Nuclear responses for double beta decay and muon capture

. The existence of the neutrinoless double beta (0 νββ ) decay is one of the most intriguing open questions in the neutrino physics ﬁeld. Despite many large-scale experiments have aimed to measure the reaction for decades, it has not yet been observed. Therefore, accurate theoretical calculations on 0 νββ are crucial. To describe the double beta decay processes reliably one needs a possibility to test the involved virtual transitions against experimental data. In this work we manifest how to utilise the charge-exchange and ordinary muon capture (OMC) data in the study of 0 νββ decay


INTRODUCTION
The neutrinoless double beta (0νββ) decay of atomic nuclei is a lepton number violating process that has not been observed, despite lots of effort has been directed to detecting it (see References [1,2,3]).Since 0νββ decay is challenging to study both experimentally and theoretically, we need some complementary tests in order to accurately calculate the involved nuclear matrix elements and to design the large-scale experiments.
ββ decays take place between two even-even nuclei of the isobaric chain trough virtual states of the intermediate odd-odd nucleus.0νββ-decay runs trough all possible multipolarities J π of the intermediate nucleus.These intermediate J π states of 0νββ decay can be studied by utilising the corresponding β − (β + ) transitions of the mother(daughter) nucleus, which correspond to the left(right)-branch virtual transitions of the 0νββ decay.
Another interesting tool to probe 0νββ decay is the ordinary muon capture (OMC).By studying OMC one can probe the right-branch of the virtual transitions of the 0νββ decay.The involved large momentum transfer, q ≈ 50−100 MeV, corresponds to the one of 0νββ decay, which makes it a promising tool to probe 0νββ decay.Furthermore, due to the large mass of the muon, the OMC can populate final nuclear states that are both highly excited and of high multipolarity J π , quite like the intermediate virtual states of 0νββ decay.

CHARGE-EXCHANGE REACTIONS AS A PROBE
In [4,5] the energetics and strength distributions of isovector spin-dipole transitions (IVSD), corresponding to the left-branch virtual transitions of the 0νββ decay, are studied using pnQRPA theory with large no-core single-particle bases.The particle-hole parameter g ph is fitted to reproduce the data on isovector spin-dipole (IVSD) J π = 2 − giant resonances.Traditionally g ph has been fitted to Gamow-Teller (GT) giant resonances.We refer to the differently fitted parameter values as g ph (SD2 − ) and g ph (GT), respectively.
The nuclear matrix elements (NMEs) of 0νββ decay are computed using three different models: In Model 1 g ph (J π ) = g ph (GT) for all J π , in Model 2 g ph (J π ) = g ph (GT) for J π 2 − and g ph (2 − ) = g ph (SD2 − ), and in Model 3 g ph (J π ) = g ph (SD2 − ) for J π 1 + and g ph (1 + ) = g ph (GT).We also compare the obtained results with the earlier study of Hyvärinen et al. [6], where the 0νββ NMEs were computed in much smaller single-particle bases without access to the isovector spin-dipole data.The results are presented in Table 1.As we can see, most of the deviations are due to the extension of the single-particle space of pnQRPA, while the effect of adjusting the particle-hole interaction to data on spin-dipole resonances is relatively smaller.[6], where smaller single-particle bases were used.

ORDINARY MUON CAPTURE AS A PROBE
Ordinary muon capture (OMC) is a weak interaction process quite like electron capture, the main difference being the large mass of the captured muon, which is about 200 times the electron mass.The OMC process we are interested in can be written as where the muon (µ − ) is captured by the 0 + ground state of the even-even nucleus X of mass number A and atomic number Z leading to the J π states of its odd-odd isobar Y of atomic number Z − 1.At the same time a muon neutrino ν µ is emitted.The energy release is about 100 MeV, of which the largest fraction is donated to the released neutrino, being the lightest object participating in the process.The involved large momentum exchange, q ≈ 50 − 100 MeV, allows highly forbidden transitions as well as highly excited final states with high multipolarities J π , which makes it a good probe for 0νββ decay.

Muon Capture Formalism
The ordinary muon capture rates are based on Morita-Fujii formalism [7].The partial muon capture rate to a J π final state can be written as where A denotes the mass number of the initial and final nuclei, J i (J f ) the angular momentum of the initial (final) nucleus, M the average nucleon rest mass, m µ the bound muon mass, m µ the muon reduced mass in the parent µmesonic atom, Z the atomic number of the initial nucleus, α the fine-structure constant and q the Q value of the OMC [7].For the heavy nuclei the atomic orbit of the muon penetrates the nucleus and the capture rate has to be corrected for the muonic screening.Here we follow the Primakoff procedure [8] where the capture rate has been corrected by the factor (Z eff /Z) 4 , where the effective atomic number is obtained from the work of Ford and Wills [9].The term P in Eq. ( 2) has a complex structure containing all the nuclear matrix elements as well as weak couplings, and some geometric factors and Racah coefficients.The exact form can be found in [7].P can be expanded in terms of a small quantity 1/M 2 as P = P 0 + P 1 , where P 0 is obtained by neglecting all terms containing 1/M 2 (except for terms containing g 2 P , which is large compared with the other coupling constants), and P 1 includes all terms of the order 1/M 2 .The leading order term P 0 is the explicit form that can be found in [7].The next-to-leading-order term  P 1 is sometimes important for weak OMC transitions usually to high-lying states.For that reason we extended the original Morita-Fujii formalism by including the P 1 term in the calculations [10].

Muon Capture Rate Distribution in 100 Nb
For the first time, OMC giant resonance was observed in 100 Nb [11].Inspired by the observation, we computed the muon capture rate distribution in 100 Nb in pnQRPA formalism with large no-core single-particle basis and compared the obtained spectrum with the experimental one [12].Both the experimental OMC spectrum and the pnQRPAcomputed one show a giant resonance at around 10-12.5 MeV, and tails at higher energies (see Fig. 1).
However, the obtained total capture rate value W tot = 17.7 × 10 6 1/s obtained using parameters g A = 0.8 and g P = 10 is a lot larger than the corresponding Primakoff estimate W Prim. = 7.7 × 10 6 (see Eq. (4.53) of the review article [13]).This suggests a strongly quenched axial vector coupling constant g A ≈ 0.5.

Muon Capture Rate Distributions on the Daughter Nuclei of ββ Decay Triplets
The ordinary muon captures on the daughter nuclei, 76 Se, 82 Kr, 96 Mo, 100 Ru, 116 Sn, 128 Xe, 130 Xe and 136 Ba, of the key double beta decay triplets leading to the excited states of the corresponding intermediate nuclei, were computed in the pnQRPA framework using large no-core single particle bases as in the case of 100 Mo.The corresponding OMC The low-energy part (E < 1.1 MeV) of the computed spectrum for the transitions 76 Se(0 + g.s.)+µ − → 76 As(J π )+ν µ can be compared with measured rates deduced from the recent results of Zinatulina et al. [14].The capture rates to each J π multipole below 1.1 MeV are summed, and the obtained experimental and pnQRPA-computed values are presented in Table 2 (for details, see Ref. [10]).In the pnQRPA calculations we used the parameter values g A = 0.8 and g P = 7.0.The obtained capture rates are generally surprisingly close to each other, but pnQRPA seems to underestimate the capture rates for transitions to 0 + states.

CONCLUSIONS
Neutrinoless double beta decay has not yet been measured despite a lot of effort has been directed to observing it.Thus, the 0νββ calculations need some complementary tests in order to reliably describe the intermediate states, and finally to probe the half-lives of 0νββ decays.
Both charge-exchange reactions and ordinary muon capture serve as useful detours to study the intermediate states of 0νββ decay, and we have studied how to utilise those reactions in the study of 0νββ decay.
By extending the experiments and calculations on OMC to other 0νββ-decaying nuclei we could shed light on the effective values of the axial-vector (g A ) and induced pseudoscalar (g P ) couplings, and the NMEs related to 0νββ decay and astro-(anti)neutrino interactions.

FIGURE 1 .FIGURE 2 .
FIGURE 1.Comparison of experimental and theoretical relative muon capture rate distributions in 100 Nb.

TABLE 1 .
The 0νββ nuclear matrix elements for different transitions computed using the different g ph models.The table has been cut in two and appears as left and right halves.
*The result is obtained from