Reliability of nuclear matrix
elements of neutrinolessdouble-β decay by QRPA
J. Terasaki, Czech Tech. Univ. in Prague
1. Brief introduction
2. Check of application of QRPA to 136Xe→136Ba
3. Brief discussion on gA
4. Prospect
Feb. 15, 2020
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Abbreviations and notation
QRPA: quasiparticle random-phase approximation,
an approximation for obtaining nuclear wave
functions.
GT: Gamow-Teller, GT transition is a charge-changetransition by the weak interaction involving spin operator
gA: axial-vector current coupling, strength of the GTcomponent of the weak interaction.
0νββ: neutrinoless double-β2νββ: two-neutrino double-β
NME: nuclear matrix element

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Introduction
My motivation: to obtain the neutrino mass
Background of my study0νββ decay of nucleus gives one of the limited
methods for this aim.
If the neutrino is a Majorana particle, and the half-life ismeasured, the effective neutrino mass can be obtained.
Theory needs to supply the NME and the phase-spacefactor.
The latter is OK.
The NME is distributed in the range of a factor of 2-3
depending on method.
Investigation of the reliability of NME calculation.
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Examinations of NME calculation of 136Xe →136Ba that I
have done so far
Comparison with relevant exp. data
● Higher-order terms of 2νββ NME● GT strength of 136Xe →136Cs● others
Self check of calculation● GT sum rule
● Convergence of 0νββ NME with respect
to intermediate states.
● others

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Higher-order term of NME of 2vββ decay

F. Šimkovic et al., PRC 97, 034315 (2018): intermediate-state energy,: final-state energy,: initial-state energy

5/17Higher-order term of NME of 2vββ decay

Statistical analysis
with the exp. data ofββ spectrum of 136Xe
A. Gando et al., PRL 122,192501 (2019)

My point (red circle),ୣ୤୤is used஺Exp. 2νββ half-life

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Running sum of 2νββ NME

My calculation

No adjustment of interaction
A. Gando et al., PRL 122, 192501(2019)
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GT– strength of 136Xe →136Cs
D. Frekers et al., Nucl. Phys. A 916, 219 (2013)

The authors obtained the GT strengths from the crosssection of 136Xe(3He,t)136Cs reaction and calibrated themusing the logft values of electron capture.
Textbook of J. Suhonen, (Springer, Berlin, 2007),I calculated
decay half-life.

using

0.49 reproducing the exp.

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Exp. data and calculation of gA2 x GT‒ strength

Excitation energy of 136Cs• The tendency of the exp. level distribution is well
reproduced.
• The calculation with gA from the exp. data of 2νββdecay is consistent with the electron capture in terms of
the GT strength.
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Exp. data and calculation of gA2 x GT‒ strength

GT sum rule is satisfied.

ΣGT–(136Xe→136Cs) ‒ ΣGT+(136Xe→136I)= 85.145 ‒ 1.138 = 84.007.
Analytical value is 3(N‒Z) = 84.
→The single-particle space is large enough for the sum rule.
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Running sum of 0vββ NME

Convergence around Eexc = 60 MeV.
A much larger energy region is necessary than for GT
sum rule and 2νββ NME.
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0νββ NME
ଵଶ

ᇲ

ᇲ

ᇲ

ᇲ

Final state

௕

76Ge

Initial state
Intermediate state

r12/R
M. Doi et al., Prog. Theor. Phys.
83, 1 (1985).
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Effective gA
Looking around data and calculations of nuclear physics, thefollowing tendency is seen:
The appreciable quenching is necessary for the GT transition.
This is confirmed in E < 10-15 MeV roughly. Enhancement may benecessary in the higher-energy region at least for the result of QRPA.

The isobaric-analog transition does not need a quenching
factor; vector current coupling gV = 1.
Electric transition can be reproduced approximately well byusing the bare charge in QRPA for magic, near-magic andwell-deformed nuclei.
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epν
π0nIs the vertex correction more important, if the transition
operator ● includes the spin operator? If so, why?
Comments are welcome.

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Is effective gA for 0vββ equal to that for other weakphenomena?
The difference between the 0νββ and other weakphenomena is the neutrino potential, which is obtainedby the Fourier transformation of the neutrino
propagator + correction terms. The integrand has apeak around 200 MeV/c.
If different, there are possible two reasons.
1. If the single-particle space is not sufficiently large,
obviously gA for 0vββ ≠ gA for others.
This is not the case for my calculation.
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2. If the vertex correction depends on the major valueof q, gA for 0vββ might be different from gA forothers.
eepepp

ν

+qnen

ν
π0p

qnn

These diagrams are embedded in a nucleus.

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Prospect
Advantage of QRPA
Very large single-particle space can be used, so thatthe convergence of the 0vββ NME is possible to obtainfor all candidate nuclei up to 150Nd of the 0vββ decay.
Disadvantage
Effect of the many-particle many-hole correlations isincluded only perturbatively. The importance of thiseffect depend on nucleus.
My short-term goal
To find the candidate nuclei with the minimum
disadvantage of QRPA.
Currently 136Xe is one of the best candidates.
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Reserved

Nuclear matrix element of 2vββ decay
This matrix element includes௝ : energy of intermediate states,

which are virtual state௙ : final-state energy,
௜ : initial-state energy

QRPA has two sets of intermediate-state energies.
One is obtained from the initial state. Another isobtained from the final state.
If the two energy sets give close NMEs, the approximationis good. J.T. PRC 100, 034325 (2019)11/27