Electron spectra in forbidden β decays and the quenching of the weak axial-vector coupling constant gA

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I. INTRODUCTION
The observation of the neutrinoless mode of double beta decay (0νββ) would be groundbreaking since it would prove that the neutrino is its own antiparticle. The half-life of 0νββ decay is proportional to the fourth power of the weak axial-vector coupling constant g A [1,2], but it is not quite clear what value to use for it when calculating theoretical predictions for the decay rates [3,4]. Theoretical predictions for the half-lives are crucial when designing optimal experimental setups to detect 0νββ events. In this context it is crucial to conceive complementary ways to address the g A problem of 0νββ decay. One alternative is the study of β decays, both allowed and forbidden, since experimental data is available to compare with the calculated physical observables. The related quenching of the computed nuclear matrix elements (NMEs) was addressed already for the allowed Gamow-Teller [4] and first-forbidden [3] transitions in medium-heavy even-mass nuclei. In the present work we extend these studies to first-, second-, third-, fourth-and fifth-forbidden β − decays of odd-mass (odd-A) nuclei by studying the shape evolution of the associated electron spectra and by calculating the involved nuclear matrix elements (NMEs) in the framework of the microscopic quasiparticle-phonon model (MQPM). In this way we hope to shed light on the effective value of g A in forbidden β transitions, which play also a prominent role in the virtual transitions mediating the 0νββ decay. * joel.j.kostensalo@student.jyu.fi † jouni.suhonen@phys.jyu.fi The weak interaction is parity nonconserving, which is reflected in the fact that the hadronic current can be written as a mixture of vector and axial-vector components [5][6][7]. The weak coupling constants g V and g A appear in the theory of β decay as means to renormalize the hadronic current, when moving from the quark level to nucleons [8]. The conserved vector-current hypothesis (CVC) and the partially conserved axial-vector-current hypothesis (PCAC) of the standard model can be used to derive the free-nucleon values g V = 1.00 and g A = 1.27 [9]. In nuclear matter, however, the value of g A is affected by many-nucleon correlations, and so in practical calculations a quenched effective value could give results which are closer to experimental results [10]. In the practical calculations the shortcomings in the treatment of the many-body quantum mechanics could also be absorbed into the value of g A [11,12].
In earlier studies the effective value of the weak axial-vector coupling constant was probed by comparing the experimental half-lives of Gamow-Teller and first-forbidden unique decays to those predicted by the proton-neutron quasiparticle randomphase approximation (pnQRPA) [3,4,[13][14][15]. These studies show consistently that a quenched value of g A is needed to reproduce the experimental data. In Ref. [16] a systematic study of high-forbidden unique decays showed similar features as the earlier studies supporting the fact that a quenched effective value of g A might need to be used for the highforbidden decay branches as well.
In a recent study (see Ref. [17]) an interesting new feature was found regarding the weak coupling constants g V and g A . The shape of the electron spectra of the fourth-forbidden nonunique ground-state-to-ground-state β − decays of 113 Cd and 115 In depend strongly on the value of g V and g A due to TABLE I. Change in angular momentum J and parity π in a Kth forbidden β decay. The decays with J = K (in the case of K = 1 also J = 0) are known as nonunique, while those with = K + 1 are known as unique.
the complicated shape factor featuring vector, axial-vector, and mixed vector-axial-vector parts containing a number of different NMEs and phase-space factors. The shapes of the theoretical and experimental spectra can be compared to find the effective values of the coupling constants in a new method called the spectrum-shape method (SSM). The study [17] used two nuclear models; namely, the MQPM [18,19] and the nuclear shell model (NSM) [20][21][22]. When the computed electron spectra of 113 Cd were compared with the experimental spectrum of Ref. [23], a close match was found by using the effective values g A ≈ 0.90 and g V = 1.0 (equal to its CVC value!). This study was extended in Ref. [24] to include also the NMEs calculated within the microscopic interacting bosonfermion model (IBFM-2) [25,26]. The remarkable result of this study was that the three models MQPM, NSM, and IBFM-2 yielded a consistent result of g A ≈ 0.92 by comparison with the experimental electron spectrum of Ref. [23]. This is the more surprising when considering the very different theory frameworks of the three models. The qualitative behavior of the MQPM, NSM, and IBFM-2 spectra was remarkably similar for both the 113 Cd and 115 In decays, suggesting that the details of the nuclear residual Hamiltonian and the many-body methods to solve the associated eigenvalue problem do not affect much the the evolution of the electron spectra and thus the SSM itself. Based on this, it is reasonable to conjecture that just one of these models could be used to further explore new, potentially interesting β decays and their basic features. In this paper we set out to find if the β spectra of other forbidden decays depend sensitively on the value of g A using the MQPM, which can be relatively easily applied to a large number of odd-A nuclei. We will explore unique and nonunique first-and second-forbidden decays and third-, fourth-, and fifth-forbidden nonunique decays. The identification of the degree of forbiddeness of a β-decay transition is presented in Table I. Most of these transitions and the associated electron spectra can be measured in the present and future low-background (underground) experiments. Some of the considered higher-forbidden transitions, with competing allowed transitions, cannot be easily measured. They were, however, included to better access the possible systematic features in transitions of increasing forbiddeness. As in Refs. [17] and [24] we take into account the next-to-leading-order terms in the shape factor, and so the spectra of unique-forbidden decays also depend on the axial-vector coupling constant in a nontrivial way. Based on the conclusions of Ref. [17], we adopt the CVC value g V = 1.00 in the calculations. In the analyzes we use the normalized electron spectra (the normalized spectra are also used when comparing with available experimental data) for which the integrated area under the spectrum curve is unity. While the half-life is affected by the absolute values of the weak coupling constants, the normalized electron spectra depend only on the ratio g A /g V , so varying only g A is sufficient.
The microscopic quasiparticle-phonon model is a fully microscopic model, which can be used to describe spherical and nearly spherical open-shell odd-A nuclei. In the MQPM all three parts of the Hamiltonian, namely the quasiparticle, phonon, and quasiparticle-phonon terms are treated in an internally consistent way. The odd-A nucleus is built from a basis of one-and three-quasiparticle states. The one-quasiparticle states are obtained by performing a BCS calculation for the neighboring even-even reference nucleus, and the threequasiparticle states by coupling the BCS quasiparticles to the QRPA phonons of Ref. [18]. Finally, the residual Hamiltonian, which contains the interaction of the odd nucleon and the even-even reference nucleus, is diagonalized in the combined one-and three-quasiparticle basis [19].
This article is organized as follows: In Sec. II we give the theoretical background behind the MQPM and the β spectrum shape. In Sec. III we describe the application of the MQPM to the spectrum shape of forbidden β − decays. In Sec. IV we present our results, and in Sec. V we draw the conclusions.

II. THEORETICAL FORMALISM
In this section we give the theoretical background behind the nuclear model MQPM used in the calculations (see Sec. II A) and present the practical aspects of the theory of forbidden β − decays and the shape of the electron spectra (see Sec. II B). The description of the MQPM summarizes the two original papers describing the model (see Refs. [18] and [19]). For a more complete description of the theory of β decays, see Ref. [27].

A. Microscopic quasiparticle-phonon model
The starting point of the MQPM is a realistic A-fermion Hamiltonian consisting of the mean-field part and the residualinteraction part. In the occupation-number representation the Hamiltonian reads [10] where c † ι and c ι are the creation and annihilation operators of the Hartree-Fock quasiparticles andv αβγ δ = αβ|v|γ δ − αβ|v|δγ is the antisymmetrized two-body matrix element. Here we adopt the notation of Baranger [28], where the Roman letter a includes the single-particle quantum numbers n a (principal), l a (orbital angular momentum), and j a (total angular momentum) and the Greek letter α, the quantum numbers a, and the magnetic quantum number m a .
The approximate ground state of the even-even reference nucleus is obtained by using BCS theory [29]. The occupation and vacancy amplitudes v a and u a emerge from the Bogoliubov-Valatin transformation [30,31], 044313-2 where a † α is the BCS quasiparticle creation operator andã α is the time-reversed BCS quasiparticle annihilation operator. After the Bogoliubov-Valatin transformation, the Hamiltonian of Eq. (1) can be written as where the indices correspond to the number of BCS creation and annihilation operators [32]. The terms H 20 and H 02 are missing from Eq. (4), since they vanish when minimizing the BCS ground-state energy. When solving the BCS equations (see formulation in, e.g., Ref. [10]), the monopole matrix elements of the two-body interaction can be scaled by pairing-strength parameters g (n) pair and g (p) pair to reproduce the semi-empirical pairing gaps. The proton and neutron pairing gaps p and n needed for solving the BCS equations can be calculated by using the three-point formulas [33,34], where S p is the proton separation energy and S n the neutron separation energy.
Once the one-quasiparticle states have been calculated by using BCS theory, the next step is to consider twoquasiparticle excitations, i.e., treat the H 22 , H 40 , and H 04 terms of the Hamiltonian (4). In the MQPM, the two-quasiparticle excitations are the quasiparticle random-phase approximation (QRPA) [28] phonons of the even-even reference nucleus defined by the QRPA phonon-creation operator where ω denotes the angular momentum J ω , parity π ω , and the index k ω which identifies the different excitations with the same J and π . The quantity A † is the corresponding annihilation operator. The forward-and backward-going amplitudes X and Y are solved by diagonalizing the QRPA matrix (see, e.g., Ref. [10]).
The basis states of the microscopic quasiparticle-phonon model are the one-quasiparticle states given by the BCS calculation and the three-quasiparticle states emerging from the coupling of QRPA phonons with BCS quasiparticles. The states of the odd-A nucleus are created by the MQPM creation operator [19] where C i n and D i aω are the amplitudes determined by the MQPM matrix equation. The use of the equations-of-motion method [35] leads to the generalized eigenvalue problem [18] where the submatrices A, A , and B are the matrix elements of H 11 , H 22 , and H 31 of equation (4), respectively. The explicit expressions for these submatrices are listed in Ref. [19].
The overlap between a one-quasiparticle state and a threequasiparticle state is always zero leading to a block-diagonal structure of the overlap matrix on the right-hand side of Eq. (8).
The overlaps between one-quasiparticle states lead to the unity matrix and the overlaps between three-quasiparticle states lead to the submatrix N of the overlap matrix. The appearance of the nondiagonal submatrix N in the overlap matrix leads, in turn, to a non-Hermitian eigenvalue problem in a nonorthogonal, often over-complete, basis set.
To solve the non-Hermitian eigenvalue problem of Eq. (8), we turn it into a Hermitian problem by writing it in a new orthogonal basis and diagonalizing it in the usual way. We start by solving the eigenvalue problem of the overlap matrix N , which reads The eigenvectors of N can be written in the basis |i = † i |QRPA , where |QRPA is the correlated ground state of the reference nucleus, as [19] The states |k with a nonzero eigenvalue n k form an orthonormal complete set. In the orthonormal basis of Eq. (10) the MQPM matrix equation (8) can be written as a real and symmetric eigenvalue problem, where The coefficients C n i of the MQPM matrix equation can be calculated from the coefficients g using the equation [19] As an example of the kind of excitation spectrum that the microscopic quasiparticle-phonon model produces, the experimental and MQPM spectra of 87 Sr are presented in Fig. 1. The experimental spectrum was obtained from Ref. [36].

B. β spectrum shape
To simplify the description of the β − decay, we assume that at the very moment of decay the decaying nucleus only interacts via the weak interaction, and the strong interaction with the A − 1 other nucleons can be neglected. In this scheme, known as the impulse approximation, the flow lines of the nucleons, i.e., the hadronic current, and the flow lines of the emitted leptons, i.e., the leptonic current, interact at a weakinteraction vertex. Since the vector boson W − has a large mass, and thus propagates only a short distance, the vertex can be considered to be point like with an effective coupling constant G F , the Fermi constant. The weak interaction is parity nonconserving, which is reflected in the fact that the hadronic current can be written at the quark level as a mixture of vector and axial-vector components as [5][6][7] where γ μ and γ 5 are the usual Dirac matrices. When moving from the quark level to the nucleon level, one must take into account renormalization effects of strong interactions. The hadronic current at the nucleon level (proton p and neutron n) can then be written as [8] where g V and g A are the weak vector and axial-vector coupling constants, respectively. The conserved vector-current hypothesis (CVC) and the partially conserved axial-vectorcurrent hypothesis (PCAC) of the standard model give the so-called bare nucleon values g V = 1.0 and g A = 1.27 [9]. In nuclear matter the value of g A is affected by many-nucleon correlations, and so the bare nucleon value might not be the one to use in practical calculations [10].
In the impulse approximation the probability of the electron being emitted with kinetic energy between W e and W e + dW e is P (W e )dW e = G F (hc) 6 1 2π 3h C(W e )p e cW e (W 0 − W e ) 2 × F 0 (Z,W e )dW e , where p e is the momentum of the electron, Z is the proton number, F 0 (Z,W e ) is the Fermi function, and W 0 is the endpoint energy of the β spectrum. The shape factor C(w e ) encodes the nuclear-structure information.
The half-life of a β decay can be expressed as whereC is the integrated shape factor and κ is a constant with value [37] κ = 2π 3h7 ln 2 m 5 e c 4 (G F cos θ C ) 2 = 6147 s, with θ C being the Cabibbo angle. For convenience, it is common to introduce unitless kinematic quantities w e = W e /m e c 2 , w 0 = W 0 /m e c 2 , and p = p e c/(m e c 2 ) = (w 2 e − 1) 1/2 . Using these quantities the integrated shape factor can be written as In Eq. (19) the kinematic factors are universal and the shape factor C(w e ) has a complicated expression including both kinematic and nuclear form factors. The choice of nuclear model (in the case of this paper, MQPM) enters the picture when calculating the one-body transition densities [38] related to the NMEs of the shape factor. The details of the shape factor and the constitution of its NMEs can be found from Refs. [27] and [38]. As in the Refs. [17] and [24] we take into account the next-to-leading-order terms of the shape function (for details see Ref. [24]). In the commonly adopted leadingorder approximation the shape of the electron spectrum of unique-forbidden decays does not depend on the NMEs and it is just scaled by g 2 A . When the next-order terms are taken into account, the shape factor of these decays depends on the computed NMEs and through this on g A in a nontrivial way [24]. This means that, at least in theory, the SSM could be applied also to unique decays.
The shape factor C(w e ) can be decomposed into vector, axial-vector, and vector-axial-vector parts. In this decomposition the shape factor reads (20) For the integrated shape factor we get the analogous expressioñ In should be noted that in Eq. (20) the shape factors C i are functions of electron kinetic energy, while the integrated shape factorsC i in Eq. (21) are real numbers.

III. NUMERICAL APPLICATION OF FORMALISM
The electron spectra of 26 forbidden β − transitions were calculated for different values of the coupling constants g A by using the NMEs produced by the MQPM model. The application of the MQPM model followed the same basic steps as the earlier studies regarding the fourthforbidden nonunique ground-state-to-ground-state β − decays of 113 Cd, 115 In [17,24,38], and 115 Cd [39], including the use of the Bonn one-boson exchange potential with G-matrix techniques [19].
The single-particle energies needed to solve the BCS equations were calculated by using the Coulomb-corrected Woods-Saxon potential with the Bohr-Mottelson parametrization [40]. For the protons of the even-even reference nuclei the valence space spanned 10 single-particle states in the range 0f 7/2 -0h 11/2 . This valence space was also used for the neutrons of the reference nuclei with A < 90, while for the heavier nuclei a larger valence space spanning the single-particle-orbitals 0f 5/2 -0i 13/2 (15 single-particle states) was used.
The BCS one-quasiparticle spectra were tuned by adjusting manually some of the key single-particle energies computed by using the Woods-Saxon potential. This was done to get a closer mach between the low-lying one-quasiparticle states and the measured experimental ones. The adjustments were kept minimal with some changes in the span of spin-orbit gap(s) close to the proton and/or neutron Fermi surfaces. The computed pairing gaps were adjusted to fit the empirical values by tuning the pairing strength parameters g p pair and g n pair for protons and neutrons, separately. The empirical values were calculated by using the three-point formulas (5) and the experimental separation energies given in Ref. [41]. The QRPA spectra of the reference nuclei were tuned by scaling the particle-hole matrix elements with the parameter g ph in order to reproduce the excitation energy of the lowest state of a given multipolarity. In the MQPM calculations a 3.0 MeV cutoff energy was used for the QRPA phonons to decrease the formidable computational burden.

IV. RESULTS AND DISCUSSION
Below we present our results: the electron spectra of 26 forbidden β − decays (Figs. 2-9) and their integrated shape factors (Table II). The electron spectra are discussed in Sec. IV A and the integrated shape factors in Sec. IV B.

A. Electron spectra and effective value of g A
The electron spectra for the studied first-forbidden decays are presented in Figs. 2-4 for g A = 0.80-1.20 (the identification of the decay type is presented in Table I). Similar figures for second-, third-, fourth-, and fifth-forbidden decays are presented in Figs. 5-9. For the nonunique decay of 125 Sb, presented in Fig. 2(a), decreasing the value of g A increases the intensity of electrons emitted with low energies (0-100 keV), while decreasing the intensity of electrons with energies 250-350 keV. The differences in the shapes of the spectra are however very small-nothing like the dramatic behavior observed for the fourth-forbidden decays of 113 Cd and 115 In in Refs. [17] and [24]. From the other first-forbidden decays very slight changes in the low-energy spectrum can be seen for the nonunique decays of 141 Ce in Fig. 2(b) and 169 Er in Fig. 3(a), as well as for the unique decay of 79 Se below it in Fig. 3(b). For the other first-forbidden decays the β spectrum seems to be independent of the value of g A . The slight variation in the spectrum of 79 Se can only be seen when the next-toleading-order terms of the shape factor are taken into account. In the lowest-order approximation the spectrum would be completely unaffected when g A is varied. The variation is minimal, but on the other hand, it is also minimal for the nonunique first-forbidden decays.  Fig. 4(a)] spectra. For all of these the spectrum shape can be described as linearly decreasing with a slight downward bend at approximately 75% of the Q value. This is remarkable, since the Q values vary from a tiny 34.1 (23) Fig. 5(a). Common to this group of decays is that the initial state has spin-parity 7/2 + . For the decays of the first group no such obvious common feature is seen.
The electron spectra of second-forbidden decays shown in Fig. 5 differ from each other much more than the firstforbidden ones. In the case of the second-forbidden unique decay of 129 I the next-to-leading-order terms of the shape factor do not make the beta spectrum, shown in Fig. 5(a), FIG. 4. The same as Fig. 2, but for the first-forbidden unique decays of 107 Pd, 125 Sb, and 137 Cs. dependent on g A in a noticeable way. Slight bending is seen for the nonunique decay of 93 Zr [ Fig. 5(b)], but the only second-forbidden transition showing heavy g A dependence is the ground-state-to-ground-state decay of 99 Tc presented in Fig. 5(c). The dependence is similar to that observed in Refs. [17,24] for 113 Cd and 115 In: when g A ≈ g V a bellshaped spectrum is produced, while otherwise the spectrum is monotonically decreasing. This decay is an excellent candidate for SSM. Comparison with experimental spectra could shed light on the effective value of g A for second-forbidden nonunique transitions, and even second-forbidden β decays in general. The shape of 135 Cs and 137 Cs spectra on the other hand does not depend on the value of axial-vector coupling constant at all, as seen from Figs. 5(d) and 5(e). Decays such as the ground-state-to-ground-state decays of 135,137 Cs, which are practically independent of g A , are also experimentally important, since they could be used to check the accuracy of the theoretical spectra predicted by the MQPM as well as by other nuclear models, such as the nuclear shell model and the microscopic interacting boson-fermion model. While doing the MQPM calculation for the 137 Cs decay, an important new feature of the β spectra was discovered. While the area under the curve depends significantly on the computed MQPM level scheme (e.g., the one shown in Fig. 1), the shape of the normalized (to unit area) electron spectrum does not. Even shifting the single-particle energies enough to change the ordering of energy levels in the odd-A nucleus did not change the shape of the normalized spectrum at all. The other studied nuclei behave similarly, although for some of them slight changes in the electron spectrum were seen. At the meanfield level, moving arbitrarily the key single-particle orbitals at the proton and/or neutron Fermi surfaces by 0.5-1.5 MeV had very little effect. However, the shape is not completely independent of the computed level scheme, and large changes in it can deform the electron spectrum considerably. In terms of practical calculations it appears that getting the low end of the level scheme to reasonably agree with data, with the right ground-state spin-parity and few low-lying states near their experimental counterparts, is sufficient to produce an electron spectrum which is largely unaffected by further fine tuning. On the other hand, the theoretical half-life predictions can change by orders of magnitude when small changes to the single-particle energies are made, especially in cases where the even-even reference nucleus is (semi-)magic [16]. This stems from the sharp Fermi surface and vanishing pairing gap, making the BCS-approach sensitive to the details of the single-particle spectrum at the Fermi surface.
The results for third-forbidden nonunique ground-state-toground-state decays of 85 Br and 87 Rb are shown in Fig. 6. The shape factor depends on g A in a very similar manner, even though the Q value for 85 Br is tenfold larger than the one for 87 Rb. Both of these decays are from a 3/2 − 044313-7 FIG. 7. The same as Fig. 2, but for the fourth-forbidden nonunique decays of 113 Cd, 115 Cd, and 115 In. initial state to a 9/2 + final state. The decay of 87 Rb is experimentally measurable and thus another candidate for application of the spectrum-shape method. Unlike in the case of 99 Tc, the dependence is fairly simple: when g A decreases, the low-energy intensity increases, and the intensity at larger energies decreases.
The results for fourth-forbidden decays, all nonunique, are presented in Figs. 7 and 8. The transitions are split into two groups according to their basic features. The first one consists of the decays of 113 Cd, 115 Cd, and 115 In shown in Fig. 7. The transitions of 113 Cd and 115 In are experimentally measurable and have been studied extensively (see Refs. [17,24,38,42]). The spectra show a distinctive hump when g A ≈ g V . Interestingly, the behavior is similar to that of 99 Tc [see Fig. 5(c)]. These decays have one common feature: the 9/2 + state is either the initial or final state. For 113 Cd and 115 In the effect of varying the weak axial-vector coupling constant is in line with the results of Ref. [24]. However, in the earlier study [17] the turning point of the MQPM spectrum of 113 Cd was found to be at g A ≈ 0.9 and for 115 In at g A ≈ 0.95. It seems that the use of a larger model space for neutrons in this study and in Ref. [24] has a notable effect on the shape of the electron spectrum. In Ref. [42] the electron spectra of these decays were calculated with the proton-neutron variant of MQPM with the free-nucleon values g V = 1.0 and g A = 1.25. The shapes of the resulting spectra are nearly identical, which further supports the results of this paper.
To see whether the strong dependence of the shape factor on the weak coupling constants of the fourth-forbidden nonunique decays is limited to the three neighboring nuclei, 113 Cd, 115 Cd, and 115 In, four further fourth-forbidden nonunique groundstate-to-ground-state transitions were studied. These were the 044313-8 ground-state-to-ground-state β − decays of 97 Zr, 101 Mo, 117 Cd, and 119 Cd, for which the spectra are presented in Fig. 8. These four decays also behave almost identically as functions of g A . For these decays the spectrum shape is rather similar when g A = 1.00-1.20, but the shape changes radically when g A is quenched below unity.
As shown thus far the dependence of the shape factor on the value of g A seems to increase when the degree of forbiddeness grows. To test this hypothesis the fifth-forbidden decay of 123 Sn to the lowest 1/2 + excited state in 123 Sb (see Fig. 9) was included in this study. This transition is not experimentally measurable, but the theoretical results are important. As seen in the figure for this decay the spectrum shape does not depend on g A in any significant way, thus invalidating the abovementioned hypothesis. The sensitivity to g A seems to connect more to the specific nuclear-structure details [like the almost magical involvement of a 9/2 + state in all the SSM-sensitive nuclei of Figs. 7, 8, and 5(c)] rather than to the forbiddeness of the decay transition.
The observation that the β spectra of many transitions behave similarly is a strong indication that significant changes in the MQPM-predicted level schemes do not affect the shape evolution of the β spectra. Similar conclusions can be drawn from Ref. [24] where the three nuclear models, MQPM, NSM, and IBFM-2 were used to study 113 Cd and 115 In decays. An obvious conclusion drawn from this is that the SSM seems to be quite insensitive to the fine details of the nuclear Hamiltonian and the different approximations adopted to solve the related eigenvalue problem. This makes the SSM very robust, and thus a reliable tool for determining the effective value of g A .

B. Integrated shape factorC
In Refs. [17] and [24] it was noticed that for the decays of 113 Cd and 115 In the componentsC V ,C A , andC VA of the decomposed integrated shape functionC [see Eq. (21)] have much larger absolute values thanC. The integrated shape function and its components are listed in Table II for each of  the presently studied transitions.   TABLE II. Unitless integrated shape functionsC of the studied transitions and their vectorC V , axial-vectorC A , and mixed components C VA . For the total integrated shape factorC the values of the coupling constants were set to g V = g A = 1.0.

Transition
TypeC VCACVAC For all transitions theC V andC A components are positive, but the sign ofC VA varies. Although the next-to-leadingorder terms are taken into account,C is dominated by the axial-vector component in the unique decays. The vector and vector-axial components add small positive corrections, which are about two orders of magnitude smaller than C A at their largest. For the nonunique first-forbidden decays the largest contribution comes fromC V . The axial-vector component introduces a small positive correction, while thẽ C VA is negative, and has an absolute value of about 50%-90% ofC V . As an exception, theC VA component is positive for 159 Gd. The second-forbidden nonunique transitions on the other hand, have very different decompositions from each other. For 93 Zr the axial-vector component dominates, and the negative correction ofC VA is about 25%. In the case of 99 Tc,C V andC A have almost identical values, whileC VA is twice as large and negative. The resulting total integrated shape functionC is two orders of magnitude smaller than the absolute values of its components. For 135 Cs and 137 Cs the results are rather similar, butC VA is positive. For the two thirdforbidden nonunique decays the vector component dominates, the axial-vector component is small, and the absolute value of the negativeC VA is about 80% of the vector component. The fourth-forbidden decays are divided into the same groups as suggested by the electron spectra. The decomposition is similar to 99 Tc for 113 Cd, 115 Cd, and 115 In. For the rest of the fourth-forbidden decaysC V dominates,C A is about 20% smaller, andC VA is negative and has an absolute value of slightly less than the sum ofC V andC A . The resulting total integrated shape factor is one order of magnitude smaller than its components. For the fifth-forbidden transition of 123 Sn,C is dominated byC V , and the corrections from other components are small.

V. CONCLUSIONS
The sensitivity of the shapes of electron spectra for 26 forbidden β − decays of odd-A nuclei were studied by using the nuclear matrix elements derived from the microscopic quasiparticle-phonon model (MQPM) and by varying the value of the axial-vector coupling constant g A . The next-to-leadingorder terms were included in the corresponding β-decay shape factors. In the spectrum-shape method (SSM) the shapes of computed electron spectra can be compared with the measured ones to access the effective value of g A . The shape of the computed electron spectrum was found to be unaffected by minor changes in the computed excitation spectra of the involved nuclei, implying stability against the variations of the details of the nuclear Hamiltonian. This is supported by the study [24] where three different nuclear Hamiltonians were used. The study of Ref. [24] suggests also that the SSM is not very sensitive to the many-body framework used to solve the Hamiltonian-related eigenvalue problem since the three nuclear-structure models produced quite compatible SSM results despite their completely different theory frameworks. At the level of the mean field, in turn, in most cases moving the single-particle energies of the underlying mean field by as much as 1.5 MeV did not affect the normalized β spectra, even though the half-life was affected significantly. Furthermore, transitions of the same type (forbiddeness, uniqueness) in neighboring nuclei produce similar spectrum shapes regardless of the Q value. The above features make the SSM a much more robust method than just comparing the theoretical and experimental half-lives to extract the effective value of g A .
The electron spectrum of the ground-state-to-ground-state decays of 99 Tc and 87 Rb depend significantly on the effective value of the axial-vector coupling constant g A . Since these decays are experimentally measurable, they can be used to gain knowledge on the effective value of g A in second-and third-forbidden nonunique beta decays by using the spectrumshape method. The decays for which the spectrum shape does not depend on the values of the weak coupling constants, such as the transitions of 135 Cs and 137 Cs, could be used to test the accuracy of the theoretical spectra.
The shape factors of the third-and especially fourthforbidden decays depend very sensitively on the value of the coupling constants. Besides the transitions in this study, there are no other medium-heavy odd-A nuclei which have a thirdor fourth-forbidden decay branch with a significant branching ratio. However, for example 50 V decays via a fourth-forbidden nonunique transition and could be a candidate for SSM when a shell-model type of framework would be used to evaluate the needed wave functions.
In general, the decomposition of the integrated shape functionsC into vector, axial-vector, and vector-axial-vector components is similar when the normalized β spectra resemble each other. However, the decompositions of the studied firstand third-forbidden nonunique decays are similar, with the exception of the one of 159 Gd, but the normalized β spectra of these decays are nothing alike.
Finally, the information gained through SSM on the effective value of g A in forbidden decay transitions could help in solving the g A problem related to the neutrinoless double beta decay, which proceeds via virtual transitions, and largely via the forbidden ones.