Two neutrino double-beta decay and effective axial-vector coupling constant

The theoretical and experimental study of the two-neutrino double-beta decay (2νββ-decay) is of crucial importance for reliable calculation of matrix elements governing the neutrinoless double-beta decay (0νββ-decay). That will allow to determine masses of neutrinos once the 0νββ-decay, which occurs if the neutrino is a massive Majorana particle and the total lepton number is not conserved, will be observed. Experiments studying the 2νββ-decay are currently approaching a qualitatively new level, where high-precision measurements are performed not only for half-lives but for all other observables of the process. In this context an improved formula for the 2νββ-decay is presented. Further, a novel approach for determining the effective axial-vector coupling constant is proposed.


Introduction
The observation of the neutrinoless double beta decay process (0νββ-decay) [1], will not only demonstrate the Majorana nature of neutrinos but it will also provide a measurement of the effective Majorana neutrino mass, Here, U ei and m i (i=1,2, and 3) are elements of the Pontecorvo-Maki-Nakagawa-Sakata neutrino mixing matrix and masses of neutrinos, respectively. The inverse value of the 0νββ-decay half-life is commonly written as T 0νββ where M 0ν , g eff A and G 0ν are the nuclear matrix element, effective axial-vector coupling constant, and phase-space factor, respectively. The calculation of M 0ν by using nuclear structure theory [2] and determination of g eff A [3] are crucial for the interpretation of the 0νββ-decay.
Understanding the two-neutrino double-beta decay (2νββ-decay), which involves the emission of two electrons and two antineutrinos, remains of major importance for reliable calculation of the 0νββ-decay nuclear matrix elements. * e-mail: simkovic@fmph.uniba.sk * * e-mail: rastonator@gmail.com * * * e-mail: dus.stefanik@gmail.com The 2νββ-decay has been a subject of experimental research for more than 60 years. It has been detected for 12 different nuclei for transition to the ground state and in two cases also to transition to 0 + excited state of the daughter nucleus [4]. This rare process is one of the major sources of background in running and planned experiments looking for a signal of the 0νββ-decay.
In this contribution a more accurate expression for the 2νββ-decay half-life is presented. In addition, a novel approach for determining the effective value of the axialvector coupling constant g eff A is suggested.

The improved 2νββ-decay rate
The inverse half-life of the 2νββ-decay transition to the 0 + ground state of the final nucleus takes the form [5] T 2νββ where G β = G F cos θ C (G F is Fermi constant and θ C is the Cabbibo angle), m e is the mass of electron and Here, , and E ν i (i = 1, 2) are the energies of initial and final nuclei, electrons and antineutrinos, correspondingly.   96 Zr, and 100 Mo. Calculation was performed within the proton-neutron QRPA with isospin restoration [8]. Results are obtained with Argonne V18 potential and for unquenched axial vector coupling constant g A = 1.269. of the Gamow-Teller nuclear matrix elements (we neglect the contribution from the double Fermi transitions to the 2νββ-decay rate), which depend on lepton energies where with Here, |0 + i (|0 + f ) is ground state of the initial (final) eveneven nuclei with energy E i (E f ), and |1 + n > are the states in the intermediate odd-odd nucleus with energies E n . The lepton energies enter in the factors The maximal value of |ε K | and |ε L | is the Q value of the process (ε K ∈ (−Q, Q)). For 2νββ-decay with energetically forbidden transition to intermediate nucleus ( The standard treatment of the 2νββ-decay [1]: The calculation of the 2νββ-decay probability is usually simplified by neglecting the factors ε K,L in energy denominators, what allows a separate calculation of the phase space factor and nuclear matrix element. Then, the inverse half-life of the 2νββ-decay can be written as where G 2ν is the phase-space factor [6]. The double Gamow-Teller (GT) matrix element is expressed in terms of sequential single β-decay Gamow-Teller transitions through virtual 1 + intermediate states.
An improved description of the 2νββ-decay [7]: A more accurate expression for the 2νββ-decay rate we get  96 Zr, and 76 Mo calculated within the QRPA with partial restoration of isospin symmetry [8]. For a given value of effective axial-vector coupling constant g eff A the strength of isoscalar interaction was adjusted to the measured 2νββ-decay half-life T 2ν−exp 1/2 [4] and the strength of isovector interaction by the fulfillment of the requirement that the 2νββ Fermi matrix vanishes [8]. P 2ν 0 and P 2ν 2 are the leading order and first order Taylor expansion contributions to 2νββ-decay width relative to the full width. G 2ν 0 and G 2ν 2 are phase-space integrals entering the expression for the 2νββ-decay rate in Eqs. (13) and (14), which were calculated with help of the exact Dirac wave functions with finite nuclear size and electron screening of the electron [6].
. By limiting our consideration to the second power in ε we end up with where partial contributions to the full 2νββ-decay width Γ 2ν associated with the leading Γ 2ν 0 , next to leading Γ 2ν 2 order in Taylor expansion are given by Nuclear matrix elements are defined as The phase-space factors are given by

A novel possibility to determine effective axial-vector coupling constant
The effective axial-vector coupling constant g eff A enters in the fourth power in both the 0νββ-and 2νββ-decay rates and is a major concern of experiments looking for a signal of the 0νββ-decay as its value might be strongly suppressed when compared to g A = 1.269 at nucleon level. In Ref. [9], it was found (g eff A ) 4 = 0.30 and 0.50 for the 2νββ-decay of 100 Mo and 116 Cd, respectively. By performing calculation within the QRPA g eff A was treated as a completely free parameter alongside g pp , which is used to renormalize particle-particle interaction of the nuclear Hamiltonian. A least-squares fit of g eff A and g pp to the βdecay rate and EC rate of the 1 + ground state in the intermediate nuclei involved in the double-beta decay in addition to the 2νββ-decay rate of the initial nuclei, led to g eff A of about 0.7 ( 100 Mo) and 0.8 ( 116 Cd). A significantly stronger quenching of the axial-vector coupling constant, namely (g eff A ) 4 (1.269 A −0.18 ) 4 = 0.063, was found within the Interacting Boson Model [10]. It has been determined by the theoretical prediction for the 2νββ-decay half-lives, which were based on the closure approximation calculated nuclear matrix element M 2ν GT , with the measured half-lives.
The value of effective axial-vector coupling constant g eff A can be determined from the measured 2νββ-decay half-life (see Eqs. (13) and (14) Within the QRPA with partial restoration of isospin symmetry [8] calculated running sums of matrix elements M 2ν GT −1 and M 2ν GT −3 as a function of the excitation energy E ex counted from the ground state are presented for the 2νββ-decay of 76 Ge, 82 Se, 96 Zr and 100 Mo in Fig. 1. The unquenched value of axial-vector coupling constant is assumed (g eff A = g A = 1.269). Recall that in the QRPA treatment of double-beta decay processes a common practice is to fix parameters of nuclear Hamiltonian by reproducing the measured 2νββ-decay half-life for a given value of g eff A . From Fig. 1 we see that matrix element M 2ν GT −3 is clearly determined by transitions through the lowest states of the intermediate nucleus unlike M 2ν GT −1 , which depends also on the transitions through higher lying states even from the region of Gamow-Teller resonance. We notice also a mutual cancellation among different contributions in the case of M 2ν GT −1 . Thus, it is expected that the calculation of M 2ν GT −3 might be more reliable when compared with evaluation of M 2ν GT −1 , especially in nuclear models which describe accurately lowest excitations of a nucleus like the interacting shell model.
In Table 1 for the 2νββ-decay of 76 Ge, 82 Se, 96 Zr and 100 Mo, which were obtained within the QRPA with partial restoration of the isospin symmetry by assuming g eff A =0.8, 1.0 and 1.269. The strengths of isocalar and isovector interactions of nuclear Hamiltonian were adjusted following the procedure in [8]. The kinematical phase-space factors G 2ν 0 and G 2ν 2 were determined by considering the exact Dirac wave functions with finite nuclear size and electron screening for emitted electrons [6]. The quantities P 2ν 0 and P 2ν 2 denote the leading and next to leading contributions of Taylor expansion in ε to 2νββ-decay width relative to the full width, respectively. From four analyzed nuclei in Table 1 the largest value of P 2ν 2 is found for 100 Mo (∼ 20%) and the smallest value for 76 Ge (∼ 3%).
By introducing the ratio ξ 2ν 13 of nuclear matrix elements M 2ν GT −3 and M 2ν GT −1 , the 2νββ-decay half-life can be written in a form without explicit dependence on M 2ν GT −1 as follows: The expression (18) can be exploited to determine the value of g eff A , if ξ 2ν 13 is determined phenomenologically from the shape of measured energy distributions of emitted electrons in the 2νββ-decay and a reliable calculation of nuclear matrix element M 2ν GT −3 is performed, e.g., within the Interacting Shell Model.
The NEMO3 experiment measured single and summed electron differential decay rates of the 2νββ-decay of 100 Mo with very high statistics of about 1 million events [11]. It allowed to obtain valuable information about the Single State Dominance (SSD) and Higher State Dominance (HSD) hypotheses [12], which discuss the importance of contributions to the 2νββ-decay NME from transitions through low-lying (SSD hypothesis [13]) and higher lying (the region of the GT resonance) intermediate nuclear states [14,15]. For these two particular cases we have ξ 2ν 13 ( 100 Mo) = 0.37 (SSD) = 0.00 (HSD).
The real value of ξ 2ν 13 ( 100 Mo) is a subject of experimental analysis within the NEMO3 collaboration. Its value can be any. The 2νββ differential decay rate is given by In Fig. 2, the effective axial-vector coupling constant g eff A calculated within the QRPA with partial restoration of isospin symmetry [8] is plotted as function of the matrix element M 2ν GT −3 for 2νββ-decay of 100 Mo. Recall that the isoscalar neutron-proton interaction of the nuclear Hamiltonian was adjusted to reproduce correctly the measured 2νββ-decay half-live of 100 Mo [4] for each g eff A . The allowed values of M 2ν GT −3 are within the range (0.064, 0.231) determined mostly by the pairing properties of involved initial and final nuclei. The SSD and HSD results are displayed as well. The experimentally determined value of ξ 2ν 13 ( 100 Mo) from the shape of the energy distribution might help to discriminate among considered scenarios and to conclude about g eff A for M 2ν GT −3 calculated within nuclear structure theory.
It is worth to note that a significant progress has been achieved in double beta decay experiments recently. The 2νββ-decay mode has been measured with high statistics in the GERDA ( 76 Ge) [16], CUORE ( 130 Te) [17], Kam-landZEN ( 136 Xe) [18], and EXO ( 136 Xe) [19] experiments. Further double beta decay experiments, which will contain significantly larger amount of double beta decay radioactive source, are in construction (SuperNEMO) or consideration (nEXO, Legend, etc.) [20]. These facts open new perspectives also for determination g eff A via the measurement of ξ 2ν 13 .

Conclusions
In summary, a more accurate expression for the 2νββdecay half-life was derived by taking into account dependence of energy denominators on lepton energies and using of a Taylor expansion, which allows to factorize calculation of phase-space factors and nuclear matrix elements. It was found that the value of additional contribution to the 2νββ-decay width corresponds to 3-20% of the full width for A=76, 81, 96 and 100 nuclear systems. The revised expression for 2νββ-decay half-life includes in addition to the well-known nuclear matrix element M 2ν GT −1 also the double Gamow-Teller matrix element M 2ν GT −3 with energy denominator to the third power. This matrix element is governed by the transitions through the lowest states of the intermediate nucleus. It was proposed that the effective axial-vector coupling constant g eff A can be determined from the measured 2νββ-decay half-life, if the ratio ξ 2ν 13 of nuclear matrix elements M 2ν GT −3 and M 2ν GT −1 is deduced phenomenologically from the shape of energy spectrum of emitted electrons in the 2νββ-decay and nuclear matrix element M 2ν GT −3 is reliably calculated, e.g., within the interacting shell model. In this way one avoids the problem of the importance of transitions through the higher lying states of the intermediate nucleus.