2008) Hoylestatein12C

Epelbaumetal,Phys.Rev.LeT.106,192501(2011)

Medium-mass nuclei 40,48 Ca [Hagen et al, Phys. Rev. Lett. 101, 092502 (2008)]

[Hagenetal,PRL104,182501(2010)]

Proton‐halostatein17F

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Lightnucleifromachiralinterac-on(N3LObyEntem&Machleidt)withno‐coreshellmodel

P.Navra-letal.,Phys.Rev.LeT.99,042501(2007),nucl‐th/0701038.Reviewofno‐coreshellmodel:Navra-l,Quaglioni,Stetcu,BarreT,arXiv:0904.0463.

12CHoylestatefromlapceEFT

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Epelbaumetal,Phys.Rev.LeT.106,192501(2011);arXiv:1101.2547

Satura-onproper-esofchiralNNinterac-onswithcoupledclusters(Λ=500MeVpoten-alfromEntem&Machleidt)

Nucleus CCSD Λ-CCSD(T) Experiment 4He 5.99 6.39 7.07 16O 6.72 7.56 7.97 40Ca 7.72 8.63 8.56 48Ca 7.40 8.28 8.67

Bindingenergypernucleon16ObyFujiietal(0908.3376)

B/A=6.62MeV(2bodyclusters)B/A=7.47MeV(3bodyclusters)

[Hagen,Papenbrock,Dean,Hjorth‐Jensen,Phys.Rev.LeT.101,092502(2008)]

(Exp‐342MeV)

(Exp:‐127.6MeV)

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Three‐bodyforcesinchiralEFTexpectedtoadd0.4MeVpernucleon?!

Es-mateformodelspacesandHamiltonianmatrixdimensions

AssumewewanttocomputethebindingenergyofanucleuswithmassnumberAinawavefunc-onbasedapproach.Assumethattheinterac-onhasamomentumcutoffΛ.

Q:Whataretheminimumrequirementsforthemodelspace?

A:1.  Thebasismustbesufficientlyextendedinposi-onspacetocapturea

nucleuswithradiusR≈1.2A⅓fm2.  Thebasismustbesufficientlyextendedinmomentumspacetocapture

thecutoffΛ.3.  THUS:weneedapproximatelyK=(RΛ/(2π))3single‐par-clestates(phase

spacevolume!)Inprac-ceK≈(RΛ/2)3~Λ3A.

ComputaQonofoxygen:Λ=4/fmandR≈2.5fmThus,ourmodelspacehasaboutK=53=125single‐par-clestates.Matrixdimension:D=K!/(K‐A)!/A!≈(K/A)A≈816≈248≈1014.

Thus,thematrixdimensionisD=K!/((K‐A)!A!),withK≈(RΛ/2)337

Someconclusions

1.  For“bare”chiralinterac-ons,matrixdiagonaliza-onispossibleonlyforlightnuclei.Oneeitherneedsamuchmoreefficientmethodoralowercutoff.

2.  ThefactorialscalingwithAisnotmatchedbyMoore’slaw(doublingofFLOPSaboutevery18monthfactor1000in15years).

3.  Forwave‐func-onbasedmethods,themosteffec-vewaytoheaviernucleiistodecreasethephase‐spacevolumeK~(ΛR)3~Λ3Abydecreasingthecutoff.

  Low‐momentuminterac-ons&similarityrenormaliza-ongrouptransforma-onsthatlowerthecutoffΛ.

Homework:Consideranoscillatorbasis.HowhasonechoosetheoscillatorfrequencyωandthenumberofoscillatorshellsNforagivenmomentumcutoffΛandmassnumberA?

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Configura-onspacemomentumspace

Hardcorehigh‐momentummodes

Bogner/Furnstahl(2007)

Momentum‐dependenceofphenomenologicalpoten-als

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Similarityrenormaliza-ongroup(SRG)transforma-onS.Glazek,K.Wilson,PRD48(1993)5863;49(1994)4214;

F.Wegner,Ann.Phys.3(1994)77

Mainidea:decouplelowfromhighmomentaviaa(unitary)similaritytransforma-on

Unitarytransforma-on

Evolu-onequa-on

Choiceofunitarytransforma-onthrough(onedoesnotneedtoconstructUexplicitly).

yieldsscale‐dependentpoten-althatbecomesmoreandmorediagonal

Note:Baker‐Campbell‐HausdorffexpansionimpliesthatSRGof2‐bodyforcegeneratesmany‐bodyforces

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SRGevolu-onofachiralpoten-al

Fig.:Bogner&Furnstahl.SeehTp://www.physics.ohio‐state.edu/~ntg/srg

Solu-onof3Hand4Hewithinducedandini-al3NF

Q:Whatistheeffectof(omiTed)4NFandforcesofevenhigherrank?

A:In4He,(shortranged)4NFyieldabout200keV(seeenergiesatsmallmomentum)Note:Thisisconsistentwithdevia-onfromexperiment! 42

Jurgenson,Navra-l&Furnstahl,Phys.Rev.LeT.103,082501(2009),arXiv:0905.1873

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Energyscalesandrelevantdegreesoffreedom

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

Chiraleffec-vefieldtheory

Energy

Effec-vetheoryofdeformednuclei

Deforma-onofatomicnuclei

Modelsrule!• BohrHamiltonian• Generalcollec-vemodel• Interac-ngbosonmodel

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

Effec-vetheory!

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Construc-onofanEFT

1.  Iden-fytherelevantdegreesoffreedomfortheresolu-onscaleofinterest

2.  Iden-fytherelevantsymmetriesoflow‐energynuclearphysicsandinves-gateifandhowtheyarebroken

3.  ConstructthemostgeneralLagrangianconsistentwiththosesymmetriesandthesymmetrybreaking.

4.  DesignanorganizaQonalscheme(powercoun-ng)thatcandis-nguishbetweenmoreandlessimportantcontribu-ons

Usefulreferences:

S.Weinberg,The Quantum Theory of Fields, Vol.II,chap.19

H.Leutwyler,Phys.Rev.D49(1994)3033,arXiv:hep‐ph/9311264

C.P.Burgess,PhysicsReports330(2000)193

172Yb

198Pt

Red: E(4+)/E(2+)=10/3Yellow: E(4+)/E(2+)=2

R4/2=E(4+)/E(2+)asameasureofdeforma-on

1.Iden-fyrelevantdegreesoffreedom

1.Iden-fyrelevantdegreesoffreedomQ:Whatfieldwouldbeabletoreproducespinsandpari-esoflow‐lyingstates?

Quadrupoledegreesoffreedomdescribespinsandparityoflow‐energyspectra

1.Iden-fyrelevantdegreesoffreedomQ:Whatfield(spin&parity)wouldbeabletoreproducespinsandpari-esoflow‐lyingstates?

2.Iden-fyrelevantsymmetriesandsymmetrybreaking

2.Iden-fyrelevantsymmetriesandsymmetrybreakingQ:Whatarethesymmetries,andaretheyspontaneouslybroken?

ξ

Ω

Separa-onofscale:ξ<<Ω

2.Iden-fyrelevantsymmetriesandsymmetrybreakingQ:Whatarethesymmetries,andaretheyspontaneouslybroken?

Symmetry:Rota-onalinvarianceVerylow‐energyexcita-ons(~Nambu‐Goldstonemodes)fromspontaneoussymmetrybreaking

ξ

Ω

Separa-onofscale:ξ<<Ω

2.Iden-fyrelevantsymmetriesandsymmetrybreakingQ:Whatarethesymmetries,andaretheyspontaneouslybroken?

Spontaneousbreakingofrota-onalsymmetry

Rota-onalsymmetry AxialsymmetrySO(3) SO(2)

3generators 1generator

Therewillbe3‐1=2Nambu‐Goldstonebosons

P.vanIsacker2008

“In a general theory of rota<on, symmetry plays a central role. Indeed, the very occurrence of collec<ve rota<onal degrees of freedom may be said to originate in a breaking of rota<onal invariance, which introduces a “deforma<on” that makes it possible to specify an orienta<on of the system. Rota<on represents the collec<ve mode associated with such a spontaneous symmetry breaking (Goldstone boson).”AageBohr,NobelLecture(1975)

3.ConstructthemostgeneralHamiltonianconsistentwiththesymmetryandthesymmetrybreaking

Assumegroundstateisinvariantonlyunderrota-onsaroundthez‐axis.Rota-onsofthatstatecannotbedis-nguishediftheydifferonlybyarota-onaroundthez‐axis.Nambu‐GoldstonemodesparameterizethecosetspaceSO(3)/SO(2)~S2,i.e.thetwo‐sphere:

NonlinearrealizaQon:Theunitvectorn(α,β)transformslinearlyunderrota-ons,buttheangles(ourNambu‐Goldstonedegreesoffreedom)transformnonlinearly.

Followingthisprocedureisabittechnical(wemightdothislater).Let’sfirstfollowasimplerandgeometricapproach,seee.g.,[H.Leutwyler,Phys.Rev.D49(1994)3033,arXiv:hep‐ph/9311264]

NonlinearrealizaQonof(rotaQonal)symmetry[Weinberg1967;Coleman,Callan,Wess&Zumino1969]

PhysicsofNambu‐Goldstonemodes

RotaQonalbandsarequanQzedNambu‐Goldstonemodes.Low‐energyconstantC0ismomentofiner-aandfittodata.

Lagrangian

Hamiltonian

Quan-za-on

Spectrum

Next‐to‐leadingorderterm(scalarinNGmodesthatwecanwritedown)

L=(C2/4)()2Q1:Whatareit’sdimensionsofC2inpowersofenergy?

Letusfirstunderstanddimensionalanalysis

1.  Low‐energyscaleisξ

2.  Leading‐orderLagrangianmustscaleas:L~ξ

3.  Energy‐-meuncertaintyimplies(ħ=1):dt~ξ

4.  ThusC0~1/ξ

Next‐to‐leadingorderterm(scalarinNGmodesthatwecanwritedown)

L=(C2/4)()2~ξ(ξ/Ω)2<<ξ

A1:Itmusthavedimensionsofenergy‐3(Wehavetwoenergyscalesξ<<Ω)

A2:C2/C0~energy‐2andisduetoomiTedphysicsatahigh‐energyscaleΩThus:C2/C0~Ω‐2(“naturalness”argument)

Q2:HowshouldthetermC2scaleprecisely?A2:ξ‐3,ξ‐2Ω‐1,ξ‐1Ω‐2,Ω‐3…?

4.Powercoun-ngandnext‐to‐leadingorder

4.Powercoun-ngandnext‐to‐leadingorder

Spectrum:J(J+1)/(2C0)–(J(J+1))2(C2/4C04)

 Bohr&MoTelson(ofcourse!)

Lagrangianatnext‐to‐leading

L=(C0/2)+(C2/4)()2

Q:Whendoestheeffec-vetheorybreaksdown?HowlargecanJbe?

A1:energycorrec-on<<leading‐orderenergy:C2/C03J(J+1)<<1A2:Thus:J<<Ω/ξ

172Yb:Rela-veerrorinLOandNLO

Smallparameter:ξ/Ω≈79/1049≈1/13

Nucleiwithfiniteground‐statespins:WessZuminoterms

Afiniteground‐statespinbreaks-mereversalinvariance. Considertermsthatarefirstorderinthe-mederiva-ve Nosuchtermsareinvariantunderrota-ons.BUT:underrota-ons,Ezchangesbyatotalderiva-ve.Ac-onremainsessen-allyinvariant(Wess‐Zuminoterm)

Lagrangian

Hamiltonian

Eigenvaluesandeigenfunc-ons(Iden-fyqwithground‐statespin!)

(WignerDfunc-ons)

173Yb:Rela-veerrorinLOandNLO

Smallparameter:ξ/Ω≈79/350≈1/4.5

BeyondNGmodes:couplingtovibra-ons

Higherenerge-cdegreesoffreedomneedtobeincluded.

Quadrupolefieldexhibitsspontaneoussymmetrybreaking.

 5DoF–2NG=3DoF

ξ

Ω

Separa-onofscale:ξ<<Ω

Couplingstovibra-ons:powercoun-ng

LowenergyscaleξHighenergyscaleΩ>>ξ

Dimensionalanalysis

Poten-alexpandedaroundminiumum

Powercoun-ng:largeamplitudesφ0≈vrestorerota-onalsymmetrybreakdownofEFT

Leadingorder~O(Ω)Lagrangianatleadingorder:

Spectrum

Leadingorderyieldsthebandheads

Lagrangianasfunc-onofEx,Ey,Dtφ0,Dtϕ2,φ0,ϕ2,needstobeformallyinvariantunderSO(2)(axialsymmetryonly).

Next‐to‐leadingorder~O(ξ)

Lagrangian

Hamiltonian(kine-cenergy)

Spectrum:arota-onalbandoneveryvibra-onalbandhead

Correc-ons~ξofbandheadsduetoanharmonici-esinthepoten-alneglected.

Innext‐to‐leadingorder,theresultsoftherota-onal‐vibra-onalmodelarereproduced.

66

SummaryConstruc-onofanEFTfor(certain)heavynuclei

1.  Iden-fytherelevantdegreesoffreedomfortheresolu-onscaleofinterest:Quadrupolephonons

2.  Iden-fytherelevantsymmetriesoflow‐energynuclearphysicsandinves-gateifandhowtheyarebroken:SpontaneouslybrokenrotaQonalsymmetry

3.  ConstructthemostgeneralLagrangianconsistentwiththosesymmetriesandthesymmetrybreaking.NonlinearrealizaQonofrotaQonalsymmetry

4.  Designanorganiza-onalscheme(powercoun-ng)thatcandis-nguishbetweenmoreandlessimportantcontribu-ons:SeparaQonofscalebetweenrotaQonalandvibraQonalmodes

Results:[TP,Nucl.Phys.852,36(2011),arXiv:1011.5026]

1.  Innext‐to‐leadingorder,theresultsoftherota-onal‐vibra-onalmodelarereproduced.2.  Nucleiwithfiniteground‐statespinstreatedonequalfoo-ngWess‐Zuminoterms

Outlook

Enthusias-candlivelyfield

Intellectuallys-mula-ngproblems

Movingtowardsaunifieddescrip-onofallatomicnuclei

Plentyofopportuni-esandchallenges

Yourideasandingenuitywillshapethefuture!