ニュートリノレス二重ベータ崩壊の λ 機構
岩田順敬（関西大学）
Yoritaka Iwata (Kansai University)Ref.) Y. Iwata, N. Shimizu et al. Phys. Rev. Lett. 2016
J. Terasaki, Y. Iwata, Phys. Rev. C 2019
S. Sarkar, Y. Iwata, P. K. Raina, Phys. Rev. C 2020

Half-life &nuclear matrix element (NME)

半減期NME

Axial vector current
coupling constant

位相空間因子位相空間因子← 放出される電子の波動関数
正確な計算が可能

原子核行列要素 ← 原子核の波動関数近似が不可欠
正確な計算は位相空間因子より難しい

me : 電子質量

Theoretical methods
nuclear matrix element (NME)ISM （殻模型計算＝相互作用するフェルミオン模型）
IBM （相互作用するボソン模型）
QRPA （準粒子乱雑位相近似）
HFB, EDF ( 対相関入り平均場計算 )GCM （生成座標の方法）

Feassler, J. Phys. Conf. Ser. 2012.

Shell model calculation for DBD of Ca48
- Large scale calculation by Tokyo group -✗✗Right handed weak boson ?

✗Summary

Brown, Horoi, Senkov, Phys. Rev. Lett., 2014

Recent trend on ISM calculations:①“more structure” = new paths]→ DGT,
2ν,0νExperiments
At the same timeNeacsu, Horoi, Phys. Rev. C, 2018

Recent trend on ISM calculations:②Hadronic current by EFT

Ordinary case
[λ mechanism] see also ... Simkovic et al., Front Phys. 2017

the left-right symmetric modelwith R-parity-violating SUSY model→ for contributions ofνL , νR, WL , WR

Yao et al., Phys. Rev. Lett., 2020

Recent trend on ISM+ calculations:③Ab-initio (IMSRG usable for ISM)

M0ν ＝ 0.61
GCM

A. Gando et al., Phys. Rev. Lett. 2019

Recent trend on NME calculations:

Based on2ν experiment

Reliability of NME values by diﬀerent models

Y. Iwata, N. Shimizu et al. Phys. Rev. Lett. 2016

Large scale ISM calculations
One major shell (0hw excitation) → Two major shells (up to 2hw excitation)
Including 2 major shells (Tokyo)
Ca (G.S.)48

A. Brown, lecture note

Ordinary onemajor shell calc.

1p1/2
1p3/2



1p1/2 (2)1p3/2 (4)0f5/2

0f5/2 (6)0f7/2

0f7/2 (8)1s1/2

1s1/2 (2)0d3/2

0d3/2 (4)0d5/2

0d5/2 (6)pf

Ti (G.S.)

48

pfsd



1p1/2
1p3/2



1p1/2 (2)1p3/2 (4)0f5/2

0f5/2 (6)0f7/2

0f7/2 (8)1s1/2

1s1/2 (2)0d3/20d3/2 (4)0d5/2

0d5/2 (6)pf1) Adjustment of interaction for “double beta decays”[EXPERIMENT] F. Videbaek et al., NPA (1986)
2nd 0+ state of 48Ca was pointed out to be proton-excitation state
proton excitation included in 2nd 0+ state of 48Ca:lle
sh GXPF1B: 0.00
1llse
sh SDPFMU: 0.22
2Ca (G.S.)48

Proton excitation
number

1p1/2
1p3/2



1p1/2 (2)1p3/2 (4)0f5/2

0f5/2 (6)0f7/2

0f7/2 (8)1s1/2

1s1/2 (2)0d3/2

0d3/2 (4)0d5/2

0d5/2 (6)“0.22” is still too small to be pronounced as the proton-excitation statecf.) the parity diﬀerence between the sd- and pf- orbits.
Our idea is to adjust the gap between the sd- and pf- shells
to reproduce the experimental excitation energy of 2nd 0+ state of 48Ca

48

Ca (g.s.)

1p1/2



1p3/2
0f5/2

To conserve the isospin symmetry

1p1/2
pf



1p3/2
0f5/2

pf

0f7/2

0f7/2

1s1/2

1s1/2

0d3/2 sd

0d3/2 sd

0d5/2

0d5/2

Two neutrino process[Experiment]: Yako et al. PRL (2009)

Ca → Sc

Ti → Sc
One major shell(GXPF1A)

Two major shell(SDPF-MU)

Terasaki, Phys. Rev. C 2018
Clarifies “it is IVSM”
Contribution from IVSM (isovector spin monopole) should be included inexperiment. However, it is not quantitatively well known.

Y. Iwata, N. Shimizu et al. Phys. Rev. Lett. 2016

NME value in large model spacecancellati
on

pne e
sdg (40-70)
pf (20-40)
sd (8-20)

82
50
2820
82M0 (1 shell)

0.833

M0 (2 shells)

1.118

Inclusion rate of 2nd major shell components ：48Ca (22%), 48Ti (33%)
sd + pf
Ca (~2%), 48Ti(~2%) pf + sdg
This result shows that
It should be necessary to take into account sd shell48

34.2 %
increased

Due to (1/1.34)2 ~ 0.56,
it means thatthe half-life is almost halved
for the same neutrino mass.

Summary of NME for 0 ofCa

48

Comparison of neutrinoless double beta decay NME (with ranges)

Previous GCM

RMF by Yao et al. 2015

M0ν ＝ 0.61

GCM by Vaquero et al. 2013
Barea et al. 2015

J. Menendez et al. 2009

Senkov-Horoi 2013

SDPFMU-db

Our results
SDPFMU

Simkovic et al. 2013
MBPT by Holt et al. 2014

Present status for DBD candidates

48

A. Feassler, J. Phys.: Conf. Ser. 337, 012065 (2012)

136 150
76 82 96 116
124
100
130

J. Engel and J. Menéndez, Rep. Prog.
Phys. 80 (2017) 046301

There has been no significant difference for these 5 years.
~ 2 to 3 times difference still exists

λ mechanism
WL- WL exchange

WL- WR exchange

λ mechanism

Calculation ofnuclear matrix elementCalculation ofneutrino pot part
Closure approx.

Y. Iwata, Nucl. Phys. Rev. (2017)
Top 10 amplitude

Non-closure:●Running non-closure●MixedS. Sarkar, Y. Iwata, P. K. Raina, Phys. Rev. C 2020

Matrix elements [L-L type]
SRC = Short range correlation ( 短距離相関 )Total

S. Sarkar, Y. Iwata, P. K. Raina, Phys. Rev. C 2020

Matrix elements [ω type]

Total
The amplitude of matrix element for L-R exchange
are comparable to the cases with L-L exchange.

S. Sarkar, Y. Iwata, P. K. Raina, Phys. Rev. C 2020

Matrix elements [q type]

Total

The amplitude of matrix element for L-R exchange
are relatively large compared to the cases with L-L exchange.

S. Sarkar, Y. Iwata, P. K. Raina, Phys. Rev. C 2020

Spin parity decomposition- intermediate state -CaSc

Ti

S. Sarkar, Y. Iwata, P. K. Raina, Phys. Rev. C 2020

Spin parity decomposition npnp

- initial and ﬁnal states -S. Sarkar, Y. Iwata, P. K. Raina, Phys. Rev. C 2020

Cutoﬀ dependence- energy -G.S. contribution
is large

S. Sarkar, Y. Iwata, P. K. Raina, Phys. Rev. C 2020

Cutoﬀ dependence- number of states -S. Sarkar, Y. Iwata, P. K. Raina, Phys. Rev. C 2020

Closure-energy dependence[constant] no signiﬁcant change is noticed in several
settings
In closure approximation

total

x1

x5

x 1/2

Effect should not be negligible.

*1 (std)large~*10 small~1/10~*1C 1st : enlarged amplitude (*5)
C 2nd : comparable amplitude (*1/2)
C 3rd : enlarged amplitude (*2.5)

Conclusion
we have foundthe large WR-WL eﬀect
almost 2 times larger than WL-WL

Previous shell model calculation did notcalculate/find the importance of 2nd and 3rd terms
Not calculated inHoroi, Neascu, PRC 2018

Refs. for studying on this direction
mechanismmechanism (mainly by RPA calculations)

Review article (e.g. hadronic current):Summary✗

ISM research [overview]+ nuclear structure, + hadronic current, +ab-initio→right-handed neutrino, right-handed W-boson

Shell model calculation for DBD of Ca48
- Large scale calculation by Tokyo group -✗✗Right handed weak bosons ?

✗(Right handed neutrino) --- sterile neutrinos ?