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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.
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Multipole expansion of NME (QRPA)
spin-parities of nuclear intermediate states
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Part 1
Quenching of spin-dependent transitions
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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
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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.
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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
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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:
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Quenching of M4 strengths (1)
A perturbative calculation of M4 transition strength in 1
6O with G-matrix.
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Quenching of M4 strengths (2)
first order second order
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Quenching of M4 strengths (3)
Reductions in amplitude (%):
at q = qpeak
at q = 100 MeV/c
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Part 2
Violation of isospin symmetry
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Multipole expansion of NME (QRPA)
The large 0+ contribution in QRPA calculationsis due to isospin symmetry breaking.
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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
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BCS formalism (2)
The variation gives
the BCS equations
pairing interaction
two-body interactionbetween valence nucleons
s.p.e. for thecore nucleus
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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.
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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.
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Part 3
Nuclear monopole interaction
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Definition of single-particle energies (1)
Prescription by Baranger Nucl. Phys. A149 (1970) 225
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Definition of single-particle energies (2)
monopole
interaction with the core nucleons interaction with the valence nucleons
the same form as the BCS formalism
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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.
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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
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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.
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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.
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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.
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0 decay transition operators
Double Gamow-Teller ME (magnetic type)
Double Fermi ME (electric type)
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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
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J= 2+ Component
Electric Quadrupole Transitions
Systematic analyses of E2 transitions have shown that
Isoscalar transitions are enhanced,
Isovector strengths have no renormalization.
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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.