# A formalism to assess the accuracy of nuclear-structure weak interaction
effects in precision $\beta$-decay studies

Ayala Glick-Magid  The Racah Institute of Physics, The Hebrew University of
Jerusalem, Givat Ram, Jerusalem, 9190401  Doron Gazit
<EMAIL_ADDRESS>The Racah Institute of Physics, The Hebrew
University of Jerusalem, Givat Ram, Jerusalem, 9190401

###### Abstract

Multiple high precision $\beta$-decay measurements are being carried out these
days on various nuclei, in search of beyond the Standard Model signatures.
These measurements necessitate accurate standard model theoretical predictions
to be compared with. Motivated by the experimental surge, we present a
formalism for such a calculation of $\beta$-decay observables, with controlled
accuracy, based on a perturbative analysis of the theoretical observables
related to the phenomena, including high order nuclear recoil and shape
corrections. The accuracy of the corrections is analyzed by identifying a
hierarchy of small parameters, related to the low momentum transfer
characterizing $\beta$-decays. Furthermore, we show that the sub-percent
uncertainties, targeted by on-going and planned experiments, entail an
accuracy of the order of 10% for the solution of the nuclear many body
problem, which is well within the reach of modern nuclear theory for light to
medium mass nuclei.

##  I Introduction

According to the Standard Model (SM), the weak interaction of quarks and weak
probes has a $V-A$ structure [1, 2], i.e., a polar-vector minus an axial-
vector current, with the same magnitude but opposite signs. This has been
experimentally tested in many ways, including by studying correlations in the
directions of the outgoing leptons in nuclear $\beta$-decays [3, 4]. However,
there are pieces of evidence showing that the SM is incomplete, even much
below the Planck scale, e.g., the existence of the neutrino mass. Since the
neutrino, as a low mass neutral lepton, interacts almost exclusively via the
weak force, this might hint that the way we understand the weak interaction is
incomplete.

These days, angular correlation and spectrum measurements in nuclear
$\beta$-decays are carefully designed to probe for signatures of deviations of
the weak interaction from the $V-A$ symmetry dictated by the SM, originating
in interactions with scalar, pseudo-scalar, or tensor symmetries, which are
a-priori theoretically allowed in Lorentz invariant theories [5]. Such
signature would be an indication for Beyond the Standard Model (BSM) physics.
Dimensional analysis shows that the order of magnitude of such deviation is
inversely proportional to the square of the new physics energy scale, e.g., a
1% deviation from the SM values of those experimental observables suggests new
physics at the TeV-scale. Thus, such measurements can be viewed as low-energy
counterparts for LHC measurements. New $\beta$-decay experiments are designed
to have a per-mill level accuracy [6]. Such new efforts include, for example,
studying the decays of ${}^{6}\text{He}$, ${}^{16}\text{N}$, and
${}^{17-23}\mbox{Ne}$ isotopes at SARAF accelerator (Israel) [7, 8],
${}^{6}\text{He}$ decay studies at the National Superconducting Cyclotron
Laboratory (USA) [9, 10, 11], Laboratoire Physique Corpusculaire de CAEN
(France) [6], the University of Washington CENPA (USA) by the He6-CRES
Collaboration [12], and more (for an overview of on-going and planned
experiments see [13, 6]). However, deviations from the simplistic textbook
formulae are also due to finite momentum transfer, radiative corrections, and
nuclear structure effects (see e.g., [14, 15] and references therein). Pin-
pointing these effects demands a detailed calculation of the nuclear dynamics
of the weak decay. In this paper we present a general formalism, to calculate
the nuclear matrix element of these observables, within the Standard Model,
including e.g., recoil and (spectrum-)shape. The main advantage of the
presented formalism is the possibility to assess a controlled theoretical
accuracy, based on an identification of the parameters governing this
accuracy.

This paper is structured as follows: We begin by identifying small parameters
involved in the $\beta$-decay formalism (sec. II). Next, we introduce the
$\beta$-decay multipole expansion, analyze its properties (sec. III), and
employ it to formulate a general perturbative expansion of the observables,
including high order corrections, for any $\beta$-decay transition (sec. IV).
Following that, we demonstrate the corrections for allowed $\beta$-decays
(sec. V), and for first forbidden decays (sec. VI). Finally, we discuss the
expected order of magnitude of BSM signatures, then consider the accuracy
required for SM calculations to detect BSM deviations in nuclear $\beta$-decay
measurements (sec. VII), and summarize (sec. VIII).

##  II Small parameters governing nuclear structure effects of $\beta$-decays
of finite nuclei

The differential distribution of a $\beta$-electron of energy $\epsilon$,
momentum $\vec{k}$ and direction $\vec{\beta}=\frac{\vec{k}}{\epsilon}$, and a
neutrino $\nu$ of momentum $\vec{\nu}$, in a $\beta^{\mp}$-decay process, can
be written as [16]:

$\displaystyle\frac{d^{5}\omega}{d\epsilon\frac{d\Omega_{k}}{4\pi}\frac{d\Omega_{\nu}}{4\pi}}$
$\displaystyle=\frac{4}{\pi^{2}}\left(\epsilon_{0}-\epsilon\right)^{2}k\epsilon
F^{\mp}\left(Z_{f},\epsilon\right)C_{\text{corr.}}\frac{1}{2J_{i}+1}\Theta\left(q,\vec{\beta}\cdot\hat{\nu}\right)\text{.}$
(1)

Here
$\left(\epsilon_{0},\vec{q}\right)=\left(\epsilon,\vec{k}\right)+\left(\nu,\vec{\nu}\right)$
is the momentum transfer in the process, i.e., the difference between the
initial momentum and final momentum of the nuclear states. $J_{i}$ is the
total angular momentum of the decaying nucleus, and $Z_{f}$ is the charge of
the nucleus after the decay. The deformation of the beta particle wave
function, due to the long-range electromagnetic interaction with the nucleus,
is taken into account at the leading order by the Fermi function
$F^{\pm}\left(Z_{f},\epsilon\right)$ for a $\beta^{\pm}$-decay.

Corrections which do not originate purely in the weak matrix element, such as
radiative corrections, finite mass and electrostatic finite size effects, and
atomic and chemical effects, are represented in eq. (1) by $C_{\text{corr.}}$.
As these corrections are assumed known in the literature (see, e.g., [14]), we
will not focus on them, but on those effects originating from the fact that
the weak interaction of a probe with a finite nucleus is affected by the
complex nuclear structure of the decaying and final nuclei. These are absorbed
inside $\Theta\left(q,\vec{\beta}\cdot\hat{\nu}\right)$. Current experimental
efforts are divided into spectrum (shape) measurements or angular correlation
measurements [6], both affected by the nuclear-structure. Our discussion will
distinguish between these corrections.

In order to compare the experimental results to the theory, there is a need
for a good understanding of the theoretical framework. As the momentum
transfer $q$ is limited in $\beta$-decays, usually, up to few tens MeV/c,
typically up to 10 MeV/c, and small compared to other energy or momentum
scales in the nuclear problem, this framework is simplified by an expansion in
a set of small parameters. For example, $\epsilon_{qr}\equiv qR$ (for an
endpoint of $\approx 2\text{MeV}$, $\epsilon_{qr}\approx 0.01A^{\frac{1}{3}}$,
where $A$ is the mass number of the nucleus), with $R$ the nucleus’s nuclear
radius. In addition, the non-relativistic character of the nuclear problem
introduces,

$\displaystyle\epsilon_{\text{NR}}$
$\displaystyle\equiv\frac{P_{\text{Fermi}}}{m_{N}}\approx 0.2,$
$\displaystyle\epsilon_{\text{recoil}}$ $\displaystyle\equiv\frac{q}{m_{N}},$
(2)

where $P_{\text{Fermi}}$ is the Fermi momentum, and $m_{N}$ is the nucleon
mass. These are the non-relativistic expansion small parameter
$\epsilon_{\text{NR}}$, and the recoil correction $\epsilon_{\text{recoil}}$
(e.g., for an endpoint of $\approx 2$ MeV, $\epsilon_{\rm recoil}\approx
0.002$), both related to the low energy assumption taken in the calculation of
the currents.

A different source of deviation from the simplistic $\beta$-decay formulas
arises from the Coulomb force between the $\beta$-particle and the nucleus,
which deforms the wave function of the $\beta$-particle from a plane-wave
[17]. This results in corrections dominated by $\epsilon_{c}\equiv\alpha
Z_{f}$ [18], where $\alpha\approx\frac{1}{137}$ is the fine structure
constant. Those can be divided, as described in Ref. [19], into two:
corrections of the spectrum, reflected in the Fermi function and in additional
spectrum corrections whose precise form depends upon the specific
$\beta$-transition, and corrections to the recoil form factors. The latter can
also be separated into two: corrections reminiscent of the spectrum
corrections, and a correction of the electron energy, following the presence
of the Coulomb potential.

The Fermi function is considered known and we will not discuss it here. The
non-energy corrections to the form factors, are of
$\epsilon_{c}\epsilon_{qr}\epsilon_{\rm recoil}$ order of magnitude [17], and
therefore negligible at this stage. The remaining corrections are those that
depend on the transition, and the correction of the electron energy.

The correction of the electron energy adds the Coulomb displacement energy to
the maximal electron energy [20] appearing in the multipole operators (see
appendix). This Coulomb displacement energy is usually taken by its
approximation for a homogeneously charged sphere $\Delta
E_{c}\approx\frac{6}{5}\frac{\alpha Z_{f}}{R}$ [20], but this is a very crude
approximation, and for comparison with experimental measurements we suggest
using the experimental value, that can be found in the literature for most
nuclei (see, e.g., [21]).

The last type of Coulomb corrections is the spectrum corrections which are
unique to each $\beta$-transition. Those are of the order of
$\epsilon_{c}^{2}$ or $\epsilon_{c}\epsilon_{qr}$ [22]. While for particularly
light nuclei these are negligible, for light-medium nuclei this is already a
substantial contribution and there is a need for neat work summarizing them
for the different transitions in a form useful for Ab initio calculations. As
current experimental efforts focus on light nuclei, we postpone a more
detailed discussion in these corrections to future work (see Refs. [23, 20],
and Refs. [14, 24] for allowed transitions).

The nuclear model is a source of uncertainty, as it is a low energy effective
model of QCD. It is generally solved in a few steps. First, a model is written
to describe the Hamiltonian and currents representing the quantum process in
the nuclear regime. There are several approaches to cope with this problem. A
modern one is based on the effective field theory (EFT) approach. EFT
expansion relies on scale separation in the low energy nuclear momentum-
energy. As $\beta$-decays and nuclei are characterized by low-energies
compared to QCD scales, they can be described effectively by an EFT which
includes nucleons and pions, an EFT named chiral EFT. EFT is a systematic
expansion of the Hamiltonian of the fundamental theory, order-by-order, in a
small parameter $\epsilon_{\text{EFT}}=Q/\Lambda_{b}$ [6], where $\Lambda_{b}$
is the breakdown scale of the EFT. In chiral EFT, there is more than just the
nucleons and pions as effective degrees of freedom, it is combined with a
specific approach to non-relativistically expand the nuclear theory, a
specific cutoff regularization and more. We note that this procedure includes
a systematic uncertainty, $\epsilon_{\text{EFT}}^{n}$, where $n$ is the order
of the EFT expansion. However, there is an additional systematic error in
doing so, $\epsilon_{\text{model}}$. Finally, the resulting equations are
solved numerically, introducing a third source of uncertainty, the convergence
error $\epsilon_{\text{conv}}$. The combination of all these is denoted by
$\epsilon_{\text{NM}}$ (NM is nuclear model). In recent years it has been
shown that this procedure reaches $\epsilon_{\text{NM}}$ of a few percents
[25].

In the following, we trace the scaling of different nuclear structure related
corrections to the aforementioned dimensionless small parameters.

##  III $\beta$-decay multipole expansion formalism

Assuming the Standard Model $V-A$ coupling, the function
$\Theta\left(q,\vec{\beta}\cdot\hat{\nu}\right)$ depends on the nuclear wave
functions, and is conveniently written using a multipole expansion [26],

$\Theta\left(q,\vec{\beta}\cdot\hat{\nu}\right)=\sum_{J=1}^{\infty}\left[\left(1-\left(\hat{\nu}\cdot\hat{q}\right)\left(\vec{\beta}\cdot\hat{q}\right)\right)\left(\left|\left\langle\left\|\hat{E}_{J}\right\|\right\rangle\right|^{2}+\left|\left\langle\left\|\hat{M}_{J}\right\|\right\rangle\right|^{2}\right)\pm\hat{q}\cdot\left(\hat{\nu}-\vec{\beta}\right)2\mathfrak{Re}\left(\left\langle\left\|\hat{E}_{J}\right\|\right\rangle\left\langle\left\|\hat{M}_{J}\right\|\right\rangle^{*}\right)\right]+\\\
+\sum_{J=0}^{\infty}\left[\left(1-\vec{\beta}\cdot\hat{\nu}+2\left(\hat{\nu}\cdot\hat{q}\right)\left(\vec{\beta}\cdot\hat{q}\right)\right)\left|\left\langle\left\|\hat{L}_{J}\right\|\right\rangle\right|^{2}+\left(1+\vec{\beta}\cdot\hat{\nu}\right)\left|\left\langle\left\|\hat{C}_{J}\right\|\right\rangle\right|^{2}-\hat{q}\cdot\left(\hat{\nu}+\vec{\beta}\right)2\mathfrak{Re}\left(\left\langle\left\|\hat{L}_{J}\right\|\right\rangle\left\langle\left\|\hat{C}_{J}\right\|\right\rangle^{*}\right)\right]\text{,}\\\
$ (3)

where $\left\langle\left\|\hat{O}_{J}\right\|\right\rangle$ is the reduced
matrix element of a rank $J$ spherical tensor operator $\hat{O}_{J}$, between
the initial- and final-nuclear state wave functions. We use a multipole
operator decomposition of the nuclear current, viz. the Coulomb, longitudinal,
electric, and magnetic operators, defined as:

$\displaystyle\hat{C}_{J}\left(q\right)$ $\displaystyle\equiv$
$\displaystyle\int
d^{3}rj_{J}\left(qr\right)Y_{J}\left(\hat{r}\right)J_{0}\left(\vec{r}\right),$
$\displaystyle\hat{L}_{J}\left(q\right)$ $\displaystyle\equiv$
$\displaystyle\frac{i}{q}\int
d^{3}r\left[\vec{\nabla}\left(j_{J}\left(qr\right)Y_{J}\left(\hat{r}\right)\right)\right]\cdot\vec{J}\left(\vec{r}\right),$
$\displaystyle\hat{E}_{J}\left(q\right)$ $\displaystyle\equiv$
$\displaystyle\frac{1}{q}\int
d^{3}r\left[\vec{\nabla}\times\left(j_{J}\left(qr\right)\vec{Y}_{JJ1}\left(\hat{r}\right)\right)\right]\cdot\vec{J}\left(\vec{r}\right),$
$\displaystyle\hat{M}_{J}\left(q\right)$ $\displaystyle\equiv$
$\displaystyle\int
d^{3}rj_{J}\left(qr\right)\vec{Y}_{JJ1}\left(\hat{r}\right)\cdot\vec{J}\left(\vec{r}\right)\mbox{,}$
(4)

where $\vec{J}\left(\vec{r}\right)$ and $J_{0}\left(\vec{r}\right)$ are the
nuclear, current and charge, coupling to the probe. $j_{J}$ are the spherical
Bessel functions, and $Y_{J}$ and $\vec{Y}_{Jl1}^{M}$ are the spherical
harmonics and the vector spherical harmonics, respectively. The nuclear wave
functions are commonly calculated numerically using an expansion in a complete
basis. In some cases, analytical expressions can be derived, for example, for
an expansion in harmonic oscillator wave functions, the nuclear matrix
elements of the multipole operators can be calculated using Ref. [27],
limiting the interaction of the weak probes to a single nucleon, and
neglecting the interaction with many-body clusters of nucleons in the nucleus.

For a vanishing parameter, the spherical Bessel functions behave as
$j_{J}\left(\rho\right)\propto\rho^{J}$ (for $\rho\ll 1$). As the momentum
transfer $q$ is small for $\beta$-decays, compared to any nuclear property,
$\epsilon_{qr}\equiv qR$ is a small parameter. Explicit analysis shows that
$\hat{C}_{J},\hat{M}_{J}\propto\epsilon_{qr}^{J}$, while
$\hat{L}_{J},\hat{E}_{J}\propto\epsilon_{qr}^{J-1}$.

Further simplification is a result of the parity of the multipole operators.
Since the spherical harmonics $Y_{J}$ and $\vec{Y}_{JJ1}^{M}$ have parity of
$\left(-\right)^{J}$, the basic parities of the multipole operators (before
taking into consideration the nuclear current involved) are
$\left(-\right)^{J}$ for the $\hat{C}_{J}$ and $\hat{M}_{J}$ operators, and
$\left(-\right)^{J+1}$ for $\hat{L}_{J}$ and $\hat{E}_{J}$. To these, we add
the parity of the nuclear currents. While the weak axial charge $J_{0}^{A}$
and the vector current $\vec{J}^{V}$ have a $\frac{\vec{p}}{2m_{N}}$
dependence at their leading order, which leads to an internal negative parity,
the weak vector charge $J_{0}^{V}$ and the axial current $\vec{J}^{A}$ have no
momentum dependence, leading to an internal positive parity at their leading
order. In conclusion, $\hat{C}_{J}^{V}$, $\hat{L}_{J}^{V}$, $\hat{E}_{J}^{V}$
and $\hat{M}_{J}^{A}$ (the superscript $V$ ($A$) denotes multipole operators
calculated with the vector (axial) symmetry contribution to the weak nuclear
current) have a parity of $\left(-\right)^{J}$, while $\hat{C}_{J}^{A}$,
$\hat{L}_{J}^{A}$, $\hat{E}_{J}^{A}$ and $\hat{M}_{J}^{V}$ have a parity of
$\left(-\right)^{J+1}$. A detailed non-relativistic expansion of the one-body
polar-vector and axial-vector currents in powers of $\epsilon_{\text{NR}}$ can
be found in the appendix, along with a detailed derivation of the resulting
multipole operators and their accuracy estimation.

The leading dependence of the nuclear matrix element in the momentum transfer
$q$ leads also to a characterization of $\beta$-decays into allowed and
forbidden transitions. The allowed transitions are, at their leading order,
independent of the momentum transfer $q$, and characterized by no change in
the orbital angular momentum $L$, and therefore no change in parity. A Fermi
(Gamow-Teller) transition is an allowed transition in which $J$, the change in
the total angular momentum of the nucleus, is 0 (1). Other transitions are
suppressed by a $q^{L}$ dependence, with $L\geq 1$, and therefore are much
slower, and are called, for historical reasons, forbidden transitions. Those
are divided into different $L^{\text{th}}$-forbidden transitions, each with a
change of orbital angular momentum $L$, resulting in a parity change of
$\left(-\right)^{L}$. $\beta$-decays in which the emitted leptons pair have a
total spin angular momentum of $S=1$, and an orbital angular momentum $L=J-1$,
are called unique $L^{\text{th}}$-forbidden transitions.

The multipole operators depend upon the nuclear model via the nuclear current
operators, and connect between the initial (decaying) and final nucleus. This
is the source of the $\epsilon_{\text{NM}}$ uncertainty described in sec. II.
We further elaborate on this in sec. VII.

##  IV A general expression for nuclear shape and recoil corrections

Assuming the Standard Model V-A coupling, a $\beta$-decay transition, with a
$J_{i}^{\pi_{i}}\rightarrow J_{f}^{\pi_{f}}$ angular momentum and parity
change, contains all the integer total angular momentum changes which uphold
the selection rule $\left|J_{i}-J_{f}\right|\leq J\leq J_{i}+J_{f}$, with the
exact parity $\Delta\pi\equiv\pi_{i}\cdot\pi_{f}$:

$\displaystyle\Theta\left(q,\vec{\beta}\cdot\hat{\nu}\right)$
$\displaystyle=\sum_{J=\left|J_{i}-J_{f}\right|}^{J_{i}+J_{f}}\Theta^{J^{\Delta\pi}}\left(q,\vec{\beta}\cdot\hat{\nu}\right)\text{.}$
(5)

The next step is to present
$\Theta^{J^{\Delta\pi}}\left(q,\vec{\beta}\cdot\hat{\nu}\right)$ for each
$J^{\Delta\pi}$. Starting from $J=0$, Fermi transition ($J^{\Delta\pi}=0^{+}$)
expression, including shape and recoil next-to-leading order (NLO)
corrections, can be written as

$\displaystyle\Theta^{0^{+}}\left(q,\vec{\beta}\cdot\hat{\nu}\right)$
$\displaystyle=\left(1+\delta_{1}^{0^{+}}\right)\left[1+a_{\beta\nu}^{0^{+}}\vec{\beta}\cdot\hat{\nu}+b_{\text{F}}^{0^{+}}\frac{m_{e}}{\epsilon}\right]\left|\left\langle\left\|\hat{C}_{0}^{V}\right\|\right\rangle\right|^{2}\text{,}$
(6)

with the NLO spectrum-shape correction:

$\displaystyle\delta_{1}^{0^{+}}$ $\displaystyle=$
$\displaystyle-\frac{\epsilon_{0}}{q}2\mathfrak{Re}\frac{\left\langle\left\|\hat{L}_{0}^{V}\right\|\right\rangle}{\left\langle\left\|\hat{C}_{0}^{V}\right\|\right\rangle}+\mathcal{O}\left(\epsilon_{\text{recoil}}^{2},\epsilon_{qr}^{2}\epsilon_{\text{NR}}^{2}\right)\text{.}$
(7)

The multipole operator $\hat{C}_{0}^{V}\propto 1$ is the Fermi leading order,
and
$\hat{L}_{0}^{V}\propto\epsilon_{\text{recoil}},\epsilon_{qr}\epsilon_{\text{NR}}$
111$\epsilon_{\text{recoil}}$ and $\epsilon_{qr}\epsilon_{\text{NR}}$ are
about the same order of magnitude for an endpoint of $\approx 2\text{MeV}$,
and $\hat{L}_{0}^{V}$ consists of these two forms of terms is its NLO recoil
correction.

One of the parameters playing an important role in beyond the standard model
experimental searches, is the Fierz interference term, which is the
coefficient of the $\frac{m_{e}}{\epsilon}$ term in the differential
distribution of a $\beta$-decay, and can be extracted from electron energy
spectrum measurements. Fierz term vanishes in the known $V-A$ differential
distribution of allowed $\beta$-decay transitions. Taking into account NLO
recoil corrections, Fierz interference term at the Fermi transition, gets the
form

$\displaystyle b_{\text{F}}^{0^{+}}=\delta_{b}^{0^{+}}$
$\displaystyle\equiv\frac{m_{e}}{q}2\mathfrak{Re}\frac{\left\langle\left\|\hat{L}_{0}^{V}\right\|\right\rangle}{\left\langle\left\|\hat{C}_{0}^{V}\right\|\right\rangle}+\mathcal{O}\left(\frac{m_{e}}{\epsilon}\epsilon_{\text{recoil}}^{2},\frac{m_{e}}{\epsilon}\epsilon_{qr}^{2}\epsilon_{\text{NR}}^{2}\right)\text{.}$
(8)

Another parameter of interest for BSM searches, is the angular correlation
between the emitted electron and neutrino, $a_{\beta\nu}$, which is the
coefficient of the $\vec{\beta}\cdot\hat{\nu}$ term in the differential
distribution of a $\beta$-decay. The $V-A$ structure of the weak interaction
entails that for a Fermi transition the $\beta-\nu$ correlation is exactly
$a_{\beta\nu}^{0^{+}}=1$.

As for $J^{\Delta\pi}=0^{-}$, which is a non-unique first-forbidden
transition, its $\Theta^{0^{-}}\left(q,\vec{\beta}\cdot\hat{\nu}\right)$
expression takes the exact form

$\Theta^{0^{-}}\left(q,\vec{\beta}\cdot\hat{\nu}\right)=\left(1+\vec{\beta}\cdot\hat{\nu}\right)\left[\left|\left\langle\left\|\hat{C}_{0}^{A}\right\|\right\rangle\right|^{2}+\left|\left\langle\left\|\hat{L}_{0}^{A}\right\|\right\rangle\right|^{2}-\frac{\epsilon_{0}}{q}2\mathfrak{Re}\left(\left\langle\left\|\hat{L}_{0}^{A}\right\|\right\rangle\left\langle\left\|\hat{C}_{0}^{A}\right\|\right\rangle^{*}\right)\right]+\\\
+\frac{m_{e}^{2}}{q\epsilon}2\mathfrak{Re}\left(\left\langle\left\|\hat{L}_{0}^{A}\right\|\right\rangle\left\langle\left\|\hat{C}_{0}^{A}\right\|\right\rangle^{*}\right)+2\frac{\epsilon\left(\epsilon_{0}-\epsilon\right)}{q^{2}}\left[\beta^{2}-\left(\vec{\beta}\cdot\hat{\nu}\right)^{2}\right]\left|\left\langle\left\|\hat{L}_{0}^{A}\right\|\right\rangle\right|^{2}\text{,}$
(9)

with $\hat{C}_{0}^{A}\propto\epsilon_{\text{NR}}$ and
$\hat{L}_{0}^{A}\propto\epsilon_{qr}$. For this $J^{\Delta\pi}=0^{-}$ first-
forbidden transition, the angular correlation coefficient can be easily
recognized as $a_{\beta\nu}^{0^{-}}=1$, and the Fierz term can be extracted,

$\displaystyle b_{\text{F}}^{0^{-}}=$
$\displaystyle\frac{\frac{m_{e}}{q}2\mathfrak{Re}\left(\left\langle\left\|\hat{L}_{0}^{A}\right\|\right\rangle\left\langle\left\|\hat{C}_{0}^{A}\right\|\right\rangle^{*}\right)}{\left|\left\langle\left\|\hat{C}_{0}^{A}\right\|\right\rangle\right|^{2}+\left|\left\langle\left\|\hat{L}_{0}^{A}\right\|\right\rangle\right|^{2}-\frac{\epsilon_{0}}{q}2\mathfrak{Re}\left(\left\langle\left\|\hat{L}_{0}^{A}\right\|\right\rangle\left\langle\left\|\hat{C}_{0}^{A}\right\|\right\rangle^{*}\right)}\text{.}$
(10)

In order to discuss the
$\Theta^{J^{\Delta\pi}}\left(q,\vec{\beta}\cdot\hat{\nu}\right)$ expressions
for $J$’s grater than $0$, we will distinguish between two types of
$0<J^{\Delta\pi}$ transitions: $\Delta\pi=\left(-\right)^{J}$, and
$\Delta\pi=\left(-\right)^{J-1}$. The first type, $J^{\left(-\right)^{J}}$,
presents non-unique $J^{\rm th}$ forbidden transitions. The second type,
$J^{\left(-\right)^{J-1}}$, presents, for $J=1$, the allowed Gamow-Teller
transition, and for $J>1$, unique $(J-1)^{\rm th}$ forbidden transitions
(together we will refer to them as unique transitions).

Let us start with the unique transitions (e.g., Gamow–Teller transition, which
is $J^{\Delta\pi}=1^{+}$). A general expression, including shape and recoil
NLO corrections, for any unique transition, i.e., a decay with
$0<J^{\left(-\right)^{J-1}}$, in the sum above (eq. (5)), can be written as:

$\displaystyle\Theta^{J^{\left(-\right)^{J-1}}}\left(q,\vec{\beta}\cdot\hat{\nu}\right)$
$\displaystyle=\frac{2J+1}{J}\left(1+\delta_{1}^{J^{\left(-\right)^{J-1}}}\right)\left\\{1+a_{\beta\nu}^{J^{\left(-\right)^{J-1}}}\hat{\nu}\cdot\vec{\beta}+b_{\rm
F}^{J^{\left(-\right)^{J-1}}}\frac{m_{e}}{\epsilon}+\right.$
$\displaystyle\left.+\frac{J-1}{2J+1}\frac{\epsilon\left(\epsilon_{0}-\epsilon\right)}{q^{2}}\left(1+\tilde{\delta}_{\beta^{2}}^{J^{\left(-\right)^{J-1}}}\right)\left[\vec{\beta}^{2}-\left(\hat{\nu}\cdot\vec{\beta}\right)^{2}\right]\right\\}\left|\left\langle\left\|\hat{L}_{J}^{A}\right\|\right\rangle\right|^{2}\text{,}$
(11)

with the shape and recoil NLO corrections:

$\displaystyle\delta_{1}^{J^{\left(-\right)^{J-1}}}$
$\displaystyle=\frac{2}{2J+1}\mathfrak{Re}\left[-J\frac{\epsilon_{0}}{q}\frac{\left\langle\left\|\hat{C}_{J}^{A}\right\|\right\rangle}{\left\langle\left\|\hat{L}_{J}^{A}\right\|\right\rangle}\pm\sqrt{J\left(J+1\right)}\frac{\epsilon_{0}-2\epsilon}{q}\frac{\left\langle\left\|\hat{M}_{J}^{V}\right\|\right\rangle}{\left\langle\left\|\hat{L}_{J}^{A}\right\|\right\rangle}+\sqrt{J\left(J+1\right)}\frac{\left\langle\left\|\hat{E}_{J}^{A\left(\rm
res\right)}\right\|\right\rangle}{\left\langle\left\|\hat{L}_{J}^{A}\right\|\right\rangle}\right]+$
$\displaystyle+\mathcal{O}\left(\epsilon_{qr}^{2}\epsilon_{\rm
NR}^{2},\epsilon_{\text{recoil}}^{2}\right),$
$\displaystyle\tilde{\delta}_{\beta^{2}}^{J^{\left(-\right)^{J-1}}}$
$\displaystyle=-\frac{2}{2J+1}\mathfrak{Re}\left[-J\frac{\epsilon_{0}}{q}\frac{\left\langle\left\|\hat{C}_{J}^{A}\right\|\right\rangle}{\left\langle\left\|\hat{L}_{J}^{A}\right\|\right\rangle}\pm\sqrt{J\left(J+1\right)}\frac{\epsilon_{0}-2\epsilon}{q}\frac{\left\langle\left\|\hat{M}_{J}^{V}\right\|\right\rangle}{\left\langle\left\|\hat{L}_{J}^{A}\right\|\right\rangle}+\frac{3J}{J-1}\sqrt{J\left(J+1\right)}\frac{\left\langle\left\|\hat{E}_{J}^{A\left(\rm
res\right)}\right\|\right\rangle}{\left\langle\left\|\hat{L}_{J}^{A}\right\|\right\rangle}\right]+$
$\displaystyle+\mathcal{O}\left(\epsilon_{qr}^{2}\epsilon_{\rm
NR}^{2},\epsilon_{\text{recoil}}^{2}\right)\text{,}$ (12)

where $\pm$ are for $\beta^{\mp}$ decays. Here the multipole operator
$\hat{L}_{J}^{A}\propto\epsilon_{qr}^{J-1}$ is the leading order operator,
while
$\frac{\hat{C}_{J}^{A}}{\hat{L}_{J}^{A}},\frac{\hat{M}_{J}^{V}}{\hat{L}_{J}^{A}}\propto\epsilon_{\text{recoil}},\epsilon_{qr}\epsilon_{\text{NR}}$
222Both multipoles, $\hat{C}_{J}^{A}$ and $\hat{M}_{J}^{V}$, consist of these
two forms of terms. $\hat{C}_{J}^{A}$ has an additional smaller term,
proportional to $\epsilon_{m_{\pi}}\equiv\frac{\epsilon_{0}q}{m_{\pi}^{2}}$
are NLO recoil corrections operators. To eliminate the electric multipole
operator $\hat{E}_{J}$, we used the relation

$\displaystyle\hat{E}_{J}=\sqrt{\frac{J+1}{J}}\hat{L}_{J}-i\sqrt{\frac{2J+1}{J}}\int
d^{3}rj_{J+1}\left(qr\right)\vec{Y}_{J,J+1,1}\left(\hat{r}\right)\cdot\vec{J}\left(\vec{r}\right)\equiv\sqrt{\frac{J+1}{J}}\hat{L}_{J}+\hat{E}_{J}^{\left(\rm
res\right)}\text{,}$ (13)

that leaves a residual correction $\hat{E}_{J}^{\left(\rm res\right)}$ of
$\hat{E}_{J}$ regarding $\hat{L}_{J}$, and for the unique transitions is
reflected in another NLO correction of $\frac{\hat{E}_{J}^{A\left(\rm
res\right)}}{\hat{L}_{J}^{A}}\propto\epsilon_{qr}^{2}$.

The $V-A$ structure of the weak interaction entails, for any specific unique
transition, that the $\beta-\nu$ correlations leading order will be
$a_{\beta\nu}^{J^{\left(-\right)^{J-1}}}=-\frac{1}{2J+1}$. Adding the NLO
corrections, the $\beta-\nu$ correlation becomes

$\displaystyle a_{\beta\nu}^{J^{\left(-\right)^{J-1}}}$
$\displaystyle=-\frac{1}{2J+1}\left(1+\tilde{\delta}_{a}^{J^{\left(-\right)^{J-1}}}\right)\text{,}$
(14)

with

$\displaystyle\tilde{\delta}_{a}^{J^{\left(-\right)^{J-1}}}$
$\displaystyle=\frac{2J}{2J+1}2\mathfrak{Re}\left[\left(J+1\right)\frac{\epsilon_{0}}{q}\frac{\left\langle\left\|\hat{C}_{J}^{A}\right\|\right\rangle}{\left\langle\left\|\hat{L}_{J}^{A}\right\|\right\rangle}\pm\sqrt{J\left(J+1\right)}\frac{\epsilon_{0}-2\epsilon}{q}\frac{\left\langle\left\|\hat{M}_{J}^{V}\right\|\right\rangle}{\left\langle\left\|\hat{L}_{J}^{A}\right\|\right\rangle}+\sqrt{J\left(J+1\right)}\frac{\left\langle\left\|\hat{E}_{J}^{A\left(\rm
res\right)}\right\|\right\rangle}{\left\langle\left\|\hat{L}_{J}^{A}\right\|\right\rangle}\right]+$
$\displaystyle+\mathcal{O}\left(\epsilon_{qr}^{2}\epsilon_{\rm
NR}^{2},\epsilon_{\text{recoil}}^{2}\right)\text{.}$ (15)

As for the Fierz term, that vanishes for unique transitions in the $V-A$
structure of the weak interaction, it now gets the form

$\displaystyle
b_{\text{F}}^{J^{\left(-\right)^{J-1}}}=\delta_{b}^{J^{\left(-\right)^{J-1}}}$
$\displaystyle\equiv\frac{1}{2J+1}\frac{m_{e}}{q}2\mathfrak{Re}\left\\{J\frac{\left\langle\left\|\hat{C}_{J}^{A}\right\|\right\rangle}{\left\langle\left\|\hat{L}_{J}^{A}\right\|\right\rangle}\pm\sqrt{J\left(J+1\right)}\frac{\left\langle\left\|\hat{M}_{J}^{V}\right\|\right\rangle}{\left\langle\left\|\hat{L}_{J}^{A}\right\|\right\rangle}\right\\}+\mathcal{O}\left(\frac{m_{e}}{\epsilon}\epsilon_{qr}^{2}\epsilon_{\rm
NR}^{2},\frac{m_{e}}{\epsilon}\epsilon_{\text{recoil}}^{2}\right)\text{.}$
(16)

For the case of non-unique transitions, i.e., a decay with
$0<J^{\left(-\right)^{J}}$, the
$\Theta^{J^{\left(-\right)^{J}}}\left(q,\vec{\beta}\cdot\hat{\nu}\right)$
expression can be written as:

$\Theta^{J^{\left(-\right)^{J}}}\left(q,\vec{\beta}\cdot\hat{\nu}\right)=\left\\{1+\frac{1}{J}\frac{\epsilon_{0}^{2}}{q^{2}}\left[1-\left(J+1\right)2\mathfrak{Re}\delta^{J^{\left(-\right)^{J}}}\right]+2\frac{\epsilon_{0}}{q^{2}}\frac{m_{e}^{2}}{\epsilon}\right\\}\left|\left\langle\left\|\hat{C}_{J}^{V}\right\|\right\rangle\right|^{2}+\left|\left\langle\left\|\hat{M}_{J}^{A}\right\|\right\rangle\right|^{2}+\\\
\pm\sqrt{\frac{J+1}{J}}\left(\frac{\epsilon_{0}^{2}}{q^{2}}-2\frac{\epsilon_{0}\epsilon}{q^{2}}+\frac{\epsilon_{0}}{q^{2}}\frac{m_{e}^{2}}{\epsilon}\right)2\mathfrak{Re}\left[\left\langle\left\|\hat{C}_{J}^{V}\right\|\right\rangle\left\langle\left\|\hat{M}_{J}^{A}\right\|\right\rangle^{*}\left(1-\delta^{J^{\left(-\right)^{J}}}\right)\right]+\\\
+\vec{\beta}\cdot\hat{\nu}\left\\{\left[1-\frac{2J+1}{J}\frac{\epsilon_{0}^{2}}{q^{2}}\left(1-\frac{J+1}{2J+1}2\mathfrak{Re}\delta^{J^{\left(-\right)^{J}}}\right)\right]\left|\left\langle\left\|\hat{C}_{J}^{V}\right\|\right\rangle\right|^{2}-\left|\left\langle\left\|\hat{M}_{J}^{A}\right\|\right\rangle\right|^{2}+\right.\\\
\left.\mp\sqrt{\frac{J+1}{J}}\left(\frac{\epsilon_{0}^{2}}{q^{2}}-2\frac{\epsilon_{0}\epsilon}{q^{2}}\right)2\mathfrak{Re}\left[\left\langle\left\|\hat{C}_{J}^{V}\right\|\right\rangle\left\langle\left\|\hat{M}_{J}^{A}\right\|\right\rangle^{*}\left(1-\delta^{J^{\left(-\right)^{J}}}\right)\right]\right\\}+\\\
+\frac{\epsilon\left(\epsilon_{0}-\epsilon\right)}{q^{2}}\left[\beta^{2}-\left(\vec{\beta}\cdot\hat{\nu}\right)^{2}\right]\left[\frac{J-1}{J}\frac{\epsilon_{0}^{2}}{q^{2}}\left(1+\frac{J+1}{J-1}2\mathfrak{Re}\delta^{J^{\left(-\right)^{J}}}\right)\left|\left\langle\left\|\hat{C}_{J}^{V}\right\|\right\rangle\right|^{2}-\left|\left\langle\left\|\hat{M}_{J}^{A}\right\|\right\rangle\right|^{2}\right]\text{,}$
(17)

with the NLO correction

$\displaystyle\delta^{J^{\left(-\right)^{J}}}$
$\displaystyle=-\frac{q}{\epsilon_{0}}\sqrt{\frac{J}{J+1}}\frac{\left\langle\left\|\hat{E}_{J}^{V\left(\rm
res\right)}\right\|\right\rangle}{\left\langle\left\|\hat{C}_{J}^{V}\right\|\right\rangle}+\mathcal{O}\left(\epsilon_{qr}^{2}\epsilon_{\text{NR}}^{2}\right)\text{.}$
(18)

Using the vector current conservation hypothesis (exact to relevant orders in
chiral EFT), which eliminates
$\hat{L}_{J}^{V}=\frac{\epsilon_{0}}{q}\hat{C}_{J}^{V}$ for $J>0$ [26], the
multipole operators involved in the leading order are only
$\hat{C}_{J}^{V},\hat{M}_{J}^{A}\propto\epsilon_{qr}^{J}$, and the NLO
operator is $\frac{\hat{E}_{J}^{V\left(\rm
res\right)}}{\hat{C}_{J}^{V}}\propto\epsilon_{qr}\epsilon_{\text{NR}}$ (the
vector current $\vec{J}^{V}\left(\vec{r}\right)\propto\epsilon_{\text{NR}}$).
For non-unique forbidden decays, the Fierz term, and the angular correlation
coefficient, are not well defined.

##  V Nuclear shape and recoil corrections for allowed transitions

For example, let us consider the most general allowed transition, e.g., mixed
Fermi ($J^{\Delta\pi}=0^{+}$) and Gamow-Teller ($J^{\Delta\pi}=1^{+}$)
transitions. This requires $\Delta\pi=+$ and $J_{i}=J_{f}>0$, so, following
eq. (1) and (5), the decay rate, including NLO corrections, will contain a sum
of $\Theta^{0^{+}}$ (eq. (6)) and $\Theta^{1^{+}}$ (eq. (IV)) (other
$\Theta^{J^{+}}$’s, with $J^{+}$s up to $J_{i}+J_{f}$, also participate in the
sum, but they contribute only higher orders):

$\frac{d^{5}\omega}{d\epsilon\frac{d\Omega_{k}}{4\pi}\frac{d\Omega_{\nu}}{4\pi}}=\frac{4}{\pi^{2}}\frac{1}{2J_{i}+1}\left(\epsilon_{0}-\epsilon\right)^{2}k\epsilon
F^{\mp}\left(Z_{f},\epsilon\right)C_{\text{corrections}}\cdot\\\
\cdot\left\\{\left(1+\delta_{1}^{0^{+}}\right)\left[1+\vec{\beta}\cdot\hat{\nu}+b_{\text{F}}^{0^{+}}\frac{m_{e}}{\epsilon}\right]\left|\left\langle\left\|\hat{C}_{0}^{V}\right\|\right\rangle\right|^{2}+3\left(1+\delta_{1}^{1^{+}}\right)\left[1-\frac{1}{3}\vec{\beta}\cdot\hat{\nu}\left(1+\tilde{\delta}_{a}^{1^{+}}\right)+b_{\text{F}}^{1^{+}}\frac{m_{e}}{\epsilon}\right]\left|\left\langle\left\|\hat{L}_{1}^{A}\right\|\right\rangle\right|^{2}\right\\}=\\\
=\frac{4}{\pi^{2}}\frac{1}{2J_{i}+1}\left(\epsilon_{0}-\epsilon\right)^{2}k\epsilon
F^{\mp}\left(Z_{f},\epsilon\right)C_{\text{corrections}}\left(\left|\left\langle\left\|\hat{C}_{0}^{V}\right\|\right\rangle\right|^{2}+3\left|\left\langle\left\|\hat{L}_{1}^{A}\right\|\right\rangle\right|^{2}\right)\cdot\\\
\cdot\left(1+\delta_{1}^{\text{F+GT}}\right)\left[1+\vec{\beta}\cdot\hat{\nu}\left(1+\tilde{\delta}_{a}^{\text{F+GT}}\right)\frac{\left|\left\langle\left\|\hat{C}_{0}^{V}\right\|\right\rangle\right|^{2}-\left|\left\langle\left\|\hat{L}_{1}^{A}\right\|\right\rangle\right|^{2}}{\left|\left\langle\left\|\hat{C}_{0}^{V}\right\|\right\rangle\right|^{2}+3\left|\left\langle\left\|\hat{L}_{1}^{A}\right\|\right\rangle\right|^{2}}+b_{F}^{\text{F+GT}}\frac{m_{e}}{\epsilon}\right]\text{,}$
(19)

with the NLO corrections:

$\displaystyle\delta_{1}^{\text{F+GT}}$
$\displaystyle=\frac{\delta_{1}^{0^{+}}\left|\left\langle\left\|\hat{C}_{0}^{V}\right\|\right\rangle\right|^{2}+3\delta_{1}^{1^{+}}\left|\left\langle\left\|\hat{L}_{1}^{A}\right\|\right\rangle\right|^{2}}{\left|\left\langle\left\|\hat{C}_{0}^{V}\right\|\right\rangle\right|^{2}+3\left|\left\langle\left\|\hat{L}_{1}^{A}\right\|\right\rangle\right|^{2}},$
$\displaystyle\tilde{\delta}_{a}^{\text{F+GT}}$
$\displaystyle=\frac{\delta_{1}^{0^{+}}\left|\left\langle\left\|\hat{C}_{0}^{V}\right\|\right\rangle\right|^{2}-\left(\delta_{1}^{1^{+}}+\tilde{\delta}_{a}^{1^{+}}\right)\left|\left\langle\left\|\hat{L}_{1}^{A}\right\|\right\rangle\right|^{2}}{\left|\left\langle\left\|\hat{C}_{0}^{V}\right\|\right\rangle\right|^{2}-\left|\left\langle\left\|\hat{L}_{1}^{A}\right\|\right\rangle\right|^{2}}\text{,}$
(20)

where $\delta_{1}^{0^{+}}$ is given in eq. (7), and $\delta_{1}^{1^{+}}$ and
$\tilde{\delta}_{a}^{1^{+}}$ can be found easily from eq. (IV) and (IV):

$\displaystyle\delta_{1}^{1+}$
$\displaystyle=\frac{2}{3}\mathfrak{Re}\left[-\frac{\epsilon_{0}}{q}\frac{\left\langle\left\|\hat{C}_{1}^{A}\right\|\right\rangle}{\left\langle\left\|\hat{L}_{1}^{A}\right\|\right\rangle}\pm\sqrt{2}\frac{\epsilon_{0}-2\epsilon}{q}\frac{\left\langle\left\|\hat{M}_{1}^{V}\right\|\right\rangle}{\left\langle\left\|\hat{L}_{1}^{A}\right\|\right\rangle}+\sqrt{2}\frac{\left\langle\left\|\hat{E}_{1}^{A\left(\rm
res\right)}\right\|\right\rangle}{\left\langle\left\|\hat{L}_{1}^{A}\right\|\right\rangle}\right]+\mathcal{O}\left(\epsilon_{qr}^{2}\epsilon_{\rm
NR}^{2},\epsilon_{\text{recoil}}^{2}\right)$
$\displaystyle\tilde{\delta}_{a}^{1+}$
$\displaystyle=\frac{4}{3}\mathfrak{Re}\left[\frac{2\epsilon_{0}}{q}\frac{\left\langle\left\|\hat{C}_{1}^{A}\right\|\right\rangle}{\left\langle\left\|\hat{L}_{1}^{A}\right\|\right\rangle}\pm\sqrt{2}\frac{\epsilon_{0}-2\epsilon}{q}\frac{\left\langle\left\|\hat{M}_{1}^{V}\right\|\right\rangle}{\left\langle\left\|\hat{L}_{1}^{A}\right\|\right\rangle}+\sqrt{2}\frac{\left\langle\left\|\hat{E}_{1}^{A\left(\rm
res\right)}\right\|\right\rangle}{\left\langle\left\|\hat{L}_{1}^{A}\right\|\right\rangle}\right]+\mathcal{O}\left(\epsilon_{qr}^{2}\epsilon_{\rm
NR}^{2},\epsilon_{\text{recoil}}^{2}\right)\text{.}$ (21)

Here the multipole operators $\hat{C}_{0}^{V},\hat{L}_{1}^{A}\propto 1$ are
the Fermi and Gamow–Teller leading orders, while
$\hat{L}_{0}^{V},\hat{C}_{1}^{A},$$\hat{M}_{1}^{V}\propto\epsilon_{\text{recoil}},\epsilon_{qr}\epsilon_{\text{NR}}$
and $\hat{E}_{1}^{A\left(\rm res\right)}\propto\epsilon_{qr}^{2}$ are their
NLO recoil corrections.

As mentioned in section IV, Fierz term, which vanishes for $V-A$ allowed
decays, does not vanish when taking into account these corrections. For a pure
Fermi transition, its new form already described in eq. (8), while for a pure
Gamow–Teller, it is (from eq. (16))

$\displaystyle
b_{\text{F}}^{1^{+}}=\delta_{b}^{1+}=\frac{2m_{e}}{3q}\mathfrak{Re}\left\\{\frac{\left\langle\left\|\hat{C}_{1}^{A}\right\|\right\rangle}{\left\langle\left\|\hat{L}_{1}^{A}\right\|\right\rangle}\pm\sqrt{2}\frac{\left\langle\left\|\hat{M}_{1}^{V}\right\|\right\rangle}{\left\langle\left\|\hat{L}_{1}^{A}\right\|\right\rangle}\right\\}+\mathcal{O}\left(\frac{m_{e}}{\epsilon}\epsilon_{qr}^{2}\epsilon_{\rm
NR}^{2},\frac{m_{e}}{\epsilon}\epsilon_{\text{recoil}}^{2}\right).$ (22)

As for the angular correlation coefficient, it is, as mentioned, exactly 1 for
a pure Fermi transition, and gets the corrected form
$a_{\beta\nu}^{1^{+}}=-\frac{1}{3}\left(1+\tilde{\delta}_{a}^{1^{+}}\right)$
for a pure Gamow–Teller (eq. (14)). Finally, for a mixed Fermi and
Gamow–Teller transition one gets:

$\displaystyle a_{\beta\nu}^{\text{F+GT}}$
$\displaystyle=\frac{\left|\left\langle\left\|\hat{C}_{0}^{V}\right\|\right\rangle\right|^{2}-\left|\left\langle\left\|\hat{L}_{1}^{A}\right\|\right\rangle\right|^{2}}{\left|\left\langle\left\|\hat{C}_{0}^{V}\right\|\right\rangle\right|^{2}+3\left|\left\langle\left\|\hat{L}_{1}^{A}\right\|\right\rangle\right|^{2}}\left(1+\tilde{\delta}_{a}^{\text{F+GT}}\right),$
$\displaystyle b_{\text{F}}^{\text{F+GT}}$
$\displaystyle=\frac{\delta_{b}^{0+}\left|\left\langle\left\|\hat{C}_{0}^{V}\right\|\right\rangle\right|^{2}+3\delta_{b}^{1+}\left|\left\langle\left\|\hat{L}_{1}^{A}\right\|\right\rangle\right|^{2}}{\left|\left\langle\left\|\hat{C}_{0}^{V}\right\|\right\rangle\right|^{2}+3\left|\left\langle\left\|\hat{L}_{1}^{A}\right\|\right\rangle\right|^{2}}\text{,}$
(23)

with $\delta_{b}^{0+}$ and $\delta_{b}^{1+}$ defined above.

##  VI Nuclear shape and recoil corrections for first-forbidden transitions

As mentioned at section III, first-forbidden transitions are transitions with
a $q^{1}$ leading order dependence. Those will be transitions involving
$J^{\Delta\pi}=0^{-},1^{-},2^{-}$. Let as now assume the most general first-
forbidden transition possible, including all the first-forbidden
$J^{\Delta\pi}$’s. That will be any transition with $\Delta\pi=-$ and
$J_{i}=J_{f}\geq 1$ (e.g., the transition $1^{+}\rightarrow 1^{-}$). This way,
following eq. (1) and (5), the leading order of its decay rate, with
multipoles proportional to $\epsilon_{\text{NR}}$ and $\epsilon_{qr}$, should
include a sum over $J^{\Delta\pi}=0^{-},1^{-},2^{-}$ (other $J^{-}$’s can also
participate in the sum, up to $J_{i}+J_{f}$, but they will contribute only
higher orders):

$\frac{d^{5}\omega}{d\epsilon\frac{d\Omega_{k}}{4\pi}\frac{d\Omega_{\nu}}{4\pi}}=\frac{4}{\pi^{2}}\frac{1}{2J_{i}+1}\left(\epsilon_{0}-\epsilon\right)^{2}k\epsilon
F^{\mp}\left(Z_{f},\epsilon\right)C_{\text{corrections}}\cdot\\\
\cdot\left\\{\left(1+\vec{\beta}\cdot\hat{\nu}\right)\left[\left|\left\langle\left\|\hat{C}_{0}^{A}\right\|\right\rangle\right|^{2}+\left|\left\langle\left\|\hat{L}_{0}^{A}\right\|\right\rangle\right|^{2}-\frac{\epsilon_{0}}{q}2\mathfrak{Re}\left(\left\langle\left\|\hat{L}_{0}^{A}\right\|\right\rangle\left\langle\left\|\hat{C}_{0}^{A}\right\|\right\rangle^{*}\right)\right]+\right.\\\
+\frac{m_{e}^{2}}{q\epsilon}2\mathfrak{Re}\left(\left\langle\left\|\hat{L}_{0}^{A}\right\|\right\rangle\left\langle\left\|\hat{C}_{0}^{A}\right\|\right\rangle^{*}\right)+2\frac{\epsilon\left(\epsilon_{0}-\epsilon\right)}{q^{2}}\left(\beta^{2}-\left(\vec{\beta}\cdot\hat{\nu}\right)^{2}\right)\left|\left\langle\left\|\hat{L}_{0}^{A}\right\|\right\rangle\right|^{2}+\\\
+\left(1+\frac{\epsilon_{0}^{2}}{q^{2}}\left[1-4\mathfrak{Re}\left(\delta^{1^{-}}\right)\right]+2\frac{\epsilon_{0}}{q^{2}}\frac{m_{e}^{2}}{\epsilon}\right)\left|\left\langle\left\|\hat{C}_{1}^{V}\right\|\right\rangle\right|^{2}+\left|\left\langle\left\|\hat{M}_{1}^{A}\right\|\right\rangle\right|^{2}+\\\
\pm\sqrt{2}\left(\frac{\epsilon_{0}^{2}}{q^{2}}-2\frac{\epsilon_{0}\epsilon}{q^{2}}+\frac{\epsilon_{0}}{q^{2}}\frac{m_{e}^{2}}{\epsilon}\right)2\mathfrak{Re}\left(\left\langle\left\|\hat{C}_{1}^{V}\right\|\right\rangle\left\langle\left\|\hat{M}_{1}^{A}\right\|\right\rangle^{*}\left(1-\delta^{1^{-}}\right)\right)+\\\
+\vec{\beta}\cdot\hat{\nu}\left[\left(1-3\frac{\epsilon_{0}^{2}}{q^{2}}\left(1-\frac{4}{3}\mathfrak{Re}\delta^{1^{-}}\right)\right)\left|\left\langle\left\|\hat{C}_{1}^{V}\right\|\right\rangle\right|^{2}-\left|\left\langle\left\|\hat{M}_{1}^{A}\right\|\right\rangle\right|^{2}+\right.\\\
\left.\mp\sqrt{2}\left(\frac{\epsilon_{0}^{2}}{q^{2}}-2\frac{\epsilon_{0}\epsilon}{q^{2}}\right)2\mathfrak{Re}\left(\left\langle\left\|\hat{C}_{1}^{V}\right\|\right\rangle\left\langle\left\|\hat{M}_{1}^{A}\right\|\right\rangle^{*}\left(1-\delta^{1^{-}}\right)\right)\right]+\\\
-\frac{\epsilon\left(\epsilon_{0}-\epsilon\right)}{q^{2}}\left[\beta^{2}-\left(\vec{\beta}\cdot\hat{\nu}\right)^{2}\right]\left|\left\langle\left\|\hat{M}_{1}^{A}\right\|\right\rangle\right|^{2}+\\\
\left.+\frac{5}{2}\left(1+\delta_{1}^{2^{-}}\right)\left\\{1-\frac{1}{5}\left(1+\delta_{a}^{2^{-}}\right)\hat{\nu}\cdot\vec{\beta}+\delta_{b}^{2^{-}}\frac{m_{e}}{\epsilon}+\frac{1}{5}\frac{\epsilon\left(\epsilon_{0}-\epsilon\right)}{q^{2}}\left(1+\tilde{\delta}_{\beta^{2}}^{2^{-}}\right)\left[\vec{\beta}^{2}-\left(\hat{\nu}\cdot\vec{\beta}\right)^{2}\right]\right\\}\left|\left\langle\left\|\hat{L}_{2}^{A}\right\|\right\rangle\right|^{2}\right\\}\text{,}$
(24)

with the shape and recoil NLO corrections:

$\displaystyle\delta^{1^{-}}$
$\displaystyle=-\frac{1}{\sqrt{2}}\frac{q}{\epsilon_{0}}\frac{\left\langle\left\|\hat{E}_{1}^{V\left(\rm
res\right)}\right\|\right\rangle}{\left\langle\left\|\hat{C}_{1}^{V}\right\|\right\rangle}+\mathcal{O}\left(\epsilon_{qr}^{2}\epsilon_{\text{NR}}^{2}\right),$
$\displaystyle\delta_{1}^{2^{-}}$
$\displaystyle=\frac{2}{5}\mathfrak{Re}\left[-\frac{2\epsilon_{0}}{q}\frac{\left\langle\left\|\hat{C}_{2}^{A}\right\|\right\rangle}{\left\langle\left\|\hat{L}_{2}^{A}\right\|\right\rangle}\pm\sqrt{6}\frac{\epsilon_{0}-2\epsilon}{q}\frac{\left\langle\left\|\hat{M}_{2}^{V}\right\|\right\rangle}{\left\langle\left\|\hat{L}_{2}^{A}\right\|\right\rangle}+\sqrt{6}\frac{\left\langle\left\|\hat{E}_{2}^{A\left(\rm
res\right)}\right\|\right\rangle}{\left\langle\left\|\hat{L}_{2}^{A}\right\|\right\rangle}\right]+\mathcal{O}\left(\epsilon_{qr}^{2}\epsilon_{\rm
NR}^{2},\epsilon_{\text{recoil}}^{2}\right),$
$\displaystyle\tilde{\delta}_{\beta^{2}}^{2^{-}}$
$\displaystyle=-\frac{2}{5}\mathfrak{Re}\left[-\frac{2\epsilon_{0}}{q}\frac{\left\langle\left\|\hat{C}_{2}^{A}\right\|\right\rangle}{\left\langle\left\|\hat{L}_{2}^{A}\right\|\right\rangle}\pm\sqrt{6}\frac{\epsilon_{0}-2\epsilon}{q}\frac{\left\langle\left\|\hat{M}_{2}^{V}\right\|\right\rangle}{\left\langle\left\|\hat{L}_{2}^{A}\right\|\right\rangle}+6\sqrt{6}\frac{\left\langle\left\|\hat{E}_{2}^{A\left(\rm
res\right)}\right\|\right\rangle}{\left\langle\left\|\hat{L}_{2}^{A}\right\|\right\rangle}\right]+\mathcal{O}\left(\epsilon_{qr}^{2}\epsilon_{\rm
NR}^{2},\epsilon_{\text{recoil}}^{2}\right),$
$\displaystyle\tilde{\delta}_{a}^{2^{-}}$
$\displaystyle=\frac{8}{5}\mathfrak{Re}\left[\frac{3\epsilon_{0}}{q}\frac{\left\langle\left\|\hat{C}_{2}^{A}\right\|\right\rangle}{\left\langle\left\|\hat{L}_{2}^{A}\right\|\right\rangle}\pm\sqrt{6}\frac{\epsilon_{0}-2\epsilon}{q}\frac{\left\langle\left\|\hat{M}_{2}^{V}\right\|\right\rangle}{\left\langle\left\|\hat{L}_{2}^{A}\right\|\right\rangle}+\sqrt{6}\frac{\left\langle\left\|\hat{E}_{2}^{A\left(\rm
res\right)}\right\|\right\rangle}{\left\langle\left\|\hat{L}_{2}^{A}\right\|\right\rangle}\right]+\mathcal{O}\left(\epsilon_{qr}^{2}\epsilon_{\rm
NR}^{2},\epsilon_{\text{recoil}}^{2}\right)\text{.}$ (25)

Here the multipole operators $\hat{C}_{0}^{A}\propto\epsilon_{\text{NR}}$ and
$\hat{L}_{0}^{A},\hat{C}_{1}^{V},\hat{M}_{1}^{A},\hat{L}_{2}^{A}\propto\epsilon_{qr}$
are the first-forbidden leading orders, while $\frac{\hat{E}_{1}^{V\left(\rm
res\right)}}{\hat{C}_{1}^{V}}\propto\epsilon_{qr}\epsilon_{\text{NR}}$ ,
$\frac{\hat{C}_{2}^{A}}{\hat{L}_{2}^{A}},\frac{\hat{M}_{2}^{V}}{\hat{L}_{2}^{A}}\propto\epsilon_{qr}\epsilon_{\text{NR}},\epsilon_{\text{recoil}}$
and $\frac{\hat{E}_{2}^{A\left(\rm
res\right)}}{\hat{L}_{2}^{A}}\propto\epsilon_{qr}^{2}$ are their NLO
corrections. As we showed at [30], the unique first-forbidden transition,
$J^{\Delta\pi}=2^{-}$, is of a great interest for Beyond the Standard Model
searches. Its Fierz term, including NLO corrections, will be (from eq. (16))
$b_{\text{F}}^{2^{-}}=\frac{2m_{e}}{5q}\mathfrak{Re}\left\\{2\frac{\left\langle\left\|\hat{C}_{2}^{A}\right\|\right\rangle}{\left\langle\left\|\hat{L}_{2}^{A}\right\|\right\rangle}\pm\sqrt{6}\frac{\left\langle\left\|\hat{M}_{2}^{V}\right\|\right\rangle}{\left\langle\left\|\hat{L}_{2}^{A}\right\|\right\rangle}\right\\}+\mathcal{O}\left(\frac{m_{e}}{\epsilon}\epsilon_{qr}^{2}\epsilon_{\rm
NR}^{2},\frac{m_{e}}{\epsilon}\epsilon_{\text{recoil}}^{2}\right)$, and its
angular correlation coefficient will be (eq. (14))
$a_{\beta\nu}^{2^{-}}=-\frac{1}{5}\left(1+\tilde{\delta}_{a}^{2^{-}}\right)$
333For $J^{\Delta\pi}=0^{-}$ Fierz term and the angular correlation are in eq.
(10) and above it, while for $J^{\Delta\pi}=1^{-}$ they are not well defined..

##  VII From nuclear structure corrections to BSM signatures

The matrix elements appearing in the formulae in the previous sections depend
upon the nuclear interactions, i.e., the Hamiltonian and the coupling to the
weak probe. Nuclear interactions are the low-energy reflection of the
fundamental QCD forces. Thus, one uses an effective approach to describe the
nuclear Hamiltonian and the nuclear currents excited by the weak probe. A most
common approach is the EFT approach, which systematically builds, order by
order, the nuclear Hamiltonian and currents (the scattering operators), based
on the symmetries of the fundamental theory. As EFT creates an expansion in a
small parameter, $Q/\Lambda_{b}$, it is also a source of systematic
uncertainty, which we coined $\epsilon_{\text{EFT}}$ in sec. II, of the order
of $\epsilon_{\text{EFT}}^{(n+1)}$, with $n$ the order of the expansion.

Pragmatically, many calculations use the so called “impulse” approximation for
the nuclear currents, which describes the interaction of the weak probe with
the strongly interacting nucleons, and neglects the interaction of the probe
with two or more nucleons simultaneously. In terms of an EFT calculation, this
is a leading order approximation, which entails an
${\mathcal{O}}(\epsilon_{EFT})$ accuracy, usually due to the leading magnetic
multipole. This is due to the fact that such currents arise at next-to-leading
order for the Magnetic multipole of the polar-vector current. Other
significant multipoles receive two-body corrections at higher orders. The one-
body currents arise consistently at leading order, and their operator
structure has been developed already from a phenomenological approach, using
Lorentz symmetry considerations, and thus will be the starting point for
theoretical predictions. In this section we outline these impulse
approximation currents, and analyse their expected accuracy, from the EFT
point of view. We emphasize that adding two-body currents has the potential to
significantly increase the accuracy.

The one-body currents include low-energy coefficients, related to the
symmetries of the probe-nucleon interactions. These are denoted by $C_{V}$ and
$C_{A}$ for the polar-vector and axial-vector SM currents, and $C_{S}$,
$C_{P}$ and $C_{T}$ for the scalar, pseudo-scalar and tensor BSM currents.
These symmetry coefficients couple to the nuclear charges $g_{\text{sym}}$
($\text{sym}\in\left\\{S,P,V,A,T\right\\}$).

The vector and axial vector one-body currents are (respectively)[32]:

$\displaystyle\left\langle
p\left(p_{p}\right)\left|\bar{u}\gamma_{\mu}d\right|n\left(p_{n}\right)\right\rangle$
$\displaystyle=\bar{u}_{p}\left(p_{p}\right)\left[g_{V}\left(q^{2}\right)\gamma_{\mu}-i\frac{\tilde{g}_{T\left(V\right)}\left(q^{2}\right)}{2m_{N}}\sigma_{\mu\nu}q^{\nu}+\frac{\tilde{g}_{S}\left(q^{2}\right)}{2m_{N}}q_{\mu}\right]u_{n}\left(p_{n}\right),$
$\displaystyle\left\langle
p\left(p_{p}\right)\left|\bar{u}\gamma_{\mu}\gamma_{5}d\right|n\left(p_{n}\right)\right\rangle$
$\displaystyle=\bar{u}_{p}\left(p_{p}\right)\left[g_{A}\left(q^{2}\right)\gamma_{\mu}-i\frac{\tilde{g}_{T\left(A\right)}\left(q^{2}\right)}{2m_{N}}\sigma_{\mu\nu}q^{\nu}+\frac{\tilde{g}_{P}\left(q^{2}\right)}{2m_{N}}q_{\mu}\right]\gamma_{5}u_{n}\left(p_{n}\right).$
(26)

In the Standard Model, $g_{V}=1$, up to second order corrections in isospin
breaking [33, 34], as a result of the conservation of the vector current, and
$g_{A}\approx 1.276g_{V}$ [35, 36]. $\tilde{g}_{S}$ and
$\tilde{g}_{T\left(A\right)}$, known as second class currents, do not exist in
the Standard Model, $\tilde{g}_{S}$ due to current conservation, and
$\tilde{g}_{T\left(A\right)}$ because of G-parity considerations. The scalar,
pseudo-scalar and tensor currents are (respectively)[32]:

$\displaystyle\left\langle
p\left(p_{p}\right)\left|\bar{u}d\right|n\left(p_{n}\right)\right\rangle$
$\displaystyle=$ $\displaystyle
g_{S}\left(0\right)\bar{u}_{p}\left(p_{p}\right)u_{n}\left(p_{n}\right)+\mathcal{O}\left(\epsilon_{\text{recoil}}^{2}\right),$
$\displaystyle\left\langle
p\left(p_{p}\right)\left|\bar{u}\gamma_{5}d\right|n\left(p_{n}\right)\right\rangle$
$\displaystyle=$ $\displaystyle
g_{P}\left(0\right)\bar{u}_{p}\left(p_{p}\right)\gamma_{5}u_{n}\left(p_{n}\right)+\mathcal{O}\left(\epsilon_{\text{recoil}}^{2}\right),$
$\displaystyle\left\langle
p\left(p_{p}\right)\left|\bar{u}\sigma_{\mu\nu}d\right|n\left(p_{n}\right)\right\rangle$
$\displaystyle=$ $\displaystyle
g_{T}\left(0\right)\bar{u}_{p}\left(p_{p}\right)\sigma_{\mu\nu}u_{n}\left(p_{n}\right)+\mathcal{O}\left(\epsilon_{\text{recoil}}^{2}\right)\text{,}$
(27)

where the scalar nuclear charge,
$g_{S}=g_{V}\frac{\left(M_{n}-M_{p}\right)^{QCD}}{m_{d}-m_{u}}\approx
0.8-1.2$, the pseudo-scalar one is
$g_{P}=g_{V}\frac{M_{n}+M_{p}}{m_{d}+m_{u}}=349(9)$ (the pseudo-scalar
contraction, $\bar{u}_{p}\gamma_{5}u_{n}$, is
$\mathcal{O}\left(\epsilon_{\text{recoil}}\right)$, as opposed to the others,
which are $\mathcal{O}\left(1\right)$[32]), and according to lattice QCD, the
tensor nuclear charge $g_{T}$ has the same order of magnitude as $g_{A}$[37].
Consequently, all these nuclear charges have the same orders of magnitude. The
suppression of the symmetry coefficients results from the effective theory’s
coefficients,
$\epsilon_{\text{sym}}\propto\left(\frac{m_{W}}{\Lambda}\right)^{n}$. These
originate in the effective weak interaction Lagrangian, where $\Lambda$
represents new physics scale, and $n=0$ for $\text{sym}=V,A$, and $n\geq 2$
for $\text{sym}\neq V,A$. Since it is an effective theory, it will be
surprising if it is accurate and contains only vector and axial currents. New
experiments will have a per-mill level precision, sensitive to new physics at
the TeV scale. For the simplest BSM operator ($n=2$), a TeV scale means
$\epsilon_{\text{sym}}\sim 10^{-3}$, so the needed accuracy of the calculation
is about $10^{-3}-10^{-4}$.

As we have shown, the presented formalism reaches an accuracy of at least
$\mathcal{O}\left(\epsilon_{qr}^{2}\epsilon_{\rm NR}^{2}\right)$ or
$\mathcal{O}\left(\epsilon_{\text{recoil}}^{2}\right)$. For an endpoint of
$\approx 2\text{MeV}$, $\epsilon_{qr}\approx 0.01A^{\frac{1}{3}}$,
$\epsilon_{NR}\approx 0.2$ and $\epsilon_{\text{recoil}}\approx 0.002$, this
entails a an order of 10% accuracy for the solution of the nuclear many bady
problem ($\epsilon_{\text{solver}}$).

##  VIII Summary

In this paper, we present a formalism to calculate $\beta$-decay rates and
observables, including high order recoil and shape corrections, required for
on going and planned experiments in search for BSM physics, which aim at a
per-mill level of accuracy. The formalism shows that nuclear structure
corrections induce a finite Fierz interference term, a fact which affects the
analysis of measurements of the angular correlation between the beta particle
and the (anti-)neutrino. In addition, We express the nuclear structure related
corrections to the angular correlation coefficient. Both Fierz term and the
angular correlation are of interest for ongoing experiments, and thus these
corrections are essential in order to distinguish BSM signatures from high
orders effects within the Standard Model.

We identify different small parameters involved in the corrections, that allow
us to robustly assess the accuracy of the theoretical calculations. These
expansion parameters originate in the low-energy character of $\beta$-decays,
the non-relativistic character of nuclear wave function, and the coulomb
interaction between the electron and the nucleus, defined as follows:

$\displaystyle\epsilon_{qr}$ $\displaystyle\equiv$ $\displaystyle qR,$
$\displaystyle\epsilon_{\text{NR}}$ $\displaystyle\equiv$
$\displaystyle\frac{P_{\text{Fermi}}}{m_{N}},$
$\displaystyle\epsilon_{\text{recoil}}$ $\displaystyle\equiv$
$\displaystyle\frac{q}{m_{N}},$ $\displaystyle\epsilon_{c}$
$\displaystyle\equiv$ $\displaystyle\alpha Z_{f}.$ (28)

In addition we define parameters that characterize the precision of the
solution of the nuclear problem, $\epsilon_{\rm NM}$, which contains the EFT
expansion order $\epsilon_{\text{EFT}}=Q/\Lambda_{b}$, its specific
application, i.e., the details related to regularization of the effective
theory, the non-relativistic expansion, etc. ($\epsilon_{\text{model}}$), and
the numerical accuracy $\epsilon_{\text{conv}}$. These enable an analysis of
the accuracy of the calculations, and demonstrate that a solution of the
nuclear many body problem accurate only to about 10%, can be used to
significantly constrain these corrections, for the needs of future
experiments.

The application of the presented formalism to specific nuclei is already
underway. The first one is 6He [38], following its role in several ongoing, or
soon to be initiated, precision $\beta$-decay experiments. The second is 23Ne
[39], in which a joint theoretical-experimental effort enabled us to reanalyze
experimental data using new branching ratio measurements, and establish bounds
on the presence of exotic tensor interactions.

###### Acknowledgements.

We would like to thank the discussions in the ECT* workshop “Precise beta
decay calculations for searches for new physics” in Trento, and especially
Leendert Hayen, for helpful comments. We wish to acknowledge the support of
the Israel Science Foundation grant no. 1446/16. AGM’s research was partially
supported by a scholarship sponsored by the Ministry of Science & Technology,
Israel.

## Appendix: nuclear currents and multipole operators at leading orders

To be able to discuss the nuclear dependent part, we need to look at the
nuclear current. In the traditional nuclear physics picture, the electroweak
current is constructed from the properties of free nucleons. In the leading
order, we can refer to the nuclear currents as one body currents. Neglecting
the second-class currents, we rewrite eq. (VII) in a more detailed form:

$\displaystyle\left\langle\vec{p}^{\prime},\sigma^{\prime},\rho^{\prime}\left|J_{\mu}^{V}\left(0\right)\right|\vec{p},\sigma,\rho\right\rangle$
$\displaystyle=\bar{u}\left(\vec{p^{\prime}},\sigma^{\prime}\right)\eta_{\rho^{\prime}}^{+}\left[g_{V}\left(q^{2}\right)\gamma_{\mu}-i\frac{\tilde{g}_{T\left(V\right)}\left(q^{2}\right)}{2m_{N}}\sigma_{\mu\nu}q^{\nu}\right]\tau^{\pm}\eta_{\rho}u\left(\vec{p},\sigma\right),$
$\displaystyle\left\langle\vec{p}^{\prime},\sigma^{\prime},\rho^{\prime}\left|J_{\mu}^{A}\left(0\right)\right|\vec{p},\sigma,\rho\right\rangle$
$\displaystyle=\bar{u}\left(\vec{p^{\prime}},\sigma^{\prime}\right)\eta_{\rho^{\prime}}^{+}\left[g_{A}\left(q^{2}\right)\gamma_{\mu}+\frac{\tilde{g}_{P}\left(q^{2}\right)}{2m_{N}}q_{\mu}\right]\gamma_{5}\tau^{\pm}\eta_{\rho}u\left(\vec{p},\sigma\right).$
(29)

Here
$u\left(\vec{p},\sigma\right)=\sqrt{\frac{\epsilon_{p}+m_{N}}{2\epsilon_{p}}}\left(\begin{array}[]{c}1\\\
\frac{\vec{\sigma}\cdot\vec{p}}{\epsilon_{p}+m_{N}}\end{array}\right)\chi_{\sigma}$
is Dirac spinor for a free nucleon of mass $m_{N}$,
$\epsilon_{p}=\sqrt{p^{2}+m_{N}^{2}}$ is the energy of the particle,
$\chi_{\sigma}$ is two-component Pauli spinor for a spin up and down along the
$q$ axis, $\eta_{\rho}$ are two-component Pauli isospinors, and
$\tau^{\pm}=\mp\frac{1}{2}\left(\tau_{x}\pm i\tau_{y}\right)$ are the isospin
raising and lowering operators, that turns a proton into neutron and vice
versa.

After substituting the mentioned explicit form of Dirac spinors (using the
convention $\bar{u}=u^{+}\gamma_{0}$, so that $u^{+}u=1$), we make a non-
relativistic expansion, expanding the matrix element consistently in powers of
$\epsilon_{\text{NR}}\equiv\frac{P_{\text{fermi}}}{m_{N}}$, as momenta are
assumed here up to few hundred MeV’s, and find the required matrix elements
(here $P_{\mu}=p_{\mu}+p_{\mu}^{{}^{\prime}}$,
$q_{\mu}=p_{\mu}-p_{\mu}^{{}^{\prime}}$ and $\epsilon_{0}\equiv q_{0}$):

$\displaystyle\left\langle\vec{p}^{\prime},\sigma^{\prime},\rho^{\prime}\left|J_{0}^{V}\left(0\right)\right|\vec{p},\sigma,\rho\right\rangle$
$\displaystyle=g_{V}\chi_{\sigma^{\prime}}^{+}\eta_{\rho^{\prime}}^{+}\tau^{\pm}\eta_{\rho}\chi_{\sigma}+\mathcal{O}\mathrm{\left(\epsilon_{\text{NR}}^{2}\right)},$
$\displaystyle\left\langle\vec{p}^{\prime},\sigma^{\prime},\rho^{\prime}\left|\vec{J}^{V}\left(0\right)\right|\vec{p},\sigma,\rho\right\rangle$
$\displaystyle=\chi_{\sigma^{\prime}}^{+}\eta_{\rho^{\prime}}^{+}\frac{1}{2m_{N}}\left[g_{V}\vec{P}+\left(g_{V}+\tilde{g}_{T\left(V\right)}\right)i\vec{q}\times\vec{\sigma}\right]\tau^{\pm}\eta_{\rho}\chi_{\sigma}+\mathcal{O}\mathrm{\left(\epsilon_{\text{NR}}^{2}\right)},$
$\displaystyle\left\langle\vec{p}^{\prime},\sigma^{\prime},\rho^{\prime}\left|J_{0}^{A}\left(0\right)\right|\vec{p},\sigma,\rho\right\rangle$
$\displaystyle=\chi_{\sigma^{\prime}}\eta_{\rho^{\prime}}^{+}\frac{1}{2m_{N}}\left[g_{A}\vec{P}\cdot\vec{\sigma}+\frac{\tilde{g}_{P}}{2m_{N}}\epsilon_{0}\vec{q}\cdot\vec{\sigma}\right]\tau^{\pm}\eta_{\rho}\chi_{\sigma}+\mathcal{O}\mathrm{\left(\epsilon_{\text{NR}}^{2}\right)},$
$\displaystyle\left\langle\vec{p}^{\prime},\sigma^{\prime},\rho^{\prime}\left|\vec{J}^{A}\left(0\right)\right|\vec{p},\sigma,\rho\right\rangle$
$\displaystyle=g_{A}\chi_{\sigma^{\prime}}\eta_{\rho^{\prime}}^{+}\vec{\sigma}\tau^{\pm}\eta_{\rho}\chi_{\sigma}+\mathcal{O}\mathrm{\left(\epsilon_{\text{NR}}^{2}\right)}\text{.}$
(30)

Then, we use the definition of the (second quantization)
$J\left(\vec{x}\right)$ current matrix element as a sum over first
quantization currents $\hat{\mathcal{J}}^{\left(1\right)}$:
$\left\langle\vec{p}^{\prime},\sigma^{\prime},\rho^{\prime}\left|J\left(\vec{r}\right)\right|\vec{p},\sigma,\rho\right\rangle=\int
d^{3}y\phi_{\vec{p}^{\prime}\sigma^{\prime}\rho^{\prime}}^{+}\left(\vec{y}\right)\left[\hat{\mathcal{J}}^{\left(1\right)}\left(\vec{y}\right)\delta^{\left(3\right)}\left(\vec{r}-\vec{y}\right)\right]\phi_{\vec{p}\sigma\rho}\left(\vec{y}\right)$.
Evaluated at $\vec{r}=0$, we find out that
$\left\langle\vec{p}^{\prime},\sigma^{\prime},\rho^{\prime}\left|J\left(0\right)\right|\vec{p},\sigma,\rho\right\rangle=\phi_{\vec{p}^{\prime}\sigma^{\prime}\rho^{\prime}}^{+}\left(0\right)\hat{\mathcal{J}}^{\left(1\right)}\left(0\right)\phi_{\vec{p}\sigma\rho}\left(0\right)$,
what permits the identification of the nuclear density operators in first
quantization (from eq. (Appendix: nuclear currents and multipole operators at
leading orders)). Finally, using the current density operator in the first
quantization,
$J\left(\vec{r}\right)=\sum_{j=1}^{A}\hat{\mathcal{J}}^{\left(1\right)}\left(j\right)\delta^{\left(3\right)}\left(\vec{r}-\vec{r}_{j}\right)$,
and under the first quantization’s assumption that there is no dependency on
the location, so
$\hat{\mathcal{J}}^{\left(1\right)}\left(j\right)=\hat{\mathcal{J}}^{\left(1\right)}\left(0\right)\left(j\right)$,
one gets the following currents:

$\displaystyle J_{0}^{V}\left(\vec{r}\right)$
$\displaystyle=g_{V}\sum_{j=1}^{A}\tau_{j}^{\pm}\delta^{\left(3\right)}\left(\vec{r}-\vec{r}_{j}\right)+\mathcal{O}\mathrm{\left(\epsilon_{\text{NR}}^{2}\right)},$
$\displaystyle\vec{J}^{V}\left(\vec{r}\right)$
$\displaystyle=\frac{1}{2m_{N}}\sum_{j=1}^{A}\left[g_{V}\left\\{\vec{p}_{j},\delta^{\left(3\right)}\left(\vec{r}-\vec{r}_{j}\right)\right\\}+\left(g_{V}+\tilde{g}_{T\left(V\right)}\right)\vec{\nabla}\times\vec{\sigma}_{j}\delta^{\left(3\right)}\left(\vec{r}-\vec{r}_{j}\right)\right]\tau_{j}^{\pm}+\mathcal{O}\mathrm{\left(\epsilon_{\text{NR}}^{2}\right)},$
$\displaystyle J_{0}^{A}\left(\vec{r}\right)$
$\displaystyle=\frac{1}{2m_{N}}\sum_{j=1}^{A}\left[g_{A}\left\\{\vec{p}_{j},\delta^{\left(3\right)}\left(\vec{r}-\vec{r}_{j}\right)\right\\}-i\frac{\tilde{g}_{P}}{2m_{N}}\left(\epsilon_{0}\pm\Delta
E_{c}\right)\vec{\nabla}\delta^{\left(3\right)}\left(\vec{r}-\vec{r}_{j}\right)\right]\cdot\vec{\sigma}_{j}\tau_{j}^{\pm}+\mathcal{O}\mathrm{\left(\epsilon_{\text{NR}}^{2}\right)},$
$\displaystyle\vec{J}^{A}\left(\vec{r}\right)$
$\displaystyle=g_{A}\sum_{j=1}^{A}\vec{\sigma}_{j}\tau_{j}^{\pm}\delta^{\left(3\right)}\left(\vec{r}-\vec{r}_{j}\right)+\mathcal{O}\mathrm{\left(\epsilon_{\text{NR}}^{2}\right)}.$
(31)

Here we made the operator replacements
$\vec{P}\rightarrow\left\\{\vec{p}_{j},\delta^{\left(3\right)}\left(\vec{r}-\vec{r}_{j}\right)\right\\}$
, and $\vec{q}\rightarrow-i\vec{\nabla}$, the last one based on a partial
integration of Fourier transform of the transition matrix element of the
current, $\int e^{-i\vec{q}\cdot\vec{r}}\left\langle
f\left|J_{\mu}\left(\vec{r}\right)\right|i\right\rangle$, with localized
densities. We also corrected here the maximal electron energy $\epsilon_{0}$
with the Coulomb replacement energy $\Delta E_{c}$, as mentioned in sec. II.

Positioning eq. (Appendix: nuclear currents and multipole operators at leading
orders) into the multipole operators definition (eq. (III)), leads to the
explicit expressions for the vector and axial currents multipole operators:

$\displaystyle\hat{C}_{J}^{V}\left(q\right)$
$\displaystyle=g_{V}\sum_{j=1}^{A}M_{J}\left(q\vec{r}_{j}\right)\tau_{j}^{\pm}+\mathcal{O}\left(\epsilon_{qr}^{J}\epsilon_{\text{NR}}^{2}\right),$
$\displaystyle\hat{L}_{J}^{V}\left(q\right)$
$\displaystyle=-\frac{q}{2m_{N}}g_{V}\sum_{j=1}^{A}\left\\{M_{J}\left(q\vec{r}_{j}\right)-2\left[\frac{1}{q}\vec{\nabla}M_{J}\left(q\vec{r}_{j}\right)\right]\cdot\frac{1}{q}\vec{\nabla}\right\\}\tau_{j}^{\pm}+\mathcal{O}\left(\epsilon_{qr}^{J-1}\epsilon_{\text{NR}}^{2}\right),$
$\displaystyle\hat{E}_{J}^{V}\left(q\right)$
$\displaystyle=\frac{q}{m_{N}}\sum_{j=1}^{A}\left\\{-ig_{V}\left[\frac{1}{q}\vec{\nabla}\times\vec{M}_{JJ1}\left(q\vec{r}_{j}\right)\right]\cdot\frac{1}{q}\vec{\nabla}+\frac{g_{V}+\tilde{g}_{T\left(V\right)}}{2}\vec{M}_{JJ1}\left(q\vec{r}_{j}\right)\cdot\vec{\sigma}_{j}\right\\}\tau_{j}^{\pm}+\mathcal{O}\left(\epsilon_{qr}^{J-1}\epsilon_{\text{NR}}^{2}\right),$
$\displaystyle\hat{M}_{J}^{V}\left(q\right)$
$\displaystyle=-\frac{iq}{m_{N}}\sum_{j=1}^{A}\left\\{g_{V}\vec{M}_{JJ1}\left(q\vec{r}_{j}\right)\cdot\frac{1}{q}\vec{\nabla}+i\frac{g_{V}+\tilde{g}_{T\left(V\right)}}{2}\left[\frac{1}{q}\vec{\nabla}\times\vec{M}_{JJ1}\left(q\vec{r}_{j}\right)\right]\cdot\vec{\sigma}_{j}\right\\}\tau_{j}^{\pm}+\mathcal{O}\left(\epsilon_{qr}^{J}\epsilon_{\text{NR}}^{2}\right),$
$\displaystyle\hat{C}_{J}^{A}\left(q\right)$
$\displaystyle=-\frac{iq}{m_{N}}\sum_{j=1}^{A}\left\\{g_{A}M_{J}\left(q\vec{r}_{j}\right)\vec{\sigma}_{j}\cdot\frac{1}{q}\vec{\nabla}+\frac{1}{2}\left[g_{A}-\frac{\tilde{g}_{P}}{2m_{N}}\left(\epsilon_{0}\pm\Delta
E_{c}\right)\right]\left[\frac{1}{q}\vec{\nabla}M_{J}\left(q\vec{r}_{j}\right)\right]\cdot\vec{\sigma}_{j}\right\\}\tau_{j}^{\pm}+\mathcal{O}\left(\epsilon_{qr}^{J}\epsilon_{\text{NR}}^{2}\right),$
$\displaystyle\hat{L}_{J}^{A}\left(q\right)$
$\displaystyle=ig_{A}\sum_{j=1}^{A}\left[\frac{1}{q}\vec{\nabla}M_{J}\left(q\vec{r}_{j}\right)\right]\cdot\vec{\sigma}_{j}\tau_{j}^{\pm}+\mathcal{O}\left(\epsilon_{qr}^{J-1}\epsilon_{\text{NR}}^{2}\right),$
$\displaystyle\hat{E}_{J}^{A}\left(q\right)$
$\displaystyle=g_{A}\sum_{j=1}^{A}\left[\frac{1}{q}\vec{\nabla}\times\vec{M}_{JJ1}\left(q\vec{r}_{j}\right)\right]\cdot\vec{\sigma}_{j}\tau_{j}^{\pm}+\mathcal{O}\left(\epsilon_{qr}^{J-1}\epsilon_{\text{NR}}^{2}\right),$
$\displaystyle\hat{M}_{J}^{A}\left(q\right)$
$\displaystyle=g_{A}\sum_{j=1}^{A}\vec{M}_{JJ1}\left(q\vec{r}_{j}\right)\cdot\vec{\sigma}_{j}\tau_{j}^{\pm}+\mathcal{O}\left(\epsilon_{qr}^{J}\epsilon_{\text{NR}}^{2}\right)\text{,}$
(32)

where

$\displaystyle M_{J}\left(q\vec{r}\right)$ $\displaystyle\equiv$
$\displaystyle j_{J}\left(qr\right)Y_{J}\left(\hat{r}\right),$
$\displaystyle\vec{M}_{JL1}\left(q\vec{r}\right)$ $\displaystyle\equiv$
$\displaystyle j_{L}\left(qr\right)\vec{Y}_{JL1}\left(\hat{r}\right)\mbox{,}$
(33)

which hold to the identities [40]:

$\displaystyle\frac{1}{q}\vec{\nabla}M_{J}\left(q\vec{r}\right)$
$\displaystyle=\sqrt{\frac{J+1}{2J+1}}\vec{M}_{J,J+1,1}\left(q\vec{r}\right)+\sqrt{\frac{J}{2J+1}}\vec{M}_{J,J-1,1}\left(q\vec{r}\right),$
$\displaystyle\frac{1}{q}\vec{\nabla}\times\vec{M}_{JJ1}\left(q\vec{r}\right)$
$\displaystyle=-i\sqrt{\frac{J}{2J+1}}\vec{M}_{J,J+1,1}\left(q\vec{r}\right)+i\sqrt{\frac{J+1}{2J+1}}\vec{M}_{J,J-1,1}\left(q\vec{r}\right)\text{.}$
(34)

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