July 29-30, 2010, Dresden 1

Forbidden Beta Transitionsin Neutrinoless Double Beta Decay

Kazuo MutoDepartment of Physics, Tokyo Institute of Technology

1. Quenching of spin-dependent transitions

2. Violation of isospin symmetry

3. Nuclear monopole interaction

July 29-30, 2010, Dresden 2

The mass term of 0 decay

There appear three nuclear matrix elements,

VAVA

The momentum integral of the virtual neutrino gives rise to a neutrino potential, which acts on the nuclear wave functions, being a long-range Yukawa-type (“range” ~ 20 fm).

A • AA • A V • V

with two-body nuclear transition operators.

July 29-30, 2010, Dresden 3

Multipole expansion of NME (QRPA)

spin-parities of nuclear intermediate states

July 29-30, 2010, Dresden 4

Part 1

Quenching of spin-dependent transitions

July 29-30, 2010, Dresden 5

Renormalization of operators due to model space truncation

Nuclear structure calculations (QRPA and shell models) in a finite model space

Renormalization of effects of coupling: model space and outside the model spaceNN interaction

(eg. G-matrix)

Transition operators

First-order approximation by effective coupling constant

July 29-30, 2010, Dresden 6

Quenching of GT Strength

Systematic analysis of GT beta decays in sd-shell nuclei

The experimental data are well reproduced with a quenching factor of 0.77, in (sd)A-18 calculation.

July 29-30, 2010, Dresden 7

The strength distribution, deduced from the charge-exchange (p,n) reaction, extends to high-excitation energy region, far beyond the giant resonance.

T. Wakasa et al., Phys. Rev. C55, 2909 (1997)

outside the model space

GT-strength Distribution

July 29-30, 2010, Dresden 8

Magnetic Stretched States

(J= 4, J= 6, J= 8) Transitions between single-

particle orbits with the largest

in respective major shells Unique configuration

within excitation The observed strengths are

quenched considerably, compared with the s.p. strength, probably due to coupling with higher excitations:

July 29-30, 2010, Dresden 9

Quenching of M4 strengths (1)

A perturbative calculation of M4 transition strength in 1

6O with G-matrix.

July 29-30, 2010, Dresden 10

Quenching of M4 strengths (2)

first order second order

July 29-30, 2010, Dresden 11

Quenching of M4 strengths (3)

Reductions in amplitude (%):

at q = qpeak

at q = 100 MeV/c

July 29-30, 2010, Dresden 12

Part 2

Violation of isospin symmetry

July 29-30, 2010, Dresden 13

Multipole expansion of NME (QRPA)

The large 0+ contribution in QRPA calculationsis due to isospin symmetry breaking.

July 29-30, 2010, Dresden 14

BCS formalism (1)

Ansatz for the BCS ground state

with

Variation with respect to the occupation amplitudes

for the modified Hamiltonian

with constraints for expectation values of the nucleon numbers

July 29-30, 2010, Dresden 15

BCS formalism (2)

The variation gives

the BCS equations

pairing interaction

two-body interactionbetween valence nucleons

s.p.e. for thecore nucleus

July 29-30, 2010, Dresden 16

Isospin symmetry in BCS

The proton and neutron systems are coupled through the proton-neutron interaction.

Isospin symmetry is conserved, if

(1) the s.p.e. spectra of the proton and neutron systems are the same (or a constant shift) for the N = Z core nucleus,

(2) s.p.e. are calculated with the two-body interaction.

July 29-30, 2010, Dresden 17

Isospin violation in BCS

In QRPA calculations, we usually replace the s.p.e. by energy eigenvalues of a nucleon in a Woods-Saxon potential.

This introduces a violation of isospin symmetry.

Shell model and self-consistent HF(B) calculations conserve the isospin symmetry, or a small violation.

July 29-30, 2010, Dresden 18

Part 3

Nuclear monopole interaction

July 29-30, 2010, Dresden 19

Definition of single-particle energies (1)

Prescription by Baranger Nucl. Phys. A149 (1970) 225

July 29-30, 2010, Dresden 20

Definition of single-particle energies (2)

monopole

interaction with the core nucleons interaction with the valence nucleons

the same form as the BCS formalism

July 29-30, 2010, Dresden 21

Monopole interaction

The monopole interaction is defined as the lowest-rank term of multipole expansion of two-body NN interaction.

Proton-neutron interaction

Like-nucleon interaction

exchange

monopole interaction

monopole interaction

: exactly the same quantity that appears in s.p. energies.

July 29-30, 2010, Dresden 22

Universality of pn interactionwhen normalized by the monopole

J.P. Schiffer and W.M. True,Rev. Mod. Phys. 48, 191 (1976)

particle-particle particle-hole

July 29-30, 2010, Dresden 23

Roles of C, T， LS interactions

j<j>

j’>

j’<

central + tensor+ LS

When both spin-orbit parners, j< and j>, are filled with nucleons,

For s.p. energies of j’< and j’> ，

(1) Central forces give the same gain,

(2) Tensor forces give no change,

(3) Spin-orbit forces enlarge the splitting.

July 29-30, 2010, Dresden 24

G-matrix is not good!

USB: filled symbols

G-matrix: open symbols

“G-matrix is good except the monopole”

The monopole strengths are accumulated in s.p.e., especially in a calculation with a large model space.

July 29-30, 2010, Dresden 25

Conclusions

Spin-dependent transitions are quenched by a factor of about 0.75 in amplitudes in a truncated model space due to coupling to higher-lying configurations. The quenching factor seems to be independent of the multipoles.

Approximations in the commonly used QRPA model violate the isospin symmetry, which overestimates the 0+ component of the 0 NME to a large extent.

Improvement is necessary in the monopole component of effective NN interactions.

A more reliable prediction of the 0 NME requires detailed comparison between results of QRPA, shell- model and IBM calculations.

July 29-30, 2010, Dresden 26

July 29-30, 2010, Dresden 27

0 decay transition operators

Double Gamow-Teller ME (magnetic type)

Double Fermi ME (electric type)

July 29-30, 2010, Dresden 28

J= 1- Component

Isovector Electric Dipole Transitions

E1 excitation strengths in the same nucleus well satisfy the TRK sum rule.

Highly collective No renomalization of the c

oupling constant

July 29-30, 2010, Dresden 29

J= 2+ Component

Electric Quadrupole Transitions

Systematic analyses of E2 transitions have shown that

Isoscalar transitions are enhanced,

Isovector strengths have no renormalization.

July 29-30, 2010, Dresden 30

J= 0+ Component The largest component of

the double Fermi ME

about 1/3 of A shell-model calculation

(for 48Ca) gives almost 0. This large value is possibl

y due to violation of isospin symmetry in QRPA calculations.