Second-forbidden nonunique β^- decays of ^{59,60}Fe: Possible candidates for g_A sensitive electron spectral-shape measurements
aa r X i v : . [ nu c l - t h ] J a n Shell-model description of the electron spectra for the second-forbidden nonunique β − decays of the f p -shell nuclei Sc and , Fe Anil Kumar ∗ , Praveen C. Srivastava † , and Jouni Suhonen ‡ Department of Physics, Indian Institute of Technology Roorkee, Roorkee 247667, India University of Jyvaskyla, Department of Physics, P.O. Box 35 (YFL), FI-40014,University of Jyvaskyla, Finland (Dated: January 11, 2021)In the present work, we have computed the shape factors and electron spectra for the second-forbidden nonunique β − -decay transitions of Sc, and , Fe in the framework of the nuclear shellmodel. We have performed the shell-model calculations of all the involved wave functions in the β -decay rate by using the KB3G and GXPF1A interactions in the full fp model space. Whencompared with the available data, these effective interactions predict the low-energy spectra andelectromagnetic properties of the involved nuclei quite successfully. This success paves the way fora reliable computation of the β -decay properties, and comparison with data. We use the spectrum-shape method (SSM), including the next-to-leading-order corrections in the shape factor, in orderto compute the electron spectral shapes as functions of the weak axial coupling constant g A . Wehave also constrained the value of the relativistic vector matrix element, V M (0) KK − , using theconserved vector-current hypothesis (CVC) and found that this procedure influences the electronspectral shapes. Based on the “CVC-inspired” SSM calculations we find that the spectral shapesof , Fe depend strongly on the value of g A , thus making these nuclei as perfect test cases of therevised SSM for future β -decay experiments that are able to resolve electron spectral shapes. PACS numbers: 21.60.Cs - shell model, 23.40.-s - β -decay I. INTRODUCTION
The nuclear β decay can be considered as a mutual in-teraction between the hadronic and leptonic current me-diated by a massive W ± vector bosons. These currentscan be expressed as mixtures of the vector and axial-vector contributions. The values of the weak couplingconstants enter the theory of β -decay when the hadroniccurrent is renormalized at the nucleon level [1]. The free-nucleon value of the vector coupling constant g V = 1 . g A = 1 .
27 yield fromthe conserved vector-current (CVC) hypothesis and thepartially conserved axial-vector-current (PCAC) hypoth-esis, respectively [2]. The value of g A is affected insidenuclear matter by nuclear many-body, delta-nucleon andmesonic correlations. These corrections to the bare valueof g A can be represented as g effA = qg A , where q is aquenching factor. The quenched or enhanced value of g A plays an important role when data on astrophysicalprocesses, single beta decays and double beta decays areto be reproduced by nuclear many-body calculations. Inthe single β decays, the decay rate depends on the sec-ond power of g A , while to the forth power for ββ decays[3, 4]. A comprehensive review of the g A problem in β and ββ decays is reported in Ref. [5]. A recent reviewof the theoretical and experimental status for the singleand double β decay is given in [6]. ∗ [email protected] † Corresponding author: [email protected] ‡ jouni.t.suhonen@jyu.fi Different methods have been used to extract the infor-mation on the effective value of g A . One possibility is thehalf-life method where the computed and experimental(partial) β -decay half-lives are matched by varying thevalue of g A . This method was used for the allowed, for-bidden, and two-neutrino double β decays in the frame-work of proton-neutron quasiparticle random-phase ap-proximation (pnQRPA) [7–14], the nuclear shell model(NSM) [15–22], and the interacting boson model (IBM)[23–25]. All these studies have shown that a quenchedvalue of g A is needed to reproduced the experimental ob-servations.In Ref. [26], for the first time, another method todetermine the effective value of g A was introduced; thismethod was coined the spectrum-shape method (SSM).In the SSM, the shapes of the computed electron spectraare compared with the experimental one, in order to findthe effective g A for which the computed spectral shapematches the experimental one. This method is applica-ble to forbidden nonunique β decays since the associatedelectron spectra depend on the details of nuclear struc-ture. In Ref [26] also the next-to-leading-order (NLO)corrections were included in the shape factor.In Ref. [26] the shape of electron spectrum forthe forth-forbidden nonunique β − decay of Cd wascomputed under the framework of the microscopicquasiparticle-phonon model (MQPM) and NSM. Thiswork was extended in Ref. [27] to include a compari-son with the results of the third nuclear model, IBM. In[27] the computed spectral shapes were compared withthe measured one of Belli et al. [28], and the closestmatch was found for the ratio of g A /g V ≈ .
92 for allthree nuclear models.In continuation, electron spectral shapes for several ex-perimentally interesting odd- A nuclei (MQPM and NSMcalculations) and even- A nuclei (NSM calculations) werestudied for their g A dependence in Refs. [29, 30]. In Refs.[26, 29, 30] it was found that the spectral shapes for the β decays of Rb, Nb, Tc, Tc,
Cd, and
In de-pend strongly on the effective value of g A , thus renderingthese decays as excellent candidates for applications ofthe SSM.Recently, in Ref. [31] the electron spectral shapes ofthe second-forbidden nonunique β − decays of Na and Cl were studied. It was found that the potential of theSSM could be enhanced by constraining the magnitudeof the small relativistic vector matrix element V M (0)211 bythe CVC (conserved vector-current hypothesis) and themeasured decay half-life. This then leads to a consistenttreatment of both the half-life and the spectral shape.This improved approach was also used for the second-forbidden nonunique β − decay of F [32, 33], and in thiswork we will resort to the same CVC-based enhancementof the SSM.In this paper, our aim is to find new candidates for theapplication of the SSM in the f p shell with well estab-lished Hamiltonians. In this region of the nuclear chartwe found the possible candidate nuclei, Sc and , Fe,for the second-forbidden nonunique β − decay. A sum-mary of the relevant experimental information for thesethree studied transitions are given in Table I. For thispurpose, we have studied the electron spectral shapes forthe second-forbidden nonunique β − decays of Sc, and , Fe by using the nuclear shell model with two-bodyinteractions GXPF1A [34, 35] and KB3G [36], suitablefor full pf -shell calculations. In the present shell-modelcalculations, the magnitude of the relativistic vector ma-trix element is constrained to reproduce the experimentalpartial half-life. This is a modification of the CVC-based(conserved vector-current hypothesis) method of deter-mining the value of this small matrix element from alarger, more stable vector matrix element, as advocatedin [37]. Here we have tested the role of this matrix el-ement in the context of the electron spectral shapes ofthe studied transitions. We find that constraining thevalue of this key matrix element in this way influencesthe electron spectral shapes. TABLE I: Experimental information on the studied transi-tions in the fp shell. All the experimental data are takenfrom [38].Transitions Q (MeV) BR (%) t / (yr) Sc(4 + ) → Ti(2 + ) 1.477 3.6 (7) × − × Fe(3 / − ) → Co(7 / − ) 1.565 0.18 (4) 67.72 Fe(0 + ) → Co(2 + ) 0.178 100 2.62 × To test the predictive power of our computed nuclearwave functions, we have calculated the energy spectra forlow-lying states as well as other spectroscopic properties of both the parent and daughter nuclei involved in thestudied β -decay transitions. All these computed quanti-ties turn out to be in good agreement with the availabledata. After all these comparisons we finally compute theelectron spectral shapes as functions of g A . It turns outthat some of these spectral shapes are strongly depen-dent on the value of g A , thus making the revised SSMa powerful probe for these cases, and a strong incentiveto measure these shapes in the future. It is importantto mention here that the results of Fe electron spectracorresponding to GXPF1A interaction are previously re-ported in Ref. [39]. However, there the calculations weredone in a very restricted model space and the drawn con-clusion about the sensitivity to the SSM analysis differsfrom the conclusions of the present calculations. Further-more, for comparison, we have also report here resultsobtained with the KB3G interaction.The current article is organized as follows. In Sec. IIwe give a short overview of theoretical formalism for β -decay. Results and discussions are reported in Sec. III.Finally, in Sec. IV we conclude. II. THEORETICAL FORMALISM
In Sec. II A, we discuss the theory of forbiddennonunique β − decays, and the shape of the electron spec-tra. In Sec. II B we give the details about the valencespace and effective Hamiltonian used in the present work. A. β -decay theory In the literature, the theoretical framework of the β decay is well established in the book by Behrens andB¨uhring [37] (see also Ref. [40]). We have used thestreamlined version of the formalism for the forbiddennonunique β − -decay theory from Refs. [27, 41]. To sim-plify the β − -decay theory, we have used the impulse ap-proximation, in which at the exact moment of the decay,only the decaying nucleon feels the weak interaction andthe strong interactions with the remaining A − β − -decay process, described asan effective point-like interaction vertex with an effectivecoupling constant G F , called Fermi coupling constant,the probability of the emitted electron to have a kineticenergy between W e and W e + dW e is given by P ( W e ) dW e = G ( ~ c ) π ~ C ( W e ) p e cW e ( W − W e ) × F ( Z, W e ) dW e , (1)where W is the endpoint energy of the β spectrum, thefactor F ( Z, W e ) is the Fermi function, and Z is the pro-ton number of the daughter nucleus. The p e and W e are the momentum and energy of the emitted electron,respectively. Furthermore, the shape factor C ( W e ) con-tains the nuclear-structure information.The partial half-life of the decay process can be writtenas t / = ln(2) R W m e c P ( W e ) dW e = κ ˜ C , (2)where m e is the rest mass of the electron, ˜ C is the di-mensionless integrated shape function, and the updated[43] value of constant κ is κ = 2 π ~ ln(2) m e c ( G F Cos θ C ) = 6289 s , (3)where the θ C is the Cabibbo angle. To simplify theformalism it is usual to adopt the unitless scaled kine-matics quantities w = W /m e c , w e = W e /m e c , and p = p e c/m e c = p ( w e − C = Z w C ( w e ) pw e ( w − w e ) F ( Z, w e ) dw e . (4)The general form of the shape factor C ( w e ) can beexpressed as C ( w e ) = X k e ,k ν ,K λ k e h M K ( k e , k ν ) + m K ( k e , k ν ) − γ k e k e w e M K ( k e , k ν ) m K ( k e , k ν ) i , (5)where the k e and k ν (both are running through 1, 2, 3,...)are the positive integers emerging from the partial-waveexpansion of the leptonic wave functions, and the K is theorder of the forbiddenness. The quantities M K ( k e , k ν )and m K ( k e , k ν ) contain all the nuclear-structure infor-mation in the form of different nuclear matrix elements(NMEs) and leptonic phase-space factors. More infor-mation on these expressions can be found in Ref. [37](also given in Ref. [27]). Here γ k e = p k e − ( αZ ) and the quantity λ k e = F k e − ( Z, w e ) /F ( Z, w e ) is theCoulomb function, where F k e − ( Z, w e ) is the generalizedFermi function [27, 41].The NMEs contains all the nuclear-structure informa-tion in the form V/A M ( N ) KLS ( pn )( k e , m, n, ρ )= √ π b J i X pn V/A m ( N ) KLS ( pn )( k e , m, n, ρ )(Ψ f || [ c † p ˜ c n ] K || Ψ i ) , (6)where the quantities V/A m ( N ) KLS ( pn )( k e , m, n, ρ ) are calledthe single-particle matrix elements (SPMEs), which char-acterizes the properties of the transition operators, and they are the same for all the nuclear models. Inour calculations, the SPMEs are computed from theharmonic-oscillator wave functions [27, 41]. The quan-tities (Ψ f || [ c † p ˜ c n ] K || Ψ i ) are the one-body transition den-sities (OBTDs) between the initial (Ψ i ) and final (Ψ f )states. The OBTDs contain the nuclear-structure infor-mation and they must be evaluated separately for eachnuclear model. The summation runs over the proton( p ) and neutron ( n ) single-particle states and the “hat-notation” reads b J i = √ J i + 1.The shape factor depends on the weak coupling con-stants g V and g A . So the shape factor can be decomposedinto vector, axial-vector and mixed vector-axial-vectorcomponents [26, 27, 29, 30] based on the weak couplingconstants they contain. In this spirit we can write C ( w e ) = g C V ( w e ) + g C A ( w e ) + g V g A C VA ( w e ) . (7)After integrating Eq. (7) with respect to electron ki-netic energy, we obtain a decomposition of the dimen-sionless integrated shape function (4) in the form.˜ C = g ˜ C V + g ˜ C A + g V g A ˜ C VA . (8)The shape factors C i in Eq. (7) depend on the electronkinetic energy, while after the integration the shape func-tions ˜ C i in Eq. (8) are just constant numbers. B. Adopted model space and Hamiltonians
For the calculations of the OBTDs, needed for the eval-uation of the NMEs contained in the β -decay amplitudes,we need to choose a nuclear model. In the present work,the wave functions of the initial and final states werecomputed by using the nuclear shell model (NSM). Theshell-model wave functions and OBTDs were computedusing the nuclear shell-model code NuShellX@MSU [44]with the well-known effective interactions KB3G andGXPF1A in the full f p model space. In these calcu-lations, we have not made any truncations in terms ofthe included many-body configurations. III. RESULTS AND DISCUSSIONS
In this section we present our computed results oflow-lying energy spectra, spectroscopic properties, shapefactors and electron spectra for the second-forbiddennonunique β − -decay transitions Sc(4 + ) → Ti(2 + ), Fe(3 / − ) → Co(7 / − ), and Fe(0 + ) → Co(2 + ).A computed electron spectrum for the decay of Sc isnot available in the literature, while for Fe there ex-ists a spectral shape computed in Ref. [30] by the NSM,but in a truncated model space. In our calculations, wehave used the full f p model space with well-establishedinteractions. As in the works [26, 27, 29, 30, 39], we haveincluded the next-to-leading-order (NLO) corrections tothe shape factor in the present calculations. In this way,the number of NMEs increases drastically. In the case ofthe second-forbidden nonunique β − decay, the number ofNMEs is increased from 8 to 27 (see the full details aboutNLO in Refs. [26, 27]).Below we present low-lying energy spectra (Figs. 1, 2,and 3), spectroscopic properties (Table II and III), thecomputed NMEs (Table IV and Table V), electron spec-tral shapes as function of electron kinetic energy (Figs.5-7) and decomposition of the integrated shape function(Table VI). A. Low-lying energy spectra and spectroscopicproperties
We performed shell-model calculations for the groundstate (g.s.) and a few excited states of the β − -decayparent and daughter nuclei of studied transitions by usingthe KB3G and GXPF1A interactions in full f p modelspace.In Fig. 1, we show the low-lying energy spectra of Sc and Ti. The experimental ground states are cor-rectly reproduced by both interactions. In the case of Sc, the ordering of the low-energy states up to 7 + isexactly reproduced by both interactions. In general, thecomputed low-energy states are in good agreement withexperimental data. The computed low-lying energy spec-tra of Fe and Co are presented in Fig. 2 and com-pared with the available experimental data. As seen inFig. 2, the computed g.s. are correctly reproduced bythe KB3G and GXPF1A interactions for Fe and Co.Also the computed excited states are in the right energyregions, though some inversions in the relative orderingof the states occur for both interactions. In Fig. 3, weshow the low-energy spectra of Fe and Co obtainedfrom the use of KB3G and GXPF1A interactions in com-parison with the experimental data. As seen in Fig. 3,the computed first 2 + state for Fe is obtained at 0.788and 0.817 MeV corresponding to KB3G and GXPF1Ainteractions, respectively, while the experimental valueis 0.823 MeV. For Co, the KB3G and GXPF1A inter-actions give 2 + as the g.s. while the experimental g.s.is 5 + . From the KB3G calculation, the first excited 5 + state is obtained at 0.034 MeV while the GXPF1A call-culation places it at 0.162 MeV. For both Fe and Cothe computed spectra contain the spin-parities of the ex-perimental spectra in roughly the right energy ranges,but sometimes in inverted orderings.The computed results of quadrupole and magnetic mo-ments are shown in Table II and compared with availableexperimental data. In most cases, both moments are wellreproduced by both interactions. The computed B ( E pf -shell nuclei are fairly wellreproduced by both the adopted effective interactions.This gives us confidence for a successful computation ofthe β -decay properties, discussed in the following section. B. Nuclear matrix elements
Once the shell-model description of low-energy spec-tra and spectroscopic properties of the involved nucleiis now under control, we are ready to use the result-ing wave functions to compute the OBTDs needed in theNMEs for the β -decay-rate calculations. The KB3G- andGXPF1A-computed NMEs for the studied decay transi-tions are presented in Table IV. The relativistic vectornuclear matrix element V M (0) KK − becomes identicallyzero due to the limitation of our adopted single-particlemodel space (see more details in Refs. [31, 32]). As men-tioned in the introduction, we constrained the value ofthis matrix element for each g A using the experimentalpartial half-life. In the case of Sc, the partial half-lifedoes not depend on this matrix element for either shell-model interactions. For this reason we are unable to re-produce the exact experimental partial half-life by tun-ing this matrix element. In this situation, we have usedthe value of this matrix element computed from the pureshell-model interactions. For the transitions of , Fe,we are able to reproduce the exact experimental partialhalf-lives corresponding to the values of the nuclear ma-trix element V M (0) KK − presented in Table V. As seen inTable IV for the transition of Fe, the magnitudes of thecomputed axial-vector matrix elements are found to belarger for the GXPF1A interaction than for the KB3Ginteraction, leading to strong differences in the shapefactors. This difference can be traced back to the differ-ences in the behavior of the cumulative sums of the vectorand axial-vector matrix elements, exemplified by the vec-tor matrix element V M (0) KK (1,1,1,1) and the axial-vectormatrix element A M (0) KK (1,1,1,1) in Fig. 4. In this figurethe cumulative sum of these matrix elements is plottedas a function of the contributing proton-neutron orbitals.As seen in the figure, for the vector matrix element thereis a shift in the contributions of the proton-neutron or-bital pairs from the very beginning such that these shiftsconspire to produce a similar total vector matrix elementfor both interactions. For the axial-vector matrix ele-ments this shift realizes only for the p − f proton-neutronorbital contributions. This difference in the cumulativebehavior is driven by the differences in the values of thevactor-type and axial-vector-type single-particle transi-tion matrix elements V/A m ( N ) KLS ( pn ) in Eq. 6.For the further calculations, we have used three dif-ferent values of the axial-vector coupling constant: Thebare-nucleon value g A = 1 .
27 and the moderatelyquenched shell-model type of value g A = 1 .
00 for allstudied transitions, and we have also used A -dependent EXPT KB3G GXPF1A0 . . . . . . . . . E n e r g y ( M e V ) Sc + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + EXPT KB3G GXPF1A01234 E n e r g y ( M e V ) Ti + + + + + + + + + + + + + + + + + + + + + + + + FIG. 1: Comparison of the KB3G- and GXPF1A-computed energy spectra with the experimental [38] one for low-lying energyspectra of Sc and Ti.
EXPT KB3G GXPF1A0 . . . . . . . E n e r g y ( M e V ) Fe / − / − / − (3 / − )(5 / − )7 / − / − / − / − / − / − / − / − / − / − / − / − / − EXPT KB3G GXPF1A0 . . . . . E n e r g y ( M e V ) Co / − / − / − / − / − / − / − / − / − / − / − / − / − / − / − / − / − / − / − / − / − / − / − / − FIG. 2: Comparison of the KB3G- and GXPF1A-computed energy spectra with the experimental [38] one for low-lying energyspectra of Fe and Co.TABLE II: Comparison of the computed quadrupole and magnetic moments with available experimental data. For the calcu-lations, we have used the effective charges e p = 1 . e and e n = 0 . e , and bare g factors ( g eff = g free ). The experimental valuesare taken from [38]. Q ( eb ) µ ( µ N )Expt. KB3G GXPF1A Expt. KB3G GXPF1A Sc 4 + +0.119 (6) +0.002 +0.007 +3.03 (2) +3.11 +2.88 Ti 2 + -0.21 (6) -0.17 -0.13 +0.98 (24) +0.57 +0.82 Fe 3 / − N/A +0.21 +0.24 -0.3358 (4) -0.24 -0.08 Co 7 / − +0.42 (3) +0.38 +0.44 +4.627 (9) +4.51 +4.59 Fe 2 + N/A -0.28 -0.30 N/A +1.06 +1.11 Co 5 + +0.44 (5) +0.44 +0.51 +3.799 (8) +3.66 +3.952 + +0.3 (4) +0.23 +0.26 +4.40 (9) +4.60 +4.26 values of g A , based on the systematics compiled in Ref. [5]. There, the effective value of g A = 0 .
94 for the region
EXPT KB3G GXPF1A0 . . . . . . . . . E n e r g y ( M e V ) Fe + + + + + + + + + + + + + + + + + + + + + + + + + + + EXPT KB3G GXPF1A0 . . . . . . E n e r g y ( M e V ) Co + + + + + + + + + + + + + + + + + + + + + + + + + + + FIG. 3: Comparison of the KB3G- and GXPF1A-computed energy spectra with the experimental [38] one for low-lying energyspectra of Fe and Co.TABLE III: Comparison of the computed and experimental B ( E
2) values in W. u. The effective charges e p = 1 . e and e n = 0 . e were used. The experimental values are taken from[38]. Transitions Expt. KB3G GXPF1A Sc B ( E
2; 6 + → + ) 3.2 (3) 2.65 2.53 Ti B ( E
2; 2 + → + ) 19.5 (6) 13.21 12.74 B ( E
2; 4 + → + ) 20.2 (13) 17.44 16.19 Fe B ( E
2; 7 / − → / − ) N/A 12.36 14.42 Co B ( E
2; 3 / − → / − ) 9.2 (20) 6.45 0.47 Fe B ( E
2; 2 + → + ) 13.6 (14) 15.53 18.89 Co B ( E
2; 4 + → + ) N/A 0.69 0.57 A = 41 −
50 (e.g. Sc) and the value g A = 0 .
84 for theregion A = 52 −
67 (e.g. , Fe) were reported for the f p model space, used in the present calculations. In all ourcalculations of the β -decay half-lives and electron spectrathe CVC-compatible value g V = 1 .
00 of the vector cou-pling, as well as measured Q values, presented in TableI, are adopted. C. Electron spectral shapes and the effective valueof g A In the SSM, the shape of the β spectrum could beused to extract the effective values of the weak couplingconstants by comparing the computed spectrum with themeasured one for forbidden nonunique β decays. In ourshell model calculations the potential of the SSM could beenhanced by using the measured half-life to constrain thevalue of the vector-matrix element V M (0) KK − for each g A separately. Based on this“CVC-inspired” SSM, we havepresented the shape of the electron spectra of the second- forbidden nonunique β − decays of Sc and , Fe byvarying the value of the axial-vector coupling g A withina physically relevant interval. It is to be noted that allthe transitions have been observed experimentally butthe electron spectra are not yet available. For the β − decays of Sc and Fe the measured branching ratio isless than 1% and for the transition of Fe we have a 100%branching ratio. It is worth noting that for the first time,we present here the computed electron spectra of Sc.In general, our computed electron spectra are very usefulfor the prospective future experiments: In the spirit ofthe revised, CVC-inspired SSM, our computed shapes ofthe electron spectra can be compared with measured onesto extract the effective values of g A .The shape factors (left panels) and electron spectra(right panels) of Sc and , Fe obtained from our cal-culations are shown in Figs. 5 −
7. In these figures arepresented the shape factor of Eq. (5) and the electronspectra corresponding to the integrand of Eq. (4) as func-tions of the electron kinetic energy, computed separatelyfor the KB3G and GXPF1A interactions. We have cho-sen to normalize the area under each curve to unity inorder to facilitate easy comparison with the potential fu-ture data. For the first time the electron spectra of Sc,shown in Fig. 5, are obtained from shell-model calcu-lations using the KB3G and GXPF1A interactions. Asseen in the figure, the computed electron spectra dependonly weakly on the value of g A for the KB3G interaction,while they become independent of the value of g A for theGXPF1A interaction. For Fe, the computed electronspectra are clearly g A dependent for both interactions:With increasing value of g A the intensity of low-energyelectrons ( ≈ . − . ≈ . − . g A de-pendence of the electron spectral shape is qualitativelysimilar. FIG. 4: Cumulative sum of the vector matrix element V M (0) KK (1,1,1,1) (left panel) and the axial-vector matrix element A M (0) KK (1,1,1,1) (right panel) for the decay of Fe. The horizontal axis lists the contributing proton-neutron orbital pairs. . . . . Sh a p e F a c t o r( a r b . un i t s ) (a) KB3G g A =0.94 g A =1.00 g A =1.27 . . . . . . . . . . . I n t e n s i t y ( a r b . un i t s ) (b) . . . . Sh a p e F a c t o r( a r b . un i t s ) (c) GXPF1A g A =0.94 g A =1.00 g A =1.27 . . . . . . . . . . . I n t e n s i t y ( a r b . un i t s ) (d) FIG. 5: The KB3G- and GXPF1A-computed shape factors (left panels) and electron spectra (right panels) as functions ofthe electron kinetic energy for the second-forbidden nonunique β − transition Sc(4 + ) → Ti(2 + ). The dashed vertical linesrepresent the end-point energy of the transition. The area under each curve is normalized to unity. The electron spectra of the β − -decay transition of Fe have previously been computed by Kostensalo et al . in
TABLE IV: Leading-order (LO) and next-to-leading-order (NLO) nuclear matrix elements (NMEs) of the second-forbiddennonunique β − decays of Sc and , Fe, computed by the use of the KB3G and GXPF1A interactions. The Coulomb-correctedNMEs are indicated by ( k e , m, n, ρ ), when such elements exist. The blank spaces denote vanishing NMEs.Transition Sc(4 + ) → Ti(2 + ) Fe(3 / − ) → Co(7 / − ) Fe(0 + ) → Co(2 + )NMEs KB3G GXPF1A KB3G GXPF1A KB3G GXPF1A LO V M (0) KK − V M (0) KK -1.1186 0.0136 13.2021 11.7172 -20.3265 -21.6968(1 , , ,
1) -1.3856 -0.0332 15.3084 13.6063 -23.6439 -25.2589(2 , , ,
1) -1.3310 -0.0423 14.4779 12.8727 -22.3782 -23.9122 A M (0) KK -0.2693 -0.0320 7.7745 7.7344 -13.3677 -19.7997(1 , , ,
1) -0.2652 -0.0021 8.8039 8.7879 -15.2409 -22.8508(2 , , ,
1) -0.2404 0.0059 8.2773 8.2694 -14.3539 -21.5871 A M (0) K +1 K −− −− NLO V M (1) KK − , , ,
1) 0.0070 0.0029 -0.1321 -0.1197 0.1856 0.1968(2 , , ,
1) 0.0062 0.0026 -0.1187 -0.1074 0.1664 0.1752(1 , , ,
1) -0.0015 -0.0007 0.0242 0.0223 -0.0351 -0.0395(2 , , ,
1) -0.0011 -0.0005 0.0169 0.0156 -0.0245 -0.0276(1 , , ,
1) 0.0086 0.0036 -0.1563 -0.1420 0.2207 0.2363(2 , , ,
1) 0.0084 0.0035 -0.1527 -0.1387 0.2154 0.2303(1 , , ,
2) 0.0094 0.0039 -0.1782 -0.1613 0.2498 0.2635(2 , , ,
2) 0.0089 0.0037 -0.1711 -0.1547 0.2396 0.2520 A M (1) KK -0.4821 -0.1626 9.7384 9.5861 -16.3829 -23.3149 V M (0) KK +11 -0.4703 -0.2026 9.6599 8.8069 -13.8826 -15.0793(1 , , ,
1) -0.5147 -0.2175 11.0414 10.0325 -15.7677 -16.8896(2 , , ,
1) -0.4808 -0.2022 10.4128 9.4538 -14.8478 -15.8523 V M (0) K +1 K +11 −− −− (1 , , ,
1) 12.9834 14.0469 5.3167 4.9067 −− −− (2 , , ,
1) 12.2775 13.2732 4.9860 4.6078 −− −− A M (0) K +1 K +11 −− −− (1 , , ,
1) 2.1027 2.1478 2.8225 1.8565 −− −− (2 , , ,
1) 2.0008 2.0436 2.7069 1.7847 −− −−
TABLE V: Values of the relativistic vector matrix element V M (0) KK − for the studied transitions of , Fe. For these valuesthe calculations reproduce the measured half-life for each value of g A . The uncertainties in the NME stem from the uncertaintiesof the experimental half-lives. Fe(3 / − ) → Co(7 / − ) Fe(0 + ) → Co(2 + ) g A KB3G GXPF1A KB3G GXPF1A0.84 − . +0 . − . − . +0 . − . − . ± . − . ± . − . +0 . − . − . +0 . − . − . ± . − . ± . − . +0 . − . − . +0 . − . − . ± . − . ± . Ref. [30]. For this calculation, they have used the oldHorie-Ogawa interaction, constrained to orbitals π f / , ν p / , ν f / and ν p / . They have performed thecalculations for the electron spectra by using the pureshell-model-predicted matrix elements, without the CVC correction. From this interaction, the shape of electronspectra turns out to be independent of g A . In our case,we have used the full f p model space with the well-established KB3G and GXPF1A interactions, and alsoconstrained the vector matrix element for each g A value . . . . Sh a p e F a c t o r( a r b . un i t s ) (a) KB3G+CVC g A = 0 . g A = 1 . g A = 1 . . . . . . . . . . . . I n t e n s i t y ( a r b . un i t s ) (b) . . . . Sh a p e F a c t o r( a r b . un i t s ) (c) GXPF1A+CVC g A = 0 . g A = 1 . g A = 1 . . . . . . . . . . . . I n t e n s i t y ( a r b . un i t s ) (d) FIG. 6: The same as in Fig. 5 for the β − -decay transition Fe(3 / − ) → Co(7 / − ). separately using the experimental half-life information.Interestingly, in the figures of Fe, the predicted shapefactors and electron spectra corresponding to g A = 0 . g A = 1 .
27 are found to be mir-ror images of each other for both interactions, while theyare found to be similar for g A = 1 .
00. As seen in TableIV, the magnitudes of the axial-vector matrix elementsfor the GXPF1A interaction are found to be larger thanthose corresponding to the KB3G interaction. Hence, forthe GXPF1A interaction these matrix elements are in aposition to influence strongly the electron spectrum withvarying value of g A . This strong difference between theresults of the two interactions can only be resolved byfuture data on the electron spectral shape. In any case,the computed electron spectral shapes for the discussed β − -decay transitions of , Fe dependent sensitively onthe g A values, thus rendering themselves as excellent can-didates for future “CVC-inspired” SSM analyses. D. Decomposition of the integrated shape function ˜ C We have decomposed the integrated shape function tosee the individual effects of the vector ˜ C V , axial-vector˜ C A , and mixed vector-axial-vector ˜ C VA components [seeEq. (8)]. The integrated shape function ˜ C and its de-composed components for the studied decay transitions,computed with the KB3G and GXPF1A interactions, arepresented in Table VI. In the case of Sc, the compo-nents are presented using the pure shell-model-calculatedmatrix elements, while for the other two transitions wehave applied the revised CVC-inspired theory, denotedhere by “SM+CVC”. For all the studied transitions, thesign of the vector ˜ C V and axial-vector ˜ C A componentsare positive for both interactions. For the KB3G interac-tion, the sign of the mixed vector-axial-vector component˜ C VA is negative for all the studied decay transitions. Butin the case of the GXPF1A interaction, the sign of ˜ C VA for the decay of Sc is positive, while for the other twodecay transitions the signs are negative. For the decayof Sc, the KB3G-computed axial-vector component ˜ C A .
00 0 .
05 0 .
10 0 .
15 0 . Sh a p e F a c t o r( a r b . un i t s ) (a) KB3G+CVC g A = 0 . g A = 1 . g A = 1 . .
00 0 .
05 0 .
10 0 .
15 0 . I n t e n s i t y ( a r b . un i t s ) (b) .
00 0 .
05 0 .
10 0 .
15 0 . Sh a p e F a c t o r( a r b . un i t s ) (c) GXPF1A+CVC g A = 0 . g A = 1 . g A = 1 . .
00 0 .
05 0 .
10 0 .
15 0 . I n t e n s i t y ( a r b . un i t s ) (d) FIG. 7: The same as in Fig. 5 for the β − -decay transition Fe(0 + ) → Co(2 + ).TABLE VI: Decomposition of the dimensionless integrated shape function into vector ˜ C V , axial-vector ˜ C A , and mixed vector-axial-vector ˜ C VA components for the studied transitions and for g V = g A = 1 . Sc(4 + ) → Ti(2 + )Interactions ˜ C V ˜ C A ˜ C VA ˜ C KB3G 2.284 × − × − -8.094 × − × − GXPF1A 1.173 × − × − × − × − Fe(3 / − ) → Co(7 / − )˜ C V ˜ C A ˜ C VA ˜ C KB3G+CVC 9.641 × − × − -8.807 × − × − GXPF1A+CVC 9.632 × − × − -8.779 × − × − Fe(0 + ) → Co(2 + )˜ C V ˜ C A ˜ C VA ˜ C KB3G+CVC 8.225 × − × − -3.349 × − × − GXPF1A+CVC 4.462 × − × − -5.307 × − × − is dominant and about twice as large as the vector com- ponent ˜ C V , and the mixed component ˜ C VA contributes1little to ˜ C . In the case of the GXPF1A interaction theaxial-vector component is dominant while the vector andmixed components contribute little to the total ˜ C . Theelectron spectral shape of Sc is g A independent for theGXPF1A interaction since the three components of thetotal integrated shape factor ˜ C have the same sign andno interference occurs between the components. For thetransition of Fe, the decomposition is surprisingly sim-ilar for both interactions: The vector component ˜ C V islargest and the axial-vector component ˜ C A is about 22%of ˜ C V . For the transition of Fe, the axial-vector com-ponent is dominant for the KB3G interaction, while thevector component dominant for the GXPF1A interac-tion. These major differences between the componentspredicted by the used interactions stem from the differ-ences in the magnitudes of the axial-vector matrix ele-ments, as visible in Table IV. These differences in thedominant components influence strongly also the elec-tron spectral shapes. One also notices that in the caseof the GXPF1A interaction the mixed component ˜ C VA is almost the sum of the ˜ C V and ˜ C A components, andnegative. IV. CONCLUSIONS
Based on the “CVC-inspired” SSM, we have com-puted shape factors and electron spectra for the second-forbidden nonunique β − -decay transitions Sc(4 + ) → Ti(2 + ), Fe(3 / − ) → Co(7 / − ), and Fe(0 + ) → Co(2 + ) in the full f p single-particle space using thewell-established effective shell-model interactions KB3Gand GXPF1A. We include also the next-to-leading-ordercorrections to the β -decay shape factor.To test the predictive power of the adopted interac-tions, we have calculated the low-lying energy spectraof the parent and daughter nuclei participating in thestudied β − transitions and compared the results withthe available data. The low-lying energy spectra nicelyagree with the experimental data. We have correctlyreproduced the ground state of the parent and daugh-ter nuclei and only for Co we are unable to obtainthe correct ground state. These effective interactionssuccessfully predict the spectroscopic properties of these f p -shell nuclei. The obtained wave functions have been used for further calculations of the β -decay rates. In or-der to calculate the shape factors and electron spectra,we have adopted a “CVC-inspired” approach and con-strained the value of the small relativistic vector matrixelement V M (0) KK − for each g A by the experimental half-life. However, in the case of the Sc decay this revisedSSM does not work, and we have used the uncorrectedshell-model-predicted value of this matrix element.The evolution of the shape factors and electron spectrafor the three second-forbidden nonunique β − -decay tran-sitions was studied within the interval of g A = 0 . − . V M (0) KK − . Inour revised SSM, the shape of the electron spectra de-pends notably on the value of g A for the transitions of , Fe, thus opening up the possibility to use the “CVC-inspired” SSM to access the effective value of g A in thesecases. For Fe the decay transition has a branchingratio of 100%, making it a perfect candidate for futurespectral-shape measurements and a subsequent applica-tion of the revised SSM. In order to see the origin ofthese variations in the electron spectral shapes we havedecomposed the total integrated shape function ˜ C intoits vector ˜ C V , axial-vector ˜ C A , and mixed vector-axial-vector ˜ C VA components. The relative sizes and signs ofthese components lead to sensitivity or non-sensitivityof the spectral shapes to the value of g A . We hope thatour theoretical work strongly encourages future measure-ments of electron spectral shapes. V. ACKNOWLEDGMENTS
We acknowledge for the financial support from theMinistry of Human Resource Development (MHRD),Government of India and the Science and Engineer-ing Research Board (SERB), Government of India,CRG/2019/000556. This work was partially supportedby the Academy of Finland under the Academy projectno. 318043. [1] K. Zuber, “
Neutrino Physics ”, (Institue of Physics Pub-lishing Ltd., London 2004).[2] E. D. Commins “
Weak Interactions ” (McGraw-Hill, NewYork, 2007).[3] J. Suhonen and O. Civitarese, “Weak-interaction andnuclear-structure aspects of nuclear double beta decay”,Phys. Rep. , 123 (1998).[4] J. Maalampi and J. Suhonen, “Neu-trinoless Double β + /EC Decays”,Adv. High Energy Phys. , 505874 (2013). [5] J. Suhonen, “Value of the axial-vector cou-pling strength in β and ββ decays: a review”,Frontier in Physics , 55 (2017).[6] H. Ejiri, J. Suhonen, and K. Zuber, “Neutrino-nuclear re-sponses for astro-neutrinos, single beta decays and doublebeta decays ”, Phys. Rep. , 1 (2019).[7] H. Ejiri, N. Soukouti, and J. Suhonen, “Spin-dipole nu-clear matrix elements for double beta decays and astro-neutrinos”, Phys. Lett. B , 27 (2014).[8] H. Ejiri and J. Suhonen, “GT neutrino-nuclear re- sponses for double beta decays and astro neutrinos”,J. Phys. G: Nucl. Part. Phys. , 055201 (2015).[9] J. Suhonen and O. Civitarese, “Probing the quench-ing of g A by single and double beta decays”,Phys. Lett. B , 153 (2013).[10] J. Suhonen and O. Civitarese, “Single and double betadecays in the A = 100, A = 116 and A = 128 triplets ofisobars”, Nucl. Phys. A , 1 (2014).[11] A. Faessler, G. L. Fogli, E. Lisi, V. Rodin, A. M.Rotunno, and F. ˇSimkovic, “Overconstrained estimatesof neutrinoless double beta decay within the QRPA”,J. Phys. G: Nucl. Part. Phys. , 075104 (2008)[12] D. S. Delion and J. Suhonen, “Effective axial-vector strength and β -decay systematics”,Eur. Phys. Lett. , 52001 (2014).[13] P. Pirinen and J. Suhonen, “Systematic approach to β and 2 νββ decays of mass A = 100 −
136 nuclei”,Phys. Rev. C , 054309 (2015).[14] F. F. Deppisch and J. Suhonen, “Statistical analysis of β decays and the effective value of g A in the proton-neutronquasiparticle random-phase approximation framework”,Phys. Rev. C , 055501 (2016).[15] B. H. Wildenthal, M. S. Curtin, and B. A. Brown, “Pre-dicted features of the beta decay of neutron-rich sd-shellnuclei”, Phys. Rev C , 1343 (1983).[16] G. Mart´ınez-Pinedo, A. Poves, E. Caurier, andA. P. Zuker, “Effective g A in the pf shell”,Phys. Rev. C. , R2602 (1996).[17] E. Caurier, F. Nowacki, and A. Poves, “ShellModel description of the ββ decay Xe”,Phys. Lett. B , 62 (2012).[18] V. Kumar, P.C. Srivastava, and H. Li, “Nuclear β − -decay half-lives for fp and fpg shell nuclei”Jour. Phys. G: Nucl. and Part. Phys. , 105104 (2016).[19] V. Kumar and P.C. Srivastava, “Shell model descrip-tion of Gamow-Teller strengths in pf -shell nuclei”Eur. Phys. J. A , 181 (2016).[20] A. Saxena, P.C. Srivastava and T. Suzuki, “Ab initiocalculations of Gamow-Teller strengths in the sd shell”,Phys. Rev. C , 024310 (2018).[21] A. Kumar, P.C. Srivastava and T. Suzuki, “Shell modelresults for nuclear β − -decay properties of sd-shell nuclei”,Prog. Theo. Expt. Phys. , 033D01 (2020).[22] J. Kostensalo and J. Suhonen, “Consistent large-scale shell-model analysis of the two-neutrino ββ and single β branchings in Ca and Zr”,Phys. Lett. B , 135192 (2020).[23] J. Barea, J. Kotila, and F. Iachello, “0 νββ and 2 νββ nu-clear matrix elements in the interacting boson model withisospin restoration”, Phys. Rev. C , 034304 (2015).[24] J. Barea, J. Kotila, and F. Iachello, “Nu-clear matrix elements for double- β decay”,Phys. Rev. C , 014315 (2015).[25] N. Yoshida and F. Iachello, “Two-neutrino double- β decay in the interacting boson-fermion model”,Prog. Theo. Expt. Phys. , 043D01 (2013).[26] M. Haaranen, P. C. Srivastava, and J. Suhonen “For-bidden nonunique β decays and effective values of weak coupling constants”, Phys. Rev. C β -decay shape factor”, Phys. Rev. C β decay of Cd”,Phys. Rev. C , 064603 (2007).[29] J. Kostensalo, M. Haaranen, and Jouni Suhonen, “Elec-tron spectra in forbidden β decays and the quench-ing of the weak axial-vector coupling constant g A ”,Phys. Rev. C , 044313 (2017).[30] J. Kostensalo and J. Suhonen, “g A -driven shapes of elec-tron spectra of forbidden β decays in the nuclear shellmodel”, Phys. Rev. C , 024317 (2017).[31] A. Kumar, P. C. Srivastava, J. Kostensalo and J.Suhonen, “Second-forbidden nonunique β − decays of Na and Cl assessed by the nuclear shell model”,Phys. Rev. C. , 064304 (2020).[32] O. S. Kirsebom et. al., “Measurement of the 2 + → + ground-state transition in the β decay of F”,Phys. Rev. C. , 065805 (2019).[33] O. S. Kirsebom et. al., “Discovery of an Excep-tionally Strong β -Decay Transition of F and Im-plications for the Fate of Intermediate-Mass Stars”,Phys. Rev. Lett. , 262701 (2019).[34] M. Honma, T. Otsuka, B. A. Brown, and T. Mizusaki,“New effective interaction for pf -shell nuclei and its im-plications for the stability of the N = Z = 28 closedcore”, Phys. Rev. C pf -shell nuclei with a new effective interaction GXPF1”,Eur. Phys. J. A
499 (2005).[36] A. Poves, J. S´anchez-Solano, E. Caurier, and F. Nowacki,“Shell model study of the isobaric chains A = 50, A =51 and A = 52”, Nucl. Phys. A
157 (2001).[37] H. Behrens and W. B¨uhring,
Electron Radial Wave Func-tions and Nuclear Beta-Decay β decay and axialstrength”, Frontier in Physics , 29 (2019).[40] H. F. Schopper, Weak Interaction and Nuclear Beta De-cay (North-Holland, Amsterdam, 1966).[41] M. T. Mustonen, M. Aunola, and J. Suho-nen, “Theoretical description of the fourth-forbidden non-unique β decays of Cd and
In”, Phys. Rev. C From Nucleons to Nucleus: Concept of Mi-croscopic Nuclear Theory , (Springer, Berlin 2007).[43] C. Patrignani and Particle Data Group, “Review of Par-ticle Physics”, Chinese Phys. C , 100001 (2016).[44] B. A. Brown and W. D. M. Rae, “The Shell-Model CodeNuShellX@MSU”, Nucl. Data Sheets120