# Momentum dependent flavor radiative corrections to the coherent elastic
neutrino-nucleus scattering for the neutrino charge-radius determination

M. Atzori Corona<EMAIL_ADDRESS>Dipartimento di Fisica,
Università degli Studi di Cagliari, Complesso Universitario di Monserrato -
S.P. per Sestu Km 0.700, 09042 Monserrato (Cagliari), Italy  Istituto
Nazionale di Fisica Nucleare (INFN), Sezione di Cagliari, Complesso
Universitario di Monserrato - S.P. per Sestu Km 0.700, 09042 Monserrato
(Cagliari), Italy  M. Cadeddu<EMAIL_ADDRESS>Istituto Nazionale di
Fisica Nucleare (INFN), Sezione di Cagliari, Complesso Universitario di
Monserrato - S.P. per Sestu Km 0.700, 09042 Monserrato (Cagliari), Italy  N.
Cargioli<EMAIL_ADDRESS>Dipartimento di Fisica, Università degli
Studi di Cagliari, Complesso Universitario di Monserrato - S.P. per Sestu Km
0.700, 09042 Monserrato (Cagliari), Italy  Istituto Nazionale di Fisica
Nucleare (INFN), Sezione di Cagliari, Complesso Universitario di Monserrato -
S.P. per Sestu Km 0.700, 09042 Monserrato (Cagliari), Italy  F. Dordei
<EMAIL_ADDRESS>Istituto Nazionale di Fisica Nucleare (INFN), Sezione
di Cagliari, Complesso Universitario di Monserrato - S.P. per Sestu Km 0.700,
09042 Monserrato (Cagliari), Italy  C. Giunti<EMAIL_ADDRESS>Istituto Nazionale di Fisica Nucleare (INFN), Sezione di Torino, Via P. Giuria
1, I–10125 Torino, Italy

( , )

###### Abstract

Despite being neutral particles, neutrinos can have a non-zero charge radius,
which represents the only non-null neutrino electromagnetic property in the
standard model theory. Its value can be predicted with high accuracy and its
effect is usually accounted for through the definition of a radiative
correction affecting the neutrino couplings to electrons and nucleons at low
energy, which results effectively in a shift of the weak mixing angle.
Interestingly, it introduces a flavour-dependence in the cross-section.
Exploiting available neutrino-electron and coherent elastic neutrino-nucleus
scattering (CE$\nu$NS) data, there have been many attempts to measure
experimentally the neutrino charge radius. Unfortunately, the current
precision allows one to only determine constraints on its value. In this work,
we discuss how to properly account for the neutrino charge radius in the
CE$\nu$NS cross-section including the effects of the non-null momentum-
transfer in the neutrino electromagnetic form factor, which have been usually
neglected when deriving the aforementioned limits. We apply the formalism
discussed to a re-analysis of the COHERENT cesium iodide and argon samples and
the NCC-1701 germanium data from the Dresden-II nuclear power plant. We
quantify the impact of this correction on the CE$\nu$NS cross-section and we
show that, despite being small, it can not be neglected in the analysis of
data from future high-precision experiments. Furthermore, this momentum
dependence can be exploited to significantly reduce the allowed values for the
neutrino charge radius determination.

##  I Introduction

Over the years, the Standard Model (SM) electroweak theory has been carefully
studied, both theoretically and experimentally, to accurately determine the
couplings between interacting particles. The quest for increasingly higher
precision brought to the introduction of the so-called radiative corrections
Erler and Su (2013), due to higher-order vertex contributions. Available
formalisms have been developed such that most of these radiative corrections
are rather universal in neutral current interactions at low energy, meaning
that they do not depend on the particular particles involved in the process.
In this work, we discuss neutrino interactions, focusing in particular on the
coherent elastic neutrino-nucleus scattering (CE$\nu$NS) process Freedman
(1974). In this case, another non-universal radiative correction has to be
accounted for, the so-called neutrino charge radius (NCR) radiative
correction. The latter represents the only flavor-dependent contribution to
the CE$\nu$NS cross-section, which otherwise would be a completely flavor-
blind process.

Despite the experimental CE$\nu$NS era having just started Akimov  _et al._
(2017), the precision already achieved in the current measurements Akimov  _et
al._ (2022, 2021); Colaresi  _et al._ (2022) permits to put stringent
constraints on different beyond the SM (BSM) quantities (see e.g. Refs. Atzori
Corona  _et al._ (2022, 2023); Cadeddu  _et al._ (2023)), requiring thus a
careful treatment of the radiative corrections affecting this specific
process. In this work, we revisit the canonical description of the neutrino
charge radius radiative correction to account for its momentum dependence, due
to the non-null momentum transfer in the neutrino electromagnetic form factor,
which has been so far neglected in CE$\nu$NS experimental and phenomenological
studies. We quantify the impact of this correction and we later apply it to
the available CE$\nu$NS data analyses, namely the COHERENT cesium iodide (CsI)
Akimov  _et al._ (2022) and argon (Ar) Akimov  _et al._ (2021) samples and the
NCC-1701 germanium (Ge) data from the Dresden-II nuclear power plant Colaresi
_et al._ (2022). Nevertheless, the formalism discussed in this work would
impact also other low-energy neutrino interactions, like e.g. neutrino elastic
scattering processes on electrons ($\nu$ES).

##  II Theoretical framework

###  II.1 Radiative corrections to the coherent elastic neutrino-nucleus
scattering cross section

Low-energy neutrino elastic scattering processes on both nuclei and electrons
are of fundamental interest in modern electroweak physics. The SM theory well
describes the cross-section for both processes. Focusing on the SM CE$\nu$NS
interactions, the differential cross section as a function of the nuclear
kinetic recoil energy $T_{\mathrm{nr}}$ for a neutrino $\nu_{\ell}$
($\ell=e,\mu,\tau$) scattering off a spin-zero nucleus $\mathcal{N}$ with $Z$
protons and $N$ neutrons is given by Freedman (1974); Drukier and Stodolsky
(1984); Barranco  _et al._ (2005); Patton  _et al._ (2012)

$\dfrac{d\sigma_{\nu_{\ell}\text{-}\mathcal{N}}}{dT_{\mathrm{nr}}}(E,T_{\mathrm{nr}})=\dfrac{G_{\text{F}}^{2}M}{\pi}\left(1-\dfrac{MT_{\mathrm{nr}}}{2E^{2}}\right)\left[g_{V}^{p}\left(\nu_{\ell}\right)ZF_{Z}\left(|\vec{q}|^{2}\right)+g_{V}^{n}NF_{N}\left(|\vec{q}|^{2}\right)\right]^{2},$
(1)

where $G_{\text{F}}$ is the Fermi constant, $E$ is the neutrino energy and $M$
the nuclear mass, while $F_{Z}\left(|\vec{q}|^{2}\right)$ and
$F_{N}\left(|\vec{q}|^{2}\right)$ are the proton and neutron form factors,
respectively, which account for the contribution of the nuclear structure to
the process. In particular, they represent the Fourier transform of the
nuclear proton and neutron density distributions, so that they account for the
spatial distribution of nucleons inside the target nucleus. In Eq. (1),
$g_{V}^{p}$ and $g_{V}^{n}$ are the neutrino couplings to protons and
neutrons, respectively, whose tree-level values are given by

$\displaystyle
g_{V}^{p}=\frac{1}{2}-2\;\textrm{sin}^{2}\;\vartheta_{\text{W}}\simeq
0.0227,\quad\quad g_{V}^{n}=-\frac{1}{2}\,,$ (2)

where $\vartheta_{\text{W}}$ is the so-called weak mixing angle, also known as
the Weinberg angle, a key parameter of the electroweak theory whose value
evaluated at zero-momentum transfer is equal to
$\textrm{sin}^{2}\vartheta_{\text{W}}(q^{2}=0)=\hat{s}^{2}_{0}=0.23863(5)$
Group (2022). These values are evaluated neglecting all the higher-order
vertex corrections to the CE$\nu$NS process. More precise values are
determined by taking into account the radiative corrections in the Minimal
Subtractions ($\overline{\mathrm{MS}}$) scheme, following Refs. Erler and Su
(2013); Group (2022); Atzori Corona  _et al._ (2023), with a well-defined
subtraction of singular terms arising in dimensional regularization, giving
rise to expressions with a logarithmic energy scale dependence that is
governed by a renormalization group equation (RGE). Choosing this energy scale
equal to the momentum transfer of the process under consideration will, in
general, avoid spurious logarithms Erler and Ramsey-Musolf (2005). Thus,
accounting for radiative corrections, the couplings become

$\displaystyle g_{V}^{p}(\nu_{\ell})$
$\displaystyle=\rho\left(\frac{1}{2}-2\;\textrm{sin}^{2}\vartheta_{\text{W}}\right)+2\;{\raisebox{-1.0pt}{\rotatebox{90.0}{\Bowtie}}}_{WW}+\Box_{WW}-2\phi_{\nu_{\ell}W}+\rho(2\boxtimes_{ZZ}^{uL}+\boxtimes_{ZZ}^{dL}-2\boxtimes_{ZZ}^{uR}-\boxtimes_{ZZ}^{dR})$
(3) $\displaystyle g_{V}^{n}$
$\displaystyle=-\frac{\rho}{2}+2\Box_{WW}+{\raisebox{-1.0pt}{\rotatebox{90.0}{\Bowtie}}}_{WW}+\rho(2\boxtimes_{ZZ}^{dL}+\boxtimes_{ZZ}^{uL}-2\boxtimes_{ZZ}^{dR}-\boxtimes_{ZZ}^{uR}).$
(4)

The quantities $\Box_{WW}$,
${\raisebox{-1.0pt}{\rotatebox{90.0}{\Bowtie}}}_{WW}$ and
$\boxtimes_{ZZ}^{fX}$, are the radiative corrections associated with the so-
called $WW$ box diagram, the $WW$ crossed-box and the $ZZ$ box respectively,
while $\rho=1.00063$ represents a low-energy correction for neutral-current
processes. These contributions have the form

$\Box_{WW}=-{\hat{\alpha}_{Z}\over
2\pi\hat{s}_{Z}^{2}}\left[1-{\hat{\alpha}_{s}(M_{W})\over
2\pi}\right],\qquad\qquad\hskip
3.0pt{\raisebox{-1.0pt}{\rotatebox{90.0}{\Bowtie}}}_{WW}={\hat{\alpha}_{Z}\over
8\pi\hat{s}_{Z}^{2}}\left[1+{\hat{\alpha}_{s}(M_{W})\over\pi}\right],$ (5)
$\boxtimes_{ZZ}^{fX}=-{3\hat{\alpha}_{Z}\over
8\pi\hat{s}_{Z}^{2}\hat{c}_{Z}^{2}}(g_{LX}^{\nu_{\ell}f})^{2}\left[1-{\hat{\alpha}_{s}(M_{Z})\over\pi}\right],$
(6)

where $M_{W}$ is the $W$ boson mass, $\hat{\alpha}_{Z}$ is the electromagnetic
coupling constant evaluated at the $Z$ boson mass, $\hat{\alpha}_{s}$ is the
strong coupling constant, $f\in\\{u,d\\}$ and $X\in\\{L,R\\}$ is the parity.
Note that in Eq. (6) all the $(g_{LX})^{\nu_{\ell}f}$ couplings are evaluated
at the lowest order of perturbation theory and the weak mixing angle is
evaluated at the $Z$-pole, i.e.
$\textrm{sin}^{2}\vartheta_{\text{W}}(M_{Z})=\hat{s}^{2}_{Z}=0.23122(4)$ Group
(2022). The remaining radiative term in Eq. (3), namely $\phi_{\nu_{\ell}W}$,
is referred to as the neutrino charge radius contribution. In the SM, it is
defined as Erler and Su (2013)

$\phi_{\nu_{\ell}W}=-{\alpha\over 6\pi}\left(\ln{M_{W}^{2}\over
m_{\ell}^{2}}+{3\over 2}\right),$ (7)

where $\alpha$ is the low-energy limit of the electromagnetic coupling, and
$m_{\ell}$ is the mass of the charged lepton with flavour $\ell$. Clearly,
this radiative contribution depends on the neutrino flavor, thus introducing a
flavor dependence in the neutrino-proton coupling alone. Numerically, the
values of the couplings correspond to

$\displaystyle g_{V}^{p}(\nu_{e})\simeq 0.0381,\quad\quad
g_{V}^{p}(\nu_{\mu})\simeq 0.0299,\quad\quad g_{V}^{p}(\nu_{\tau})\simeq
0.0255,\quad\quad g_{V}^{n}\simeq-0.5117\,.$ (8)

In the following, we will analyse in detail the different formalisms used to
derive the NCR correction and we will focus in particular on its momentum
dependence.

###  II.2 Neutrino charge radius radiative correction

Neutrinos are neutral particles, therefore, they can not couple directly with
photons. However, even if the electric charge is null, the electric form
factor, $\mathbb{f}_{Q}(q^{2})$, carries non-trivial information about the
neutrino electric properties, like the neutrino charge radius, millicharge and
magnetic moment. In principle, a neutral particle can be characterized by a
superposition of two different charge distributions of opposite signs
described by an electric form factor which is nonzero only for momentum
transfers different from zero, $q^{2}\neq 0$ Giunti and Studenikin (2015).
Expanding the form factor in a series of powers of $q^{2}$ we get

$\mathbb{f}_{Q}(q^{2})=\mathbb{f}_{Q}(0)+q^{2}\dfrac{d\mathbb{f}_{Q}(q^{2})}{dq^{2}}\Big{|}_{q^{2}=0}+\dots\,.$
(9)

In the “Breit frame”, the charge form factor depends only on $|\vec{q}|$ and
it can be interpreted as the Fourier transform of a spherically symmetric
charge distribution, $\rho(r)$, so that Giunti and Studenikin (2015)

$\mathbb{f}_{Q}(q^{2})=\int\rho(r)e^{-i\vec{q}\cdot\vec{r}}d^{3}r=\int\rho(r)\dfrac{\sin(qr)}{qr}d^{3}r\,.$
(10)

From this interpretation, it is easy to understand that the first term in the
expansion in Eq. (9), $\mathbb{f}_{Q}(0)$, has to be zero, since neutrinos are
neutral, while the second term is identified as the neutrino charge radius,
i.e. the radius of the electric charge distribution. By deriving the
expression in Eq. (10) with respect to $q^{2}$ and taking the limit for
$q^{2}\rightarrow 0$, we obtain

$\lim_{q^{2}\rightarrow
0}\dfrac{d\mathbb{f}_{Q}(q^{2})}{dq^{2}}=\int\rho(r)\dfrac{r^{2}}{6}d^{3}r=\dfrac{\langle
r^{2}\rangle}{6}\,,$ (11)

where we introduced the squared neutrino charge radius $\langle r^{2}\rangle$
which is given by

$\langle r^{2}\rangle\equiv
6\dfrac{d\mathbb{f}_{Q}(q^{2})}{dq^{2}}\Big{|}_{q^{2}=0}\,.$ (12)

Let us note that $\langle r^{2}\rangle$ has no defined sign because $\rho(r)$
is not a positively defined quantity.

Figure 1: Diagrams describing the neutrino charge radius contributions to the
photon-neutrino vertices.

Practically, the charge radius of a neutrino is generated by a loop insertion
into the $\nu_{\ell}$ line, where $W$ boson(s) and charged lepton(s) $\ell$
can enter, as shown by the $WW\ell$ loop (left diagram) and the $\ell\ell W$
loop (right diagram) in Fig. 1. These diagrams’ contribution can be
calculated, and according to Refs. Bernabeu  _et al._ (2000, 2002, 2004) the
NCR corresponds to a physical observable, being finite and gauge invariant. In
particular, the SM calculation gives

$\langle r^{2}_{\nu_{\ell}}\rangle_{\rm
SM}=-\dfrac{G_{F}}{2\sqrt{2}\pi^{2}}\Big{[}3-2\ln\Big(\dfrac{m_{\ell}^{2}}{M_{W}^{2}}\Big{missing})\Big{]}\,,$
(13)

where the sum of two contributions appears. Namely, one constant term arises
from the $WW\ell$ loop diagram, where the photon interacts with the $W$ boson,
and another is generated by the photon’s interaction with the lepton $\ell$.
The latter generates an electroweak logarithm, which is divergent in the UV
range, that is regularized at the lepton mass, $m_{\ell}$, thus introducing a
dependence of the NCR from the lepton flavor Erler and Su (2013); Bernabeu
_et al._ (2004). In the SM, the numerical values of the neutrino charge radius
for the different neutrino flavors are

$\displaystyle\langle r^{2}_{\nu_{e}}\rangle$ $\displaystyle\simeq-8.3\times
10^{-33}\,\mathrm{cm}^{2}\,,$ (14) $\displaystyle\langle
r^{2}_{\nu_{\mu}}\rangle$ $\displaystyle\simeq-4.8\times
10^{-33}\,\mathrm{cm}^{2}\,,$ (15) $\displaystyle\langle
r^{2}_{\nu_{\tau}}\rangle$ $\displaystyle\simeq-3.0\times
10^{-33}\,\mathrm{cm}^{2}\,.$ (16)

The neutrino charge radius affects the scattering of neutrinos with charged
particles. Therefore, it contributes to the $\nu$ES process, whereas in the
case of CE$\nu$NS it contributes only to the neutrino-proton coupling, and not
to the neutron one, as visible in Eqs. (3) and (4). There, the NCR
contribution modifies the proton coupling constant Giunti and Studenikin
(2015); Atzori Corona  _et al._ (2022); Cadeddu  _et al._ (2020)

$g_{V}^{p}(\nu_{\ell})\rightarrow\tilde{g}_{V}^{p}(\nu_{\ell})-\dfrac{2}{3}M_{W}^{2}\langle
r^{2}_{\nu_{\ell}}\rangle\sin^{2}\vartheta_{W}=\tilde{g}_{V}^{p}(\nu_{\ell})-\dfrac{\sqrt{2}\pi\alpha}{3G_{F}}\langle
r^{2}_{\nu_{\ell}}\rangle\,,$ (17)

where $\tilde{g}_{V}^{p}(\nu_{\ell})\simeq 0.0184$ is the neutrino-proton
coupling without the contribution of the SM NCR. The latter appears thus as a
radiative correction, as shown in Eqs. (3) and (7), that practically generates
an effective shift of $\sin^{2}\\!\vartheta_{W}$. In the case of
$\bar{\nu}_{\ell}\text{-}\mathcal{N}$ scattering, it is sufficient to operate
the substitutions: $g_{V}^{p,n}\to-g_{V}^{p,n}$ and
$\langle{r}_{\nu_{\ell}}\rangle\to\langle{r}_{\bar{\nu}_{\ell}}\rangle=-\langle{r}_{\nu_{\ell}}\rangle$.
Therefore, the NCR of neutrinos and antineutrinos contribute with the same
sign to the shift of the weak mixing angle in the CE$\nu$NS cross-section.

Despite a measurement of the NCR would represent a fundamental test of the SM,
the available data are still insufficient to provide a first determination. So
far, only constraints have been put on its value Atzori Corona  _et al._
(2022); De Romeri  _et al._ (2023); Deniz  _et al._ (2010); Ahrens  _et al._
(1990); Group (2022). One has to keep in mind that when considering BSM
effects that affect the NCR, it is also possible to have so-called off-
diagonal flavor-changing contributions, which are often referred to as
transition charge radii. Using available data, limits on these quantities have
been derived, see e.g. Ref. Atzori Corona  _et al._ (2022). Nevertheless, in
the following, we will focus only on diagonal NCRs. In this way, we probe the
values of the NCR in the SM. However, since it is also likely that BSM physics
generates off-diagonal neutrino charge radii that are much smaller than the
diagonal ones and that can thus be neglected in a first approximation, new
physics models can also be tested in this scenario.

Only recently, the RGE formalism was developed to make the weak mixing angle
value not dependent on the particular scattering particle, such that the NCR
is incorporated as an additional radiative correction to the coupling constant
as shown in Eq. (17) Erler and Ramsey-Musolf (2005); Erler and Ferro-Hernández
(2018). Historically, the NCR was considered as one of the radiative
corrections contributing to the running of the weak mixing angle in the case
of neutrino scattering processes Marciano and Parsa (2003); Sehgal (1985). In
this scenario, the running is evaluated only at the one-loop level using an
effective form factor $k_{\nu_{\ell}}(q^{2})$ which allows one to define the
weak mixing angle at a certain energy scale starting from its value at the
$Z$-mass energy scale, namely

$\sin^{2}\vartheta_{W}(q^{2})=k_{\nu_{\ell}}(q^{2})\sin^{2}\vartheta_{W}(M_{Z})\equiv
k_{\nu_{\ell}}(q^{2})\hat{s}^{2}_{Z}\,,$ (18)

where $k_{\nu_{\ell}}(q^{2})$ is defined by

$\displaystyle k_{\nu_{\ell}}(q^{2})$
$\displaystyle=1-\dfrac{\alpha}{2\pi\hat{s}^{2}_{Z}}\Big{[}2\sum_{f}(T_{3f}Q_{f}-2\hat{s}^{2}_{Z}Q_{f}^{2})J_{f}(q^{2})++\dfrac{\hat{c}^{2}_{Z}}{3}+\dfrac{1}{\hat{c}^{2}_{Z}}\Big{(}\dfrac{19}{8}+\dfrac{17}{4}\hat{s}^{2}_{Z}+3\hat{s}_{Z}^{4}\Big{)}$
(19)
$\displaystyle-\Big{(}\dfrac{7}{2}\hat{c}_{Z}^{2}+\dfrac{1}{12}\Big{)}\ln\hat{c}^{2}_{Z}\Big{]}-\dfrac{\alpha}{\pi\hat{s}^{2}_{Z}}\Big{[}-R_{\ell}(q^{2})+\dfrac{1}{4}\Big{]}\,,$

and we defined $\hat{s}^{2}_{Z}\equiv\sin^{2}\vartheta_{W}(M_{Z})$ and
$\hat{c}^{2}_{Z}\equiv\cos^{2}\vartheta_{W}(M_{Z})$. $T_{3f}$ is the fermion
weak isospin third component, $Q_{f}$ the fermion charge, and the sum runs
over all the fermions $f$. It is crucial to notice that while the
$J_{f}(q^{2})$ term is relative to all the fermions $f$, the $R_{\ell}(q^{2})$
one depends on the lepton flavor $\ell$ of the neutrino involved in the
scattering process. These two terms are defined by the following integrals

$\displaystyle J_{f}(q^{2})$
$\displaystyle=\int_{0}^{1}dx\,x(1-x)\ln[\dfrac{m_{f}^{2}-q^{2}x(1-x)}{M_{Z}^{2}}\Big{]}\,,$
(20) $\displaystyle R_{\ell}(q^{2})$
$\displaystyle=\int_{0}^{1}dx\,x(1-x)\ln[\dfrac{m_{\ell}^{2}-q^{2}x(1-x)}{M_{W}^{2}}\Big{]}\,,$
(21)

showing a very similar structure but for the different masses in the
denominator. If no neutrino is involved in the scattering process, the last
term in Eq. (19), which involves also $R_{\ell}(q^{2})$, vanishes. In this
case one retrieves the definition of the weak mixing angle running for a
generic scattering process, which can thus be compared with the RGE running
(see also Ref. Erler and Ramsey-Musolf (2005)) in the zero momentum limit,
obtaining

$k_{\nu_{\ell}}(q^{2}=0)\hat{s}^{2}_{Z}-\hat{s}_{0}^{2}(\mathrm{RGE})=-\dfrac{2\alpha}{9\pi}+\mathcal{O}(\alpha^{2})\,.$
(22)

It can be noticed that the difference of the weak mixing angle values consists
only in a constant term (see also Appendix A of Ref. Davoudiasl  _et al._
(2023)). Therefore, the last term in Eq. (19), in the $q^{2}\rightarrow 0$
limit, reduces to the NCR flavor-dependent radiative correction as defined in
the RGE formalism (see Eq. (7)), which can thus be also written as

$\phi_{\nu_{\ell}W}=-\dfrac{\alpha}{\pi}\left(-R_{\ell}(0)+\dfrac{1}{4}\right)\quad\quad\mathrm{with}\quad\quad
R_{\ell}(0)=\dfrac{1}{6}\ln\dfrac{m_{\ell}^{2}}{M_{W}^{2}}\,.$ (23)

However, the interesting aspect of the latter definition is that it can be
easily extended to consider the case of a nonzero momentum transfer just by
letting the momentum in the $R_{\ell}$ integral be different from zero. This
represents a more realistic scenario since all the experiments are run at a
nonzero momentum transfer. By doing so, we are explicitly distinguishing
between the NCR radiative correction and the actual physical NCR. In fact, the
NCR is by definition independent of the momentum transfer, as it represents
the value of the form factor derivative at zero momentum transfer, as shown in
Eq. (12), while the NCR radiative correction has a momentum dependence, as it
enters as a correction to the coupling.
Thus, we can introduce such a momentum dependence in the NCR radiative
correction by defining

$\displaystyle\phi_{\nu_{\ell}W}^{\mathrm{eff}}(q^{2})$
$\displaystyle=-\dfrac{\alpha}{\pi}\left(-R_{\ell}(q^{2})+\dfrac{1}{4}\right)$
$\displaystyle=-\dfrac{\alpha}{\pi}\left(-\int_{0}^{1}dx\,x(1-x)\ln[\dfrac{m_{\ell}^{2}-q^{2}x(1-x)}{M_{W}^{2}}\Big{]}+\dfrac{1}{4}\right)\,,$
(24)

which should then be used in the experimental extraction of the neutrino
charge radius whenever $q^{2}\neq 0$. Practically speaking, one should
carefully correct for the momentum dependence, and this can be done
introducing a “neutrino charge radius form factor”. We define this form factor
by isolating the momentum-dependent NCR with respect to the SM picture, so
basically as

$\mathcal{F}_{\nu_{\ell}}(T_{\rm nr})=\dfrac{\langle
r^{2}_{\nu_{\ell}}\rangle^{\rm eff}(T_{\rm nr})}{\langle
r^{2}_{\nu_{\ell}}\rangle^{\rm eff}(0)}\,\equiv\dfrac{\langle
r^{2}_{\nu_{\ell}}\rangle^{\rm eff}(T_{\rm nr})}{\langle
r^{2}_{\nu_{\ell}}\rangle^{\rm SM}}\,,$ (25)

where we introduced an effective NCR111Since the NCR is a well-defined
physical quantity, as also shown in Eq. (12), it is worth stressing that this
effective definition has not to be considered as the actual definition of the
neutrino charge radius. The charge radius is by definition evaluated at zero-
momentum transfer. This effective definition can be thus viewed as a form
factor which incorporates the momentum dependence, in analogy to what is
commonly discussed for the nuclear form factor Atzori Corona  _et al._
(2023)., namely

$\langle r_{\nu_{\ell}}^{2}\rangle^{\rm
eff}=\dfrac{6G_{F}}{\sqrt{2}\pi\alpha}\phi^{\rm
eff}_{\nu_{\ell}W}(q^{2})=-\dfrac{G_{F}}{2\sqrt{2}\pi^{2}}\Big{[}3-12R_{\ell}(q^{2})\Big{]}\,,$
(26)

so that, we can obtain the neutrino couplings by using this effective NCR
radiative correction instead of the classical one inside Eq. (17) and
analogously for the case of the neutrino-electron coupling. In this manner,
the NCR form factor is naturally normalized to unity for a zero-momentum
transfer and decreases as the momentum grows, with a different steepness
depending on the considered neutrino flavor. The momentum dependence of
$\phi_{\nu_{\ell}W}^{\rm eff}(q^{2})$ comes from $R_{\ell}(q^{2})$ as defined
in Eq. (21), and its impact becomes relevant for momenta larger than the mass
of the corresponding charged lepton $\ell$ that enters the loops in Fig. 1,
i.e. $q^{2}\gtrsim m_{\ell}^{2}$. Thus, for $\nu_{e}$ processes the correction
to the couplings becomes visible for $q\gtrsim 0.5\,\mathrm{MeV}$, while for
$\nu_{\mu}$ only above $\sim 100\,\mathrm{MeV}$. In the case of $\nu_{\tau}$
an even higher momentum transfer is needed to appreciate any difference, which
is not relevant for the typical momenta of the $\nu$ES and CE$\nu$NS
experiments that we are considering in this work. We remind here that the
momentum transfer is derived as a function of the nuclear or electron recoil
energy as

$Q^{2}=-q^{2}\simeq 2m_{\rm tar}T\,,$ (27)

where $m_{\rm tar}$ is the target mass, so either the nuclear or the electron
mass, and $T$ is the nuclear, $T_{\rm nr}$, or electron, $T_{e}$, recoil
energy, depending on the case. In Fig. 2, we compare the SM value of the
electron and muon NCRs with its effective value as a function of the nuclear
recoil energy (considering a cesium nucleus). Moreover, in the lower panel of
the figure we show the behaviour of the NCR form factor,
$\mathcal{F}_{\nu_{\ell}}$, as a function of the nuclear recoil energy for
both flavors.

Figure 2:  (a) (Upper) Comparison between the value of the NCR at zero-
momentum transfer (solid lines) and the effective definition (dashed lines) in
the case of the muonic flavor (green/lime) and the electron one (orange/red)
as a function of the nuclear recoil energy for a cesium target. (Lower)
Effective form factor as a function of the nuclear recoil energy for the muon
flavor (green) and the electron one (orange). (b) Neutrino-proton coupling for
the CE$\nu$NS process on cesium nuclei as a function of the nuclear recoil
energy. The solid lines refer to the couplings considering a constant NCR
radiative correction, while the dashed ones to the momentum-dependent neutrino
charge radius radiative correction case. The vertical light blue lines and the
shaded area indicate the typical momentum transfers of CE$\nu$NS experiments.

We can now consider the effect of the NCR correction on the neutrino-proton
coupling. In Fig. 2 we show the variation of the latter as a function of the
nuclear recoil energy by comparing the two NCR radiative correction
definitions. If we consider the typical momentum transfer (and thus recoil
energy) of CE$\nu$NS experiments, $Q\sim 10-100\,\mathrm{MeV}$, as shown by
the shaded light blue area, we can notice that the variation of the coupling
is clearly non-negligible for the $\nu_{e}$ case. The effect on the
$\nu_{\mu}$ proton coupling is, as expected, significant for higher energies,
so that, for $\nu_{\mu}$ CE$\nu$NS at the current precision one can still
employ a constant NCR radiative correction. Quantitatively, considering the
recoil energy region of interest for CE$\nu$NS, the variation of the $\nu_{e}$
proton coupling due to the NCR correction momentum dependence is between $\sim
10-20\%$, while we see that there is almost no effect for the other two
neutrino flavors.

The overall effect on the CE$\nu$NS cross section is small. Indeed, this
effect involves mostly $\nu_{e}$, so that in the case of COHERENT only one
flux component is affected. Moreover, the variation affects only the coupling
to protons, which is naturally suppressed by the weak mixing angle, so in
practice, we are dealing with a minor effect. We estimate that the systematic
bias of the $\nu_{e}-\mathcal{N}$ scattering cross section is around $1-2\%$,
which is an effect of about 20% with respect to the current systematic
uncertainties affecting CE$\nu$NS for the CsI COHERENT dataset Akimov  _et
al._ (2022). For reference, the current statistical precision on the flux
averaged CE$\nu$NS cross section measured by COHERENT is about 15-18% for the
CsI data Akimov  _et al._ (2022) and about $\sim 32\%$ for the Ar one Akimov
_et al._ (2021). Nevertheless, for future measurements, it will become
imperative to account for the momentum dependence of the NCR correction,
especially as the community is putting a great effort into reaching the high
precision frontier. Moreover, if one wants also to perform the first neutrino
charge radius measurement, this effect will heavily mislead its extracted
value, as the quantity that one measures, in particular in the case of the
electron neutrinos, is the effective NCR, which has to be corrected for the
momentum dependence in order to extract the physical NCR.

For completeness, we briefly discuss here also the comparison with the
radiative corrections calculated in Ref. Tomalak  _et al._ (2021), in which an
Effective Field Theory (EFT) approach has been employed to determine them for
the CE$\nu$NS process. The formalism is rather different with respect to the
one considered in our work, however, we estimate that the two descriptions
should be equivalent. In the EFT approach, the flavor-dependent contribution,
given by the neutrino charge radius, has been evaluated through a polarization
diagram, which is indeed equivalent to the integral defined in Eq. (21). In
Ref. Tomalak  _et al._ (2021), the authors themselves compared the
prescription commonly adopted in literature Bernabeu  _et al._ (2004); Cadeddu
_et al._ (2018) with their EFT approach, considering however only the zero-
momentum limit, so that a more complete and careful comparison is left for
future studies.

##  III Results

In this section, we will present the results of the analysis of the diagonal
neutrino charge radii, $\langle r^{2}_{\nu_{e}}\rangle$ and $\langle
r^{2}_{\nu_{\mu}}\rangle$, obtained using the latest COHERENT cesium iodide
and argon data-set, both alone and in combination, with the germanium NCC-1701
data from the Dresden-II nuclear reactor power plant222To model the
antineutrino spectra from the Dresden-II reactor we have considered the so-
called HMVE parametrization, obtained by combining the expected spectra for
antineutrino energies above 2 MeV from Ref. Mueller  _et al._ (2011), that we
indicate as HM, with the low energy part determined by Ref. Vogel and Engel
(1989), that we indicate as VE. Nevertheless, the impact of the particular
antineutrino flux on the results obtained is negligible. Moreover, we use the
Fef quenching factor Colaresi  _et al._ (2022).. To analyse the data, we
follow the same procedures described in Ref. Atzori Corona  _et al._ (2022).
In particular, we focus on the comparison between the neutrino charge radii
results obtained from CE$\nu$NS data that are present in literature (see e.g.
Ref. Atzori Corona  _et al._ (2022) for a strict comparison, given that the
same formalism has been used, but also Ref. De Romeri  _et al._ (2023)), which
are obtained considering the NCR correction without any momentum dependence
(i.e. $\mathcal{F}_{\nu_{\ell}}\equiv 1$), with our reanalysis in which we
include the impact of a momentum-dependent form factor (i.e.
$\mathcal{F}_{\nu_{\ell}}(T_{\rm nr})$). In the latter case, we are attempting
to extract the physical value of the NCR at zero momentum transfer, whereas,
neglecting the momentum dependence, one is measuring only an average effective
neutrino charge radius relative to the typical momentum transfer of the
experiment.

Figure 3: Allowed regions at 90% CL from the analysis of the latest COHERENT
CsI and Ar data (COH), Dresden-II (DII) and their combination (COH+DII) in the
$\langle r^{2}_{\nu_{e}}\rangle$ vs $\langle r^{2}_{\nu_{\mu}}\rangle$ plane
in case of momentum-independent (left) and -dependent (right) neutrino charge
radii correction. The red cross indicates the SM values in Eqs. (14) and (15).

By using the form factor $\mathcal{F}_{\nu_{\ell}}$ definition in Eq. (25) and
shown in Fig. 2, we assume that possible BSM contributions affect only the
value of the neutrino charge radius defined at zero-momentum transfer and the
momentum dependence of the NCR radiative correction is the same as in the SM.
Therefore, the neutrino-proton coupling defined in Eq. (17) is modified by

$g_{V}^{p}(\nu_{\ell},T_{\rm
nr})\rightarrow\tilde{g}_{V}^{p}(\nu_{\ell})-\dfrac{\sqrt{2}\pi\alpha}{3G_{F}}\langle
r^{2}_{\nu_{\ell}}\rangle\mathcal{F}_{\nu_{\ell}}(T_{\rm nr})\,,$ (28)

where $\langle r^{2}_{\nu_{\ell}}\rangle$ is the physical value of the
neutrino charge radius at zero-momentum transfer that we aim to measure
through the data.

The results from the analysis of COHERENT CsI, Ar and Dresden-II Ge data are
shown in Fig. 3 at $90\%$ CL, where the left plot shows the
$\mathcal{F}_{\nu_{\ell}}\equiv 1$ limit, while in the right plot a momentum-
dependent charge radius form factor $\mathcal{F}_{\nu_{\ell}}(T_{\rm nr})$ has
been used. Since reactors provide only a flux of electron antineutrinos, the
analysis of the Dresden-II data results in two degenerate bands, in
correspondence of the SM $\langle r^{2}_{\nu_{e}}\rangle$ value and for a
large negative one, which produces a degenerate value of the CE$\nu$NS cross-
section as defined in Eq. (1). Instead, the COHERENT CsI and Ar data analyses
produce 4 closed contours, since they are also sensitive to the muonic flavor.
These allowed regions correspond to the SM values of the electron and muon
neutrino charge radii and to large negative charge radius values which produce
a degenerate cross-section.

Figure 4: Allowed contours at 90% CL from the combined analysis of COHERENT
and Dresden-II data in the $\langle r^{2}_{\nu_{e}}\rangle$ vs $\langle
r^{2}_{\nu_{\mu}}\rangle$ plane (allowing only for diagonal NCR
contributions). The red cross indicates the SM value of the neutrino charge
radii as reported in Eqs. (14) and (15). We compare the results with the
current best limits on $\langle r^{2}_{\nu_{\mu}}\rangle$ from BNL-E7342
Ahrens  _et al._ (1990) (gold) and $\langle r^{2}_{\nu_{e}}\rangle$ from
TEXONO2 Deniz  _et al._ (2010) (brown).

By comparing the two figures in Fig. 3 we observe that the effect of the NCR
form factor leads, as expected, to a small shift of the Dresden-II bands,
while the closed contours allowed by the COHERENT CsI and Ar data are more
significantly affected. This can be understood considering that COHERENT and
Dresden-II data refer to rather different momentum transfer regimes. Namely,
the Dresden-II data come from a reactor experiment, which is operated at a
much lower energy. Moreover, all of the obtained contours are slightly
enlarged by the introduction of the form factor.
The main impact of accounting for the NCR form factor is that, by combining
different measurements, is possible to significantly reduce the allowed
regions in the parameter space. Indeed, the different momentum transfer
regimes relative to the various data, produce a reduced overlap between the
allowed regions for non-SM values of $\langle r^{2}_{\nu_{e}}\rangle$, as
shown in Fig. 4, where the current best limits from BNL-E734 Ahrens  _et al._
(1990) on $\langle r^{2}_{\nu_{\mu}}\rangle$ and TEXONO Deniz  _et al._ (2010)
on $\langle r^{2}_{\nu_{e}}\rangle$ are also shown. Interestingly, the SM
predicted values for the neutrino charge radii fall within the allowed regions
from all the experimental data.

Finally, in Fig. 5, we show the marginal $\Delta\chi^{2}$’s for $\langle
r^{2}_{\nu_{e}}\rangle$ (left) and $\langle r^{2}_{\nu_{\mu}}\rangle$ (right)
obtained from the combined COHERENT and Dresden-II data in the two NCR form
factor regimes. As expected, the momentum dependence impacts the constraints
only for the electron flavor, leaving the results for the muon flavor
practically unchanged. In particular, the presence of the NCR form factor
decreases the significance of the allowed $\langle r^{2}_{\nu_{e}}\rangle$
values particularly different from the SM prediction.

Figure 5: Marginal $\Delta\chi^{2}$’s for $\langle{r}_{\nu_{e}}^{2}\rangle$
(left) and $\langle{r}_{\nu_{\mu}}^{2}\rangle$ (right) obtained from the
analysis of the COHERENT (CsI and Ar) data in combination with Dresden-II
considering $\mathcal{F}_{\nu_{\ell}}=1$ (dot-dashed magenta) and
$\mathcal{F}_{\nu_{\ell}}(T_{\rm nr})$ (solid blue). The red line near the
origin indicates the SM predictions. The short vertical dashed lines show the
lower and upper 90% bounds on $\langle{r}_{\nu_{e}}^{2}\rangle$ obtained in
the TEXONO Deniz  _et al._ (2010) (left) and on
$\langle{r}_{\nu_{\mu}}^{2}\rangle$ BNL-E734 Ahrens  _et al._ (1990) (right)
experiments.

The numerical results obtained from our reanalysis are summarized in Tab. 1.
In particular, we obtain a competitive limit on $\langle
r^{2}_{\nu_{e}}\rangle$ at 90% CL, namely

$-9.5<\langle{r_{\nu_{e}}^{2}}\rangle\,[10^{-32}\,\mathrm{cm}^{2}]<5.5,$ (29)

and on $\langle r^{2}_{\nu_{\mu}}\rangle$,

$-59.2<\langle{r_{\nu_{\mu}}^{2}}\rangle\,[10^{-32}\,\mathrm{cm}^{2}]<-51.0\quad\mathrm{and}\quad-5.9<\langle{r_{\nu_{\mu}}^{2}}\rangle[10^{-32}\,\mathrm{cm}^{2}]<4.1.$
(30)

While the bound on $\langle{r_{\nu_{\mu}}^{2}}\rangle$ shown in Fig. 5 (right)
is practically unchanged with respect to the result for
$\mathcal{F}_{\nu_{\ell}}(T_{\rm nr})=1$ (dot-dashed curve), where no momentum
dependence is considered, the constraint set on $\langle
r^{2}_{\nu_{e}}\rangle$ in Fig. 5 (left) is largely affected, as the
$\Delta\chi^{2}$’s minimum centred at large negative NCR values is disfavoured
at more than $2\sigma$ CL, while it is favored at about $1\sigma$ CL for
$\mathcal{F}_{\nu_{\ell}}(T_{\rm nr})=1$. Our results have to be compared to
the current best constraints at 90% CL set by TEXONO Deniz  _et al._ (2010),
namely
$-4.2<\langle{r_{\nu_{e}}^{2}}\rangle\,[10^{-32}\,\mathrm{cm}^{2}]<6.6$, and
by BNL-E734 Ahrens  _et al._ (1990),
$-5.7<\langle{r_{\nu_{\mu}}^{2}}\rangle\,[10^{-32}\,\mathrm{cm}^{2}]<1.1$.333Both
TEXONO and BNL-E734 results have been corrected by a factor of two due to a
different convention, see Ref. Cadeddu  _et al._ (2018). Moreover, for the
latter, we use the corrected value in Ref. Hirsch  _et al._ (2003).
Interestingly, we are able to improve the best upper bound limit for
$\langle{r_{\nu_{e}}^{2}}\rangle$ previously set by TEXONO.

##  IV Conclusions

To conclude, in this work we discuss the need to properly account for the non-
null momentum transfer of the experiments in the calculation of the neutrino
charge radius radiative correction, focusing in particular on the coherent
elastic neutrino-nucleus scattering case. We show that the impact of this
correction to the cross-section is small, but non-negligible, especially when
considering the contribution of electron neutrinos. Moreover, when aiming to
measure the neutrino charge radius, the inclusion of such a momentum
dependence results in a significant reduction of the allowed parameter space,
already for available data. We thereby stress the importance of using this
treatment in particular when analysing future CE$\nu$NS and neutrino-electron
scattering data, for a more accurate interpretation of the data.

| $1\sigma$ | $90\%$ | $2\sigma$ | $3\sigma$
---|---|---|---|---
COHERENT (CsI+Ar)
$\langle{r}_{\nu_{e}}^{2}\rangle$ | $(-95.0,-77.4)$ | $(-100.0,-69.8)$ | $(-102.6,-64.8)$ | $(-110.4,30.7)$
$(0.09,12.8)$ | $(-8.6,19.1)$ | $(-13.9,22.2)$
$\langle{r}_{\nu_{\mu}}^{2}\rangle$ | $(-6.8,0.5)$ | $(-57.6,-48.9)$ | $(-59.2,-47.1)$ | $(-63.3,-42.3)$
$(-9.3,2.9)$ | $(-10.7,4.2)$ | $(-15.2,8.1)$
Dresden-II
$\langle{r}_{\nu_{e}}^{2}\rangle$ | $(-62.5,-53.7)$ | $(-65.7,-48.5)$ | $(-67.2,-45.0)$ | $(-71.1,9.7)$
$(-9.0,1.8)$ | $(-13.8,4.5)$ | $(-17.2,6.0)$
COHERENT (CsI+Ar) + Dresden-II
$\langle{r}_{\nu_{e}}^{2}\rangle$ | $(-5.8,3.1)$ | $(-9.5,5.5)$ | $(-11.6,6.8)$ | $(-70.3,-47.7)$
$(-19.8,10.1)$
$\langle{r}_{\nu_{\mu}}^{2}\rangle$ | $(-56.8,-53.3)$ | $(-59.2,-51.0)$ | $(-60.4,-49.9)$ | $(-63.8,-46.8)$
$(-4.0,2.1)$ | $(-5.9,4.1)$ | $(-6.9,5.3)$ | $(-9.9,8.7)$
Table 1:  Bounds on the diagonal neutrino charge radii in units of
$10^{-32}~{}\text{cm}^{2}$ obtained from our reanalysis of the COHERENT
(CsI+Ar) and Dresden-II data alone and their combination at different
confidence levels, considering the momentum dependence in the NCR form factor.

###### Acknowledgements.

The authors are sincerely thankful to J. Erler, M. Gorchtein, H. Spiesberger,
R. Ferro-Hernandez and A. Pisano for the fruitful discussions during the
completion of this work.

## References

  * Erler and Su (2013) J. Erler and S. Su, Prog. Part. Nucl. Phys. 71, 119 (2013), arXiv:1303.5522 [hep-ph] .
  * Freedman (1974) D. Z. Freedman, Phys. Rev. D 9, 1389 (1974).
  * Akimov  _et al._ (2017) D. Akimov _et al._ (COHERENT), Science 357, 1123 (2017), arXiv:1708.01294 [nucl-ex] .
  * Akimov  _et al._ (2022) D. Akimov _et al._ (COHERENT), Phys. Rev. Lett. 129, 081801 (2022), arXiv:2110.07730 [hep-ex] .
  * Akimov  _et al._ (2021) D. Akimov _et al._ (COHERENT), Phys. Rev. Lett. 126, 012002 (2021), arXiv:2003.10630 [nucl-ex] .
  * Colaresi  _et al._ (2022) J. Colaresi, J. I. Collar, T. W. Hossbach, C. M. Lewis, and K. M. Yocum, Phys. Rev. Lett. 129, 211802 (2022), arXiv:2202.09672 [hep-ex] .
  * Atzori Corona  _et al._ (2022) M. Atzori Corona, M. Cadeddu, N. Cargioli, F. Dordei, C. Giunti, Y. F. Li, C. A. Ternes, and Y. Y. Zhang, JHEP 09, 164 (2022), arXiv:2205.09484 [hep-ph] .
  * Atzori Corona  _et al._ (2023) M. Atzori Corona, M. Cadeddu, N. Cargioli, F. Dordei, C. Giunti, and G. Masia, Eur. Phys. J. C 83, 683 (2023), arXiv:2303.09360 [nucl-ex] .
  * Cadeddu  _et al._ (2023) M. Cadeddu, F. Dordei, and C. Giunti, EPL 143, 34001 (2023), arXiv:2307.08842 [hep-ph] .
  * Drukier and Stodolsky (1984) A. Drukier and L. Stodolsky, Phys. Rev. D 30, 2295 (1984).
  * Barranco  _et al._ (2005) J. Barranco, O. G. Miranda, and T. I. Rashba, JHEP 12, 021 (2005), arXiv:hep-ph/0508299 .
  * Patton  _et al._ (2012) K. Patton, J. Engel, G. C. McLaughlin, and N. Schunck, Phys. Rev. C 86, 024612 (2012), arXiv:1207.0693 [nucl-th] .
  * Group (2022) P. D. Group, Progress of Theoretical and Experimental Physics 2022, 083C01 (2022), https://academic.oup.com/ptep/article-pdf/2022/8/083C01/49175539/ptac097.pdf .
  * Erler and Ramsey-Musolf (2005) J. Erler and M. J. Ramsey-Musolf, Phys. Rev. D 72, 073003 (2005).
  * Giunti and Studenikin (2015) C. Giunti and A. Studenikin, Rev. Mod. Phys. 87, 531 (2015), arXiv:1403.6344 [hep-ph] .
  * Bernabeu  _et al._ (2000) J. Bernabeu, L. G. Cabral-Rosetti, J. Papavassiliou, and J. Vidal, Phys. Rev. D 62, 113012 (2000), arXiv:hep-ph/0008114 .
  * Bernabeu  _et al._ (2002) J. Bernabeu, J. Papavassiliou, and J. Vidal, Phys. Rev. Lett. 89, 101802 (2002), [Erratum: Phys.Rev.Lett. 89, 229902 (2002)], arXiv:hep-ph/0206015 .
  * Bernabeu  _et al._ (2004) J. Bernabeu, J. Papavassiliou, and J. Vidal, Nucl. Phys. B 680, 450 (2004), arXiv:hep-ph/0210055 .
  * Cadeddu  _et al._ (2020) M. Cadeddu, F. Dordei, C. Giunti, Y. F. Li, and Y. Y. Zhang, Phys. Rev. D 101, 033004 (2020), arXiv:1908.06045 [hep-ph] .
  * De Romeri  _et al._ (2023) V. De Romeri, O. G. Miranda, D. K. Papoulias, G. Sanchez Garcia, M. Tórtola, and J. W. F. Valle, JHEP 04, 035 (2023), arXiv:2211.11905 [hep-ph] .
  * Deniz  _et al._ (2010) M. Deniz _et al._ (TEXONO), Phys. Rev. D 81, 072001 (2010), arXiv:0911.1597 [hep-ex] .
  * Ahrens  _et al._ (1990) L. A. Ahrens, S. H. Aronson, P. L. Connolly, B. G. Gibbard, M. J. Murtagh, S. J. Murtagh, S. Terada, D. H. White, J. L. Callas, D. Cutts, M. V. Diwan, J. S. Hoftun, B. W. Hughlock, R. E. Lanou, T. Shinkawa, Y. Kurihara, K. Amako, S. Kabe, Y. Nagashima, Y. Suzuki, S. Tatsumi, Y. Yamaguchi, K. Abe, E. W. Beier, D. C. Doughty, L. S. Durkin, S. M. Heagy, M. Hurley, A. K. Mann, F. M. Newcomer, H. H. Williams, T. York, D. Hedin, M. D. Marx, and E. Stern, Phys. Rev. D 41, 3297 (1990).
  * Erler and Ferro-Hernández (2018) J. Erler and R. Ferro-Hernández, JHEP 03, 196 (2018), arXiv:1712.09146 [hep-ph] .
  * Marciano and Parsa (2003) W. J. Marciano and Z. Parsa, J. Phys. G 29, 2629 (2003), arXiv:hep-ph/0403168 .
  * Sehgal (1985) L. Sehgal, Physics Letters B 162, 370 (1985).
  * Davoudiasl  _et al._ (2023) H. Davoudiasl, K. Enomoto, H.-S. Lee, J. Lee, and W. J. Marciano, Phys. Rev. D 108, 115018 (2023), arXiv:2309.04060 [hep-ph] .
  * Tomalak  _et al._ (2021) O. Tomalak, P. Machado, V. Pandey, and R. Plestid, JHEP 02, 097 (2021), arXiv:2011.05960 [hep-ph] .
  * Cadeddu  _et al._ (2018) M. Cadeddu, C. Giunti, K. A. Kouzakov, Y.-F. Li, Y.-Y. Zhang, and A. I. Studenikin, Phys. Rev. D 98, 113010 (2018), [Erratum: Phys.Rev.D 101, 059902 (2020)], arXiv:1810.05606 [hep-ph] .
  * Mueller  _et al._ (2011) T. A. Mueller _et al._ , Phys. Rev. C 83, 054615 (2011), arXiv:1101.2663 [hep-ex] .
  * Vogel and Engel (1989) P. Vogel and J. Engel, Phys. Rev. D 39, 3378 (1989).
  * Hirsch  _et al._ (2003) M. Hirsch, E. Nardi, and D. Restrepo, Phys. Rev. D 67, 033005 (2003), arXiv:hep-ph/0210137 .