Some notes about 2 -decay (NMEs) Both and operators connect the - - PowerPoint PPT Presentation

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Some notes about 2 -decay (NMEs) Both and operators connect the - - PowerPoint PPT Presentation

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TRIUMF DBD workshop Interfacing theory and experiment for reliable DBD NMEs calculation Vancouver, Canada, May 11-13, 2016 2 bb-decay is the key for reliable calculation of 0 bb-decay NMEs Fedor imkovic 5/11/2016 Fedor Simkovic 1

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5/11/2016 Fedor Simkovic 1

2bb-decay is the key for reliable calculation of 0bb-decay NMEs

Fedor Šimkovic

TRIUMF DBD workshop

Interfacing theory and experiment for reliable DBD NMEs calculation

Vancouver, Canada, May 11-13, 2016

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5/11/2016 Fedor Simkovic 2

OUTLINE

  • I. Some notes about 2 -decay
  • II. The DBD Nuclear Matrix Elements

and the SU(4) symmetry

  • III. QRPA for description of states of

multiphonon origin

  • IV. How many 0 -decay NMEs

we need to calculate?

November 1984, Dubna We need reliable calculation of DBD NMEs

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5/11/2016 Fedor Simkovic 3

Some notes about 2-decay (NMEs)

Both  and  operators connect the same states. Both change two neutrons into two protons. Explaining -decay is necessary but not sufficient There is no reliable calculation of the 2-decay NMEs Calculation via intermediate nuclear states: QRPA (sensitivity to pp-int.) ISM (quenching, truncation of model space, spin-orbit partners) Calculation via closure NME: IBM, PHFB No calculation: EDF

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5/11/2016 Fedor Simkovic 4

-decay nuclear matrix elements

Differencies in NME: by factor ~ 10 Deduced from measured T1/2

2

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5/11/2016 Fedor Simkovic 5

The cross sections of (t,3He) and (d,2He) reactions give B(GT±) for  and , product of the amplitudes (B(GT)1/2) entering the numerator of M2

GT

Closure 2-decay NME SSD hypothesis

Grewe, …Frekers at al, PRC 78, 044301 (2008)

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5/11/2016 Fedor Simkovic 6

Single State Dominance ( 100Mo, 106Cd,116Cd, 128Te …)

HSD, higher levels

contribute to the decay

SSD, 1 level

dominates in the decay

(Abad et al., 1984,

  • Ann. Fis. A 80, 9)

100Mo

0

100Tc

1

Ei-Ef= -0.343 MeV Ei-Ef= -0.041 MeV Ei-Ef= 0.705 MeV

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5/11/2016 Fedor Simkovic 7

SSD – theoretical studies

Isotope f.s. T1/2(SSD)[y] T1/2(exp.)[y] 

100Mo

0g.s. 6.8 1018 6.8 1018 01 4.2 1020 6.1 1018

116Cd

0g.s. 1.1 1019 2.6 1019

128Te

0g.s. 1.1 1025 2.2 1024 EC/EC

106Cd

0g.s. >4.4 1021 >5.8 1017

130Ba

0g.s. 5.0 1022 4.0 1021

Domin, Kovalenko, Šimkovic, Semenov, NPA 753, 337 (2005)

SSD

common approx.

E1-Ei ≈ 0 or neg.  sensitivity to lepton energies in energy denominators  SSD and HSD offer different differential characteristics

Šimkovic, Šmotlák, Semenov

  • J. Phys. G, 27, 2233, 2001
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SSD differential characteristics 2ECdecay 2decay

100Mo → 100Ru

Do not depend

  • n

MiMf

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5/11/2016 Fedor Simkovic 9

MO100, EE-int, Emin ENRGY RAW-BGR spectrum and MTCA 2b2n 25 50 75 100 125 150 175 200 250 500 750 1000 1250 1500 1750 2000 E single, keV Events / 24 keV

Šimkovic, Šmotlák, Semenov

  • J. Phys. G, 27, 2233, 2001

Single electron spectrum different between SSD and HSD

100Mo 22: Experimental Study of SSD Hypothesis

22 HSD Monte Carlo

HSD

higher levels

Background subtracted

  • Data

MO100, EE-int, Emin ENRGY RAW-BGR spectrum and MTCA 2b2n 25 50 75 100 125 150 175 200 250 500 750 1000 1250 1500 1750 2000 E single, keV Events / 24 keV

22 SSD Monte Carlo Background subtracted

  • Data

SSD

Single State

HSD: T1/2 = 8.61 ± 0.02 (stat) ± 0.60 (syst)  1018 y SSD: T1/2 = 7.72 ± 0.02 (stat) ± 0.54 (syst)  1018 y

100Mo 22 single energy distribution

in favour of Single State Dominant (SSD) decay 4.57 kg.y

E1 + E2 > 2 MeV

4.57 kg.y

E1 + E2 > 2 MeV

/ndf = 139. / 36 /ndf = 40.7 / 36

Esingle (keV) Esingle (keV)

Esingle (keV)

NEMO 3 exp.

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5/11/2016 Fedor Simkovic 10

2-decay rate

In the limit

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5/11/2016 Fedor Simkovic 11

2-decay within the field theory

Weak interaction Hamiltonian 2nbb-decay amplitude Hadron part of amplitude

F.Š., G. Pantis, Phys. Atom. Nucl. 62 (1999) 585

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5/11/2016 Fedor Simkovic 12

Integral representation of MGT Completeness: n |n><n|=1

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Double beta decay is a two-body process

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Operator Expansion Method and DBD NMEs

C.R. Ching, T.H. Ho, Commun. Theor. Phys. 10, 45 (1988); 11, 433 (1989); 11, 495 (1989)

  • F. Š., JINR Commun. 39, 21 (1989); M. Gmitro, F. Š., Izv. AN SSR 54, 1780 (1990);
  • F. Š., G. Pantis, Czech. J. Phys. B 48, 235 (1998); A. Faessler, F. Š., J. Phys. G 24, 2139 (1998)

Convergence of a series? This problem does not appear?

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5/11/2016 Fedor Simkovic 15

2 NME within the OEM

Nuclear Hamiltonian Effective Coulomb int. due to different ground states Central and tensor nuclear interactions

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If central and tensor interactions are neglected we end up with closure NME with <En-(Ei+Ef)/2> = Ei – Ef = 

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The DBD Nuclear Matrix Elements and the SU(4) symmetry

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Suppression of the DBD NMEs and their sensitivity to particle particle interaction strength Suppression of the Two Neutrino Double Beta Decay by Nuclear Structure Effects

  • P. Vogel, M.R. Zirnbauer, PRL (1986) 3148
  • O. Civitarese, A. Faessler, T. Tomoda,

PLB 194 (1987) 11

  • E. Bender, K. Muto, H.V. Klapdor,

PLB 208 (1988) 53 …

About 30 years ago The isospin is known to be a good approximation in nuclei In heavy nuclei the SU(4) symmetry is strongly broken by the spin-orbit splitting. What is beyond this behavior? Is it an artifact of the QRPA?

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5/11/2016 Fedor Simkovic 19

gpair- strength of isovector like nucleon pairing (L=0, S=0, T=1, MT=±1) gpp

T=1- strength of isovector spin-0 pairing (L=0, S=0, T=1, MT=0

gpp

T=0- strength of isoscalar spin-1 pairing (L=0, S=1, T=0)

gph- strength of particle-hole force HI violates SU(4) symmetry s.p. mean-field MF and MGT do not depend on the mean-field part of H and are governed by a weak violation

  • f the SU(4) symmetry by the

particle-particle interaction of H Conserves SU(4) symmetry

  • D. Štefánik, F.Š., A. Faessler, PRC 91, 064311 (2015)
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Energies of excited states for the case of conserved SU(4) symmetry MF=0, MGT=0 (see SU(4) multiplets)

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MGT up to the second order of perturbation theory due to violation of the SU(4) symmetry by the particle-particle interaction of H

  • D. Štefánik, F.Š., A. Faessler, PRC 91, 064311 (2015)
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5/11/2016 Fedor Simkovic 22

Results confirm dependence of MF and MGT on gpp

T=0 and gpp T=1 by the QRPA

  • D. Štefánik, F.Š., A. Faessler, PRC 91, 064311 (2015)
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M2

F depends strongly on gpp T=1

M2

GT does not depend on gpp T=1 F.Š., V. Rodin, A. Faessler, and P. Vogel, PRC 87, 045501 (2013)

QRPA with Isospin restoration

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5/11/2016 Fedor Simkovic 24

By assuming a fixed violation of the SU(4) symmetry by particle-particle int. MGT decreases by increase of isospin of the ground state

  • D. Štefánik, F.Š., A. Faessler, PRC 91, 064311 (2015)
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Energy weighted sum rules

  • f =2 nuclei
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What is the meaning of quantity (2En=1-Ei-Ef)?

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11/6/2015 27 MEDEX 2015

QRPA for description of states of multiphonon origin

  • A. Smetana, F.Š., M. Macko, AIP Conf. Proc. 1686, 020022 (2015)

and to be submitted

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β- transitions in the standard QRPA

11/6/2015 28 MEDEX 2015

Calculate what can be confronted with experiment.

  • low-lying states are expected

to be important for 2νββ decay

  • we need improvement in this region
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Limitations of the standard QRPA

11/6/2015 29 MEDEX 2015

We want to fix the following limitations of the standard QRPA:

  • 1. Due to the QBA Pauli principle is broken and the QRPA

colapses for the higher values of coupling parameters, which might be of physical interest.

  • 2. Excited states of multi-phonon structure are neglected.

Only the linear terms in phonon operator are considered.

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Schematic model

11/6/2015 30 MEDEX 2015

single J-shell with semidegeneracy Use exactly solvable model to test your ideas. We demonstrate the insufficiency of the multi-phonon approx. by comparison with the exact solution.

pn—Lipkin model

has the structure of the realistic hamiltonian. parametrizes particle-particle and parametrizes particle-hole interactions

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Schematic model – exact solution

11/6/2015 31 MEDEX 2015

The even and odd states do not mix! Results are obtained from diagonalization of Hamiltonian. basis of states:

  • dd and even eigenstates:
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Schematic model – energy spectrum

11/6/2015 32 MEDEX 2015

exact Fermion model vs exact QBA model vs multi-phonon approximation colapse of QRPA

  • multi-phonon approach gives poor

agreement for higher excited states

  • standard QRPA is built for the first

excited state only QBA

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Schematic model – β- transitions

11/6/2015 33 MEDEX 2015

The multi-phonon approximation cannot reproduce the exact solution! zero in multi-phonon approx. non-negligible contribution in the physical region

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Idea of nonlinear phonon operator

11/6/2015 34 MEDEX 2015

Desired first goal: the first and higher excited states described by single QRPA equation state of 3 phonon (lin. op.) origin

We introduce non-linear phonon operator: QBA

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QRPA with non-linear phonon operator

11/6/2015 35 MEDEX 2015

The QRPA equation: Even in QBA approximation the norm matrix has not the standard form. The RPA vacuum gets very complicated!!! Need for further approximations and for constructing a closed iterative procedure. the first 4 terms in the expansion of the RPA vacuum:

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QRPA with non-linear phonon operator

11/6/2015 36 MEDEX 2015

1.

convert the norm matrix to its standar form… …obtaining the parameters: its rotational angle & eigenvalues In every step of iteration we do:

  • 2. which are used to „rotate“ the system into the standard QRPA form
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QRPA with non-linear phonon operator

11/6/2015 37 MEDEX 2015

introducing F-operators to write the phonon operator in its „linear“ form: where: assuming QBA for F-operators it allows us to construct standard-like RPA vacuum: …and we „linearize“ the procedure

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QRPA with non-linear phonon operator

11/6/2015 38 MEDEX 2015

ITERATION PROCEDURE: „linearized“ RPA vacuum RPA vacuum expansion

QRPA eq‘n standardize QRPA rotational param‘s solving QRPA

initial values amplitudes and energies if the iteration converges

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Results

11/6/2015 39 MEDEX 2015

  • Energies of the first and the third excited states
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Results

11/6/2015 40 MEDEX 2015

  • beta transition operators
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Is there a scaling factor between - and 2-decay NMEs?

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  • F. Š., Nucl.Part.Phys.Proc. 265-266 (2015) 19-24

M2/R M0

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How many -decay NMEs have to be calculated?

MF, MGT, MT …

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The -decay with emission of electrons in s1/2 and p1/2 wave state

Exact relativ. electron w.f. Higher order terms

  • f nucleon current

with nucleon recoil

  • D. Štefánik, R. Dvornický, F.Š., Nuclear Theory 33 (2014) 115
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0-decay rate with p1/2 electrons (2 additional NMEs and 5 phase-space factors)

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Calculated phase-space factors for 0-decay with emission of s1/2 and p1/2 electrons (m mechanism)

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Effect of p1/2 wave is below 10%.

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 NMEs EDF, PHFB IBM, ISM QRPA, RQRPA

Understanding of the 2bb-decay is the key for reliable calculation

  • f 0bb-decay NMEs