aa r X i v : . [ nu c l - t h ] N ov Ingredients of nuclear matrix elementfor two-neutrino double-beta decay of Ca Y. I wata , N. S himizu , Y. U tsuno , M. H onma , T. A be , and T. O tsuka , , Center for Nuclear Study, School of Science, The University of Tokyo, Japan Japan Atomic Energy Agency, Japan Center for Mathematical Sciences, University of Aizu, Japan Department of Physics, School of Science, The University of Tokyo, Japan National Superconducting Cyclotron Laboratory, Michigan State University, USAE-mail: [email protected] (Received July 31, 2014)Large-scale shell model calculations including two major shells are carried out, and the ingredients ofnuclear matrix element for two-neutrino double beta decay are investigated. Based on the comparisonbetween the shell model calculations accounting only for one major shell ( p f -shell) and those for twomajor shells ( sd p f -shell), the e ff ect due to the excitation across the two major shells is quantitativelyevaluated. KEYWORDS:
Double beta decay, Nuclear matrix element, Shell model calculation
1. Introduction
There are two types of double-beta decay processes depending on whether neutrinos are emittedor not. The former one is referred to two-neutrino double-beta decay, and the latter one to neutrino-less double beta decay. Only the two-neutrino double-beta decay is admissible if the neutrino is theDirac particle, while both types can take place if the neutrino is the Majorana particle. In this senseexperimental observation of neutrino-less double beta process has an impact on determining one ofthe most fundamental properties of neutrino.In this article ingredients of the nuclear matrix element for two-neutrino double-beta decayof Ca are calculated based on large-scale shell model calculations (KSHELL [1]). Two-neutrinodouble-beta decay processes consist mostly of double Gamow-Teller transition processes, so that twokinds of experiments are associated with the two-neutrino double-beta decay; one is the double-beta-decay half-life experiment, and the other is the Gamow-Teller transition experiments. The half-lifewas measured to be (4 . ± . × yr [2] (eval. 2006), which is relatively well reproduced by shellmodel calculations accounting only for one major shell [3]. Meanwhile, according to the experimentsby Yako et al. [4], shell model calculations including only one major shell [3] possibly underestimatethe Gamow-Teller transition strength from Ca to Sc and that from Sc to Ti (Fig. 1). Notethat the transition strength shown in [4] may or may not include isovector spin monopole transitionstrength [5] in addition to the Gamow-Teller transition strength.The ultimate goal of our research project is to present the matrix element of neutrino-less doublebeta decay process with the highest accuracy so far, and to predict both possible half-life of neutrino-less double beta process and the mass of neutrino. Among others here we concentrate on establishing aframework of shell model calculation (i.e. the e ff ective nuclear force), which well describes Gamow-Teller transition processes around Ca. dB(GT − )/ dE dB(GT + )/ dE Fig. 1. (color online) GT ± transition strength based on a shell model calculation (blue curves) accountingonly for one major shell (GXPF1A) and experimental data (red bars) [4]. Horizontal axis means the excitationenergy of 1 + c state of Sc measured from the ground state of Sc (see Eq. (2)). The Gaussian with its squareroot of the variance: σ = − transition strength from Ca to Sc ( σ = + transition strength from Ti to Sc ( σ =
2. Nuclear matrix elements of two-neutrino double beta-decay
We consider two-neutrino double-beta decay process: Ca → Ti + e − + ν, where Ca and Ti correspond to the initial and final nuclei respectively. In particular Sc plays arole of providing intermediate virtual states. The inverse of the half-life is represented by[ T / ] − = G ν | M ν (GT) | , (1)where G ν is the phase space factor, and M ν (GT) denotes the nuclear matrix elements due to theGamow-Teller (GT) transition. Note that transitions other than the GT transition are negligible, asfar as two-neutrino double-beta decay processes are concerned [6]. Although the value of G ν hasnot been fixed so far (e.g., see [7]), here we take G ν = . × − yr − MeV [8]. This valuecorresponds to the value adopted in relevant papers [3, 4]. The nuclear matrix element is representedby M ν (GT) = c max X c = < + f || ( τσ ) − || + c >< + c || ( τσ ) − || + i > E c − E i + Q ββ / (2)where states | i > , | f > and | c > stand for the 0 + ground state of the initial nucleus with the energy E i , 0 + ground state of the final nucleus with the energy E f , and 1 + intermediate virtual states with theenergy E c , respectively. Q ββ = E i − E f denotes the Q -value of the double beta decay, and the valueof Q ββ is almost precisely determined in recent experiments (e.g., Q ββ = . τσ ) ± means the GT ± transition operator, and it is replaced by the e ff ective GT ± transitionoperator ( τσ ) ± e ff = q ( τσ ) ± = . τσ ) ± in shell model calculations [10]. Even though a constant c max is equal to ∞ in rigorous treatments, it is replaced by a finite value if the transition through 1 + c ( > c max ) [1/MeV] [MeV] 0 5 10 15 20 25 300.00.10.20.30.40.5 [1/MeV] [MeV]Experimentsigma=0.51sigma=0.34sigma=0.17 dB(GT − )/ dE dB(GT + )/ dE Fig. 2. (color online) GT ± transition strength based on shell model calculations (blue curves) accounting fortwo major shells (SDPFMU-2 ~ ω ) and experimental data (red bars) [4]. Description manner follows from Fig.1. state is negligible. Using the experimental half-life ((4 . ± . × yr [2]) and Eq. (1), the matrixelement for two-neutrino double-beta decay of Ca is M ν (GT) = . ± . − , while the corresponding value obtained by shell model calculation accounting only for one majorshell is 0.0539 MeV − [3]. The nuclear matrix element is seemingly well reproduced by a shell modelcalculation. However it is notable that cancellations may lead to the unexpected coincidence betweenexperimental and theoretical values for the nuclear matrix element, since the sign of numerator inEq. (2) for a certain c is not necessarily positive. The claim of the discrepancy in GT transitionstrength [4] implies the reality of such unexpected cancellations.
3. Results
The GT transition strengths dB (GT − ) / dE and dB (GT + ) / dE are essentially regarded as the mainingredients of the nuclear matrix element (cf. the numerator in Eq. (2)): B (GT − ; c ) = | < + c || ( τσ ) − || + i > | , B (GT + ; c ) = | < + f || ( τσ ) + || + c > | . A shell model calculation including only one-major shell (employing GXPF1A [11]) is comparedwith the GT transition experiments in Fig. 1. Remarkable discrepancies are noticed in dB (GT − ) / dE for energies E ∗ > . dB (GT + ) / dE for energies E ∗ > . E ∗ denotes the excitation energy of 1 + c state of Sc measured from the ground state of Sc. That is,we see no significant di ff erence in low energies satisfying E ∗ < . ff erence in low energies can be found in dB (GT + ) / dE values compared to dB (GT − ) / dE values.Among several reasons for the discrepancy in higher energies, the most crucial missing contributionis expected to arise from the excitations across the major shells. By employing SDPF-MU interaction [12], shell model calculations including two major shells( sd p f -shell consisting of sd and p f shells) are carried out. Because of the limited computational able I. Proton and neutron excitations across the major shells; the number of excited protons and neutrons(in this order) from sd -shell to p f -shell are shown in each cell. Ca (0 + ) Ca (0 + ) Ti (0 + ) Ti (0 + ) Sc (1 + ) Sc (1 + )SDPFMU-2 ~ ω power, it is impossible to fully take into account all the configurations allowed in the sd p f -shellmodel space (the corresponding m -scheme dimension for the diagonalization > ). Therefore, afteremploying the Lanczos strength function method [13], we truncate the model space in a reasonablemanner (the corresponding m -scheme dimension ∼ ) in which the excitation is limited up to 2 ~ ω type excitations (SDPFMU-2 ~ ω ). A shell model calculation including two major shell is comparedwith the GT transition experiments in Fig. 2. The discrepancy between experimental and theoreticalvalues becomes smaller mainly for energy region 12.5 < E ∗ < − transition, but thereare still significant discrepancies in high energy regions E ∗ > . + transition.
4. Concluding remark
We have obtained a better description of Gamow-Teller transitions around Ca by introducing thetwo major shells, but there are still missing higher energy contributions. The numbers of excited nu-cleons across the major shells are smaller than expected (Table I); indeed Ca (0 + ) has been claimedto be proton-excitation state [14], so that the corresponding number of excited proton is expected tobe close to 2.00. One reason is that the interaction (SDPF-MU) is not su ffi cient to describe GT transi-tions around Ca within the truncated sd p f -model space. In fact the excitation energies of Ca (0 + )and Ti (0 + ) are 5.099 and 4.097 MeV for SDPFMU-2 ~ ω calculations, while the corresponding ex-perimental values are 4.283 and 2.997 MeV respectively [15]. Note that the corresponding values forGXPF1A calculations are 5.275 and 4.048 MeV respectively. Further investigation of nuclear inter-action is in progress for the better description.This work was supported by HPCI Strategic Programs for Innovative Research Field 5 “The originof matter and the universe”. Authors are grateful to Prof. Yako for fruitful comments. References [1] N. Shimizu, arXiv:1310.5431 (2013).[2] List of Adopted Double Beta Decay Values, NNDC database; http: // / bbdecay / list.html.[3] M. Horoi, S. Stoica and B. A. Brown, Phys. Rev. C 75 et al. , Phys. Rev. Lett. (2000) 265.[7] J. Kotila and F. Iachello, Phys. Rev.
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