Ab initio neutrinoless double-beta decay matrix elements for 48Ca, 76Ge, and 82Se
A. Belley, C. G. Payne, S. R. Stroberg, T. Miyagi, J. D. Holt
AAb initio neutrinoless double-beta decay matrix elements for Ca, Ge, and Se A. Belley,
1, 2, 3
C. G. Payne,
1, 3, ∗ S. R. Stroberg, T. Miyagi, and J. D. Holt
1, 2 TRIUMF 4004 Wesbrook Mall, Vancouver BC V6T 2A3, Canada Department of Physics, McGill University, 3600 Rue University, Montr´eal, QC H3A 2T8, Canada Department of Physics & Astronomy, University of British Columbia, Vancouver, British Columbia V6T 1Z1, Canada Department of Physics, University of Washington, Seattle, WA 98195, USA
We calculate basis-space converged neutrinoless ββ decay nuclear matrix elements for the lightestcandidates: Ca, Ge and Se. Starting from initial two- and three-nucleon forces, we apply theab initio in-medium similarity renormalization group to construct valence-space Hamiltonians andconsistently transformed ββ -decay operators. We find that the tensor component is non-negligiblein Ge and Se, and resulting nuclear matrix elements are overall 25-45% smaller than thoseobtained from the phenomenological shell model. While a final matrix element with uncertaintiesstill requires substantial developments, this work nevertheless opens a path toward a true first-principles calculation of neutrinoless ββ decay in all nuclei relevant for ongoing large-scale searches. Neutrinoless double-beta (0 νββ ) decay is a hypothe-sized nuclear-weak process in which two neutrons trans-form into two protons by emitting two electrons [1]. Thekey feature of this decay is that it produces two lep-tons (the electrons) without any anti-leptons, thus vi-olating lepton-number conservation. For such a decayto occur, the neutrino must be Majorana, i.e. its ownanti-particle [2, 3]. Furthermore, under standard light-neutrino exchange, the rate of the process can be relatedto the effective neutrino mass (cid:104) m ββ (cid:105) [4]:[ T ν / ] − = G ν | M ν | (cid:18) (cid:104) m ββ (cid:105) m e (cid:19) , (1)where G ν is a phase-space factor whose value is gener-ally agreed upon [5, 6]. Thus an observation could de-termine the absolute neutrino mass, its Majorana/Diraccharacter, and most importantly, provide an observationof lepton-number violation, which would have deep impli-cations for the matter-antimatter asymmetry puzzle [7].From Eq. 1, we see that the rate cannot be directly con-nected to neutrino masses without first having knowledgeof the non-observable nuclear matrix element (NME), M ν , governing the decay. As large-scale searches world-wide will soon enter a ton-scale era probing the invertedneutrino mass hierarchy [8–12], a reliable NME with rig-orous theoretical uncertainty estimates is imperative notonly to pin down m ββ , should a discovery be made, butalso to interpret evolving experimental lifetime limits interms of excluded neutrino mass scales.Calculations of the NME have proven to be tremen-dously challenging for nuclear theory, as they require aconsistent treatment of nuclear and electroweak forces,as well as an accurate solution of the nuclear many-bodyproblem in heavy systems. To date, almost all calcu-lations of 0 νββ decay have been based on uncontrollednuclear models, but since no 0 νββ -decay data exist, un-surprisingly a spread in results (up to factors of three)has emerged [4, 13–17]. This spread is not a true uncer-tainty, however, as all models are known to neglect es- sential physics. Since experimental expectations for ma-terial and timescale requirements are based on the cur-rently available spread, they may need to be reevaluatedshould improved values lie well outside the existing range.Therefore it is crucial to have next-generation NMEsfor the most prominent experimental candidates – Ge,
Te and
Xe – to guide next-generation searches.Chiral effective field theory [18, 19] in principle pro-vides a prescription for the consistent treatment of nu-clear forces and electroweak currents relevant for 0 νββ decay [20–22]. While first calculations have been car-ried out in the lightest nuclei [23, 24], the only calcula-tions of experimental 0 νββ -decay candidates from chiralforces have been in a perturbative shell-model effective-interaction framework [25–28]. While results were en-couraging, order-by-order convergence was unclear. Withthe advent of nonperturbative theories capable of reach-ing at least A = 100 [29–32], the primary bottleneck hasbeen the computational resources needed to obtain con-verged results and the treatment of deformed systems.With ongoing advances in the field, the first ab initiocalculations of 0 νββ -decay are within reach, and indeedvery recently NMEs were reported for Ca in the in-medium generator coordinate method (IM-GCM) [33].In this Letter we extend ab initio calculations of theNMEs to the three lightest ββ -decay nuclei Ca, Geand Se using the valence-space formulation of the inmedium similarity renormalization group (VS-IMSRG)[32, 34–38]. We first demonstrate convergence in termsof the single-particle basis size and truncations imposedon three-nucleon (3N) forces. In contrast to phenomenol-ogy, we find that the tensor operator is non-negligible for Ge and Se, and is approximately the same magnitudeas the Fermi term. As seen in Fig. 1, the overall matrixelements are smaller than standard shell model calcula-tions by approximately 25% in Ca, 30% in Ge, and45% in Se, but in remarkably good agreement with IM-GCM in Ca when starting from the same input forces.The 0 νββ -decay operator under light neutrino ex-change is given by the sum of the allowed Gamow-Teller a r X i v : . [ nu c l - t h ] A ug M Ca E = 20 SMIM-GCM0.50.60.7 M G T M F e max0.200.150.10 M T Ge E = 24 SM1230.30.40.50.6 6 8 10 12 14 e max0.50.40.3 1.01.52.02.53.0 Se E = 24 SMParent ref.Daughter ref.1230.30.40.50.6 6 8 10 12 14 e max0.50.40.3 FIG. 1. NMEs for the 0 νββ -decay of Ca, Ge, and Se as a function of e max , at a fixed E . The bands represent theuncertainty from the choice of reference in the ENO procedure. We also show the convergence of the GT, F, and T operatorsseparately. In addition we compare to phenomenological shell-model (SM) results for each decay and to complementary abinitio IM-GCM values [33] (the blue band) in Ca, which agree remarkably well. (GT), Fermi (F), and tensor (T) transitions [13]: M ν = M νGT − (cid:16) g V g A (cid:17) M νF + M νT (2)where g V = 1 and g A = 1 .
27 are the unquenched vec-tor and axial coupling constants, respectively. Explicitexpressions and details for the matrix elements can befound in Refs. [4, 22, 39–41]. To avoid explicit sums overintermediate states, we use the standard closure approx-imation, with “closure energy” ¯ E ≈ E k − ( E i + E f ) / E changes the NME by less than 1% per MeV [41]. Tofacilitate benchmarking with previous calculations, weused a value ¯ E = 7 .
72 MeV for Ca and ¯ E = 9 . V = 850 MeV andΛ A = 1086 MeV [44], and multiply the NMEs by thenuclear radius R = 1 . A / fm to make them dimen-sionless [4]. The necessity of a leading-order short-rangecontact term has recently been discovered [21, 22]. As-sessment of its importance is currently underway, and wewill discuss the impact in a future publication.We calculate NMEs from two-nucleon (NN) plus 3Nforces inspired by chiral effective field theory. In particu-lar we use 1.8/2.0(EM) from a well-established familyof Hamiltonians [45–47], where 3N couplings are con-strained only by the the biding energy of H and thecharge radii of He. This interaction globally reproduces ground-state energies to the tin isotopes, including theproton and neutron driplines in the light- and medium-mass regions, albeit while giving radii that are systemat-ically too small compared to experiment [31, 47, 48].We begin in a harmonic-oscillator (HO) basis with (cid:126) ω = 16 MeV and e = 2 n + l ≤ e max with a cut of e + e + e ≤ E on 3N matrix elements. We trans-form the Hamiltonian and 0 νββ -decay operator to theHartree-Fock (HF) basis, where we account for 3N forcesbetween valence nucleons via ensemble normal-ordering(ENO) [38]. Previous limitations on E were roughly16 or 18, but with recent advances, we are now ableto routinely calculate with E = 24 or higher [49],putting heavy nuclei well within reach.We then use the Magnus formulation of the IMSRG[35, 50] to derive an approximate unitary transformationto decouple a valence-space Hamiltonian [32, 34, 36], andobtain consistently-transformed operators [51]. We workin the IMSRG(2) approximation in which all operatorsare truncated at the normal-ordered two-body level. Wetake the standard pf -shell valence space for Ca and the p / , p / , f / , g / proton and neutron orbits outside a Ni core for Ge and Se. The valence-space diagonal-ization is done using the KSHELL shell model code [52].Before addressing 0 νββ decay, we must first validateand benchmark in as many relevant electroweak processesas possible. For the longstanding puzzle of g A quenchingin nuclei, which still persists in discussions of 0 νββ de- E x c i t a t i o n E n e r g y ( M e V ) + Exp. + + + + VS-IMSRG + + + -416.00 MeV -416.70 MeV Ca + Exp. + + + + + + + VS-IMSRG + + + + + + -418.70 MeV -419.75 MeV Ti E x c i t a t i o n E n e r g y ( M e V ) + Exp. + + + + VS-IMSRG + + + + -661.60 MeV -672.89 MeV Ge + Exp. + + + VS-IMSRG + + + + -662.07 MeV -676.31 MeV Se FIG. 2. Excitation spectra of Ca/Ti and Ge/Se from theVS-IMSRG using the 1.8/2.0(EM) interaction compared toexperimental values [53, 54]. Certain states have been high-lighted to help guide the comparison. cay, we have recently shown that across a wide range ofnuclei from light to heavy regions, when two-body cur-rents consistent with input Hamiltonians are included inab initio many-body treatments, experimental GT tran-sitions are largely reproduced with no modification of g A [55]. We have also calculated 2 νββ decay withouttwo-body currents and find a preliminary NME of 0.029for Ca, consistent with the experimental value, but anin-depth comparison is currently underway with coupled-cluster theory [56]. Finally we have benchmarked ficti-tious 0 νββ -decay rates in light nuclei with those from no-core shell model, coupled-cluster theory and IM-GCM,generally finding good agreement [57]. Therefore, it ap-pears that the physics expected to be relevant for 0 νββ decay is largely enough under control to begin first ab
10 12 14 16 18 200.50.60.7 M Ca e max = 12 Parent ref.Daughter ref.
10 12 14 16 18 20 22 242.02.5 M Ge e max = 12
10 12 14 16 18 20 22 24 E M Se e max = 12 FIG. 3. Convergence of the NMEs as we vary the size of the3-body storage truncation E at fixed e max . As we see,convergence is obtained at E = 20 in Ca and E =24 for the heavier isotopes, validating the choices in Fig. 1. initio explorations in heavier experimental candidates.In Fig. 2 we show the excitation spectra for both parentand daughter nuclei compared to the experimental valuesfor the Ca and Ge transitions (the spectrum of Seis similar to that of Ge). We see that for the A = 48cases, the computed spectra are in good agreement withexperiment, similar to the IM-GCM [33]. Only the firstexcited state in Ca is several hundred keV too high, butthe IMSRG(2) approximation is known to produce toohigh first excited states in doubly magic systems [47, 58].Otherwise the spectrum of Ti is very well reproduced,implying the collective nature of the nucleus is adequatelycaptured, similar to observations in the sd shell [37]. Forthe heavier cases, however, the computed spectra are toospread compared to experiment, likely due to missingcollectivity. Further benchmarks are underway, but fromIM-GCM studies, only a weak correlation was seen be-tween NMEs and ( E
2) strength [33]. For the A = 48systems, the calculated ground-state energies agree withexperiment to better than 1% and the Q -value to 300keV,while for A = 76 ,
82 the ground states agree to 2% and Q -values to 3 MeV and 4 MeV, respectively.Turning to our 0 νββ -decay results, Fig. 1 shows thecomputed NMEs of Ca, Ge and Se. Here we seeclear convergence by e max = 14 for the total matrix ele-ment as well as the three components of the decay. Sincethe ENO procedure takes a specific nucleus as the ref-erence, we also examine this reference-state dependence.While it is negligible in Ca, there can be changes of upto 10% in the heavier nuclei. We also note that ordering Ca Ge SeHO HF IMSRG HO HF IMSRG HO HF IMSRGGT 0 . . . . . . . . . . . . . . . . . . − . − . − . − . − . − . − . − . − . . . . . . . . . . Ca, Ge, Se into their Gamow-Teller (GT), Fermi (F) and tensor (T) part at e max = 14 and E = 20 for Ca and E = 24 for Ge and Se. For the Fermi part, the factor of − ( g v g a ) as beenincluded. We present the values for the operator in the HO and HF bases with the IMSRG-evolved wavefunctions as well asthe fully evolved IMSRG results (IMSRG). The uncertainty represents the range due to the choice of reference state. of HF single-particle levels can change with increasing e max , which changes the occupations taken for the ENOprocedure, as observed between e max = 6 − Se.The reference-state dependence is expected to be reducedwith the introduction of three-body operators in the VS-IMSRG(3). In Fig. 3 we show convergence with E .While Ca is well converged to better than 0.01 in theoverall matrix element by E = 16, perhaps some-what unexpectedly E = 20 is necessary to achievethe same level of convergence in both Ge and Se.Taking a more detailed look at the NME values, werefer to Table I, where we break down the GT, F, and Tcomponents for the unevolved 0 νββ -decay operator inboth the HO and HF bases with (albeit inconsistent)VS-IMSRG wavefunctions, as well as the final IMSRG-transformed operator consistent with the wavefunctions.In Ca, we find that the tensor part of the NME, whichhas been largely neglected in the past [59] or found to benegligible by phenomenological methods [60], accountsfor 20% of the total matrix element, a modest increasefrom its contribution in the HO and HF frames.For Ge and Se we observe very similar patterns.In previous phenomenological studies, the tensor compo-nent is taken or shown to be negligible, which is whatwe find for HO and HF pictures. However, the IMSRGrenormalization induces a significant tensor component,reducing the value of the total NME by roughly 15-20%.While the F part is largely unaffected in all cases, there isalso a significant reduction in the GT component. Takentogether these effects reduce the NME by a factor of morethan two, and thus we see that correlations do not alwayshave a consistent effect for different transitions.The fact that the evolution of the two-body operatorleads to such a significant change in the final NMEs high-lights the need to investigate the effects of three-bodyoperators. Indeed, we expect the contribution of many-body operators to diminish as we increase particle num-ber, but since there is no one-body term to compare, dueto the use of the closure energy approximation,estimatingthe magnitude of three-body terms is crucial to ensurethat the two-body term is dominant. Therefore, beforeclaiming final results for the NME, we must first assessthe importance of three-body terms in IMSRG(3). Comparing our NMEs to those from the phenomeno-logical shell model, we see an overall reduction: 25% in Ca, 30% in Ge, and 45% in Se, making the NMEspresented here among the smallest ever reported for thesethree nuclei. This appears to be an emerging picture fromcomplementary ab initio theories. Starting from the same1.8/2.0(EM) interaction, and employing the same IM-SRG(2) approximation, our NME for Ca is completelyconsistent with the IM-GCM findings in Ref. [33] (seen inFig. 1), as well as upcoming results from coupled clustertheory [61]. Furthermore the NME for Ge again ap-pears to be consistent with preliminary IM-GCM resultsat the same level of many-body approximation [62].In conclusion, we have computed 0 νββ -decay NMEsfor Ca, Ge and Se, finding convergence by e max =14 and E = 20 with overall smaller values comparedto the phenomenological shell model by 25-45%. While Ca is not a primary experimental candidate, its rel-atively light mass and doubly magic nature make it avaluable benchmark for various ab initio theories goingforward. With the VS-IMSRG advances presented herewe have now provided ab-initio NME computations ofthe first of three major players in experimental searches: Ge. With capabilities to perform calculations at high E , we are already poised to provide NMEs for Teand
Xe at the same level as in this work.Significant work remains to be done assessing all rele-vant sources of theoretical uncertainty before any claimsto a final NME can be made. We must first study a widerange of input NN+3N forces and corresponding currentsto establish an interaction uncertainty, including an esti-mation of the leading order contact [21]. We have imple-mented consistent free-space SRG evolution of the 0 νββ -decay operator and are currently investigating the possi-ble importance of induced three-body terms [49]. Finallydevelopment of the IMSRG(3) is underway, which willprovide handle on many-body uncertainties and bringthe IMSRG to the analogous level of approximation as incoupled-cluster theory. Only once this has been accom-plished and complementary many-body approaches caneach produce independent predictions with uncertaintyestimates, can the field give a firm statement on NMEsfor experimental 0 νββ searches.We thank J. Engel, G. Hagen, H. Hergert, M. Horoi,B. Hu, J. Men´endez, P. Navr´atil, S. Novario, T. Papen-brock, N. Shimizu, and J. M. Yao for enlightening dis-cussions and extensive benchmarking, and K. Hebeler forproviding momentum-space inputs for generation of the3N forces used in this work. The IMSRG code used inthis work makes use of the Armadillo
C++ library [63].TRIUMF receives funding via a contribution through theNational Research Council of Canada. This work wasfurther supported by NSERC, the Arthur B. McDon-ald Canadian Astroparticle Physics Research Institute,the Canadian Institute for Nuclear Physics, and the USDepartment of Energy (DOE) under contract DE-FG02-97ER41014. Computations were performed with an allo-cation of computing resources on Cedar at WestGrid andCompute Canada, and on the Oak Cluster at TRIUMFmanaged by the University of British Columbia depart-ment of Advanced Research Computing (ARC). ∗ Present address: Institut f¨ur Kernphysik and PRISMA + Cluster of Excellence, Johannes Gutenberg-Universit¨atat Mainz, 55128 Mainz, Germany[1] F. T. Avignone, S. R. Elliott, and J. Engel, Rev. Mod.Phys. , 481 (2008).[2] W. H. Furry, Phys. Rev. , 56 (1938).[3] J. Schechter and J. W. F. Valle, Phys. Rev. D , 774(1982).[4] J. Engel and J. Men´endez, Rep. Prog. Phys. , 046301(2017).[5] J. Kotila and F. Iachello, Phys. Rev. C , 034316 (2012).[6] J. Suhonen and O. Civitarese, Phys. Rep. , 123(1998).[7] M. Fukugita and T. Yanagida, Phys. Lett. B174 , 45(1986).[8] S. A. Kharusi et al. (nEXO), (2018), arXiv:1805.11142[physics.ins-det].[9] J. Myslik (LEGEND) (2018) arXiv:1812.08191[physics.ins-det].[10] J. Paton (SNO+), in
Prospects in Neutrino Physics (Nu-Phys2018) London, United Kingdom, December 19-21,2018 (2019) arXiv:1904.01418 [hep-ex].[11] Y. Gando (KamLAND-Zen), in
Prospects in NeutrinoPhysics (NuPhys2018) London, United Kingdom, De-cember 19-21, 2018 (2019) arXiv:1904.06655 [physics.ins-det].[12] R. Arnold et al. , EPJ C , 927 (2010).[13] J. Men´endez, A. Poves, E. Caurier, and F. Nowacki,Nucl. Phys. A , 139 (2009).[14] F. Simkovic, V. Rodin, A. Faessler, and P. Vogel, Phys.Rev. C , 045501 (2013).[15] R. A. Sen’kov and M. Horoi, Phys. Rev. C , 051301(2014).[16] N. L´opez Vaquero, T. R. Rodr´ıguez, and J. L. Egido,Phys. Rev. Lett. , 142501 (2013).[17] J. Barea, J. Kotila, and F. Iachello, Phys. Rev. C ,034304 (2015).[18] E. Epelbaum, H.-W. Hammer, and U.-G. Meißner, Rev.Mod. Phys. , 1773 (2009). [19] R. Machleidt and D. R. Entem, Phys. Rep. , 1 (2011).[20] J. Men´endez, D. Gazit, and A. Schwenk, Phys. Rev.Lett. , 062501 (2011).[21] V. Cirigliano, W. Dekens, J. De Vries, M. L. Graesser,E. Mereghetti, S. Pastore, and U. Van Kolck, Phys. Rev.Lett. , 202001 (2018).[22] V. Cirigliano, W. Dekens, E. Mereghetti, and A. Walker-Loud, Phys. Rev. C , 065501 (2018).[23] V. Cirigliano, W. Dekens, J. De Vries, M. Graesser,E. Mereghetti, S. Pastore, M. Piarulli, U. Van Kolck,and R. Wiringa, Phys. Rev. C , 055504 (2019).[24] X. B. Wang, A. C. Hayes, J. Carlson, G. X. Dong,E. Mereghetti, S. Pastore, and R. B. Wiringa, Phys.Lett. B , 134974 (2019).[25] J. D. Holt and J. Engel, Phys. Rev. C , 064315 (2013).[26] A. Kwiatkowski et al. , Phys. Rev. C , 045502 (2014).[27] C. F. Jiao, J. Engel, and J. D. Holt, Phys. Rev. C ,054310 (2017).[28] L. Coraggio, A. Gargano, N. Itaco, R. Mancino, andF. Nowacki, Phys. Rev. C , 044315 (2020).[29] G. Hagen, T. Papenbrock, M. Hjorth-Jensen, and D. J.Dean, Rep. Prog. Phys. , 096302 (2014).[30] H. Hergert, S. K. Bogner, T. D. Morris, A. Schwenk, andK. Tsukiyama, Phys. Rept. , 165 (2016).[31] T. D. Morris, J. Simonis, S. R. Stroberg, C. Stumpf,G. Hagen, J. D. Holt, G. R. Jansen, T. Papenbrock,R. Roth, and A. Schwenk, Phys. Rev. Lett. , 152503(2018).[32] S. R. Stroberg, S. K. Bogner, H. Hergert, and J. D. Holt,Ann. Rev. Nucl. Part. Sci. , 307 (2019).[33] J. M. Yao, B. Bally, J. Engel, R. Wirth, T. R. Rodrguez,and H. Hergert, Phys. Rev. Lett. , 232501 (2020).[34] K. Tsukiyama, S. K. Bogner, and A. Schwenk, Phys.Rev. C , 061304 (2012).[35] H. Hergert, S. K. Bogner, T. D. Morris, A. Schwenk, andK. Tsukiyama, Phys. Rep. , 165 (2016).[36] S. K. Bogner, H. Hergert, J. D. Holt, A. Schwenk,S. Binder, A. Calci, J. Langhammer, and R. Roth, Phys.Rev. Lett. , 142501 (2014).[37] S. R. Stroberg, H. Hergert, J. D. Holt, S. K. Bogner, andA. Schwenk, Phys. Rev. C , 051301(R) (2016).[38] S. R. Stroberg, A. Calci, H. Hergert, J. D. Holt, S. K.Bogner, R. Roth, and A. Schwenk, Phys. Rev. Lett. ,032502 (2017).[39] M. Doi, T. Kotani, and E. Takasugi, PTP Supplement , 1 (1985).[40] T. Tomoda, Rep. Prog. Phys. , 53 (1991).[41] M. Horoi and S. Stoica, Phys. Rev. C , 024321 (2010).[42] R. A. Sen’kov and M. Horoi, Phys. Rev. C , 064312(2013).[43] W. Haxton, Prog. Part. Nucl. Phys. , 409 (1984).[44] F. ˇSimkovic, A. Faessler, H. M¨uther, V. Rodin, andM. Stauf, Phys. Rev. C , 055501 (2009).[45] K. Hebeler, S. K. Bogner, R. J. Furnstahl, A. Nogga, andA. Schwenk, Phys. Rev. C , 031301 (2011).[46] J. Simonis, K. Hebeler, J. D. Holt, J. Men´endez, andA. Schwenk, Phys. Rev. C , 011302 (2016).[47] J. Simonis, S. R. Stroberg, K. Hebeler, J. D. Holt, andA. Schwenk, Phys. Rev. C , 014303 (2017).[48] J. D. Holt, S. Stroberg, A. Schwenk, and J. Simonis,(2019), arXiv:1905.10475 [nucl-th].[49] T. Miyagi et al. , in preparation.[50] T. D. Morris, N. M. Parzuchowski, and S. K. Bogner,Phys. Rev. C , 034331 (2015). [51] N. M. Parzuchowski, S. R. Stroberg, P. Navr´atil, H. Herg-ert, and S. K. Bogner, Phys. Rev. C , 034324 (2017).[52] N. Shimizu, T. Mizusaki, Y. Utsuno, and Y. Tsunoda,Comput. Phys. Commun. , 372 (2019).[53] M. Wang, G. Audi, F. G. Kondev, W. Huang, S. Naimi,and X. Xu, Chin. Phys. C , 030003 (2017).[54] M. Wang, G. Audi, A. H. Wapstra, F. G. Kondev,M. MacCormick, X. Xu, and B. Pfeiffer, Chin. Phys.C , 1603 (2012).[55] P. Gysbers et al. , Nature Phys. , 428 (2019). [56] A. Belley, S. Novario, et al. , in preparation ().[57] A. Belley, J. M. Yao, et al. , in preparation ().[58] R. Taniuchi et al. , Nature , 53 (2019).[59] M. Kortelainen and J. Suhonen, Phys. Rev. C , 051303(2007).[60] R. A. Sen’kov and M. Horoi, Phys. Rev. C93