J-PARC-TH-0240, KUNS-2860

1]Department of Physics, Hokkaido University, Sapporo 060-0810, Japan

2]Research Center for Nuclear Physics, Osaka University, Ibaraki, Osaka
567-0047, Japan 3]J-PARC Branch, KEK Theory Center, Institute of Particle and
Nuclear Studies (KEK) and Theory Group, Particle and Nuclear Physics Division,
J-PARC Center, Tokai, Ibaraki, 319-1106, Japan

4]Faculty of Science and Engineering, Kyushu Sangyo University, Fukuoka
813-8503, Japan

5]Department of Physics, Kyoto University, Kyoto 606-8502, Japan

# Beta-decay formulas revisited (I): Gamow–Teller and spin-dipole
contributions to allowed and first-forbidden transitions

Wataru Horiuchi  [  Toru Sato  [  [  Yuichi Uesaka  [  Kenichi Yoshida  [

###### Abstract

We propose formulas of the nuclear beta-decay rate that are useful in a
practical calculation. The decay rate is determined by the product of the
lepton and hadron current densities. A widely used formula relies upon the
fact that the low-energy lepton wave functions in a nucleus can be well
approximated by a constant and linear to the radius for the $s$-wave and
$p$-wave wave functions, respectively. We find, however, the deviation from
such a simple approximation is evident for heavy nuclei with large $Z$ by
numerically solving the Dirac equation. In our proposed formulas, the neutrino
wave function is treated exactly as a plane wave, while the electron wave
function is obtained by iteratively solving the integral equation, thus we can
control the uncertainty of the approximate wave function. The leading-order
approximation gives a formula equivalent to the conventional one and
overestimates the decay rate. We demonstrate that the next-to-leading-order
formula reproduces well the exact result for a schematic transition density as
well as a microscopic one obtained by a nuclear energy-density functional
method.

##  1 Introduction

The physics of exotic nuclei away from the stability line has been a major
subject in nuclear physics. The lifetime of neutron-rich nuclei is governed by
beta decay. Since the beta decay determines the time scale of the rapid-
neutron-capture process ($r$-process) and the production of heavy elements
together with the beta-delayed neutron(s) emission, the beta-decay rates of
exotic nuclei are an important microscopic input for the simulation of
nucleosynthesis lan03 . The multi-messenger observations from a binary neutron
star merger abb17a ; abb17b  imply that heavy neutron-rich nuclei that are
even close to the drip line are involved in the $r$-process. Thus, the Coulomb
effect on the beta particle (emitted electron) should be carefully examined
under the extreme environment where the $Q$ value for the beta decay,
$Q_{\beta}$, is high and the nuclear charge $Z$ is large.

A careful analysis of the Coulomb effect is also useful for a precision test
of the standard model to find a signal of new physics. For example, the effect
on spectra of the beta particle and angular correlation as well as beta-decay
rates has been studied to test the unitarity of the Cabibbo–Kobayashi–Maskawa
(CKM) matrix, the scalar and tensor interactions, and the effect of neutrino
mass in the allowed and first-forbidden transitions Ando:2009jk ; Glick-
Magid:2016rsv ; Gonzalez-Alonso:2018omy ; Cirgiliano:2019nyn .

The formulation of nuclear beta decay within the distorted-wave impulse
approximation of the electron Coulomb interaction has been matured
stech1964nuclear ; schulke1964nuclear ; Behrens:1971rbq ; sch66 ;
Schopper:1969jkp ; Morita:1963zz ; morita1973beta ; Koshigiri:1979zd . The
crucial part is how to handle the electron Coulomb wave function with a
potential of the finite-size nuclear-charge distribution. Using the Maclaurin
expansion of the nuclear radius $r$, the exact electron wave function was
included in Ref. buhring1963beta . An iterative solution of the integral
equation was found to have a better convergence by Behrens and Bühling
rose1951note ; Behrens:1971rbq . The formula is arranged in the order of
${\cal O}(r^{a}V_{C}^{b}E_{e}^{c}m_{e}^{d})$, where $V_{C}$, $E_{e}$ and
$m_{e}$ represent the Coulomb potential, the energy and the mass of an
electron, respectively. It has been widely used in the calculations such as in
Refs. gov71 ; war88 ; war91  and in the recent application to the $r$-process
nuclei eng99 ; mol03 ; bor00 ; bor03 ; bor06 ; cue07 ; suz12 ; niu12 ; zhi13 ;
mus16 ; mar16 ; ney20 . In most of those works, however, the leading-order
approximation of the formula in Refs. Behrens:1971rbq ; Schopper:1969jkp  is
adopted. Instead of expanding the lepton wave functions, one can incorporate
the numerical solution of the charged lepton wave functions thanks to the
advance of the computational ability. The muon capture morita1960theory  and
the beta decay Koshigiri:1979zd  are formulated suitable for this purpose. In
this formulation, the nuclear matrix element is defined in a transparent way
and appears similarly in Refs. Nakamura:2000vp ; walecka2012semileptonic  for
the semi-leptonic nuclear processes and electron scattering DeForest:1966ycn .
It is thus straightforward to apply it to the charged-current neutrino
reaction and lepton capture reaction. Developing an analytic formula of beta
decay based on Ref. Koshigiri:1979zd  would also contribute to a precise
understanding of the neutrino-nucleus reactions to extract neutrino properties
from neutrino experiments as discussed in Ref. nak17 ; Alvarez-Ruso:2017oui .

The high-energy forbidden transitions occur under the exotic environment with
high $Q_{\beta}$ value yos17 . Therefore, in this work we revisit the
formulation of beta decay for not only the allowed but the first-forbidden
transitions induced by the Gamow–Teller and spin-dipole type operators. We
provide a simple way to improve the widely used formula in the nuclear beta-
decay study to apply for nuclei with large $Z$ and away from the stability
line. We start from the formulation of Koshigiri et al. Koshigiri:1979zd  and
use iterative solutions of the integral equation rose1951note ;
Behrens:1971rbq . In the previous formalism, one often expands the electron
and neutrino wave functions in the long wave-length approximation and collect
terms in a systematic way. Here we avoid this expansion of the neutrino wave
function. We use an analytic form of the LO and NLO electron wave functions
combined with the numerical table of the electron wave function at the origin.
This makes the formula of the beta-decay rate simple and easy to use.

This paper is organized in the following way. We start from the formulation of
the beta decay with the partial wave expansion for the lepton wave functions
in Sect. 2. We provide an explicit expression of the first (LO) and the second
(NLO) iteration of the integral equation for an electron wave function in
Sect. 3. Formulas of the beta-decay rate are given and compared with the
widely used one in Sect. 4. The formulas of LO and NLO are examined in Sect.
5, using a schematic transition density that is given by a sum of two
Gaussians. We then in Sect. 6 apply the formulas to the neutron-rich Ni and Sn
isotopes where the transition densities are microscopically obtained by a
nuclear energy-density functional (EDF) method. Summary and perspectives are
given finally in Sect. 7.

##  2 Formalism

###  2.1 Effective Hamiltonian

An effective Hamiltonian for a low-energy charged-current reaction is given as

$\displaystyle H_{\mathrm{eff}}=\frac{G_{F}V_{ud}}{\sqrt{2}}\int
d\bm{x}[\bar{\ell}(\bm{x})\gamma^{\mu}(1-\gamma_{5})\nu_{\ell}(\bm{x})J_{\mu}(\bm{x})+\bar{\nu}_{\ell}(\bm{x})\gamma^{\mu}(1-\gamma_{5})\ell(\bm{x})J_{\mu}^{\dagger}(\bm{x})],$
(1)

where $\ell(\bm{x})$ represents either the electron, muon, or tau field and
$\bar{\psi}=\psi^{\dagger}\gamma^{0}$. The hadron current $J^{\mu}(\bm{x})$ is
given by the vector and axial vector currents

$\displaystyle J^{\mu}(\bm{x})=V^{\mu}(\bm{x})-A^{\mu}(\bm{x}),$ (2)

where $G_{\mathrm{F}}=1.166\times 10^{-5}\mathrm{GeV}^{-2}$ is the Fermi
coupling constant and $V_{ud}=0.9737$ is the CKM matrix Zyla:2020zbs . Here we
take natural units $\hbar=c=1$.

The effective Hamiltonian describes semi-leptonic nuclear weak processes such
as lepton capture, neutrino reaction, and $\beta^{\pm}$ decay. For $\beta^{-}$
decay, $i\rightarrow e^{-}(p_{e})+\bar{\nu}_{e}(p_{\nu})+f$, where $i$ and $f$
respectively denote the initial and final nuclear states, and $p_{\ell}$ is
the lepton momentum, the transition matrix element is given as

$\displaystyle\left<e^{-}(p_{e})\bar{\nu}_{e}(p_{\nu})f\right|H_{\mathrm{eff}}\left|i\right>=\frac{G_{F}V_{ud}}{\sqrt{2}}\int
d\bm{x}\bar{\psi}_{e^{-},p_{e},s_{e}}^{(-)}(\bm{x})\gamma^{\mu}(1-\gamma_{5})v_{s_{\nu}}(\bm{p}_{\nu})e^{-i\bm{p}_{\nu}\cdot\bm{x}}\left<f\right|J_{\mu}(\bm{x})\left|i\right>,$
(3)

and for $\beta^{+}$ decay

$\displaystyle\left<e^{+}(p_{e})\nu_{e}(p_{\nu})f\right|H_{\mathrm{eff}}\left|i\right>=\frac{G_{F}V_{ud}}{\sqrt{2}}\int
d\bm{x}\bar{u}_{s_{\nu}}(\bm{p}_{\nu})e^{-i\bm{p}_{\nu}\cdot\bm{x}}\gamma^{\mu}(1-\gamma_{5})\psi_{e^{+},p_{e},s_{e}}^{(+)}(\bm{x})\left<f\right|J_{\mu}^{\dagger}(\bm{x})\left|i\right>,$
(4)

where $u$ and $v$ are the Dirac spinors of the neutrino and antineutrino,
respectively. The electron scattering wave functions $\psi$ with the
superscript $(-)$ and $(+)$ satisfy the incoming and outgoing boundary
conditions, respectively.

###  2.2 Multipole Expansion of the Effective Hamiltonian

The standard formulation of the beta decay adopts the partial wave expansion
of both neutrino and electron wave functions. We use the following electron
(charged lepton in general) scattering wave function

$\displaystyle\psi_{e,p_{e},s_{e}}^{(\mp)}(\bm{x})=\sum_{\kappa_{e},m_{e},\mu_{e}}(4\pi)i^{l_{\kappa_{e}}}(l_{\kappa_{e}},m_{e},1/2,s_{e}|j_{\kappa_{e}},\mu_{e})Y^{*}_{l_{\kappa_{e}},m_{e}}(\hat{p}_{e})e^{\mp
i\Delta_{\kappa_{e}}}\left(\begin{array}[]{c}G_{\kappa_{e}}(r)\chi_{\kappa_{e}}^{\mu_{e}}\\\
iF_{\kappa_{e}}(r)\chi_{-\kappa_{e}}^{\mu_{e}}\end{array}\right).$ (7)

Here, $(j_{1},m_{1},j_{2},m_{2}|J,M)$ is the Clebsch–Gordan coefficient
de1974nuclear ; rose1957ang ; condon1935theory . For positron,
$G_{\kappa_{e}},F_{\kappa_{e}}$ are calculated by replacing $Z$ of the Coulomb
interaction by $-Z$. $\Delta_{\kappa_{e}}$ is the Coulomb phase. The
normalization of the scattering wave function in the plane wave expansion
$\psi_{e}(\bm{x})\rightarrow u(p_{e})\exp(i\bm{p}_{e}\cdot\bm{x})$ is given as

$\displaystyle
u_{s_{e}}(p_{e})=\sqrt{\frac{E_{e}+m_{e}}{2E_{e}}}\left(\begin{array}[]{c}1\\\
\frac{\bm{\sigma}\cdot\bm{p}_{e}}{E_{e}+m_{e}}\end{array}\right)\chi_{s_{e}}.$
(10)

It is noticed the electron wave functions $(G_{\kappa},F_{\kappa})$ in Ref.
Koshigiri:1979zd  are defined by multiplying $e^{i\Delta_{\kappa_{e}}}$ to
ours, while those of Refs. buhring1963beta ; Behrens:1971rbq  are given by
multiplying $\sqrt{2}p_{e}$ to ours.

The neutrino and antineutrino wave functions are respectively expanded as

$\displaystyle u_{s_{\nu}}(\bm{p}_{\nu})e^{i\bm{p}_{\nu}\cdot\bm{r}}$
$\displaystyle=\sum_{\kappa_{\nu},m_{\nu},\mu_{\nu}}\frac{4\pi}{\sqrt{2}}i^{l_{\kappa_{\nu}}}Y_{l_{\kappa_{\nu}}m_{\nu}}^{*}(\hat{p}_{\nu})(l_{\kappa_{\nu}},m_{\nu},1/2,s_{\nu}|j_{\kappa_{\nu}},\mu_{\nu})\left(\begin{array}[]{c}g_{\kappa_{\nu}}(r)\chi_{\kappa_{\nu}}^{\mu_{\nu}}\\\
if_{\kappa_{\nu}}(r)\chi_{-\kappa_{\nu}}^{\mu_{\nu}}\end{array}\right),$ (13)
$\displaystyle v_{s_{\nu}}(\bm{p}_{\nu})e^{-i\bm{p}_{\nu}\cdot\bm{r}}$
$\displaystyle=\sum_{\kappa_{\nu},m_{\nu},\mu_{\nu}}\frac{4\pi}{\sqrt{2}}i^{-l_{\kappa_{\nu}}}Y_{l_{\kappa_{\nu}}m_{\nu}}^{*}(\hat{p}_{\nu})(l_{\kappa_{\nu}},m_{\nu},1/2,-s_{\nu}|j_{\kappa_{\nu}},\mu_{\nu})(-1)^{1/2-s_{\nu}}$
$\displaystyle\times\left(\begin{array}[]{c}-if_{\kappa_{\nu}}(r)\chi_{-\kappa_{\nu}}^{\mu_{\nu}}\\\
g_{\kappa_{\nu}}(r)\chi_{\kappa_{\nu}}^{\mu_{\nu}}\end{array}\right)$ (16)

with

$\displaystyle g_{\kappa}(r)$ $\displaystyle=j_{l_{\kappa}}(p_{\nu}r),$ (17)
$\displaystyle f_{\kappa}(r)$
$\displaystyle=S_{\kappa}j_{\bar{l}_{\kappa}}(p_{\nu}r),$ (18)

where $j_{l}(x)$ is the spherical Bessel function of order $l$,
$S_{\kappa}={\rm sgn}(\kappa)$ is the sign of $\kappa$, and
$\bar{l}_{\kappa}=l_{-\kappa}$.

With the partial wave expansion of the electron and neutrino wave functions,
one obtains the following form Koshigiri:1979zd :

$\displaystyle
H_{\mathrm{eff}}=\frac{G_{F}V_{ud}}{\sqrt{2}}\sqrt{\frac{(4\pi)^{3}}{2}}\sum_{el}\sum_{neu}\sum_{L,J}(j_{e},-\mu_{e},j_{\nu},\mu_{\nu}|J,M)(-1)^{1/2-\mu_{e}}\Xi_{JLM}(\kappa_{e},\kappa_{\nu}),$
(19)

and

$\displaystyle\Xi_{JLM}(\kappa_{e},\kappa_{\nu})$
$\displaystyle=S_{\kappa_{e}}\int d\bm{r}$ $\displaystyle\times\left\\{\mp
Y_{JM}(\hat{r})V_{0}(\bm{r})\delta_{L,J}(G_{\kappa_{e}}(r)g_{\kappa_{\nu}}(r)S_{0JJ}(\kappa_{e},\kappa_{\nu})+F_{\kappa_{e}}(r)f_{\kappa_{\nu}}(r)S_{0JJ}(-\kappa_{e},-\kappa_{\nu}))\right.$
$\displaystyle\pm
i[Y_{L}(\hat{r})\otimes\bm{V}(\bm{r})]_{JM}(G_{\kappa_{e}}(r)f_{\kappa_{\nu}}(r)S_{1LJ}(\kappa_{e},-\kappa_{\nu})-F_{\kappa_{e}}(r)g_{\kappa_{\nu}}(r)S_{1LJ}(-\kappa_{e},\kappa_{\nu}))$
$\displaystyle+iY_{JM}(\hat{r})A_{0}(\bm{r})\delta_{L,J}(G_{\kappa_{e}}(r)f_{\kappa_{\nu}}(r)S_{0JJ}(\kappa_{e},-\kappa_{\nu})-F_{\kappa_{e}}(r)g_{\kappa_{\nu}}(r)S_{0JJ}(-\kappa_{e},\kappa_{\nu}))$
$\displaystyle-\left.[Y_{L}(\hat{r})\otimes\bm{A}(\bm{r})]_{JM}(G_{\kappa_{e}}(r)g_{\kappa_{\nu}}(r)S_{1LJ}(\kappa_{e},\kappa_{\nu})+F_{\kappa_{e}}(r)f_{\kappa_{\nu}}(r)S_{1LJ}(-\kappa_{e},-\kappa_{\nu}))\right\\},$
(20)

where $[\mathcal{O}_{k_{1}}\otimes\mathcal{O}_{k_{2}}^{\prime}]_{k_{3}m_{3}}$
denotes the tensor product. Here we adopt the following simplified notation
for $\sum_{el}$ and $\sum_{neu}$:

$\displaystyle\sum_{el}$
$\displaystyle=\sum_{\kappa_{e},\mu_{e}}i^{-l_{\kappa_{e}}}e^{i\Delta_{\kappa_{e}}}(l_{\kappa_{e}},m_{e},1/2,s_{e}|j_{\kappa_{e}},\mu_{e})Y_{l_{\kappa_{e}}m_{e}}(\hat{p}_{e}),$
(21) $\displaystyle\sum_{neu}$
$\displaystyle=\sum_{\kappa_{\nu},\mu_{\nu}}{}i^{-l_{\kappa_{\nu}}}(-1)^{1/2-s_{\nu}}$
$\displaystyle\times[Y_{l_{\kappa_{\nu}}m_{\nu}}^{*}(\hat{p}_{\nu})(l_{\kappa_{\nu}},m_{\nu},1/2,-s_{\nu}|j_{\kappa_{\nu}},\mu_{\nu})\pm
Y_{\bar{l}_{\kappa}m_{\nu}}^{*}(\hat{p}_{\nu})(\bar{l}_{\kappa_{\nu}},m_{\nu},1/2,-s_{\nu}|j_{\kappa_{\nu}},\mu_{\nu})],$
(22)

and

$\displaystyle S_{KLJ}(\kappa^{\prime},\kappa)$
$\displaystyle=\sqrt{2(2j_{\kappa}+1)(2j_{\kappa^{\prime}}+1)(2l_{\kappa}+1)(2l_{\kappa^{\prime}}+1)(2K+1)}(l_{\kappa},0,l_{\kappa^{\prime}},0|L,0)\left\\{\begin{array}[]{ccc}l_{\kappa^{\prime}}&1/2&j_{\kappa^{\prime}}\\\
l_{\kappa}&1/2&j_{\kappa}\\\ L&K&J\end{array}\right\\}.$ (26)

Using the above form of the effective Hamiltonian, the beta-decay rate is
given by integrating the scattering angles of the neutrino and electron as

$\displaystyle\Gamma=\frac{(G_{F}V_{ud})^{2}}{\pi^{2}}\int_{m_{e}}^{E_{0}}dE_{e}p_{e}E_{e}(E_{0}-E_{e})^{2}\sum_{J,L,\kappa_{e},\kappa_{\nu}}\frac{1}{2J_{i}+1}|\left<f\right\|\Xi_{JL}(\kappa_{e},\kappa_{\nu})\left\|i\right>|^{2},$
(27)

where $J_{i}$ is the angular momentum of the initial state. Neglecting the
mass of a neutrino, the maximum energy of an electron $E_{0}$ is the $Q$ value
of the nuclear transition, $Q_{\beta}$. See the Appendix A for the derivation.

##  3 Electron Coulomb Wave function

###  3.1 Parametrization of Lepton Wave Function

The general formula given in Eq. (20) is ready for the use of any allowed and
forbidden transition rates by evaluating the nuclear transition density.
However, an explicit formula for the allowed and first-forbidden transitions
helps extract nuclear structure information from the beta-decay observables.
Since the electron Coulomb wave function is rather involved in evaluating the
beta-decay rate, we briefly describe the derivation of the expression of
charged lepton wave functions by iterating the integral equation following
Refs. rose1951note ; Behrens:1971rbq .

A Dirac wave function of an electron is given as

$\displaystyle[\bm{\alpha}\cdot\bm{p}_{e}+\beta
m_{e}+V_{C}(r)]\psi_{e}(\bm{r})=E_{e}\psi_{e}(\bm{r}).$ (28)

The electron wave functions $G_{\kappa},F_{\kappa}$ satisfy the coupled first-
order differential equation with the Coulomb potential $V_{C}$:

$\displaystyle\frac{dG_{\kappa}}{dr}+\frac{1+\kappa}{r}G_{\kappa}-(m_{e}+E_{e}-V_{C})F_{\kappa}$
$\displaystyle=0,$ (29)
$\displaystyle\frac{dF_{\kappa}}{dr}+\frac{1-\kappa}{r}F_{\kappa}-(m_{e}-E_{e}+V_{C})G_{\kappa}$
$\displaystyle=0.$ (30)

Throughout this paper we keep the electron mass explicit so that in future we
can use the formula for the muon neutrino reactions. Electron wave functions
are parametrized by taking into account the behavior of the wave function at
the origin $r\sim 0$ Behrens:1971rbq  as

$\displaystyle G_{-k}(r)$
$\displaystyle=\alpha_{-k}\frac{(p_{e}r)^{k-1}}{(2k-1)!!}[H_{k}(r)-h_{k}(r)],$
(31) $\displaystyle F_{k}(r)$
$\displaystyle=\alpha_{k}\frac{(p_{e}r)^{k-1}}{(2k-1)!!}[H_{k}(r)+h_{k}(r)],$
(32) $\displaystyle G_{k}(r)$
$\displaystyle=\alpha_{k}\frac{(p_{e}r)^{k-1}}{(2k-1)!!}\frac{r}{R}[D_{k}(r)+d_{k}(r)],$
(33) $\displaystyle F_{-k}(r)$
$\displaystyle=-\alpha_{-k}\frac{(p_{e}r)^{k-1}}{(2k-1)!!}\frac{r}{R}[D_{k}(r)-d_{k}(r)].$
(34)

Here $k>0$ and $H_{k}(0)=1$ and $h_{k}(0)=0$. The normalization of the
electron wave functions are determined by constants $\alpha_{\kappa}$. This
parametrization of $G_{\kappa},F_{\kappa}$ incorporates the boundary condition
of the wave function at the origin. Then the following set of coupled integral
equations is obtained:

$\displaystyle H_{k}(r)$
$\displaystyle=1+\int_{0}^{r}\frac{r^{\prime}}{R}[(-E_{e}+V_{C}(r^{\prime}))D_{k}(r^{\prime})+m_{e}d_{k}(r^{\prime})]dr^{\prime},$
(35) $\displaystyle h_{k}(r)$
$\displaystyle=\int_{0}^{r}\frac{r^{\prime}}{R}[m_{e}D_{k}(r^{\prime})+(-E_{e}+V_{C}(r^{\prime}))d_{k}(r^{\prime})]dr^{\prime},$
(36) $\displaystyle\frac{r}{R}D_{k}(r)$
$\displaystyle=\int_{0}^{r}\left(\frac{r^{\prime}}{r}\right)^{2k}[(E_{e}-V_{C}(r^{\prime}))H_{k}(r^{\prime})+m_{e}h_{k}(r^{\prime})]dr^{\prime},$
(37) $\displaystyle\frac{r}{R}d_{k}(r)$
$\displaystyle=\int_{0}^{r}\left(\frac{r^{\prime}}{r}\right)^{2k}[m_{e}H_{k}(r^{\prime})+(E_{e}-V_{C}(r^{\prime}))h_{k}(r^{\prime})]dr^{\prime}.$
(38)

At this stage $R$ is just a parameter of dimension length. We take $R$ as the
nuclear radius though the final formulas are independent of the choice of $R$.

###  3.2 Iterative Solution of Integral Equation

Taking into account the boundary condition, $H_{k},h_{k},D_{k}$, and $d_{k}$
are expanded according to the number of iteration as

$\displaystyle H_{k}(r)$
$\displaystyle=1+H_{k}^{(2)}(r)+H_{k}^{(4)}(r)+\cdots,$ (39) $\displaystyle
h_{k}(r)$ $\displaystyle=h_{k}^{(2)}(r)+h_{k}^{(4)}(r)+\cdots,$ (40)
$\displaystyle D_{k}(r)$ $\displaystyle=D_{k}^{(1)}(r)+D_{k}^{(3)}(r)+\cdots,$
(41) $\displaystyle d_{k}(r)$
$\displaystyle=d_{k}^{(1)}(r)+d_{k}^{(3)}(r)+\cdots.$ (42)

The first iteration of the integral equation gives

$\displaystyle\frac{r}{R}D_{k}^{(1)}(r)$
$\displaystyle=\int_{0}^{r}\left(\frac{r^{\prime}}{r}\right)^{2k}(E_{e}-V_{C}(r^{\prime}))dr^{\prime},$
(43) $\displaystyle\frac{r}{R}d_{k}^{(1)}(r)$
$\displaystyle=\int_{0}^{r}\left(\frac{r^{\prime}}{r}\right)^{2k}m_{e}dr^{\prime},$
(44)

and further iterations give

$\displaystyle H_{k}^{(2n)}(r)$
$\displaystyle=\int_{0}^{r}\left[(-E_{e}+V_{C}(r^{\prime}))\frac{r^{\prime}}{R}D_{k}^{(2n-1)}(r^{\prime})+m_{e}\frac{r^{\prime}}{R}d_{k}^{(2n-1)}(r^{\prime})\right]dr^{\prime},$
(45) $\displaystyle h_{k}^{(2n)}(r)$
$\displaystyle=\int_{0}^{r}\left[m_{e}\frac{r^{\prime}}{R}D_{k}^{(2n-1)}(r^{\prime})+(-E_{e}+V_{C}(r^{\prime}))\frac{r^{\prime}}{R}d_{k}^{(2n-1)}(r^{\prime})\right]dr^{\prime},$
(46)

and

$\displaystyle\frac{r}{R}D_{k}^{(2n+1)}(r)$
$\displaystyle=\int_{0}^{r}\left(\frac{r^{\prime}}{r}\right)^{2k}\left[(E_{e}-V_{C}(r^{\prime}))H_{k}^{(2n)}(r^{\prime})+m_{e}h_{k}^{(2n)}(r^{\prime})\right]dr^{\prime},$
(47) $\displaystyle\frac{r}{R}d_{k}^{(2n+1)}(r)$
$\displaystyle=\int_{0}^{r}\left(\frac{r^{\prime}}{r}\right)^{2k}\left[m_{e}H_{k}^{(2n)}(r^{\prime})+(E_{e}-V_{C}(r^{\prime}))h_{k}^{(2n)}(r^{\prime})\right]dr^{\prime}$
(48)

for $n=1,2,\dots$. The exact electron wave functions in terms of
$E_{e},m_{e}$, and $V_{C}$ are obtained from the iterative solution of the
above equations.

###  3.3 LO and NLO electron wave functions

We denote the leading order (LO) electron wave function as

$\displaystyle H_{k}^{\mathrm{LO}}(r)$ $\displaystyle=1,$ (49) $\displaystyle
h_{k}^{\mathrm{LO}}(r)$ $\displaystyle=0,$ (50)
$\displaystyle\frac{r}{R}D_{k}^{\mathrm{LO}}(r)$
$\displaystyle=\frac{r}{R}D_{k}^{(1)}(r)=\frac{E_{e}r}{2k+1}+V_{D1}(r),$ (51)
$\displaystyle\frac{r}{R}d_{k}^{\mathrm{LO}}(r)$
$\displaystyle=\frac{r}{R}d_{k}^{(1)}(r)=\frac{m_{e}r}{2k+1}$ (52)

with

$\displaystyle V_{D1}(r)$
$\displaystyle=-\int_{0}^{r}\left(\frac{r^{\prime}}{r}\right)^{2k}V_{C}(r^{\prime})dr^{\prime}.$
(53)

Adding the next-to-leading order (NLO), the NLO wave function is given as

$\displaystyle H_{k}^{\mathrm{NLO}}(r)$ $\displaystyle=1+H_{k}^{(2)}(r),$ (54)
$\displaystyle h_{k}^{\mathrm{NLO}}(r)$ $\displaystyle=h_{k}^{(2)}(r),$ (55)
$\displaystyle\frac{r}{R}D_{k}^{\mathrm{NLO}}(r)$
$\displaystyle=\frac{r}{R}(D_{k}^{(1)}(r)+D_{k}^{(3)}(r)),$ (56)
$\displaystyle\frac{r}{R}d_{k}^{\mathrm{NLO}}(r)$
$\displaystyle=\frac{r}{R}(d_{k}^{(1)}(r)+d_{k}^{(3)}(r)),$ (57)

where

$\displaystyle H_{k}^{(2)}(r)$
$\displaystyle=-\frac{p_{e}^{2}r^{2}}{2(2k+1)}+V_{H2}(r),$ (58) $\displaystyle
h_{k}^{(2)}(r)$ $\displaystyle=V_{h2}(r),$ (59)
$\displaystyle\frac{r}{R}D_{k}^{(3)}(r)$
$\displaystyle=-\frac{p_{e}^{2}E_{e}r^{3}}{2(2k+1)(2k+3)}+V_{D3}(r),$ (60)
$\displaystyle\frac{r}{R}d_{k}^{(3)}(r)$
$\displaystyle=-\frac{p_{e}^{2}m_{e}r^{3}}{2(2k+1)(2k+3)}+V_{d3}(r)$ (61)

with

$\displaystyle V_{H2}(r)$
$\displaystyle=\int_{0}^{r}\left[(V_{C}(r^{\prime})-E_{e})V_{D1}(r^{\prime})+E_{e}\frac{r^{\prime}}{2k+1}V_{C}(r^{\prime})\right]dr^{\prime},$
(62) $\displaystyle V_{h2}(r)$
$\displaystyle=m_{e}\int_{0}^{r}\left[V_{D1}(r^{\prime})+\frac{r^{\prime}}{2k+1}V_{C}(r^{\prime})\right]dr^{\prime},$
(63) $\displaystyle V_{D3}(r)$
$\displaystyle=\int_{0}^{r}\left(\frac{r^{\prime}}{r}\right)^{2k}\left[(E_{e}-V_{C}(r^{\prime}))V_{H2}(r^{\prime})+m_{e}V_{h2}(r^{\prime})+\frac{p_{e}^{2}{r^{\prime}}^{2}}{2(2k+1)}V_{C}(r^{\prime})\right]dr^{\prime},$
(64) $\displaystyle V_{d3}(r)$
$\displaystyle=\int_{0}^{r}\left(\frac{r^{\prime}}{r}\right)^{2k}[m_{e}V_{H2}(r^{\prime})+(E_{e}-V_{C}(r^{\prime}))V_{h2}(r^{\prime})]dr^{\prime}.$
(65)

The explicit expressions of $H_{k}^{(2)},h_{k}^{(2)},D_{k}^{(i)}$, and
$d_{k}^{(i)}$ for the uniform charge distribution are given in the Appendix B.

##  4 Decay rate and comparison with the conventional formula

###  4.1 Decay rate

The beta-decay rate is usually expressed in terms of the Fermi function
$F(Z,E_{e})$ and the shape correction factor $C(E_{e})$ as morita1973beta ;
Schopper:1969jkp

$\displaystyle\Gamma$
$\displaystyle=\frac{(G_{F}V_{ud})^{2}}{2\pi^{3}}\int_{m_{e}}^{E_{0}}dE_{e}p_{e}E_{e}(E_{0}-E_{e})^{2}F(Z,E_{e})C(E_{e}).$
(66)

Using the matrix element of the effective operator $\Xi_{JLM}$, we obtain

$\displaystyle F(Z,E_{e})C(E_{e})$
$\displaystyle=\sum_{J,L,\kappa_{e},\kappa_{\nu}}\frac{2\pi}{2J_{i}+1}|\left<f\right\|\Xi_{JL}(\kappa_{e},\kappa_{\nu})\left\|i\right>|^{2},$
(67)

and $F(Z,E_{e})=\alpha_{-1}^{2}+\alpha_{1}^{2}$.

###  4.2 Allowed and first-forbidden transitions of axial vector current

We focus on the transition rate due to the space component of the axial vector
current. In the impulse approximation, the axial vector current is given as

$\displaystyle\bm{A}(\bm{r})=g_{A}\sum_{\tau,\tau^{\prime}}\sum_{\sigma,\sigma^{\prime}}\psi^{\dagger}(\bm{r}\sigma\tau)\psi(\bm{r}\sigma^{\prime}\tau^{\prime})\langle\tau|\tau^{\mp}|\tau^{\prime}\rangle\langle\sigma|\bm{\sigma}|\sigma^{\prime}\rangle$
(68)

with the nucleon field operators $\psi,\psi^{\dagger}$, at position $\bm{r}$,
spin $\sigma$, and isospin $\tau$. The transition density $\rho_{JL}(r)$
represented in the radial coordinate is defined as

$\displaystyle g_{\rm A}\rho_{JL}(r)$ $\displaystyle=\left<f\right\|\int
d\Omega_{r}[Y_{L}(\hat{r})\otimes\bm{A}(\bm{r})]_{J}\left\|i\right>.$ (69)

The reduced matrix element of the effective operator $\Xi_{JLM}$ is given in
terms of the radial integral of the transition density $\rho_{JL}(r)$
multiplied by combination of the electron and neutrino wave functions with the
coefficients $c_{g}$ and $c_{f}$ given in the Appendix C:

$\displaystyle\left<f\right\|\Xi_{JL}(\kappa_{e},\kappa_{\nu})\left\|i\right>$
$\displaystyle=\left<f\right\|\int
d\bm{r}[Y_{L}(\hat{r})\otimes\bm{A}(\bm{r})]_{J}[c_{g}G_{\kappa_{e}}(r)g_{\kappa_{\nu}}(r)+c_{f}F_{\kappa_{e}}(r)f_{\kappa_{\nu}}(r)]\left\|i\right>$
$\displaystyle=g_{\rm
A}\int_{0}^{\infty}drr^{2}\rho_{JL}(r)[c_{g}G_{\kappa_{e}}(r)g_{\kappa_{\nu}}(r)+c_{f}F_{\kappa_{e}}(r)f_{\kappa_{\nu}}(r)].$
(70)

The leading-order formula by Behrens–Bühring (LOB) of Ref. Behrens:1971rbq
conventionally used in the nuclear structure calculations can be derived by
using approximate lepton wave functions in Eqs. (67) and (70). We take the LO
electron wave function and the leading-order approximation of the neutrino
wave function. For the allowed transition with $\Delta J^{\pi}=1^{+}$, we
approximate the $s$-wave wave functions as a constant number:

$\displaystyle G_{-1}(r)\sim\alpha_{-1},$ $\displaystyle\quad g_{-1}(r)\sim
1,$ (71) $\displaystyle F_{1}(r)\sim\alpha_{1},$ $\displaystyle\quad
f_{1}(r)\sim 1,$ (72)

and neglect all other partial waves. For the spin-dipole transition with
$\Delta J^{\pi}=0^{-},1^{-}$, and $2^{-}$, in addition to the above
approximation to the $s$-wave wave function, we use the following leading-
order approximation for the $p$-wave wave functions

$\displaystyle
G_{1}(r)\sim\alpha_{1}\dfrac{r}{3}[E_{e}+m_{e}+\frac{3V_{D1}(r)}{r}],$
$\displaystyle\quad g_{1}(r)\sim\dfrac{p_{\nu}r}{3},$ (73) $\displaystyle
F_{-1}(r)\sim-\alpha_{-1}\dfrac{r}{3}[E_{e}-m_{e}+\frac{3V_{D1}(r)}{r}],$
$\displaystyle\quad f_{-1}(r)\sim-\dfrac{p_{\nu}r}{3},$ (74) $\displaystyle
G_{-2}(r)\sim\alpha_{-2}\dfrac{p_{e}r}{3},$ $\displaystyle\quad
g_{-2}(r)\sim\dfrac{p_{\nu}r}{3},$ (75) $\displaystyle
F_{2}(r)\sim\alpha_{2}\dfrac{p_{e}r}{3},$ $\displaystyle\quad
f_{2}(r)\sim\dfrac{p_{\nu}r}{3}.$ (76)

For the allowed $\Delta J^{\pi}=1^{+}$ transition, two partial waves of
leptons $(\kappa_{e},\kappa_{\nu})=(-1,-1)$ and $(1,1)$ contribute within the
LOB,

$\displaystyle\sum_{\kappa_{e},\kappa_{\nu}}\left|\left<f\right\|\Xi_{J=1,L=0}\left\|i\right>\right|^{2}$
$\displaystyle\sim 2g_{\rm
A}^{2}\left\\{\left|\int_{0}^{\infty}drr^{2}\rho_{10}(r)\left[G_{-1}(r)g_{-1}(r)+\frac{1}{3}F_{-1}(r)f_{-1}(r)\right]\right|^{2}\right.$
$\displaystyle+\left.\left|\int_{0}^{\infty}drr^{2}\rho_{10}(r)\left[\frac{1}{3}G_{1}(r)g_{1}(r)+F_{1}(r)f_{1}(r)\right]\right|^{2}\right\\}$
(77) $\displaystyle\sim 2g_{\rm
A}^{2}(\alpha_{-1}^{2}+\alpha_{1}^{2})\left|\int_{0}^{\infty}drr^{2}\rho_{10}(r)\right|^{2}.$
(78)

In the last step, we use the approximation for the lepton wave functions. As a
result the shape correction factor is given as

$\displaystyle C(E_{e})=g_{\rm
A}^{2}\left|\int_{0}^{\infty}drr^{2}\sqrt{4\pi}\rho_{10}(r)\right|^{2}.$ (79)

The second example is the first-forbidden transition $\Delta J^{\pi}=0^{-}$.
The leading-order partial waves are $(\kappa_{e},\kappa_{\nu})=(-1,1)$ and
$(1,-1)$. We then obtain

$\displaystyle\sum_{\kappa_{e},\kappa_{\nu}}|\left<f\right\|\Xi_{J=0,L=1}\left\|i\right>|^{2}$
$\displaystyle\sim 2g_{\rm
A}^{2}\left\\{\left|\int_{0}^{\infty}drr^{2}\rho_{01}(r)[G_{-1}(r)g_{1}(r)-F_{-1}(r)f_{1}(r)]\right|^{2}\right.$
$\displaystyle+\left.\left|\int_{0}^{\infty}drr^{2}\rho_{01}(r)[G_{1}(r)g_{-1}(r)-F_{1}(r)f_{-1}(r)]\right|^{2}\right\\}$
(80) $\displaystyle\sim\frac{2}{9}g_{\rm
A}^{2}\left\\{\alpha_{-1}^{2}\left|\int_{0}^{\infty}drr^{3}\rho_{01}(r)\left[p_{\nu}+E_{e}-m_{e}+3\frac{V_{D1}(r)}{r}\right]\right|^{2}\right.$
$\displaystyle+\left.\alpha_{1}^{2}\left|\int_{0}^{\infty}drr^{3}\rho_{01}(r)\left[p_{\nu}+E_{e}+m_{e}+3\frac{V_{D1}(r)}{r}\right]\right|^{2}\right\\}.$
(81)

In order to compare our formula with LOB, for example, Eq. (10.56) of Ref.
sch66 , introducing nuclear matrix elements

$\displaystyle\omega$ $\displaystyle=g_{\rm
A}\sqrt{4\pi}\int_{0}^{\infty}drr^{3}\rho_{01}(r),$ (82)
$\displaystyle\xi\omega^{\prime}$ $\displaystyle=g_{\rm
A}\sqrt{4\pi}\int_{0}^{\infty}drr^{2}\rho_{01}(r)V_{D1}(r),$ (83)

we obtain

$\displaystyle
C(E_{e})=\zeta_{0}^{2}+\frac{\omega^{2}m_{e}^{2}}{9}-\frac{2}{3}\frac{\mu_{1}\gamma_{1}m_{e}^{2}}{E_{e}}\zeta_{0}\omega,$
(84)

where

$\displaystyle\zeta_{0}$
$\displaystyle=\frac{E_{0}\omega}{3}+\xi\omega^{\prime},$ (85)
$\displaystyle\gamma_{k}$ $\displaystyle=\sqrt{k^{2}-(\alpha Z)^{2}},$ (86)
$\displaystyle\mu_{k}$
$\displaystyle=\frac{k}{\gamma_{k}}\frac{E_{e}}{m_{e}}\frac{\alpha_{-k}^{2}-\alpha_{k}^{2}}{\alpha_{-1}^{2}+\alpha_{1}^{2}}.$
(87)

Here $\alpha$ is the fine structure constant. A similar comparison can be done
for the transitions to $1^{-}$ and $2^{-}$ states, and we can confirm the use
of the approximate lepton wave function within our formalism leads to the
‘conventional’ formula of the decay rate.

##  5 Analysis with a schematic model

In the following, we examine the validity of the approximation for the
electron wave function proposed in this work by using a schematic model of
transition density. Three sets of treatment of the lepton wave function (i)
exact, (ii) LO, and (iii) NLO are defined. By (i) ‘exact’, we use the electron
wave function obtained by a numerical solution of the Dirac equation and the
spherical Bessel function for the neutrino wave function. In (ii) LO and (iii)
NLO, we approximate the electron wave function by the LO and NLO wave
functions described in the previous section. Notice that we do not expand the
neutrino wave function. We use the uniform charge distribution for the nuclear
charge with a charge radius $R_{A}=1.2\times A^{1/3}$ fm, and a transition
density given by a sum of two Gaussians. The analytic expressions for the LO
and NLO terms of the electron wave functions are summarized in the Appendix B.
We found that the numerical results of LO are very close to those of
‘conventional’ formula LOB.

The explicit forms of the LO and NLO approximation of the electron wave
functions of $\kappa_{e}=-1$ with the $s$-wave large component ($G_{-1}$) and
the $p$-wave small component ($F_{-1}$) are given as

$\displaystyle G_{-1}^{\rm LO}(r)=$ $\displaystyle\alpha_{-1},$ (88)
$\displaystyle F_{-1}^{\rm LO}(r)=$
$\displaystyle-\alpha_{-1}r\left[\frac{E_{e}-m_{e}}{3}+\xi s_{1}(x)\right],$
(89) $\displaystyle G_{-1}^{\rm NLO}(r)=$ $\displaystyle G_{-1}^{\rm
LO}(r)+\alpha_{-1}r^{2}\left[-\frac{p_{e}^{2}}{6}+\xi(E_{e}s_{2}(x)-m_{e}h_{2}(x))+\xi^{2}t_{2}(x)\right],$
(90) $\displaystyle F_{-1}^{\rm NLO}(r)=$ $\displaystyle F_{-1}^{\rm
LO}(r)-\alpha_{-1}r^{3}\left[-\frac{p_{e}^{2}(E_{e}-m_{e})}{30}+\xi(p_{e}^{2}s_{3}(x)+m_{e}(m_{e}-E_{e})t_{3}(x))\right.$
$\displaystyle\left.+\xi^{2}(E_{e}w_{3}(x)-m_{e}z_{3}(x))+\xi^{3}y_{3}(x)\right],$
(91)

where $x=r/R_{A}$ and $\xi=\alpha Z/(2R_{A})$.

Figure 1 shows the electron wave functions $G_{-1}$ and $F_{-1}$ at $E_{e}=10$
MeV for $Z=82$ and $A=208$. The ‘exact’ and the ‘LO’ wave functions are shown
by the solid and short-dashed curves, respectively. The deviation of the LO
wave function from the exact one grows as $r$ increases. One can see that the
deviation is larger for an $s$-wave than a $p$-wave wave function. By taking
into account the NLO correction, the wave functions are greatly improved, but
a slight deviation from the ‘exact’ wave function still remains for a larger
$r$ region. For the uniform charge distribution, the Coulomb potential for
$r>R_{A}$ agrees with the point Coulomb potential. Therefore, by connecting
the NLO wave function with the combination of the analytic form of the regular
and irregular point Coulomb wave functions, we can obtain the electron wave
functions for $r>R_{A}$ with improved accuracy (NLO∗). This is indeed the
case, as shown in the blue dashed curves in Fig. 1. Figure 2 is the same as
Fig. 1 but for $Z=28,A=80$. One sees that the effects of the NLO correction
are smaller than in the $Z=82$ case, though the deviation of the LO wave
function is distinct for the $s$-wave.

For $\beta^{+}$ decay, the sign changes for the odd power terms of $\xi$. The
Coulomb potential enters in the Dirac equation in the form of $E_{e}-V_{C}$ as
given in Eqs. (29) and (30). The Coulomb effect is constructive to $E_{e}$ for
an electron, while it is destructive for a positron. For $E_{e}>|V_{C}|$, the
deviation from the LO wave function becomes smaller for a positron than for an
electron.

The $E_{e}$ and $Z$ dependence of the NLO correction is parametrized
essentially by two non-dimensional parameters $R_{A}E_{e}$ and
$R_{A}\xi=\alpha Z/2$. Figure 3 shows the deviation of the approximate LO and
NLO electron wave functions $G_{-1}$ at the nuclear surface $r=R_{A}$,
$[G_{-1}({\rm approx.})/G_{-1}({\rm exact})-1]\times 100$, as a function of
$R_{A}E_{e}$ and $Z$. One sees a considerable deviation for a larger nuclear
charge $Z$ and a higher $R_{A}E_{e}$ value. The LO approximation overestimates
the amplitude of the wave function at the nuclear surface. By including the
NLO correction, an error is notably reduced. For $E_{e}=10$ MeV, the case of
$Z=82,A=208$ and $Z=28,A=80$ corresponds to $R_{A}E_{e}\sim 0.30,R_{A}\xi\sim
0.36$ and $R_{A}E_{e}\sim 0.26,R_{A}\xi\sim 0.10$, respectively.

Figure 1:  Electron wave functions $G_{-1}$ (left) and $F_{-1}$ (right) for
$Z=82,A=208$, $\kappa_{e}=-1$ and $E_{e}=10$ MeV. The NLO wave function of
this work and the LO wave function approximately corresponding to
‘conventional’ (LOB) are compared. The NLO∗ denotes the one obtained by
connecting the NLO wave function with the point Coulomb regular and irregular
wave functions at $r=R_{A}$.

Figure 2: Same as Fig. 1 but for $Z=28$ and $A=80$.

Figure 3: Relative deviation $[G_{-1}({\rm approx.})/G_{-1}({\rm
exact})-1]\times 100$ as a function of $R_{A}E_{e}$ and $Z$. The LO wave
function (left) and NLO wave function (right) is used for the approximate
electron wave function.

The difference between the LO and ‘exact’ lepton wave functions observed above
certainly affects the beta-decay rate. The magnitude of the effect depends on
the transition density of nuclear weak currents. To examine the effects on the
beta-decay rate, we take the following simple form of the transition density
for the Gamow–Teller and spin-dipole transitions:

$\displaystyle\rho_{\rm tr}={\cal N}[a\
e^{-(r-r_{1})^{2}/b^{2}}+e^{-(r-r_{2})^{2}/b^{2}}].$ (92)

Here we take $r_{1}=0.9R_{A},b=R_{A}/4$ and $r_{2}=3r_{1}/4$. By varying
$-1\leq a\leq-0.2$, we investigate the validity of the approximation for the
electron wave function on the decay rate. The transition density multiplied by
$r^{2}$, $(r/R_{A})^{2}\rho_{\rm tr}(r)$, is shown in Fig. 4 with $a$ as a
parameter. For $a=-0.2$, the contribution of the $r<R_{A}$ region is
predominant for the transition matrix element, while for $a=-1$, the $r\sim
R_{A}$ region gives a prevailing contribution to the matrix element. For
$a=-0.6$, a strong suppression of the matrix element would take place.

Figure 4: Transition density used in the present simple model analysis
multiplied by $(r/R_{A})^{2}$ with ${\cal N}=1$ as a function of the scaled
radial coordinate $r/R_{A}$. See text for details. The dotted horizontal line
indicates zero.

In what follows, we investigate the decay rate using the transition density
(92). First, we use the exact electron wave function and set $Z=82\pm 1$ for
$\beta^{\mp}$ decay and $A=208$. The decay rate shown in Fig. 5 with open
symbols for $\beta^{-}$ decay is given in arbitrary unit normalized to unity
at $a=-0.8$. The lines in the figure are guide for the eye. Strong suppression
of the transition rate of the allowed Gamow–Teller transition is seen around
$a=-0.6$, while it happens around $a=-0.5$ for the $0^{-}$ transition. For the
first-forbidden transition, an extra factor $r$ of the operator moves slightly
the minimum position of the matrix element. We obtain a similar $a$ dependence
of the $\beta^{+}$ decay rate for the normalized rate as shown by filled
symbols.

We then study the validity of the approximation of the electron wave function
thoroughly. Figure 6 shows the ratio of the $\beta^{-}$ decay rate calculated
with the LO wave function to that with the exact one, $\Gamma({\rm
LO})/\Gamma({\rm exact})$, drawn by the dashed curve with open symbols (filled
symbols for $\beta^{+}$ decay), and compares it with the ratio of the rate
calculated with the NLO wave function to that with the exact one, $\Gamma({\rm
NLO})/\Gamma({\rm exact})$, depicted by the solid curve with open symbols
(filled symbols for $\beta^{+}$ decay). A large deviation is suspected to
occur with the use of the approximate lepton wave functions in particular when
a delicate cancellation of the radial integral takes place. The deviation from
the exact calculation is large for $a<-0.6$, where the contribution at $r\sim
R_{A}$ is more important than $r<R_{A}$. The use of the augmented NLO electron
wave function significantly improves for $a<-0.7$, and works reasonably well
even when a severe cancellation between the inner and the outer contribution
of the integration takes place around $a=-0.6$.

Figure 5: $a$ dependence of the decay rate for the Gamow–Teller (GT) $\Delta
J^{\pi}=1^{+}$ (left) and spin-dipole (SD) $\Delta J^{\pi}=0^{-}$ (right)
transitions of $\beta^{\pm}$ decay. Decay rates are calculated with the
‘exact’ wave function for $Z=82\pm 1$, $A=208$, $E_{0}=10$ MeV and normalized
as unity at $a=-0.8$. The lines are guide for the eye.

Figure 6: Ratio of the transition rate for the Gamow–Teller (GT) $\Delta
J^{\pi}=1^{+}$ (left) and spin-dipole (SD) $\Delta J^{\pi}=0^{-}$ (right)
transitions. The NLO $\Gamma({\rm NLO})/\Gamma({\rm exact})$ and the LO
$\Gamma({\rm LO})/\Gamma({\rm exact})$ for $\beta^{\pm}$ decay are compared.
The decay rate is calculated for $Z=82\pm 1$, $A=208$, and $E_{0}=10$ MeV. The
lines are guide for the eye.

At the end of the study with the schematic model, we investigate the $Z$
dependence of the NLO correction. Figure 7 shows the $Z$ dependence of the
$\beta^{-}$ decay rate for $E_{0}=10$ MeV calculated by the LO, $\Gamma({\rm
LO})/\Gamma({\rm exact})$, and by the NLO, $\Gamma({\rm NLO})/\Gamma({\rm
exact})$, for the Gamow–Teller and spin-dipole transitions. Here the
transition density with a moderate cancellation of the matrix element with
$a=-0.8$ is used. One sees a simple use of the LO or conventional (LOB)
formula overestimates the exact rate by about 50–100% for heavy nuclei. This
suggests that the $B(GT)$ value extracted from the beta-decay rate using the
LO can be underestimated for the transition involving heavy neutron-rich
nuclei. However, it is apparent that our NLO approximation works well for a
wide range of the nuclear charge.

Figure 7: Ratio of the $\beta^{-}$ decay rate, $\Gamma({\rm NLO})/\Gamma({\rm
exact})$ and $\Gamma({\rm LO})/\Gamma({\rm exact})$ for the Gamow–Teller (GT)
$\Delta J^{\pi}=1^{+}$, and the spin-dipole (SD) $\Delta J^{\pi}=0^{-}$,
$1^{-}$ and $2^{-}$ transitions. The decay rate is calculated for $E_{0}=10$
MeV and $A=2Z$. The dotted horizontal line indicates unity.

##  6 EDF transition density and NLO electron wave function

To investigate the validity of our formalism in realistic cases, we use the
transition densities microscopically calculated by a nuclear energy-density
functional (EDF) method. Since the details of the formalism can be found in
Ref. yos13 , here we recapitulate the basic equations relevant to the present
study. In the framework of the nuclear EDF method we employ, the ground state
of a mother nucleus is described by solving the Kohn–Sham–Bogoliubov (KSB)
equation dob84

$\displaystyle\begin{bmatrix}h^{q}(\bm{r}\sigma)-\lambda^{q}&\tilde{h}^{q}(\bm{r}\sigma)\\\
\tilde{h}^{q}(\bm{r}\sigma)&-h^{q}(\bm{r}\sigma)+\lambda^{q}\end{bmatrix}\begin{bmatrix}\varphi^{q}_{1,\alpha}(\bm{r}\sigma)\\\
\varphi^{q}_{2,\alpha}(\bm{r}\sigma)\end{bmatrix}=E_{\alpha}\begin{bmatrix}\varphi^{q}_{1,\alpha}(\bm{r}\sigma)\\\
\varphi^{q}_{2,\alpha}(\bm{r}\sigma)\end{bmatrix},$ (93)

where the KS potentials $h$ and $\tilde{h}$ are given by the EDF. An explicit
expression of the potentials can be found for example in the Appendix of Ref.
kas21 . The chemical potential $\lambda$ is determined so as to give the
desired nucleon number as an average value. The superscript $q$ denotes n
(neutron, $\tau_{z}=1$) or p (proton, $\tau_{z}=-1$).

The excited states $|f;J^{\pi}\rangle$ in a daughter nucleus are described as
one-phonon excitations built on the ground state $|i\rangle$ of the mother
nucleus as

$\displaystyle|f;J^{\pi}\rangle$
$\displaystyle=\Gamma^{\dagger}_{f}|i\rangle,$ (94)
$\displaystyle\Gamma^{\dagger}_{f}$
$\displaystyle=\sum_{\alpha\beta}\left\\{X_{\alpha\beta}^{f}a^{\dagger}_{\alpha,{\rm
n}}a^{\dagger}_{\beta,{\rm p}}-Y_{\alpha\beta}^{f}a_{\beta,{\rm
p}}a_{\alpha,{\rm n}}\right\\},$ (95)

where $a^{\dagger}_{\rm n}(a^{\dagger}_{\rm p})$ and $a_{\rm n}(a_{\rm p})$
are the neutron (proton) quasiparticle (labeled by $\alpha$ and $\beta$)
creation and annihilation operators that are defined in terms of the solutions
of the KSB equation (93) with the Bogoliubov transformation. The phonon
states, the amplitudes $X^{f},Y^{f}$ and the vibrational frequency
$\omega_{f}$, are obtained in the proton–neutron quasiparticle-random-phase
approximation (pnQRPA). The residual interactions entering into the pnQRPA
equation are given by the EDF self-consistently. With the solutions of the
pnQRPA equation, the transition density is given as

$\displaystyle g_{A}\delta\bm{\rho}_{f;J^{\pi}}(\bm{r})=\langle
f;J^{\pi}|\bm{A}(\bm{r})|i\rangle=\langle
i|[\Gamma_{f},\bm{A}(\bm{r})]|i\rangle$ (96)

in a standard quasi-boson approximation. One obtains the transition density in
the radial coordinate as

$\displaystyle\rho_{JLK}(r)=\int
d\Omega_{r}[Y_{L}(\hat{r})\otimes\delta\bm{\rho}(\bm{r})]_{JK},$ (97)

which is independent of $K$ in the present case for spherical systems. Thus,
the input transition density is obtained by
$\rho_{JL}(r)=\sqrt{2J+1}\rho_{JL0}(r)$.

We apply our formula for the medium-heavy Ni and Sn isotopes. Since a
considerable contribution of the first-forbidden transition is predicted in
the Sn isotopes mus16 , we take 160Sn as an example in the present study.
Furthermore, an interplay between the allowed and first-forbidden transitions
has been discussed around 78Ni yos19 , and we thus take 80Ni as a target of
the present study as well and employ the same Skyrme and pairing EDF as in
Ref. yos19 . Within the pnQRPA, the maximum electron energy is given as
$E_{0}=B(Z+1,N-1)-B(Z,N)+(m_{\rm n}-m_{\rm p}-m_{e})\simeq\lambda^{\rm
n}-\lambda^{\rm p}-\omega+0.78$ MeV for $\beta^{-}$ decay eng99 .

The transition densities $\rho_{JL}$ of the $J^{\pi}=1^{+}$ ($E_{0}=12.1$ MeV)
and $0^{-}$ (16.1 MeV) states in 160Sn are shown in Fig. 8. Those states give
the largest contribution to the transition rate for each $J^{\pi}$. One sees
there are nodes similarly to the transition densities of the schematic model.
In such a case, the contribution around the nuclear surface $r\sim R_{A}$ is
important.

Using the transition densities microscopically calculated by the EDF method,
we evaluate the half-life of $\beta^{-}$ decay of the allowed Gamow–Teller and
the first-forbidden spin-dipole transitions of 80Ni and 160Sn. The $\beta$
decay rates are calculated within the impulse approximation for the space
component of the axial vector current only. Here we use the effective axial
vector coupling constant $g_{\rm A}=1$. The half-life is calculated using the
‘exact’ formula without approximation for the lepton wave functions. The
contribution of all the states up to $E_{0}\sim 14$ MeV (16 MeV) for 80Ni
(160Sn) are included. Shown in Tab. 1 is the half-life thus calculated for
each $J^{\pi}$. We show the ratios of the half-life
$t_{1/2}$(LO)/$t_{1/2}$(exact) and $t_{1/2}$(NLO)/$t_{1/2}$(exact) in the
table as well. As suspected the LO overestimates the transition rate by about
5 to 15% depending on the type of the transition and nuclide. Therefore, the
half-lives are underestimated. The deviation from the ‘exact’ calculation is
larger for Sn than for Ni. Introducing the NLO correction, those errors are
nicely restored, as shown in the third column of Tab. 1. We can therefore
argue that our simple NLO formula is very effective in realistic calculations.

Figure 8:  Transition density $r^{2+L}\rho_{JL}$ of (left) Gamow–Teller transition $J^{\pi}=0^{+}\rightarrow 1^{+}$ ($E_{0}=12.1$ MeV) multiplied by $10^{3}$ and (right) spin-dipole transition $J^{\pi}=0^{+}\rightarrow 0^{-}$ ($E_{0}=16.1$ MeV) of 160Sn. The dotted horizontal line indicates zero. Table 1: Half life $t_{1/2}$ of the $\beta^{-}$ decay of 80Ni and 160Sn to the daughter nucleus with the states $J^{\pi}$. The ratios of the half life $t_{1/2}$(LO)/$t_{1/2}$(exact) and $t_{1/2}$(NLO)/$t_{1/2}$(exact) are shown in the second and the third column. |  80Ni |  |  160Sn
---|---|---|---
$J^{\pi}$ | $t_{1/2}({\rm s})$ | LO | NLO |  | $t_{1/2}({\rm s})$ | LO | NLO
$1^{+}$ | 3.50 $\times 10^{-1}$ | 0.962 | 1.00 |  | 2.39$\times 10^{-3}$ | 0.874 | 1.00
$0^{-}$ | 1.08 | 0.928 | 1.00 |  | 1.34$\times 10^{-2}$ | 0.874 | 1.00
$1^{-}$ | 3.02 | 0.943 | 1.00 |  | 5.18$\times 10^{-2}$ | 0.895 | 1.00
$2^{-}$ | 2.18 | 0.942 | 1.00 |  | 1.42$\times 10^{-1}$ | 0.857 | 1.00

##  7 Summary

We have investigated the Coulomb effects on the beta-decay rate. The decay
rate is determined by the product of the lepton and hadron current densities.
A widely used formula relies on the fact that the low-energy lepton wave
functions in a nucleus can be well approximated by a constant and linear to
the radius for the $s$-wave and $p$-wave wave functions, respectively. We
found, however, the Coulomb wave function is conspicuously different from such
a simple approximation for heavy nuclei with large $Z$ by numerically solving
the Dirac equation. We then have proposed formulas of the nuclear beta-decay
rate that are useful in a practical calculation.

In our proposed formulas, the neutrino wave function is treated exactly as a
plane wave, while the electron wave function is obtained by iteratively
solving the integral equation; thus, we can control the uncertainty of the
approximate electron wave function order by order. The leading-order
approximation gives a formula that is almost equivalent to the widely used one
and overestimates the decay rate by about 50–100% for heavy nuclei with $Z\sim
80$. We demonstrated that the next-to-leading-order formula reproduces well
the exact result for a schematic transition density as well as a microscopic
one obtained by a nuclear energy-density functional method. For the beta decay
involving heavy neutron-rich nuclei, the NLO will be needed for the
determination of the Gamow–Teller strength from the beta-decay rate.

We considered only the space component of the axial vector currents and kept
only the lowest multipoles. The time components as well as the vector currents
can have a comparable contribution to the decay rate, and we plan to present
these improvements in a sequel to the present article. The beta decay provides
a unique spectroscopic tool of exotic nuclei, that is, the angular correlation
contains rich information of nuclear structure. Furthermore, the electron/muon
capture is an important process in the application to astrophysics and
fundamental physics. It is straightforward to extend our formalism towards
these directions.

## Acknowledgment

We would like to thank Prof. K. Koshigiri for useful discussions. This work
was in part supported by the JSPS KAKENHI Grants Nos. JP18H01210, JP18H04569,
JP18K03635, JP19H05104, JP19H05140, and JP19K03824, the Collaborative Research
Program 2019–2020, Information Initiative Center, Hokkaido University, and the
JSPS/NRF/NSFC A3 Foresight Program “Nuclear Physics in the 21st Century.” The
nuclear EDF calculation was performed on CRAY XC40 at the Yukawa Institute for
Theoretical Physics, Kyoto University.

##  Appendix A Derivation of the decay rate

In this appendix, we show the derivation of the decay rate Eq. (27) for
$\beta^{-}$ decay. In the present case, it is useful to expand the effective
Hamiltonian in terms of the angular momentum. With the partial wave expansion
of the neutrino wave function, we get

$\displaystyle\left(1-\gamma_{5}\right)v^{s_{\nu}}\left(p_{\nu}\right)e^{-i\bm{p}_{\nu}\cdot\bm{r}}=$
$\displaystyle
4\pi\sum_{neu}\begin{pmatrix}-if_{\kappa_{\nu}}(r)\chi_{-\kappa_{\nu}}^{\mu_{\nu}}\\\
g_{\kappa_{\nu}}(r)\chi_{\kappa_{\nu}}^{\mu_{\nu}}\end{pmatrix},$ (98)

where we use the abbreviated notation (upper sign) defined in Eq. (22). Since
the neutrino mass is negligible, we used here
$g_{\kappa}(r)=-S_{\kappa}f_{-\kappa}(r)$ and
$f_{\kappa}(r)=S_{\kappa}g_{-\kappa}(r)$. We also have an alternative
expression:

$\displaystyle\left(1-\gamma_{5}\right)v^{s_{\nu}}\left(p_{\nu}\right)e^{-i\bm{p}_{\nu}\cdot\bm{r}}=$
$\displaystyle
4\pi\sum_{neu}\begin{pmatrix}-g_{\kappa_{\nu}}(r)\chi_{-\kappa_{\nu}}^{\mu_{\nu}}\\\
if_{\kappa_{\nu}}(r)\chi_{\kappa_{\nu}}^{\mu_{\nu}}\end{pmatrix}.$ (99)

Thus, we obtain two equivalent expressions:

$\displaystyle\psi^{s_{e}(-)}_{p_{e}}\left(\bm{r}\right)\gamma^{\nu}\left(1-\gamma_{5}\right)v^{s_{\nu}}\left(p_{\nu}\right)e^{-i\bm{p}_{\nu}\cdot\bm{r}}$
$\displaystyle=$
$\displaystyle\left(4\pi\right)^{2}\sum_{el}\sum_{neu}\left(G_{\kappa_{e}}\left(r\right)\chi_{\kappa_{e}}^{\mu_{e}\dagger},iF_{\kappa_{e}}\left(r\right)\chi_{-\kappa_{e}}^{\mu_{e}\dagger}\left(\hat{r}\right)\right)\gamma^{\nu}\begin{pmatrix}-if_{\kappa_{\nu}}(r)\chi_{-\kappa_{\nu}}^{\mu_{\nu}}\\\
g_{\kappa_{\nu}}(r)\chi_{\kappa_{\nu}}^{\mu_{\nu}}\end{pmatrix}$ (100)
$\displaystyle=$
$\displaystyle\left(4\pi\right)^{2}\sum_{el}\sum_{neu}\left(G_{\kappa_{e}}\left(r\right)\chi_{\kappa_{e}}^{\mu_{e}\dagger},iF_{\kappa_{e}}\left(r\right)\chi_{-\kappa_{e}}^{\mu_{e}\dagger}\left(\hat{r}\right)\right)\gamma^{\nu}\begin{pmatrix}-g_{\kappa_{\nu}}(r)\chi_{-\kappa_{\nu}}^{\mu_{\nu}}\\\
if_{\kappa_{\nu}}(r)\chi_{\kappa_{\nu}}^{\mu_{\nu}}\end{pmatrix},$ (101)

where we used Eq. (21).

As in Eq. (2), the hadron current is composed of the vector and axial vector
components. We can thus write

$\displaystyle\psi^{s_{e}(-)}_{p_{e}}\left(\bm{r}\right)\gamma^{\nu}\left(1-\gamma_{5}\right)v^{s_{\nu}}\left(p_{\nu}\right)e^{-i\bm{p}_{\nu}\cdot\bm{r}}J_{\nu}$
$\displaystyle=$
$\displaystyle\left(4\pi\right)^{2}\sum_{el}\sum_{neu}\left(G_{\kappa_{e}}\left(r\right)\chi_{\kappa_{e}}^{\mu_{e}\dagger}\left(\hat{r}\right),iF_{\kappa_{e}}\left(r\right)\chi_{-\kappa_{e}}^{\mu_{e}\dagger}\left(\hat{r}\right)\right)$
$\displaystyle\times\left\\{\left(\gamma^{0}V_{0}-\bm{\gamma}\cdot\bm{V}\right)\begin{pmatrix}-g_{\kappa_{\nu}}(r)\chi_{\kappa_{\nu}}^{\mu_{\nu}}\left(\hat{r}\right)\\\
if_{\kappa_{\nu}}(r)\chi_{-\kappa_{\nu}}^{\mu_{\nu}}\left(\hat{r}\right)\end{pmatrix}-\left(\gamma^{0}A_{0}-\bm{\gamma}\cdot\bm{A}\right)\begin{pmatrix}-if_{\kappa_{\nu}}(r)\chi_{-\kappa_{\nu}}^{\mu_{\nu}}\\\
g_{\kappa_{\nu}}(r)\chi_{\kappa_{\nu}}^{\mu_{\nu}}\end{pmatrix}\right\\}.$
(102)

The products of the two-component spinors are given as

$\displaystyle\chi_{\kappa}^{\mu\dagger}\left(\hat{r}\right)\chi_{\kappa^{\prime}}^{\mu^{\prime}}\left(\hat{r}\right)=$
$\displaystyle\frac{1}{\sqrt{4\pi}}\sum_{L,M}\left(j_{\kappa},-\mu,j_{\kappa^{\prime}},\mu^{\prime}|L,M\right)(-1)^{1/2-\mu}S_{\kappa}S_{0LL}(\kappa,\kappa^{\prime})Y_{LM}\left(\hat{r}\right),$
(103)
$\displaystyle\chi_{\kappa}^{\mu\dagger}\left(\hat{r}\right)\sigma^{i}\chi_{\kappa^{\prime}}^{\mu^{\prime}}\left(\hat{r}\right)=$
$\displaystyle\frac{-1}{\sqrt{4\pi}}\sum_{J,L,M}\left(j_{\kappa},-\mu,j_{\kappa^{\prime}},\mu^{\prime}|J,M\right)(-1)^{1/2-\mu}S_{\kappa}S_{1LJ}(\kappa,\kappa^{\prime})\left[Y_{L}\left(\hat{r}\right)\otimes\left(\epsilon^{i}\right)\right]_{JM},$
(104)

where $\epsilon$ is a unit vector. Using these relations, we obtain the
effective Hamiltonian Eqs. (19) and (20). According to the Wigner–Eckart
theorem, the $M$-dependence of the spherical tensor $\Xi_{JLM}$, Eq. (20), is
known as

$\displaystyle\Braket{f}{\Xi_{JLM}\left(\kappa_{e},\kappa_{\nu}\right)}{i}=\frac{\left(J_{i},s_{i},J,M|J_{f},s_{f}\right)}{\sqrt{2J_{f}+1}}\Braket{f}{\Xi_{JL}\left(\kappa_{e},\kappa_{\nu}\right)}{i},$
(105)

where the reduced matrix element
$\Braket{f}{\Xi_{JL}\left(\kappa_{e},\kappa_{\nu}\right)}{i}$ is independent
of $M$, and $J_{f}$ is the angular momentum of the final nuclear state.

With the obtained Hamiltonian $H_{\mathrm{eff}}$, the decay rate is given by

$\displaystyle\Gamma$
$\displaystyle=\frac{1}{2J_{i}+1}\sum_{s_{i}}\sum_{s_{f},s_{e},s_{\nu}}\int\frac{d^{3}p_{\nu}}{\left(2\pi\right)^{3}}\frac{d^{3}p_{e}}{\left(2\pi\right)^{3}}(2\pi)\delta\left(E_{\nu}+E_{e}-E_{0}\right)\left|H_{\mathrm{eff}}\right|^{2}$
$\displaystyle=\frac{G_{F}^{2}V_{ud}^{2}\left(4\pi\right)^{3}}{4\left(2\pi\right)^{5}}\int_{m_{e}}^{E_{0}}dE_{e}p_{e}E_{e}\left(E_{0}-E_{e}\right)^{2}\frac{1}{\left(2J_{i}+1\right)\left(2J_{f}+1\right)}\sum_{s_{f},s_{i}}\sum_{s_{e},s_{\nu}}\int
d\Omega_{e}d\Omega_{\nu}$
$\displaystyle\times\left|\sum_{el}\sum_{neu}\sum_{J,L,M}(-1)^{1/2-\mu_{e}}\left(j_{\kappa_{e}},-{\mu_{e}},j_{\kappa_{\nu}},\mu_{\nu}|J,M\right)\left(J_{i},s_{i},J,M|J_{f},s_{f}\right)\Braket{f}{\Xi_{JL}\left(\kappa_{e},\kappa_{\nu}\right)}{i}\right|^{2},$
(106)

where $J_{i}$ is the angular momentum of the initial nuclear state. For an
arbitrary function $X\left(\kappa,\mu\right)$, we have

$\displaystyle\int
d\Omega_{e}\sum_{s_{e}}\left|\sum_{el}X\left(\kappa_{e},\mu_{e}\right)\right|^{2}=\sum_{\kappa_{e},\mu_{e}}\left|X\left(\kappa_{e},\mu_{e}\right)\right|^{2},$
(107)

and

$\displaystyle\int
d\Omega_{\nu}\sum_{s_{\nu}}\left|\sum_{neu}X\left(\kappa_{\nu},\mu_{\nu}\right)\right|^{2}=2\sum_{\kappa_{\nu},\mu_{\nu}}\left|X\left(\kappa_{\nu},\mu_{\nu}\right)\right|^{2}.$
(108)

Applying Eqs. (107) and (108), and the orthonormal relation of the
Clebsh–Gordan coefficients to Eq. (106), one can perform all the angular
integral and summation. We then arrive at Eq. (27).

##  Appendix B Explicit formula for the uniform charge distribution

For the uniform charge distribution of nuclei with radius $R_{A}$, the Coulomb
potential for an electron is given as

$\displaystyle V_{C}(r)$ $\displaystyle=-\frac{\alpha
Z}{2R_{A}}\left[\theta(1-x)(3-x^{2})+\theta(x-1)\frac{2}{x}\right],$ (109)

where $\alpha$ is the fine structure constant and $x=r/R_{A}$. The electron
wave functions $D^{(i)},d^{(i)},H^{(2)}$, and $h^{(2)}$ are given as

$\displaystyle\frac{r}{R}D_{k}^{(1)}(r)$
$\displaystyle=r\left[\frac{E_{e}}{2k+1}+\xi s_{1}(x)\right],$ (110)
$\displaystyle\frac{r}{R}d_{k}^{(1)}(r)$
$\displaystyle=r\left[\frac{m_{e}}{2k+1}\right],$ (111) $\displaystyle
H^{(2)}(r)$ $\displaystyle=r^{2}\left[-\frac{p_{e}^{2}}{2(2k+1)}+E_{e}\xi
s_{2}(x)+\xi^{2}t_{2}(x)\right],$ (112) $\displaystyle h^{(2)}(r)$
$\displaystyle=r^{2}[m_{e}\xi h_{2}(x)],$ (113)
$\displaystyle\frac{r}{R}D_{k}^{(3)}(r)$
$\displaystyle=r^{3}\left[-\frac{p_{e}^{2}E_{e}}{2(2k+1)(2k+3)}+p_{e}^{2}\xi
s_{3}(x)+m_{e}^{2}\xi t_{3}(x)+E_{e}\xi^{2}w_{3}(x)+\xi^{3}y_{3}(x)\right],$
(114) $\displaystyle\frac{r}{R}d_{k}^{(3)}(r)$
$\displaystyle=r^{3}\left[-\frac{p_{e}^{2}m_{e}}{2(2k+1)(2k+3)}+m_{e}E_{e}\xi
t_{3}(x)+m_{e}\xi^{2}z_{3}(x)\right],$ (115)

where $\xi=\alpha Z/(2R_{A})$ for an electron and $\xi=-\alpha Z/(2R_{A})$ for
a positron.

The functions $s_{a},t_{a},w_{a},y_{a},$ and $z_{a}$ for $k=1$ are given as

$\displaystyle s_{1}(x)$
$\displaystyle=\theta(1-x)\left(1-\frac{x^{2}}{5}\right)+\theta(x-1)\frac{1}{x}\left(1-\frac{1}{5x^{2}}\right),$
(116) $\displaystyle s_{2}(x)$
$\displaystyle=\theta(1-x)\left(-1+\frac{2x^{2}}{15}\right)+\theta(x-1)\frac{1}{x}\left(-\frac{5}{3}+\frac{1}{x}-\frac{1}{5x^{2}}\right),$
(117) $\displaystyle t_{2}(x)$
$\displaystyle=\theta(1-x)\left(-\frac{3}{2}+\frac{2x^{2}}{5}-\frac{x^{4}}{30}\right)+\theta(x-1)\frac{1}{x^{2}}\left(-\frac{14}{15}-\frac{1}{5x^{2}}-2\ln
x\right),$ (118) $\displaystyle w_{2}(x)$
$\displaystyle=\theta(1-x)\frac{x^{2}}{30}+\theta(x-1)\frac{1}{x}\left(\frac{1}{3}-\frac{1}{2x}+\frac{1}{5x^{2}}\right),$
(119) $\displaystyle s_{3}(x)$
$\displaystyle=\theta(1-x)\left(-\frac{3}{10}+\frac{3x^{2}}{70}\right)+\theta(x-1)\frac{1}{x}\left(-\frac{1}{2}+\frac{1}{3x}-\frac{1}{10x^{2}}+\frac{1}{105x^{4}}\right),$
(120) $\displaystyle t_{3}(x)$
$\displaystyle=\theta(1-x)\left(-\frac{1}{5}+\frac{x^{2}}{42}\right)+\theta(x-1)\frac{1}{x}\left(-\frac{1}{3}+\frac{1}{6x}-\frac{1}{105x^{4}}\right),$
(121) $\displaystyle w_{3}(x)$
$\displaystyle=\theta(1-x)\left(-\frac{9}{10}+\frac{9x^{2}}{35}-\frac{x^{4}}{54}\right)$
$\displaystyle+\theta(x-1)\frac{1}{x^{2}}\left(-\frac{6}{5}+\frac{1}{x}-\frac{3}{5x^{2}}+\frac{131}{945x^{3}}-\frac{2\ln
x}{3}\right),$ (122) $\displaystyle y_{3}(x)$
$\displaystyle=\theta(1-x)\left(-\frac{9}{10}+\frac{27x^{2}}{70}-\frac{x^{4}}{18}+\frac{x^{6}}{330}\right)$
$\displaystyle+\theta(x-1)\frac{1}{x^{3}}\left(\frac{1}{15}-\frac{439}{693x^{2}}-2\ln
x-\frac{2\ln x}{5x^{2}}\right),$ (123) $\displaystyle z_{3}(x)$
$\displaystyle=\theta(1-x)\left(-\frac{3}{10}+\frac{x^{2}}{14}-\frac{x^{4}}{135}\right)$
$\displaystyle+\theta(x-1)\frac{1}{x^{2}}\left(\frac{2}{15}-\frac{1}{2x}+\frac{1}{5x^{2}}-\frac{131}{1890x^{3}}-\frac{2\ln
x}{3}\right).$ (124)

Similar formulas for $k=2$ are given as

$\displaystyle s_{1}(x)$
$\displaystyle=\theta(1-x)\left(\frac{3}{5}-\frac{x^{2}}{7}\right)+\theta(x-1)\frac{1}{x}\left(\frac{1}{2}-\frac{3}{70x^{4}}\right),$
(125) $\displaystyle s_{2}(x)$
$\displaystyle=\theta(1-x)\left(-\frac{3}{5}+\frac{3x^{2}}{35}\right)+\theta(x-1)\frac{1}{x}\left(-\frac{9}{10}+\frac{2}{5x}-\frac{1}{70x^{4}}\right),$
(126) $\displaystyle t_{2}(x)$
$\displaystyle=\theta(1-x)\left(-\frac{9}{10}+\frac{9x^{2}}{35}-\frac{x^{4}}{42}\right)+\theta(x-1)\frac{1}{x^{2}}\left(-\frac{271}{420}-\frac{3}{140x^{4}}-\ln
x\right),$ (127) $\displaystyle w_{2}(x)$
$\displaystyle=\theta(1-x)\frac{x^{2}}{70}+\theta(x-1)\frac{1}{x}\left(\frac{1}{10}-\frac{1}{10x}+\frac{1}{70x^{4}}\right),$
(128) $\displaystyle s_{3}(x)$
$\displaystyle=\theta(1-x)\left(-\frac{9}{70}+\frac{13x^{2}}{630}\right)+\theta(x-1)\frac{1}{x}\left(-\frac{11}{60}+\frac{2}{25x}-\frac{1}{140x^{4}}+\frac{4}{1575x^{6}}\right),$
(129) $\displaystyle t_{3}(x)$
$\displaystyle=\theta(1-x)\left(-\frac{3}{35}+\frac{x^{2}}{90}\right)+\theta(x-1)\frac{1}{x}\left(-\frac{2}{15}+\frac{3}{50x}-\frac{2}{1575x^{6}}\right),$
(130) $\displaystyle w_{3}(x)$
$\displaystyle=\theta(1-x)\left(-\frac{27}{70}+\frac{13x^{2}}{105}-\frac{23x^{4}}{2310}\right)$
$\displaystyle+\theta(x-1)\frac{1}{x^{2}}\left(-\frac{943}{2100}+\frac{1}{5x}-\frac{1}{20x^{4}}+\frac{157}{5775x^{5}}-\frac{\ln
x}{5}\right),$ (131) $\displaystyle y_{3}(x)$
$\displaystyle=\theta(1-x)\left(-\frac{27}{70}+\frac{13x^{2}}{70}-\frac{23x^{4}}{770}+\frac{x^{6}}{546}\right)$
$\displaystyle+\theta(x-1)\frac{1}{x^{3}}\left(-\frac{83}{420}-\frac{87}{2860x^{4}}-\frac{\ln
x}{2}-\frac{3\ln x}{70x^{4}}\right),$ (132) $\displaystyle z_{3}(x)$
$\displaystyle=\theta(1-x)\left(-\frac{9}{70}+\frac{x^{2}}{30}-\frac{4x^{4}}{1155}\right)$
$\displaystyle+\theta(x-1)\frac{1}{x^{2}}\left(-\frac{103}{2100}-\frac{1}{20x}+\frac{1}{140x^{4}}-\frac{157}{23100x^{5}}-\frac{\ln
x}{5}\right).$ (133)

##  Appendix C Table of the coefficients in $\Xi_{JLM}$

For the axial vector space component, the effective operator is expressed as
Eq. (70):

$\displaystyle\Xi_{JLM}(\kappa_{e},\kappa_{\nu})=\int
d\bm{r}[Y_{L}(\hat{r})\otimes\bm{A}(\bm{r})]_{JM}[c_{g}G_{\kappa_{e}}(r)g_{\kappa_{\nu}}(r)+c_{f}F_{\kappa_{e}}(r)f_{\kappa_{\nu}}(r)].$

Tables 2 and 3 list the explicit numbers of the coefficients of each
$(\kappa_{e},\kappa_{\nu})$ for the Gamow–Teller and spin-dipole transitions,
respectively.

Table 2: Coefficients in Eq. (70) for the Gamow–Teller transition ($L=0,\Delta J^{\pi}=1^{+}$). $(\kappa_{e},\kappa_{\nu})$ | $(c_{g},c_{f})$ |  | $(\kappa_{e},\kappa_{\nu})$ | $(c_{g},c_{f})$
---|---|---|---|---
$(-1,-1)$ | $(\sqrt{2},\sqrt{2}/3)$ |  | $(-2,1)$ | $(4/3,0)$
$(1,1)$ | $(-\sqrt{2}/3,-\sqrt{2})$ |  | $(2,-1)$ | $(0,-4/3)$
$(-2,-2)$ | $(-2\sqrt{5}/3,-2/\sqrt{5})$ |  | $(1,-2)$ | $(4/3,0)$
$(2,2)$ | $(2/\sqrt{5},2\sqrt{5}/3)$ |  | $(-1,2)$ | $(0,-4/3)$
Table 3: Same as Tab. 2 but for the spin-dipole transition ($L=1,\Delta J^{\pi}=0^{-},1^{-},2^{-}$). |  | $J=0$ | $J=1$ | $J=2$
---|---|---|---|---
$(\kappa_{e},\kappa_{\nu})$ |  | $(c_{g},c_{f})$ | $(c_{g},c_{f})$ | $(c_{g},c_{f})$
$(-1,1)$ |  | $(-\sqrt{2},\sqrt{2})$ | $(2/\sqrt{3},2/\sqrt{3})$ |
$(1,-1)$ |  | $(-\sqrt{2},\sqrt{2})$ | $(-2/\sqrt{3},-2/\sqrt{3})$ |
$(-2,2)$ |  | $(2,-2)$ | $(-4\sqrt{2/15},-4\sqrt{2/15})$ | $(\sqrt{2}/5,-\sqrt{2}/5)$
$(2,-2)$ |  | $(2,-2)$ | $(4\sqrt{2/15},4\sqrt{2/15})$ | $(\sqrt{2}/5,-\sqrt{2}/5)$
$(-2,-1)$ |  |  | $(\sqrt{2/3},\sqrt{2/3})$ | $(\sqrt{2},\sqrt{2}/5)$
$(2,1)$ |  |  | $(-\sqrt{2/3},-\sqrt{2/3})$ | $(-\sqrt{2}/5,-\sqrt{2})$
$(-1,-2)$ |  |  | $(-\sqrt{2/3},-\sqrt{2/3})$ | $(\sqrt{2},\sqrt{2}/5)$
$(1,2)$ |  |  | $(\sqrt{2/3},\sqrt{2/3})$ | $(-\sqrt{2}/5,-\sqrt{2})$

##  Appendix D Tables of electron and positron wave functions

We provide numerical tables of the four constants
$\alpha_{1},\alpha_{-1},\alpha_{2},$ and $\alpha_{-2}$ needed to construct the
electron and positron wave functions in this paper. Since these constants are
strongly dependent on the electron momentum $p_{e}$ and charge number $Z$ of a
nucleus, we rewrite them to $L_{0},\lambda_{2},\mu_{1}$, and $\mu_{2}$
according to Ref. Schopper:1969jkp :

$\displaystyle F(Z,E_{e})$
$\displaystyle=\alpha_{-1}^{2}+\alpha_{1}^{2}=F_{0}L_{0},$ (134)
$\displaystyle\lambda_{2}$
$\displaystyle=\frac{\alpha_{-2}^{2}+\alpha_{2}^{2}}{\alpha_{-1}^{2}+\alpha_{1}^{2}},$
(135) $\displaystyle\mu_{k}$
$\displaystyle=\frac{kE_{e}}{\gamma_{k}m_{e}}\frac{\alpha_{-k}^{2}-\alpha_{k}^{2}}{\alpha_{-k}^{2}+\alpha_{k}^{2}},$
(136)

where

$\displaystyle F_{0}(Z,E)$
$\displaystyle=4(2p_{e}R_{A})^{-2(1-\gamma_{1})}e^{\pi\nu}\left|\frac{\Gamma(\gamma_{1}+i\nu)}{\Gamma(2\gamma_{1}+1)}\right|^{2},$
(137) $\displaystyle\gamma_{k}$ $\displaystyle=\sqrt{k^{2}-(\alpha Z)^{2}},$
(138) $\displaystyle\nu$ $\displaystyle=\frac{\alpha ZE_{e}}{p_{e}}$ (139)

for the uniform nuclear charge distribution with radius $R_{A}$. Actually,
these variables have a milder momentum and charge dependence than $\alpha_{\pm
1,2}$. First, we calculate $\alpha_{\pm 1,2}$ by numerically solving the Dirac
equation and convert them into $L_{0},\lambda_{2},\mu_{1}$, and $\mu_{2}$ at
various $p_{e}/m_{e}$ and $Z$. These generated tables are respectively
interpolated by assuming the following polynomial function at three regions,
$p_{e}/m_{e}=$0.01–1, 1–10, and 10–100:

$\displaystyle
P(p_{e}/m_{e},Z)=\sum_{t=-m}^{n}\left[\sum_{s=-m}^{n}W_{st}\left(\frac{p_{e}}{m_{e}}\right)^{s}\right]Z^{t}.$
(140)

The weights of the polynomial $W_{st}$ are determined by the least-square
method. Finally, we reconstruct $\alpha_{\pm 1,2}$ from
$L_{0},\lambda_{2},\mu_{1}$, and $\mu_{2}$. Since all $\alpha_{\pm 1,2}$
values are positive, the reconstruction can be made easily. We confirm that
the resulting numerical tables are accurate more than 3–4 digits with
$(m,n)=(3,4)$.

For the convenience of a user, we provide a FORTRAN program code to generate
$\alpha_{\pm 1,2}$ with a given $p_{e}/m_{e}(=$0.01–100) and $Z(=1$–90) for
$\beta^{\mp}$ decay as supplemental material. The nuclear charge radius
$R_{A}$ is set to be $1.2A^{1/3}$ fm with the nuclear mass number $A$. To
cover stable and neutron-rich unstable nuclei for the $\beta^{-}$ decay, a
variation of the charge radius can be considered among five options:
$A=2Z,2.5Z,3Z,3.5Z$, and $4Z$.

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