# Recent Progress in Low Energy Neutrino Scattering Physics and Its
Implications for the Standard and Beyond the Standard Model Physics

V. Pandey<EMAIL_ADDRESS>Fermi National Accelerator Laboratory, Batavia,
Illinois 60510, USA

###### Abstract

Neutrinos continue to provide a testing ground for the structure of the
standard model of particle physics as well as hints towards the physics beyond
the standard model. Neutrinos of energies spanning over several orders of
magnitude, originating in many terrestrial and astrophysical processes, have
been detected via various decay and interaction mechanisms. At MeV scales,
there has been one elusive process, until a few years ago, known as coherent
elastic neutrino-nucleus scattering (CEvNS) that was theoretically predicted
over five decades ago but was never observed experimentally. The recent
experimental observation of the CEvNS process by the COHERENT collaboration at
a stopped pion neutrino source has inspired physicists across many subfields.
This new way of detecting neutrinos has vital implications for nuclear
physics, high-energy physics, astrophysics, and beyond. CEvNS, being a low-
energy process, provides a natural window to study light, weakly-coupled, new
physics in the neutrino sector. Leveraging orders of magnitude higher CEvNS
cross section, new physics can be searched with relatively small detectors.

In this review, we intend to provide the current status of low energy neutrino
scattering physics and its implications for the standard and beyond the
standard model physics. We discuss low energy sources of neutrinos with a
focus on neutrinos from the stopped pions. Stopped pion sources cover energies
in the tens of MeVs and are almost optimal for studying CEvNS. Several
worldwide experimental programs have been or are being set up to detect CEvNS
and new physics signals in the near future with complementary detection
technologies and physics goals. We discuss the general formalism of
calculating the tree-level CEvNS cross section and the estimated theoretical
uncertainties on the CEvNS cross section stemming from different sources. We
also discuss the inelastic scattering of tens of MeV neutrinos that have
implications for supernova detection in future neutrino experiments. The
stopped-pion facilities are also a near-ideal tens of MeV neutrino source to
study inelastic neutrino-nucleus cross sections. We discuss how the CEvNS
experiments can be used as a testing ground for the Standard Model (SM) weak
physics as well as in searching for the Beyond the Standard Model (BSM)
physics signals. Any deviation from the SM predicted event rate either with a
change in the total event rate or with a change in the shape of the recoil
spectrum, could indicate new contributions to the interaction cross-section.
The SM implications include the study of weak nuclear form factor and weak
mixing angle. The BSM studies include non-standard interactions, neutrino
electromagnetic properties, and sterile neutrino searches. Stopped pion
facilities are also a copious source of neutral and changed mesons that allow
study of several dark sector physics scenarios such as vector portal models,
leptophobic dark matter as well as axion-like particle searches.

††preprint: FERMILAB-PUB-23-245-ND

###### Contents

  1. I Introduction
  2. II Low Energy Neutrino Sources
    1. II.1 Stopped Pion Source
  3. III Coherent Elastic Neutrino Scattering off Nuclei
    1. III.1 Tree-level Cross Section
    2. III.2 Uncertainty on the Cross Section
    3. III.3 Input from Parity Violating Electron Scattering
  4. IV Inelastic Neutrino Scattering off Nuclei
  5. V Experimental Landscape
  6. VI Implications for the Standard Model Physics
    1. VI.1 Weak Nuclear Form Factor
    2. VI.2 Weak Mixing Angle
  7. VII Implications for Beyond the Standard Model Physics
    1. VII.1 Non-standard Interactions of Neutrinos
    2. VII.2 Neutrino Electromagnetic Properties
    3. VII.3 Sterile Neutrino Oscillations
    4. VII.4 Accelerator Produced Light Dark Matter
  8. VIII Summary

##  I Introduction

Neutrinos, often referred to as elusive or ghostly elementary particles, are
fascinating. Starting from their postulation as a mere theoretical idea of an
undetectable particle by Wolfgang Pauli in 1930, to now known as the most
abundant matter particle in the Universe, neutrinos have played a prominent
role in our understanding of the nature of the Universe. In recent years,
neutrinos have not only provided a testing ground for the structure of the
Standard Model (SM) of particle physics but also continue to provide us hints
towards the physics beyond the SM. One of the most prominent one of those is
the discovery of neutrino mixing and oscillation that imply that neutrinos can
no longer be considered as massless particles as described in the Standard
Model. SM provides the framework describing how neutrinos interact with
leptons and quarks through weak interactions but it does not answer
fundamental questions about neutrinos. What is the origin of neutrino mass and
why are they orders of magnitude smaller compared to other SM particles? We
don’t know if neutrinos are Dirac particles or Majorana particles. Are there
more neutrinos than three flavors, consistent with leptons, or are more of
them as some experimental anomalies suggest? Neutrinos continue to provide
both a testing ground for the SM and direct evidence for physics beyond the SM
Huber:2022lpm ; Balantekin:2022jrq ; deGouvea:2022gut ; Acharya:2023swl .

Neutrinos originate via various mechanisms in many terrestrial and
astrophysical processes, covering energies as low as from sub-eV scale to as
high as EeV scale. We have detected neutrino from a variety of astrophysical
(e.g., solar, supernova) and terrestrial (e.g., reactors and accelerator)
sources Davis:1968cp ; Kamiokande-II:1987idp ; Bionta:1987qt ; IceCube:2018cha
using a variety of interaction processes ranging from inverse beta decay to
scattering off quarks, nucleons, and nuclei. At MeV scale energies, which is
the focus of this article, neutrinos have been detected via several distinct
interaction channels including neutrino-electron elastic scattering, as well
as neutral and charged current inelastic interactions on nucleons and nuclei
Formaggio:2012cpf  and via inverse-beta decay process. Among them there has
been one elusive process, until a few years ago, known as Coherent Elastic
Neutrino Nucleus Scattering (CEvNS) that was first postulated over nearly five
decades ago.

CEvNS was suggested soon after the experimental discovery of the weak neutral
current in neutrino interactions Stodolsky:1966zz ; Freedman:1973yd ;
Kopeliovich:1974mv . In his 1974 article, Freedman suggested that “if there is
a weak neutral current, then the elastic process $\nu+A\rightarrow\nu+A$
should have a sharp coherent forward peak just as the $e+A\rightarrow e+A$
does” Freedman:1973yd . Freedman went ahead and declared that the experimental
detection of CEvNS would be an “act of hubris” due to the associated “grave
experimental difficulties”. The experimental difficulty that Freedman referred
to was despite the fact that the CEvNS cross section is larger due to the
$\sim N^{2}$ enhancement it receives. The only experimental signature of the
coherent elastic process is the kinetic energy $T$ of the recoiling nucleus.
The maximum recoil energy is limited by the kinematics of the elastic
scattering

$T_{\rm max}=\frac{E_{\nu}}{1+M_{A}/(2E_{\nu})}$ (1)

where $E_{\nu}$ is the incoming neutrino energy and $M_{A}$ is the mass of the
target nuclei. For tens of MeV incident neutrino energies, where CEvNS cross
section is supposed to dominate, and for medium-sized nuclei, the recoil
energy amounts to several tens of keV, making it experimentally challenging to
detect.

Over after nearly four decades of its predictions by Freedman, the CEvNS
signal was finally detected by the COHERENT collaboration in 2017
COHERENT:2017ipa . The necessary keV scale low-threshold needed to detect
CEvNS signal benefited from the recent developments in the detector
technologies that are primarily driven by dark sector searches that also rely
on tiny nuclear recoils. Typically, the recoil energy is collected in the form
of scintillation photons or ionized charge, depending on the detector
technology. The COHERENT collaboration announced the detection of the first
CEvNS signal using a stopped–pion neutrino source at the Spallation Neutron
Source (SNS) at Oak Ridge National Laboratory with a CsI[Na] scintillating
crystal detector, an experimental discovery of CEvNS signal at the $6.7\sigma$
confidence level COHERENT:2017ipa ; COHERENT:2018imc . In the following years,
COHERENT collaboration presented another CEvNS measurement with a single-phase
liquid argon detector COHERENT:2019iyj ; COHERENT:2020iec , and a follow-up
CsI[Na] COHERENT:2021xmm  measurement with a larger exposure.

This new way of detecting neutrinos has wider implications for border
communities that span nuclear physics, particle physics, astrophysics, and
beyond. Leveraging orders of magnitude higher CEvNS cross section, one could
do groundbreaking searches with relatively small detectors as opposed to the
typically large detector size needed for most neutrino experiments. CEvNS,
being a low-energy process, provides a natural window to study light, weakly-
coupled, new physics in the neutrino sector Barranco:2005yy ; Scholberg:2005qs
; Barranco:2007tz ; Dutta:2015nlo ; Lindner:2016wff ; Coloma:2017ncl ;
Farzan:2017xzy ; Billard:2018jnl ; AristizabalSierra:2018eqm ; Brdar:2018qqj ;
Abdullah:2018ykz ; AristizabalSierra:2019zmy ; Miranda:2019skf ; Bell:2019egg
; AristizabalSierra:2019ufd ; Cadeddu:2019eta ; Coloma:2019mbs ; Canas:2019fjw
; Dutta:2019eml ; Denton:2020hop ; Skiba:2020msb ; Cadeddu:2020nbr ;
Canas:2018rng .

The remainder of this article is organized as follows. In Sec. II, we discuss
low-energy sources of neutrinos with a focus on neutrinos from the stopped
pion sources. In Sec. III, we lay out the general formalism of calculating the
tree-level CEvNS cross section and discuss the estimated theoretical
uncertainties on the CEvNS cross section stemming from different sources. Tens
of MeV neutrinos also scatter via inelastic neutrino-nucleus scattering, we
discuss those in Sec. IV. These processes have implications for supernova
detection in future neutrino experiments. The observable final-state particles
of these inelastic scattering have typical energies of the same order as the
incident neutrino energies. CEvNS experiments are, in principle, sensitive to
inelastic processes as well if they have the dynamic range. In Sec. V, we
briefly review current and proposed CEvNS experimental facilities. In Sec. VI,
we discuss how the CEvNS experiments can be used as a testing ground for the
SM weak physics. We continue to discuss the implications of CEvNS physics for
the global efforts of the BSM physics searches in Sec. VII. CEvNS provides a
natural window to study light, weakly-coupled, beyond the standard model
physics in the neutrino sector. Finally, we summarize in Sec. VIII.

##  II Low Energy Neutrino Sources

The coherent elastic process dominates only at tens of MeV neutrino energies.
The tens of MeV neutrinos come from several sources including nuclear
reactors, accelerator produced decay at rest sources, as well as astrophysical
sources such as supernova and solar neutrinos. Neutrinos from reactors have
been detected using the inverse beta decay reaction,
$\bar{\nu}_{e}+p\rightarrow e^{+}+n$, by observing both the outgoing positron
and coincident neutron. Nuclear reactors are copious sources of electron anti-
neutrinos therefore, reactors have long been the sources of choice for CEvNS
searches. However, since the typical reactor neutrino energies are of the
order of few MeV and even if the coherence condition for the recoil is largely
preserved, the scattering signal is sub-kev energy scale nuclear recoil making
it even harder to detect even for many sensitive detection technologies.
Furthermore, the total CEvNS cross section scales as a function of incident
neutrino energy, therefore, higher energies are beneficial up until the point
at which CEvNS is strongly dominated relative to inelastic scattering.
Neutrinos from stopped pions sources cover energies in the tens of MeV scale
and are almost optimal for studying CEvNS, finding a sweet spot where the
CEvNS rate is high enough and recoil energies are more easily detectable above
the threshold. So far, CEvNS is observed only at the decay at rest sources.
Therefore most of the discussions in this paper focus on neutrinos from a pion
decay at rest source. We comment on other low-energy neutrino sources where
appropriate.

###  II.1 Stopped Pion Source

Figure 1: Standard neutrino energy distribution of pion decay at rest
neutrinos with a 29.8 MeV monoenergetic $\nu_{\mu}$, while the energies of
$\nu_{e}$s and $\bar{\nu}_{\mu}$s ranges upto $m_{\mu}/2$.

An intense beam of protons accelerated to hundreds of MeV to GeV scale,
directed to collide with a target, producing a copious number of secondary
hadrons. Protons with energies $>$ 300 MeV will produce large numbers of
pions, these pions can lose energy in dense material, stop and decay after
coming to rest. Negative pions are often captured by nuclei. To produce clean
stopped-pion neutrinos: (a) the optimum proton energies are of the order of 1
GeV or less Alonso:2010fs , this suppresses the decay-in-flight component that
is heavily used in typical accelerator-based short- and long-baseline neutrino
facilities, (b) the target is preferred to be dense to allow the pions to stop
and decay at rest.

The dominant neutrino production from stopped pions is from the weak-
interaction two-body prompt decay

$\pi^{+}\rightarrow\mu^{+}+\nu_{\mu}~{}~{}(\text{decay time:}~{}\tau\sim
26~{}{\text{n}s})$ (2)

followed by a three-body delayed decay of muons

$\mu^{+}\rightarrow e^{+}+\nu_{e}+\bar{\nu}_{\mu}~{}~{}(\text{decay
time:}~{}\tau\sim 2.2~{}\mu{\text{s}})$ (3)

producing a well know spectrum shape. The spectral functions are given by

$\Phi_{\nu_{\mu}}(E_{\nu})=\frac{2m_{\pi}}{m_{\pi}^{2}-m_{\mu}^{2}}\delta\left(1-\frac{2E_{\nu}m_{\pi}}{m_{\pi}^{2}-m_{\mu}^{2}}\right)$
(4)
$\Phi_{\nu_{e}}(E_{\nu})=\frac{192}{m_{\mu}}\left(\frac{E_{\nu}}{m_{\mu}}\right)^{2}\left(\frac{1}{2}-\frac{E_{\nu}}{m_{\mu}}\right)$
(5)
$\Phi_{\bar{\nu}_{\mu}}(E_{\nu})=\frac{64}{m_{\mu}}\left(\frac{E_{\nu}}{m_{\mu}}\right)^{2}\left(\frac{3}{4}-\frac{E_{\nu}}{m_{\mu}}\right)\\\
\\\ $ (6)

For a pion decay at rest source, $E_{\nu}^{\text{max}}=m_{\mu}/2$ where
$m_{\mu}=105.65$ MeV is the muon mass. The well-known energy spectrum is shown
in Fig. 1 with a 29.8 MeV monoenergetic $\nu_{\mu}$ while the energies of
$\nu_{e}$s and $\bar{\nu}_{\mu}$s range upto $m_{\mu}/2$. Fig. 2 shows the
standard timing distribution with a prompt $\nu_{\mu}$ and delayed $\nu_{e}$
and $\bar{\nu}_{\mu}$ signal. The pulsed time structure gives a strong handle
on suppressing the background.

There are a few percent of radiative corrections on this flux decaying from
pions and muons, these are evaluated in Ref. Tomalak:2021lif  by comparing the
tree-level neutrino energy spectra with the $\mathcal{O}(\alpha)$
contributions. Radiative effects modify the expected neutrino fluxes from
around the peak region by 3–4 permille.

Figure 2: Standard timing distribution of pion decay at rest neutrinos with a
prompt $\nu_{\mu}$ and delayed $\nu_{e}$ and $\bar{\nu}_{\mu}$ signal.

##  III Coherent Elastic Neutrino Scattering off Nuclei

The coherent elastic neutrino-nucleus scattering process occurs when a
neutrino scatters off an entire nucleus, exchanging a $Z^{0}$ boson,
transferring some of its momenta to the nucleus as a whole, but creating no
internal excitations of the nucleus or ejected particles. It’s elastic in the
sense that no new particles are created in the scattering and the residual
nucleus stays in its ground state. For neutrinos carrying a few tens of MeV
energies and scattering off medium-sized nuclei, a dominant fraction of
interactions are expected to be of coherent type.

Figure 3: Diagrammatic representation of the CEvNS process where a single
$Z^{0}$ boson is exchanged between the neutrino and the target nucleus. The
nucleus stays in its ground state and a keV scale nuclear recoil energy is
deposited in the detector.

###  III.1 Tree-level Cross Section

A neutrino with four momentum $k_{i}=(E_{i},\vec{k}_{i})$ scatters off the
nucleus, which is initially at rest in the lab frame with
$p_{A}=(M_{A},\vec{0})$, exchanging a $Z^{0}$ boson. The neutrino scatters
off, carrying away four momentum $k_{f}=(E_{f},\vec{k}_{f})$ while the nucleus
remains in its ground state and receives a small recoil energy $T$, so that
$p^{\prime}_{A}=(M_{A}+T,\vec{p}^{\prime}_{A})$ with
$|\vec{p}^{\prime}_{A}|=\sqrt{(M_{A}+T)^{2}-M_{A}^{2}}$ and $T=q^{2}/2M_{A}$.
Here, $M_{A}$ is the rest mass of the nucleus, $q=|\vec{q}|$ is the absolute
value of the three–momentum transfer which is of the order of keV for neutrino
energies of tens of MeV, $Q^{2}\approx q^{2}=|\vec{k}_{f}-\vec{k}_{i}|^{2}$,
and the velocity dependent factor in the denominator refers to the relative
velocity of the interacting particles. The process is schematically shown in
Fig. 3.

The initial elementary expression for the cross section reads

$\displaystyle\mathrm{d}^{6}\sigma$
$\displaystyle=\frac{1}{\left|\vec{v}_{i}-\vec{v}_{A}\right|}\frac{m_{i}}{E_{i}}\frac{m_{f}}{E_{f}}\frac{\mathrm{d}^{3}\vec{k}_{f}}{(2\pi)^{3}}\frac{M_{A}}{M_{A}+T}\frac{\mathrm{d}^{3}\vec{p}^{\prime}_{A}}{(2\pi)^{3}}$
(7)
$\displaystyle\times(2\pi)^{4}\overline{\sum}_{fi}\left|\mathcal{M}\right|^{2}\delta^{(4)}(k_{i}+p_{A}-k_{f}-p^{\prime}_{A}).$

This expression can be integrated to yield the expression for the cross
section differential in neutrino scattering angle $\theta_{f}$:

$\displaystyle\frac{\mathrm{d}\sigma}{\mathrm{d}\cos{\theta_{f}}}$
$\displaystyle=\frac{m_{i}}{E_{i}}\frac{m_{f}}{E_{f}}\frac{M_{A}}{M_{A}+T}\frac{E_{f}^{2}}{2\pi}f_{rec}^{-1}\overline{\sum}_{fi}\left|\mathcal{M}\right|^{2}.$
(8)

The recoil factor reads

$f_{rec}=\frac{E_{i}}{E_{f}}\frac{M_{A}}{M_{A}+T}.$ (9)

Working out the Feynman amplitude one gets

$\overline{\sum}_{fi}\left|\mathcal{M}\right|^{2}=\frac{G_{F}^{2}}{2}L_{\mu\nu}W^{\mu\nu},$
(10)

with the nuclear tensor $W^{\mu\nu}$ reading

$W^{\mu\nu}=\overline{\sum}_{fi}(\mathcal{J}^{\mu}_{nuc})^{\dagger}\mathcal{J}^{\nu}_{nuc}.$
(11)

The summation symbols in these expressions denote summing and averaging over
initial and final polarizations, respectively. The nuclear tensor depends on
the nuclear current transition amplitudes:

$\mathcal{J}^{\mu}_{nuc}=\langle\Phi_{\textrm{0}}|\widehat{J}^{\mu}(\vec{q})|\Phi_{\textrm{0}}\rangle.$
(12)

Under the assumption that the nuclei of interest are spherically symmetric
with $J^{\pi}=0^{+}$ and taking the z–axis to be along the direction of
$\vec{q}$, one only needs to take into account the zeroth and third component
of the nuclear current’s vector part, which are furthermore connected through
vector current conservation (CVC):

$q^{\mu}\widehat{J}_{\mu}(\vec{q})=0.\\\ $ (13)

Through performing the necessary algebra, one arrives at the final expression

$\frac{\mathrm{d}\sigma}{\mathrm{d}\cos{\theta_{f}}}=\frac{G_{F}^{2}}{2\pi}\frac{E_{f}^{3}}{E_{i}}\left[\frac{Q^{4}}{q^{4}}(1+\cos{\theta_{f}})|\mathcal{J}^{V}_{0}|^{2}\right]$
(14)

where $\mathcal{J}^{V}_{0}$ is the transition amplitude induced by the nuclear
current. One can then safely approximate $\frac{Q^{4}}{q^{4}}\approx 1$ and
express the differential cross section as a function of the neutrino
scattering angle $\theta_{f}$ as:

$\frac{\mathrm{d}\sigma}{\mathrm{d}\cos{\theta_{f}}}=\frac{G_{F}^{2}}{2\pi}\frac{E_{f}^{3}}{E_{i}}(1+\cos{\theta_{f}})\frac{Q_{W}^{2}}{4}F_{W}^{2}(Q^{2})$
(15)

where $G_{F}$ is the Fermi coupling constant, and $Q_{W}$ the tree-level weak
nuclear charge:

$Q^{2}_{W}=[g_{p}^{V}Z+g_{n}^{V}N]^{2}=[(1-4\sin^{2}\theta_{\text{W}})Z-N]^{2}$
(16)

with coupling constants $g_{n}^{V}=-1$ and
$g_{p}^{V}=(1-4\sin^{2}\theta_{\text{W}})$. $N$ and $Z$ are the nucleus’
neutron and proton number, and $\theta_{W}$ is the weak mixing angle. The
value is such that $\sin^{2}{\theta_{W}}=0.23857$, which is valid at low
momentum transfers Ishikawa:2018rlv .

Here we have introduced the elastic form factor, $F_{W}^{2}(Q^{2})$, which we
will discuss later in this subsection. In elastic scattering, the entire
nuclear dynamics is encoded in this form factor. Equivalently one can express
the differential cross section as a function of the nuclear recoil $T$, which
reads:

$\frac{\mathrm{d}\sigma}{\mathrm{d}T}=\frac{G^{2}_{F}}{\pi}M_{A}\left(1-\frac{T}{E_{i}}-\frac{M_{A}T}{2E^{2}_{i}}\right)~{}\frac{Q^{2}_{W}}{4}~{}F_{W}^{2}(Q^{2}).$
(17)

In Eq. (15) and (17), we have expressed the CEvNS kinematic distribution both
in neutrino scattering angle, $\theta_{f}$, and in nuclear recoil energy $T$.
$T=q^{2}/(2M)=E_{\nu}-E_{\nu}^{\prime}$ is the nuclear recoil energy (taking
values in $[0,2E_{\nu}^{2}/(M+2E_{\nu})]$). Terms of order $T/E_{\nu}\lesssim
2E_{\nu}/M_{A}$ are usually neglected since they will be negligible for
neutrino energies $E_{\nu}\lesssim 50$ MeV accessible at the stopped pion
sources. The cross section represents the truly “coherent” contribution, in
the sense that the nuclear structure physics that enter the definition of weak
form factor $F_{\text{W}}$, indeed scale with $Z$ and $N$.

In most experiments, the only signal of a CEvNS event is a nuclear recoil
energy deposition. In principle, future experiments with more advanced
detector technologies may be able to detect both nuclear recoil and angular
distribution simultaneously. Such capabilities are already being explored in
some dark-matter experiments and will significantly enhance the physics
capabilities of future CEvNS experiments Abdullah:2020iiv . The cross section
can also be expressed in terms of the direction of the recoil, converting the
recoil to an angular spectrum. This is referred to in the literature as the
Directional Recoil Spectrum (DRS) Abdullah:2020iiv  where the angles are those
of the scattered nucleus measured with respect to the incident neutrino
direction and can be written as

$\frac{\text{d}^{2}R}{\text{d}\Omega\text{d}T}=\frac{1}{2\pi}\left.\frac{\text{d}\sigma}{\text{d}T}\right|_{E_{\nu}=\varepsilon}\,\frac{\varepsilon^{2}}{E_{\nu}^{\text{min}}}\left.\frac{\text{d}\Phi}{\text{d}E_{\nu}}\right|_{E_{\nu}=\varepsilon}\,,$
(18)

where $\text{d}\Phi/\text{d}E_{\nu}$ is the differential neutrino flux,
$E_{\nu}^{\text{min}}=\sqrt{MT/2}$, and

$\frac{1}{\varepsilon}=\frac{\cos\theta}{E_{\nu}^{\text{min}}}-\frac{1}{M}\,.$
(19)

To switch variables directly between $T$ and $\Omega$ one can use the
following relation and the associated Jacobian:

$T=\frac{2ME_{\nu}^{2}\cos^{2}\theta}{(E_{\nu}+M)^{2}-E_{\nu}^{2}\cos^{2}\theta}\,.$
(20)

The directional and energy double differential cross section can be written by
noting that the scattering has azimuthal symmetry about the incoming neutrino
direction. Integrating over outgoing nuclear recoil energy gives

$\frac{\text{d}\sigma}{\text{d}\Omega}=\frac{G_{F}^{2}}{16\pi^{2}}Q_{\text{W}}^{2}E_{\nu}(1+\cos\theta)\big{[}F_{\text{W}}(q^{2})\big{]}^{2}\,,$
(21)

where the angle is defined as

$\text{d}\Omega=2\pi\cos\theta\text{d}\theta$ (22)

and $\theta$ is the scattering angle between the direction of the incoming and
outgoing neutrino.

The scattering process’ cross section is proportional to the squared magnitude
of the transition amplitude induced by the nuclear current. Since the relevant
ground state to ground state transition for spherically symmetrical nuclei is
$0^{+}\rightarrow 0^{+}$, only the vector part of the current will contribute.
The amplitude can be expressed as

$\displaystyle\mathcal{J}^{V}_{0}$
$\displaystyle=\langle\Phi_{0}|\widehat{J}_{0}^{V}(\vec{q})|\Phi_{0}\rangle$
(23) $\displaystyle=\int
e^{i\vec{q}\cdot\vec{r}}\langle\Phi_{0}|\widehat{J}_{0}^{V}(\vec{r})|\Phi_{0}\rangle$
$\displaystyle=\frac{1}{2}\left[\left(1-4\sin^{2}{\theta_{W}}\right)f_{p}(\vec{q})F_{p}(Q^{2})\right.$
$\displaystyle-\left.f_{n}(\vec{q})F_{n}(Q^{2})\right],$

where we have inserted the impulse approximation (IA) expression for the
nuclear current, as a sum of single–body operators:

$\widehat{J}_{0}^{V}(\vec{r})=\sum_{i}F^{Z}(Q^{2},i)\delta^{(3)}(\vec{r}-\vec{r}_{i}),$
(24)

with

$\displaystyle F^{Z}(Q^{2},i)$
$\displaystyle=\left(\frac{1}{2}-\sin^{2}{\theta_{W}}\right)(F_{p}-F_{n})\tau_{3}(i)$
(25) $\displaystyle-\sin^{2}{\theta_{W}}(F_{p}+F_{n}),$

where we used the convention $\tau_{3}(i)=+1$ for proton, -1 for neutrons.
Furthermore, $f_{p}(\vec{q})$ and $f_{n}(\vec{q})$ are the Fourier transforms
of the proton and neutron densities, respectively. $F_{p}$ and $F_{n}$ are
proton and neutron form factors, for which we adopt the standard Galster
parametrization. Note that using a more sophisticated parametrization of the
form factor, other than Galster, will not affect the results at the energies
relevant to this work. The overall structure of the transition amplitude
consists of products of the weak charge with two factors: the nuclear form
factor, determined by the spatial distribution of the nucleons in the nucleus,
as well as the nucleon form factor. We arrive at the expression:

$\displaystyle F_{W}(Q^{2})$
$\displaystyle=\frac{1}{Q_{W}}\left[\left(1-4\sin^{2}{\theta_{W}}\right)f_{p}(\vec{q})F_{p}(Q^{2})\right.$
(26)
$\displaystyle\left.-f_{n}(\vec{q})F_{n}(Q^{2})\right]=\frac{2}{Q_{W}}\mathcal{J}^{V}_{0},$

such that the form factor becomes 1 in the static limit. Note that in writing
down the functional dependence, we can make use of the non–relativistic
approximation $Q\approx|\vec{q}|$, valid in the energy regime considered.

###  III.2 Uncertainty on the Cross Section

At tree-level, the theoretical uncertainty on the CEvNS cross section is
driven by the uncertainty on the weak form factor of the nucleus. Although, in
deriving CEvNS cross section in the previous section, a number of subtleties
have been ignored that including subleading kinematic effects, axial-vector
contributions and radiative corrections. In this subsection, we will first
discuss the uncertainty on the tree-level cross section driven by weak form-
factor and then briefly discuss other subleading uncertainties.

The CEvNS cross section is proportional to the weak form factor of Eq. (26).
In general, the form factor can be reasonably approximated by several
different functional forms. The simplest way is to denote neutron and proton
form factors in the Eq. (26) as Fourier transforms of neutron and proton
densities considering the nucleus to be spherically symmetric.

$F_{n}(Q^{2})=\frac{4\pi}{N}\int
dr~{}r^{2}~{}\frac{\sin(Qr)}{Qr}~{}\rho_{n}(r)$ (27)

$F_{p}(Q^{2})=\frac{4\pi}{Z}\int
dr~{}r^{2}~{}\frac{\sin(Qr)}{Qr}~{}\rho_{p}(r)$ (28)

where $\rho_{n}(r)$ and $\rho_{p}(r)$ are neutron and proton density
distributions normalized to the neutron and proton numbers. The value of the
nuclear form factors in the limit for q $\rightarrow$ 0 is 1. A small value of
the coefficient of the proton form factor in Eq. (26) makes the weak form
factor and hence CEvNS is mainly sensitive to neutron density distribution.
The charge density of a nucleus is strongly dominated by the protons and has
been extensively studied with impressive precision in elastic electron
scattering experiments started in the late 1950’s Hofstadter:1956qs  followed
by subsequent refinements over the decades DeVries:1987atn ; Fricke:1995zz ;
Angeli:2013epw . On the other hand, the neutron density distributions are hard
to determine, and various efforts using hadronic probes were plagued by
uncontrolled model–dependent uncertainties associated with the strong
interaction Thiel:2019tkm . Electroweak processes such as parity-violating
electron scattering (PVES) Donnelly:1989qs  and CEvNS have long been
considered clean and model-independent probes for extracting ground-state
neutron densities. Both of these, though long considered experimentally
challenging, are becoming a reality in recent years.

Phenomenological form factors, such as Helm Helm:1956  and Klein-Nystrand
KN:1999 , are widely used in the CEvNS community where density distributions
are represented by analytical expressions. The empirical value of proton rms
radius, extracted from elastic electron scattering data, is often used to
evaluate the proton form factor, and the same parameterization (or a variation
of that) is assumed for the neutron form factor.

In the Helm approach Helm:1956 , the nucleonic density distribution is
described as a convolution of a uniform density with radius $R_{0}$ and a
Gaussian profile characterized by the folding width $s$, accounting for the
surface thickness and the form factor is expressed as:

$F_{\text{Helm}}(q^{2})=\frac{3j_{1}(qR_{0})}{qR_{0}}e^{-q^{2}s^{2}/2}$ (29)

where $j_{1}(x)=\sin(x)/x^{2}-\cos(x)/x$ is a spherical Bessel function of the
first kind. $R_{0}$ is an effective nuclear radius given as:
$R_{0}^{2}=(1.23A^{1/3}-0.6)^{2}+\frac{7}{3}\pi^{2}r_{0}^{2}-5s^{2}$ with
$r_{0}$ = 0.52 fm and $s$ = 0.9 fm, fitted Duda:2006uk ; Lewin:1995rx  to muon
spectroscopy and electron scattering data compiled in Fricke:1995zz .

The Klein–Nystrand (KN) form factor, adapted by the COHERENT Collaboration, is
obtained from the convolution of a short-range Yukawa potential with range
$a_{k}$ = 0.7 fm over a Woods–Saxon distribution approximated as a hard sphere
with radius $R_{A}=1.23A^{1/3}$ fm KN:1999 . The resulting form factor is
expressed as:

Figure 4: Relative differences in the 40Ar weak form factor predictions of
Payne et al. Payne:2019wvy , Yang et al. Yang:2019pbx , Hoferichter et al.
Hoferichter:2020osn , Helm Helm:1956 , Klein–Nystrand KN:1999  and the adapted
Klein–Nystrand AristizabalSierra:2019zmy ; Papoulias:2019xaw , all with
respect to HF calculations of Van Dessel et al. VanDessel:2020epd . Figure
adapted from Ref. VanDessel:2020epd .

$F_{\text{KN}}(q^{2})=\frac{3j_{1}(qR_{A})}{qR_{A}}\left[\frac{1}{1+q^{2}a_{k}^{2}}\right].$
(30)

An adapted version of the KN form factor, (ad.) KN form factor, is often used
where $R_{A}$ is defined as $R_{A}=\sqrt{\frac{5}{3}r_{0}^{2}-10a_{k}^{2}}$
utilizing measured proton rms radii $r_{0}$ of the nucleus
AristizabalSierra:2019zmy ; Papoulias:2019xaw . The measured proton rms radii
of 40Ar is, $r_{0}=3.427$ fm Angeli:2013epw .

More involved nuclear structure calculations which describe a more accurate
picture of the nuclear ground state such as first-principles calculation using
coupled–cluster theory from first principles of Payne et al. Payne:2019wvy ,
shell-model calculations of Hoferichter et al. Hoferichter:2020osn  where form
factors are calculated using a large–scale nuclear shell model, relativistic
mean–field method of Yang et al. Yang:2019pbx  where form factors predictions
are informed by properties of finite nuclei and neutron star matter, and
Hartree–Fock approach of Van Dessel et al. VanDessel:2020epd  where form
factors are computed in a mean-field using Skyrme potential, has been reported
in recent years. In order to quantify differences between different form
factors and the CEvNS cross section due to different underlying nuclear
structure details, we can consider quantities that emphasize the relative
differences between the results of different calculations, arbitrarily using
Hartree–Fock (HF) as a reference calculation, as follows:

Figure 5: Relative differences in the 40Ar CEvNS cross section predictions of
Payne et al. Payne:2019wvy , Yang et al. Yang:2019pbx , Hoferichter et al.
Hoferichter:2020osn , Helm Helm:1956 , Klein–Nystrand KN:1999  and the adapted
Klein–Nystrand AristizabalSierra:2019zmy ; Papoulias:2019xaw , all with
respect to HF calculations of Van Dessel et al. VanDessel:2020epd . Figure
adapted from Ref. VanDessel:2020epd .

$|\Delta
F_{\text{W}}^{i}(q)|~{}=~{}\frac{|F_{\text{W}}^{i}(q)-F_{\text{W}}^{\text{HF}}(q)|}{|F_{\text{W}}^{\text{HF}}(q)|},$
(31)

$|\Delta\sigma_{\text{W}}^{i}(E_{\nu})|~{}=~{}\frac{|\sigma_{\text{W}}^{i}(E_{\nu})-\sigma_{\text{W}}^{\text{HF}}(E_{\nu})|}{|\sigma_{\text{W}}^{\text{HF}}(E_{\nu})|},$
(32)

where $i$ refers to calculations from different approaches as discussed above.

The relative differences are shown in Fig. 4 and Fig. 5. We show only the
low–momentum part of the weak form factor to a maximum value of $q$ = 0.5 fm-1
($\sim$ 100 MeV) that corresponds to a maximum incoming neutrino energy of E
$\sim$ 50 MeV. The relative differences are shown on a linear scale. At
smaller energies, the momentum transfer is low and hence the differences
between form factors are also small. For higher energies, the available
momentum transfer increases and therefore, the differences between the form
factors become more prevalent. The differences in model predictions amount to
$<7.5\%$ over the entire momentum transfer range. The differences rise rapidly
at the higher end of the $q$ range. This translates into relative differences
in CE$\nu$NS cross sections, $\Delta\sigma(E)$, of $<5\%$ over the whole
energy range, where $E\lesssim 55$ MeV, relevant for neutrinos from pion
decay-at-rest.

In writing down the CEvNS cross section, Eq. (17), only the vector operators
were considered. In principle, the axial-vector operator adds an additional
contribution that is not coherently enhanced, including this, modifies the
cross section to the form

$\frac{\text{d}\sigma}{\text{d}T}=\frac{G_{F}^{2}M}{4\pi}\bigg{(}1-\frac{MT}{2E_{\nu}^{2}}-\frac{T}{E_{\nu}}\bigg{)}Q_{\text{W}}^{2}\big{[}F_{\text{W}}(q^{2})\big{]}^{2}+\frac{G_{F}^{2}M}{4\pi}\bigg{(}1+\frac{MT}{2E_{\nu}^{2}}-\frac{T}{E_{\nu}}\bigg{)}F_{A}(q^{2})\,,$
(33)

with an axial-vector form factor $F_{A}(q^{2})$ Hoferichter:2020osn . The
axial-vector form factor depends on the axial charges and radii of the
nucleon. This contribution vanishes for spin-zero nuclei such as 40Ar.

The CEvNS cross section expression of Eq. (17) holds true at tree-level, in
which case $Q_{\text{W}}$ are flavor universal and apply both to neutrino and
electron scattering. Once including radiative corrections, process- and
flavor-dependent contributions arise, in such a way that separate weak charges
need to be defined. Electrons and muons running in loops introduce a non-
trivial dependence on the momentum transfer due to their relatively light
masses. These break the flavor universality because of mass-dependent
electromagnetic radiative corrections. For CEvNS, the corresponding radiative
corrections have been studied in Ref. Tomalak:2020zfh . At next-to-leading
order (NLO) in the electromagnetic coupling constant $\alpha$, photon-mediated
scattering takes place and the cross section inherits a flavor-dependent
contribution entering with a charge form factor of the nucleus.

$\frac{\mathrm{d}\sigma_{\nu_{\ell}}}{\mathrm{d}T}=\frac{\mathrm{G}_{\mathrm{F}}^{2}M_{\mathrm{A}}}{4\pi}\left(1-\frac{T}{E_{\nu}}-\frac{M_{\mathrm{A}}T}{2E_{\nu}^{2}}\right)\left(\mathrm{F}_{\mathrm{W}}\left(Q^{2}\right)+\frac{\alpha}{\pi}[\delta^{\nu_{\ell}}+\delta^{\text{QCD}}]\mathrm{F}_{\mathrm{ch}}(Q^{2})\right)^{2},$
(34)

The expression depends on the weak, $\mathrm{F}_{\mathrm{W}}$, and charge,
$\mathrm{F}_{\mathrm{ch}}$, nuclear form factors. The charge form factor
enters multiplied by $\delta^{\nu_{\ell}}$ and $\delta^{\text{QCD}}$ which are
radiative corrections. The corrections induced by hadronic and/or quark loops,
proportional to $\delta^{\text{QCD}}$, are flavor independent, whereas the
corrections from charged leptons, proportional to $\delta^{\nu_{\ell}}$,
depend on the neutrino flavor $\ell$.

A detailed total theoretical uncertainty on the CEvNS cross sections 40Ar
nucleus was estimated Ref. Tomalak:2020zfh , and is shown in Tab. 1. The
estimated error budget accounts for uncertainties stemming from a variety of
sources including nuclear, nucleon, and quark levels. At higher energies, the
main source of uncertainty for the CEvNS cross section comes from nuclear
physics. In fact, this can be traced down to the error of the neutron
distribution inside the nucleus.

Incident Neutrino Energy |  Nuclear Level Uncertainty on ${}^{40}\mathrm{Ar}$ |  Total Estimated Theoretical Uncertainty on ${}^{40}\mathrm{Ar}$
---|---|---
10 (MeV) | 0.04% | 0.58%
30 (MeV) | 1.5% | 1.65%
50 (MeV) | 4.0% | 4.05%
Table 1: Estimated theoretical errors budget on the CEvNS cross section on
on$~{}^{40}\mathrm{Ar}$ target. Table adapted from Ref. Tomalak:2020zfh .

###  III.3 Input from Parity Violating Electron Scattering

CEvNS and Parity Violating Electron Scattering (PVES) are intimately connected
to each other. From the formal point of view, both processes are described in
first order perturbation theory via the exchange of an electroweak gauge boson
between a lepton and a nucleus. While in CEvNS the lepton is a neutrino and a
$Z^{0}$ boson is exchanged, in PVES the lepton is an electron, but measuring
the asymmetry allows one to select the interference between the $\gamma$ and
$Z^{0}$ exchange. As a result, both the CEvNS cross section and the PVES
asymmetry depend on the weak form factor $F_{W}(Q^{2})$, which is mostly
determined by the neutron distribution within the nucleus. The latter builds
an even stronger anchor between CEvNS and PVES.

The key experimental observable in the elastic scattering of longitudinally
polarized electrons from the unpolarized spin-0 nucleus is the parity-
violating asymmetry ${A}_{\mathrm{PV}}$. The parity-violating asymmetry arises
from the interference of $\gamma$-mediated and $Z$-mediated scattering
diagrams. The asymmetry ${A}_{\mathrm{PV}}$ is determined from the fractional
difference in cross sections between the scattering of positive and negative
helicity electrons

$A_{pv}=\frac{d\sigma/d\Omega_{+}-d\sigma/d\Omega_{-}}{d\sigma/d\Omega_{+}+d\sigma/d\Omega_{-}}$
(35)

where $\pm$ refers to the polarization of the electron. In the Born
approximation at low momentum transfer, $\mathrm{A}_{\mathrm{PV}}$ is
proportional to the ratio of the weak to the charge form factors of the
nucleus

$A_{pv}=\frac{G_{F}q^{2}|Q_{W}|}{4\pi\alpha\sqrt{2}Z}\frac{F_{W}(q^{2})}{F_{ch}(q^{2})}.$
(36)

For a given nucleus, if $\mathrm{F}_{\text{ch}}(Q^{2})$ is already known from
the elastic electron scattering experiment, one can extract
$\mathrm{F}_{\text{W}}(Q^{2})$ from measured $\mathrm{A}_{\mathrm{PV}}$ in at
the momentum transfer of the experiment after accounting for radiative
corrections and Coulomb distortion effects not considered in the Born
approximation Horowitz:1999fk . Coulomb distortions can be theoretically
calculated by solving the Dirac equation for an electron moving in a nuclear
potential Yennie:1954zz ; Yennie:1965zz ; Kim:1996ua ; Kim:2001sq  and are
relatively well understood Horowitz:1998vv .

The PREX experiment at the Jefferson Lab (JLab) has recently provided the
first model-independent determination of the weak-charge form factor of 208Pb
at $\mathrm{F}_{\text{W}}(\langle Q^{2}\rangle)=0.204\pm 0.028$ at the average
momentum transfer of the experiment $\langle Q^{2}\rangle\approx
8800~{}\mathrm{MeV}^{2}$ Abrahamyan:2012gp ; Horowitz:2012tj . The follow-up
PREX-II experiment is underway to improve the precision of that measurement.
Another PVES experiment at JLab, CREX, is planned to measure the weak-charge
form factor of 48Ca Kumar:2020ejz . In practice, however, both PREX-II and
CREX measurements will make weak form factor measurements on a single value of
the momentum transfer and is not expected to perform measurements at several
values of the momentum transfer. Future facilities such as the MESA facility
in Mainz, envisioned to start operations in a few years, will also be suited
for high-precision parity-violating experiments Becker:2018ggl . Tab. 2
summarizes current and near-future PVES experiments. It is worth noting that
CEvNS can be used to probe the weak form factor only at low momentum transfers
where the process remains coherent, but accesses a continuum of four-momentum
transfers. In contrast, PVES experiments are usually carried out at a single
value of the momentum transfer at a time. A combination of measurements from
these two independent and complementary scattering techniques is ideal since
systematic uncertainties are largely uncorrelated. This will then provide an
empirical extraction of a nucleus’ weak form factor in a clean and model-
independent fashion.

Experiment | Target |  $q^{2}$ (GeV2) |  $A_{pv}$ (ppm) |  $\pm\delta R_{n}$ (%)
---|---|---|---|---
PREX at JLab |  208Pb | 0.00616 | $0.550\pm 0.018$ | 1.3
CREX at JLab |  48Ca | 0.0297 |  | 0.7
Qweak at JLab |  27Al | 0.0236 | $2.16\pm 0.19$ | 4
MREX at MESA |  208Pb | 0.0073 |  | 0.52
Table 2: Parity violating elastic electron scattering experiments.

In principle, parity-violating electron scattering experiments offer the least
model-dependent and most precise approach to experimentally probing the
neutron distribution. Any result that will come from the PVES program with the
goal of pinning down the neutron-skin thickness will help improve our
understanding of the weak form factor and hence influence CEvNS. However,
CEvNS has also been proposed as an alternative and attractive opportunity in
the future to constrain the neutron distribution and the neutron radius in
nuclei Amanik:2009zz ; Patton:2012jr ; Cadeddu:2017etk , provided that enough
statistics can be reached.

The main difference lies in the choice of the nuclear target, which is
determined by practical considerations. In the case of PVES, the targets need
to be stable (or almost stable) neutron-rich nuclei, such as 208Pb and 48Ca,
that do not present low-lying excited states that would contribute to the
background noise. In the case of CEvNS, isotopes of sodium, argon, germanium,
cesium and iodine will be used, as the low cost allows to build large
detectors with these materials. Because various electroweak observable
correlate Yang:2019pbx  with each other, theoretical calculations will help to
further connect the various nuclear targets and the two endeavors of CEvNS and
PVES. For example, we can expect that constraints experimentally determined on
the neutron-skin thickness of one nuclear target will affect the prediction of
the weak form factor of another target. CEvNS experiments also prefer detector
materials with low scintillation or ionization thresholds in order to
efficiently measure low-energy nuclear recoils. Quite the contrary is needed
as target material in parity violation electron scattering experiments: in
this case, the highest the excited state of the nucleus, the lower the
contamination of the elastic asymmetries by inelastic contributions from the
excited state. In addition, due to the high intensity of the electron beam, a
high melting temperature of the target material is also desirable.

##  IV Inelastic Neutrino Scattering off Nuclei

Figure 6: Diagrammatic representation of the inelastic neutrino-nucleus
scattering where a single $W^{+}$ (CC) or $Z^{0}$ (NC) boson is exchanged
between neutrino and target nucleus, exciting the nucleus into low-lying
nuclear states, followed by nuclear de-excitation products, $X$ (gamma or a
nucleon), with energies of the same order as the incident neutrino energies.

CEvNS experiments at stopped–pion sources are also sensitive to inelastic
neutrino-nucleus interactions. For neutrino energies less than about $\sim$100
MeV, the CEvNS interaction channel dominates the neutrino-nucleus cross
section over inelastic charged-current (CC) and neutral-current (NC) neutrino-
nucleus interactions. In the inelastic NC or CC scattering, shown in Fig. 6,
the neutrino excites the target nucleus to a low-lying nuclear state, followed
by nuclear de-excitation products such as gamma rays or ejected nucleon. The
interaction cross sections for these processes lack the $N^{2}$ enhancement
associated with CEvNS and, therefore, tend to be at least one order of
magnitude smaller than that of CEvNS process, as shown in Fig. 7. The
observable final-state particles of these inelastic scattering have typical
energies of the same order as the incident neutrino energies.

Figure 7: CEvNS cross section compared with CC and NC inelastic scattering
cross section on argon.

The inelastic neutrino-nucleus scattering process is schematically shown in
Fig. 6. A neutrino with four-momentum $k_{i}=(E_{i},\vec{k}_{i})$ scatters off
the nucleus, which is initially at rest in the lab frame, exchanging a $W^{+}$
(CC) or a $Z^{0}$ (NC) boson. The nucleus receives four momentum
$Q=(\omega,\vec{q})$, where $\omega=E_{i}-E_{f}$ and
$\vec{q}=\vec{k}_{i}-\vec{k}_{f}$, while the scattered lepton carries away
four momentum $k_{f}=(E_{f},\vec{k}_{f})$. For an inclusive process, the
hadronic part of the final states is integrated out. The inelastic neutrino-
nucleus differential cross section of this process can be written as

$\displaystyle\frac{\mathrm{d}^{3}\sigma}{\mathrm{d}\omega\mathrm{d}\Omega}=$
$\displaystyle\sigma_{W}E_{f}k_{f}\zeta^{2}(Z^{\prime},E_{f})$ (37)
$\displaystyle\times\left(v^{\mathcal{M}}R^{\mathcal{M}}+v^{\mathcal{L}}R^{\mathcal{L}}+v^{\mathcal{ML}}R^{\mathcal{ML}}\right.$
$\displaystyle+\left.v^{T}R^{T}+hv^{TT}R^{TT}\right),$

with the Mott-like cross section prefactor $\sigma_{W}$ defined as

$\sigma_{W}^{CC}=\left(\frac{G_{F}\cos{\theta_{c}}}{2\pi}\right)^{2},~{}\sigma_{W}^{NC}=\left(\frac{G_{F}}{2\pi}\right)^{2},$

where $G_{F}$ is the Fermi constant and $\cos{\theta_{c}}$ the Cabibbo angle.
The factor $\zeta^{2}(Z^{\prime},E_{f})$ is introduced in order to take into
account the distortion of the scattered lepton wave function in the Coulomb
field of the final nucleus with $Z^{\prime}$ protons, in the case of CC
interaction VanDessel:2019obk ; Pandey:2014tza . In the NC case
$\zeta^{2}(Z,E_{f})$ equals $1$. The influence of the lepton helicity on the
cross section is encoded in $h$ which is $+$ for neutrinos and $-$ for
antineutrinos.

The $v$–factors are leptonic functions that are entirely determined by lepton
kinematics. The $R$–factors are the nuclear response functions that depend on
the energy and momentum transfer ($\omega$, $q$) and contain all the nuclear
information involved in this process. The indices $L$ and $T$ correspond to
longitudinal and transverse contributions, relative to the direction of the
momentum transfer. The leptonic coefficients $v^{\mathcal{M}}$,
$v^{\mathcal{L}}$, $v^{\mathcal{M\mathcal{L}}}$, $v^{T}$, and $v^{TT}$ are
given as

Figure 8: Charged current neutrino-argon differential cross section for
neutrino energy 30 and 50 MeV shown as a function of the energy transferred to
the nucleus ($\omega$).

$\displaystyle
v^{\mathcal{M}}=\left[1+\frac{\kappa_{f}}{\varepsilon_{f}}\cos\theta\right],$
(38) $\displaystyle
v^{\mathcal{L}}=\left[1+\frac{\kappa_{f}}{\varepsilon_{f}}\cos\theta-\frac{2\varepsilon_{i}\varepsilon_{f}}{|\vec{q}|^{2}}{\left(\frac{\kappa_{f}}{\varepsilon_{f}}\right)}^{2}\sin^{2}\theta\right],$
(39) $\displaystyle
v^{\mathcal{M}\mathcal{L}}=\left[\frac{\omega}{|\vec{q}|}\left(1+\frac{\kappa_{f}}{\varepsilon_{f}}\cos\theta\right)+\frac{m_{l}^{2}}{\varepsilon_{f}|\vec{q}|}\right],$
(40) $\displaystyle
v^{T}=\left[1-\frac{\kappa_{f}}{\varepsilon_{f}}\cos\theta+\frac{\varepsilon_{i}\varepsilon_{f}}{|\vec{q}|^{2}}{\left(\frac{\kappa_{f}}{\varepsilon_{f}}\right)}^{2}\sin^{2}\theta\right],$
(41) $\displaystyle
v^{TT}=\left[\frac{\varepsilon_{i}+\varepsilon_{f}}{|\vec{q}|}\left(1-\frac{\kappa_{f}}{\varepsilon_{f}}\cos\theta\right)-\frac{m_{l}^{2}}{\varepsilon_{f}|\vec{q}|}\right],$
(42)

and response functions $R^{\mathcal{M}}$, $R^{\mathcal{L}}$,
$R^{\mathcal{ML}}$, $R^{T}$, and $R^{TT}$ are defined as

$\displaystyle R^{\mathcal{M}}=|\langle
J_{f}||\widehat{\mathcal{M}}_{J}(|\vec{q}|)||J_{i}\rangle|^{2},$ (43)
$\displaystyle R^{\mathcal{L}}=|\langle
J_{f}||\widehat{\mathcal{L}}_{J}(|\vec{q}|)||J_{i}\rangle|^{2},$ (44)
$\displaystyle R^{\mathcal{ML}}=~{}\mathcal{R}\left[\langle
J_{f}||\widehat{\mathcal{L}}_{J}(|\vec{q}|)||J_{i}\rangle\langle
J_{f}||\widehat{\mathcal{M}}_{J}(|\vec{q}|)||J_{i}\rangle^{\ast}\right],$ (45)
$\displaystyle R^{T}=\left[|\langle
J_{f}||\widehat{\mathcal{J}}_{J}^{mag}(|\vec{q}|)||J_{i}\rangle|^{2}+|\langle
J_{f}||\widehat{\mathcal{J}}_{J}^{el}(|\vec{q}|)||J_{i}\rangle|^{2}\right],$
$\displaystyle R^{TT}=~{}\mathcal{R}\left[\langle
J_{f}||\widehat{\mathcal{J}}_{J}^{mag}(|\vec{q}|)||J_{i}\rangle\langle
J_{f}||\widehat{\mathcal{J}}_{J}^{el}(|\vec{q}|)||J_{i}\rangle^{\ast}\right].$

Here $\widehat{\mathcal{M}}_{J}$, $\widehat{\mathcal{L}}_{J}$,
$\widehat{\mathcal{J}}_{J}^{mag}$ and $\widehat{\mathcal{J}}_{J}^{el}$ are the
Coulomb, longitudinal, transverse magnetic, and transverse electric operators,
respectively OConnell:1972edu ; Walecka:1995 .

Figure 9: Neutral current neutrino-argon differential cross section for
neutrino energy 30 and 50 MeV shown as a function of the energy transferred to
the nucleus ($\omega$).

The nuclear responses are function of the transition amplitude,
${J}_{\mu}^{nucl}(\omega,q)$, between the initial $|\Phi_{\textrm{0}}\rangle$
and final $|\Phi_{\textrm{f}}\rangle$ state:

${J}_{\mu}^{nucl}(\omega,q)=\langle\Phi_{\textrm{f}}|\hat{J}_{\mu}(q)|\Phi_{\textrm{0}}\rangle,$
(48)

where the nuclear current, $\hat{J}_{\mu}({q})$, is the Fourier transform of
the nuclear current operator in coordinate space:

$\hat{J}_{\mu}(q)=\int\mathrm{d}{x}e^{i{x}\cdot{q}}\hat{J}_{\mu}({x}).$ (49)

Nuclear responses are computed within a nuclear model. Fig. 8 and 9 show
inelastic CC and NC cross-section on 40Ar as a function of $\omega$ for
incoming neutrino energy of 30 and 50 MeV, calculated within a microscopic
many-body nuclear theory approach of Refs. Jachowicz:2002rr ; Pandey:2014tza ;
Pandey:2016jju ; VanDessel:2019atx . With the stopped-pion flux, only
$\nu_{e}$ CC interactions are accessible, given that $\nu_{\mu}$ and
$\bar{\nu}_{\mu}$ are below the CC threshold of $\sim$ 110 MeV, needed to
create a muon. While NC interactions are available for all neutrino types,
$\nu_{e}$, $\nu_{\mu}$ and $\bar{\nu}_{\mu}$ at the stopped-pion facility. The
experimental requirements for CEvNS and inelastic signals are quite different.
Larger masses are needed for inelastics, as well as the dynamic range to
record MeV-scale energy depositions, while very low thresholds are not
required.

Reaction Channel | Experiment | Measurement ($10^{-42}$ cm2)
---|---|---
12C($\nu_{e},e^{-}$)12Ng.s. | KARMEN | $9.1\pm 0.5{\rm(stat)}\pm 0.8{\rm(sys)}$
| E225 | $10.5\pm 1.0{\rm(stat)}\pm 1.0{\rm(sys)}$
| LSND | $8.9\pm 0.3{\rm(stat)}\pm 0.9{\rm(sys)}$
12C($\nu_{e},e^{-}$)12N∗ | KARMEN | $5.1\pm 0.6{\rm(stat)}\pm 0.5{\rm(sys)}$
| E225 | $3.6\pm 2.0{\rm(tot)}$
| LSND | $4.3\pm 0.4{\rm(stat)}\pm 0.6{\rm(sys)}$
12C($\nu_{\mu},\nu_{\mu}$)12C∗ | KARMEN | $3.2\pm 0.5{\rm(stat)}\pm 0.4{\rm(sys)}$
12C($\nu,\nu$)12C∗ | KARMEN | $10.5\pm 1.0{\rm(stat)}\pm 0.9{\rm(sys)}$
56Fe($\nu_{e},e^{-}$) 56Co | KARMEN | $256\pm 108{\rm(stat)}\pm 43{\rm(sys)}$
127I($\nu_{e},e^{-}$)127Xe | LSND | $284\pm 91{\rm(stat)}\pm 25{\rm(sys)}$
127I($\nu_{e},e^{-}$)X | COHERENT | $920^{+2.1}_{-1.8}$
natPb($\nu_{e},Xn$) | COHERENT | – –
Table 3: Flux-averaged cross-sections measured at stopped pion facilties on
various nuclei. Experimental data gathered from the LAMPF Willis:1980pj ,
KARMEN KARMEN:1998xmo ; KARMEN:1991vkr ; Maschuw:1998qh ; Zeitnitz:1994kz ,
E225 Krakauer:1991rf , LSND LSND:2001fbw ; LSND:2002oco ; Distel:2002ch , and
COHERENT COHERENT:2023ffx ; COHERENT:2022eoh  experiments. Table adapted from
the Ref. Formaggio:2012cpf .

The detection of the burst of 10s of MeV neutrinos from the galactic core-
collapse supernova is one of the primary physics goals of the future DUNE
experiment DUNE:2020zfm ; DUNE:2023rtr , as stated in the DUNE’s TDR
DUNE:2020lwj ; DUNE:2020ypp ; DUNE:2020mra ; DUNE:2020txw  , “Detect and
measure the nue flux from a core-collapse supernova within our galaxy, should
one occur during the lifetime of the DUNE experiment. Such a measurement would
provide a wealth of unique information about the early stages of core
collapse, and could even signal the birth of a black hole”. Detecting
supernova will provide unique insight into the properties of neutrinos, as
well as into the astrophysics of core-collapse supernova. DUNE’s capabilities
of supernova neutrino detection in the relevant tens-of-MeV neutrino energy
range as well as the physics to be learned from a DUNE supernova burst
detection will be limited by the lack of knowledge of the inelastic neutrino-
argon cross section. The inelastic neutrino-argon cross sections in this
energy range have never been measured. In the absence of experimental data,
the uncertainties in the theoretical calculations are not quantified at all.
The theory predictions, in fact, differ by orders of magnitude, see e.g. Fig.
6 in Ref. DUNE:2023rtr . In order to reconstruct the energy of the incoming
neutrinos from a supernova, the energy of all final state particles needs to
be known. These will include nuclear de-excitation products such as
$\gamma$-rays and potential nuclear fragments (neutrons, protons, deuterons,
etc). In a recent analysis performed by the DUNE collaboration, Ref.
DUNE:2023rtr , reports that the total inelastic neutrino-argon cross section
needs to be known at about 5% (in the absence of any external constraints) for
a measurement of the integrated neutrino luminosity with less than 10% bias
with DUNE.

The well-understood stopped-pion neutrino spectrum is a near-ideal 10s of MeV
neutrino source Scholberg:2012id  which can provide a unique opportunity to
measure neutrino-argon cross sections in this energy regime. Inelastic
interactions of neutrinos with nuclei are still poorly understood: theory is
sparse and experiments have large error bars. There are very few existing
measurements, none at better than the 10% uncertainty level, they are
summarized in Table 3 Formaggio:2012cpf . So far, there are no measurements on
the argon nucleus performed to date. Because inelastic neutrino interactions
have big uncertainties, in the future it will be crucial to measure inelastic
electron scattering cross sections at energies below the 50 MeV mark and use
those data to calibrate theoretical models for the neutrino scattering
process. Theoretical understanding of these processes is also relatively poor,
due to the strong dependence of the interaction rates on the specific initial-
and final-state nuclear wavefunctions. Inelastic neutrino-argon cross sections
shown in Figs. 7, Figs. 8 and Figs. 9 have never been measured before. CEvNS
experiments at decay at rest sources, such as COHERENT COHERENT:2020iec  and
Coherent CAPTAIN-Mills CCM  experiments are well suited to make those
measurements. The technical challenge is that the experiment has to have a
dynamic range of detecting keV energy recoil (signal for CEvNS), and MeV
energy nuclear deexcitation and nuclear fragment products (signal for
inelastic scattering) in the same detector. COHERENT experiment has recently
demonstrated their capabilities of measuring inelastic cross section by
performing two measurements, 127I($\nu_{e},e^{-}$)X and natPb($\nu_{e},Xn$)
COHERENT:2023ffx ; COHERENT:2022eoh .

##  V Experimental Landscape

Experiment | Nuclear Target | Detector Technology | Mass (kg) | Distance from source (m) | Dates
---|---|---|---|---|---
COHERENT | CsI[Na] | Scintillating crystal | 14 | 19.6 | 2015-2019
(ORNL) | Pb, Fe | Liquid scintillator | 1,000 | 19.0 | 2015-
| NaI[Tl] | Scintillating crystal | 185 | 21.0 | 2016-
| LAr | Noble scintillator | 24 | 27.5 | 2017-
| D2O | Cherenkov | 600 | 22.0 | 2022-
| Ge | HPGe PPC | 18 | 21.0 | 2022-
| NaI[Tl] | Scintillating crystal | 3,388 | 24.0 | 2022-
CCM | LAr | Noble scintillator | 10,000 | 23.0 | 2019 -
(LANL) |  |  |  |  |
Table 4: Current CEvNS experiments at the stopped pion sources.

Several experimental programs have been or are being set up to detect CEvNS
and BSM signals in the near future using stopped–pion neutrino sources as well
as with reactor sources. It all started with the COHERENT collaboration
reporting on the first detection of the CEvNS process in 2017. The measurement
was performed with an exposure of 14.6-308 kg-days, the COHERENT collaboration
identified nuclear recoil events from CEvNS viewed by a single photomultiplier
tube (PMT). The measurement was well in excess of the expected background
events for this exposure. A likelihood analysis considering the signal and
background shapes in time and PE yielded a result of 134 $\pm$ 22 CEvNS
events, with the uncertainty being primarily statistical, the SM prediction
for this analysis is 178 $\pm$ 43 CEvNS events. The observed event rate was
consistent with the SM prediction within uncertainties COHERENT:2017ipa . This
led to a flurry of proposals and experiments worldwide with complementary
detection technologies and physics goals. In Table 4, we list currently
running CEvNS experiments at stopped pion sources. There are several proposed
experiments at existing and planned facilities that are not included in the
table but we discuss them below. For the sake of completeness, we also list
CEvNS experiments at reactors in Table 5. These include CONNIE CONNIE , MINER
MINER , $\nu$GEN vGEN , NUCLEUS NUCLEUS , RICOCHET RICOCHET , TEXONO TEXONO ,
NEON NEON  and vIOLETA vIOLETA  experiments. The current theme of reactor
experiments is the observation of neutrino-nucleus elastic scattering at the
kinematic regime where complete quantum-mechanical coherency is expected.

SNS at ORNL: The Spallation Neutron Source at the Oak Ridge National
Laboratory has the most ambitious CEvNS-based experimental program
Barbeau:2021exu ; Asaadi:2022ojm . SNS is consistently running at 1 GeV proton
energy and 1.4 MW beam power. By 2024 after the next round of upgrades, it
will be running with 1.3 GeV proton energy and 2 MW beam power. The SNS First
Target Station (FTS) proton beam consists of a linear $H^{-}$ ion accelerator,
an accumulator ring, and a proton target. The proton target employs liquid
mercury contained inside a double-walled stainless steel vessel
Henderson:2014paa ; Haines:2014kna . The SNS generates 400-nanosecond bursts
of protons on target at 60 Hz frequency allowing for a highly effective
suppression of backgrounds and simultaneous measurement of neutrino signal and
backgrounds. A second target station (STS) with a solid tungsten target is
planned for the SNS. For this stage, the total beam power will be increased to
2.8 MW and the proton beam will be split between two targets with 45 Hz to the
first target and 15 Hz to the second, creating even more favorable conditions
to suppress steady-state backgrounds. The COHERENT collaboration continued
pursuing several additional detector technologies for CEvNS, to span a range
of N values, as well as detectors to address additional physics goals.

Lujan at LANL: The Lujan Center at the Los Alamos National Laboratory is a
prolific source of neutrinos from decays of stopped pions and muons created by
an 800 MeV proton beam. An 800MeV protons are delivered at a rate of 20Hz in a
280 ns triangular pulse from the LANSCE beamline and interact in a thick
tungsten target, copiously producing charged and neutral mesons. A 10 ton
liquid argon scintillation detector, Coherent CAPTAIN-Mills (CCM), is
currently operating. The CCM upright cylindrical cryostat 2.58 m in diameter
and 2.25 m high. A ton-scale mass and a keV-range energy threshold allow the
CCM detector to possess leading sensitivity to potential dark-sector physics
signals CCM ; CCM:2021yzc ; CCM:2021lhc .

Experiment | Detector Technology | Location | Source
---|---|---|---
CONNIE | Si CCDs | Brazil | Reactor
CONUS | HPGe | Germany | Reactor
MINER | Ge/Si cryogenic | USA | Reactor
NuCleus | Cryogenic CaWO4, Al2O3 calorimeter array | Europe | Reactor
$\nu$GEN | Ge PPC | Russia | Reactor
RED-100 | LXe dual phase | Russia | Reactor
Riochet | Ge,Zn | France | Reactor
TEXONO | p-PCGe | Taiwan | Reactor
NCC-1701 | p-PCGe | Germany | Reactor
Table 5: A list of reactor based CEvNS experiments.

JSNS2 at JPARC: The Japan Spallation Neutron Source of J-PARC is featured by a
1 MW beam of 3 GeV protons incident on a mercury target, creating an intense
neutrino flux from the stopped-pion and stopped-muon decays. The JSNS2 (J-PARC
Sterile Neutrino Search at J-PARC Spallation Neutron Source) experiment aims
to search for the existence of neutrino oscillations and to offer the ultimate
test of the LSND anomaly at a 17-ton fiducial volume Gd-dopped liquid
scintillation detector, new detector is being planned to study not only CEvNS
but potential low-mass dark-matter signals Ajimura:2017fld ; Ajimura:2020qni .

ESS: The European Spallation Source (ESS), sited in Sweden, will combine the
world’s most powerful superconducting proton linac with an advanced hydrogen
moderator, generating the most intense neutron beams for multi-disciplinary
science. It will also generate the largest pulsed neutrino flux suitable for
the detection of CEvNS. The ESS aims to achieve the power of 5 MW and proton
energy of 2 GeV. Several detector technologies sensitive to keV energy nuclear
recoils are being considered, these include a cryogenic undoped CsI
scintillator array, silicon Charge Coupled Devices (CCDs), and high-pressure
gaseous xenon detectors Baxter:2019mcx .

PIP2-BD at FNAL: The Proton Improvement Project II (PIP-II) is the first phase
of a major transformation of the accelerator complex underway at Fermilab to
prepare the lab to host the Deep Underground Neutrino Experiment (DUNE). The
completion of the PIP-II superconducting LINAC at Fermilab as a proton driver
for DUNE/LBNF in the late 2020s creates an attractive opportunity to build
such a dedicated beam dump facility at Fermilab, this will require the
addition of an accumulator ring to bunch the PIP-II beam current into short
proton pulses. A unique feature of this Fermilab beam dump facility is that it
can be optimized from the ground up for HEP. Thus, relative to spallation
neutron facilities dedicated to neutron physics and optimized for neutron
production operating at a similar proton beam power, a HEP-dedicated beam dump
facility would allow for better sensitivity to various physics goals. The
facility could also accommodate multiple, 100-ton-scale HEP experiments
located at different distances from the beam dump and at different angles with
respect to the beam Toups:2022yxs .

##  VI Implications for the Standard Model Physics

Since the uncertainty on the SM predicted CEvNS cross sections is relatively
small, CEvNS cross section measurement allows testing of SM weak physics. The
experiments measure the number of events $N$ generated by neutrinos of a given
flavor $\alpha$ and collected by a detector with a finite threshold

$\frac{dN}{dT}=N_{t}\sum_{\alpha=\nu_{e},\nu_{\mu},\bar{\nu}_{\mu}}\int^{E_{\nu}^{\text{max}}}_{E_{\nu}^{\text{min}}}\Phi_{\alpha}(E_{\nu})~{}\frac{d\sigma}{dT}~{}dE_{\nu}$
(50)

where $N_{t}$ is a normalization constant that depends on the number of
protons on target, the neutrino yield per proton, the mass of the detector,
detection efficiency and the distance of the detector from the source. Any
deviation from the SM predicted event rate of Eq. (50), either with a change
in the total event rate or with a change in the shape of the recoil spectrum,
could indicate new contributions to the interaction cross-section. More
generally, what can be probed is the weak nuclear form factor of a nucleus and
weak mixing angle (see, Eq. (17)). There are many important results that can
be extracted from the CEvNS measurements, recent work has considered CEvNS as
a percent-level probe of SM physics Scholberg:2005qs ; Miranda:2019skf ;
Cadeddu:2019eta ; Canas:2018rng ; Papoulias:2019xaw ; Baxter:2019mcx ;
Huang:2019ene ; Bernabeu:2002nw ; Bernabeu:2002pd ; Papavassiliou:2005cs ;
Cadeddu:2018dux . Note that for a given stopped pion production facility,
experiments have control over choosing the baseline (the distance from the
source to the detector), the angular placement of the detector with respect to
the beam axis, and on the nuclear target employed in the detector. These can
be exploited to increase the sensitivity of the primary physics goal of the
experiment. An additional advantage of the stopped pion source is that one
could exploit both timing and energy data. The timing profile, See Fig. 2,
allows the separation of the prompt neutrino flavor from the delayed neutrino
flavor.

###  VI.1 Weak Nuclear Form Factor

The modest loss of coherence at stopped-pion energies can be valuable for the
understanding of the nuclear structure, given that one can probe the form
factor for a given nucleus as a function of $Q$. A precise measurement of the
CEvNS cross section can be used to extract the weak form factor, using Eq.
(17), given one measures recoil spectrum shape. Observed recoil energy T can
be used to determine Q; therefore, the observed CEvNS recoil energy spectrum
allows one to map the effect of the weak form factor of the nucleus at low
momentum transfer.

As discussed in Sec. III, the weak nuclear charge is strongly dominated by its
neutron content. The observation of CEvNS can, therefore, further provide
important nuclear structure information through the determination of the weak
form factor, which constrains the neutron density distribution, at least at
low momentum transfers where the process remains coherent
AristizabalSierra:2019zmy ; Payne:2019wvy ; Hoferichter:2020osn ; Yang:2019pbx
; VanDessel:2020epd ; Patton:2012jr ; Cadeddu:2017etk ; Co:2020gwl ;
Ciuffoli:2018qem ; Papoulias:2019lfi . Furthermore, since proton density
distributions are generally well understood, a measure of the mean radius of
the neutron distribution (the “neutron radius”) enables the determination of
the “neutron skin” of a nucleus — the difference between the larger neutron
radius and the proton radius. These measurements complement PVES experiments
not only due to additional data but also due to different energy ranges and
nuclear targets, which could be used to calibrate nuclear-structure
calculations. Furthermore, improved measurements of the neutron skin would
have important consequences for the equation of the state of neutron-rich
matter, which plays an essential role in understanding the structure and
evolution of neutron stars Fattoyev:2017jql ; Reed:2021nqk ; Lattimer:2012xj ;
Hebeler:2013nza ; Hagen:2015yea . With more ambitious precision measurements,
axial-vector contributions to the weak nuclear response can also be
determined, in principle, for nuclei with non-zero spin.

However, arguably one of the most intricate aspects of nuclear-structure input
concerns searches for physics beyond the SM. In principle, CEvNS cross
sections provide constraints on the combination of nuclear responses and the
BSM effects. Therefore external independent experimental information for the
neutron responses, such as from the PVES experiment would be vital. In fact,
in order to derive BSM constraints beyond the level at which current nuclear-
structure calculations constrain the neutron distribution, a combined analysis
of multiple targets and momentum transfers is required to distinguish between
nuclear structure and potential BSM contributions Abdullah:2022zue .

###  VI.2 Weak Mixing Angle

In quantum field theory, the weak mixing angle, $\theta_{W}$, depends on the
energy scale at which it is measured. There exists an experimental anomaly
with respect to the SM predictions for neutrino-nucleon scattering at the
$Q\sim$ GeV/c scale NuTeV:2001whx . Since CEvNS cross section and, therefore,
the event rate depends on the weak mixing angle, Eq. (17) and Eq. (16), the
measured CEvNS event counts can be used to infer the weak mixing angle. A
change in $\theta_{W}$ will result in eventual event rate scaling.

Furthermore, measurements on multiple nuclear targets will further enhance the
sensitivity to extracting weak mixing angles from weak nuclear charge
Scholberg:2005qs ; Miranda:2019skf ; Canas:2018rng ; Papoulias:2019xaw ;
Baxter:2019mcx ; Huang:2019ene . As CEvNS measurements become more precise in
the near-future, one could extract the weak mixing angle at $Q$ values of a
few tens of MeV/c; that will be competitive with other methods for determining
$\theta_{W}$ at low $Q$ from parity-violating electron-proton scattering
Qweak:2018tjf , Moller scattering MOLLER:2014iki  and atomic parity violation
Roberts:2014bka .

##  VII Implications for Beyond the Standard Model Physics

CEvNS, being a low-energy process, provides a natural window to study light,
weakly-coupled, beyond the standard model physics in the neutrino sector.
Several extensions of the SM can be explored at low energy Barranco:2005yy ;
Scholberg:2005qs ; Barranco:2007tz ; Lindner:2016wff ; Coloma:2017ncl ;
Farzan:2017xzy ; AristizabalSierra:2018eqm ; Brdar:2018qqj ; Abdullah:2018ykz
; AristizabalSierra:2019zmy ; Miranda:2019skf ; Bell:2019egg ;
AristizabalSierra:2019ufd ; Cadeddu:2019eta ; Coloma:2019mbs ; Canas:2019fjw ;
Dutta:2019eml ; Denton:2020hop ; Skiba:2020msb ; Cadeddu:2020nbr ;
Abdullah:2020iiv ; Papoulias:2019xaw ; Baxter:2019mcx ;
AristizabalSierra:2017joc ; Giunti:2019xpr ; Liao:2017uzy ; Dent:2017mpr ;
Dent:2019ueq ; Miranda:2020syh ; Suliga:2020jfa ; Dutta:2020che ;
Dutta:2019nbn . Since CEvNS cross section is orders of magnitude higher at low
energies, the BSM searches can be done with relatively small detectors instead
of the typical large neutrino detectors.

Any significant deviations between the SM theory predictions and the
experiment event rate of Eq. (50) will indicate the presence of new physics
and can be expressed as some conventional definition for the low-energy
property. The deviation from the SM predictions can either be reflected as a
change in the total event rate or a change in the shape of the recoil
spectrum. For a given stopped pion production facility, experiments have
control over choosing the baseline (the distance from the source to the
detector), the angular placement of the detector with respect to the beam
axis, and the nuclear target employed in the detector. These can be exploited
to increase the sensitivity of a particular BSM scenario. An additional
advantage of the stopped pion source is that one could exploit the timing
structure of the neutrino background, see Fig. 2, for a given BSM signal.

###  VII.1 Non-standard Interactions of Neutrinos

Figure 10: A diagrammatic illustration of the neutral current non-standard
neutrino interactions process where $f$ refers to SM fermions.

In the context of neutrino physics, the term Non-Standard Interactions (NSI)
usually refers to the inclusion of four-fermion operators leading to
modifications of the Wilson coefficients already present in the SM
Farzan:2017xzy ; Wolfenstein:1977ue ; Proceedings:2019qno . NSIs appear in
several appealing SM extensions and provide a general effective field theory
(EFT) framework to parameterize new physics in the neutrino sector. NSIs can
be charged current or neutral current interactions with matter particles e, u,
and/or d. Charged-current NSI leads to modification of both the production and
detection of neutrinos but also leads to charge-lepton flavor violation while
NC NSI does not modify production or detection when charged leptons are
involved.

In a model-independent approach, a useful parametrization of the possible
effects of new physics at low energies is through the addition of higher-
dimensional operators to the SM Lagrangian that respect the SM gauge group.
The allowed set of operators includes four-fermion operators affecting
neutrino production, propagation, and detection processes. For example,
operators of the form

$\mathcal{L}_{NSI}^{CC}~{}=~{}2\sqrt{2}G_{F}~{}\varepsilon_{\alpha\beta}^{ff^{\prime},P}~{}\bar{\nu}_{\alpha}\gamma^{\mu}(1-\gamma_{5})l_{\beta}~{}\bar{f^{\prime}}\gamma_{\mu}f$
(51)

would induce non-standard charged-current (CC) production and detection
processes for neutrinos of flavor $\alpha$, while operators such as

$\mathcal{L}_{NSI}^{NC}~{}=~{}-2\sqrt{2}G_{F}~{}\varepsilon_{\alpha\beta}^{fP}~{}\bar{\nu}_{\alpha}\gamma^{\mu}(1-\gamma_{5})\nu_{\beta}~{}\bar{f}\gamma_{\mu}f$
(52)

would lead to flavor-changing neutral-current (NC) interactions of neutrinos
with other fermions (if $\alpha=\beta$), or to a modified NC interaction rate
with respect to the SM expectation (if $\alpha\neq\beta$). Here, $f$ and
$f^{\prime}$ refer to SM fermions. The parameters $\varepsilon$ describe the
size of non-standard interactions relative to standard charged or neutral
current weak interactions.

In the SM, the weak charge of a nucleus only depends on the SM vector
couplings to protons and neutrons, Eq. (16), and is independent of the
neutrino flavor. In the presence of NC NSI, this effective charge gets
modified by the new operators introduced as

$\displaystyle Q^{2}_{W,NSI}$ $\displaystyle=$
$\displaystyle[(g_{n}^{V}+2\varepsilon_{\alpha\alpha}^{uV}+\varepsilon_{\alpha\alpha}^{dV})N+(g_{p}^{V}+\varepsilon_{\alpha\alpha}^{uV}+2\varepsilon_{\alpha\alpha}^{dV})Z]^{2}$
(53)
$\displaystyle+\sum_{\beta\neq\alpha}[(\varepsilon_{\alpha\beta}^{uV}+2\varepsilon_{\alpha\beta}^{dV})N+(2\varepsilon_{\alpha\beta}^{uV}+\varepsilon_{\alpha\beta}^{dV})Z]^{2}$

Any deviation from SM predicted rates (plus the form factor uncertainty)
signals a probing non-standard interaction of neutrinos. Vector couplings are
characterized by the spin-independent combination while axial couplings are
characterized by the orthogonal spin-dependent combination. Since typically in
the CEvNS process, the axial contribution is negligible in comparison to the
vector contribution (due to spin cancellation), we assume that spin-dependent
axial NSI contributions are small. The effect of non-zero values of epsilons,
which can be either positive or negative, can be either an enhancement or
suppression of the CEvNS rate. Of course, some combinations of NSI parameter
values for a given $N$ and $Z$ can result in the SM CEvNS rate. Therefore, a
combination of CEvNS measurements on targets with different $N$ and $Z$ values
can break any such accidental degeneracies (as well as cancel flux-related
uncertainties). A stopped-pion source contains both electron and muon flavor;
hence a CEvNS measurement gives direct access to all coupling coefficients
except $\varepsilon_{\tau\tau}$. Furthermore, because at the SNS neutrino
flavors can be separated by timing, see Fig. 2, one can in principle probe
electron and muon NSIs separately Coloma:2017ncl ; Liao:2017uzy ; Dent:2017mpr
.

The flavor-changing NSI in neutrino oscillation experiments leads to the
appearance of new degeneracies involving standard oscillation parameters and
NSI operators. These can affect DUNE’s capability in extracting the CP-
violation phase. Data from the CEvNS experiment to the global fits from
oscillation breaks the degeneracies involving flavor-diagonal NSI, since CEvNS
experiments can directly measure the neutrino-nucleus interactions for both
electron and muon neutrinos. Thus constraining NSI parameters with CEvNS
experiments can significantly improve the extraction of the CP-violation phase
in DUNE Coloma:2016gei ; Coloma:2017egw .

###  VII.2 Neutrino Electromagnetic Properties

In the SM, neutrino electromagnetic interactions are negligible. BSM processes
can induce significant electromagnetic effects, these can be probed in CEvNS
experiments. Since neutrinos oscillate and therefore have mass, it provides
the best motivation for the existence of non-trivial neutrino electromagnetic
properties such as a neutrino magnetic moment or a neutrino charge radius.

The SM, minimally extended to allow massive neutrinos, predicts a very small
magnetic moment of neutrino ParticleDataGroup:2020ssz

$\mu_{\nu}=3.2\times 10^{-19}\mu_{B}\left(\frac{m_{\nu}}{{\rm eV}}\right).$
(54)

Within the minimum neutrino mass range of $10^{-4}$ to $1$ eV, the SM electron
neutrino magnetic moment ranges [$3\times 10^{-21},3\times 10^{-19}$]
$\mu_{B}$ for Dirac and [$5\times 10^{-25},8\times 10^{-23}$] $\mu_{B}$ for
Majorana case Balantekin:2013sda . The differential cross section in the
presence of a neutrino magnetic moment adds incoherently to the Standard Model
cross section due to the required spin-flip

$\left(\frac{d\sigma}{dT}\right)_{\mathrm{tot}}=\left(\frac{d\sigma}{dT}\right)_{\mathrm{SM}}+\left(\frac{d\sigma}{dT}\right)_{\mathrm{EM}}$
(55)

where the EM contribution has a characteristic $1/T$ dependence, while its
strength is controlled by the size of the neutrino magnetic moment
Vogel:1989iv

$\left(\frac{d\sigma}{dT}\right)_{\mathrm{EM}}=\frac{\pi\alpha^{2}\mu_{\nu}^{2}\,Z^{2}}{m_{e}^{2}}\left(\frac{1-T/E_{\nu}}{T}+\frac{T}{4E_{\nu}^{2}}\right)F_{\text{ch}}^{2}(q^{2})\,\\\
$ (56)

where $m_{e}$ is the mass of the electron, $\alpha$ is the fine structure
constant. $F_{\text{ch}}$, normalized as $F_{\text{ch}}(0)=1$, is the charge
form factor of the nucleus that is known with high precision for many nuclei.
In the presence of neutrino magnetic moments, the neutrino interaction cross
section will be modified. The low-energy cross section is more sensitive to
these small changes. Any measurement of magnetic moment larger than this would
be a signature of BSM physics. Larger neutrino magnetic moments would also
hint that neutrinos are Majorana particles Bell:2006wi . The current best
limits on neutrino magnetic moment have been set by solar and astrophysical
neutrinos ParticleDataGroup:2020ssz ; Borexino:2017fbd . The strongest
magnetic moment limits on $\nu_{\mu}$ are from LSND experiment LSND:2001akn ;
Kosmas:2015sqa . The experimental signature would be an enhancement of the
rate at low recoil energy, both for scattering on electrons and on nuclei
Vogel:1989iv ; Kosmas:2015sqa ; Dodd:1991ni . Therefore, to measure neutrino
magnetic moment, the detector need to have a low energy threshold and good
energy resolution.

The impact of the neutrino charge radius, being a helicity-preserving
quantity, is taken as a shift on the weak mixing angle according to

$\sin^{2}\theta_{W}\rightarrow\sin^{2}\theta_{W}+\frac{\sqrt{2}\pi\alpha}{3G_{F}}\langle
r_{\nu_{\alpha}}^{2}\rangle\,.$ (57)

The neutrino charge radius has a small flavor-dependent effect on the CEvNS
cross section. These can be measured in CEvNS experiments at stopped pion
sources where muon- and electron-neutrino’s CEvNS event rates can be separated
using the timing structure, see Fig. 2. The effects is expected to be of
percent level Papavassiliou:2005cs ; Cadeddu:2018dux .

###  VII.3 Sterile Neutrino Oscillations

One possible extension of the SM is the existence of sterile neutrinos, new
neutrino states that do not interact with the SM weak interactions. These new
gauge singlet fermions can be included as a minimal extension of the SM. There
are several experimental anomalies, called short-baseline neutrino anomalies,
that hint at the existence of sterile neutrinos. There are a number of
experiments underway testing short-baseline anomalies. CEvNS is also an
excellent tool for the search for sterile neutrino oscillations by setting up
multiple identical detectors at different baselines from the neutrino
production point. Flavor-blind neutral-currents can be used to probe the
disappearance of active neutrinos. The signal would be both a distortion of
recoil spectra and an overall rate suppression.

Oscillation probabilities are simplest for monoenergetic sources and decay at
rest provides a high-intensity monoenergetic source of $\nu_{\mu}$s, thus
being a natural candidate for carrying out high statistics searches for
$\nu_{\mu}$ disappearance. In this context, CEvNS has been recognized as being
advantageous due to its relatively large cross section Anderson:2012pn ;
Formaggio:2011jt ; Kosmas:2017zbh . Working at lower energies allows for
shorter baselines with equivalent L/E and consequently higher fluxes as
compared to e.g., the decay in-flight experiments. In particular, the
sensitivity to sterile neutrinos is maximized when deploying multiple
detectors at different distance baselines in the range $\sim 20$–$40$ m
Anderson:2012pn . This configuration can probe parameter space that is
consistent with the $\sim$ eV mass-scale hinted at by LSND and MiniBooNE,
thereby providing an independent test of sterile neutrino parameter space
Anderson:2012pn ; Blanco:2019vyp .

###  VII.4 Accelerator Produced Light Dark Matter

Figure 11: A diagrammatic representation of the Dark Matter particle, $\chi$,
scattering off a nucleus, $N$, mediated via a dark photon $A^{\prime}$.

The stopped-pion facilities produce a copious amount of neutral and charged
mesons as well as photons. These can decay into dark sector particles that can
either scatter off or decay in the detector material via kinetic mixing with
the SM particles Dutta:2019nbn ; deNiverville:2015mwa ; deNiverville:2016rqh .
Dark matter candidate particle masses can be probed in the few to few hundred
MeV range at these facilities. For example, within a dark photon model, the
dark photon $A^{\prime}$ undergoes kinetic mixing with the SM photon and can
be described by the Lagrangian:

$\mathcal{L}\supset g_{D}A^{\prime}_{\mu}\bar{\chi}\gamma^{\mu}\chi+e\epsilon
Q_{q}A^{\prime}_{\mu}\bar{q}\gamma^{\mu}q$ (58)

where $g_{D}$ is the dark coupling constant, $\epsilon$ is the mixing
parameter, $Q_{q}$ is quark’s electric charge. The dark photon can be produced
at the stopped pion facilities via the processes of pion capture, pion decay,
and the photons emerging from the cascades:

$\displaystyle\pi^{-}+p$ $\displaystyle\rightarrow$ $\displaystyle
n+A^{\prime}$ (59) $\displaystyle\pi^{+}+n$ $\displaystyle\rightarrow$
$\displaystyle p+A^{\prime}$ (60) $\displaystyle\pi^{0}$
$\displaystyle\rightarrow$ $\displaystyle\gamma+A^{\prime}$ (61)

The dark photons then decay into dark matter candidates:
$A^{\prime}\rightarrow\chi\bar{\chi}$. The signature in the detector is either
elastic scattering with specific keV energy nuclear recoil signatures or
inelastic scattering with specific MeV energy nuclear deexcitation gammas
Dutta:2022tav ; Dutta:2023fij . The process is schematically shown in Fig. 11.

The time structure of the beam is especially valuable for separating SM
neutrino scattering recoils from the DM-induced recoils. Since the DM signal
is expected to be prompt, the delayed neutrino signal can provide a powerful
constraint on the neutrino background in the prompt window. The DM signal will
also have a distinctive dependence on the direction with respect to the
source, whereas CEvNS signals will be isotropic. For dark sector searches,
event rate scaling, recoil spectrum shape, timing, and direction with respect
to source are all helpful.

Recent results from the COHERENT and Coherent CAPTAIN-Mills experiments have
demonstrated how detectors capable of measuring coherent elastic neutrino-
nucleus scattering (CEvNS) can also be used to set limits on vector portal and
leptophobic DM at proton beam dumps. They also provide excellent opportunities
to search for axion-like particles (ALPs) Dent:2019ueq .

##  VIII Summary

Neutrinos continue to provide a testing ground for the structure of the
standard model and hints toward the physics beyond the standard model.
Neutrinos of energies spanning over several orders of magnitude have been
detected via various mechanisms ranging from inverse beta decay to scattering
off quarks, nucleons, and nuclei. At MeV scales, there has been one elusive
process, until a few years ago, known as coherent elastic neutrino-nucleus
scattering that was theoretically predicted for over five decades ago but was
never observed experimentally. The recent experimental observation of CEvNS by
the COHERENT collaboration at a stopped pion neutrino source has inspired
physicists across many subfields. This new way of detecting neutrinos has
wider implications for border communities that span nuclear physics, particle
physics, astrophysics, and beyond. Leveraging orders of magnitude higher CEvNS
cross section, new physics can be searched with relatively small detectors.
CEvNS, being a low-energy process, provides a natural window to study light,
weakly-coupled, new physics in the neutrino sector.

Neutrinos from stopped pions sources cover energies in the tens of MeVs and
are almost optimal for studying CEvNS, finding a sweet spot where the CEvNS
rate is high enough and recoil energies are more easily detectable above the
threshold. So far, CEvNS is observed only at the decay at rest sources. In
addition, the pulsed time structure of the beam source provides a strong
handle for suppressing the background for new physics searches. Several
worldwide experimental programs have been or are being set up to detect CEvNS
and BSM signals in the near future at stopped–pion neutrino sources (as well
as with reactor sources where the CEvNS process is yet to be detected) with
complementary detection technologies and physics goals, making it an emerging
exciting avenue.

Disentangling new physics signals in these experiments requires a precise
understanding of the CEvNS SM scattering rate. At tree level, the theoretical
uncertainty on the CEvNS cross section is driven by the uncertainty on the
weak form factor of the nucleus. The charge density of a nucleus is strongly
dominated by protons and has been extensively studied with impressive
precision in elastic electron scattering experiments. While the neutron
density, which CEvNS is most sensitive to, leads the overall uncertainties on
the CEvNS rate. For non-zero spin nuclei, the axial-vector part adds an
additional contribution that is often not included in CEvNS estimation. The
CEvNS process also receives a few percent of radiative corrections, from
electrons and muons running in loops introducing a non-trivial dependence on
the momentum transfer due to their relatively light masses. Parity-violating
electron scattering experiments offer complementary input and provide a
precise approach to experimentally probing weak form factors and neutron
distribution. Although, the choice of the nuclear targets, so far in PVES and
CEvNS experiments is not the same, since they are both driven by different
physics motivations and have varied technical needs.

CEvNS experiments at stopped pion sources are also sensitive to tens of MeV
inelastic CC and NC neutrino-nucleus scattering processes. These processes
have implications for supernova detection in future neutrino experiments. The
interaction cross sections for these processes lack the $N^{2}$ enhancement
associated with CEvNS and, therefore, tend to be at least one order of
magnitude smaller than that of the CEvNS rate. The detectable final-state
particles of these inelastic scattering have typical energies of the same
order as the incident neutrino energies. The experimental requirements for
CEvNS and inelastic signals are quite different, larger masses are needed for
inelastic, as well as the dynamic range to record MeV-scale energy
depositions, while very low thresholds are not required. DUNE’s capabilities
of supernova neutrino detection in the relevant tens-of-MeV neutrino energy
range as well as the physics to be learned from a DUNE supernova burst
detection will be limited by the lack of knowledge of the inelastic neutrino-
argon cross section. The well-understood stopped-pion neutrino spectrum is a
near-ideal tens of MeV neutrino source which provides a unique opportunity to
study inelastic neutrino-nucleus cross sections at tens of MeVs.

Since the uncertainty on the SM predicted CEvNS cross sections is relatively
small, CEvNS cross section measurement allows testing of SM weak physics and
in probing new physics signals. For a given stopped pion production facility,
experiments can in principle choose the baseline, the angular placement of the
detector with respect to the beam axis, and the nuclear target employed in the
detector to optimize the sensitivity of the primary physics goal of the
experiment. An additional advantage of the stopped pion source is that one
could exploit both timing and energy data. Any deviation from the SM predicted
event rate either with a change in the total event rate or with a change in
the shape of the recoil spectrum, could indicate new contributions to the
interaction cross-section. In particular, the weak nuclear form factor of a
nucleus and weak mixing angle can be probed. CEvNS, being a low-energy
process, provides a natural window to study light, weakly-coupled, beyond the
standard model physics in the neutrino sector. Several extensions of the SM
can be explored at low energy. Since CEvNS cross section is orders of
magnitude higher at low energies, the BSM searches can be done with the
relatively small detector. In particular, NSIs, neutrino electromagnetic
properties in terms of neutrino magnetic moment and neutrino charge radius,
and sterile neutrinos can be studied. Stopped pion facilities are also a
copious source of neutral and changed mesons as well as photons that allows
proving several dark sector physics scenarios such as vector portal models,
leptophobic dark matter as well as axion-like particle searches.

## Acknowledgements

V.P. thanks Oleksandr Tomalak, Pedro Machado and Ryan Plestid for discussions
on Ref. Tomalak:2020zfh ; Nils Van Dessel, Heather Ray and Natalie Jachowicz
for discussion on Ref. VanDessel:2020epd ; Bhaskar Dutta, Wei-Chih Huang and
Jayden Newstead for discussion on Ref. Dutta:2022tav ; colleagues from the
Coherent CAPTAIN-Mills experiment for various discussions on the experimental
scope of the CEvNS experiments - all of which have motivated the content of
this review. This manuscript has been authored by Fermi Research Alliance, LLC
under Contract No. DE-AC02-07CH11359 with the U.S. Department of Energy,
Office of Science, Office of High Energy Physics.

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