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Spin-tensor decomposition of the effectiveinteraction for the shell model

Naofumi Tsunoda

Department of Physics, The University of Tokyo

EFES-NSCL meeting“Perspectives on the modern shell model and related experimental

topics”, 4-6 Feb. 2010, Michigan State University

CollaboratorsTakaharu Otsuka(Tokyo), Koshiroh Tsukiyama(Tokyo)

Morten Hjourth-Jensen(Oslo), Toshio Suzuki(Nihon Univ.)Yutaka Utsuno(JAEA), Michio Homma(Aizu Univ.)

Introduction Effective interaction Tensor force Three body force Summary 1/ 32

Outline

1 Introduction

2 Review on theories of effective interaction

3 Tensor force in effective interaction

4 Three body force contribution to effective interaction

5 Summary


Tensor force and shell structure

Tensor force has opposite sign between j>and j

Realistic nuclear force

Argonne potentials: written in operator formFor example, Argonne V14, with operators Op

V14(r) =14

∑

p=1

vp(r)Op, vp(r) = vpπ (r) + v

pI (r) + v

pS (r)

Long range vπ(r): One Pion Exchange Potential (OPEP)

Intermediate range vI (r): phenomenological fit

Short range vS(r): phenomenological fit

Operators included in Argonne v18, v14 and v8’ potentialCIB CSB operators

v14 No No {1, (σ1 · σ2), S12,L · S,L2,L2σ1 · σ2,L · S}⊗{1, (τ1 · τ2)}

v18 Yes Yes

{1, (σ1 · σ2), S12,L · S,L2,L2σ1 · σ2,L · S}⊗{1, (τ1 · τ2)}

,T12, (σ1 · σ2)T12, S12T12, (τz1 + τz2)v8’ No No {1, (σ1 · σ2), S12, L · S}

⊗{1, (τ1 · τ2)}v8’ potential is a reduction of v18 potential.


Central: VC (r) = V1(r)+ (σ1 ·σ2)V2(r)+ (τ1 · τ2)V3(r)+ (σ1 ·σ2)(τ1 · τ2)V4(r)

Tensor: VT (r) = V1(r) + (τ1 · τ2)V2(r)

Central force: strong short range repulsion

Tensor froce: long range nature


Review on theories of effective interaction


Theory of effective interaction

Original eigenvalue problem: H|Ψi 〉 = Ei |Ψi 〉Effective interaction in P-space is defined as

PH̃P|φi 〉 = Ei |φi 〉, |φi 〉 = P|Ψi 〉.This is accomplished by the following similarity transformation withdecoupling property.

H̃ = e−ωHeω, QωP = ω, PH̃Q = 0Formal solution is

ω =

d∑

i=1

Q|Ψi 〉〈φ̃i |P, |Ψi 〉 = |φi 〉 + ω|φi 〉

If the unperturbed model space is degenerate H0|φi 〉 = E0|φi 〉,

V(n)eff

= Q̂(E0) +∑

m

Q̂m(E0){V (n−1)eff }m

Q̂(E0) ≡ PH1P + PH11

E0 − QHQQH1P, Q̂m(E0) ≡

1

m!

dmQ̂(E0)

dEm0Q̂(E0): Q-box


Low-momentum interaction Vlowk Bogner, Kuo, Schwenk (2001)

Solve decoupling equation in momentum space ( ω =d

X

i=1

Q|Ψi 〉〈φ̃i |P )

( 1S0 channel )

Conserve all thelow-momentumobservables and wavefunctions.

Coupling betweenhigh momentum andlow momentumdecreases.

Low momentumattraction increases.

Λ : boundary between low momentum and high momentum.→ cutoff parameter


Similarity renoramalizaton group (SRG) Bonger, Furnstahl and Perry (2007)

Another method to obtain low-momentum effective interactionFlow equation

H(s) = U(s)HU†(s) ≡ Trel + V (s)dH(s)

ds= [η,H(s)], η(s) =

dU(s)

dsU†(s) = −η†(s)

If we choose η(s) = [Trel, H(s)]

dV (k , k ′)

ds= −(k2 − k ′2)2V (k , k ′) +

∫

q2dq(k2 + k ′2 − 2q2)V (k , q)V (q, k ′)

Figures are taken from http://www.physics.ohio-state.edu/ ntg/srg/

Channel: 3S1 − 3S1 λ = s−4 ≈ Λ in VlowkIntroduction Effective interaction Tensor force Three body force Summary 9/ 32

Tensor force in effective interaction


Analyzing the tensor component (Spin-tensor decomposition)

Transform jj-coupled matrix elements to LS-coupled matrix elements

〈ABLSJT |V |CDL′S ′JT 〉 = [(1 + δAB)(1 + δCD)]1/2∑

ja,jb,jc ,jd

la12 ja

lb12 jb

L S J

lc12 jc

ld12 jd

L′ S ′ J

×[(1 + δab)(1 + δcd)]1/2〈abJT |V |cdJT 〉

Twobody interaction: V =∑

p

Vp =∑

p

Up · X p

Up: rank p operator in coordinate space

Xp: rank p operator in spin space

p = 0: centralp = 1: spin-orbit

p = 2: tensor

Spin-tensor decomposition

〈ABLS |Vp|CDL′S ′〉J′T = (−1)J′

p̂

{

L S J ′

S ′ L′ p

}

×∑

J

(−1)J Ĵ{

L S J

S ′ L′ p

}

〈ABLS |V |CDL′S ′〉JT

Uniquely decomposed into central, spin-orbit and tensor component


Tensor force in low-momentum interaction (sd-shell)

Tensor-force monopole Central-force monopole

Tensor-force monopole is not affected much even in thelow-momentum interaction Vlowk , whereas the central force changemuch by renormalization.


Tensor force in low-momentum interaction (pf -shell)

Tensor-force monopole

Tensor forcesurvives in pf -shellalso

PRL.104,012501(2010)


In momentum space

V (k ; 3S1, k′; 3D1)

3S1 − 3D1 channel: only tensor forceLong-range nature of the tensor force


Effect of renormalization of the tensor force

Vlowk from Av8’ Tensor subtracted (Av8’ TS)Vlowk from Av8’ (Av8 full)

Central-force monopole

(Λ = 2.1fm−1)Short range part of the tensor force is renormalized topredominantly the central force.The process shown is expected to have the largest contribution.


Schrodinger equation for deuteron:

− ~2

M

d2u(r)

dr2+ VCu(r) +

√8VTw(r) = Edu(r)

− ~2

M

d2w(r)

dr2+

(

6~2

Mr2+ VC − 2VT − 3VLS

)

w(r) +√

8VTu(r) = Edw(r)

Effective central force

Veff(r ;3S1) = VC (r ;

3S1) + ∆Veff(r ;3S1), ∆Veff(r ;

3S1) ≡√

8VT (r)w(r)

u(r).

→ This leads additional attraction to the central forceFeshbach’s formal solution of the effective interaction

PHP|Ψ〉 + PHQ 1E − QHQ QHP|Ψ〉 = E |Ψ〉, Heff = PHP + PHQ

1

E − QHQ QHP

Vlowk

P-space: low-momentum

Q-space: high-momentum

Effective central force

P-space: S-wave wavefunction

Q-space: D-wave wavefunction

→ can be understood on the same footingIntroduction Effective interaction Tensor force Three body force Summary 16/ 32

Tensor force in another potential

We calculated tensor-force monopole of Vlowk starting fromanother potential (χN3LO)

sd-shell pf -shell

Similar result is obtained when the original potential is N3LO→ result does not depends on spcific model of the nuclear force


Tensor force in the effective interaction for the shell model

Effective interaction for shell model

Starting from Vlowk

Q-box and its folded diagrams is taken into account

Effective interaction for sd-shell and pf -shell

Harmonic oscillator basis

Diagrams included in 2nd order Q-box


Tensor-force monopole (sd-shell) Tensor-force monopole (pf -shell)

Tensor force survives even in the effective interaction for shellmodel, in both sd-shell and pf -shell

complicated and specific structure of the tensor force

PRL.104,012501(2010)


Evaluation of the Q-box

core-polarization particle-particle ladder hole-hole ladder

Predominantly the central force

Tensor component is not coherent

Only exchange diagram contribute toTensor-force monopole→ always has the same sign→ not predominantly the tensor force(tensor force has strong state dependence)


Comparison with empirical interactions for shell model

Good agreement in T = 0channel of SDPF-M

USD families has weakertensor force

T = 1 channel includeseffects from neutron richnuclei including dripline


Monopole universal interaction VMU

When the Tensor-force monopole is π + ρ exchange, remaining canbe well approximated by simple Gaussian central force.

PRL.104,012501(2010)


Similar observation also in T = 1 matrix monopole.


Tensor force in VSRG

Tensor-force monopole V (k ;3S1, k

′; 3D1)

Tensor force in VSRG is also similar to that of original Av8’ tensorforce.


Three body force contribution toeffective interaction


Fujita-Miyazawa three-body force

Three-body force

Virtual excitation to the ∆(1232): lowest excitedstate of the nucleons

exchange π meson two times

Renormalization of single particle

energies affected by the Pauli’s

exclusion principle in nuclear

medium

This effect is included automaticallyif we consider exchange diagram(Delta-hole diagram)

→ effective two-body force

→ we call this effective twobody force comes from ∆ hole diagramFM-twobody forcewe calculate the multipole of FM-twobody force in T = 1 channel


Interaction between N and ∆ Green (1976)

Interaction between N and ∆ is similar in structure to OPEP

V πNN(k) =f 2πNNmπ

(σ1 · k)(σ2 · k)m2π + k

2τ1 · τ2

V πN∆(k) =fπNNfπN∆

mπ

(σ · k)(S · k)m2π + k

2τ · T

S,T is spin and isospin operators which convert nucleon to ∆(spin = 32 , isospin =

32)

For simplicity, we neglect ρ meson contribution and others


Spin-tensor decomposition of FM-twobody force

sd-shell pf -shell

Predominantly central force

Weak LS and tensor component (other than p1-p1 monopole)

Relatively strong ALS(anti-symmetric spin-orbit) component

→ ALS component does not exist realistic nuclear force→ consequence of renormalization in nuclear medium


Spin-tensor decomposition (revisited)

Transform jj-coupled matrix elements to LS-coupled matrix elements

〈ABLSJT |V |CDL′S ′JT 〉 = [(1 + δAB)(1 + δCD)]1/2∑

ja,jb,jc ,jd

la12 ja

lb12 jb

L S J

lc12 jc

ld12 jd

L′ S ′ J

×[(1 + δab)(1 + δcd)]1/2〈abJT |V |cdJT 〉

Twobody interaction: V =∑

p

Vp =∑

p

Up · X p

Up: rank p operator in coordinate space

Xp: rank p operator in spin space

p = 0: centralp = 1: spin-orbit

p = 2: tensor

Spin-tensor decomposition

〈ABLS |Vp|CDL′S ′〉J′T = (−1)J′

p̂

{

L S J ′

S ′ L′ p

}

×∑

J

(−1)J Ĵ{

L S J

S ′ L′ p

}

〈ABLS |V |CDL′S ′〉JT

Uniquely decomposed into central, spin-orbit and tensor component


Anti-symmetric spin-orbit force

In LS-coupled matrix element 〈ABLS |Vp|CDL′S ′〉J′T , S 6= S ′component is called anti-symmetric spin-orbit (ALS) force.

Consider a two-body problem.If S 6= S ′, this force is automatically the rank 1 force.Moreover, from the Pauli’s exclusion principle, orbital angularmomenta which have different parity must be mixed, if isospin isconserved.

→ such a component does not exist realistic nuclear force→ consequence of renormalization of nuclear force in nuclearmedium


Let us consider diagrams included in 2nd order Q-box

In pp-ladder and hh-lader diagramsinteractions does not change totalspin S in each vertex→ does not induce ALS componentto the effective interaction

Core-polarization diagram is different instructure→ total spin of two nucleons can be changed(Delta-hole diagram has a similar structure)

Similar state dependence

Comparable in strength


Summary

Tensor force is not affected much by renormalization, at leastin its monopole part.

Tensor-force monopole in Vlowk is quite similar to that oforiginal potential.Tensor-force monopole in the effective interaction for the shellmodel calculated by the Q-box expansion is also similar to thatof original potential.These results do not rely on the spcific model of the nuclearforce.

→ Tensor force survives both of the two-step renormalization.Effective two-body force produced by Fujita-Miyazawathree-body force can be described by a simple picture, centralforce and anti-symmetric spin-orbit force (ALS).The strength of ALS is comparable to that ofcore-polarization diagram.


Summary

Tensor force is not affected much by renormalization, at leastin its monopole part.

Tensor-force monopole in Vlowk is quite similar to that oforiginal potential.Tensor-force monopole in the effective interaction for the shellmodel calculated by the Q-box expansion is also similar to thatof original potential.These results do not rely on the spcific model of the nuclearforce.

→ Tensor force survives both of the two-step renormalization.Effective two-body force produced by Fujita-Miyazawathree-body force can be described by a simple picture, centralforce and anti-symmetric spin-orbit force (ALS).The strength of ALS is comparable to that ofcore-polarization diagram.


IntroductionTensor force

Review on theories of effective interactionRealistic nuclear forceFormal theory for the effective interactionLow-momentum interactionSimilarity renormalization group (SRG)

Tensor force in effective interactionSpin-tensor decompositionTensor force in low-momentum interactionTensor force in the effective interaction for the shell modelTensor force in VSRG

Three body force contribution to effective interactionFM-twobody forceAnti-symmetric spin-orbit force

Summary

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