Shell-model study of weak β-decays relevant to astrophysical processes

Shell-model studies on the weak $\beta$-decay in nuclei relevant to astrophysical processes are carried out. The $\beta$-decay rates, as well as electron-capture rates in the $sd$-$pf$ shell induced by Gamow–Teller (GT) transition, are evaluated in astrophysical environments. The weak rates for the Urca pair of nuclei with $A$ = 31 in the island of inversion, which are important for the nuclear Urca processes in neutron star crusts, are investigated by shell-model calculations in the $sd$–$pf$ shell. The GT strength is evaluated in the $sd$–$pf$ shell for selected $\beta$-decays in the $sd$-shell nuclei, and the effects of the expansion of the configuration space on the quenching of the axial–vector coupling are examined. $\beta$-decay rates induced by first-forbidden (FF) transitions are studied by the Behrens–Bühring (BB) method for the isotones with $N$ = 126 and compared with the Walecka method. The important role of the electron distortions in the $\beta$-decays of ²⁰⁶Hg and ²⁰⁷Tl is pointed out.

1 Introduction

Weak transition rates in stellar environments relevant to astrophysical processes in stars were evaluated with new shell-model Hamiltonians in the $sd$ shell [1] and $pf$ shell [2–4], which can describe spin responses in nuclei quite well. Electron-capture and $\beta$-decay rates thus obtained were applied to study nuclear Urca processes in ONeMg cores of stars with 8–10 ${\text{M}}_{\odot }$ [5–7] and nucleosynthesis of iron-group elements in type Ia supernova (SN) explosions [8, 9]. New shell-model calculations lead to remarkable improvements in the weak rates induced by GT transitions. The quenching of the axial–vector coupling constant is introduced to take into account the effects of the truncation of the shell-model space as well as the coupling to non-nucleonic degrees of freedom such as ${\mathrm{\Delta }}_{33}$ resonance.

Neutron-rich nuclei in the island of inversion ($sd$–$pf$ shell) [10] have been studied by shell-model [11] calculations with phenomenological interactions whose cross-shell part is constructed based on monopole-based universal interactions [12]. One of such interactions, SDPF-M [13], which induces a large admixture of $pf$-shell components, was successful in reproducing reduced excitation energies of $2_1$ states and enhanced B (E2) values. However, it failed to explain low-lying levels of ³¹Mg. The new effective interaction, EEdf1 [14, 15], obtained by the extended Kuo–Krenciglowa (EKK) method [16], is shown to be successful in explaining the structure of ³¹Mg. The weak rates for nuclei in the island of inversion are investigated in the $sd$–$pf$ shell with the use of the effective interaction, EEdf1, especially for the pair of nuclei with A = 31, ³¹Al-³¹Mg, which are important for the nuclear Urca processes in neutron star crusts [17]. The $\beta$-decay rates for $sd$-shell nuclei induced by GT transitions are evaluated by shell-model calculations in the $sd$–$pf$ shell using an effective interaction obtained by the EKK method. The effects of the extension of the configuration space on the quenching factor of $g_A$ are investigated.

$\beta$-decay and e-capture rates induced by second-forbidden transitions in ²⁰F-²⁰Ne were evaluated with the Behrens–Bühring (BB) [18, 19] and Walecka [20, 21] methods. The difference between the two methods was found to be insignificant as far as the conserved vector-current (CVC) condition was taken into account [22]. A possible important role of double e-capture reactions in ²⁰Ne on the heating of the ONeMg cores in the late stages of star evolution was discussed [22–25]. The e-capture rates induced by first-forbidden transitions in ⁷⁸Ni were studied with both the BB and the Walecka methods. The effect of electron distortion was found to be rather minor for the nucleus [22]. $\beta$-decay rates induced by first-forbidden transitions were studied with the BB method for the isotones with $N$ = 126 and applied to r-process nucleosynthesis [26–28].

Here, the $\beta$-decay rates induced by first-forbidden (FF) transitions in ²⁰⁶Hg and ²⁰⁷Tl are investigated with both the BB and the Walecka methods, and the two methods are compared. The effects of the electron distortion are examined.

2 $\beta$-decay and e-capture rates induced by GT transitions

2.1 Weak rates in stellar environments

The $\beta$-decay rate at finite density and temperature is given as follows in the multipole expansion method by Walecka [20, 21]:

$\begin{array}{l}{\lambda }^{\beta }\left(T\right)=\frac{V_{ud}^2{g}_V^{2}c}{{\pi }^2{\left(\hslash c\right)}^3}{\displaystyle \sum _i}{\displaystyle \sum _f}{\int }_{m_e{c}^2}^{Q_{if}}{S}_{f,i}\left(E_e,T\right)E_e{p}_{e}c{\left(Q_{if}-E_e\right)}^2\left(1-f\left(E_e\right)\right)d{E}_e{S}_{f,i}\left(E_e,T\right)\\ =\frac{\left(2J_i+1\right)e^{-E_i/kT}}{{\sum }_j\left(2J_j+1\right)e^{-E_j/kT}}\frac{G_F^2}{2\pi }F\left(Z+1,E_e\right)C_{f,i}\left(E_e\right)C_{f,i}\left(E_e\right)\\ =\int \frac{1}{4\pi }d{\mathrm{\Omega }}_{\nu }\int d{\mathrm{\Omega }}_k\frac{1}{2J_i+1}\left({\displaystyle \sum _{J\ge 1}}\left\{\left(1-\left(\widehat{\vec{\nu }}\cdot \widehat{\vec{q}}\right)\left(\vec{\beta }\cdot \widehat{\vec{q}}\right)\left[{|\langle J_f\Vert T_J^{\mathit{mag}}\Vert J_i\rangle |}^2\right.\right.\right.\right.\\ +\left.\left.{|\langle J_f\Vert T_J^{\mathit{elec}}\Vert J_i\rangle |}^2\right]+2\widehat{\vec{q}}\cdot \left(\widehat{\vec{\nu }}-\vec{\beta }\right)\mathrm{R}\mathrm{e}\langle J_f\Vert T_J^{\mathit{mag}}\Vert J_i\rangle \langle J_f\Vert T_J^{\mathit{elec}}\Vert J_i{\rangle }^{*}\right\}\\ +{\displaystyle \sum _{J\ge 0}}\left\{\left(1-\widehat{\vec{\nu }}\cdot \vec{\beta }\right)+2\left(\widehat{\vec{\nu }}\cdot \widehat{\vec{q}}\right)\left(\vec{\beta }\cdot \widehat{\vec{q}}\right)|\langle J_F\Vert L_J\Vert J_i\rangle {|}^2\right.\\ +\left(1+\widehat{\vec{\nu }}\cdot \vec{\beta }\right)|\langle J_f\Vert M_J\Vert J_i\rangle {|}^2-\\ \left.\left.2\widehat{\vec{q}}\cdot \left(\widehat{\vec{\nu }}+\vec{\beta }\right)\mathrm{R}\mathrm{e}\langle J_f\Vert L_J\Vert J_i\rangle \langle J_f\Vert M_J\Vert J_i{\rangle }^{*}\right\}\right),\end{array}(1)$

where $V_{ud}=\cos {\theta }_C$ is the up–down element in the Cabibbo–Kobayashi–Maskawa quark mixing matrix with ${\theta }_C$ the Cabibbo angle; $g_V=1$ the weak vector coupling constant; $E_e$ and $p_e$ are electron energy and momentum, respectively; and $f(E_e)$ is the Fermi–Dirac distribution for the electron. $G_F$ is the Fermi coupling constant, $F(Z+1,E_e)$ is the Fermi function, and $\vec{q}$ = $\vec{k}+\vec{\nu }$ with $\vec{\nu }$ and $\vec{k}$ are the neutrino and electron momenta, respectively, $\widehat{\vec{q}}$ and $\widehat{\vec{\nu }}$ are the corresponding unit vectors, and $\vec{\beta }$ = $\vec{k}/E_e$. $E_i$ $(J_i)$ and $E_f$ $(J_f)$ are the excitation energies (spins) of initial and final nuclear states, respectively. The $Q$ value is determined from $Q_{if}=M_i-M_f$, where $M_i$ and $M_f$ are the masses of parent and daughter nuclei, respectively. The Coulomb, longitudinal, transverse magnetic, and electric multipole operators with multipolarity $J$ are denoted as $M_J$, $L_J$, $T_J^{\mathit{mag}}$, and $T_J^{\mathit{elec}}$, respectively, and the factor $1-f(E_e)$ denotes the blocking of the decay by electrons in high-density matter.

In the case of an allowed GT transition, the sum of the axial electric dipole and axial longitudinal dipole terms contribute to the rate, and the shape factor $C_{f,i}(E_e)$ becomes independent of the electron energy.

$C_{f,i}\left(E_e\right)=B_{if}\left(\text{GT}\right)={\left(g_A/g_V\right)}^2\frac{1}{2J_i+1}|⟨f\Vert {\displaystyle \sum _k}{\sigma }^k{t}_{-}^k\Vert i⟩{|}^2,(2)$

where $J_i$ is the total spin of the initial state and $t_{-}|n\rangle =|p\rangle$. This formula for the allowed transition given by Eq. 2 is equivalent to that in [3, 4, 29], which is based on the Behrens–Bühring method [18].

The e-capture rate at finite density and temperature is given by changing the integral in the first line of Eq. 1 as [20, 21], ${\int }_{E_{th}}^{\infty }{S}_{f,i}(E_e,T)E_e{p}_{e}c{E}_{\nu }^{2}f(E_e)d{E}_e,$, where $E_{th}$ is the threshold energy for the electron capture and $E_{\nu }=E_e+Q_{if}+E_i-E_f$ is the neutrino energy. $F(Z+1,E_e)$ is replaced by $F(Z,E_e)$ in the second line of Eq. 1. The shape factor $C_{f,i}(E_e)$ is expressed in the same way as shown in Eq. 1, except that an integral $\frac{1}{4\pi }\int d{\mathrm{\Omega }}_k$ is replaced by 1. $\vec{q}=\vec{\nu }-\vec{k}$ is the momentum transfer, and the phase of the lepton matrix elements in the interference term of magnetic and electric form factors is reversed. In the nuclear transition matrix, $t_{-}$ is replaced by $t_{+}$ and $t_{+}|p\rangle =|n\rangle$.

Electron-capture and $\beta$-decay rates in the $sd$ shell were evaluated with the USDB Hamiltonian [1], with the quenching of the axial–vector coupling ($q_A$ = $g_A^{\mathit{eff}}/g_A^{\mathit{free}}$ = 0.764 [30]) at high temperatures ($T$ = $10^8$ -$10^{10}$ K) and high densities ($\rho Y_e$ = $10^8$ -$10^{10}$ g ${\text{cm}}^{-3}$ with $Y_e$ the electron fraction) and applied to nuclear Urca processes in ONeMg cores. The e-capture rates increase, while the $\beta$-decay rates decrease, as the density increases due to the increase in electron chemical potential at high densities. Both the weak rates coincide at a certain density, called an Urca density, almost independent of temperatures. Both $\nu$ and $\bar{\nu }$ are emitted at the Urca density, thus taking away the energy from the star, which results in a drastic cooling of the core of the star. This mechanism, called the nuclear Urca process, occurs quite efficiently for the nuclear pairs with $A$ = 23 and 25, where the transitions between the ground states (g.s.s) are GT ones [5, 6]. The weak rates for the nuclear pairs, ²³Na–²³Mg and ²⁵Mg–²⁵Na, and the cooling of the ONeMg core of a star with 8.8 ${\text{M}}_{\odot }$ were studied in Refs [5, 7].

2.2 Weak rates of nuclei in the island of inversion

Urca processes for nuclear pairs in the island of inversion [10] such as ³¹Mg–³¹Al and ³³Mg–³³Al pairs have been pointed out to be important for the cooling of neutron star crusts [17]. We discuss the weak rates of the ³¹Mg–³¹Al pair. The SDPF-M interaction fails to reproduce the energy levels of ³¹Mg, that is, ${7/2}^{-}$ state becomes the g.s., while the experimental g.s. is ${1/2}^{+}$. The Urca density cannot be clearly assigned for the weak rates for SDPF-M, as the transitions between the g.s.’s are forbidden. This shortcoming can be improved for the effective interaction obtained by the EKK (extended Kuo–Krenciglowa) method [14], starting from the chiral EFT ${\text{N}}^3$LO [31] and Fujita–Miyazawa 3N interaction [32]. The EKK method can treat Q-box calculations in two major shells without divergence problems [16]. For this interaction, referred to as EEdf1 [15], neutron effective single-particle energies between $sd$-shell and $pf$-shell orbits become much closer in the neutron-rich region, $Z$ = 10–12, compared with the conventional $sd$–$pf$ shell Hamiltonian, SDPF-M [13]. This results in larger admixtures of $pf$-shell components for the EEdf1. Including up to 6p–6h excitations, energy levels of ³¹Mg can be well-explained by the EEdf1 [14, 15]. The g.s. of ³¹Mg is calculated to be ${1/2}^{+}$, which is consistent with the experimental observation [33]. The first excited state is predicted to be ${3/2}^{+}$, which is very close to the g.s. ${1/2}^{+}$. As the g.s. of ³¹Al is ${5/2}^{+}$, the GT transition between the ${3/2}^{+}$ state in ³¹Mg and ${5/2}_{g.s.}^{+}$ in ³¹Al gives the main contribution to the e-capture and $\beta$-decay rates for the $A$ = 31 pair. The weak rates in stellar environments obtained with the EEdf1 are shown in Figure 1. The GT transitions between ³¹Mg (${3/2}^{+}$, ${1/2}^{+}$) and ³¹Al (${5/2}^{+}$, ${1.2}^{+}$, ${3/2}^{+}$) are taken into account. The free value for $g_A$ is used as the shell-model space is large. There exists an Urca density at ${\text{log}}_{10}$($\rho Y_e$) = 10.14, as shown in Figure 1 (left panel) for the EEdf1, since the excitation energy of the ${3/2}^{+}$ state in ³¹Mg is as small as 0.05 MeV. If the g.s. of ³¹Mg is taken to be ${7/2}^{-}$, there does not exist an Urca density, as shown in Figure 1 (right panel), because of the non-existence of GT transitions between low-lying states. The transitions between the g.s.’s of ³¹Mg (${1/2}^{+}$) and ³¹Al (${5/2}^{+}$) are second-forbidden transitions. Their rates can be evaluated with the method explained in Refs [22, 23], and their contributions to the weak rates prove to be quite tiny and negligible in contrast to the case for the ²⁰Ne ($0^{+}$)-²⁰F ($2^{+}$) pair.

Figure 1

Figure 1. $\beta$-decay and e-capture rates for the nuclear pair, ³¹Mg–³¹Al, as a function of density ${\text{log}}_{10}$ $(\rho Y_e)$ for various temperatures; ${\text{log}}_{10}$(T) = 8.0, 8.10–8.85 (in steps of 0.15), 8.95, 9.05, and 9.15. The $\beta$-decay (e-capture) rates decrease (increase) as the density increases. The left figure shows the rates evaluated with the EEdf1 interaction obtained by the EKK method [14]. The right figure shows the rates obtained with the SDPF-M Hamiltonian [13].

2.3 $\beta$-decay strengths of $sd$-shell nuclei in $sd$–$pf$ shell configurations

Although $\beta$-decay rates in $sd$-shell nuclei are usually evaluated within the $sd$ shell with a quenching for the axial–vector coupling, $q_A$ = 0.764 for USDB [30]; for example, we study here $\beta$-decay strengths of $sd$-shell nuclei in an extended shell-model space, that is, in $sd$-$pf$ shell. An effective interaction obtained with the EKK method is used. A modified version of EEdf1, which will be referred to as EEdf2 [34], is used. In EEdf2, the chiral ${\text{N}}^2$LO three-nucleon interaction [35] is adopted instead of the Fujita–Miyazawa force. The following $\beta$-decay transitions treated in Ref. [36] except for ³⁴P $\to$³⁴S and four additional ones with A = 21 and 23, ²¹Na (${3/2}^{+}$) $\to$²¹Ne (${3/2}^{+}$), ²³Mg (${3/2}^{+}$) $\to$²³Na (${3/2}^{+}$), ²³Mg (${3/2}^{+}$) $\to$²³Na (${5/2}^{+}$), and ²³Ne (${5/2}^{+}$) $\to$²³Na (${3/2}^{+}$) are examined. The quenching factor for $g_A$ is obtained by chi-squared fittings to the experimental data of the GT matrix element, which is defined as

$\begin{array}{ccc}\hfill M\left(GT\right)& =\hfill & \sqrt{\left(2J_i+1\right)B\left(GT\right)}\hfill \\ \hfill B\left(GT\right)& =\hfill & \frac{1}{2J_i+1}|⟨f\Vert {\displaystyle \sum _k}{\sigma }^k{t}_{\mp }^k\Vert i⟩{|}^2,\hfill \end{array}(3)$

where $J_i$ is the spin of the initial state. The quenching factor for $g_A$ is obtained to be $q_A$ = 0.86 $\pm$ 0.06 for the EEdf2 for the configurations including up to 2p–2h excitations outside the $sd$ shell. The quenching factor is obtained to be $q_A$ = 0.81 $\pm$ 0.02 for the USDB in the $sd$ shell. Calculated $M(GT)$ for the EEdf2 and USDB as well as the experimental data [37, 38] are shown in Figure 2 and Table 1. The quenching factor for $g_A$ in the $sd$–$pf$ shell is found to become closer to $q_A$ = 1, compared with the case within the $sd$ shell. The inclusion of more transitions is in progress. When about 90 more transitions in nuclei with $A$ = 19–34 are included, $q_A$ for EEdf2 remains higher than that for USDB by $\sim$0.05, while the latter comes close to $q_A$ = 0.77, which is consistent with the value reported in Ref. [30] for USDB [39].

Figure 2

Figure 2. Gamow–Teller matrix elements $M(GT)$ obtained with the EEdf2 in the $sd$–$pf$ shell and the USDB in the $sd$ shell are compared with the experimental values. Filled triangles and circles show the EEdf2 and USDB cases, respectively. Horizontal axis values correspond to the calculated values with the quenching factors of ${\text{q}}_A$ = 0.86 (0.81) for the EEdf2 (USDB), while the vertical axis values denote the experimental values. Solid, dashed, and dash-dotted lines show the cases with $q_A$ = 0.86, 0.81, and 1.0, respectively.

Table 1

Table 1. Calculated values of the Gamow–Teller matrix elements (Eq. 3) obtained by the EEdf2 and USDB interactions as well as the experimental values. Numbers in the parentheses for the experimental values (EXP.) show experimental errors.

An ab initio calculation with the valence-space in-medium renormalization group (VS-IMSRG) approach gives $q_A$ = 0.89 $\pm$ 0.04 and $q_A$ = 0.96 $\pm$ 0.06 for the case without and with the two-body current contributions, respectively [36]. The quenching factor would come closer to $q_A$ = 1 with the two-body current contributions.

3 $\beta$-decay rates induced by first-forbidden transitions

The shape factors in the low momentum transfer limit obtained by the Walecka method are given as follows [22]:

$\begin{array}{c}{C}_{\beta }^{0^{-}}={\left({\xi }^{\prime }v+\frac{1}{3}w{W}_0\right)}^2,C_{\beta }^{1^{-}}={\left[{\xi }^{\prime }y+\frac{1}{3}\left(u-x\right)W_0\right]}^2+\frac{1}{18}{W}_0^2{\left(u+2x\right)}^2\hfill \\ +W\left[-\frac{4}{3}{\xi }^{\prime }yu-\frac{W_0}{9}\left(4x^2+5u^2\right)\right]+\frac{W^2}{9}\left(4x^2+5u^2\right)\hfill \\ C_{\beta }^{2^{-}}=\frac{1}{3}{z}^2\left\{{\left(W_0-W\right)}^2+W^2-1\right\},\hfill \end{array}(4)$

where

$\begin{array}{l}{\xi }^{\prime }v=-\frac{\sqrt{3}}{\sqrt{2J_i+1}}{g}_A\langle f\Vert \frac{1}{M}{\left[\vec{\sigma }\times \vec{\nabla }\right]}^{\left(0\right)}\Vert i\rangle ,\hfill \\ w=-\frac{\sqrt{3}}{\sqrt{2J_i+1}}{g}_A\langle f\Vert r{\left[C^1\left(\mathrm{\Omega }\right)\times \vec{\sigma }\right]}^{\left(0\right)}\Vert i\rangle {\xi }^{\prime }y=\frac{1}{\sqrt{2J_i+1}}\langle f\Vert \frac{\vec{\nabla }}{M}\Vert i\rangle ,x=\frac{1}{\sqrt{2J_i+1}}\langle f\Vert r{C}^1\left(\mathrm{\Omega }\right)\Vert i\rangle \hfill \\ u=\frac{\sqrt{2}}{\sqrt{2J_i+1}}{g}_A\langle f\Vert r{\left[C^1\left(\mathrm{\Omega }\right)\times \vec{\sigma }\right]}^{\left(1\right)}\Vert i\rangle ,z=\frac{1}{\sqrt{2J_i+1}}{g}_A\langle f\Vert r{\left[C^1\left(\mathrm{\Omega }\right)\times \vec{\sigma }\right]}^{\left(2\right)}\Vert i\rangle ,\hfill \end{array}(5)$

with $W$ as the electron energy (=$E_e$). Here, $W_0=|Q|$, where $Q$ is the $Q$-value for the reaction and $J_i$ is the angular momentum of the initial state. The matrix elements, $w$, $u$, and $z$, are contributions from spin-dipole transitions. $x$, ${\xi }^{\prime }y$, and ${\xi }^{\prime }v$ are Coulomb, transverse electric, and ${\gamma }_5$ terms, respectively. The relation, ${\xi }^{\prime }y$ = $\mathrm{\Delta }{E}_{fi}x$ with $\mathrm{\Delta }{E}_{fi}=E_f-E_i$, is satisfied from the CVC.

In the Behrens–Bühring (BB) method, distorted electron wave functions are used, which results in extra interference terms between the operators and the electron wave functions: ${\xi }^{\prime }v\to {\xi }^{\prime }v+\xi w^{\prime }$, where $\xi$ = $\alpha Z/2R$ with $\alpha$ the fine structure constant, for ${\lambda }^{\pi }$ = $0^{-}$, and ${\xi }^{\prime }y\to {\xi }^{\prime }y-\xi (u^{\prime }+x^{\prime })$ for ${\lambda }^{\pi }$ = $1^{-}$ (see Refs [18, 22] for the details). When these distortion effects are added to Walecka’s formulas, Eqs 4, 5, the method will be referred to as “Walecka with distortion.” Moreover, the following higher-order terms are usually added in the BB method. They can become important when dominant terms cancel to each other.

$\begin{array}{c}\delta C_{{\beta }^{-}}^{0^{-}}=-\frac{2}{3}{\mu }_1{\gamma }_1\left({\xi }^{\prime }v+\xi w^{\prime }+\frac{1}{3}w{W}_0\right)w/W+\frac{1}{9}{w}^2\hfill \\ \delta C_{{\beta }^{-}}^{1^{-}}=\frac{1}{9}{\left(x+u\right)}^2-\frac{{\lambda }_2}{18}{\left(2x-u\right)}^2-\frac{4}{9}{\mu }_1{\gamma }_{1}u\left(x+u\right)\hfill \\ +\frac{1}{18}{W}^2\left({\lambda }_2-1\right){\left(2x-u\right)}^2+\frac{2}{3}{\mu }_1{\gamma }_1\left({\xi }^{\prime }y-\xi \left(u^{\prime }+x^{\prime }\right)\right)\left(x+u\right)/W\hfill \\ \delta C_{{\beta }^{-}}^{2^{-}}=\frac{1}{3}{z}^2\left({\lambda }_2-1\right)\left(W^2-1\right),\hfill \end{array}(6)$

where ${\gamma }_1$ = $\sqrt{1-{(\alpha Z)}^2}$ and ${\lambda }_2$ and ${\mu }_1$ are distortion parameters, which are usually taken to be 1.0. The values of ${\lambda }_2$ and ${\mu }_1$ are close to 1, but ${\lambda }_2$ can become as small as 0.7 in the low electron momentum region for Z $\approx$ 80 [40]. $x^{\prime }$, $u^{\prime }$, and $w^{\prime }$ are modified from x, u, and w, respectively, by taking account of the finite-size effect of the nucleus. The $\beta$-decay rate $\lambda$ is obtained from the shape factors, and the half-life is given by $t_{1/2}=\frac{ln2}{\lambda }$.

The shape factors and log $ft$ values are evaluated by (A) the BB method, (B) BB method with ${\lambda }_2$ = ${\mu }_1$ = 1.0, and (C) the BB method with ${\lambda }_2$ = ${\mu }_1$ = 1.0, but without the subdominant term (Eq. 6), which is equivalent to the Walecka method with distortion effects added ($\xi \ne$ 0): “Walecka with distortion” and (D) Walecka method (without the distortion ($\xi$ = 0); Eqs 4, 5), and they are compared to each other. Calculated results of the averaged shape factors [26] and log $ft$ values for $\beta$-decays in ²⁰⁶Hg and ²⁰⁷Tl are shown in Table 2. Shell-model calculations are performed with the same modified G-matrix and model space, as used in Refs [26, 28]. A closed $N$ = 126 core is assumed for the parent nucleus. For proton holes, full configurations with the 0$h_{11/2}$, 0$g_{7/2}$, 1$d_{5/2}$, 1$d_{3/2}$, and 2$s_{1/2}$ orbits are taken into account. The quenching factors for the axial-vector and vector coupling constants are taken to be $q_A$ = 0.34 and $q_V$ = 0.68, respectively [26, 41], and the enhancement factor for the ${\gamma }_5$ term in $0^{-}$ transition is taken to be $q_A$ = 1.75 [26, 42]. Similar large quenching of $g_A$ and $g_V$ in $1^{-}$ and $2^{-}$ transitions was also reported in Ref. [27].

Table 2

Table 2. Calculated square roots of the averaged shape factors and log $ft$ values for the $\beta$-decays in²⁰⁶Hg and²⁰⁷Tl obtained by the BB method, the BB method with an approximation with ${\lambda }_2$ = ${\mu }_1$ = 1, the Walecka method with electron distortion effects, and the Walecka method without the distortion effects.

As we can see from Table 2, the approximation to use ${\lambda }_2$ = ${\mu }_1$ = 1 is good enough, and the Walecka method with the electron distortion, $\xi \ne 0$, is satisfactory, while the deviation from the results of the BB method becomes large when the distortion is switched off in the Walecka method.

4 Summary and discussion

The new effective interaction in the $sd$–$pf$ shell obtained by the EKK method [14, 16] from fundamental interactions [31, 32, 35] proves to be successful in the description of the structure in the island of inversion [10] and is used to evaluate the $\beta$-decay and e-capture rates for the nuclear pair, ³¹Mg–³¹Al, in stellar environments. The Urca density for the pair can be assigned because dominant transitions between low-lying states are induced by GT transition. This leads to nuclear Urca processes in neutron star crusts [43]. The quenching of the axial–vector coupling constant in selected $sd$-shell nuclei is examined with the use of the effective interaction in the $sd$-$pf$ shell. The extension of the model space to the $sd$–$pf$ shell is found to enhance the quenching factor by $\sim$0.05 compared to the conventional Hamiltonians within the $sd$ shell. More systematic studies including more $sd$-shell nuclei with contributions from two-body currents [36] are an interesting future issue.

$\beta$-decays in ²⁰⁶Hg and ²⁰⁷Tl induced by first-forbidden transitions are studied with both the Behrens–Bühring (BB) [18] and the Walecka [20, 21] methods. The Walecka method with electron distortion corrections is shown to give results close to those of the BB method for the averaged shape factors and log $ft$ values. Unless accidental cancellations among the dominant terms take place, the Walecka method with the distortion corrections, simpler and more accessible than the BB method, can be a useful approximation with enough accuracy even in the Z $\approx$ 80 region. It would be interesting to find out to what extent this statement is valid.

Data availability statement

The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation.

Author contributions

TS: writing–original draft and writing–review and editing. NS: methodology, software, and writing–review and editing.

Funding

The authors declare that financial support was received for the research, authorship, and/or publication of this article. NAT Research Center; evaluation of the weak rates, shell-model calculations, publication charge, MEXT, Japan (JPMXP1020230411) “Program for promoting research on supercomputer FUGAKU,” and KAKENHI 24H00239; shell-model calculations, publication charge.

Acknowledgments

The authors would like to acknowledge the support from the “Program for promoting research on the supercomputer Fugaku,” MEXT, Japan (JPMXP1020230411), and KAKENHI budget 24H00239. TS would also like to acknowledge support from the NAT Research Center.

Conflict of interest

Author TS was employed by NAT Corporation.

The remaining author declares that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Publisher’s note

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