Electron spectra in forbidden $\beta$ decays and the quenching of the weak axial-vector coupling constant $g_A$

Evolution of the electron spectra with the effective value of the weak axial-vector coupling constant $g_{\mathrm{A}}$ was followed for 26 first-, second-, third-, fourth- and fifth-forbidden ${\beta }^{-}$ decays of odd-$A$ nuclei by calculating the involved nuclear matrix elements (NMEs) in the framework of the microscopic quasiparticle-phonon model (MQPM). The next-to-leading-order terms were included in the $\beta$-decay shape factor of the electron spectra. The spectrum shapes of third- and fourth-forbidden nonunique decays were found to depend strongly on the value of $g_{\mathrm{A}}$, while first- and second-forbidden decays were mostly unaffected by the tuning of $g_{\mathrm{A}}$. The $g_{\mathrm{A}}$-driven evolution of the normalized $\beta$ spectra was found to be quite universal, largely insensitive to the small changes in the nuclear mean field and the adopted residual many-body Hamiltonian producing the excitation spectra of the MQPM. This makes the comparison of experimental and theoretical electron spectra, coined “the spectrum-shape method” (SSM), a robust tool for extracting information on the effective values of the weak coupling constants. In this exploratory work two new experimentally interesting decays for the SSM treatment were discovered: the ground-state-to-ground-state decays of ${}^{99}\mathrm{Tc}$ and ${}^{87}\mathrm{Rb}$. Comparing the experimental and theoretical spectra of these decays could shed light on the effective values of $g_{\mathrm{A}}$ and $g_{\mathrm{V}}$ for second- and third-forbidden nonunique decays. The measurable decay transitions of ${}^{135}\mathrm{Cs}$ and ${}^{137}\mathrm{Cs}$, in turn, can be used to test the SSM in different many-body formalisms. The present work can also be considered as a (modest) step towards solving the $g_{\mathrm{A}}$ problem of the neutrinoless double beta decay.

Figure 1

Experimental and MQPM excitation spectra for the nucleus ${}^{87}\mathrm{Sr}$.

Figure 2

The $\beta$ spectra for first-forbidden nonunique ${\beta }^{-}$ decays of the ground states of ${}^{125}\mathrm{Sb}, {}^{141}\mathrm{Ce}, {}^{159}\mathrm{Gd}$, and ${}^{161}\mathrm{Tb}$. The color coding represents the value of the weak axial-vector coupling constant $g_{\mathrm{A}}$. For the vector coupling constant $g_{\mathrm{V}}$ the value 1.00 was adopted.

Figure 3

The same as Fig. 2, but for the first-forbidden nonunique decay of ${}^{169}\mathrm{Er}$ and the unique decays of ${}^{79}\mathrm{Se}, {}^{85}\mathrm{Kr}$, and ${}^{89}\mathrm{Sr}$.

Figure 4

The same as Fig. 2, but for the first-forbidden unique decays of ${}^{107}\mathrm{Pd}, {}^{125}\mathrm{Sb}$, and ${}^{137}\mathrm{Cs}$.

Figure 5

The same as Fig. 2, but for the second-forbidden unique decay of ${}^{129}\mathrm{I}$ and the nonunique decays of ${}^{93}\mathrm{Zr}, {}^{99}\mathrm{Tc}, {}^{135}\mathrm{Cs}$, and ${}^{137}\mathrm{Cs}$.

Figure 6

The same as Fig. 2, but for the third-forbidden nonunique decays of ${}^{85}\mathrm{Br}$ and ${}^{87}\mathrm{Rb}$.

Figure 7

The same as Fig. 2, but for the fourth-forbidden nonunique decays of ${}^{113}\mathrm{Cd}, {}^{115}\mathrm{Cd}$, and ${}^{115}\mathrm{In}$.

Figure 8

The same as Fig. 2, but for the fourth-forbidden nonunique decays of ${}^{97}\mathrm{Zr}, {}^{101}\mathrm{Mo}, {}^{117}\mathrm{Cd}$, and ${}^{119}\mathrm{In}$.

Figure 9

The same as Fig. 2, but for the fifth-forbidden nonunique decay of ${}^{123}\mathrm{Sn}$.