Spectrum-shape method and the next-to-leading-order terms of the $\beta$-decay shape factor

Effective values of the axial-vector coupling constant $g_{\mathrm{A}}$ have lately attracted much attention due to the prominent role of $g_{\mathrm{A}}$ in determining the half-lives of double $\beta$ decays, in particular their neutrinoless mode. The half-life method, i.e., comparing the calculated half-lives to the corresponding experimental ones, is the most widely used method to access the effective values of $g_{\mathrm{A}}$. The present paper investigates the possibilities offered by a complementary method: the spectrum-shape method (SSM). In the SSM, comparison of the shapes of the calculated and measured $\beta$ electron spectra of forbidden nonunique $\beta$ decays yields information on the magnitude of $g_{\mathrm{A}}$. In parallel, we investigate the impact of the next-to-leading-order terms of the $\beta$-decay shape function and the radiative corrections on the half-life method and the SSM by analyzing the fourfold forbidden decays of ${}^{113}\mathrm{Cd}$ and ${}^{115}\mathrm{In}$ by using three nuclear-structure theory frameworks; namely, the nuclear shell model, the microscopic interacting boson-fermion model, and the microscopic quasiparticle-phonon model. The three models yield a consistent result, $g_{\mathrm{A}}\approx 0.92$, when the SSM is applied to the decay of ${}^{113}\mathrm{Cd}$ for which $\beta$-spectrum data are available. At the same time the half-life method yields results which are in tension with each other and the SSM result.

Figure 1

Partial half-lives of the (a) ${}^{113}\mathrm{Cd}$ and (b) ${}^{115}\mathrm{In}$ decay branches as a function of the axial-vector coupling constant $g_{\mathrm{A}}$. Vector coupling constant is set to $g_{\mathrm{V}}=1$, and the partial half-lives are calculated up to second order. Experimental half-lives are given as gray horizontal lines.

Figure 2

Dependence of the next-to-leading-order contributions on the axial-vector coupling constant $g_{\mathrm{A}}$. Graphs represent the relative correction to the partial half-life stemming from the second-order terms of the shape factor.

Figure 3

$\beta$ spectra of the decays of (a)–(e) ${}^{113}\mathrm{Cd}$ and (f)–(j) ${}^{115}\mathrm{In}$ for $g_{\mathrm{V}}=1.0$. Selected values of the axial-vector coupling are used to illustrate the dependence of the shape of the spectra on the value of $g_{\mathrm{A}}$. To perform a comparison between the three nuclear models, the areas under each curve are normalized to unity.

Figure 4

The extreme effects of the next-to-leading-order terms of the shape factor on the shapes of the $\beta$ spectra of ${}^{113}\mathrm{Cd}$. The adopted values of $g_{\mathrm{A}}$ (indicated in each figure) are extracted from Fig. 2.

Figure 5

Decomposed $\beta$ spectra of the ${}^{113}\mathrm{Cd}$ decay. In the upper panels the vector ($C_{\mathrm{V}}$), axial-vector ($C_{\mathrm{A}}$) and mixed terms ($C_{\mathrm{VA}}$) of the shape factor are presented including the second-order corrections. The lower panels represent the summed $\beta$ spectra by setting the values of the weak coupling constants to unity. Abbreviation $2^{\mathrm{nd}}$ corresponds to calculations up to second order, and $1^{\mathrm{st}}$ up to first order. The scales on the $y$ axes are chosen to enable an easier visual comparison. Notice that the sign of the mixed term $C_{\mathrm{VA}}$ is negative.

Figure 6

Decomposed $\beta$ spectra of the ${}^{115}\mathrm{In}$ decay (see caption of Fig. 5).

Figure 7

Comparison of the computed $\beta$ spectra of ${}^{113}\mathrm{Cd}$ with the experiment. Computations are done including the second-order corrections, and only the best matches are shown in the figure. The canonical value $g_{\mathrm{V}}=1.0$ is used for the vector coupling constant. Notice that the $\beta$-spectrum calculations are done with a $Q$ value of $343.1\pm 0.6$ keV to enable comparison with the experimental data of Ref. [41]. The areas under the curves are normalized to unity.