Probingproton-neutronpairingwithGamow-Tellerstrengthsintwo-

nucleonconfigura8ons

YutakaUtsunoAdvancedScienceResearchCenter,JapanAtomicEnergyAgency

CenterforNuclearStudy,UniversityofTokyo

Y.UtsunoandY.Fujita,arXiv1701.00869

Introduc8on:Fujita-san’sseminarinJan.2016•  Systema8csofGamow-Tellerdistribu8onsforN=Z+2→N=Znuclei

–  Usingchargeexchangereac8on(3He,t)

ObservedGTstrength•  Concentra8oninthe1+1levelfor

42Ca→42Sc

–  B(GT;0+1→1+1)=2.2

Y.Fujitaetal.,Phys.Rev.Le`.112,112502(2014).

“low-energysuperGTstate”

Ikedasumrule•  WhenB(GT)isdefinedas𝐵(GT↑± ;𝑖→𝑓)= |𝛼↓𝑓 𝐽↓𝑓  |𝜎𝑡↑± | 𝛼↓𝑖 𝐽↓𝑖  |↑2 /2𝐽↓𝑖 +1 =∑𝜇𝑀↓𝑓 ↑▒|𝛼↓𝑓 𝐽↓𝑓 𝑀↓𝑓 𝜎↓𝜇 𝑡↑±  𝛼↓𝑖 𝐽↓𝑖 𝑀↓𝑖  |↑2  ,∑𝑓↑▒𝐵(GT↑− ;𝑖→𝑓)− ∑𝑓↑▒𝐵(GT↑+ ;𝑖→𝑓)=3(𝑁−𝑍) issa8sfied.(with 𝑡↑− | 𝑛⟩=| 𝑝⟩)

•  IfGT+transi8onishinderedbyPauliblocking,∑𝑓↑▒𝐵(GT↑− ;𝑖→𝑓) iscloseto3(𝑁−𝑍).

–  ∑𝑓↑▒𝐵(GT↑− ;𝑖→𝑓) for42Cashouldbecloseto6,ifprotonexcita8ontothepfshellissmall.

proton

neutron

SU(4)?single-par8cletransi8on?•  Thesitua8onissimilartowhatisexpectedfromtheSU(4)

symmetry(nospinorisospindependentforces).

•  SU(4)isstronglybrokenbytheexistenceofspin-orbitsplifng.

•  Single-par8clestructure?

f7/2→f7/2f7/2→f5/2

ε(f5/2)-ε(f7/2)

B(GT)distribu8onwithpureconfigura8ons

•  Concentra8oninasinglestatecannotbeaccountedforbyasimpleshell-modelargument.

Fromthesimplesingle-par8clepicture

Experimentalstrength

0 10

Essen8alroleofconfigura8onmixing

Y.Fujitaetal.,Phys.Rev.C91,064316(2015)

•  RPAcalcula8on–  Isoscalarpairingisessen8al.

•  CoherenceintheGTtransi8onisalsoobtainedbyshell-modelcalc.

•  Whyiscollec8vitycreated?

C.L.Baietal.,Phys.Rev.C90,054335(2014).Shell-modelanalysis

Two-par8clevs.two-holeconfigura8ons•  Aques8onbysomeone(sorry,Idonotrememberwhoraised)

– Whatabouttwoholesystems?

twohole：14C→14NlogA=9.1

p3/2

p1/2

two-par8cle：6He→6Li

logA=2.9

protons neutrons

B(GT) Two-par0cle Two-holep sd pf p sd

A=6 A=18 A=42 A=14 A=381+1 4.7 3.1 2.2 3.5×10-6 0.0601+2 0.13 0.10 2.8 1.5

Distribu8onofknownlogA

6He→6Li 14C→14N

Num

bero

fcases

Ques8ons•  Whydoesthe“low-energysuperGTstate”appearfortwo-par8cle

configura8ons?

•  WhyisthecorrespondingB(GT)valuesfortwo-holeconfigura8onsverysmall?

•  Whatdoesthoseproper8estellusaboutisovectorandisoscalarpairingproper8es,sincecorrelatedtwonucleonsforma“Cooperpair”?

Unifiedtreatmentofp-pandh-hsystems•  Thefollowingtwodescrip8onsareiden8cal

–  Par8cleHamiltonian𝐻=∑𝑖↑▒𝜀↓𝑖 𝑎↓𝑖 ↓↑† 𝑎↓𝑖  + 1/4 ∑𝑖,𝑗,𝑘,𝑙↑▒𝑣↓𝑖𝑗;𝑘𝑙  𝑎↓𝑖 ↓↑† 𝑎↓𝑗 ↓↑† 𝑎↓𝑙 𝑎↓𝑘 

–  HoleHamiltonian𝐻=∑𝑖↑▒𝜀 ↓𝑖 𝑏↓𝑖 ↓↑† 𝑏↓𝑖  + 1/4 ∑𝑖,𝑗,𝑘,𝑙↑▒𝑣 ↓𝑖𝑗;𝑘𝑙  𝑏↓𝑖 ↓↑† 𝑏↓𝑗 ↓↑† 𝑏↓𝑙 𝑏↓𝑘 

when𝑣↓𝑖𝑗;𝑘𝑙 = 𝑣 ↓𝑖𝑗;𝑘𝑙 and 𝜀 ↓𝑖 =𝐸( 𝑖↑−1 )aresa8sfied.

Two-bodymatrixelementsarethesame,butthesingle-par8cleenergiesareinthereversedorderforthehole-holeHamiltonian.

p3/2

p1/2

GTstrengthin2pand2hconf.:p-shellcase•  Asshowninthelastslide,boththe6He→6Li(2p)and14C→14N

(2h)decayscanbedescribedastwo-nucleonsystemswiththesametwo-bodymatrixelements.

•  Theonlydifferencebetweenthemisthesingle-par8clesplifngΔ𝜀↓𝑝 =𝜀( 𝑝↓1/2 ) −𝜀( 𝑝↓3/2 ):–  Forthepar8clecase:Δ𝜀↓𝑝 =0.1 MeV

–  Fortheholecase:Δ𝜀↓𝑝 =−6.3 MeV

•  Itisinteres8ngtoplottheGamow-Tellermatrixelement𝑀(𝐺𝑇)= 𝐽↓𝑓 |𝜎𝑡↑− | 𝐽↓𝑖  asafunc8onofΔ𝜀↓𝑝 forgiventwo-bodyinterac8onsinordertoseethedependenceonthesingle-par8cleenergies.

TakenfromtheCohen-Kurath’sCKIIinterac8on

(1)Cohen-Kurath’sCKIIinterac8on•  Oneofthemostpopularempiricalinterac8onsforthepshell.

–  15two-bodymatrixelementsarededucedbyfifngenergylevelsofA=8-16nuclei.

(2)Pairinginterac8on•  Isoscalar-andisovector-pairinginterac8onsaredefinedas𝑉↑IVpair = 𝐺↑IV ∑𝜇↑▒𝑃↓𝜇↑†  𝑃↓𝜇 𝑉↑ISpair = 𝐺↑IS ∑𝜇↑▒𝐷↓𝜇↑†  𝐷↓𝜇 where𝑃↓𝜇↑† =√1/2 ∑𝑛𝑙↑▒(−1)↑𝑙 √2𝑙+1 [𝑎↓𝑛𝑙↑† × 𝑎↓𝑛𝑙↑† ] ↓𝑀↓𝐿 =0,𝑀↓𝑆 =0,𝑀↓𝑇 =𝜇↑𝐿=0,𝑆=0,𝑇=1 and �𝐷↓𝜇↑† =√1/2 ∑𝑛𝑙↑▒(−1)↑𝑙 √2𝑙+1 [𝑎↓𝑛𝑙↑† × 𝑎↓𝑛𝑙↑† ] ↓𝑀↓𝐿 =0,𝑀↓𝑆 =𝜇, 𝑀↓𝑇 =0↑𝐿=0,𝑆=1,𝑇=0 .

•  Isovector-paircrea8onoperator 𝑃↓𝜇↑† andisoscalar-paircrea8onoperators 𝐷↓𝜇↑† aresymmetricintermsofSandT.

M(GT)withthepairinginterac8on

•  M(GT):EnhancedforsmallΔ𝜀↓𝑝 andnovanishingforΔ𝜀↓𝑝 <0.–  Largerthansingle-par8clelimitsonthebothsides

•  StrengthsofGISandGIVaredeterminedsoastoreproducethemeansquare(J,T)=(1,0)and(0,1)matrixelementsofCKII.

p3/2limit:√10/3 ≈1.83

p1/2limit:√2/3 ≈0.82

CoherenceoftheGTmatrixelement•  Whentwo-nucleonwavefunc8onsaredecomposedintobasis

statesas| 0↓𝑘↑+ ⟩=∑𝑎𝑏↑▒𝛼↓𝑎𝑏↑IV (𝑘) | 𝑎𝑏 𝐽=0 𝑇=1⟩and| 1↓𝑘↑+ ⟩=∑𝑎𝑏↑▒𝛼↓𝑎𝑏↑I𝑆 (𝑘) | 𝑎𝑏 𝐽=1 𝑇=0⟩,theGamow-Tellermatrixelementiswri`enas𝑀(𝐺𝑇;1↓𝑘↑+ )=∑𝑎𝑏𝑐𝑑↑▒𝑚↓𝑎𝑏𝑐𝑑 ( 1↓𝑘↑+ ) where𝑚↓𝑎𝑏𝑐𝑑 (1↓𝑘↑+ )= 𝛼↓𝑎𝑏↑I𝑆 ↑∗ (𝑘)𝛼↓𝑐𝑑↑IV (1)𝑎𝑏 𝐽=1 𝑇=0|𝜎𝑡↑− | 𝑐𝑑 𝐽=0 𝑇=1 .

•  Signsof𝑚↓𝑎𝑏𝑐𝑑 (1↓𝑘↑+ )–  Construc8veordestruc8veinterference—essen8alfordetermininglargeor

smallB(GT)

Signsof𝑚↓𝑎𝑏𝑐𝑑 (1↓𝑘↑+ ):CKII

•  Destruc8veinterferenceforΔ𝜀↓𝑝 =−5 MeV.

𝑚↓𝑎𝑏𝑐𝑑 (1↓𝑘↑+ )= 𝛼↓𝑎𝑏↑I𝑆 ↑∗ (𝑘)𝛼↓𝑐𝑑↑IV (1)𝑎𝑏 𝐽=1 𝑇=0|𝜎𝑡↑− | 𝑐𝑑 𝐽=0 𝑇=1 .

Δ𝜀↓𝑝 =5 MeV Δ𝜀↓𝑝 =−5 MeV

Signsof𝑚↓𝑎𝑏𝑐𝑑 (1↓𝑘↑+ ):pairing

•  Itlooksthatthepairinginterac8onalwaysgivesaconstruc8veinterference,butrealis8cinterac8onsdonot.

𝑚↓𝑎𝑏𝑐𝑑 (1↓𝑘↑+ )= 𝛼↓𝑎𝑏↑I𝑆 ↑∗ (𝑘)𝛼↓𝑐𝑑↑IV (1)𝑎𝑏 𝐽=1 𝑇=0|𝜎𝑡↑− | 𝑐𝑑 𝐽=0 𝑇=1 .

Δ𝜀↓𝑝 =5 MeV Δ𝜀↓𝑝 =−5 MeV

Theoremforthesignsof𝑚↓𝑎𝑏𝑐𝑑 (1↓𝑘↑+ )

Whenthe(J,T)=(0,1)and(1,0)two-bodymatrixelementsaregivenbytheisovector-andisoscalar-pairinginterac8onswith

nega8veGIVandGIS,respec8vely,allthesignsof𝑚↓𝑎𝑏𝑐𝑑 (1↓𝑘↑+ )intwo-nucleonsystemsarethesameforanyvalenceshell

(includingmul8-jshell)andforanysingle-par8clesplifng.

𝐻↑pair =∑𝑖↑▒𝜀↓𝑖 𝑎↓𝑖 ↓↑† 𝑎↓𝑖  + 𝑉↑pair 

𝑚↓𝑎𝑏𝑐𝑑 (1↓𝑘↑+ )= 𝛼↓𝑎𝑏↑I𝑆 ↑∗ (𝑘)𝛼↓𝑐𝑑↑IV (1)𝑎𝑏 𝐽=1 𝑇=0|𝜎𝑡↑− | 𝑐𝑑 𝐽=0 𝑇=1 ≥0

𝑉↑pair =𝐺↑IV ∑𝜇↑▒𝑃↓𝜇↑†  𝑃↓𝜇 + 𝐺↑IS ∑𝜇↑▒𝐷↓𝜇↑†  𝐷↓𝜇 

Proofofthetheorem(1)•  Writedownj-jcoupledtwo-bodymatrixelements:𝑎𝑏 𝐽=0 𝑇=1𝑉↑pair  𝑐𝑑 𝐽=0 𝑇=1 = 𝐺↑IV 𝜒↓𝑎𝑏↑IV 𝜒↓𝑐𝑑↑IV 𝑎𝑏 𝐽=1 𝑇=0𝑉↑pair  𝑐𝑑 𝐽=1 𝑇=0 = 𝐺↑I𝑆 𝜒↓𝑎𝑏↑IS 𝜒↓𝑐𝑑↑IS with𝜒↓𝑎𝑏↑IV = (−1)↑𝑙↓𝑎  √𝑗↓𝑎 +1/2 𝛿↓𝑎𝑏 𝜒↓𝑎𝑏↑IS =√2/1+ 𝛿↓𝑎𝑏   (−1)↑𝑗↓𝑎 −1/2 √(2𝑗↓𝑎 +1)(2𝑗↓𝑏 +1) {█1/2&𝑗↓𝑎 &𝑙↓𝑎 @𝑗↓𝑏 &1/2&1 }𝛿↓𝑛↓𝑎 𝑛↓𝑏  𝛿↓𝑙↓𝑎 𝑙↓𝑏  

•  Onecaneasilyshowthatthesignof𝜒↓𝑎𝑏↑IV is(−1)↑𝑙↓𝑎  andthatthatof𝜒↓𝑎𝑏↑IS is(−1)↑𝑗↓𝑏 −1/2 byusingtheexactformof{█1/2&𝑗↓𝑎 &𝑙↓𝑎 @𝑗↓𝑏 &1/2&1 }.

•  Whentheconven8onsof| 𝑎𝑏 𝐽=0 𝑇=1⟩ = (−1)↑𝑙↓𝑎  | 𝑎𝑏 𝐽=0 𝑇=1⟩and | 𝑎𝑏 𝐽=1 𝑇=0⟩ = (−1)↑𝑗↓𝑏 −1/2 | 𝑎𝑏 𝐽=1 𝑇=0⟩aretaken,allthetwo-bodymatrixelementsarenonposi8ve.

Proofofthetheorem(2)•  MatrixelementoftheHamiltonian𝐻↓𝑖𝑗↑pair = 𝛿↓𝑖𝑗 ( 𝜀↓𝑎(𝑖) + 𝜀↓𝑏(𝑖) )+ 𝑉↓𝑖𝑗↑pair 

•  Alltheoff-diagonalmatrixelementsarenonposi8vewhenthepresentphaseconven8on.

•  FromaversionofthePerron-Frobeniustheorem,thecomponentsofthelowesteigenvectorarecompletelyofthesamesign.

Foramatrix[█ℎ↓11 &⋯&ℎ↓1𝑛 @⋮&⋱&⋮@ℎ↓𝑛1 &⋯&ℎ↓𝑛𝑛  ]with ℎ↓𝑖𝑗 ≤0(𝑖≠𝑗),thelowesteigenvector𝑣↓1  =[█𝛼↓1 @⋮@𝛼↓𝑛  ]sa8sfies 𝛼↓𝑖 ≥0forany𝑖.

Proofofthetheorem(3)•  Gamow-Tellermatrixelementsfortwo-nucleonconfigura8ons

Condon-Shortleyconven8on

presentconven8on

Completelythesamesign!

Proofofthetheorem(4)•  Wegobackto𝑀(𝐺𝑇;1↓𝑘↑+ )=∑𝑎𝑏𝑐𝑑↑▒𝑚↓𝑎𝑏𝑐𝑑 ( 1↓𝑘↑+ ) with 𝑚↓𝑎𝑏𝑐𝑑 (1↓𝑘↑+ )= 𝛼↓𝑎𝑏↑I𝑆 ↑∗ (𝑘)𝛼↓𝑐𝑑↑IV (1)𝑎𝑏 𝐽=1 𝑇=0|𝜎𝑡↑− | 𝑐𝑑 𝐽=0 𝑇=1 .

•  Allof𝛼↓𝑎𝑏↑I𝑆 ↑∗ (1), 𝛼↓𝑐𝑑↑IV (1),and 𝑎𝑏 𝐽=1 𝑇=0|𝜎𝑡↑− | 𝑐𝑑 𝐽=0 𝑇=1 havefixedsignswhenthephaseconven8onsof| 𝑎𝑏 𝐽=0 𝑇=1⟩ = (−1)↑𝑙↓𝑎  | 𝑎𝑏 𝐽=0 𝑇=1⟩and | 𝑎𝑏 𝐽=1 𝑇=0⟩ = (−1)↑𝑗↓𝑏 −1/2 | 𝑎𝑏 𝐽=1 𝑇=0⟩aretaken.Therefore,thesignsof𝑚↓𝑎𝑏𝑐𝑑 (1↓1↑+ )arethesameforallpossible(𝑎,𝑏,𝑐,𝑑).

•  Thisistheoriginofthecoherence(“low-energysuperGTstate”)obtainedfortwo-par8cleconfigura8ons.

(rough)ProofofthePerron-Frobeniustheorem•  Considerareal-symmetricmatrixwith𝐻=( ℎ↓𝑖𝑗 )with∀ ℎ↓𝑖𝑗 ≤0

(𝑖≠𝑗)andlet𝑣↓1  =(𝛼↓1 , …, 𝛼↓𝑛 )bethelowesteigenvector.•  Since 𝑣↓1  isthelowesteigenvector,thereisnovectorthat

sa8sfies 𝑣𝐻�𝑣 < 𝑣↓1  𝐻� 𝑣↓1  .•  If 𝑣↓1  containsposi8veandnega8vecomponents,onecan

generallyassume 𝛼↓1 ≥0,…,𝛼↓𝑘 ≥0, 𝛼↓𝑘+1 <0,…,𝛼↓𝑛 <0.Let 𝑣↓1 ′  be(𝛼↓1 , … 𝛼↓𝑘 , − 𝛼↓𝑘+1 , −𝛼↓𝑛 ).

•  Since 𝑣↓1  𝐻� 𝑣↓1  =∑𝑖𝑗↑▒𝛼↓𝑖  ℎ↓𝑖𝑗 𝛼↓𝑗 ,onegets𝑣↓1  𝐻� 𝑣↓1  − 𝑣↓1 ′ 𝐻� 𝑣↓1 ′ =∑𝑖≤𝑘, 𝑗>𝑘↑▒2 𝛼↓𝑖 ℎ↓𝑖𝑗 𝛼↓𝑗 >0.Thiscontradictsthat𝑣↓1  isthelowesteigenvector.

GraphicalimageoftheP-Ftheorem•  Representanoff-diagonalmatrixelementℎ↓𝑖𝑗 

asabondthatconnectsthesite𝑖and𝑗.•  Foreachsite𝑖,aposi8ve(nega8ve)𝛼↓𝑖 is

representedasanupward(downward)arrow.

•  Therightfigureshowsfavoredsigns,whichareanalogoustoferromagne8smandan8ferromagne8sm.

•  Thesitua8onofP-Ftheoremislikeparallelspinalignmentinferromagne8smthatmakesenergystable.

⊝𝑖 𝑗

⊕𝑖 𝑗

⊝

⊝

⊝

⊝⊝

⊝

Goingbacktophysics•  ThecoherenceofGTmatrixelementsisduetothe“alignment”of

allthetwo-nucleonconfigura8ons,wherealignmentstandsforthatcoefficientsofbasisvectorsareofthesamesign.

•  Sincethisisindependentofsingle-par8cleenergies,suchan“aligned”statemustbemoststablealsofortwo-holeconfigura8ons.

•  Theactualsitua8onisdifferent.Thismeansthatsomeoftheoff-diagonalmatrixelementsinthe(J,T)=(1,0)and/or(0,1)channelsareopposite.

Realis8c(J,T)=(0,1)and(1,0)matrixelements•  Ques8ons

–  Isovectororisoscalar?–  Whichmatrixelementsaredifferent?

–  Anyruleaboutthedifferentsigns?

•  Examiningoff-diagonalmatrixelementsinrealis8cinterac8ons–  Empiricalandmicroscopic(Gmatrix)

–  Phaseconven8onsof| 𝑎𝑏 𝐽=0 𝑇=1⟩ = (−1)↑𝑙↓𝑎  | 𝑎𝑏 𝐽=0 𝑇=1⟩and | 𝑎𝑏 𝐽=1 𝑇=0⟩ = (−1)↑𝑗↓𝑏 −1/2 | 𝑎𝑏 𝐽=1 𝑇=0⟩

CKII Kuop

-1.56 -1.75

-3.55 -5.06

+1.70 +2.31

CKII Kuop

-4.86 -3.96

(J,T)=(0,1)off-diagonal (J,T)=(1,0)off-diagonal

sdshell

USD Kuosd

-0.72 -1.62

-2.54 -3.17

+0.57 +0.04

+1.10 +0.24

-1.18 -0.60

-1.71 -1.91

-2.10 -1.71

+0.40 +0.80

-0.03 +0.21

-1.25 -0.31

USD Kuosd

-3.19 -3.79

-1.32 -0.97

-1.08 -0.74

(J,T)=(0,1)off-diagonal (J,T)=(1,0)off-diagonal

Someisoscalarmatrixelementshavetheoppositesign.

Pairingvs.realis8cinterac8ons:p-shellcase

•  Samesignsforisovectormatrixelements.

•  Oppositesignfor 𝑝↓3/2 𝑝↓1/2  𝑉� 𝑝↓1/2 𝑝↓1/2  inthe(J,T)=(1,0)channel.Thiscauses“frustra8on”,whichcannotuniquelydeterminethesigns.Theactualsignsdependonthediagonalmatrixelements.–  Two-par8cleconfig.:coherentduetothedominanceof𝑣↓1 and 𝑣↓2 –  Two-holeconfig.:noncoherentduetodominanceof𝑣↓2 and 𝑣↓3 .

Originofdifferenceinsign•  Weconsiderthematrixelementsofthedeltainterac8onasashort-

rangecentralinterac8on.

•  Thej-jcoupledmatrixelementsarewri`enas𝑎𝑏𝐽𝑇 𝛿(𝑟) 𝑐𝑑𝐽𝑇 = 𝐶↓0 ∑𝐿𝑆 (𝐿+𝑆+𝑇=odd)↑▒𝜂↓𝑎𝑏 (𝐿𝑆𝐽)𝜂↓𝑐𝑑 (𝐿𝑆𝐽) with 𝜂↓𝑎𝑏 (𝐿𝑆𝐽)=√1− (−1)↑𝑆+𝑇 /1+ 𝛿↓𝑎𝑏   𝛾↓𝐿𝑆↑(𝐽) (𝑎,𝑏)(−1)↑𝑛↓𝑎 + 𝑛↓𝑏  √(2𝑙↓𝑎 +1)(2𝑙↓𝑏 +1) (█𝐿&𝑙↓𝑎 &𝑙↓𝑏 @0&0&0 )and 𝛾↓𝐿𝑆↑(𝐽) (𝑎,𝑏)=√(2𝑗↓𝑎 +1)(2𝑗↓𝑏 +1 )(2𝐿+1)(2𝑆+1) {█𝑙↓𝑎 &1/2&𝑗↓𝑎 @𝑙↓𝑏 &1/2&𝑗↓𝑏 @𝐿&𝑆&𝐽 }.

•  𝐶↓0 dependson(𝑛↓𝑎 , 𝑙↓𝑎 )etc.,butweassumeaconstant𝐶↓0 whichiscalledthesurfacedeltainterac8on.

•  Inthiscase,the𝐿=0contribu8onsareexactlythesameasthepairingmatrixelements.

L=0and2contribu8ons•  𝑎𝑏𝐽𝑇 𝑉↑SDI  𝑐𝑑𝐽𝑇 =𝐶↓0 ∑𝐿𝑆 (𝐿+𝑆+𝑇=odd)↑▒𝜂↓𝑎𝑏 (𝐿𝑆𝐽)𝜂↓𝑐𝑑 

(𝐿𝑆𝐽) •  Restric8ons

–  𝐽 = 𝐿 + 𝑆 ;𝑆=0or1.

–  𝐿isonlyevenwhenoneconsidersasingle-majorshell,since𝜂↓𝑎𝑏 (𝐿𝑆𝐽)∝(█𝐿&𝑙↓𝑎 &𝑙↓𝑏 @0&0&0 ).

•  Possible𝐿𝑆–  Only𝐿𝑆=00for(𝐽,𝑇)=(0,1)

–  𝐿𝑆=01and21for(𝐽,𝑇)=(1,0)

Onemusttakethe𝐿𝑆=21termintoaccountwhenfullyevalua8ngthematrixelementsofshort-rangecentralforces.

Cancella8onduetoL=2•  𝑎𝑏𝐽𝑇 𝑉↑SDI  𝑐𝑑𝐽𝑇 =𝐶↓0 ∑𝐿𝑆 (𝐿+𝑆+𝑇=odd)↑▒𝜂↓𝑎𝑏 (𝐿𝑆𝐽)𝜂↓𝑐𝑑 

(𝐿𝑆𝐽) •  Exactexpressionof 𝜂↓𝑎𝑏 (𝐿 𝑆=1 𝐽=1)dividedby𝑠= (−1)↑𝑗↓𝑏 

−1/2 

•  Clearly,cancella8onduetoL=2occursfor 𝑗↓> 𝑗↓>  𝑉� 𝑗↓> 𝑗↓<  and 𝑗↓< 𝑗↓<  𝑉� 𝑗↓> 𝑗↓<  .

𝑙↑′ =𝑙−2

Quan8ta8veargument•  𝑎𝑏 𝐽=1 𝑇=0𝑉↑SDI  𝑐𝑑 𝐽=1 𝑇=0 /𝑙withtheconven8onof(−1)↑𝑗↓𝑏 −1/2 | 𝑎𝑏 𝐽=1 𝑇=0⟩↓Condon−Shortley 

•  𝑗↓< 𝑗↓<  𝑉� 𝑗↓> 𝑗↓<  withlow-𝑙arestronglycancelledbyL=2.•  Finite-rangeinterac8onscanmake𝑗↓< 𝑗↓<  𝑉� 𝑗↓> 𝑗↓<  posi8ve.

Frustra8oncausedbyΔl=2mixing•  Matrixelementsconcerning| 𝑗↓<  𝑗↓>↑′ ⟩with 𝑙↑′ =𝑙−2(suchas| 𝑑↓3/2 𝑠↓1/2 ⟩)doesnotappearwiththeL=0terms,andcanbeanaddi8onalsourceoffrustra8on.

•  Thesignof ℎ↓12 ℎ↓23 ℎ↓31 isinvariantunderphasetransforma8on.Whenitisposi8ve,thefrustra8onoccursamong 𝑣↓1 , 𝑣↓2 and 𝑣↓3 .𝜂↓𝑎𝑏 (𝐿 𝑆=1 𝐽=1)/ (−1)↑𝑗↓𝑏 −1/2 

⊝⊕

⊝

( 𝑗↓<  𝑗↓>↑′ )

( 𝑗↓<  𝑗↓>↑ )( 𝑗↓>  𝑗↓>↑ )

Tensor-forcecontribu8ons

Matrixelement(MeV)

diagonal +1.40

+0.59

-0.78

off-diagonal +1.16

-0.22

+0.43

Matrixelement(MeV)

diagonal +1.05

+0.51

-0.23

off-diagonal +0.76

-0.25

+0.21

•  PointedoutbyTalmiintermsoftheoriginoflonglife8meof14C.

•  Evalua8onwiththeπ+ρmesonexchangetensorforceusingthe(−1)↑𝑗↓𝑏 −1/2 | 𝑎𝑏 𝐽=1 𝑇=0⟩↓Condon−Shortley conven8on

pshell sdshell

Pairtransfermatrixelements•  Itiso{endiscussedthatpairtransferprobabilityisagoodmeasure

ofpairingcorrela8on.

•  Isoscalarpaircrea8onandremovalprobabili8es:|𝐽↓𝑓 𝑇↓𝑓  ||𝐷↑† || 𝐽↓𝑖 𝑇↓𝑖  |↑2 and |𝐽↓𝑓 𝑇↓𝑓  ||𝐷|| 𝐽↓𝑖 𝑇↓𝑖  |↑2 

Paircrea0ononvacuum

Pairremovalfromclosure

p 8.1 5.3×10-3

sd 15.7 1.7

Possibleeffectsonpaircondensate•  OnthebasisofstrongT=0a`rac8veforce,isoscalarpair

condensatesareexpectedtooccurespeciallyaroundN=Znuclei,similartowell-knownisovectorpaircondensates.–  | 𝛹⟩= (𝛬↑† )↑𝑁 | −⟩withtheCooperpair𝛬↑† =∑𝑎𝑏↑▒𝜆↓𝑎𝑏 [𝑎↓𝑖↑† 

×𝑎↓𝑗↑† ]↑𝐽=1 𝑇=0  

•  Thereseemstobenostrongevidenceforisoscalarpaircondensatesintheactualnuclei.

•  Asshowninthistalk,thesignsoftheisoscalarpairarenotuniquelydeterminedbecauseoffrustra8on.Thisisapossiblereasonwhyisoscalarpaircondensatesarenotwellestablished.

Summary•  WehaveexactlyproventhattheGTmatrixelementsintwo-nucleon

configura8ons(2pand2h)arealwaysinphasewiththeisovetor-andisoscalar-pairinginterac8ons,whichaccountsfor“low-energysuperGTstates”.

•  TheobservedhindranceoftheGTmatrixelementsintwo-holeconfigura8onsisdueto“noncoherence”ofrealis8c(J,T)=(1,0)matrixelements,whichcannotgivedefinitesignsin”Cooperpairs”.

•  Differenceinsignbetweenisoscalar-pairingandrealis8cinterac8onsispredominantlycausedbyL=2centralforcesandtensorforces.

•  Thiseffectcanpreventcorrelatedproton-neutronpairfromformingisoescalar-paircondensates,whicharenotestablishedinexperiement.