# Pseudo-neutrino versus recoil formalism for 4-body phase space and
applications to nuclear decay

Chien-Yeah Seng1,2 1Facility for Rare Isotope Beams, Michigan State
University, East Lansing, MI 48824, USA  2Department of Physics, University of
Washington, Seattle, WA 98195-1560, USA

(August 28, 2024)

###### Abstract

It is well-known that the traditional treatment of radiative corrections that
utilizes the “true” neutrino momentum $\vec{p}_{\nu}$ in the differential
decay rate formula could lead to a $\sim\alpha/\pi$ systematic error in
certain observables due to the mistreatment of 4-body kinematics. We
investigate the theory structure of one of the proposed solutions, the
“$\nu^{\prime}$-formalism”, in the non-recoil limit appropriate for neutron
and nuclear beta decays. We derive an elegant master formula for the 4-body
phase space and use it to re-analyze the spectrum-dependent “outer” radiative
corrections to the beta decay of a polarized spin-half nucleus; a complete set
of analytic expressions is provided for readers to straightforwardly obtain
the final numerical results. We compare it to the “recoil formalism” where the
energy of the recoil nucleus is fixed.

##  I Introduction

Neutron and nuclear beta decays provide an excellent avenue for precision
tests of the Standard Model (SM) Brodeur et al. (2023); Alarcon et al. (2023).
Apart from the decay lifetime that determines the Cabibbo-Kobayashi-Maskawa
(CKM) matrix $V_{ud}$, the study of the beta decay spectrum and various
correlations constructed from spin and momentum vectors set stringent
constraints on coupling strengths induced by physics beyond the Standard Model
(BSM). The first comprehensive analysis of various terms in the beta decay
differential rate in terms of the parameters in the most general Lee-Yang
Lagrangian Lee and Yang (1956) was done by Jackson, Treiman and Wyld Jackson
et al. (1957a, b). Taking neutron decay as an example, the well-known tree-
level differential decay rate of a polarized neutron (with unpolarized final
states) reads:

$\left(\frac{d\Gamma}{dE_{e}d\Omega_{e}d\Omega_{\nu}}\right)_{\text{tree}}\propto
1+a\vec{\beta}\cdot\hat{p}_{\nu}+b\frac{m_{e}}{E_{e}}+\hat{s}_{n}\cdot\left[A\vec{\beta}+B\hat{p}_{\nu}+D\vec{\beta}\times\hat{p}_{\nu}\right]\leavevmode\nobreak\
,$ (1)

where $\hat{p}_{\nu}$ is the unit neutrino momentum vector,
$\vec{\beta}=\vec{p}_{e}/E_{e}=\beta\hat{p}_{e}$ is the electron velocity
vector, and $\hat{s}_{n}$ is the neutron unit polarization vector. The SM
provides well-defined predictions for the correlation coefficients $a$, $b$,
$A$ and $B$ and $D$; experimental confirmation of any deviation from such
predictions will be an unambiguous signal of BSM physics.

As the present experimental precision of beta decay observables have reached
$10^{-4}$, one needs to carefully account for the higher-order SM corrections
to the tree-level expression; the most prominent ones are recoil effects and
the radiative corrections (RC). The latter consists of the Fermi function
Fermi (1934) and “outer corrections” that are functions of $E_{e}$ and the
angles, and “inner corrections” that only serve to renormalize weak coupling
constants; the first two are more important for the determination of
correlation coefficients.

Pioneering works by Sirlin Sirlin (1967), Shann Shann (1971) and Garcia-Maya
Garcia and Maya (1978) established the $\mathcal{O}(\alpha)$ outer corrections
to the polarized neutron differential rate, which takes a very elegant form:

$\delta\left(\frac{d\Gamma}{dE_{e}d\Omega_{e}d\Omega_{\nu}}\right)_{\text{outer}}\propto\frac{\alpha}{2\pi}\left\\{\delta_{1}+a\vec{\beta}\cdot\hat{p}_{\nu}(\delta_{1}+\delta_{2})+\hat{s}_{n}\cdot\left[A\vec{\beta}(\delta_{1}+\delta_{2})+B\hat{p}_{\nu}\delta_{1}\right]\right\\}\leavevmode\nobreak\
,$ (2)

where $\delta_{1}$, $\delta_{2}$ are known functions of $E_{e}$. This equation
serves as an important theory input for the study of the beta spectrum and the
beta asymmetry parameter $A$. On the other hand, for observables such as the
neutrino-electron correlation coefficient $a$ and the neutrino asymmetry
parameter $B$, it was noticed long ago Toth (1984) that the representation
above, which we call the neutrino ($\nu$)-formalism, contains a serious
drawback as it depends on the neutrino momentum $\vec{p}_{\nu}$ which is not
directly measured. In a three-body decay $\phi_{i}\rightarrow\phi_{f}e\nu$, it
could be deduced from the momentum conservation
$\vec{p}_{\nu}=-\vec{p}_{f}-\vec{p}_{e}$, where both $\vec{p}_{f}$ and
$\vec{p}_{e}$ are measurable quantities. However, this relation breaks down in
the bremsstrahlung process where an extra real photon is emitted. in fact,
since in most experiments the neutrino and the extra photon(s) are not
directly detected, there is no way to determine the neutrino solid angle
$\Omega_{\nu}$ using the momenta of $\phi_{f}$ and $e$. Putting it in other
words, using the $\nu$-formalism for the differential rate together with the
3-body momentum conservation relations could lead to a theoretical error of
the order $\alpha/\pi$ for the $\hat{p}_{\nu}$-dependent observables due to
the mistreatment of the 4-body kinematics.

Several alternative approaches were proposed to overcome the drawback in
Eq.(2). An important example is the “recoil formalism”, namely to fix the
momenta $\vec{p}_{e}$ and $\vec{p}_{f}$ (instead of $\vec{p}_{e}$ and
$\vec{p}_{\nu}$) in the calculation; in this way one may choose the variables
as $E_{e},E_{f}$ or $E_{e},\cos\theta_{ef}\equiv\hat{p}_{e}\cdot\hat{p}_{f}$.
This is analogous to the treatment of RC in semileptonic kaon decays Ginsberg
(1966, 1968, 1967, 1970). The recoil formalism was used to study decays of
unpolarized Toth et al. (1986); Gluck (1993) and polarized Gluck and Toth
(1990, 1992); Glück (1998) baryons. The complicated bremsstrahlung phase space
integrals were performed either semi-analytically Gluck (1993); Gluck and Toth
(1992) or numerically Toth et al. (1986); Gluck and Joo (1996, 1997); Gluck
(1997). Recently, the need to apply the correct recoil formalism (instead of
the $\nu$-formalism) to the decay of free neutrons was reiterated Glück
(2023), and subsequent re-analysis was performed to the aCORN Wietfeldt et al.
(2023) and aSPECT Beck et al. (2023) data.

Another approach is the pseudo-neutrino ($\nu^{\prime}$)-formalism. It defines
a “pseudo-neutrino” momentum
$\vec{p}_{\nu}^{\prime}\equiv-\vec{p}_{e}-\vec{p}_{f}$ which is a direct
experimental observable; this definition holds with or without extra photons.
Subsequently, one expresses the differential rate in terms of
$E_{e},\Omega_{e},\Omega_{\nu}^{\prime}$ instead of
$E_{e},\Omega_{e},\Omega_{\nu}$. This approach possesses several advantageous
comparing to the recoil formalism:

  * •

It is the most natural generalization to the traditional $\nu$-formalism and
has the closest resemblance to the latter, which makes cross-checking easier.
It is also the most natural formalism to describe decays of polarized nuclei.

  * •

It is much more natural when combined with non-recoil approximation (unlike
the recoil formalism where the variation of $E_{f}$ cannot be discussed
without involving recoil effects). It is sometimes beneficial to study both
the kinematical dynamical recoil corrections simultaneously within a unified
framework, e.g. using methods like the heavy baryon expansion Jenkins and
Manohar (1991), to avoid possible double-counting. So it is desirable to split
recoil effects completely from RC, and the $\nu^{\prime}$-formalism is best
suited for this purpose.

  * •

The analytic expressions of the phase space integrals in the
$\nu^{\prime}$-formalism are usually much more elegant, and only a minimal
amount of numerical integration is required.

Existing literature that discussed the $\nu^{\prime}$-formalism includes
Refs.Toth and Gluck (1989); Gluck and Toth (1990, 1992); Gluck (1993), with
applications to decays of unpolarized and polarized neutron and hyperons.
Despite their comprehensiveness, we failed to identify a complete treatment of
the outer RC to the differential rate of a polarized parent nucleus in the
non-recoil limit (appropriate for neutron and nuclear decays) with the
$\nu^{\prime}$-formalism which, in principle, should display a similar degree
of theory elegance as Eq.(2). Also, analytic expressions of many important
intermediate quantities (e.g. expansion coefficients and results of phase-
space integrals) are not explicitly shown, which makes their results not so
straightforwardly applicable in numerical analysis; furthermore, we also find
that some numerical results in these earlier papers are not completely
accurate. In the meantime, while many recent computer packages are capable to
produce reliable numerical results Glück (2023), they are practically a black
box to outsiders.

The purpose of this paper is to provide a transparent, elegant and self-
contained theoretical description of the $\nu^{\prime}$-formalism in the non-
recoil limit. As a simple application, We study the outer RC to the
differential rate of the $\beta^{\pm}$ decay of a polarized spin-1/2 nucleus,
but the final result is equally applicable to spin-0 systems. Therefore, our
analysis covers a wide range of interesting beta decay processes including
semileptonic pion decay, free neutron decay, superallowed $0^{+}\rightarrow
0^{+}$ decays Hardy and Towner (2020), spin-half nuclear mirror transitions
Naviliat-Cuncic and Severijns (2009) (e.g. 19Ne$\rightarrow$19F), etc. We
provide the full set of analytic expressions of all intermediate functions
needed in the analysis, so that our quoted numerical results are fully
transparent and can be easily reproduced by any interested reader, without the
need of complicated numerical packages. For completeness, we also provide a
similar set of analytic formula for unpolarized nuclei in the recoil
formalism.

##  II 3-body phase space

We start by briefly reviewing the total rate formula of a 3-body decay
$\phi_{i}(p_{i})\rightarrow\phi_{f}(p_{f})+e(p_{e})+\nu(p_{\nu})$111The term
“electron” throughout this paper may refer to an electron or a positron
depending on the decay mode, and similar for the neutrino.:

$\Gamma_{3}=\int d\Pi_{3}|\mathcal{M}|_{\text{3-body}}^{2}$ (3)

with the 3-body phase space:

$\int
d\Pi_{3}=\frac{1}{2m_{i}}\int\frac{d^{3}p_{f}}{(2\pi)^{3}2E_{f}}\frac{d^{3}p_{e}}{(2\pi)^{3}2E_{e}}\frac{d^{3}p_{\nu}}{(2\pi)^{3}2E_{\nu}}(2\pi)^{4}\delta^{(4)}(p_{i}-p_{f}-p_{e}-p_{\nu})\leavevmode\nobreak\
.$ (4)

We are interested in beta decay processes where $m_{i},m_{f}\gg m_{i}-m_{f}$.
The evaluation of this phase space in the $\nu$-formalism consists of first
integrating out $\vec{p}_{f}$ using the spatial delta function. The remaining
temporal delta function reads:

$\delta(m_{i}-E_{f}-E_{e}-E_{\nu})\approx\delta(E_{m}-E_{e}-E_{\nu})$ (5)

which only set bounds on the energy variables but not the angles; here we have
taken the non-recoil approximation, $E_{f}\approx m_{f}$ and
$E_{m}\equiv(m_{i}^{2}-m_{f}^{2}+m_{e}^{2})/(2m_{i})\approx m_{i}-m_{f}$ is
the electron end-point energy. Of course, in practice one needs to account for
recoil corrections to the tree-level 3-body rate for precision, but that is
not the focus of this paper. This delta function is then used to integrate out
the neutrino energy, which leaves us with:

$\int d\Pi_{3}\approx\frac{1}{512\pi^{5}m_{i}m_{f}}\int
d\Omega_{e}d\Omega_{\nu}\int_{m_{e}}^{E_{m}}dE_{e}|\vec{p}_{e}|E_{\nu}\leavevmode\nobreak\
,$ (6)

where $E_{\nu}=E_{m}-E_{e}$; the upper limit of $E_{e}$ is set by the temporal
delta function: $E_{m}-E_{e}=E_{\nu}\geq 0$. Eq.(6) is the underlying reason
why one utilized $E_{e}$, $\Omega_{e}$ and $\Omega_{\nu}$ as free variables in
the traditional analysis of beta decays.

To bypass the conceptual deficit in the $\nu$-formalism that $\Omega_{\nu}$ is
not a practical observable, we turn to the $\nu^{\prime}$-formalism by
defining the pseudo-neutrino momentum:

$p_{\nu}^{\prime}\equiv
p_{i}-p_{f}-p_{e}=(E_{\nu}^{\prime},\vec{p}_{\nu}^{\prime})$ (7)

which is an experimental observable, because both $e$ and $\phi_{f}$ can be
directly detected. In the non-recoil limit, $E_{\nu}^{\prime}=E_{m}-E_{e}$ is
only a function of electron energy. In this formalism, we change the
integration over $\vec{p}_{f}$ to that over $\vec{p}_{\nu}^{\prime}$:

$\int d^{3}p_{f}d^{3}p_{e}=\int
d^{3}p_{\nu}^{\prime}d^{3}p_{e}\leavevmode\nobreak\ ,$ (8)

and rewrite Eq.(4) in the non-recoil limit as:

$\int d\Pi_{3}\approx\frac{1}{(2\pi)^{6}8m_{i}m_{f}}\int
d^{3}p_{\nu}^{\prime}\frac{d^{3}p_{e}}{E_{e}}\frac{d^{3}p_{\nu}}{(2\pi)^{3}2E_{\nu}}(2\pi)^{4}\delta^{(4)}(p_{\nu}^{\prime}-p_{\nu})\leavevmode\nobreak\
,$ (9)

and integrate out $\vec{p}_{\nu}$, instead of $\vec{p}_{f}$, using the spatial
delta function. The remaining temporal delta function reads
$\delta(E_{\nu}^{\prime}-|\vec{p}_{\nu}^{\prime}|)$, which is then used to
integrate out $|\vec{p}_{\nu}^{\prime}|$. This yields:

$\int d\Pi_{3}\approx\frac{1}{512\pi^{5}m_{i}m_{f}}\int
d\Omega_{e}d\Omega_{\nu}^{\prime}\int_{m_{e}}^{E_{m}}dE_{e}|\vec{p}_{e}|E_{\nu}^{\prime}\leavevmode\nobreak\
.$ (10)

This is identical to Eq.(6) upon replacing $E_{\nu}^{\prime}\rightarrow
E_{\nu}$ and $\Omega_{\nu}^{\prime}\rightarrow\Omega_{\nu}$, which do not
change the physics since $p_{\nu}^{\prime}=p_{\nu}$ in 3-body decay. The real
distinction between the two formalisms only shows up when extra photons are
emitted.

To facilitate the discussion later, we also provide the representation of the
3-body phase space in the recoil formalism, i.e. using $E_{e}$ and $E_{f}$ as
variables. It is most convenient to adopt the notations used in semileptonic
decays of kaons Ginsberg (1966, 1968, 1967, 1970), where the differential
decay rate is expressed in terms of the following dimensionless Lorentz
scalars:

$x\equiv\frac{p_{\nu}^{\prime 2}}{m_{i}^{2}}\leavevmode\nobreak\
,\leavevmode\nobreak\ y\equiv\frac{2p_{i}\cdot
p_{e}}{m_{i}^{2}}\leavevmode\nobreak\ ,\leavevmode\nobreak\
z\equiv\frac{2p_{i}\cdot p_{f}}{m_{i}^{2}}\leavevmode\nobreak\ ,$ (11)

with $y=2E_{e}/m_{i}$ and $z=2E_{f}/m_{i}$ energy parameters in the parent’s
rest frame. In 3-body decay, $p_{\nu}^{\prime 2}=p_{\nu}^{2}=0$ assuming
massless neutrino, and we obtain the following exact relation:

$\int
d\Pi_{3}=\frac{m_{i}}{256\pi^{3}}\int_{\mathcal{D}_{3}}dydz\leavevmode\nobreak\
,$ (12)

where the integration region $\mathcal{D}_{3}$ can be represented in two
equivalent ways:

$\displaystyle a(y)-b(y)<z<a(y)+b(y)\leavevmode\nobreak\ ,$ $\displaystyle
2\sqrt{r_{e}}<y<1+r_{e}-r_{f}$ $\displaystyle
a(y)=\frac{(2-y)(1+r_{f}+r_{e}-y)}{2(1+r_{e}-y)}\leavevmode\nobreak\ ,$
$\displaystyle
b(y)=\frac{\sqrt{y^{2}-4r_{e}}(1+r_{e}-r_{f}-y)}{2(1+r_{e}-y)}\leavevmode\nobreak\
,$ (13)

or

$\displaystyle c(z)-d(z)<y<c(z)+d(z)\leavevmode\nobreak\ ,$ $\displaystyle
2\sqrt{r_{f}}<z<1+r_{f}-r_{e}$ $\displaystyle
c(z)=\frac{(2-z)(1+r_{e}+r_{f}-z)}{2(1+r_{f}-z)}\leavevmode\nobreak\ ,$
$\displaystyle
d(z)=\frac{\sqrt{z^{2}-4r_{f}}(1+r_{f}-r_{e}-z)}{2(1+r_{f}-z)}\leavevmode\nobreak\
,$ (14)

with $r_{f}\equiv m_{f}^{2}/m_{i}^{2}$, $r_{e}\equiv m_{e}^{2}/m_{i}^{2}$.

##  III 4-body phase space

The contribution from the bremsstrahlung process
$\phi_{i}(p_{i})\rightarrow\phi_{f}(p_{f})+e(p_{e})+\nu(p_{\nu})+\gamma(k)$ to
the total decay rate is:

$\Gamma_{4}=\int d\Pi_{4}|\mathcal{M}|^{2}_{\text{4-body}}\leavevmode\nobreak\
,$ (15)

where the 4-body phase space reads:

$\int
d\Pi_{4}=\frac{1}{2m_{i}}\int\frac{d^{3}p_{f}}{(2\pi)^{3}2E_{f}}\frac{d^{3}p_{e}}{(2\pi)^{3}2E_{e}}\frac{d^{3}p_{\nu}}{(2\pi)^{3}2E_{\nu}}\frac{d^{3}k}{(2\pi)^{3}2E_{k}}(2\pi)^{4}\delta^{(4)}(p_{i}-p_{f}-p_{e}-p_{\nu}-k)\leavevmode\nobreak\
.$ (16)

Let us again evaluate the phase space first in the traditional
$\nu$-formalism. We integrate out $\vec{p}_{f}$ using the spatial delta
function, which leaves the temporal part as:

$\delta(m_{i}-E_{f}-E_{e}-E_{\nu}-E_{k})\approx\delta(E_{m}-E_{e}-E_{\nu}-E_{k})\leavevmode\nobreak\
.$ (17)

Here the non-recoil approximation can be safely taken, because in
neutron/nuclear beta decays a recoil correction on top of a RC gives a
$\sim(\alpha/\pi)(m_{i}-m_{f})/m_{i}\lesssim 10^{-5}$ correction to the total
decay rate, which exceeds the experimental precision and can be neglected. The
remaining delta function is then used to integrate out $E_{\nu}$, which gives:

$\int d\Pi_{4}\approx\frac{1}{512\pi^{5}m_{i}m_{f}}\int
d\Omega_{e}d\Omega_{\nu}\int_{m_{e}}^{E_{m}}dE_{e}|\vec{p}_{e}|\int_{E_{k}<E_{m}-E_{e}}\frac{d^{3}k}{(2\pi)^{3}2E_{k}}E_{\nu}\leavevmode\nobreak\
,$ (18)

where $E_{\nu}=E_{m}-E_{e}-E_{k}$. The unobserved photon momentum $\vec{k}$ is
then integrated out, leaving $E_{e}$, $\Omega_{e}$ and $\Omega_{\nu}$ as
variables in the differential decay rate formula.

As we stressed in the introduction, since in Eq.(18) the unobserved photon
momentum is integrated out, further applying the 3-body momentum conservation
$p_{\nu}=p_{i}-p_{f}-p_{e}$ to the 4-body phase space in the $\nu$-formalism
leads to an error. This issue is non-existent in the recoil formalism, since
the variables $\\{x,y,z\\}$ are experimentally measurable. In this formalism,
one can derive the following exact formula of the 4-body phase space Seng et
al. (2021):

$\displaystyle\int d\Pi_{4}$ $\displaystyle=$
$\displaystyle\frac{m_{i}^{3}}{512\pi^{4}}\left\\{\int_{\mathcal{D}_{3}}dydz\int_{0}^{\alpha_{+}(y,z)}dx+\int_{\mathcal{D}_{4-3}}dydz\int_{\alpha_{-}(y,z)}^{\alpha_{+}(y,z)}dx\right\\}\int\frac{d^{3}k}{(2\pi)^{3}2E_{k}}\frac{d^{3}p_{\nu}}{(2\pi)^{3}2E_{\nu}}$
(19)
$\displaystyle\times(2\pi)^{4}\delta^{(4)}(p_{\nu}^{\prime}-p_{\nu}-k)\leavevmode\nobreak\
,$

where

$\alpha_{\pm}(y,z)\equiv
1-y-z+r_{f}+r_{e}+\frac{yz}{2}\pm\frac{1}{2}\sqrt{y^{2}-4r_{e}}\sqrt{z^{2}-4r_{f}}\leavevmode\nobreak\
,$ (20)

and the region $\mathcal{D}_{4-3}$ can also be expressed in two equivalent
ways:

$2\sqrt{r_{f}}<z<a(y)-b(y)\leavevmode\nobreak\ ,\leavevmode\nobreak\
2\sqrt{r_{e}}<y<1-\sqrt{r_{f}}+\frac{r_{e}}{1-\sqrt{r_{f}}}$ (21)

or

$2\sqrt{r_{e}}<y<c(z)-d(z)\leavevmode\nobreak\ ,\leavevmode\nobreak\
2\sqrt{r_{f}}<z<1-\sqrt{r_{e}}+\frac{r_{f}}{1-\sqrt{r_{e}}}\leavevmode\nobreak\
.$ (22)

This formula is extremely useful, because it turns out the $p_{\nu}$\- and
$k$-integration with respect to the squared amplitude is always analytically
doable Ginsberg (1967), so we are left with at most one single numerical
integration over $x$, if the differential rate is expressed as $d\Gamma/dydz$.
The recoil formalism has also been applied to baryon decays Toth et al.
(1986); Gluck (1993); Gluck and Toth (1990, 1992); Glück (1998); in
particular, the non-overlapping regions $\mathcal{D}_{3}$ and
$\mathcal{D}_{4-3}$ in Eq.(19) are just what known as the “in” and “out”
region respectively in Ref.Glück (2023). However, since $\\{y,z\\}$ are all
energy and not angular variables, the connection to the traditional
representation (1), (2) is not straightforward.

To overcome the aforementioned shortcomings, we re-evaluate the 4-body phase
space in the $\nu^{\prime}$-formalism, keeping close analogy to the 3-body
case as well as the kaon representation, in order to retain the advantages in
both methods. Notice that the definition of the pseudo-neutrino momentum
$p_{\nu}^{\prime}$ in Eq.(7) remains unchanged in 4-body, and the relation
$E_{\nu}^{\prime}=E_{m}-E_{e}$ still holds. We first write:

$\int d^{3}p_{f}\frac{d^{3}p_{e}}{E_{e}}=\int
d^{3}p_{\nu}^{\prime}\frac{d^{3}p_{e}}{E_{e}}=\int
d\Omega_{e}d\Omega_{\nu}^{\prime}\int dE_{e}|\vec{p}_{e}|\int
d|\vec{p}_{\nu}^{\prime}||\vec{p}_{\nu}^{\prime}|^{2}\leavevmode\nobreak\ ,$
(23)

and deduce the boundaries of each variable. First, the boundaries of $E_{e}$,
$\Omega_{e}$ and $\Omega_{\nu}^{\prime}$ must be exactly the same as in the
3-body case; an easy way to understand this is that it must be so to ensure
the exact cancellation of the infrared (IR)-divergence between the 3-body
(one-loop) and 4-body (tree-level) contributions to the differential decay
rate $d\Gamma/dE_{e}d\Omega_{e}d\Omega_{\nu}^{\prime}$ Bloch and Nordsieck
(1937); Yennie et al. (1961); Kinoshita and Sirlin (1959); Kinoshita (1962);
Lee and Nauenberg (1964). The boundaries of $|\vec{p}_{\nu}^{\prime}|$ are
determined by the delta function:

$p_{\nu}^{\prime 2}=E_{\nu}^{\prime
2}-|\vec{p}_{\nu}^{\prime}|^{2}=(p_{\nu}+k)^{2}=(E_{\nu}+E_{k})^{2}-|\vec{p}_{\nu}+\vec{k}|^{2}\geq
0\leavevmode\nobreak\ ,$ (24)

so we have $|\vec{p}_{\nu}^{\prime}|\leq E_{\nu}^{\prime}$, with the equal
sign occurring when $\vec{p}_{\nu}\parallel\vec{k}$. Meanwhile, for any given
$\vec{p}_{e}$, a $\vec{p}_{\nu}^{\prime}=\vec{0}$ configuration can always be
achieved by setting $\vec{p}_{f}=-\vec{p}_{e}$ (which does not affect the
energy $E_{f}\approx m_{f}$ in the non-recoil limit), with the remaining
energy of the system carried away by a $\nu\gamma$-pair with equal and
opposite momenta. So, the lower and upper limits of the
$|\vec{p}_{\nu}^{\prime}|$-integration are 0 and $E_{\nu}^{\prime}$
respectively. Thus, we obtain the $\nu^{\prime}$-representation of the 4-body
phase space in the non-recoil limit:

$\displaystyle\int d\Pi_{4}$ $\displaystyle\approx$
$\displaystyle\frac{1}{512\pi^{6}m_{i}m_{f}}\int
d\Omega_{e}d\Omega_{\nu}^{\prime}\int_{m_{e}}^{E_{m}}dE_{e}|\vec{p}_{e}|\int_{0}^{E_{\nu}^{\prime}}d|\vec{p}_{\nu}^{\prime}||\vec{p}_{\nu}^{\prime}|^{2}\int\frac{d^{3}k}{(2\pi)^{3}2E_{k}}\frac{d^{3}p_{\nu}}{(2\pi)^{3}2E_{\nu}}$
(25)
$\displaystyle\times(2\pi)^{4}\delta^{(4)}(p_{\nu}^{\prime}-p_{\nu}-k)\leavevmode\nobreak\
.$

Eq.(25) is the first central result of this work. It possesses the following
advantages:

  1. 1.

The differential rate is expressible in terms of the experimentally-measurable
energy and angular variables $E_{e}$, $\Omega_{e}$ and
$\Omega_{\nu}^{\prime}$, which solves the problem in the traditional
$\nu$-formalism.

  2. 2.

The mathematical structure of Eq.(25) is much more elegant than the recoil
formalism, Eq.(19). For instance, here we do not need to separate the so-
called “in” and “out” region as two different terms, and the boundaries are
also much simpler.

  3. 3.

The full $p_{\nu}$\- and $k$-integrations are retained as in the kaon
formalism. This allows one to directly make use of the known analytic formula
of such integrations in the latter.

Finally, one can easily check the correctness of Eq.(25) by comparing it to
Eq.(19). For instance, after removing the $p_{\nu}$\- and $k$-measures and the
delta function, we evaluate the two expressions numerically by substituting
the neutron decay parameters, and find that their difference is at the level
0.1% which is the typical size of a recoil correction
$\sim(m_{n}-m_{p})/m_{n}$. This shows that Eq.(25) is indeed the correct
4-body phase space formula in the non-recoil limit.

##  IV Radiative corrections to nuclear decay in $\nu^{\prime}$-formalism

As an important application of the $\nu^{\prime}$-formalism, we study the so-
called “outer RC” in a $J_{i}=J_{f}=1/2$ decay of a polarized parent nucleus
(with unmeasured final spins) at the $\mathcal{O}(\alpha/\pi)$ level. This
includes well-known one-loop corrections and the one-photon bremsstrahlung
process. The former does not distinguish between the $\nu$\- and
$\nu^{\prime}$-method, so we focus on the latter.

We start by writing down the hadronic and leptonic charged weak current
operator222In this work we adopt the experimentalists’ convention, $g_{A}<0$.:

$J_{\text{had}}^{\mu}=\left\\{\begin{array}[]{ccc}\bar{n}\gamma^{\mu}(g_{V}+g_{A}\gamma_{5})p&,&\beta^{+}\\\
\bar{p}\gamma^{\mu}(g_{V}+g_{A}\gamma_{5})n&,&\beta^{-}\end{array}\right.\leavevmode\nobreak\
,\leavevmode\nobreak\
J_{\text{lep}}^{\mu}=\left\\{\begin{array}[]{ccc}\bar{\nu}\gamma^{\mu}(1-\gamma_{5})e&,&\beta^{+}\\\
\bar{e}\gamma^{\mu}(1-\gamma_{5})\nu&,&\beta^{-}\end{array}\right.\leavevmode\nobreak\
.$ (26)

The Fermi (F) and Gamow-Teller (GT) matrix element of $J_{\text{had}}^{\mu}$
are defined through:

$\langle f|\int
d^{3}xJ_{\text{had}}^{0}(\vec{x})|i\rangle=g_{V}M_{\text{F}}\mathbbm{1}\leavevmode\nobreak\
,\leavevmode\nobreak\ \langle f|\int
d^{3}x\vec{J}_{\text{had}}(\vec{x})|i\rangle=\frac{g_{A}}{\sqrt{3}}M_{\text{GT}}\vec{\sigma}\leavevmode\nobreak\
.$ (27)

With this we can define a “generalized” axial-to-vector ratio:

$\lambda\equiv\frac{M_{\text{GT}}}{\sqrt{3}M_{\text{F}}}\frac{g_{A}}{g_{V}}\leavevmode\nobreak\
,$ (28)

which reduces to $g_{A}/g_{V}$ for neutron decay ($M_{\text{F}}=1$ and
$M_{\text{GT}}=\sqrt{3}$). Taking the non-recoil limit at the amplitude level,
which corresponds to setting $p_{f}\approx p_{i}\equiv p=(m,\vec{0})$ with
$m=m_{i}$ for simplicity, we can write down the tree-level squared amplitude
as:

$|\mathcal{M}|_{\text{tree}}^{2}\approx
16G_{V}^{2}m^{2}E_{e}E_{\nu}^{\prime}\left\\{1+3\lambda^{2}+(1-\lambda^{2})\vec{\beta}\cdot\hat{p}_{\nu}^{\prime}+\hat{s}\cdot\left[2\lambda(\lambda\zeta-1)\vec{\beta}-2\lambda(\lambda\zeta+1)\hat{p}_{\nu}^{\prime}\right]\right\\}\leavevmode\nobreak\
,$ (29)

where $\zeta=+1(-1)$ for $\beta^{+}(\beta^{-})$-decay, $G_{V}\equiv
G_{F}V_{ud}g_{V}M_{F}$ is the effective weak vector coupling, and $\hat{s}$ is
the unit vector of the parent’s polarization. Plugging it into Eq.(10) gives
the tree-level differential rate in the non-recoil limit (taking $m^{2}\approx
m_{i}m_{f}$):

$\displaystyle\left(\frac{d\Gamma}{dE_{e}d\Omega_{e}d\Omega_{\nu}^{\prime}}\right)_{\text{tree}}$
$\displaystyle\approx$
$\displaystyle\frac{G_{V}^{2}}{(2\pi)^{5}}|\vec{p}_{e}|E_{e}E_{\nu}^{\prime
2}\Bigl{\\{}1+3\lambda^{2}+(1-\lambda^{2})\vec{\beta}\cdot\hat{p}_{\nu}^{\prime}$
(30)
$\displaystyle+\hat{s}\cdot\left[2\lambda(\lambda\zeta-1)\vec{\beta}-2\lambda(\lambda\zeta+1)\hat{p}_{\nu}^{\prime}\right]\Bigr{\\}}\leavevmode\nobreak\
.$

###  IV.1 The bremsstrahlung squared amplitude

Figure 1:  Bremsstrahlung diagrams.

We now write down the bremsstrahlung amplitude deduced from the Feynman
diagrams in Fig.1. We express it in a relativistic form, but again take take
the non-recoil limit to the amplitude, which gives:

$\displaystyle\mathcal{M}$ $\displaystyle\approx$
$\displaystyle\frac{G_{V}e}{\sqrt{2}}\zeta\left(\frac{p\cdot\varepsilon^{*}}{p\cdot
k}-\frac{p_{e}\cdot\varepsilon^{*}}{p_{e}\cdot
k}\right)H_{\mu}L^{\mu}-\frac{G_{V}e}{\sqrt{2}}\left(\frac{\zeta(k^{\mu}\varepsilon^{*\nu}-k^{\nu}\varepsilon^{*\mu})+i\epsilon^{\mu\nu\alpha\beta}k_{\alpha}\varepsilon_{\beta}^{*}}{2p_{e}\cdot
k}\right)H_{\mu}L_{\nu}$ (31) $\displaystyle\equiv$
$\displaystyle\mathcal{M}_{\text{I}}+\mathcal{M}_{\text{II}}\leavevmode\nobreak\
.$

where

$H_{\mu}=\bar{u}_{f}(p)\gamma_{\mu}(1+\lambda\gamma_{5})u_{i}(p)\leavevmode\nobreak\
,\leavevmode\nobreak\
L_{\mu}=\left\\{\begin{array}[]{ccc}\bar{u}_{\nu}(p_{\nu})\gamma_{\mu}(1-\gamma_{5})v_{e}(p_{e})&,&\beta^{+}\\\
\bar{u}_{e}(p_{e})\gamma_{\mu}(1-\gamma_{5})v_{\nu}(p_{\nu})&,&\beta^{-}\end{array}\right.$
(32)

are the hadronic and leptonic matrix elements of the charged weak current,
respectively. Summing over the final spins leads to the following hadronic and
leptonic tensors:

$\displaystyle H_{\mu\nu}$ $\displaystyle\equiv$
$\displaystyle\sum_{s_{f}}H_{\mu}H_{\nu}^{*}=\text{Tr}[(\not{p}+m)\gamma_{\mu}(1+\lambda\gamma_{5})\Sigma_{i}(\not{p}+m)\gamma_{\nu}(1+\lambda\gamma_{5})]$
$\displaystyle L_{\mu\nu}$ $\displaystyle\equiv$
$\displaystyle\sum_{s_{e},s_{\nu}}L_{\mu}L_{\nu}^{*}=8\left\\{-g_{\mu\nu}p_{e}\cdot
p_{\nu}+p_{e\mu}(p_{\nu})_{\nu}+p_{e\nu}(p_{\nu})_{\mu}-i\zeta\epsilon_{\alpha\beta\mu\nu}p_{e}^{\alpha}p_{\nu}^{\beta}\right\\}\leavevmode\nobreak\
,$ (33)

where $\Sigma_{i}\equiv(1+\gamma_{5}\not{s})/2$ is the parent’s spin
projection operator, with $s^{\mu}=(0,\hat{s})$ the parent’s spin vector. In
what follows, the sum over all final spins is always assumed in the squared
amplitude.

The square of $\mathcal{M}_{\text{I}}$ needs some extra care:

$|\mathcal{M}_{\text{I}}|^{2}=-\frac{G_{V}^{2}e^{2}}{2}\left(\frac{p}{p\cdot
k}-\frac{p_{e}}{p_{e}\cdot
k}\right)^{2}H_{\mu\nu}L^{\mu\nu}\equiv|\mathcal{M}_{\text{I}}|^{2}_{a}+|\mathcal{M}_{\text{I}}|^{2}_{b}\leavevmode\nobreak\
,$ (34)

where we split the “true” neutrino momentum as $p_{\nu}=p_{\nu}^{\prime}-k$,
and $p_{\nu}^{\prime}(-k)$ gives the $a(b)$ term respectively; it turns out
that the $(a)$ term is the only piece that gives rise to an IR-divergence upon
the phase-space integration. So, we can split the full squared amplitude into
an IR-divergent piece and a regular piece:
$|\mathcal{M}|^{2}=|\mathcal{M}_{\text{I}}|^{2}_{a}+|\mathcal{M}|^{2}_{\text{reg}}$,
where

$|\mathcal{M}|^{2}_{\text{reg}}=|\mathcal{M}_{\text{I}}|^{2}_{b}+2\mathfrak{Re}\\{\mathcal{M}_{\text{I}}\mathcal{M}_{\text{II}}^{*}\\}+|\mathcal{M}_{\text{II}}|^{2}\leavevmode\nobreak\
.$ (35)

###  IV.2 “Analytic” part of the outer RC

Next we plug the squared amplitude into the 4-body decay rate formula in the
$\nu^{\prime}$-formalism, i.e. Eqs.(15), (25). To evaluate the IR-divergent
bremsstrahlung contribution we spell out the dot products in
$|\mathcal{M}_{\text{I}}|_{a}^{2}$:

$\displaystyle|\mathcal{M}_{\text{I}}|_{a}^{2}$ $\displaystyle=$
$\displaystyle-16G_{V}^{2}e^{2}m^{2}E_{e}E_{\nu}^{\prime}\left(\frac{p}{p\cdot
k}-\frac{p_{e}}{p_{e}\cdot
k}\right)^{2}\left\\{1+3\lambda^{2}+(1-\lambda^{2})\left(1+\frac{|\vec{p}_{\nu}^{\prime}|-E_{\nu}^{\prime}}{E_{\nu}^{\prime}}\right)\vec{\beta}\cdot\hat{p}_{\nu}^{\prime}\right.$
(36)
$\displaystyle\left.+\hat{s}\cdot\left[2\lambda(\lambda\zeta-1)\vec{\beta}-2\lambda(\lambda\zeta+1)\left(1+\frac{|\vec{p}_{\nu}^{\prime}|-E_{\nu}^{\prime}}{E_{\nu}^{\prime}}\right)\hat{p}_{\nu}^{\prime}\right]\right\\}\leavevmode\nobreak\
,$

where
$\hat{p}_{\nu}^{\prime}\equiv\vec{p}_{\nu}^{\prime}/|\vec{p}_{\nu}^{\prime}|$
is the unit pseudo-neutrino momentum vector. We plug it into Eqs.(15), (25),
and the explicit dependence of $p_{\nu}$, $k$ and $|\vec{p}_{\nu}^{\prime}|$
in the expression above allows us to integrate them analytically using the
formula in Appendix A. That gives the following correction to the differential
rate :

$\displaystyle\delta\left(\frac{d\Gamma}{dE_{e}d\Omega_{e}d\Omega_{\nu}^{\prime}}\right)_{(\text{I}.a)}$
$\displaystyle=$
$\displaystyle\frac{G_{V}^{2}}{(2\pi)^{5}}|\vec{p}_{e}|E_{e}E_{\nu}^{\prime
2}\frac{\alpha}{2\pi}\left\\{(1+3\lambda^{2})\delta_{\text{I}a1}+(1-\lambda^{2})(\delta_{\text{I}a1}+\delta_{\text{I}a2})\vec{\beta}\cdot\hat{p}_{\nu}^{\prime}\right.$
(37)
$\displaystyle\left.+\hat{s}\cdot\left[2\lambda(\lambda\zeta-1)\delta_{\text{I}a1}\vec{\beta}-2\lambda(\lambda\zeta+1)(\delta_{\text{I}a1}+\delta_{\text{I}a2})\hat{p}_{\nu}^{\prime}\right]\right\\}\leavevmode\nobreak\
.$

The function $\delta_{\text{I}a1}$ contains an IR-divergence regularized by a
fictitious photon mass $m_{\gamma}$, which is exactly canceled with that in
the virtual outer correction, see Appendix B. Combining these two pieces
yields the “analytic” part of the outer RC:

$\displaystyle\delta\left(\frac{d\Gamma}{dE_{e}d\Omega_{e}d\Omega_{\nu}^{\prime}}\right)_{v}+\delta\left(\frac{d\Gamma}{dE_{e}d\Omega_{e}d\Omega_{\nu}^{\prime}}\right)_{(\text{I}.a)}$
(38) $\displaystyle=$
$\displaystyle\frac{G_{V}^{2}}{(2\pi)^{5}}|\vec{p}_{e}|E_{e}E_{\nu}^{\prime
2}\frac{\alpha}{2\pi}\left\\{(1+3\lambda^{2})\delta_{\text{an}}+(1-\lambda^{2})(\delta_{\text{an}}+\delta_{\text{I}a2}+\delta_{v2})\vec{\beta}\cdot\hat{p}_{\nu}^{\prime}\right.$
$\displaystyle\left.+\hat{s}\cdot\left[2\lambda(\lambda\zeta-1)(\delta_{\text{an}}+\delta_{v2})\vec{\beta}-2\lambda(\lambda\zeta+1)(\delta_{\text{an}}+\delta_{\text{I}a2})\hat{p}_{\nu}^{\prime}\right]\right\\}\leavevmode\nobreak\
,$

where the IR-finite function $\delta_{\text{an}}$ reads:

$\displaystyle\delta_{\text{an}}(E_{e},c^{\prime})$ $\displaystyle=$
$\displaystyle\delta_{\text{I}a1}(E_{e},c^{\prime})+\delta_{v1}(E_{e})$ (39)
$\displaystyle=$ $\displaystyle-2\left(4-\ln\frac{4E_{\nu}^{\prime
2}}{m_{e}^{2}}\right)\left(\frac{1}{\beta}\tanh^{-1}\beta-1\right)+\frac{3}{2}\ln\frac{m^{2}}{m_{e}^{2}}-\frac{11}{4}+2\ln\left(\frac{1-\beta
c^{\prime}}{1+\beta}\right)$
$\displaystyle-\frac{1}{\beta}\text{Li}_{2}\left(\frac{2\beta}{1+\beta}\right)-\frac{1}{\beta}\text{Li}_{2}\left(\frac{-2\beta}{1-\beta}\right)-\frac{2}{\beta}\text{Li}_{2}\left(\frac{\beta(c^{\prime}+1)}{1+\beta}\right)+\frac{2}{\beta}\text{Li}_{2}\left(\frac{\beta(c^{\prime}-1)}{1-\beta}\right)$
$\displaystyle-\frac{2}{\beta}(\tanh^{-1}\beta)^{2}+2(1+\beta)\tanh^{-1}\beta\leavevmode\nobreak\
,$

with
$c^{\prime}\equiv\cos\theta_{e\nu^{\prime}}=\hat{p}_{e}\cdot\hat{p}_{\nu}^{\prime}$.

###  IV.3 “Regular” bremsstrahlung contribution

Next we evaluate the the contribution from $|\mathcal{M}|_{\text{reg}}^{2}$.
After applying the replacement (76) to the quantity $s\cdot k$ in the
numerator (see the explanation in Appendix D), we can organize the squared
amplitude as:

$|\mathcal{M}|_{\text{reg}}^{2}\rightarrow 8G_{V}^{2}\alpha
m^{2}E_{e}E_{\nu}^{\prime 2}\sum_{ij}\frac{1}{(p\cdot k)^{i}(p_{e}\cdot
k)^{j}}\left\\{C_{ij}^{0}+\hat{s}\cdot\left[C_{ij}^{\beta}\vec{\beta}+C_{ij}^{\nu}\hat{p}_{\nu}^{\prime}\right]\right\\}\leavevmode\nobreak\
,$ (40)

where $i,j$ are integers, and the coefficients $C_{ij}^{X}$ ($X=0,\beta,\nu$)
are functions of $E_{e},c^{\prime},|\vec{p}_{\nu}^{\prime}|$ and $\lambda$;
their explicit expressions can be found in Appendix E. In this form, the
$p_{\nu}$\- and $k$-integrations can be performed analytically using the
formula in Appendix C, which give rise to the functions $I_{i,j}$. We are then
left with the one-fold integration over $|\vec{p}_{\nu}^{\prime}|$, which in
principle can also be done analytically with great patience; but for practical
purpose it is more efficient to just perform it numerically given that the
integration is stable. With this, the contribution of
$|\mathcal{M}|_{\text{reg}}^{2}$ to the differential decay rate in the
$\nu^{\prime}$-formalism reads:

$\delta\left(\frac{d\Gamma}{dE_{e}d\Omega_{e}d\Omega_{\nu}^{\prime}}\right)_{\text{reg}}=\frac{G_{V}^{2}}{(2\pi)^{5}}|\vec{p}_{e}|E_{e}E_{\nu}^{\prime
2}\frac{\alpha}{2\pi}\left\\{\delta_{\text{reg}}^{0}+\hat{s}\cdot\left[\delta_{\text{reg}}^{\beta}\vec{\beta}+\delta_{\text{reg}}^{\nu}\hat{p}_{\nu}^{\prime}\right]\right\\}\leavevmode\nobreak\
,$ (41)

where

$\delta_{\text{reg}}^{X}(E_{e},c^{\prime},\lambda)=\sum_{ij}\int_{0}^{E_{\nu}^{\prime}}d|\vec{p}_{\nu}^{\prime}||\vec{p}_{\nu}^{\prime}|^{2}C_{ij}^{X}(E_{e},c^{\prime},|\vec{p}_{\nu}^{\prime}|,\lambda)I_{i,j}(E_{e},c^{\prime},|\vec{p}_{\nu}^{\prime}|)\leavevmode\nobreak\
,\leavevmode\nobreak\ X=0,\beta,\nu\leavevmode\nobreak\ .$ (42)

###  IV.4 Total outer RC in the $\nu^{\prime}$-formalism

Adding Eqs.(38) and (41) returns the master formula for the total outer RC to
the differential decay rate in the $\nu^{\prime}$-formalism, which is the next
central result of this work:

$\delta\left(\frac{d\Gamma}{dE_{e}d\Omega_{e}d\Omega_{\nu}^{\prime}}\right)_{\text{outer}}=\frac{G_{V}^{2}}{(2\pi)^{5}}|\vec{p}_{e}|E_{e}E_{\nu}^{\prime
2}\frac{\alpha}{2\pi}\left\\{\delta_{\text{tot}}^{0}+\hat{s}\cdot\left[\delta_{\text{tot}}^{\beta}\vec{\beta}+\delta_{\text{tot}}^{\nu}\hat{p}_{\nu}^{\prime}\right]\right\\}\leavevmode\nobreak\
,$ (43)

where

$\displaystyle\delta_{\text{tot}}^{0}(E_{e},c^{\prime},\lambda)$
$\displaystyle=$
$\displaystyle(1+3\lambda^{2})\delta_{\text{an}}(E_{e},c^{\prime})+(1-\lambda^{2})\left(\delta_{\text{an}}(E_{e},c^{\prime})+\delta_{\text{I}a2}(E_{e})+\delta_{v2}(E_{e})\right)\beta
c^{\prime}$ $\displaystyle+\delta_{\text{reg}}^{0}(E_{e},c^{\prime},\lambda)$
$\displaystyle\delta_{\text{tot}}^{\beta}(E_{e},c^{\prime},\lambda)$
$\displaystyle=$ $\displaystyle
2\lambda(\lambda\zeta-1)\left(\delta_{\text{an}}(E_{e},c^{\prime})+\delta_{v2}(E_{e})\right)+\delta_{\text{reg}}^{\beta}(E_{e},c^{\prime},\lambda)$
$\displaystyle\delta_{\text{tot}}^{\nu}(E_{e},c^{\prime},\lambda)$
$\displaystyle=$
$\displaystyle-2\lambda(\lambda\zeta+1)\left(\delta_{\text{an}}(E_{e},c^{\prime})+\delta_{\text{I}a2}(E_{e})\right)+\delta_{\text{reg}}^{\nu}(E_{e},c^{\prime},\lambda)\leavevmode\nobreak\
,$ (44)

definitions of the individual functions at the right hand side can be found in
Eqs.(39), (42), (64) and (66). This is a fully-analytic formula (apart from a
possible numerical integration over $|\vec{p}_{\nu}^{\prime}|$ in
$\delta_{\text{reg}}^{X}$ that can be performed trivially) which is readily
applicable to the analysis of neutron and nuclear beta decay data. Advantages
of this master formula over other formalisms in literature include the
unnecessity to involve complicated numerical packages, and the explicit
cancellation of IR-divergences.

We stress that, although the results above are based on $J_{i}=J_{f}=1/2$,
they are equally applicable to $J_{i}=J_{f}=0$ by simply taking $\lambda=0$.

###  IV.5 Comparison with the $\nu$-formalism

Our master formula should be compared to the result in the traditional
$\nu$-formalism Sirlin (1967); Shann (1971); Garcia and Maya (1978):

$\displaystyle\delta\left(\frac{d\Gamma}{dE_{e}d\Omega_{e}d\Omega_{\nu}}\right)_{\text{outer}}$
$\displaystyle=$
$\displaystyle\frac{G_{V}^{2}}{(2\pi)^{5}}|\vec{p}_{e}|E_{e}E_{\nu}^{\prime
2}\frac{\alpha}{2\pi}\Bigl{\\{}(1+3\lambda^{2})\delta_{1}+(1-\lambda^{2})(\delta_{1}+\delta_{2})\beta
c$ (45)
$\displaystyle+\hat{s}\cdot\left[2\lambda(\lambda\zeta-1)(\delta_{1}+\delta_{2})\vec{\beta}-2\lambda(\lambda\zeta+1)\delta_{1}\hat{p}_{\nu}\right]\Bigr{\\}}\leavevmode\nobreak\
,$

where $c\equiv\cos\theta_{e\nu}=\hat{p}_{e}\cdot\hat{p}_{\nu}$ is to be
distinguished from $c^{\prime}$ in the previous expressions. The functions

$\displaystyle\delta_{1}(E_{e})$ $\displaystyle=$ $\displaystyle
3\ln\frac{m}{m_{e}}-\frac{3}{4}+4\left(\frac{1}{\beta}\tanh^{-1}\beta-1\right)\left(\ln\frac{2E_{\nu}^{\prime}}{m_{e}}+\frac{E_{\nu}^{\prime}}{3E_{e}}-\frac{3}{2}\right)$
$\displaystyle-\frac{4}{\beta}\text{Li}_{2}\left(\frac{2\beta}{1+\beta}\right)+\frac{1}{\beta}\tanh^{-1}\beta\left(2+2\beta^{2}+\frac{E_{\nu}^{\prime
2}}{6E_{e}^{2}}-4\tanh^{-1}\beta\right)$ $\displaystyle\delta_{2}(E_{e})$
$\displaystyle=$ $\displaystyle
2\left(\frac{1-\beta^{2}}{\beta}\right)\tanh^{-1}\beta+\frac{4E_{\nu}^{\prime}(1-\beta^{2})}{3\beta^{2}E_{e}}\left(\frac{1}{\beta}\tanh^{-1}\beta-1\right)$
(46) $\displaystyle+\frac{E_{\nu}^{\prime
2}}{6\beta^{2}E_{e}^{2}}\left(\frac{1-\beta^{2}}{\beta}\tanh^{-1}\beta-1\right)$

depend only on the electron energy and not the angles. Eqs.(43) and (45) are
just describing the same quantity in two different languages: The former
utilizes a measurable
$c^{\prime}=\cos\theta_{e\nu^{\prime}}=\hat{p}_{e}\cdot\hat{p}_{\nu}^{\prime}$
while the latter depends on an unmeasured quantity
$c=\cos\theta_{e\nu}=\hat{p}_{e}\cdot\hat{p}_{\nu}$. An important consistency
check is that the two expressions should give rise to the same result for
$\delta(d\Gamma/dE_{e}d\Omega_{e})_{\text{outer}}$; in other words, if the
recoiled daughter nucleus is not detected in the experiment (e.g. during the
measurement of the beta spectrum or the beta asymmetry parameter $A$ that
depend only on $\vec{p}_{e}$), then the $\nu$\- and $\nu^{\prime}$-formalisms
make no difference. This implies the following relations:

$\displaystyle\frac{1}{2}\int_{-1}^{+1}dc^{\prime}\delta_{\text{tot}}^{0}(E_{e},c^{\prime},\lambda)$
$\displaystyle=$ $\displaystyle(1+3\lambda^{2})\delta_{1}(E_{e})$
$\displaystyle\frac{1}{2}\int_{-1}^{+1}dc^{\prime}\left[\delta_{\text{tot}}^{\beta}(E_{e},c^{\prime},\lambda)+\frac{c^{\prime}}{\beta}\delta_{\text{tot}}^{\nu}(E_{e},c^{\prime},\lambda)\right]$
$\displaystyle=$ $\displaystyle
2\lambda(\lambda\zeta-1)\left(\delta_{1}(E_{e})+\delta_{2}(E_{e})\right)\leavevmode\nobreak\
.$ (47)

One can check numerically that the two relations above are exactly satisfied,
which is a solid proof of the correctness of all the analytic formula
presented in this paper.

Figure 2: Left: $(\alpha/2\pi)\delta_{\text{tot}}^{0}$ (blue, solid) as a function of $c^{\prime}$ versus $(\alpha/2\pi)[(1+3\lambda^{2})\delta_{1}+(1-\lambda^{2})(\delta_{1}+\delta_{2})\beta c]$ (red, dashed) as a function of $c$ at $E_{e}=0.8E_{m}$. Right: The function $r_{e\nu}$ at $E_{e}=m_{e}+0.2(E_{m}-m_{e})$ (blue) and $E_{e}=m_{e}+0.8(E_{m}-m_{e})$ (red). Solid line: Our result; dots: Discrete points extracted from Fig.8 in Ref.Glück (2023). Here we take $\lambda=-1.27$.  $c^{\prime}$ |  |  |  | $r_{e\nu}$ | (%) |  |  |
---|---|---|---|---|---|---|---|---
0.9 | 0.021 | 0.038 | 0.056 | 0.075 | 0.093 | 0.112 | 0.129 | 0.146
0.7 | 0.018 | 0.032 | 0.046 | 0.060 | 0.074 | 0.087 | 0.100 | 0.112
0.5 | 0.014 | 0.024 | 0.034 | 0.044 | 0.053 | 0.061 | 0.069 | 0.077
0.3 | 0.010 | 0.016 | 0.021 | 0.026 | 0.031 | 0.036 | 0.040 | 0.044
0.1 | 0.005 | 0.006 | 0.008 | 0.009 | 0.011 | 0.012 | 0.012 | 0.013
-0.1 | -0.001 | -0.003 | -0.005 | -0.007 | -0.009 | -0.012 | -0.014 | -0.016
-0.3 | -0.007 | -0.012 | -0.018 | -0.023 | -0.029 | -0.034 | -0.039 | -0.044
-0.5 | -0.012 | -0.021 | -0.030 | -0.039 | -0.047 | -0.055 | -0.063 | -0.071
-0.7 | -0.018 | -0.030 | -0.042 | -0.054 | -0.065 | -0.076 | -0.086 | -0.096
-0.9 | -0.024 | -0.039 | -0.054 | -0.068 | -0.082 | -0.095 | -0.108 | -0.120
$x_{e}$ | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | 0.9
Table 1: Our result of $r_{e\nu}(E_{e},c^{\prime})$ with
$E_{e}=m_{e}+x_{e}(E_{m}-m_{e})$, which should be compared to Table V in
Ref.Gluck (1993).

We argued before that, a mis-identification of $\hat{p}_{\nu}$ by
$\hat{p}_{\nu}^{\prime}$ in the $\nu$-formalism would lead to an error of the
order $\alpha/\pi\sim 10^{-3}$. To see this, let us compare the numerical
results deduced from Eqs.(43) and (45) for the case of neutron decay:
$\zeta=-1$, $m_{i}=m_{n}$, $m_{f}=m_{p}$. We start from the spin-independent
part of the outer RC, namely $(\alpha/2\pi)\delta_{\text{tot}}^{0}$ in the
$\nu^{\prime}$-formalism and
$(\alpha/2\pi)[(1+3\lambda^{2})\delta_{1}+(1-\lambda^{2})(\delta_{1}+\delta_{2})\beta
c]$ in the $\nu$-formalism; they affect the precise extraction of the
neutrino-electron correlation coefficient $a$. From the first diagram in Fig.2
we see that the dependence of these functions on their respective cosines
indeed shows a difference at the order $10^{-3}$. Also, to compare with
existing literature, in the second diagram we plot the the
$\\{E_{e},c^{\prime}\\}$ Dalitz distribution of the outer RC:

$r_{e\nu}(E_{e},c^{\prime})\equiv\frac{\delta(d\Gamma/dE_{e}dc^{\prime})_{\text{outer}}}{(d\Gamma/dE_{e}dc^{\prime})_{\text{tree}}}-\frac{\delta(d\Gamma/dE_{e})_{\text{outer}}}{(d\Gamma/dE_{e})_{\text{tree}}}\approx\frac{\alpha}{2\pi}\left[\frac{\delta_{\text{tot}}^{0}(E_{e},c^{\prime},\lambda)}{1+3\lambda^{2}+(1-\lambda^{2})\beta
c^{\prime}}-\delta_{1}(E_{e})\right]$ (48)

at two different $E_{e}$; we find excellent agreement with Fig.8 of Ref.Glück
(2023). We also notice that a table of $r_{e\nu}(E_{e},c^{\prime})$ was given
in an earlier paper Gluck (1993), but using our analytic expressions we find
that some entries in that table are off by an absolute value of $\sim 0.01\%$.
Our new results are displayed in Table 1, but for practical applications one
should of course recompute everything using our given formula instead of just
applying the table.

Figure 3:  The difference
$(\alpha/2\pi)[\bar{\delta}_{\nu}^{\prime}(E_{e},\lambda)-2\lambda(\lambda-1)\delta_{1}(E_{e})]$
as a function of $E_{e}$. Again we take $\lambda=-1.27$.

Next we study a more interesting quantity, namely the outer RC needed to
extract the neutrino asymmetry parameter $B$. At $\mathcal{O}(\alpha^{0})$
(which does not distinguish $\nu^{\prime}$ from $\nu$), a possible way to
extract $B$ is to flip the neutron spin and integrate over $\Omega_{e}$:

$\left.\left(\frac{d\Gamma}{dE_{e}d\Omega_{\nu}^{(\prime)}}\right)_{\text{tree}}\right|_{\hat{s}}-\left.(\dots)\right|_{-\hat{s}}\propto
B\hat{s}\cdot\hat{p}_{\nu}^{(\prime)}\leavevmode\nobreak\ .$ (49)

However, outer correction to the expression above takes very different forms
in the $\nu$\- and $\nu^{\prime}$-formalism:

$\displaystyle\delta\left.\left(\frac{d\Gamma}{dE_{e}d\Omega_{\nu}}\right)_{\text{outer}}\right|_{\hat{s}}-\left.(\dots)\right|_{-\hat{s}}$
$\displaystyle\propto$ $\displaystyle\frac{\alpha}{2\pi}\times
2\lambda(\lambda-1)\delta_{1}(E_{e})\hat{s}\cdot\hat{p}_{\nu}$
$\displaystyle\delta\left.\left(\frac{d\Gamma}{dE_{e}d\Omega_{\nu}^{\prime}}\right)_{\text{outer}}\right|_{\hat{s}}-\left.(\dots)\right|_{-\hat{s}}$
$\displaystyle\propto$
$\displaystyle\frac{\alpha}{2\pi}\times\frac{1}{2}\int_{-1}^{+1}dc^{\prime}\left[\delta_{\text{tot}}^{\nu}(E_{e},c^{\prime},\lambda)+\beta
c^{\prime}\delta_{\text{tot}}^{\beta}(E_{e},c^{\prime},\lambda)\right]\hat{s}\cdot\hat{p}_{\nu}^{\prime}$
(50) $\displaystyle\equiv$
$\displaystyle\frac{\alpha}{2\pi}\bar{\delta}_{\nu}^{\prime}(E_{e},\lambda)\hat{s}\cdot\hat{p}_{\nu}^{\prime}$

We see that, unlike the $\nu$-formalism that probes only the
$\hat{s}\cdot\hat{p}_{\nu}$ term in the outer correction, the
$\nu^{\prime}$-formalism probes both the $\hat{s}\cdot\hat{p}_{\nu}^{\prime}$
and $\hat{s}\cdot\vec{\beta}$ terms, because both their coefficients are
functions of $c^{\prime}$. In Fig.3 we plot the difference between the
function $(\alpha/2\pi)\bar{\delta}_{\nu}^{\prime}(E_{e},\lambda)$ (defined
above) and $(\alpha/2\pi)\times 2\lambda(\lambda-1)\delta_{1}(E_{e})$, and see
that it is again of the order $10^{-3}$. This means, a mis-identification of
$\hat{p}_{\nu}$ by $\hat{p}_{\nu}^{\prime}$ in the $\nu$-formalism would lead
to a $10^{-3}$ error in the extraction of $B$.

##  V Radiative corrections in the recoil formalism

In some experiments such as aCORN and Nab Fry et al. (2019), it is $E_{f}$
instead of $\Omega_{\nu}^{\prime}$ that is being measured; in this case the
recoil formalism, and not $\nu^{\prime}$, is the more appropriate description
for the outer RC. So, for the sake of completeness we provide also the set of
analytic expressions to compute the outer RC of the differential decay rate in
the recoil formalism, in full analogy to those in the $\nu^{\prime}$-formalism
from the previous sections.

First, it is obvious that one has no access to spin-dependent correlations if
only $E_{e}$ and $E_{f}$ are measured, because there would be no vector to
contract with $\hat{s}$ as all solid angles of momenta are integrated out.
Thus, for decays of polarized nuclei it is more natural to adopt the
$\nu^{\prime}$-formalism where the solid angles $\Omega_{e}$ and
$\Omega_{\nu}^{\prime}$ are fixed, whereas in the recoil formalism we shall
restrict ourselves to unpolarized nuclei. We start with the tree-level squared
amplitude in the non-recoil limit, Eq.(29), and recast the variables in terms
of 4-vector products:

$E_{e}=\frac{p_{i}\cdot p_{e}}{m_{i}}\leavevmode\nobreak\
,\leavevmode\nobreak\ E_{\nu}^{\prime}=\frac{p_{i}\cdot
p_{\nu}^{\prime}}{m_{i}}\leavevmode\nobreak\ ,\leavevmode\nobreak\
\vec{p}_{e}\cdot\vec{p}_{\nu}^{\prime}=\frac{p_{i}\cdot p_{e}p_{i}\cdot
p_{\nu}^{\prime}}{m_{i}^{2}}-p_{e}\cdot p_{\nu}^{\prime}\leavevmode\nobreak\
,$ (51)

which allows us to express the squared amplitude as a function of $\\{y,z\\}$
(recall that $x=0$ for 3-body decay). That gives the tree-level differential
rate:

$\left(\frac{d\Gamma}{dydz}\right)_{\text{tree}}^{\mathcal{D}_{3}}\approx\frac{G_{V}^{2}m^{5}}{(4\pi)^{3}}\left\\{(1+3\lambda^{2})y(2-y-z)+(1-\lambda^{2})\left[y(2-y-z)+2(r_{e}-r_{f}+z-1)\right]\right\\}\leavevmode\nobreak\
,$ (52)

where the superscript $\mathcal{D}_{3}$ denotes that the expression survives
only if $\\{y,z\\}\in\mathcal{D}_{3}$.

The determination of the outer RC proceeds in the exact same way as before:
For the virtual corrections, we take again the results in Refs.Sirlin (1967);
Shann (1971); Garcia and Maya (1978) and use Eq.(51) to recast them in terms
of $\\{y,z\\}$; this contribution survives only in the $\mathcal{D}_{3}$
region. For the bremsstrahlung contribution, we follow Sec.IV.1 and split the
squared amplitude into two pieces:

$|\mathcal{M}|^{2}=|\mathcal{M}_{\text{I}}|_{a}^{2}+|\mathcal{M}|_{\text{reg}}^{2}\leavevmode\nobreak\
,$ (53)

and plug it into Eq.(19); the $x$-integral is evaluated analytically for
$|\mathcal{M}_{\text{I}}|^{2}_{a}$ and numerically for
$|\mathcal{M}|^{2}_{\text{reg}}$. The IR-divergence occurs in the $x$-integral
of $|\mathcal{M}_{\text{I}}|_{a}^{2}$ in the $\mathcal{D}_{3}$ region, which
analytic result can be found in Appendix D of Ref.Seng et al. (2021); it
combines with the virtual correction to yield an IR-finite outcome. Meanwhile,
for the “regular” piece we expand the squared amplitude as:

$|\mathcal{M}|_{\text{reg}}^{2}=4G_{V}^{2}\alpha
m^{2}\sum_{ij}\frac{1}{(p\cdot k)^{i}(p_{e}\cdot
k)^{j}}d_{ij}\leavevmode\nobreak\ ,$ (54)

where the expansion coefficients are just $d_{ij}=2E_{e}E_{\nu}^{\prime
2}C_{ij}^{0}$; nevertheless, we still provide their analytic expressions in
terms of $\\{x,y,z\\}$ in Appendix F for the convenience of readers. The
$p_{\nu}$\- and $k$-integrals can be performed analytically using Appendix C.

Combining all the above, we obtain the total outer RC in the $\mathcal{D}_{3}$
and $\mathcal{D}_{4-3}$ region as follows:

$\displaystyle\delta\left(\frac{d\Gamma}{dydz}\right)_{\text{outer}}^{\mathcal{D}_{3}}$
$\displaystyle=$
$\displaystyle\frac{G_{V}^{2}m^{5}}{(4\pi)^{3}}\frac{\alpha}{2\pi}\left\\{(1+3\lambda^{2})y(2-y-z)\delta_{\text{an}}^{r}+2(1-\lambda^{2})\alpha_{+}\delta_{\text{Ia2}}^{r}\right.$
$\displaystyle\left.+(1-\lambda^{2})\left[y(2-y-z)+2(r_{e}-r_{f}+z-1)\right](\delta_{\text{an}}^{r}+\delta_{v2})+\delta_{\text{reg}}^{r,\mathcal{D}_{3}}\right\\}$
$\displaystyle\delta\left(\frac{d\Gamma}{dydz}\right)_{\text{outer}}^{\mathcal{D}_{4-3}}$
$\displaystyle=$
$\displaystyle\frac{G_{V}^{2}m^{5}}{(4\pi)^{3}}\frac{\alpha}{2\pi}\left\\{\delta_{\text{Ia2}}^{r}\ln\frac{\alpha_{+}}{\alpha_{-}}\left[(1+3\lambda^{2})y(2-y-z)+(1-\lambda^{2})\left(y(2-y-z)\right.\right.\right.$
(55)
$\displaystyle\left.\left.\left.+2(r_{e}-r_{f}+z-1)\right)\right]+2(1-\lambda^{2})(\alpha_{+}-\alpha_{-})\delta_{\text{Ia2}}^{r}+\delta_{\text{reg}}^{r,\mathcal{D}_{4-3}}\right\\}\leavevmode\nobreak\
,$

where the functions at the right hand side are (the extra superscript $r$
stands for “recoil”):

$\displaystyle\delta_{\text{an}}^{r}(y,z)$ $\displaystyle=$
$\displaystyle\left(\frac{2}{\beta}\tanh^{-1}\beta+\frac{1}{2}\right)\ln\frac{m^{2}}{m_{e}^{2}}-\frac{11}{4}+\left(\frac{2}{\beta}\tanh^{-1}\beta-1\right)\ln\frac{m^{2}\alpha_{+}^{2}}{4P_{0}^{2}}$
$\displaystyle+\ln\frac{(r_{e}-r_{f}+z-1)^{2}}{\alpha_{+}^{2}}-\frac{1}{\beta}\text{Li}_{2}\left(\frac{2\beta}{1+\beta}\right)-\frac{1}{\beta}\text{Li}_{2}\left(-\frac{2\beta}{1-\beta}\right)$
$\displaystyle-\frac{2}{\beta}\text{Li}_{2}\left(\frac{\beta}{1+\beta}\left(\frac{P_{1}}{P_{0}}+1\right)\right)+\frac{2}{\beta}\text{Li}_{2}\left(\frac{\beta}{1-\beta}\left(\frac{P_{1}}{P_{0}}-1\right)\right)$
$\displaystyle-\frac{2}{\beta}\left(\tanh^{-1}\beta\right)^{2}+2\beta\tanh^{-1}\beta$
$\displaystyle\delta_{\text{Ia2}}^{r}(y)$ $\displaystyle=$ $\displaystyle
4\left(\frac{1}{\beta}\tanh^{-1}\beta-1\right)$ $\displaystyle\delta_{v2}(y)$
$\displaystyle=$ $\displaystyle\frac{2(1-\beta^{2})}{\beta}\tanh^{-1}\beta$
$\displaystyle\delta_{\text{reg}}^{r,\mathcal{D}_{3}}(y,z)$ $\displaystyle=$
$\displaystyle\sum_{ij}\int_{0}^{\alpha_{+}}dxd_{ij}I_{i,j}$
$\displaystyle\delta_{\text{reg}}^{r,\mathcal{D}_{4-3}}(y,z)$ $\displaystyle=$
$\displaystyle\sum_{ij}\int_{\alpha_{-}}^{\alpha_{+}}dxd_{ij}I_{i,j}\leavevmode\nobreak\
,$ (56)

with

$P_{0}=\frac{m}{2}(2-y-z)\leavevmode\nobreak\ ,\leavevmode\nobreak\
P_{1}=\frac{m}{2\beta
y}\left[y(2-y-z)+2(r_{e}-r_{f}+z-1)\right]\leavevmode\nobreak\ ,$ (57)

and the electron speed $\beta=\sqrt{y^{2}-4r_{e}}/y$.

Figure 4: The function $r_{p}$ with respect to the proton kinetic energy
$T=y_{p}T_{\text{max}}=y_{p}((m_{n}-m_{p})^{2}-m_{e}^{2})/(2m_{n})$, with
$\lambda=-1.27$. Solid line: Our result. Dots: Discrete points extracted from
Fig.7 in Ref.Glück (2023).

Eq.(55) is the third central result of this work; a simple consistency check
is that it must reproduce the same outer RC to the beta spectrum as in
Sirlin’s approach, up to small recoil corrections (on top of the RC). This
implies:

$\frac{\delta\left(d\Gamma/dy\right)_{\text{outer}}}{\left(d\Gamma/dy\right)_{\text{tree}}}=\frac{\int
dz\delta\left(d\Gamma/dydz\right)_{\text{outer}}}{\int
dz\left(d\Gamma/dydz\right)_{\text{tree}}}\approx\frac{\alpha}{2\pi}\delta_{1}(E_{e})\leavevmode\nobreak\
,\leavevmode\nobreak\ 2\sqrt{r_{e}}<y<1+r_{e}-r_{f}\leavevmode\nobreak\ .$
(58)

Substituting the neutron decay parameters, we find that this relation is
satisfied to a $<0.1\%$ relative error, which proves the correctness of all
our analytic formula. As another check, we compute the relative RC $r_{p}$ to
the proton spectrum in neutron decay defined in Ref.Glück (2023), which reads:

$r_{p}(T)\equiv\frac{\delta\left(d\Gamma/dz\right)_{\text{outer}}}{\left(d\Gamma/dz\right)_{\text{tree}}}-\frac{\delta\Gamma_{\text{outer}}}{\Gamma_{\text{tree}}}\leavevmode\nobreak\
,$ (59)

where we use the proton kinetic energy $T=m_{n}z/2-m_{p}$ as the variable. The
outcome is plotted in Fig.4 and we again observe excellent agreement with the
result in Ref.Glück (2023) obtained using numerical packages.

##  VI Summary

In this paper we carefully analyze the theory structure of the so-called
$\nu^{\prime}$-formalism, which is designed to bypass the conceptual problem
in the traditional treatment of outer RC by Sirlin, Shann and Garcia-Maya. The
formalism itself has already appeared in many literature and is not at all a
new concept, but our paper provides the most elegant representation of the
4-body phase space formula consistent to the non-recoil approximation. With
this master formula, we derive the full set of analytic expressions needed for
the evaluation of the outer RC to the differential rate of a $J_{i}=J_{f}=1/2$
(or 0) nuclear $\beta^{\pm}$ decay. Our results are explicitly
$m_{\gamma}$-independent, and require only a minimal amount of numerical
integration. We compare the outcome with existing literature and discuss its
implications on $\vec{p}_{\nu}^{\prime}$-dependent observables. Similar
analytic expressions in the recoil formalism are also derived assuming
unpolarized nuclei.

While our master formula for the 4-body phase space is completely general, we
restrict ourselves to spin-half systems which allows us to make use of the
elegant spinor technique in field theory that simplifies the squared
amplitude. Generalization to generic nuclear beta decays with arbitrary
$J_{i,f}$ may involve, apart from more complicated analytic expressions for
existing terms, new spin-dependent correlation structures such as the
$c$-coefficient Jackson et al. (1957a). Its outer RC may be studied either
using traditional approaches focusing on the “convention term” contribution
Meister and Yennie (1963), or using the more recent effective field theory
description Cirigliano et al. (2023); Hill and Plestid (2023a, b). This will
be worked out in a follow-up work.

###### Acknowledgements.

We thank Ferenc Glück for many useful discussions. The work of C.-Y.S. is
supported in part by the U.S. Department of Energy (DOE), Office of Science,
Office of Nuclear Physics, under the FRIB Theory Alliance award DE-SC0013617,
and by the DOE grant DE-FG02-97ER41014. We acknowledge support from the DOE
Topical Collaboration “Nuclear Theory for New Physics”, award No. DE-
SC0023663.

##  Appendix A IR-divergent phase space integral

In the $\nu^{\prime}$-formalism, the only IR-divergent phase space integral
reads:

$\int_{0}^{E_{\nu}^{\prime}}d|\vec{p}_{\nu}^{\prime}||\vec{p}_{\nu}^{\prime}|^{2}\int\frac{d^{3}k}{(2\pi)^{3}2E_{k}}\frac{d^{3}p_{\nu}}{(2\pi)^{3}2E_{\nu}}(2\pi)^{4}\delta^{(4)}(p_{\nu}^{\prime}-p_{\nu}-k)\left(\frac{p}{p\cdot
k}-\frac{p_{e}}{p_{e}\cdot
k}\right)^{2}\equiv-\frac{E_{\nu}^{\prime}}{8\pi}\delta_{\text{I}a1}\leavevmode\nobreak\
.$ (60)

The IR-divergence arises from the $|\vec{p}_{\nu}^{\prime}|$-integral at
$|\vec{p}_{\nu}^{\prime}|\rightarrow E_{\nu}^{\prime}$. One may evaluate this
integral using the method outlined in Ref.Seng et al. (2021), namely to apply
dimensional regularization of the photon momentum Marciano and Sirlin (1975):

$\frac{d^{3}k}{(2\pi)^{3}2E_{k}}\rightarrow\mu^{4-d}\frac{d^{d-1}k}{(2\pi)^{d-1}2E_{k}}\leavevmode\nobreak\
.$ (61)

A important advantage of this method is that the massless on-shell condition
$k^{2}=0$ still holds, which greatly simplifies the intermediate steps. After
finishing the evaluation, one may choose to switch back to the photon-mass
prescription by the following matching:

$\frac{2}{4-d}-\gamma_{E}+\ln
4\pi\rightarrow\ln\frac{m_{\gamma}^{2}}{\mu^{2}}\leavevmode\nobreak\ .$ (62)

With this, we obtain the result of the integral as:

$\displaystyle\delta_{\text{I}a1}(E_{e},c^{\prime})$ $\displaystyle=$
$\displaystyle-2\left(4-\ln\frac{4E_{\nu}^{\prime
2}}{m_{\gamma}^{2}}\right)\left(\frac{1}{\beta}\tanh^{-1}\beta-1\right)+2\tanh^{-1}\beta+2\ln\left(\frac{1-\beta
c^{\prime}}{1+\beta}\right)$ (63)
$\displaystyle+\frac{1}{\beta}\text{Li}_{2}\left(\frac{2\beta}{1+\beta}\right)-\frac{1}{\beta}\text{Li}_{2}\left(\frac{-2\beta}{1-\beta}\right)-\frac{2}{\beta}\text{Li}_{2}\left(\frac{\beta(c^{\prime}+1)}{1+\beta}\right)$
$\displaystyle+\frac{2}{\beta}\text{Li}_{2}\left(\frac{\beta(c^{\prime}-1)}{1-\beta}\right)\leavevmode\nobreak\
.$

For terms proportional to $\hat{p}_{\nu}^{\prime}$ in
$|\mathcal{M}_{\text{I}}|^{2}_{a}$, the involved integral is Eq.(60) but with
an extra factor
$1+(|\vec{p}_{\nu}^{\prime}|-E_{\nu}^{\prime})/E_{\nu}^{\prime}$ multiplied to
the integrand. The integration returns
$-(E_{\nu}^{\prime}/8\pi)(\delta_{\text{I}a1}+\delta_{\text{I}a2})$, where

$\delta_{\text{I}a2}(E_{e})=4(1-\ln
4)\left(\frac{1}{\beta}\tanh^{-1}\beta-1\right)\leavevmode\nobreak\ .$ (64)

##  Appendix B Virtual outer corrections

Figure 5:  Self-energy and one-particle irreducible diagrams that contribute
to the outer RC.

The SM one-loop RC to beta decay contribute to three quantities: (1) The Fermi
function Fermi (1934) that describes the Coulomb interaction between the
outgoing electron and the daughter nucleus; (2) The “outer” RC that summarizes
all the non-Fermi and $E_{e}$-dependent corrections; (3) The “inner” RC which
only effect is to renormalize the effective weak coupling constants. In this
paper we are only interested in the outer RC, which is obtained by computing
the Feynman diagrams in Fig.5, taking the nuclei as point-like particles. For
a $J_{i}=J_{f}=1/2$ decay with polarized parent nucleus, the outcome in the
non-recoil limit reads Sirlin (1967); Shann (1971); Garcia and Maya (1978):

$\displaystyle\delta\left(\frac{d\Gamma}{dE_{e}d\Omega_{e}d\Omega_{\nu}^{\prime}}\right)_{v}$
$\displaystyle=$
$\displaystyle\frac{G_{V}^{2}}{(2\pi)^{5}}|\vec{p}_{e}|E_{e}E_{\nu}^{\prime
2}\frac{\alpha}{2\pi}\left\\{(1+3\lambda^{2})\delta_{v1}+(1-\lambda^{2})(\delta_{v1}+\delta_{v2})\vec{\beta}\cdot\hat{p}_{\nu}^{\prime}\right.$
(65)
$\displaystyle\left.+\hat{s}\cdot\left[2\lambda(\lambda\zeta-1)(\delta_{v1}+\delta_{v2})\vec{\beta}-2\lambda(\lambda\zeta+1)\delta_{v1}\hat{p}_{\nu}^{\prime}\right]\right\\}\leavevmode\nobreak\
,$

where

$\displaystyle\delta_{v1}(E_{e})$ $\displaystyle=$
$\displaystyle-2\ln\frac{m_{e}^{2}}{m_{\gamma}^{2}}\left(\frac{1}{\beta}\tanh^{-1}\beta-1\right)+\frac{3}{2}\ln\frac{m^{2}}{m_{e}^{2}}-\frac{11}{4}-\frac{2}{\beta}\text{Li}_{2}\left(\frac{2\beta}{1+\beta}\right)$
$\displaystyle-\frac{2}{\beta}(\tanh^{-1}\beta)^{2}+2\beta\tanh^{-1}\beta$
$\displaystyle\delta_{v2}(E_{e})$ $\displaystyle=$
$\displaystyle\frac{2(1-\beta^{2})}{\beta}\tanh^{-1}\beta\leavevmode\nobreak\
.$ (66)

##  Appendix C Analytic formula for $p_{\nu}$\- and $k$-integration

If we drop the parent’s spin vector $s^{\mu}$, then there are only three
vectors that can be dotted to the photon momentum $k$ in the squared
amplitude: $p$, $p_{e}$ and $p_{\nu}^{\prime}$; but momentum conservation
$p_{\nu}^{\prime}-k=p_{\nu}$ implies $2p_{\nu}^{\prime}\cdot k=p_{\nu}^{\prime
2}$ assuming massless neutrino. The $p_{\nu}$\- and $k$-integrations in
$|\mathcal{M}|^{2}_{\text{reg}}$ can be represented in terms of the functions
below:

$I_{i,j}(E_{e},c^{\prime},|\vec{p}_{\nu}^{\prime}|)\equiv\int\frac{d^{3}k}{(2\pi)^{3}2E_{k}}\frac{d^{3}p_{\nu}}{(2\pi)^{3}2E_{\nu}}(2\pi)^{4}\delta^{(4)}(p_{\nu}^{\prime}-p_{\nu}-k)\frac{1}{(p\cdot
k)^{i}(p_{e}\cdot k)^{j}}\leavevmode\nobreak\ ,$ (67)

where $i,j$ are integers. Analytic expressions of the functions needed in this
work can be found in Ref.Ginsberg (1967) (but beware of the difference in the
overall normalization):

$\displaystyle I_{0,0}$ $\displaystyle=$ $\displaystyle\frac{1}{8\pi}$
$\displaystyle I_{-1,0}$ $\displaystyle=$
$\displaystyle\frac{\alpha_{p}}{16\pi}$ $\displaystyle I_{1,0}$
$\displaystyle=$
$\displaystyle\frac{1}{8\pi\beta_{p}}\ln\frac{\alpha_{p}+\beta_{p}}{\alpha_{p}-\beta_{p}}$
$\displaystyle I_{0,1}$ $\displaystyle=$
$\displaystyle\frac{1}{8\pi\beta_{e}}\ln\frac{\alpha_{e}+\beta_{e}}{\alpha_{e}-\beta_{e}}$
$\displaystyle I_{2,0}$ $\displaystyle=$ $\displaystyle\frac{1}{2\pi
m^{2}p_{\nu}^{\prime 2}}$ $\displaystyle I_{0,2}$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi m_{e}^{2}p_{\nu}^{\prime 2}}$ $\displaystyle
I_{1,1}$ $\displaystyle=$
$\displaystyle\frac{1}{4\pi\gamma_{pe}p_{\nu}^{\prime 2}}\ln\frac{p\cdot
p_{e}+\gamma_{pe}}{p\cdot p_{e}-\gamma_{pe}}$ $\displaystyle I_{1,-1}$
$\displaystyle=$
$\displaystyle\frac{1}{8\pi}\left(\frac{(pp_{e}:p_{\nu}^{\prime})}{\beta_{p}^{2}}+\frac{p_{\nu}^{\prime
2}(p_{e}p_{\nu}^{\prime}:p)}{2\beta_{p}^{3}}\ln\frac{\alpha_{p}+\beta_{p}}{\alpha_{p}-\beta_{p}}\right)$
$\displaystyle I_{-1,1}$ $\displaystyle=$
$\displaystyle\frac{1}{8\pi}\left(\frac{(p_{e}p:p_{\nu}^{\prime})}{\beta_{e}^{2}}+\frac{p_{\nu}^{\prime
2}(pp_{\nu}^{\prime}:p_{e})}{2\beta_{e}^{3}}\ln\frac{\alpha_{e}+\beta_{e}}{\alpha_{e}-\beta_{e}}\right)$
$\displaystyle I_{2,-1}$ $\displaystyle=$
$\displaystyle\frac{1}{8\pi}\left(\frac{2(p_{e}p_{\nu}^{\prime}:p)}{m^{2}\beta_{p}^{2}}+\frac{(pp_{e}:p_{\nu}^{\prime})}{\beta_{p}^{3}}\ln\frac{\alpha_{p}+\beta_{p}}{\alpha_{p}-\beta_{p}}\right)$
$\displaystyle I_{-1,2}$ $\displaystyle=$
$\displaystyle\frac{1}{8\pi}\left(\frac{2(pp_{\nu}^{\prime}:p_{e})}{m_{e}^{2}\beta_{e}^{2}}+\frac{(p_{e}p:p_{\nu}^{\prime})}{\beta_{e}^{3}}\ln\frac{\alpha_{e}+\beta_{e}}{\alpha_{e}-\beta_{e}}\right)$
$\displaystyle I_{-2,1}$ $\displaystyle=$
$\displaystyle\frac{1}{8\pi}\left[\frac{(p_{\nu}^{\prime
2})^{2}(pp_{\nu}^{\prime}:p_{e})^{2}}{4\beta_{e}^{5}}\ln\frac{\alpha_{e}+\beta_{e}}{\alpha_{e}-\beta_{e}}+\frac{p_{\nu}^{\prime
2}(p_{e}p:p_{\nu}^{\prime})(pp_{\nu}^{\prime}:p_{e})}{\beta_{e}^{4}}+\frac{\alpha_{e}(p_{e}p:p_{\nu}^{\prime})^{2}}{2\beta_{e}^{4}}\right.$
$\displaystyle\left.+\left(\frac{\beta_{e}^{2}\beta_{p}^{2}-(p_{e}p:p_{\nu}^{\prime})^{2}}{4\beta_{e}^{4}}\right)\left(\alpha_{e}-\frac{m_{e}^{2}p_{\nu}^{\prime
2}}{2\beta_{e}}\ln\frac{\alpha_{e}+\beta_{e}}{\alpha_{e}-\beta_{e}}\right)\right]$
$\displaystyle I_{-2,2}$ $\displaystyle=$
$\displaystyle\frac{1}{8\pi}\left[\frac{p_{\nu}^{\prime
2}(pp_{\nu}^{\prime}:p_{e})^{2}}{m_{e}^{2}\beta_{e}^{4}}+\frac{p_{\nu}^{\prime
2}(pp_{\nu}^{\prime}:p_{e})(p_{e}p:p_{\nu}^{\prime})}{\beta_{e}^{5}}\ln\frac{\alpha_{e}+\beta_{e}}{\alpha_{e}-\beta_{e}}+\frac{(p_{e}p:p_{\nu}^{\prime})^{2}}{\beta_{e}^{4}}\right.$
(68)
$\displaystyle\left.-\left(\frac{\beta_{e}^{2}\beta_{p}^{2}-(p_{e}p:p_{\nu}^{\prime})^{2}}{2\beta_{e}^{4}}\right)\left(2-\frac{\alpha_{e}}{\beta_{e}}\ln\frac{\alpha_{e}+\beta_{e}}{\alpha_{e}-\beta_{e}}\right)\right]\leavevmode\nobreak\
.$

Here we have defined:

$\displaystyle\alpha_{p}\equiv p\cdot p_{\nu}^{\prime}\leavevmode\nobreak\
,\leavevmode\nobreak\ \alpha_{e}\equiv p_{e}\cdot
p_{\nu}^{\prime}\leavevmode\nobreak\ ,\leavevmode\nobreak\
\beta_{p}\equiv\sqrt{\alpha_{p}^{2}-m^{2}p_{\nu}^{\prime
2}}\leavevmode\nobreak\ ,\leavevmode\nobreak\
\beta_{e}\equiv\sqrt{\alpha_{e}^{2}-m_{e}^{2}p_{\nu}^{\prime
2}}\leavevmode\nobreak\ ,$ $\displaystyle\gamma_{pe}\equiv\sqrt{(p\cdot
p_{e})^{2}-m^{2}m_{e}^{2}}\leavevmode\nobreak\ ,\leavevmode\nobreak\
(ab:c)\equiv(a\cdot c)(b\cdot c)-c^{2}(a\cdot b)\leavevmode\nobreak\ .$ (69)

Finally, in the $\nu^{\prime}$-formalism we use the following representation
of the dot products:

$\displaystyle p^{2}=m^{2}\leavevmode\nobreak\ ,\leavevmode\nobreak\ p\cdot
p_{e}=mE_{e}\leavevmode\nobreak\ ,\leavevmode\nobreak\ p\cdot
p_{\nu}^{\prime}=mE_{\nu}^{\prime}\leavevmode\nobreak\ ,\leavevmode\nobreak\
p_{e}^{2}=m_{e}^{2}\leavevmode\nobreak\ ,\leavevmode\nobreak\ p_{e}\cdot
p_{\nu}^{\prime}=E_{e}E_{\nu}^{\prime}-|\vec{p}_{e}||\vec{p}_{\nu}^{\prime}|c^{\prime}\leavevmode\nobreak\
,$ $\displaystyle p_{\nu}^{\prime 2}=E_{\nu}^{\prime
2}-|\vec{p}_{\nu}^{\prime}|^{2}\leavevmode\nobreak\ ,$ (70)

whereas, in the recoil formalism we use:

$\displaystyle p^{2}=m^{2}\leavevmode\nobreak\ ,\leavevmode\nobreak\ p\cdot
p_{e}=\frac{m^{2}y}{2}\leavevmode\nobreak\ ,\leavevmode\nobreak\ p\cdot
p_{\nu}^{\prime}=\frac{m^{2}}{2}(2-y-z)\leavevmode\nobreak\
,\leavevmode\nobreak\ p_{e}^{2}=m^{2}r_{e}\leavevmode\nobreak\ ,$
$\displaystyle p_{e}\cdot
p_{\nu}^{\prime}=\frac{m^{2}}{2}(1+r_{f}-r_{e}-x-z)\leavevmode\nobreak\
,\leavevmode\nobreak\ p_{\nu}^{\prime 2}=m^{2}x\leavevmode\nobreak\ .$ (71)

##  Appendix D $p_{\nu}$\- and $k$-integral with the spin vector

In this work we allow the parent nucleus to be polarized. This introduces the
spin vector $s^{\mu}$ which can appear at most linearly in the squared
amplitude. So, in doing the $p_{\nu}$\- and $k$-integrations, we may also
encounter integrals of the form

$\int\frac{d^{3}k}{(2\pi)^{3}2E_{k}}\frac{d^{3}p_{\nu}}{(2\pi)^{3}2E_{\nu}}(2\pi)^{4}\delta^{(4)}(p_{\nu}^{\prime}-p_{\nu}-k)\frac{s\cdot
k}{(p\cdot k)^{i}(p_{e}\cdot k)^{j}}$ (72)

in addition to those in Appendix C. To evaluate it, we first parameterize:

$I_{i,j}^{\mu}\equiv\int\frac{d^{3}k}{(2\pi)^{3}2E_{k}}\frac{d^{3}p_{\nu}}{(2\pi)^{3}2E_{\nu}}(2\pi)^{4}\delta^{(4)}(p_{\nu}^{\prime}-p_{\nu}-k)\frac{k^{\mu}}{(p\cdot
k)^{i}(p_{e}\cdot
k)^{j}}=c_{1}p^{\mu}+c_{2}p_{e}^{\mu}+c_{3}p_{\nu}^{\prime\mu}\leavevmode\nobreak\
,$ (73)

and evaluate the coefficients $c_{1,2,3}$. Contracting Eq.(73) with $p_{\mu}$,
$p_{e\mu}$ and $(p_{\nu}^{\prime})_{\mu}$ gives rise to the following matrix
equation:

$\left(\begin{array}[]{c}c_{1}\\\ c_{2}\\\
c_{3}\end{array}\right)=M^{-1}\left(\begin{array}[]{c}I_{i-1,j}\\\
I_{i,j-1}\\\ \frac{p_{\nu}^{\prime
2}}{2}I_{i,j}\end{array}\right)\leavevmode\nobreak\ ,\leavevmode\nobreak\
M=\left(\begin{array}[]{ccc}m_{N}^{2}&p\cdot p_{e}&p\cdot p_{\nu}^{\prime}\\\
p\cdot p_{e}&m_{e}^{2}&p_{e}\cdot p_{\nu}^{\prime}\\\ p\cdot
p_{\nu}^{\prime}&p_{e}\cdot p_{\nu}^{\prime}&p_{\nu}^{\prime
2}\end{array}\right)\leavevmode\nobreak\ .$ (74)

Plugging this back to Eq.(73) and contracting to $s_{\mu}$ gives:

$\displaystyle s_{\mu}I_{i,j}^{\mu}$ $\displaystyle=$
$\displaystyle\left[(M^{-1})_{21}I_{i-1,j}+(M^{-1})_{22}I_{i,j-1}+\frac{p_{\nu}^{\prime
2}}{2}(M^{-1})_{23}I_{i,j}\right]s\cdot p_{e}$ (75)
$\displaystyle+\left[(M^{-1})_{31}I_{i-1,j}+(M^{-1})_{32}I_{i,j-1}+\frac{p_{\nu}^{\prime
2}}{2}(M^{-1})_{33}I_{i,j}\right]s\cdot p_{\nu}^{\prime}\leavevmode\nobreak\
.$

So, the new integral is expressible in terms of the old integrals $I_{i,j}$.
In fact, this result can be conveniently implemented by making the following
replacement at the squared amplitude:

$\displaystyle s\cdot k$ $\displaystyle\rightarrow$
$\displaystyle\left[(M^{-1})_{21}s\cdot p_{e}+(M^{-1})_{31}s\cdot
p_{\nu}^{\prime}\right]p\cdot k+\left[(M^{-1})_{22}s\cdot
p_{e}+(M^{-1})_{32}s\cdot p_{\nu}^{\prime}\right]p_{e}\cdot k$ (76)
$\displaystyle+\left[(M^{-1})_{23}s\cdot p_{e}+(M^{-1})_{33}s\cdot
p_{\nu}^{\prime}\right]p_{\nu}^{\prime 2}/2\leavevmode\nobreak\ .$

##  Appendix E The coefficients $C_{ij}^{X}$

Here we provide the analytic expressions of the coefficients $C_{ij}^{X}$
defined in Eq.(40).

###  E.1 $C_{ij}^{0}$

The non-zero coefficients are:

$\displaystyle C_{00}^{0}$ $\displaystyle=$
$\displaystyle\frac{16\pi(\lambda^{2}+1)}{E_{e}E_{\nu}^{\prime 2}}$
$\displaystyle C_{10}^{0}$ $\displaystyle=$ $\displaystyle\frac{8\pi
m(4E_{e}-(1+3\lambda^{2})E_{\nu}^{\prime})}{E_{e}E_{\nu}^{\prime 2}}$
$\displaystyle C_{01}^{0}$ $\displaystyle=$
$\displaystyle-\frac{4\pi}{E_{e}E_{\nu}^{\prime
2}}\left[(\lambda^{2}-1)(2|\vec{p}_{e}||\vec{p}_{\nu}^{\prime}|c^{\prime}+|\vec{p}_{\nu}^{\prime}|^{2}-E_{\nu}^{\prime
2}-2m_{e}^{2})+8E_{e}^{2}(\lambda^{2}+1)-2E_{e}E_{\nu}^{\prime}(5\lambda^{2}+3)\right]$
$\displaystyle C_{02}^{0}$ $\displaystyle=$
$\displaystyle\frac{4\pi(\lambda^{2}-1)m_{e}^{2}(|\vec{p}_{\nu}^{\prime}|^{2}-E_{\nu}^{\prime
2})}{E_{e}E_{\nu}^{\prime 2}}$ $\displaystyle C_{11}^{0}$ $\displaystyle=$
$\displaystyle\frac{4\pi(\lambda^{2}-1)m(E_{\nu}^{\prime
2}-|\vec{p}_{\nu}^{\prime}|^{2})}{E_{\nu}^{\prime 2}}$ $\displaystyle
C_{-11}^{0}$ $\displaystyle=$
$\displaystyle-\frac{16\pi(\lambda^{2}+1)(2E_{e}-E_{\nu}^{\prime})}{E_{e}E_{\nu}^{\prime
2}m}$ $\displaystyle C_{2-1}^{0}$ $\displaystyle=$
$\displaystyle\frac{8\pi(\lambda^{2}-1)m^{2}}{E_{e}E_{\nu}^{\prime 2}}$
$\displaystyle C_{-12}^{0}$ $\displaystyle=$
$\displaystyle\frac{16\pi(\lambda^{2}+1)m_{e}^{2}(E_{e}-E_{\nu}^{\prime})}{E_{e}E_{\nu}^{\prime
2}m}$ $\displaystyle C_{-21}^{0}$ $\displaystyle=$
$\displaystyle-\frac{16\pi(\lambda^{2}+1)}{E_{e}E_{\nu}^{\prime 2}m^{2}}$
$\displaystyle C_{-22}^{0}$ $\displaystyle=$
$\displaystyle\frac{16\pi(\lambda^{2}+1)m_{e}^{2}}{E_{e}E_{\nu}^{\prime
2}m^{2}}$ (77)

It is worth noticing that $C_{ij}^{0}$ is independent of $\zeta$, i.e. they
are the same for $\beta^{\pm}$ decay. This is not the case for the spin-
dependent coefficients $C_{ij}^{\beta}$ and $C_{ij}^{\nu}$.

###  E.2 $C_{ij}^{\beta}$

The non-zero coefficients are:

$\displaystyle C_{00}^{\beta}$ $\displaystyle=$
$\displaystyle-\frac{16\pi\lambda\left[|\vec{p}_{\nu}^{\prime}|(-2(\lambda\zeta+2)E_{e}+(-\lambda\zeta+1)E_{\nu}^{\prime})+|\vec{p}_{e}|c^{\prime}E_{\nu}^{\prime}(\lambda\zeta+1)\right]}{|\vec{p}_{e}|^{2}|\vec{p}_{\nu}^{\prime}|(c^{\prime
2}-1)E_{\nu}^{\prime 2}}$ $\displaystyle C_{-10}^{\beta}$ $\displaystyle=$
$\displaystyle\frac{32\pi\lambda}{|\vec{p}_{e}|^{2}(c^{\prime
2}-1)E_{\nu}^{\prime 2}m}$ $\displaystyle C_{10}^{\beta}$ $\displaystyle=$
$\displaystyle\frac{8\pi\lambda
m}{|\vec{p}_{e}|^{2}|\vec{p}_{\nu}^{\prime}|(c^{\prime 2}-1)E_{\nu}^{\prime
2}}\left[2|\vec{p}_{e}|^{2}|\vec{p}_{\nu}^{\prime}|c^{\prime
2}(\lambda\zeta-1)-|\vec{p}_{e}|c^{\prime}(\lambda\zeta+1)(|\vec{p}_{\nu}^{\prime}|^{2}+E_{\nu}^{\prime}(2E_{e}-E_{\nu}^{\prime}))\right.$
$\displaystyle\left.+2|\vec{p}_{\nu}^{\prime}|(2E_{e}^{2}(\lambda\zeta+2)+(E_{e}E_{\nu}^{\prime}+m_{e}^{2})(\lambda\zeta-1))\right]$
$\displaystyle C_{01}^{\beta}$ $\displaystyle=$
$\displaystyle\frac{-8\pi\lambda}{|\vec{p}_{e}|^{2}|\vec{p}_{\nu}^{\prime}|(c^{\prime
2}-1)E_{\nu}^{\prime
2}}\left[2|\vec{p}_{e}|^{2}|\vec{p}_{\nu}^{\prime}|c^{\prime
2}(\lambda\zeta-1)(2E_{e}-E_{\nu}^{\prime})-|\vec{p}_{e}||\vec{p}_{\nu}^{\prime}|^{2}c^{\prime}(E_{e}(\lambda\zeta+3)+E_{\nu}^{\prime}(\lambda\zeta-1))\right.$
$\displaystyle-|\vec{p}_{e}|E_{\nu}^{\prime}c^{\prime}(4E_{e}^{2}(\lambda\zeta+1)+E_{e}E_{\nu}^{\prime}(\lambda\zeta-5)-E_{\nu}^{\prime
2}(\lambda\zeta-1))$
$\displaystyle\left.+2|\vec{p}_{\nu}^{\prime}|E_{e}(4E_{e}^{2}+2E_{e}E_{\nu}^{\prime}(\lambda\zeta-1)+(3\lambda\zeta-1)m_{e}^{2})\right]$
$\displaystyle C_{20}^{\beta}$ $\displaystyle=$ $\displaystyle\frac{-8\pi
c^{\prime}E_{e}(\lambda\zeta+1)\lambda
m^{2}(|\vec{p}_{\nu}^{\prime}|^{2}-E_{\nu}^{\prime
2})}{|\vec{p}_{e}||\vec{p}_{\nu}^{\prime}|(c^{\prime 2}-1)E_{\nu}^{\prime 2}}$
$\displaystyle C_{02}^{\beta}$ $\displaystyle=$ $\displaystyle\frac{-8\pi
c^{\prime}\lambda m_{e}^{2}(|\vec{p}_{\nu}^{\prime}|^{2}-E_{\nu}^{\prime
2})(E_{e}(\lambda\zeta+1)+E_{\nu}^{\prime}(\lambda\zeta-1))}{|\vec{p}_{e}||\vec{p}_{\nu}^{\prime}|(c^{\prime
2}-1)E_{\nu}^{\prime 2}}$ $\displaystyle C_{11}^{\beta}$ $\displaystyle=$
$\displaystyle\frac{8\pi c^{\prime}E_{e}\lambda
m(|\vec{p}_{\nu}^{\prime}|^{2}-E_{\nu}^{\prime
2})(2E_{e}(\lambda\zeta+1)+E_{\nu}^{\prime}(\lambda\zeta-1))}{|\vec{p}_{e}||\vec{p}_{\nu}^{\prime}|(c^{\prime
2}-1)E_{\nu}^{\prime 2}}$ $\displaystyle C_{1-1}^{\beta}$ $\displaystyle=$
$\displaystyle-\frac{16\pi(\lambda\zeta+1)\lambda
m}{|\vec{p}_{e}|^{2}(c^{\prime 2}-1)E_{\nu}^{\prime 2}}$ $\displaystyle
C_{-11}^{\beta}$ $\displaystyle=$
$\displaystyle\frac{-16\pi\lambda}{|\vec{p}_{e}|^{2}|\vec{p}_{\nu}^{\prime}|(c^{\prime
2}-1)E_{\nu}^{\prime
2}m}\left[|\vec{p}_{e}|^{2}|\vec{p}_{\nu}^{\prime}|c^{\prime
2}(\lambda\zeta-1)-|\vec{p}_{e}||\vec{p}_{\nu}^{\prime}|^{2}c^{\prime}\right.$
$\displaystyle\left.-|\vec{p}_{e}|E_{\nu}^{\prime}c^{\prime}(E_{e}(\lambda\zeta+3)+E_{\nu}^{\prime}(\lambda\zeta-2))+|\vec{p}_{\nu}^{\prime}|(4E_{e}^{2}+E_{e}E_{\nu}^{\prime}(\lambda\zeta-1)+(\lambda\zeta+1)m_{e}^{2})\right]$
$\displaystyle C_{2-1}^{\beta}$ $\displaystyle=$ $\displaystyle-\frac{16\pi
E_{e}(\lambda\zeta+1)\lambda m^{2}}{|\vec{p}_{e}|^{2}(c^{\prime
2}-1)E_{\nu}^{\prime 2}}$ $\displaystyle C_{-12}^{\beta}$ $\displaystyle=$
$\displaystyle\frac{16\pi\lambda
m_{e}^{2}}{|\vec{p}_{e}|^{2}|\vec{p}_{\nu}^{\prime}|(c^{\prime
2}-1)E_{\nu}^{\prime
2}m}\left[|\vec{p}_{e}|^{2}|\vec{p}_{\nu}^{\prime}|c^{\prime
2}(\lambda\zeta-1)-|\vec{p}_{e}||\vec{p}_{\nu}^{\prime}|^{2}c^{\prime}\right.$
$\displaystyle\left.-|\vec{p}_{e}|E_{\nu}^{\prime}c^{\prime}(E_{e}(\lambda\zeta+1)+E_{\nu}^{\prime}(\lambda\zeta-2))+|\vec{p}_{\nu}^{\prime}|(2E_{e}^{2}+(E_{e}E_{\nu}^{\prime}+m_{e}^{2})(\lambda\zeta-1))\right]$
$\displaystyle C_{-21}^{\beta}$ $\displaystyle=$
$\displaystyle\frac{32\pi\lambda(|\vec{p}_{e}|E_{\nu}^{\prime}c^{\prime}-|\vec{p}_{\nu}^{\prime}|E_{e})}{|\vec{p}_{e}|^{2}|\vec{p}_{\nu}^{\prime}|(c^{\prime
2}-1)E_{\nu}^{\prime 2}m^{2}}$ $\displaystyle C_{-22}^{\beta}$
$\displaystyle=$ $\displaystyle\frac{32\pi\lambda
m_{e}^{2}(|\vec{p}_{\nu}^{\prime}|E_{e}-|\vec{p}_{e}|E_{\nu}^{\prime}c^{\prime})}{|\vec{p}_{e}|^{2}|\vec{p}_{\nu}^{\prime}|(c^{\prime
2}-1)E_{\nu}^{\prime 2}m^{2}}$ (78)

###  E.3 $C_{ij}^{\nu}$

The non-zero coefficients are:

$\displaystyle C_{00}^{\nu}$ $\displaystyle=$
$\displaystyle-\frac{16\pi\lambda\left[|\vec{p}_{\nu}^{\prime}|c^{\prime}(2(\lambda\zeta+2)E_{e}+(\lambda\zeta-1)E_{\nu}^{\prime})-|\vec{p}_{e}|E_{\nu}^{\prime}(\lambda\zeta+1)\right]}{|\vec{p}_{e}||\vec{p}_{\nu}^{\prime}|(c^{\prime
2}-1)E_{e}E_{\nu}^{\prime 2}}$ $\displaystyle C_{-10}^{\nu}$ $\displaystyle=$
$\displaystyle-\frac{32\pi\lambda c^{\prime}}{|\vec{p}_{e}|(c^{\prime
2}-1)E_{e}E_{\nu}^{\prime 2}m}$ $\displaystyle C_{10}^{\nu}$ $\displaystyle=$
$\displaystyle\frac{8\pi\lambda
m}{|\vec{p}_{e}||\vec{p}_{\nu}^{\prime}|(c^{\prime 2}-1)E_{e}E_{\nu}^{\prime
2}}\left[|\vec{p}_{e}|(\lambda\zeta+1)(|\vec{p}_{\nu}^{\prime}|^{2}(2c^{\prime
2}-1)+E_{\nu}^{\prime}(2E_{e}-E_{\nu}^{\prime}))\right.$
$\displaystyle\left.-2|\vec{p}_{\nu}^{\prime}|E_{e}c^{\prime}(3E_{e}(\lambda\zeta+1)+E_{\nu}^{\prime}(\lambda\zeta-1))\right]$
$\displaystyle C_{01}^{\nu}$ $\displaystyle=$
$\displaystyle\frac{-8\pi\lambda}{|\vec{p}_{e}||\vec{p}_{\nu}^{\prime}|(c^{\prime
2}-1)E_{e}E_{\nu}^{\prime
2}}\left[|\vec{p}_{e}||\vec{p}_{\nu}^{\prime}|^{2}(E_{e}(4c^{\prime
2}(\lambda\zeta+1)-3\lambda\zeta-1)+E_{\nu}^{\prime}(\lambda\zeta-1))\right.$
$\displaystyle+|\vec{p}_{e}|E_{\nu}^{\prime}(4E_{e}^{2}(\lambda\zeta+1)+E_{e}E_{\nu}^{\prime}(\lambda\zeta-5)-E_{\nu}^{\prime
2}(\lambda\zeta-1))$
$\displaystyle\left.-2|\vec{p}_{\nu}^{\prime}|c^{\prime}(E_{e}(2E_{e}^{2}+m_{e}^{2})(\lambda\zeta+1)+E_{\nu}^{\prime}(E_{e}^{2}+m_{e}^{2})(\lambda\zeta-1))\right]$
$\displaystyle C_{20}^{\nu}$ $\displaystyle=$
$\displaystyle\frac{8\pi(\lambda\zeta+1)\lambda
m^{2}(|\vec{p}_{\nu}^{\prime}|^{2}-E_{\nu}^{\prime
2})}{|\vec{p}_{\nu}^{\prime}|(c^{\prime 2}-1)E_{\nu}^{\prime 2}}$
$\displaystyle C_{02}^{\nu}$ $\displaystyle=$ $\displaystyle\frac{8\pi\lambda
m_{e}^{2}(|\vec{p}_{\nu}^{\prime}|^{2}-E_{\nu}^{\prime
2})(E_{e}(\lambda\zeta+1)+E_{\nu}^{\prime}(\lambda\zeta-1))}{|\vec{p}_{\nu}^{\prime}|(c^{\prime
2}-1)E_{e}E_{\nu}^{\prime 2}}$ $\displaystyle C_{11}^{\nu}$ $\displaystyle=$
$\displaystyle-\frac{8\pi\lambda
m(|\vec{p}_{\nu}^{\prime}|^{2}-E_{\nu}^{\prime
2})(2E_{e}(\lambda\zeta+1)+E_{\nu}^{\prime}(\lambda\zeta-1))}{|\vec{p}_{\nu}^{\prime}|(c^{\prime
2}-1)E_{\nu}^{\prime 2}}$ $\displaystyle C_{1-1}^{\nu}$ $\displaystyle=$
$\displaystyle\frac{16\pi c^{\prime}(\lambda\zeta+1)\lambda
m}{|\vec{p}_{e}|(c^{\prime 2}-1)E_{e}E_{\nu}^{\prime 2}}$ $\displaystyle
C_{-11}^{\nu}$ $\displaystyle=$
$\displaystyle\frac{-16\pi\lambda}{|\vec{p}_{e}||\vec{p}_{\nu}^{\prime}|(c^{\prime
2}-1)E_{e}E_{\nu}^{\prime
2}m}\left[|\vec{p}_{e}||\vec{p}_{\nu}^{\prime}|^{2}(c^{\prime
2}(\lambda\zeta+1)-\lambda\zeta)+|\vec{p}_{e}|E_{\nu}^{\prime}(E_{e}(\lambda\zeta+3)+E_{\nu}^{\prime}(\lambda\zeta-2))\right.$
$\displaystyle\left.-|\vec{p}_{\nu}^{\prime}|c^{\prime}(E_{e}^{2}(\lambda\zeta+3)+E_{e}E_{\nu}^{\prime}(\lambda\zeta-1)+2m_{e}^{2})\right]$
$\displaystyle C_{2-1}^{\nu}$ $\displaystyle=$ $\displaystyle\frac{16\pi
c^{\prime}(\lambda\zeta+1)\lambda m^{2}}{|\vec{p}_{e}|(c^{\prime
2}-1)E_{\nu}^{\prime 2}}$ $\displaystyle C_{-12}^{\nu}$ $\displaystyle=$
$\displaystyle\frac{16\pi\lambda
m_{e}^{2}}{|\vec{p}_{e}||\vec{p}_{\nu}^{\prime}|(c^{\prime
2}-1)E_{e}E_{\nu}^{\prime
2}m}\left[|\vec{p}_{e}||\vec{p}_{\nu}^{\prime}|^{2}(c^{\prime
2}(\lambda\zeta+1)-\lambda\zeta)+|\vec{p}_{e}|E_{\nu}^{\prime}(E_{e}(\lambda\zeta+1)+E_{\nu}^{\prime}(\lambda\zeta-2))\right.$
$\displaystyle\left.-|\vec{p}_{\nu}^{\prime}|c^{\prime}E_{e}(E_{e}(\lambda\zeta+1)+E_{\nu}^{\prime}(\lambda\zeta-1))\right]$
$\displaystyle C_{-21}^{\nu}$ $\displaystyle=$
$\displaystyle\frac{32\pi\lambda(|\vec{p}_{\nu}^{\prime}|E_{e}c^{\prime}-|\vec{p}_{e}|E_{\nu}^{\prime})}{|\vec{p}_{e}||\vec{p}_{\nu}^{\prime}|(c^{\prime
2}-1)E_{e}E_{\nu}^{\prime 2}m^{2}}$ $\displaystyle C_{-22}^{\nu}$
$\displaystyle=$ $\displaystyle\frac{32\pi\lambda
m_{e}^{2}(|\vec{p}_{e}|E_{\nu}^{\prime}-|\vec{p}_{\nu}^{\prime}|E_{e}c^{\prime})}{|\vec{p}_{e}||\vec{p}_{\nu}^{\prime}|(c^{\prime
2}-1)E_{e}E_{\nu}^{\prime 2}m^{2}}$ (79)

##  Appendix F The coefficients $d_{ij}$

Here we provide the analytic expressions of the non-zero coefficients $d_{ij}$
defined in Eq.(54) in the recoil formalism.

$\displaystyle d_{00}$ $\displaystyle=$ $\displaystyle 32\pi(\lambda^{2}+1)$
$\displaystyle d_{10}$ $\displaystyle=$ $\displaystyle 8\pi
m^{2}\left[(3\lambda^{2}+5)y+(3\lambda^{2}+1)(z-2)\right]$ $\displaystyle
d_{01}$ $\displaystyle=$ $\displaystyle 8\pi
m^{2}\left[(\lambda^{2}-1)(r_{e}+r_{f}-z+1)-(\lambda^{2}+1)(4y^{2}+2yz-4y)\right]$
$\displaystyle d_{02}$ $\displaystyle=$ $\displaystyle-8\pi
m^{4}(\lambda^{2}-1)r_{e}x$ $\displaystyle d_{11}$ $\displaystyle=$
$\displaystyle 4\pi m^{4}(\lambda^{2}-1)xy$ $\displaystyle d_{-11}$
$\displaystyle=$ $\displaystyle-16\pi(\lambda^{2}+1)(3y+z-2)$ $\displaystyle
d_{2-1}$ $\displaystyle=$ $\displaystyle 16\pi m^{2}(\lambda^{2}-1)$
$\displaystyle d_{-12}$ $\displaystyle=$ $\displaystyle 16\pi
m^{2}(\lambda^{2}+1)r_{e}(2y+z-2)$ $\displaystyle d_{-21}$ $\displaystyle=$
$\displaystyle-\frac{32\pi}{m^{2}}(\lambda^{2}+1)$ $\displaystyle d_{-22}$
$\displaystyle=$ $\displaystyle 32\pi(\lambda^{2}+1)r_{e}$ (80)

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