Correlating the 0νββ-Decay Amplitudes of ¹³⁶Xe with the Ordinary Muon Capture (OMC) Rates of ¹³⁶Ba

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

This paper builds upon the work of ref [32], which compared muon capture matrix elements to double beta decay matrix elements, computed in both the nuclear shell model and in the QRPA.  That paper focused more on the contributions from 1-body and 2-body operators and phenomenological operators, while the current paper focuses on the contributions from Fermi, Gamow-Teller, and Tensor. The authors finds significant correlations/anti-correlations between muon capture and double beta-decay matrix elements, which is useful for benchmarking calculations.

Overall these are interesting and useful results. I have to say, however, this is is really just a slightly different variation on the work of ref [32], and ideally the two papers would have been better combine. This comes close to being "least publishable units." Nonetheless, I overall support publication.

I bring up the following concerns to be addressed before final acceptance.

1. One really has to read ref [32] to understand this paper. That is broadly okay--one should not simply reproduce all the formalism here, but I would suggest expanding section 2.1 slightly.  At the very least they should reference the shell-model interaction used, and to explain that the QRPA calculations use a 'realistic' force (Bonn-A G-matrix) and the g_pp, g_ph renormalize those portions of the force. I originally had the impression they were using the old, schematic force frequently used in QRPA.

2. As far as I can tell, neither the current paper nor the precursor [32] discuss if the QRPA calculations are spherical or deformed.  For example, 136Xe is spherical, or nearly so, but 136Ba is deformed, as seen in their excitation spectra. Indeed, this mismatch between spherical and deformed is common in double-beta decay and is a large driver of reducing the nuclear matrix element. This issue should be clarified and highlighted. This could affect the correlations between muon capture and double-beta decay matrix elements.

Finally, a large number of the citations are to papers by the senior author. He is indeed an expert in the intersection of muon capture, double-beta decay, and QRPA, and when I search on Google scholar a large fraction of them indeed belong to him. Thus, although this certainly could look like excessive self-citation, the choice of references seem broadly appropriate. One relevant paper that could be included might be Simkovic et al, Phys. Rev. C 102, 034301 (2020), which compares experiment vs theory of muon capture rates. 

Author Response

This paper builds upon the work of ref [32], which compared muon capture matrix elements to double beta decay matrix elements, computed in both the nuclear shell model and in the QRPA.  That paper focused more on the contributions from 1-body and 2-body operators and phenomenological operators, while the current paper focuses on the contributions from Fermi, Gamow-Teller, and Tensor. The authors finds significant correlations/anti-correlations between muon capture and double beta-decay matrix elements, which is useful for benchmarking calculations.

Overall these are interesting and useful results. I have to say, however, this is is really just a slightly different variation on the work of ref [32], and ideally the two papers would have been better combine. This comes close to being "least publishable units." Nonetheless, I overall support publication.

Note from the Authors to Referee 1: Authors want to point out that ref. [32] i.e. Gimeno, P.; Jokiniemi, L.; Kotila, J.; Ramalho, M.; Suhonen, J. Ordinary Muon Capture on 136Ba: Comparative Study Using the Shell Model and pnQRPA. Universe 2023, 9. https://doi.org/10.3390/universe9060270 does not include any computations for NDBD NMEs, and solely focuses on OMC rates. In our manuscript, we make fresh NDBD NME computations, where for NSM we have used non-closure approximation for the very first time. This ref. is now ref [33] in the new manuscript.

I bring up the following concerns to be addressed before final acceptance.

1. One really has to read ref [32] to understand this paper. That is broadly okay--one should not simply reproduce all the formalism here, but I would suggest expanding section 2.1 slightly.  At the very least they should reference the shell-model interaction used, and to explain that the QRPA calculations use a 'realistic' force (Bonn-A G-matrix) and the g_pp, g_ph renormalize those portions of the force. I originally had the impression they were using the old, schematic force frequently used in QRPA.

Response: We agree with the point referee makes. The points has been addressed in the manuscript as per referees suggestion. The necessary text has been added in section 2.1

2. As far as I can tell, neither the current paper nor the precursor [32] discuss if the QRPA calculations are spherical or deformed.  For example, 136Xe is spherical, or nearly so, but 136Ba is deformed, as seen in their excitation spectra. Indeed, this mismatch between spherical and deformed is common in double-beta decay and is a large driver of reducing the nuclear matrix element. This issue should be clarified and highlighted. This could affect the correlations between muon capture and double-beta decay matrix elements.

Response: The point raised by the referee is a valid one. We have provided further clarification on the issue and make it clear how our calculations remain valid for the aims of the article. The necessary text has been added as last paragraph in section 2.1.

Finally, a large number of the citations are to papers by the senior author. He is indeed an expert in the intersection of muon capture, double-beta decay, and QRPA, and when I search on Google scholar a large fraction of them indeed belong to him. Thus, although this certainly could look like excessive self-citation, the choice of references seem broadly appropriate. One relevant paper that could be included might be Simkovic et al, Phys. Rev. C 102, 034301 (2020), which compares experiment vs theory of muon capture rates. 

Response: In agreement with the referee, we have now cited Simkovic et al, Phys. Rev. C 102, 034301 (2020) which appears as reference [30]

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

The paper is well written and the hypotheses clearly outlined. The references are also clear with cross-references to previous work, where it is possible to find the results used for comparison. The topic is of fundamental importance since NME evaluations still represent the pricipal uncertainty source in the extraction of parameters of fundamental importance such as the neutrino effective mass (m_ee) in the experimental study of double beta decay without neutrino emission (NDBD). The observed correlations suffer from limited statistical information and therefore should, as indicated by the authors, be expanded so as to understand whether the observed (anti)correlations can indeed be an indication of important information that can be extracted from a process such as OMC involving features more similar (momentum exchanged) to the NDBD case. The conclusions are weak and above all do not in my judgment indicate in any clear way (although one can certainly guess in what direction the authors would like to move) how they intend to possibly use these correlations to improve the quality of NME calculations. While considering the work proto for pblublication I believe that a closer examination of this last part would certainly improve its usefulness and relevance.

Author Response

The paper is well written and the hypotheses clearly outlined. The references are also clear with cross-references to previous work, where it is possible to find the results used for comparison. The topic is of fundamental importance since NME evaluations still represent the pricipal uncertainty source in the extraction of parameters of fundamental importance such as the neutrino effective mass (m_ee) in the experimental study of double beta decay without neutrino emission (NDBD). The observed correlations suffer from limited statistical information and therefore should, as indicated by the authors, be expanded so as to understand whether the observed (anti)correlations can indeed be an indication of important information that can be extracted from a process such as OMC involving features more similar (momentum exchanged) to the NDBD case. The conclusions are weak and above all do not in my judgment indicate in any clear way (although one can certainly guess in what direction the authors would like to move) how they intend to possibly use these correlations to improve the quality of NME calculations. While considering the work proto for publication I believe that a closer examination of this last part would certainly improve its usefulness and relevance.

Response: As per referees' suggestions, we considered expanding the results making the connection OMC rates and NDBD amplitudes more tangible For this we have included new figures that plot the running cumulative percentages of absolute value of 0vbb amplitudes |M(0ν)|  and average OMC "NME" |Mμ|ave  as done in Ref. Phys. Rev. C 102, 024303 (2020)  as functions of the excitation energy of the individual Jπ states below 1 MeV of excitation in 136Cs. This has been done in Fig. 3.

Author Response File: Author Response.pdf