Large-scale structure power spectrum from scalar-tensor gravity

Joseph Ntahompagaze1, Amare Abebe2 and Manasse R. Mbonye1,3,4,5

1 Department of Physics, University of Rwanda, Kigali, Rwanda
2 Center for Space Research, North-West University, Mahikeng, South Africa
3 ICTP-East Africa Institute for Fundamental Research, Kigali, Rwanda
4 Rwanda Academy of Science, Kigali, Rwanda
5 Rochester Institute of Technology, NY, USA

Correspondence<EMAIL_ADDRESS>
Abstract

This work deals with the computation of the power spectrum of large-scale
structure using the dynamical system approach for a multi-fluid universe in
scalar-tensor theory of gravity. We use the $1+3$ covariant approach to obtain
evolution equations and study the behavior of the matter power spectrum of
perturbation equations. The study is based on the equivalence between $f(R)$
theory of gravity and scalar-tensor theory of gravity. We find that, for
power-law $(R^{n})$ models, with $1<n<1.3$, we have the power spectrum
evolving above general relativistic scale-invariant line. For $n\geq 1.3$, the
power spectrum starts with constant amplitude then it experiences oscillations
and eventually saturates at finite amplitude. Such behavior is consistent with
other observations in the literature. The result supports the ongoing
investigations of the equivalence between $f(R)$ and scalar-tensor theory at
linear order.

keywords: $f(R)$ gravity — scalar-tensor — scalar field —cosmology —covariant
perturbation
PACS numbers: 04.50.Kd, 98.80.-k, 95.36.+x, 98.80.Cq; MSC numbers: 83Dxx,
83Fxx

##  1 Introduction

In studies of large scale structure based on General Relativity (GR) the
matter power spectrum has been shown to be scale invariant [1, 2]. The
modification of GR leads to different theories of gravity. In studying matter
power spectrum in such theories one seeks to know whether, or not, among other
observational constraints, the behavior of that power spectrum would be scale
invariant. Several studies have been done in the context of analyzing the
behavior of the matter power spectrum in $f(R)$ models. In [3], the authors
considered linear perturbations in two different types $f(R)$ models: power
law and exponential law. They pointed out that amplitude of the matter power
spectrum resulting from power law $f(R)$ model is bigger than that of
exponential law $f(R)$ models. The dependence of energy density perturbations
are scale-dependence for both models. In [4], the power spectrum is computed
using N-body simulation method for $f(R)$ models proposed in [5] and the
scale-dependence of the power spectrum has been shown. Several $f(R)$ models
have been treated in [6], where power spectrum was obtained through the use of
the $1+3$ covariant approach to establish the matter energy density dependence
on wave-mode $k$. In [7], power spectrum computations were used to distinguish
between two models, GR and a re-constructed $f(R)$ model. In the present
paper, based on the work done in [8], we relate scalar field and $f(R)$ theory
to study power-spectrum in scalar-tensor theory and use the results to develop
the perturbation equations.

In studying the matter power spectrum of the energy density perturbations, one
can introduce dynamical system techniques [9] in the analysis of cosmic
evolution equations. The working scheme can be summarised as follows. One
defines dimensionless variables rendering the field equations dimensionless
[9, 10, 11]. The result can be used in perturbation equations to construct the
matter power spectrum [11].

We use the above scheme to compute matter power spectrum in scalar-tensor
theories of gravity. The process involves performing harmonic decomposition of
perturbation equations to obtain the wave modes ($k$). Then the dependence on
$k$ of energy density gradient variable is used to obtain the transfer
function which can be related to matter power spectrum. The matter power
spectrum is computed at a fixed redshift, say today.

This paper is organised as follows. In the next section, we provide and
defined dimensionless variables and their evolution equations. In Section 3,
we provide perturbation equations. Section 4 is about computation of power
spectrum after making perturbation equations dimensionless. In Sections 5 and
6, we provide discussions and conclusions respectively. The adopted spacetime
signature is (-+++) and unless stated otherwise, we have used the convention
$8\pi G=c=1$, where $G$ is the gravitational constant and $c$ is the speed of
light.

##  2 The Theoretical Framework

The action for scalar-tensor theory (a Brans-Dicke sub-class) is given as [12,
13]

$I_{f(\phi)}=\frac{1}{2}\int
d^{4}x\sqrt{-g}\left[f(\phi)+\mathcal{L}_{m}\right]\;,$ (1)

where $R$ is the Ricci scalar and $\mathcal{L}_{m}$ is the matter Lagrangian.
The action that represents $f(R)$ gravity given as

$I=\frac{1}{2}\int d^{4}x\sqrt{-g}\left[f(R)+\mathcal{L}_{m}\right]\;.$ (2)

We can connect the two actions by defining the scalar field $\phi$ to be
function of Ricci-scalar as [8, 12]

$\phi=f^{\prime}-1\;.$ (3)

Here, the scalar field $\phi$ should be invertible [13, 14]. One can see that
for GR case, this scalar field vanishes. For $f(R)$ gravity, we have the
curvature fluid thermodynamic quantities for an FRLW universe as background
for zero-order are here given as

$\displaystyle\mu_{R}=\frac{1}{f^{\prime}}\left[\frac{1}{2}(Rf^{\prime}-f)-\Theta
f^{\prime\prime}\dot{R}\right]\;,$ (4) $\displaystyle
p_{R}=\frac{1}{f^{\prime}}\left[\frac{1}{2}(f-Rf^{\prime})+f^{\prime\prime}\ddot{R}+f^{\prime\prime\prime}\dot{R}^{2}+\frac{2}{3}\Theta
f^{\prime\prime}\dot{R}\right]\;,$ (5)

where $\mu_{R}$ is the curvature energy density and $p_{R}$ is the isotropic
pressure of the curvature fluid, here $\theta$ is the volume expansion
parameter. The trace equation leads to

$\frac{\mu_{m}}{f^{\prime}}-\frac{3p_{m}}{f^{\prime}}+R-\frac{2f}{f^{\prime}}-\frac{9Hf^{\prime\prime}\dot{R}}{f^{\prime}}-\frac{3f^{\prime\prime}\ddot{R}}{f^{\prime}}-\frac{3f^{\prime\prime\prime}\dot{R}^{2}}{f^{\prime}}=0\;,$
(6)

where $\mu_{m}$ is the matter energy density and $p_{m}$ is isotropic pressure
of matter fluid. The Ricci scalar is given as [11]

$R=6H^{2}\Big{(}\frac{\dot{H}}{H^{2}}+2\Big{)}\;.$ (7)

The Friedmann equation is given as

$H^{2}=\frac{R}{6}-\frac{f}{6f^{\prime}}-\frac{\dot{R}Hf^{\prime\prime}}{f^{\prime}}+\frac{\mu_{m}}{3f^{\prime}}-\frac{k}{a^{2}}\;,$
(8)

where we have used the fact that $H=\frac{1}{3}\Theta$. The Raychaudhuri
equation is given as

$\dot{H}=-H^{2}-\frac{1}{6f^{\prime}}\left(2\mu_{m}-f-6\dot{R}Hf^{\prime\prime}\right)\;.$
(9)

###  2.1 The dynamical system of Radiation-dust mixture

We can rewrite the Friedmann equation $(\ref{dynamicall0})$ as

$1-\frac{R}{6H^{2}}+\frac{f}{6f^{\prime}H^{2}}+\frac{\dot{R}f^{\prime\prime}}{f^{\prime}H}-\frac{\mu_{r}}{3f^{\prime}H^{2}}-\frac{\mu_{d}}{3f^{\prime}H^{2}}=0\;,$
(10)

where we have considered flat universe and the matter energy density $\mu_{m}$
has been decomposed into its components of dust energy density $\mu_{d}$ and
radiation energy density $\mu_{r}$. We define dimensionless parameters [9, 11]

$x\equiv\frac{\dot{R}f^{\prime\prime}}{f^{\prime}H}\;,\quad
y\equiv\frac{f}{6f^{\prime}H^{2}}-\frac{R}{6H^{2}}\;,\quad\Omega_{r}\equiv\frac{\mu_{r}}{3f^{\prime}H^{2}}\;,\quad\Omega_{d}\equiv\frac{\mu_{d}}{3f^{\prime}H^{2}}\;,$
(11)

so that the Friedmann equation can be re-expressed in dimensionless parameters
as

$1+y+x-\Omega_{d}-\Omega_{r}=0\;.$ (12)

One of the simplest and well studied models of $f(R)$ gravity theory is the
power law model [11]

$f(R)=\beta H^{2}_{0}\Big{(}\frac{R}{H^{2}_{0}}\Big{)}^{n}\;.$ (13)

In this model, the the dimensionless variables can be reduced as [9]:

$x=\frac{\dot{R}(n-1)}{RH}\;,\quad
y=\frac{R(1-n)}{6nH^{2}}\;,\quad\Omega_{r}=\frac{\mu_{r}}{3n\beta
H^{2}\Big{(}\frac{R}{H^{2}_{0}}\Big{)}^{n-1}}\;,\quad\Omega_{d}=\frac{\mu_{d}}{3n\beta
H^{2}\Big{(}\frac{R}{H^{2}_{0}}\Big{)}^{n-1}}\;.$ (14)

From equation $(\ref{dynamicall1})$, we have

$\frac{\dot{H}}{H^{2}}=-1-(\Omega_{r}+\Omega_{d})-\frac{y}{n-1}+x.$ (15)

Let us now define a logarithmic time variable as

$N=\ln(a)\;,$ (16)

with $a$ being the scale factor, such that the evolution of the dimensionless
variables $x,y,\Omega_{d}$ and $\Omega_{r}$ in Eq. (14) can be given as [9]

$\displaystyle\frac{dx}{dN}=-x-x^{2}+\frac{(4-2n+nx)y}{n-1}+\Omega_{d}\;,$
(17) $\displaystyle\frac{dy}{dN}=4y+\frac{(x+2ny)y}{n-1}\;,$ (18)
$\displaystyle\frac{d\Omega_{d}}{dN}=\Big{(}1-x+\frac{2ny}{n-1}\Big{)}\Omega_{d}\;,$
(19)
$\displaystyle\frac{d\Omega_{r}}{dN}=\Big{(}-x+\frac{2ny}{n-1}\Big{)}\Omega_{r}\;.$
(20)

The stability of the equilibrium points of the above system has been studied
in details in [9]. We can write these evolution equations as functions of
redshift $z$. To this end, we start from the relationship between the
cosmological scale factor $a$ and cosmological redshift $z$ given as [15, 16]
$a=\frac{1}{1+z}$, so that for any quantity $X(z)$, we can write
$\frac{dX}{dN}=-(1+z)\frac{dX}{dz}$. We therefore have equations
$(\ref{xN}),(\ref{yN})$, $(\ref{OmegarN})$ and $(\ref{OmegadN})$ given in
terms of redshift $z$ as [11]

$\displaystyle-(1+z)\frac{dx}{dz}=-x-x^{2}+\frac{(4-2n+nx)y}{n-1}+\Omega_{d}\;,$
(21) $\displaystyle-(1+z)\frac{dy}{dz}=4y+\frac{(x+2ny)y}{n-1}\;,$ (22)
$\displaystyle-(1+z)\frac{d\Omega_{d}}{dz}=\Big{(}1-x+\frac{2ny}{n-1}\Big{)}\Omega_{d}\;,$
(23)
$\displaystyle-(1+z)\frac{d\Omega_{r}}{dz}=\Big{(}-x+\frac{2ny}{n-1}\Big{)}\Omega_{r}\;.$
(24)

If one defines $h=\frac{H}{H_{0}}$ [11], then its evolution given as

$(1+z)\frac{dh}{dz}=\frac{h(2+ny)}{(n-1)}\;.$ (25)

##  3 Perturbation equations

In the language of the scalar-tensor theory, we write linear quantities of
curvature energy density and pressure as [17]

$\displaystyle\mu_{\phi}=\frac{1}{\phi+1}\left[\frac{1}{2}\Big{(}(\phi+1)R-f\Big{)}-\Theta\dot{\phi}-\phi^{\prime}\tilde{\nabla}^{2}R\right]\;,$
(26) $\displaystyle
p_{\phi}=\frac{1}{\phi+1}\left[\frac{1}{2}\Big{(}(f-R(\phi+1)\Big{)}+\ddot{\phi}-\frac{\dot{\phi}\dot{\phi}^{\prime}}{\phi^{\prime}}\phi^{\prime\prime}+\frac{\phi^{\prime\prime}\dot{\phi}^{2}}{\phi^{\prime
2}}+\frac{2}{3}\Theta\dot{\phi}-\frac{2\phi^{\prime}\tilde{\nabla}^{2}R}{3}\right]\;.$
(27)

We follow the $1+3$ covariant approach to define gradient variables to be used
to build perturbation equations. This approach allows decomposition of the
four-dimensional cosmological manifold into a three dimensional sub-manifold
perpendicular to a timelike vector field $u^{a}$. The derivatives (temporal
and spatial) follow [18, 19, 20, 21]. The scalar components of the
perturbations are believed to be the ones responsible for most of the large-
scale structure formation. We thus extract the scalar parts from the vector
gradient quantities using the local decomposition for a vector $X_{a}$ as [21,
22]

$a\tilde{\nabla}_{b}X_{a}=X_{ab}=\frac{1}{3}h_{ab}X+\Sigma^{X}_{ab}+X_{[ab]}\;,$
(28)

where $\Sigma^{X}_{ab}=X_{(ab)}-\frac{1}{3}h_{ab}X$ describes shear and
$X_{[ab]}$ describes vorticity. We therefore define the gradient variables as

$\Delta_{m}=\frac{a^{2}}{\mu_{m}}\tilde{\nabla}^{2}\mu_{m}\;,Z=a^{2}\tilde{\nabla}^{2}\Theta\;,\Phi=a^{2}\tilde{\nabla}^{2}\phi\;,\text{
and }\Psi=a^{2}\tilde{\nabla}^{2}\dot{\phi}\;.$ (29)

We use harmonic decomposition method to transform covariant Laplace-Beltrami
operator [19, 20, 21]. For a given quantity $X$, we write

$X=\sum_{k}X^{k}(t)Q_{k}(\vec{x}),$ (30)

where $Q_{k}$ are the eigenfunctions of the covariant Laplace-Beltrami
operator such that

$\tilde{\nabla}^{2}Q=-\frac{k^{2}}{a^{2}}Q\;,$ (31)

and the order of the scale factor dependent harmonic (wavenumber) $k$ is

$k=\frac{2\pi a}{\lambda}\;,$ (32)

where $\lambda$ is the physical wavelength of the mode and a is the scale
factor. The evolution of the above gradient variables leads to perturbation
equations given as

$\displaystyle\ddot{\Delta}^{k}_{m}$ $\displaystyle-$
$\displaystyle\Big{[}\Big{(}\frac{\dot{\phi}}{\phi+1}-\frac{2\Theta}{3}\Big{)}+w\Theta\Big{]}\dot{\Delta}^{k}_{m}-\Big{[}-w\Theta\Big{(}\frac{\dot{\phi}}{\phi+1}-\frac{2\Theta}{3}\Big{)}+\frac{w+1}{(\phi+1)}\mu_{m}$
(33) $\displaystyle-$ $\displaystyle
w\Big{(}\mu_{m}+\frac{1}{3}\Theta^{2}-\frac{f}{2(\phi+1)}-\frac{k^{2}\phi^{\prime}R}{a^{2}(\phi+1)}\Big{)}+w\dot{\Theta}-\frac{wk^{2}}{a^{2}}\Big{]}\Delta^{k}_{m}+\frac{(1+w)\Theta}{\phi+1}\dot{\Phi}^{k}$
$\displaystyle+$
$\displaystyle(1+w)\Big{[}\frac{1}{2\phi^{\prime}}+\frac{2\mu_{m}-f-2\Theta\dot{\phi}}{2(\phi+1)^{2}}+\frac{k^{2}\phi^{\prime\prime}R}{a^{2}\phi^{\prime}(\phi+1)}-\frac{k^{2}\phi^{\prime}R}{a^{2}(\phi+1)^{2}}+\frac{k^{2}}{a^{2}(\phi+1)}\Big{]}\Phi^{k}=0\;,\;\;\;\;\;\;\;$
$\displaystyle\ddot{\Phi}^{k}$ $\displaystyle-$
$\displaystyle\Big{[}\frac{\ddot{\phi}\phi^{\prime\prime}}{\phi^{\prime
2}}-\frac{2\dot{\phi}\dot{\phi}^{\prime}\phi^{\prime\prime}}{\phi^{\prime
3}}-\frac{(\phi+1)}{3\phi^{\prime}}+\frac{R}{3}-\frac{R(\phi+1)\phi^{\prime\prime}}{3\phi^{\prime
2}}-\frac{(1-3w)\phi^{\prime\prime}\mu_{m}}{3\phi^{\prime
2}}+\frac{2f\phi^{\prime\prime}}{3\phi^{\prime
2}}-\frac{\dot{R}^{2}\phi^{\prime\prime\prime}}{\phi^{\prime}}\;\;$ (34)
$\displaystyle+$
$\displaystyle\frac{\Theta\phi^{\prime\prime}\dot{\phi}}{\phi^{\prime
2}}+\frac{3\dot{\phi}^{2}\phi^{\prime\prime 2}}{\phi^{\prime
4}}+\frac{\dot{\phi}\dot{\phi}^{\prime\prime}}{\phi^{\prime
2}}-\frac{k^{2}}{a^{2}}\Big{]}\Phi^{k}-\Big{(}\frac{\dot{\phi}^{\prime}}{\phi^{\prime}}-\Theta-\frac{2\dot{\phi}\phi^{\prime\prime}}{\phi^{\prime
2}}\Big{)}\dot{\Phi}^{k}-\Big{[}\frac{w\dot{\phi}}{w+1}\Big{(}\frac{\dot{\phi}^{\prime}}{\phi^{\prime}}-\frac{2\dot{\phi}\phi^{\prime\prime}}{\phi^{\prime
2}}\Big{)}$ $\displaystyle-$
$\displaystyle\frac{2w\ddot{\phi}}{(w+1)}+\frac{(1-3w)\mu_{m}}{3}-\frac{2w\Theta\dot{\phi}}{1+w}\Big{]}\Delta^{k}_{m}-\Big{(}\frac{\dot{\phi}}{1+w}-\frac{w\dot{\phi}}{(w+1)}\Big{)}\dot{\Delta}^{k}_{m}=0\;.$

Note that we have instituted the harmonic decomposition along the way. As most
of the large-scale structures are believe to have formed during the epoch of
matter domination, this epoch rules mainly after decoupling. We therefore
consider the dust case with $w=0$ so that we write the above equations as

$\displaystyle\ddot{\Delta}^{k}_{d}$ $\displaystyle-$
$\displaystyle\Big{(}\frac{\dot{\phi}}{\phi+1}-\frac{2\Theta}{3}\Big{)}\dot{\Delta}^{k}_{d}-\frac{\mu_{d}}{(\phi+1)}\Delta^{k}_{d}+\frac{\Theta}{\phi+1}\dot{\Phi}^{k}+\Big{[}\frac{1}{2\phi^{\prime}}+\frac{2\mu_{d}-f-2\Theta\dot{\phi}}{2(\phi+1)^{2}}$
(35) $\displaystyle+$
$\displaystyle\frac{k^{2}\phi^{\prime\prime}R}{a^{2}\phi^{\prime}(\phi+1)}-\frac{k^{2}\phi^{\prime}R}{a^{2}(\phi+1)^{2}}+\frac{k^{2}}{a^{2}(\phi+1)}\Big{]}\Phi^{k}=0\;,$
$\displaystyle\ddot{\Phi}^{k}$ $\displaystyle-$
$\displaystyle\Big{[}\frac{\ddot{\phi}\phi^{\prime\prime}}{\phi^{\prime
2}}-\frac{2\dot{\phi}\dot{\phi}^{\prime}\phi^{\prime\prime}}{\phi^{\prime
3}}-\frac{(\phi+1)}{3\phi^{\prime}}+\frac{R}{3}-\frac{R(\phi+1)\phi^{\prime\prime}}{3\phi^{\prime
2}}-\frac{\phi^{\prime\prime}\mu_{d}}{3\phi^{\prime
2}}+\frac{2f\phi^{\prime\prime}}{3\phi^{\prime
2}}-\frac{\dot{R}^{2}\phi^{\prime\prime\prime}}{\phi^{\prime}}$ (36)
$\displaystyle+$
$\displaystyle\frac{\Theta\phi^{\prime\prime}\dot{\phi}}{\phi^{\prime
2}}+\frac{3\dot{\phi}^{2}\phi^{\prime\prime 2}}{\phi^{\prime
4}}+\frac{\dot{\phi}\dot{\phi}^{\prime\prime}}{\phi^{\prime
2}}-\frac{k^{2}}{a^{2}}\Big{]}\Phi^{k}-\Big{(}\frac{\dot{\phi}^{\prime}}{\phi^{\prime}}-\Theta-\frac{2\dot{\phi}\phi^{\prime\prime}}{\phi^{\prime
2}}\Big{)}\dot{\Phi}^{k}-\frac{\mu_{d}}{3}\Delta^{k}_{d}-\dot{\phi}\dot{\Delta}^{k}_{d}=0\;\;\;\;\;\;\;\;\;.$

To easily analyse the above equations, we transform them into the
dimensionless form. Each term was transformed such a way that we have the
perturbations equations written as

$\displaystyle\Delta^{{}^{\prime\prime}k}_{d}$ $\displaystyle+$
$\displaystyle\Big{[}\frac{(2+ny)}{(1+z)(n-1)}+\frac{x-1}{(1+z)}\Big{]}\Delta^{{}^{\prime}k}_{d}-\frac{3\Omega_{d}}{(1+z)^{2}}\Delta^{k}_{d}-\frac{3(1-n)^{n-1}}{(1+z)\beta
n(6nh^{2}y)^{n-1}}\Phi^{{}^{\prime}k}$ (37) $\displaystyle+$
$\displaystyle\frac{1}{(1+z)^{2}}\Big{[}\frac{(1-n)^{n-2}}{2h^{2}\beta
n(n-1)(6nh^{2}y)^{n-2}}+\frac{3(1-n)^{n-1}}{\beta
n(6nh^{2}y)^{n-1}}\Big{(}\Omega_{d}-x-\frac{y}{n-1}\Big{)}\Big{]}\Phi^{k}=0\;,\;\;\;\;\;\;\;$
$\displaystyle\Phi^{{}^{\prime\prime}k}$ $\displaystyle+$
$\displaystyle\Big{[}\frac{(2+ny)}{(1+z)(n-1)}-\frac{2}{(1+z)}-\frac{x(n-2)}{(n-1)((1+z))}\Big{]}\Phi^{{}^{\prime}k}-\frac{(n-2)}{(1+z)^{2}}\Big{[}-\frac{\Omega_{d}}{n-1}$
(38) $\displaystyle+$
$\displaystyle\frac{4(1-q)}{3n(n-1)}+\frac{(n^{2}-3n-4)x^{2}}{(n-1)^{2}}+\frac{3x}{(n-1)}-(n-2)x^{2}-\frac{\hat{k}^{2}}{(n-2)}\Big{]}\Phi^{k}$
$\displaystyle-$ $\displaystyle\frac{\Omega_{d}\beta
n(6nh^{2}y)^{n-1}}{(1+z)^{2}(1-n)^{n-1}}\Delta^{k}_{d}-\frac{\beta
xn(6nh^{2}y)^{n-1}}{(1+z)(1-n)^{n-1}}\Delta^{{}^{\prime}k}_{d}=0\;,$

where $q=\frac{ny}{n-1}+1$. In the following section, we use these two
equations to compute the power spectrum of dust energy density perturbations.

##  4 Power spectrum

The matter power spectrum in modified gravity theories was widely explored in
cosmology [11, 26, 27, 28, 5, 29, 30, 31]. In the present paper, the analysis
will be done in line with the work covered in [11]. Having said that, we can
define a transfer function $T(k)$ given as [11, 22]

$T(k)=\left\langle\left|\Delta^{k}_{m}\right|^{2}\right\rangle\;,$ (39)

where $\Delta^{k}_{m}$ is the matter density perturbation and $k$ is the wave-
number that characterizes the harmonic mode. Due to isotropy of the observable
universe, one has

$T(k)=\left|\Delta^{k}_{m}\right|^{2}\;.$ (40)

The power spectrum is a quantity $P(k)$ is proportional to the transfer
function $T(k)$

$P(k)\varpropto T(k)\;.$ (41)

This is shown by the relation [11]

$P^{f(R)}_{k}=T(k)P^{\Lambda CDM}_{k}\Big{|}_{eq}\;,$ (42)

where $P^{\Lambda CDM}_{k}\Big{|}_{eq}$ is the power spectrum during epoch of
the equality of radiation and matter in the linear regime in the $\Lambda CDM$
model. Thus one can write power spectrum $P^{\phi}(k)$ given as

$P^{\phi}(k)=\frac{P^{f(\phi)}_{k}}{P^{\Lambda
CDM}_{k}\Big{|}_{eq}}=\left|\Delta^{k}_{d}\right|^{2},$ (43)

where $\Delta^{k}_{d}$ is the pressureless dust (baryonic matter) energy
density perturbation obtained after solving the system of equations (37) and
(38). To obtain the results, we have simultaneously solved Eqs. (21)-(25)
together with the constraint Eq. (12) to obtain the redshift dependent
solutions for parameters $x(z),y(z)$ as well as $\Omega_{d}(z)$. Having
obtained such parameters, with the consideration of different values of $n$,
one solved simultaneously two equations namely Eq. (37) and Eq. (38) to obtain
$\Delta^{k}_{d}(z)$ to be used in the plotting of the Eq. (43) with different
set of initial conditions. The plotting was done at a fixed redshift $(z=0)$,
means today and the power spectrum $P(k)$ was plotted against $k$. To easily
visualize the changes and behaviors, the plots are presented with log on both
axes.

Figure 1: Power spectrum for $n=1,n=1.1,,n=1.2,n=1.3,n=1.4,n=1.5,n=1.6,n=1.7$,
initial conditions at $z_{0}=2000$ are $\Delta(z_{0})=10^{-5}$,
$\Delta^{\prime}(z_{0})=10^{-5}$, $\Phi(z_{0})=10^{-5}$ and
$\Phi^{\prime}(z_{0})=10^{-5}$, obtained from Eq. (43).

Figure 2: Power spectrum for $n=1,n=1.1,,n=1.2,n=1.3,n=1.4,n=1.5,n=1.6,n=1.7$,
initial conditions at $z_{0}=2000$ are $\Delta(z_{0})=10^{-5}$,
$\Delta^{\prime}(z_{0})=0$, $\Phi(z_{0})=10^{-5}$ and
$\Phi^{\prime}(z_{0})=0$, obtained from Eq. (43).

##  5 Discussions

We consider different values of the power law index $n$ in $R^{n}$ model
following the choice made in [22] and maintain the initial conditions used in
[11]. The plots for matter power spectrum are presented in Fig. 2, Fig. 2,
Fig. 4 and Fig. 4. The initial conditions are of four types. All of them are
considered at redshift $z_{0}=2000$. The first set of initial condition is
$I:\Delta(z_{0})=10^{-5},\Delta^{\prime}(z_{0})=10^{-5},\Phi(z_{0})=10^{-5}\text{
and }\Phi^{\prime}(z_{0})=10^{-5}$. The second set of initial condition
$II:\Delta(z_{0})=10^{-5},\Delta^{\prime}(z_{0})=0,\Phi(z_{0})=10^{-5}\text{
and }\Phi^{\prime}(z_{0})=0$. The third set of initial condition is
$III:\Delta(z_{0})=10^{-5},\Delta^{\prime}(z_{0})=10^{-8},\Phi(z_{0})=10^{-5}\text{
and }\Phi^{\prime}(z_{0})=10^{-8}$. The last set of initial condition is
$IV:\Delta(z_{0})=10^{-5},\Delta^{\prime}(z_{0})=10^{-3},\Phi(z_{0})=10^{-5}\text{
and }\Phi^{\prime}(z_{0})=10^{-3}$. The effects of setting some of the above
initial conditions are discussed in details in [11]. The background equations
take initial values from the observations. We have used observable parameters
such as the Hubble constant $H_{0}$, deceleration parameter $q_{0}$ and
density parameters taken from $\Lambda$CDM initial conditions. The
perturbations are evaluated at redshift $z=0$ that is today. The ranges of $k$
used are taken from [23]. The $\Lambda$CDM model’s cosmological parameters are
taken from the SDSS and Planck data Survey [24, 25] respectively. Each type
produces changes in amplitudes. The GR case is for $n=1$ where the power
spectrum is scale invariant under dust perturbations. For $1<n<1.3$, with the
first set of initial conditions we have power spectrum evolving above GR
invariant line. For $n\geq 1.3$, the power spectrum starts with constant
amplitude then it experiences oscillations and eventually saturates at finite
amplitude. This behavior of the power spectrum is similar to the one presented
in [22] for $f(R)$ gravity, where they have obtained three different regimes
depending on the finite amplitude-saturation of the power spectrum. We have
paid attention to $n=1.4$ since it is very clear from the power spectrum that
the values close to $n=1.4$ are behaving similarly and have the finite
amplitude. We choose values close to $n=1.4$ to see how their corresponding
power spectrum behave. We have plotted them in Fig. 6, Fig. 6, Fig. 8 and 8.
For the initial conditions of type $II$, the oscillations are frequent
compared to other initial conditions during the amplitude finite-saturation.
However, one can see that the power spectrum of $n=1.4$ is lower than others
for large values of $k$ in all sets of initial conditions. The present work
reveals a couple of information in different aspects $(i)$ the obtained
perturbations are new in comparision with the work done in [11], $(ii)$ The
obtained power spectra around $n=1.4$ have shown to saturate at finite
amplitude as well, a feature not deeply explored in [11], $(iii)$ with some
set of initial conditions, the behavior of power spectrum around $n=1.4$ shows
oscillations during the finite-saturation state. This characteristics was not
identified in the work done in [11].

Figure 3: Power spectrum for $n=1,n=1.1,,n=1.2,n=1.3,n=1.4,n=1.5,n=1.6,n=1.7$,
initial conditions at $z_{0}=2000$ are $\Delta(z_{0})=10^{-5}$,
$\Delta^{\prime}(z_{0})=10^{-8}$, $\Phi(z_{0})=10^{-5}$ and
$\Phi^{\prime}(z_{0})=10^{-8}$, obtained from Eq. (43).

Figure 4: Power spectrum for $n=1,n=1.1,,n=1.2,n=1.3,n=1.4,n=1.5,n=1.6,n=1.7$,
initial conditions at $z_{0}=2000$ are $\Delta(z_{0})=10^{-5}$,
$\Delta^{\prime}(z_{0})=10^{-3}$, $\Phi(z_{0})=10^{-5}$ and
$\Phi^{\prime}(z_{0})=10^{-3}$, obtained from Eq. (43).

Figure 5: Power spectrum for $n=1.41,n=1.4,n=1.385,n=1.398$, initial
conditions at $z_{0}=2000$ are $\Delta(z_{0})=10^{-5}$,
$\Delta^{\prime}(z_{0})=10^{-5}$, $\Phi(z_{0})=10^{-5}$ and
$\Phi^{\prime}(z_{0})=10^{-5}$, obtained from Eq. (43).

Figure 6: Power spectrum for $n=1.41,n=1.4,n=1.385,n=1.398$, initial
conditions at $z_{0}=2000$ are $\Delta(z_{0})=10^{-5}$,
$\Delta^{\prime}(z_{0})=0$, $\Phi(z_{0})=10^{-5}$ and
$\Phi^{\prime}(z_{0})=0$, obtained from Eq. (43).

Figure 7: Power spectrum for $n=1.41,n=1.4,n=1.385,n=1.398$, initial
conditions at $z_{0}=2000$ are $\Delta(z_{0})=10^{-5}$,
$\Delta^{\prime}(z_{0})=10^{-8}$, $\Phi(z_{0})=10^{-5}$ and
$\Phi^{\prime}(z_{0})=10^{-8}$, obtained from Eq. (43).

Figure 8: Power spectrum for $n=1.41,n=1.4,n=1.385,n=1.398$, initial
conditions at $z_{0}=2000$ are $\Delta(z_{0})=10^{-5}$,
$\Delta^{\prime}(z_{0})=10^{-3}$, $\Phi(z_{0})=10^{-5}$ and
$\Phi^{\prime}(z_{0})=10^{-3}$, obtained from Eq. (43).

##  6 Conclusions

We have used the $1+3$ covariant approach to obtain the evolution equations
and studied the behavior of matter power spectrum of perturbation equations.
These equations are written in dimensionless form using a dynamical system
approach. We have defined the gradient variables based on the equivalence
between $f(R)$ theory of gravity and scalar-tensor theory of gravity. The
background cosmological parameters are considered from the known values in the
$\Lambda$CDM model. We show that the behavior of the power spectrum obtained
is similar to the one obtained in [22], the authors focused on $f(R)$ theory
only. We have extended the discussion to scalar tensor theory using a
dynamical system to treat linear perturbations and compute matter power
spectrum. In the present work, we obtained newly equations based on the
gradients developed that involve the definition of the scalar field provided.
We noted that for $1<n<1.3$, with the first set of initial conditions we have
power spectrum evolving above the GR scale-invariant line. For $n\geq 1.3$,
the power spectrum starts with constant amplitude then evolves into
oscillations and eventually saturates at a finite amplitude. This behavior of
the power spectrum is similar to the one presented in [22] for $f(R)$ gravity,
where authors obtained three different regimes depending on the finite
amplitude-saturation of the power spectrum. We have paid attention to $n=1.4$
since it is very clear from the power spectrum that the values close to
$n=1.4$ are behaving similarly and have the finite amplitude. We choose values
close to $n=1.4$ to study how their corresponding power spectra behave. We
have plotted the results in Fig. 6, Fig. 6, Fig. 8 and 8. For the initial
conditions of type $II$, the oscillations are frequent compared to other
initial conditions during the amplitude finite-saturation, a feature not
previously explored in [11]. However, one can see that the power spectrum of
the $n=1.4$ case is lower than the others for large values of $k$ in all sets
of initial conditions. Such behaviors were previously observed in the
literature as well [22]. The present results, where there is a consideration
of the definition of scalar field that vanishes in GR limit, support the
ongoing investigations of the equivalence between $f(R)$ theory and scalar-
tensor theory at linear order.

## Acknowledgments

AA acknowledge that this work is based on the research supported in part by
the National Research Foundation (NRF) of South Africa (grant number 112131).
AA also acknowledges the hospitality of the Department of Physics of the
University of Rwanda, where part of this work was completed. JN and MM
acknowledge the support from International Science Program (ISP) to Rwanda
Astrophysics, Space and Climate Science Research Group (RASCSRG), University
of Rwanda (grant number RWA01).

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∎

11institutetext: O. Tomalak 22institutetext: Theoretical Division, T-2, MS
B283, LANL
87545 Los Alamos, USA
Tel.: +13312185818
22email<EMAIL_ADDRESS>
# Radiative corrections to neutron beta decay and (anti)neutrino-nucleon
scattering from low-energy effective field theory††thanks: LA-UR-23-20003

Oleksandr Tomalak

(Received: date / Accepted: date)

###### Abstract

We study radiative corrections to neutron beta decay and low-energy
(anti)neutrino-nucleon scattering within a top-down effective field theory
approach. As it was recently shown, a few electromagnetic and electroweak low-
energy coupling constants in heavy-baryon chiral perturbation theory are yet
to be determined. Performing matching to the four-fermion effective field
theory, we relate these low-energy constants to correlation functions of
vector and axial-vector currents. Such relations allow us to explicitly
clarify scheme dependence for radiative corrections to neutron decay and low-
energy charged-current (anti)neutrino scattering, provide a robust prediction
of leading in the electromagnetic coupling constant contributions, and achieve
a clear separation between short-distance and long-distance contributions.

###### Keywords:

Neutron beta decay Heavy-baryon chiral perturbation theory Four-fermion
effective theory

##  1 Introduction

Neutron and proton are the lightest baryons. Electroweak interactions trigger
the decay of the heavier isospin partner, the neutron, into the proton, the
electron, and the electron antineutrino. There are two precise experimental
ways to measure the neutron lifetime: beam method and bottle method. The beam
method counts the number of slow neutrons that decay while they travel a
certain distance. The incoming neutron flux is compared to the number of
protons, which are detected in the electric and magnetic field around the
neutron decay volume. In the bottle method, the ultracold neutrons are trapped
in a reservoir by gravitational or magnetic field. The number of neutrons
inside the reservoir is monitored after some period of time. Results that are
obtained by these two methods consistently disagree by 8 seconds, which
corresponds to 3-to-5 standard deviations Wietfeldt:2018upi ;
Castelvecchi:2021hec . Such a discrepancy is larger than the uncertainty of
the most precise measurements of the neutron lifetime from a single
experiment. We illustrate the current status of the neutron lifetime
measurements in Fig. 1 Wietfeldt:2018upi ; Castelvecchi:2021hec .

Figure 1: Latest measurements of the neutron lifetime by the beam and bottle
methods are presented Wietfeldt:2018upi ; Castelvecchi:2021hec .

The latest and most precise measurement by the bottle method was performed at
Los Alamos National Laboratory (LANL) UCNt:2021pcg . It reaches a relative
accuracy between 3 and 4 permilles, while theoretical predictions have an
error of the same order of magnitude Marciano:2005ec ; Seng:2018yzq ;
Seng:2018qru ; Czarnecki:2019mwq ; Seng:2020wjq ; Hayen:2020cxh ;
Shiells:2020fqp ; Hayen:2021iga ; Gorchtein:2021fce  and heavily rely on the
experimental value of the axial-vector coupling constant. Future measurements
by the beam method will decrease the uncertainty by a factor 10; however, we
don’t know whether that will resolve or deepen the discrepancy.

The neutron lifetime measurement in combination with precise extractions of
the axial-vector coupling constant is a promising future tool for the
determination of the Cabibbo–Kobayashi–Maskawa (CKM) matrix element $V_{ud}$
and tests of the unitarity for this matrix, after resolving the above-
mentioned discrepancy. To fully utilize modern and upcoming precise lifetime
measurements for the extraction of the physical parameters, an improvement in
the theoretical description is required.

In the Standard Model, the neutron decay is described by the exchange of the
electroweak $W$ boson (Fig. 2).

Figure 2: The leading-order diagram for neutron decay in the Standard Model.

However, energy scales for this process ($\sim\mathrm{MeV}$) are well below
the electroweak scale ($\sim 100~{}\mathrm{GeV}$). Consequently, the
historical description within the low-energy EFT is the correct physics
picture for the neutron decay. We illustrate the leading-order contribution in
Fig. 3.

Figure 3: The leading-order diagram for neutron decay in the low-energy
effective field theory.

The leading-order amplitude for the neutron decay is proportional to the Fermi
coupling constant $\mathrm{G}_{\mathrm{F}}$, which is precisely determined
from the muon decay. Charged currents with quarks introduce the CKM matrix
element $V_{ud}$ to the neutron beta ($\beta$) decay amplitude. At low
energies, the neutron decay is described by two coupling constants: the vector
$g_{\mathrm{V}}$ and the axial-vector $g_{\mathrm{A}}$. All theoretical works
rely on the precise evaluation of electroweak, quantum chromodynamics (QCD),
and long-distance hadronic contributions to these coupling constants. In this
manuscript, we provide a consistent top-down effective field theory (EFT)
description for the neutron decay, starting with the Lagrangian of the
Standard Model.

The paper is organized as follows. First, we introduce the low-energy EFT
framework for quantum electrodynamics (QED) radiative corrections to the
neutron decay in Section 2. Then, we discuss the bottom-up EFT step of the
inclusion of pion degrees of freedom Cirigliano:2022hob  in Section 3. We
provide main results for the radiative corrections to the neutron decay within
the traditional approach in Section 4. In Section 5, we describe the four-
fermion EFT with all necessary ingredients following the matching to the
Standard Model previously performed in Refs. Hill:2019xqk ; Dekens:2019ept .
The main Section 6 of this paper describes the matching from the four-fermion
EFT to the heavy-baryon chiral perturbation theory
$\mathrm{HB}\chi\mathrm{PT}$. In the following Section 7, we provide necessary
framework for the evaluation of the hadronic two-current correlation
functions. We provide results for the vector coupling and two-current
contributions to the axial-vector coupling in Section 8. Section 9 discusses
the relation of the two-current correlators to the experimental data. In
Appendix, we provide the notations for the main text of this manuscript.

##  2 Low-energy effective theory for neutron decay

The neutron decay is low-energy process, with an energy scale of the order of
the neutron-proton mass difference ($1.3~{}\mathrm{MeV}$). The mass of the
electron $m\approx 511~{}\mathrm{keV}$ is in the same energy range. The scale
of the nucleon mass $M$ is 1000 times above the energies of the decay
products. Consequently, we can safely work in the EFT considering nucleons as
massive particles. Within the bottom-up approach to EFTs, we write the most
general form for the interaction Lagrangian, with the vector $g_{V}$ and the
axial-vector $g_{\mathrm{A}}$ coupling constants Ando:2004rk ; Raha:2011aa

$\displaystyle\mathrm{{\cal{L}}}^{\mathrm{e^{2}p}}_{\mathrm{\pi
Ne}}=-\sqrt{2}\mathrm{G}_{\mathrm{F}}V_{ud}\bar{e}\gamma_{\rho}\mathrm{P}_{\mathrm{L}}\nu_{e}\bar{N}_{v}\left(g_{\mathrm{V}}v^{\rho}-2g_{\mathrm{A}}S^{\rho}\right)\mathrm{T}^{+}N_{v}+\mathrm{h.c.}+\mathrm{O}\left(\frac{m}{M},\alpha\right),$
(1)

where $N_{v}=\left(p,n\right)^{\mathrm{T}}$ is the heavy-nucleon field doublet
with the nucleon four-momentum $p^{\rho}$ and velocity
$v^{\rho}=\frac{p^{\rho}}{M}$, the nucleon spin vector
$S^{\rho}=\frac{1}{2}\left(0,\vec{\mathrm{T}}\right)$, the Pauli matrices
$\mathrm{T}^{i}$,
$\mathrm{T}^{+}=\frac{1}{2}\left(\mathrm{T}^{1}+i\mathrm{T}^{2}\right)$, and
$\alpha\approx 1/137$ is the electromagnetic coupling constant. This
Lagrangian is the first step in neutron decay studies. It has already appeared
in its covariant form in the seminal work of Sirlin Sirlin:1967zza . The same
Lagrangian describes the low-energy (anti)neutrino-nucleon scattering and is a
starting point for the evaluation of inverse $\beta$ decay cross sections.

QED radiative corrections were evaluated from this Lagrangian in Ref.
Ando:2004rk  (for a detailed study of the radiation of photons, see Ref.
Bernard:2004cm  in detail). Authors have reproduced the universal radiative
correction to low-energy decay rates, which is known as the Sirlin function
Kinoshita:1958ru ; Sirlin:1967zza , as well as corrections to the angular
correlation term Garcia:1978bq . Nucleon recoil corrections were evaluated in
Refs. Wilkinson:1982hu ; Ivanov:2020ybx . Following the bottom-up approach,
authors of Ref. Ando:2004rk  introduce counterterms to cancel ultraviolet
divergences111The same ultraviolet divergences appear in the static limit of
the (anti)neutrino-nucleon scattering Raha:2011aa ; Fukugita:2004cq ;
Tomalak:2021hec ; Tomalak:2022xup . and propose to fix them from the
experimental data.

The traditional approach to radiative corrections in the nucleon $\beta$ decay
Sirlin:1967zza  matches this EFT directly to the Standard Model in the
particular renormalization and regularization scheme without specifying them
explicitly (see Section 4).

To precisely determine the low-energy coupling constants in Eq. (1) at
$\mathrm{O}\left(\alpha\right)$, we perform two-step matching from the
Standard Model to the four-fermion EFT and to the heavy-baryon chiral
perturbation theory in this work. Contrary to the well-known determination of
the coupling constants in the meson sector by the two-step matching procedure
in Ref. Descotes-Genon:2005wrq , we work everywhere in the well-developed
$\overline{\mathrm{MS}}$ renormalization and dimensional regularization
schemes.

##  3 Role of pions

Recently, next in energy degrees of freedom (pions) were included in Ref.
Cirigliano:2022hob , where authors have evaluated the electromagnetic
corrections including the virtual pions. According to the results of this
work, the vector coupling constant $g_{V}$ does not renormalize by QED loops
in $\chi\mathrm{PT}$ and is solely determined by the same counterterm as in
Ref. Ando:2004rk . In contrast, the axial-vector coupling constant is changed
by a percent-level contribution from the loops on the $\chi\mathrm{PT}$ side.
In the $\overline{\mathrm{MS}}$ renormalization scheme, counterterms of the
leading-order Lagrangian in Eq. (1) are given by

$\displaystyle g_{\mathrm{V}}\left(\mu\right)$
$\displaystyle=\frac{\alpha}{2\pi}\hat{C}_{\mathrm{V}}\left(\mu\right),$ (2)
$\displaystyle g_{\mathrm{A}}\left(\mu\right)$
$\displaystyle=\frac{\alpha}{2\pi}Z_{\pi}\left[\frac{1+3\left(g^{\left(0\right)}_{\mathrm{A}}\right)^{2}}{2}\left(\ln\frac{\mu^{2}}{m^{2}_{\pi}}-1\right)-\left(g^{\left(0\right)}_{\mathrm{A}}\right)^{2}\right]+\frac{\alpha}{2\pi}\hat{C}_{\mathrm{A}}\left(\mu\right)g^{\left(0\right)}_{\mathrm{A}},$
(3)

with the pion mass $m_{\pi}$, the pion wave-function renormalization constant
$Z_{\pi}$, and the axial charge in the chiral limit and without
electromagnetic corrections $g^{\left(0\right)}_{\mathrm{A}}$.
$\hat{C}_{\mathrm{V}}$ and $\hat{C}_{\mathrm{A}}$ contain the short-distance
contributions and hadronic corrections. The determination of these quantities
is the main subject of this work. Authors of Ref. Cirigliano:2022hob  have
also presented the next-to-leading order contribution in $\chi\mathrm{PT}$
with nonzero renormalization of $g_{\mathrm{A}}$. The change in the axial-
vector coupling constant is relevant for the comparison of the lattice QCD
calculations of the axial-vector coupling constant to the extractions from the
experimental data.

##  4 Radiative corrections to neutron decay

Traditionally, radiative corrections to the low-energy electroweak processes
are formulated and evaluated within the current algebra approach of Ref.
Sirlin:1977sv . This approach was applied to the neutron decay in Refs.
Sirlin:1967zza ; Marciano:1985pd , where authors have separated large short-
distance electroweak logarithms from the smaller long-distance hadronic
contributions and universal QED corrections at low energies. At leading order
in $\alpha$, the vector and axial-vector coupling constants renormalize with
the same short-distance contributions as

$\displaystyle g_{\mathrm{V}}$
$\displaystyle=1+\frac{\alpha}{2\pi}\left(g+\frac{3}{2}\ln\frac{M_{W}}{M_{A}}+\frac{1}{2}\ln\frac{M_{W}}{M_{A}}+A_{g}+C_{\mathrm{V}}\right),$
(4) $\displaystyle\frac{g_{\mathrm{A}}}{g^{\left(0\right)}_{\mathrm{A}}}$
$\displaystyle=1+\frac{\alpha}{2\pi}\left(g+\frac{3}{2}\ln\frac{M_{W}}{M_{A}}+\frac{1}{2}\ln\frac{M_{W}}{M_{A}}+A_{g}+C_{\mathrm{A}}\right),$
(5)

where $g$ is properly averaged over the spectrum in the $\beta$ decay Sirlin’s
function, $A_{g}$ denotes perturbative QCD corrections, $C_{\mathrm{V}}$ and
$C_{\mathrm{A}}$ determine the long-distance hadronic contributions, and the
separation scale $M_{A}$ is of order $1~{}\mathrm{GeV}$ (modern treatments
avoid introducing this scale Shiells:2020fqp ; Gorchtein:2021fce ). Marciano
and Sirlin have also resummed the large electroweak logarithms in their
approach Marciano:1985pd . The vector contribution $C_{\mathrm{V}}$ is
expressed in terms of the $\gamma W$ box and can be evaluated with quantified
uncertainties. Assuming $C_{\mathrm{A}}\approx C_{\mathrm{V}}$
Czarnecki:2004cw , the uncertainty of the vector contribution determines the
resulting error of the theoretical prediction for the $\beta$ decay rate.
According to Ref. Czarnecki:2004cw , the relative theory uncertainty is at the
level 4-8$\times 10^{-4}$, which is 2 times above the experimental precision
for the neutron lifetime UCNt:2021pcg , and urgently requires further
reduction.

Recently, this dominant hadronic uncertainty was reduced through the inclusion
of details of the physics at low virtualities. Authors of Refs. Seng:2018yzq ;
Seng:2018qru  have provided a model for the pion-nucleon contributions and
higher-resonance excitations with consistent treatment of the resonant
background, and have connected this region to the high-energy description
within the Regge phenomenology and vector-meson dominance model. The updated
value for the hadronic contributions to the vector coupling
$\frac{\alpha}{2\pi}C_{\mathrm{V}}=3.83(11)\times 10^{-3}$ is a few standard
deviations above the result of the previous evaluation
$\frac{\alpha}{2\pi}C_{\mathrm{V}}=3.26(19)\times 10^{-3}$ Czarnecki:2004cw .
This discrepancy resulted in three new dispersive calculations of the $\gamma
W$ contribution to $g_{V}$ Czarnecki:2019mwq ; Hayen:2020cxh ; Shiells:2020fqp
, which gave results slightly closer to the recent dispersive determination
Seng:2018yzq ; Seng:2018qru ; Seng:2020wjq ; Gorchtein:2021fce ; for a concise
summary, see Ref. Cirigliano:2022yyo . The first lattice-QCD evaluations of
the hadronic contributions to the $\gamma W$ box diagram in the pion and
neutron decays are also emerging Seng:2020jtz ; Feng:2020zdc ; Ma:2021azh ;
Yoo:2022lmt . Two groups have also recently estimated the difference
$C_{\mathrm{A}}-C_{\mathrm{V}}$ to be consistent with zero within
uncertainties Gorchtein:2021fce ; Hayen:2021iga .

In the following Sections, we describe an alternative treatment of the
hadronic and short-distance contributions within the EFT framework for the
radiative corrections to neutron decay to provide right physical picture and
clear separation between different physics regimes.

##  5 Four-fermion effective field theory

The matching to the Standard Model and the determination of the semileptonic
coupling constant $c^{ud}$ in the four-fermion EFT in the neutrino sector,
with explicit scale and scheme dependencies, are described in Ref.
Hill:2019xqk . Extending the Lagrangian of the four-fermion EFT Hill:2019xqk ;
Cirigliano:2022hob  with external vector $v_{\rho}$ and axial-vector
$a_{\rho}$ sources, vector ${\textbf{q}}_{V}$, axial ${\textbf{q}}_{A}$, and
weak ${\textbf{q}}_{W}$ charge spurions, we obtain

$\displaystyle{\cal L}^{\mathrm{spurions}}_{\mathrm{4-Fermi}}$
$\displaystyle=-e\frac{1}{2}A_{\rho}\left(\overline{q}{\textbf{q}}_{V}\gamma^{\rho}q-\overline{q}{\textbf{q}}_{A}\gamma^{\rho}\gamma_{5}q\right)+\frac{1}{2}\overline{q}\left(v_{\rho}\gamma^{\rho}-a_{\rho}\gamma^{\rho}\gamma_{5}\right)q$
$\displaystyle+\frac{c^{ud}}{2\sqrt{2}\mathrm{G}_{\mathrm{F}}V_{ud}}\left(\overline{e}\gamma_{\rho}\mathrm{P}_{\mathrm{L}}\nu_{e}\overline{q}{\textbf{q}}_{W}\gamma^{\rho}\mathrm{P}_{\mathrm{L}}q+\mathrm{h.c.}\right).$
(6)

The weak spurion contains a prefactor, such that
${\textbf{q}}_{W}=-2\sqrt{2}\mathrm{G}_{\mathrm{F}}V_{ud}\mathrm{T}^{+}$ at
the physical point. For studies of the renormalization, it is convenient to
express the $\mathrm{O}\left(\alpha\right)$ counterterm contributions in the
four-fermion EFT with explicit charge factors for quarks and the electron

$\displaystyle{\cal L}^{\mathrm{CT}}_{\mathrm{4-Fermi}}$
$\displaystyle=-2\sqrt{2}\mathrm{G}_{\mathrm{F}}V_{ud}e^{2}\left(\overline{e}\gamma_{\rho}\mathrm{P}_{\mathrm{L}}\nu_{e}\overline{d}\gamma^{\rho}\mathrm{P}_{\mathrm{L}}u+\mathrm{h.c.}\right)$
$\displaystyle\times\left(g_{00}Q^{2}_{e}+g_{23}\left(Q_{u}-Q_{d}\right)^{2}+g_{03}Q_{e}Q_{u}+g_{02}Q_{e}Q_{d}\right).$
(7)

Following Ref. Descotes-Genon:2005wrq , we generalize the Lagrangian of Eq.
(7) with charge spurions:

$\displaystyle{\cal L}^{\mathrm{CT}}_{\mathrm{4-Fermi}}=$
$\displaystyle-2Q^{2}_{e}g_{00}e^{2}\overline{e}\left(i\partial\\!\\!\\!/\\!\,\,-eQ_{e}A\\!\\!\\!/\\!\,\,-m\right)e$
$\displaystyle+Q_{e}\left(\overline{e}\gamma_{\rho}\mathrm{P}_{\mathrm{L}}\nu_{e}\left(g_{02}e^{2}\overline{q}{\textbf{q}}_{W}{\textbf{q}}_{L}\gamma^{\rho}\mathrm{P}_{\mathrm{L}}q+g_{03}e^{2}\overline{q}{\textbf{q}}_{L}{\textbf{q}}_{W}\gamma^{\rho}\mathrm{P}_{\mathrm{L}}q\right)+\mathrm{h.c.}\right)$
$\displaystyle-
ig_{23}e^{2}\left(\overline{q}_{L}\left[{\textbf{q}}_{L},D^{\rho}{\textbf{q}}_{L}\right]\gamma_{\rho}q_{L}+\overline{q}_{R}\left[{\textbf{q}}_{R},D^{\rho}{\textbf{q}}_{R}\right]\gamma_{\rho}q_{R}\right),$
(8)

where
$D^{\rho}{\textbf{q}}_{L}\equiv\partial^{\rho}{\textbf{q}}_{L}-i\left[l^{\rho},{\textbf{q}}_{L}\right]$
and
$D^{\rho}{\textbf{q}}_{R}\equiv\partial^{\rho}{\textbf{q}}_{R}-i\left[r^{\rho},{\textbf{q}}_{R}\right]$
are covariant derivatives, and ultraviolet (UV) divergent contributions to the
effective couplings Descotes-Genon:2005wrq  are given by

$\displaystyle g_{00}^{\varepsilon}$
$\displaystyle=-\frac{1}{32\pi^{2}}\frac{1}{\varepsilon_{\mathrm{UV}}},$
$\displaystyle g_{23}^{\varepsilon}$
$\displaystyle=-\frac{1}{32\pi^{2}}\frac{1}{\varepsilon_{\mathrm{UV}}},$
$\displaystyle g_{02}^{\varepsilon}$
$\displaystyle=\frac{1}{16\pi^{2}}\frac{1}{\varepsilon_{\mathrm{UV}}},$
$\displaystyle g_{03}^{\varepsilon}$
$\displaystyle=-\frac{1}{4\pi^{2}}\frac{1}{\varepsilon_{\mathrm{UV}}},$ (9)

where the UV pole $\frac{1}{\varepsilon_{\mathrm{UV}}}$ in the
$\overline{\mathrm{MS}}$ renormalization scheme is related to the space-time
dimension $d=4-2\varepsilon$ as

$\displaystyle\frac{1}{\varepsilon_{\mathrm{UV}}}$
$\displaystyle=\frac{1}{\varepsilon}-\gamma_{E}+\ln\left(4\pi\right).$ (10)

The scale- and scheme-dependent finite one-loop contributions to effective
couplings are given by Hill:2019xqk ; Dekens:2019ept

$\displaystyle g_{00}^{r}$
$\displaystyle=-\frac{1}{32\pi^{2}}\ln\frac{\mu^{2}}{\lambda^{2}}+\frac{1}{64\pi^{2}},$
$\displaystyle g_{23}^{r}$
$\displaystyle=-\frac{1}{32\pi^{2}}\ln\frac{\mu^{2}}{\lambda^{2}}+\frac{1}{64\pi^{2}},$
$\displaystyle g_{02}^{r}$
$\displaystyle=\frac{1}{16\pi^{2}}\ln\frac{\mu^{2}}{\lambda^{2}}+\frac{3}{32\pi^{2}}+a\frac{1}{8\pi^{2}},$
$\displaystyle g_{03}^{r}$
$\displaystyle=-\frac{1}{4\pi^{2}}\ln\frac{\mu^{2}}{\lambda^{2}}+a\frac{1}{8\pi^{2}}-\frac{3}{16\pi^{2}},$
(11)

where we have performed the calculation in the naive dimensional
regularization (NDR) scheme for $\gamma_{5}$, and the parameter $a$ is defined
from the following contraction in $d$ dimensions Buras:1989xd ; Dugan:1990df ;
Herrlich:1994kh :

$\displaystyle\gamma^{\kappa}\gamma^{\rho}\gamma^{\sigma}\mathrm{P}_{\mathrm{L}}\otimes\gamma_{\sigma}\gamma_{\rho}\gamma_{\kappa}\mathrm{P}_{\mathrm{L}}$
$\displaystyle=4\left[1+a\left(4-d\right)\right]\gamma^{\rho}\mathrm{P}_{\mathrm{L}}\otimes\gamma_{\rho}\mathrm{P}_{\mathrm{L}}+\mathrm{E}.$
(12)

The same scheme was used for the determination of the four-fermion coupling
constant $c^{ud}$ Hill:2019xqk .

##  6 Matching of four-fermion theory to heavy-baryon $\chi\mathrm{PT}$

In this Section, we relate low-energy coupling constants to correlation
functions in the four-fermion EFT. To derive such relations, we perform the
matching by equating the matrix elements of functional derivatives from the
generating functional $w$ w.r.t. the spurion fields between $\chi\mathrm{PT}$
and four-fermion calculations, and account for the tree-level and
$\mathrm{O}\left(\alpha\right)$ contributions. Schematically,

$\displaystyle\left(\int\mathrm{d}^{d}x<N|\frac{\delta^{2}w\left({\textbf{q}}_{W},{\textbf{q}}_{V},{\textbf{q}}_{A}\right)}{\delta{\textbf{q}}_{j^{b}}\left(x\right)\delta{\textbf{q}}_{i^{a}}\left(0\right)}\Bigg{|}^{\mathrm{tree~{}level+\mathrm{O}\left(\alpha\right)~{}loops+\mathrm{ct.}}}_{{\textbf{q}}=0}|N>\right)_{\chi\mathrm{PT}}=$
$\displaystyle\left(\int\mathrm{d}^{d}x<N|\frac{\delta^{2}w\left({\textbf{q}}_{W},{\textbf{q}}_{V},{\textbf{q}}_{A}\right)}{\delta{\textbf{q}}_{j^{b}}\left(x\right)\delta{\textbf{q}}_{i^{a}}\left(0\right)}\Bigg{|}^{\mathrm{tree~{}level+\mathrm{O}\left(\alpha\right)~{}loops+\mathrm{ct.}}}_{{\textbf{q}}=0}|N>\right)_{4-\mathrm{Fermi}},$
(13)

where $i$ and $j$ stands for vector, axial, and weak, $V,~{}A$, and $W$,
respectively, and ct. stands for counterterms. Equation (13) might be extended
with possible momentum insertions and corresponding derivatives, derivatives
w.r.t. external sources, and the pion in the initial state. In Sections 6.1
and 6.2, we illustrate expressions for the tree-level contributions in
$\chi\mathrm{PT}$. In Section 6.4, we provide an explicit form for the
counterterm contribution in the 4-Fermi theory. For convenience, we perform
matching at the same scale on both $\chi\mathrm{PT}$ and four-fermion sides,
which allows us to exactly cancel the gauge dependence and the electron wave-
function renormalization.

###  6.1 Electromagnetic coupling constants

Following the derivation in Refs. Meissner:1997ii ; Gasser:2002am , we write
down the general form for the leading-order electromagnetic Lagrangian in the
heavy-baryon sector including all allowed operators and keeping only
$<{\cal{Q}}_{-}>=0$ constraint

$\displaystyle\mathrm{{\cal{L}}}^{\mathrm{e^{2}p}}_{\mathrm{\pi N}}$
$\displaystyle=e^{2}\sum\limits_{i=1}^{14}g_{i}\bar{N}_{v}O^{e^{2}p}_{i}N_{v},$
(14)

with the explicit form for the operators of interest, which might enter the
neutron $\beta$ decay,

$\displaystyle O^{e^{2}p}_{1}$
$\displaystyle=<{\cal{Q}}_{+}^{2}-{\cal{Q}}_{-}^{2}>S\cdot u,\qquad
O^{e^{2}p}_{11}=i\left[{\cal{Q}}_{+},S\cdot c^{-}\right],$ $\displaystyle
O^{e^{2}p}_{13}$ $\displaystyle=<{\cal{Q}}_{+}^{2}+{\cal{Q}}_{-}^{2}>S\cdot
u,\qquad O^{e^{2}p}_{12}=i\left[{\cal{Q}}_{-},S\cdot c^{+}\right],$
$\displaystyle O^{e^{2}p}_{2}$ $\displaystyle=<{\cal{Q}}_{+}>^{2}S\cdot
u,\qquad\qquad~{}O^{e^{2}p}_{9}=\frac{i}{2}\left[{\cal{Q}}_{+},v\cdot
c^{+}\right]+\mathrm{h.c.},$ (15)

where the standard notations are described in Appendix.

Exploiting the spurion technique from Ref. Descotes-Genon:2005wrq , we take
derivatives from the generating functional w.r.t. the spurion fields and
derive matching relations for the determination of the coupling constants
$g_{i}$. In the following, we provide a few representations for the
electromagnetic coupling constants and combinations of interest in terms of
correlation functions with and without external pion fields.

$\displaystyle\frac{i\varepsilon^{abc}}{2}v_{\rho}\frac{\partial}{\partial
r_{\rho}}\left(\int\mathrm{d}^{d}xe^{ir\cdot
x}<N|\frac{\delta^{2}w\left({\textbf{q}}_{V},{\textbf{q}}_{A}\right)}{\delta{\textbf{q}}_{V^{b}}\left(x\right)\delta{\textbf{q}}_{V^{a}}\left(0\right)}\Bigg{|}^{\mathrm{tree~{}level}}_{{\textbf{q}}=0}|N>\right)\Bigg{|}_{r_{\rho}=0}$
$\displaystyle=e^{2}g_{9}\bar{N}_{v}\mathrm{T}^{c}N_{v},$ $\displaystyle
i\varepsilon^{abc}\frac{\partial}{\partial
r_{\rho}}\left(\int\mathrm{d}^{d}xe^{ir\cdot
x}<N|\frac{\delta^{2}w\left({\textbf{q}}_{V},{\textbf{q}}_{A}\right)}{\delta{\textbf{q}}_{V^{b}}\left(x\right)\delta{\textbf{q}}_{A^{a}}\left(0\right)}\Bigg{|}^{\mathrm{tree~{}level}}_{{\textbf{q}}=0}|N>\right)\Bigg{|}_{r_{\rho}=0}$
$\displaystyle=e^{2}\left(g_{11}+g_{12}\right)\bar{N}_{v}S^{\rho}\mathrm{T}^{c}N_{v},$
$\displaystyle\frac{1}{5}\left(\delta^{ab}\delta^{cd}-\frac{1}{2}\delta^{ac}\delta^{bd}\right)\int\mathrm{d}^{d}y\mathrm{d}^{d}x<N|\frac{\delta^{3}w\left(v^{\mu},a^{\mu},{\textbf{q}}_{V},{\textbf{q}}_{A}\right)}{\delta
a^{c}_{\rho}\left(y\right)\delta{\textbf{q}}_{V^{b}}\left(x\right)\delta{\textbf{q}}_{V^{a}}\left(0\right)}\Bigg{|}^{\mathrm{tree~{}level}}_{{\textbf{q}}=0}|N>$
$\displaystyle+\frac{\delta^{cd}}{2}\int\mathrm{d}^{d}y\mathrm{d}^{d}x<N|\frac{\delta^{3}w\left(v^{\mu},a^{\mu},{\textbf{q}}_{V},{\textbf{q}}_{A}\right)}{\delta
a^{c}_{\rho}\left(y\right)\delta{\textbf{q}}_{V^{0}}\left(x\right)\delta{\textbf{q}}_{V^{0}}\left(0\right)}\Bigg{|}^{\mathrm{tree~{}level}}_{{\textbf{q}}=0}|N>$
$\displaystyle=e^{2}\left(g_{1}+g_{2}+\frac{g_{11}}{2}+g_{13}\right)\bar{N}_{v}\mathrm{T}^{d}S^{\rho}N_{v}+...,\qquad
a,b,c\neq 0,$
$\displaystyle-\delta^{ac}\int\mathrm{d}^{d}y\mathrm{d}^{d}x<N|\frac{\delta^{3}w\left(v^{\mu},a^{\mu},{\textbf{q}}_{V},{\textbf{q}}_{A}\right)}{\delta
v^{c}_{\rho}\left(y\right)\delta{\textbf{q}}_{A^{b}}\left(x\right)\delta{\textbf{q}}_{V^{a}}\left(0\right)}\Bigg{|}^{\mathrm{tree~{}level}}_{{\textbf{q}}=0}|N>$
$\displaystyle=e^{2}g_{11}\bar{N}_{v}\mathrm{T}^{b}S^{\rho}N_{v}+...,\qquad
a,b,c\neq 0,$
$\displaystyle\delta^{ab}\int\mathrm{d}^{d}y\mathrm{d}^{d}x<N|\frac{\delta^{3}w\left(v^{\mu},a^{\mu},{\textbf{q}}_{V},{\textbf{q}}_{A}\right)}{\delta
v^{c}_{\rho}\left(y\right)\delta{\textbf{q}}_{A^{b}}\left(x\right)\delta{\textbf{q}}_{V^{a}}\left(0\right)}\Bigg{|}^{\mathrm{tree~{}level}}_{{\textbf{q}}=0}|N>$
$\displaystyle=e^{2}\left(g_{11}+g_{12}\right)\bar{N}_{v}\mathrm{T}^{c}S^{\rho}N_{v}+...,\qquad
a,b,c\neq 0.$ (16)

To cancel the gauge dependence for electromagnetic coupling constants on the
level of the matching between $\chi\mathrm{PT}$ and correlation functions of
quark currents in four-fermion EFT (at least for the vector coupling), we
include also one-loop corrections on the $\chi\mathrm{PT}$ side. Note that the
coupling constant $g_{13}$ always enters the matching expressions and
observables in the same combination with the coupling constant $g_{1}$. Thus,
omitting the operator $O^{e^{2}p}_{13}$ does not introduce biases in any
physical quantity.

###  6.2 Electroweak coupling constants

Reducing the number of assumptions for coordinate-dependent spurions compared
to Ref. Cirigliano:2022hob , but still keeping $<{\cal{Q}}^{W}_{L}>=0$
constraint, we obtain the most general weak-interaction Lagrangian in the
heavy-baryon sector for the $\mathrm{SU}\left(2\right)$ isospin group

$\displaystyle\mathrm{{\cal{L}}}^{\mathrm{e^{2}p}}_{\mathrm{\pi
Ne}}=e^{2}\sum\limits_{i=1}^{6}\bar{e}\gamma_{\rho}\mathrm{P}_{\mathrm{L}}\nu_{e}\bar{N}_{v}\left({V}_{i}v^{\rho}-2{A}_{i}g^{(0)}_{A}S^{\rho}\right){{\mathrm{O}_{i}}}N_{v}+\mathrm{h.c.}$
(17)

We present $\mathrm{{\cal{L}}}^{\mathrm{e^{2}p}}_{\mathrm{\pi Ne}}$ in the
same form as the leading-order Lagrangian in Eq. (1), where the operators in
Eq. (17) are given by

$\displaystyle{\mathrm{O}_{1}}$
$\displaystyle=[{{\cal{Q}}}_{L},{\cal{Q}}^{W}_{L}],\qquad~{}~{}{\mathrm{O}_{2}}=[{{\cal{Q}}}_{R},{\cal{Q}}^{W}_{L}],$
$\displaystyle{\mathrm{O}_{3}}$
$\displaystyle=\\{{{\cal{Q}}}_{L},{\cal{Q}}^{W}_{L}\\},\qquad~{}{\mathrm{O}_{4}}=\\{{{\cal{Q}}}_{R},{\cal{Q}}^{W}_{L}\\},$
$\displaystyle{\mathrm{O}_{5}}$
$\displaystyle=<{{\cal{Q}}}_{L}{\cal{Q}}^{W}_{L}>,\qquad{\mathrm{O}_{6}}=<{{\cal{Q}}}_{R}{\cal{Q}}^{W}_{L}>.$
(18)

Our low-energy constants are related to the couplings in Ref.
Cirigliano:2022hob  by $\tilde{X}_{1}=V_{1}+V_{3}+V_{4}-A_{1}-A_{3}-A_{4}$,
$\tilde{X}_{2}=-V_{2}$, $\tilde{X}_{3}=2A_{2}g^{(0)}_{A}$,
$\tilde{X}_{4}=V_{4}+V_{6}$, and
$\tilde{X}_{5}=-2\left(A_{4}+A_{6}\right)g^{(0)}_{A}$. All coupling constants
are dimensionless.

All coupling constants in Eq. (17) are conveniently determined from the
functional derivatives w.r.t. the spurion fields in the same combination in
which they enter physical observables as

$\displaystyle\int\mathrm{d}^{d}x<e^{-}\bar{\nu}_{e}N|\left(\frac{\delta^{2}w\left({\textbf{q}}_{V},{\textbf{q}}_{A},{\textbf{q}}_{W}\right)}{\delta{\textbf{q}}_{V^{0}}\left(x\right)\delta{\textbf{q}}_{W^{a}}\left(0\right)}+\frac{\varepsilon^{abc}}{2i}\frac{\delta^{2}w\left({\textbf{q}}_{V},{\textbf{q}}_{A},{\textbf{q}}_{W}\right)}{\delta{\textbf{q}}_{V^{b}}\left(x\right)\delta{\textbf{q}}_{W^{c}}\left(0\right)}\right)\Bigg{|}^{\mathrm{tree~{}level}}_{{\textbf{q}}=0}|N>$
$\displaystyle=e^{2}\sum\limits_{i=1}^{4}\bar{e}\gamma_{\rho}\mathrm{P}_{\mathrm{L}}\nu_{e}\bar{N}_{v}\left({V}_{i}{v}^{\rho}-2{A}_{i}g^{(0)}_{A}S^{\rho}\right)\mathrm{T}^{a}N_{v}.$
(19)

###  6.3 Contribution to beta decay and (anti)neutrino scattering

Electromagnetic and electroweak low-energy coupling constants of
$\chi\mathrm{PT}$ effective Lagrangians in Eqs. (14) and (17) contribute to
the neutron decay and (anti)neutrino-proton scattering through the
modification of the vector and axial-vector coupling constants
$\hat{C}_{\mathrm{V}}$ and $\hat{C}_{\mathrm{A}}$ of the leading-order
Lagrangian in Eq. (1) as

$\displaystyle\hat{C}_{\mathrm{V}}$
$\displaystyle=8\pi^{2}\left[-\frac{X_{6}}{2}+2\left({V}_{1}+{V}_{2}+{V}_{3}+{V}_{4}\right)-g_{9}\right],$
(20) $\displaystyle\hat{C}_{\mathrm{A}}$
$\displaystyle=8\pi^{2}\left[-\frac{X_{6}}{2}+2\left({A}_{1}+{A}_{2}+{A}_{3}+{A}_{4}\right)+\frac{g_{1}+g_{2}+\frac{g_{11}}{2}+g_{13}}{g_{\mathrm{A}}^{(0)}}\right],$
(21)

where we have also added the contribution of the pure-leptonic operator and
corresponding coupling constant $X_{6}$ according to Ref. Cirigliano:2022hob .
Equations (20) and (21) generalize Ref. Cirigliano:2022hob  by including
contributions of the electroweak constants
$V_{2},~{}V_{3},~{}V_{4},~{}A_{3},~{}A_{4}$ and electromagnetic coupling
constant $g_{13}$.

###  6.4 Renormalization in four-fermion effective field theory

The evaluation of coupling constants in Eqs. (20) and (21) from the
correlation functions in the four-fermion EFT contains divergent integrals in
$d=4$. Taking functional derivatives w.r.t. to the spurion fields from the
counterterm Lagrangian of the four-fermion EFT in Eq. (8), we compute
corresponding contributions to the vector
$\hat{C}^{\mathrm{ct.}}_{\mathrm{V}}$ and axial-vector
$\hat{C}^{\mathrm{ct.}}_{\mathrm{A}}$ coupling constants:

$\displaystyle\hat{C}^{\mathrm{ct.}}_{\mathrm{V}}$
$\displaystyle=\hat{C}^{\mathrm{ct.}}_{\mathrm{A}}=8\pi^{2}\left(-g_{00}^{\varepsilon}-g_{23}^{\varepsilon}+\frac{2g_{03}^{\varepsilon}-g_{02}^{\varepsilon}}{3}\right),$
(22)

where $g_{00}$ is a purely leptonic counterterm. As it is expected from the
consistency of the renormalization, the counterterm contribution takes the
form of the physical amplitude
$g_{00}Q^{2}_{e}+g_{23}\left(Q_{u}-Q_{d}\right)^{2}+g_{03}Q_{e}Q_{u}+g_{02}Q_{e}Q_{d}$,
cf. the Lagrangian in Eq. (7).

##  7 Hadron tensor and operator product expansion

In this paper, we define the hadron tensor
${T}^{\sigma\rho}_{V^{b}V(A)^{a}}\left(L,p\right)$ in terms of the two-point
correlation function of quark currents, following Abers:1968zz , as:

$\displaystyle{T}^{\sigma\rho}_{V^{b}V(A)^{a}}\left(L,p\right)$
$\displaystyle=\frac{i}{2}\int\mathrm{d}^{d}xe^{iL\cdot
x}<N|\mathrm{T}\left[\overline{q}\mathrm{T}^{b}\gamma^{\sigma}q\left(x\right)\overline{q}\mathrm{T}^{a}\gamma^{\rho}\left(\gamma_{5}\right)q(0)\right]|N>$
$\displaystyle=\frac{1}{M}\bar{N}_{v}M^{\sigma\rho}N_{v}.$ (23)

For applications in $d$ dimensions, we generalize the tensor decomposition of
Eq. (23) Ji:1993ey ; Blumlein:1996tp ; Maul:1996dx ; Blumlein:2012bf , which
is commonly used in deep inelastic scattering literature, by following ideas
of Refs. Drell:1966kk ; Bjorken:1966jh

$\displaystyle M^{\sigma\rho}$
$\displaystyle=M^{\sigma\rho}_{S}+M^{\sigma\rho}_{A},$ (24) $\displaystyle
M^{\sigma\rho}_{S}$
$\displaystyle=\tilde{g}^{\sigma\rho}\left(\mathrm{T}_{1}(\nu,L^{2})+\frac{L\\!\\!\\!/\\!\,\,\gamma_{5}}{M}\mathrm{A}_{1}(\nu,L^{2})\right)+\frac{\tilde{p}^{\sigma}\tilde{\gamma}^{\rho}+\tilde{p}^{\rho}\tilde{\gamma}^{\sigma}}{2M}\gamma_{5}\mathrm{B}_{1}(\nu,L^{2})$
$\displaystyle+\frac{\tilde{p}^{\sigma}\tilde{p}^{\rho}}{M^{2}}\left(\mathrm{T}_{2}(\nu,L^{2})+\frac{L\\!\\!\\!/\\!\,\,\gamma_{5}}{M}\mathrm{A}_{2}(\nu,L^{2})\right)+\frac{p^{\sigma}\tilde{\gamma}^{\rho}+p^{\rho}\tilde{\gamma}^{\sigma}}{2M}\gamma_{5}\mathrm{B}_{2}(\nu,L^{2})$
$\displaystyle+\frac{L^{\sigma}L^{\rho}}{M^{2}}\left(\mathrm{T}_{4}(\nu,L^{2})+\frac{L\\!\\!\\!/\\!\,\,\gamma_{5}}{M}\mathrm{A}_{4}(\nu,L^{2})\right)$
$\displaystyle+\frac{p^{\sigma}L^{\rho}+p^{\rho}L^{\sigma}}{2M^{2}}\left(\mathrm{T}_{5}(\nu,L^{2})+\frac{L\\!\\!\\!/\\!\,\,\gamma_{5}}{M}\mathrm{A}_{5}(\nu,L^{2})\right),$
(25) $\displaystyle M^{\sigma\rho}_{A}$
$\displaystyle=\frac{1}{2M}\\{\gamma^{\sigma\rho},L\\!\\!\\!/\\!\,\,\left(\mathrm{S}_{1}(\nu,L^{2})+\frac{\gamma_{5}}{2}\mathrm{T}_{3}(\nu,L^{2})\right)+p\\!\\!\\!/\\!\,\,\mathrm{S}_{3}(\nu,L^{2})\\}$
$\displaystyle+\frac{1}{2M^{2}}\Big{(}[\gamma^{\sigma},\gamma^{\rho}]L^{2}+L^{\sigma}[\gamma^{\rho},L\\!\\!\\!/\\!\,\,]+L^{\rho}[L\\!\\!\\!/\\!\,\,,\gamma^{\sigma}]\Big{)}\mathrm{S}_{2}(\nu,L^{2}),$
(26)

with $\nu=\frac{p\cdot L}{M}$,
$\tilde{g}^{\sigma\rho}=-g^{\sigma\rho}+\frac{L^{\sigma}L^{\rho}}{L^{2}}$,
$\tilde{p}^{\sigma}=p^{\sigma}-\frac{\left(p\cdot
L\right)}{L^{2}}\,L^{\sigma}$,
$\gamma^{\sigma\rho}=\frac{1}{2}\left[\gamma^{\sigma},\gamma^{\rho}\right]$,
and
$\tilde{\gamma}^{\sigma}={\gamma}^{\sigma}-\frac{L^{\sigma}}{L^{2}}L\\!\\!\\!/\\!\,\,$.
$M^{\sigma\rho}_{S}$ and $M^{\sigma\rho}_{A}$ denote the symmetric and
antisymmetric contributions, respectively. Note that we have changed the sign
in front of the amplitude $\mathrm{T}_{3}$, as compared to Ref.
Gorchtein:2021fce .

The optical theorem relates the imaginary parts of the hadron amplitudes to
the proton structure functions as

$\displaystyle\Im\mathrm{T}_{1}\left(\nu,L^{2}\right)$ $\displaystyle=$
$\displaystyle\pi
F_{1}\left(\nu,L^{2}\right),\quad~{}~{}~{}~{}~{}~{}\Im\mathrm{T}_{2,3,4,5}\left(\nu,L^{2}\right)=\frac{\pi
M}{\nu}F_{2,3,4,5}\left(\nu,L^{2}\right),$
$\displaystyle\Im\mathrm{S}_{1,3}\left(\nu,L^{2}\right)$ $\displaystyle=$
$\displaystyle\frac{\pi
M}{\nu}g_{1,3}\left(\nu,L^{2}\right),\qquad~{}~{}\Im\mathrm{S}_{2}\left(\nu,L^{2}\right)=\frac{\pi
M^{2}}{\nu^{2}}g_{2}\left(\nu,L^{2}\right),$
$\displaystyle\Im\mathrm{S}_{3}\left(\nu,L^{2}\right)$ $\displaystyle=$
$\displaystyle\frac{\pi
M}{\nu}g_{3}\left(\nu,L^{2}\right),\qquad~{}~{}~{}\Im\mathrm{A}_{1}\left(\nu,L^{2}\right)=\frac{\pi
M}{\nu}a_{1}\left(\nu,L^{2}\right),$
$\displaystyle\Im\mathrm{A}_{2,4,5}\left(\nu,L^{2}\right)$ $\displaystyle=$
$\displaystyle\frac{\pi
M^{2}}{\nu^{2}}a_{2,4,5}\left(\nu,L^{2}\right),~{}~{}\Im\mathrm{B}_{1,2}\left(\nu,L^{2}\right)=\frac{\pi
M}{\nu}b_{1,2}\left(\nu,L^{2}\right).$ (27)

Crossing relations for isovector contributions are given by

$\displaystyle F_{1,5}\left(-\nu,~{}L^{2}\right)$
$\displaystyle=F_{1,5}\left(\nu,~{}L^{2}\right),\quad
F_{2,3,4}\left(-\nu,~{}L^{2}\right)=-F_{2,3,4}\left(\nu,~{}L^{2}\right),$
$\displaystyle g_{1,2}\left(-\nu,~{}L^{2}\right)$
$\displaystyle=-g_{1,2}\left(\nu,~{}L^{2}\right),\qquad
g_{3}\left(-\nu,~{}L^{2}\right)=g_{3}\left(\nu,~{}L^{2}\right),$
$\displaystyle a_{1,5}\left(-\nu,~{}L^{2}\right)$
$\displaystyle=a_{1,5}\left(\nu,~{}L^{2}\right),\qquad
a_{2,4}\left(-\nu,~{}L^{2}\right)=-a_{2,4}\left(\nu,~{}L^{2}\right),$
$\displaystyle b_{1,2}\left(-\nu,~{}L^{2}\right)$
$\displaystyle=-b_{1,2}\left(\nu,~{}L^{2}\right),$ (28)

while these relations for isoscalar contributions have opposite signs.

The amplitudes
$\mathrm{T}_{1},~{}\mathrm{T}_{2},~{}\mathrm{T}_{4},~{}\mathrm{T}_{5},~{}\mathrm{S}_{1},~{}\mathrm{S}_{2},~{}\mathrm{S}_{3}$
represent vector-vector correlators, while the amplitudes
$\mathrm{T}_{3},~{}\mathrm{A}_{1},~{}\mathrm{A}_{2},~{}\mathrm{A}_{4},~{}\mathrm{A}_{5},~{}\mathrm{B}_{1},~{}\mathrm{B}_{2}$
correspond to vector-axial correlators. For vanishing lepton masses,
amplitudes
$\mathrm{T}_{4},~{}\mathrm{T}_{5},~{}\mathrm{A}_{4},~{}\mathrm{A}_{5}$ do not
contribute. The only other tensor structure in Ref. Blumlein:2012bf  has a
factor of the photon momentum $L$ and, therefore, the corresponding amplitude
does not contribute at vanishing lepton masses. Amplitudes
$\mathrm{T}_{4},~{}\mathrm{T}_{5},~{}\mathrm{S}_{3},~{}\mathrm{A}_{4},~{}\mathrm{A}_{5},~{}\mathrm{B}_{2}$
vanish except for nonzero quark masses; consequently, we neglect them in the
matching. We are left with the contributions from
$\mathrm{T}_{1},~{}\mathrm{T}_{2},~{}\mathrm{T}_{3},~{}\mathrm{S}_{1},~{}\mathrm{S}_{2}$
as well as $\mathrm{A}_{1},~{}\mathrm{A}_{2}$, and $\mathrm{B}_{1}$.

To express the effects of the hadron tensor on the vector and axial-vector
coupling constants in terms of the nucleon structure functions, we separate
the nucleon pole contributions and write down the dispersion relations
accounting for the crossing properties of the invariant amplitudes

$\displaystyle\mathrm{Re}~{}\mathrm{T}_{3}(\nu,L^{2})=\mathrm{Re}~{}\mathrm{T}_{3}^{\mathrm{pole}}(\nu,L^{2})+2M\int\limits_{\nu^{\mathrm{inel}}_{\mathrm{thr}}}^{\infty}\frac{F_{3}\left(\nu^{\prime},~{}L^{2}\right)}{\nu^{\prime
2}-\nu^{2}}\mathrm{d}\nu^{\prime},$
$\displaystyle\mathrm{Re}~{}\mathrm{S}_{1}(\nu,L^{2})=\mathrm{Re}~{}\mathrm{S}_{1}^{\mathrm{pole}}(\nu,L^{2})+2M\int\limits_{\nu^{\mathrm{inel}}_{\mathrm{thr}}}^{\infty}\frac{g_{1}\left(\nu^{\prime},~{}L^{2}\right)}{\nu^{\prime
2}-\nu^{2}}\mathrm{d}\nu^{\prime},$
$\displaystyle\mathrm{Re}~{}\mathrm{S}_{2}(\nu,L^{2})=\mathrm{Re}~{}\mathrm{S}_{2}^{\mathrm{pole}}(\nu,L^{2})+2M^{2}\nu\int\limits_{\nu^{\mathrm{inel}}_{\mathrm{thr}}}^{\infty}\frac{g_{2}\left(\nu^{\prime},~{}L^{2}\right)}{\nu^{\prime
2}\left(\nu^{\prime 2}-\nu^{2}\right)}\mathrm{d}\nu^{\prime},$
$\displaystyle\mathrm{Re}\left(\nu\mathrm{S}_{2}(\nu,L^{2})\right)=\mathrm{Re}\left(\nu\mathrm{S}_{2}(\nu,L^{2})\right)^{\mathrm{pole}}+2M^{2}\int\limits_{\nu^{\mathrm{inel}}_{\mathrm{thr}}}^{\infty}\frac{g_{2}\left(\nu^{\prime},~{}L^{2}\right)}{\nu^{\prime
2}-\nu^{2}}\mathrm{d}\nu^{\prime}.$ (29)

To control the UV behavior of hadron-structure contributions to the vector and
axial-vector coupling constants, we determine the leading-twist terms of the
operator product expansion (OPE) for the nucleon structure amplitudes
Maul:1996dx ; Manohar:1992tz

$\displaystyle\mathrm{S}_{1}(\nu,L^{2})$
$\displaystyle\to\mathrm{S}_{3}(\nu,L^{2})\to\frac{M^{2}}{L^{2}}g^{\left(0\right)}_{\mathrm{A}},\quad\mathrm{T}_{3}(\nu,L^{2})\to-2\frac{M^{2}}{L^{2}},\quad~{}\mathrm{S}_{2}(\nu,L^{2})<\frac{1}{L^{3}},$
$\displaystyle\mathrm{A}_{1}(\nu,L^{2})$
$\displaystyle\to-\frac{M^{2}}{L^{2}}g^{\left(0\right)}_{\mathrm{A}},\qquad\qquad\quad~{}~{}\mathrm{B}_{1}(\nu,L^{2})\to
2\frac{M}{\nu}g^{\left(0\right)}_{\mathrm{A}},\quad\mathrm{A}_{2}(\nu,L^{2})<\frac{1}{L^{3}}.$
(30)

##  8 Vector coupling and two-point correlation functions

Evaluating $\hat{C}_{\mathrm{V}}$ and $\hat{C}_{\mathrm{A}}$ from the
amplitudes used in Eqs. (30), and adding the counterterm contributions from
Eq. (22), we obtain the expected UV poles for the counterterms of the leading-
order Lagrangian, cf. Eq. (1):

$\displaystyle\hat{C}_{\mathrm{V}}^{\mathrm{UV}}$
$\displaystyle=\hat{C}_{\mathrm{A}}^{\mathrm{UV}}=-\frac{3}{4}\frac{1}{\varepsilon_{\mathrm{UV}}}.$
(31)

With OPE expressions, we also reproduce the expected scheme dependence, which
enters through the parameter $a$.

Accounting for the scale and scheme dependence of the coupling constant in the
four-fermion EFT, we determine the counterterm for the vector coupling
constant of the leading-order Lagrangian, cf. Eq. (1), in the
$\overline{\mathrm{MS}}$ renormalization scheme:222For evaluation of low-
energy QED diagrams in the conventional to $\chi\mathrm{PT}$
$\overline{\mathrm{MS}}$ scheme Cirigliano:2022hob ; Ando:2004rk , we
substitute
$\hat{C}_{\mathrm{V},\mathrm{A}}\to\hat{C}_{\mathrm{V},\mathrm{A}}+\frac{3}{4}$.

$\displaystyle\hat{C}_{\mathrm{V}}\left(\mu\right)$
$\displaystyle=-\frac{a}{3}+\frac{1}{4}\ln\frac{\mu^{2}}{\lambda^{2}}-\ln\frac{\mu^{2}}{\mu_{0}^{2}}+\frac{1}{8}+\frac{8\pi^{2}}{e^{2}}\left(\frac{c^{ud}\left(a,\mu_{0}\right)}{2\sqrt{2}\mathrm{G}_{\mathrm{F}}V_{ud}}-1\right)$
$\displaystyle+\frac{8\pi^{2}}{3}\int\frac{i\mathrm{d}^{4}L}{\left(2\pi\right)^{4}}\frac{\nu^{2}-L^{2}}{L^{2}\left(L^{2}-\lambda^{2}\right)}\left(\frac{\mathrm{T}_{3}(\nu,L^{2})}{M^{2}}+\frac{2}{L^{2}}\right).$
(32)

The same counterterm enters the theory with electromagnetic interaction of
virtual pions. By this, we provide a well-defined framework for radiative
corrections at the nucleon level for the superallowed $0^{+}\to 0^{+}$ Fermi
transitions.

An analogous expression for the axial-vector coupling constant is given by

$\displaystyle\hat{C}_{\mathrm{A}}\left(\mu\right)$
$\displaystyle=-\frac{a}{3}+\frac{1}{4}\ln\frac{\mu^{2}}{\lambda^{2}}-\ln\frac{\mu^{2}}{\mu_{0}^{2}}+\frac{1}{8}+\frac{8\pi^{2}}{e^{2}}\left(\frac{c^{ud}\left(a,\mu_{0}\right)}{2\sqrt{2}\mathrm{G}_{\mathrm{F}}V_{ud}}-1\right)$
$\displaystyle+\frac{16\pi^{2}}{9}\int\frac{i\mathrm{d}^{4}L}{\left(2\pi\right)^{4}}\frac{\nu^{2}+2L^{2}}{L^{2}\left(L^{2}-\lambda^{2}\right)}\left(\frac{\mathrm{S}_{1}(\nu,L^{2})}{M^{2}g_{\mathrm{A}}^{(0)}}-\frac{1}{L^{2}}\right)$
$\displaystyle+\frac{16\pi^{2}}{3}\int\frac{i\mathrm{d}^{4}L}{\left(2\pi\right)^{4}}\frac{1}{L^{2}}\frac{\nu}{M}\frac{\mathrm{S}_{2}(\nu,L^{2})}{M^{2}g_{\mathrm{A}}^{(0)}}.$
(33)

Eq. (33) properly describes the scale dependence. For correct finite
contributions, this expression has to be extended by additional hadron
structure contributions from three-current correlation functions, virtual QED,
and pion loops on the $\chi\mathrm{PT}$ side.

The isoscalar amplitudes $\mathrm{A}_{1},~{}\mathrm{A}_{2}$, and
$\mathrm{B}_{1}$ contribute only to the axial-vector coupling constant as

$\displaystyle\hat{C}^{\mathrm{S}}_{\mathrm{A}}$
$\displaystyle=\frac{8\pi^{2}}{d-1}\int\frac{i\mathrm{d}^{d}L}{\left(2\pi\right)^{d}}\frac{\nu^{2}-L^{2}}{L^{2}\left(L^{2}-\lambda^{2}\right)}\frac{1}{L^{2}}\frac{\nu}{M}\left(\frac{\mathrm{B}_{1}(\nu,L^{2})}{g_{\mathrm{A}}^{(0)}}+\frac{\nu}{M}\frac{\mathrm{A}_{2}(\nu,L^{2})}{g_{\mathrm{A}}^{(0)}}\right)$
$\displaystyle+8\pi^{2}\int\frac{i\mathrm{d}^{d}L}{\left(2\pi\right)^{d}}\frac{\nu^{2}-L^{2}}{L^{2}\left(L^{2}-\lambda^{2}\right)}\frac{\mathrm{A}_{1}(\nu,L^{2})}{M^{2}g_{\mathrm{A}}^{(0)}},$
(34)

with the infrared regulator $\lambda$. However, this contribution can be
alternatively expressed in terms of the correlation functions from three-quark
currents with a help of Eqs. (16).

At one-loop level, the low-energy coupling constants have an extremely simple
form:333Coincidentally, the same constant term $-11/8$ is obtained in Ref.
Abers:1968zz .

$\displaystyle\hat{C}_{\mathrm{V}}\left(\mu\right)$
$\displaystyle=\frac{1}{4}\ln\frac{\mu^{2}}{\lambda^{2}}-\ln\frac{\mu^{2}}{M^{2}_{Z}}-\frac{11}{8}$
$\displaystyle+\frac{8\pi^{2}}{3}\int\frac{i\mathrm{d}^{4}L}{\left(2\pi\right)^{4}}\frac{\nu^{2}-L^{2}}{L^{2}\left(L^{2}-\lambda^{2}\right)}\left(\frac{\mathrm{T}_{3}(\nu,L^{2})}{M^{2}}+\frac{2}{L^{2}}\right),$
(35) $\displaystyle\hat{C}_{\mathrm{A}}\left(\mu\right)$
$\displaystyle=\frac{1}{4}\ln\frac{\mu^{2}}{\lambda^{2}}-\ln\frac{\mu^{2}}{M^{2}_{Z}}-\frac{11}{8}+\frac{16\pi^{2}}{3}\int\frac{i\mathrm{d}^{4}L}{\left(2\pi\right)^{4}}\frac{1}{L^{2}}\frac{\nu}{M}\frac{\mathrm{S}_{2}(\nu,L^{2})}{M^{2}g_{\mathrm{A}}^{(0)}}$
$\displaystyle+\frac{16\pi^{2}}{9}\int\frac{i\mathrm{d}^{4}L}{\left(2\pi\right)^{4}}\frac{\nu^{2}+2L^{2}}{L^{2}\left(L^{2}-\lambda^{2}\right)}\left(\frac{\mathrm{S}_{1}(\nu,L^{2})}{M^{2}g_{\mathrm{A}}^{(0)}}-\frac{1}{L^{2}}\right).$
(36)

Compared to the traditional framework, the hadronic contributions enter
integrals without electroweak parameters and there is no need to separate
$M_{A}$ from the long-distance contributions. Moreover, all electroweak
physics is conveniently included in logarithmic and constant terms. Assuming
the validity of the OPE above some scale $\Lambda$, the difference to the
traditional approach can be expressed as

$\displaystyle\Delta\hat{C}_{\mathrm{V}}$
$\displaystyle=\frac{1}{4}\ln\frac{M_{W}^{2}}{M^{2}_{W}+\Lambda^{2}}+\frac{8\pi^{2}}{3}\int\frac{i\mathrm{d}^{4}L}{\left(2\pi\right)^{4}}\frac{\nu^{2}-L^{2}}{L^{2}\left(L^{2}-M^{2}_{W}\right)}\frac{\mathrm{T}_{3}(\nu,L^{2})}{M^{2}}\Theta\left(\Lambda^{2}+L^{2}\right),$
(37)

with a similar difference for $\hat{C}_{\mathrm{A}}$, up to the possible
contributions discussed around Eq. (34). As expected, the difference in Eq.
(37) vanishes in the physics case $M^{2}\ll\Lambda^{2}\ll M^{2}_{W}$ and in
the more general limit $\Lambda^{2}\ll M^{2}_{W}$.

Our resulting expressions in $d=4$ are in agreement with known results
Marciano:2005ec ; Seng:2018yzq ; Seng:2018qru ; Seng:2020wjq ; Hayen:2021iga ;
Gorchtein:2021fce  in the limit of a large mass of the $W$ boson
$M_{W}\to\infty$. We have also verified our expressions in $d=4$ by
considering the decomposition into the forward scattering amplitudes for the
massless electron and in the limits $m\to 0$ and $M\to\infty$ for the
amplitudes with the massive electron. Although an extra nucleon spin-flip
amplitude is present in the case of massive electrons Tomalak:2017owk , its
effects on all observables are suppressed by the electron mass and can be
neglected for the phenomenology of the neutron decay.

##  9 Relation to the experimental data

Hadron structure in the isospin space is described by isovector amplitudes
${T}_{V^{0}V(A)^{0}}:1\otimes 1$,
${T}_{V^{0}V(A)^{a}}:1\otimes\mathrm{T}^{a}$,
${T}^{\mathrm{V}}_{V^{a}V(A)^{b}}:\mathrm{T}^{a}\otimes\mathrm{T}^{b}$ for
$a=b$ and isoscalar amplitudes
${T}^{\mathrm{S}}_{V^{a}V(A)^{b}}:\mathrm{T}^{a}\otimes\mathrm{T}^{b}$ for
$a\neq b$, where we have used the following decomposition for the tensor with
two space indices $a,b\neq 0$:
${T}_{V^{a}V^{b}}={T}^{\mathrm{V}}_{V^{a}V^{b}}\delta^{ab}+i\varepsilon^{abc}{T}^{\mathrm{S}}_{V^{a}V^{b}}\mathrm{T}^{c}$.
Only ${T}_{V^{0}A^{a}}$ and
${T}_{V^{0}V^{a}},~{}{T}^{\mathrm{S}}_{V^{a}A^{b}}$ contributions enter the
correction to the vector and axial-vector couplings, respectively.

The photon interacts with the quark current through a pure vector contribution
with the following isospin structure:

$\displaystyle\frac{1}{6}+\frac{\mathrm{T}^{3}}{2}.$ (38)

Considering the coupling of two photons to the proton and neutron lines
separately, we can express all $VV$ contributions in terms of the difference
between the proton and neutron amplitudes as

$\displaystyle\left(\frac{1}{36}{T}_{V^{0}V^{0}}+\frac{1}{4}{T}^{\mathrm{V}}_{V^{a}V^{b}}\right)+\frac{1}{6}{T}_{V^{0}V^{a}}\mathrm{T}^{a}=\frac{{T}^{p}+{T}^{n}}{2}+\frac{{T}^{p}-{T}^{n}}{2}\mathrm{T}^{a},$
(39)

with ${T}_{V^{0}V^{a}}=3\left({T}^{p}-{T}^{n}\right)$. However, such an easy
relation does not hold for $VA$ amplitudes. The amplitudes of the type
${T}_{V^{0}A^{a}}$ enter only the correlation function with $\gamma$ and $Z$
couplings, and were constrained by the parity-violating experimental data in
Refs. Seng:2018yzq  and Seng:2018qru . However, due to the sparsity of the
available data, the authors of these works had to rely on modeling for the
connection between amplitudes of different isospin. The amplitudes of the type
${T}^{\mathrm{S}}_{V^{a}A^{b}}$ only enter physical processes with the
coupling of $\gamma$ and $W$ to the nucleon line.

##  10 Conclusions and outlook

We have described the systematic EFT framework for radiative corrections to
the neutron $\beta$ decay and low-energy (anti)neutrino-nucleon scattering
with a robust separation of short- and long-distance contributions. First, we
integrate out electroweak physics and match the Standard Model to the four-
fermion EFT. Working in the $\overline{\mathrm{MS}}$ renormalization scheme
and dimensional regularization allows us to consistently resum large
perturbative logarithms. Performing matching of the four-fermion EFT to the
heavy-baryon chiral perturbation theory, we express low-energy coupling
constants in terms of correlation functions of quark currents. Compared to the
traditional approach, we explicitly specify the scale and scheme dependence in
all steps of the calculation. We have finalized the formalism for the
determination of the vector low-energy coupling constant and are working on
the determination of the axial-vector coupling. Our next steps include an
investigation of opportunities for calculating the relevant correlation
functions on the lattice, an inclusion of higher-order perturbative
contributions, and an evaluation of the long-distance hadronic contributions,
based on both the experimental data and the hadronic models.

Notations

We take the irreducible representation of $\mathrm{SU}(2)$ algebra in the form
of the Pauli matrices, i.e.,
$[\mathrm{T}^{a},\mathrm{T}^{b}]=2i\varepsilon^{abc}\mathrm{T}^{c}$,
$\\{\mathrm{T}^{a},\mathrm{T}^{b}\\}=2\delta^{ab}$, and $\mathrm{T}^{0}=1$.

The vector and axial charge spurions are expressed in terms of the left and
right charge spurions as

$\displaystyle{\textbf{q}}_{V}$
$\displaystyle={\textbf{q}}_{L}+{\textbf{q}}_{R},\qquad{\textbf{q}}_{A}={\textbf{q}}_{L}-{\textbf{q}}_{R}.$
(40)

The spurion combinations that enter the gauge-invariant expressions are

$\displaystyle{{\cal{Q}}}^{W}_{L}=u{\textbf{q}}_{W}u^{\dagger},\qquad{{\cal{Q}}}_{L}=u{\textbf{q}}_{L}u^{\dagger},\qquad{{\cal{Q}}}_{R}=u^{\dagger}{\textbf{q}}_{R}u,\qquad{{\cal{Q}}}_{\pm}=\frac{{{\cal{Q}}}_{L}\pm{{\cal{Q}}}_{R}}{2},$
(41)

where the pion fields in the isospin basis $\pi^{a}$ are combined into the
standard fields

$\displaystyle U=u^{2}=e^{\frac{i\pi^{a}\mathrm{T}^{a}}{F}},$ (42)

and $F\approx 92.4~{}\mathrm{MeV}$ is related to the pion decay constant in
the chiral limit.

We extend the standard definitions of the $\chi\mathrm{PT}$ building blocks by
external sources as

$\displaystyle u_{\rho}$ $\displaystyle=iu^{\dagger}D_{\rho}Uu^{\dagger},\quad
c_{\rho}^{\pm}=-\frac{i}{2}\left(u\left(i\partial_{\rho}{\textbf{q}}_{L}+\left[l_{\rho},{\textbf{q}}_{L}\right]\right)u^{\dagger}\pm
u^{\dagger}\left(i\partial_{\rho}{\textbf{q}}_{R}+\left[r_{\rho},{\textbf{q}}_{R}\right]\right)u\right),$
$\displaystyle u_{\rho}$
$\displaystyle=i\\{u^{\dagger}\left(\partial_{\rho}-ir_{\rho}+ie{\textbf{q}}_{R}A_{\rho}\right)u-u\left(\partial_{\rho}-il_{\rho}-i{\textbf{q}}_{W}\bar{e}\gamma_{\rho}\mathrm{P}_{\mathrm{L}}\nu_{e}+ie{\textbf{q}}_{L}A_{\rho}\right)u^{\dagger}\\},$
$\displaystyle\tilde{D}_{\rho}$
$\displaystyle=\partial_{\rho}+\frac{1}{2}\left[u^{\dagger},\partial_{\rho}u\right]-\frac{i}{2}u^{\dagger}\left(r_{\rho}-e{\textbf{q}}_{R}A_{\rho}\right)u-\frac{i}{2}u\left(l_{\rho}-e{\textbf{q}}_{L}A_{\rho}-{\textbf{q}}_{W}\bar{e}\gamma_{\rho}\mathrm{P}_{\mathrm{L}}\nu_{e}\right)u^{\dagger},$
(43)

without specifying the $\bar{\nu}_{e}\gamma_{\rho}\mathrm{P}_{\mathrm{L}}e$,
which is irrelevant for this paper.

We decompose electromagnetic charge spurions for baryons
${\textbf{q}}^{\mathrm{b}}$ and for quarks ${\textbf{q}}^{\mathrm{q}}$ into
isoscalar $q^{0}$ and isovector $q^{a}$ contributions as

${\textbf{q}}^{\mathrm{b}}=q^{0}+q^{a}\mathrm{T}^{a},\qquad{\textbf{q}}^{\mathrm{q}}=\frac{q^{0}}{3}+q^{a}\mathrm{T}^{a},$
(44)

with physical values $q_{0}=q^{a}=\frac{1}{2}$ for the left and right
spurions, and $q_{0}=q^{a}=1$ and $0$ for the vector and axial spurions,
respectively.

###### Acknowledgements.

I acknowledge many useful discussions and validations with Emanuele
Mereghetti, Vincenzo Cirigliano, Richard Hill, and Wouter Dekens, and useful
correspondence with Bastian Kubis. I thank Emanuele Mereghetti and Wouter
Dekens for reading this manuscript. This work is supported by the US
Department of Energy through the Los Alamos National Laboratory and by LANL’s
Laboratory Directed Research and Development (LDRD/PRD) program under projects
20210968PRD4 and 20210190ER. Los Alamos National Laboratory is operated by
Triad National Security, LLC, for the National Nuclear Security Administration
of U.S. Department of Energy (Contract No. 89233218CNA000001). FeynCalc
Mertig:1990an ; Shtabovenko:2016sxi , LoopTools Hahn:1998yk , and Mathematica
Mathematica  were extremely useful in this work.

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