#  Neutrino millicharge and other electromagnetic interactions with
COHERENT-2021 data

Amir N. Khan<EMAIL_ADDRESS>Max-Planck-Institut für Kernphysik,
Postfach 103980, D-69029 Heidelberg, Germany

###### Abstract

We analyze new data from the COHERENT experiment of the coherent neutrino-
nucleus scattering to investigate the electromagnetic interactions of
neutrinos. With almost double the statistics and precision now, the
statistical significance of the observed process has now enhanced to
11.6$\sigma$. We derive constraints on the electromagnetic properties of
neutrinos using the new COHERENT data. The constraints improve by more than a
factor of two compared to the previous bounds. Furthermore, we discuss the
unique behavior of the neutrino millicharge at lower energy recoils and show
its unique dependence on its interference with the standard model
contribution, inverse power of recoil energy and the mass of the target
particle in comparison to the other interactions.

###### pacs:

xxxxx

##  I Introduction

Coherent elastic neutrino-nucleus scattering (CE$\nu$NS) is a SM process that
was predicted forty years ago [1, 2, 3, 4] and was recently observed by the
COHERENT experiment [5, 6]. The importance of the process ranges from its
ability as a precision probe of the SM parameters [1, 2, 3, 4, 7, 8, 9, 10,
11, 12] to test new interactions at low momentum transfer including its
importance for the direction detection of dark matter [13, 14, 15, 16, 17, 18,
19, 20, 21, 22, 23, 24]. The COHERENT collaboration has updated the result for
the aforementioned process by doubling the statistics and the precision by
reducing the overall systematic errors to half [25]. In particular, the error
on the quenching factor improves from 25% to 4%. Compared to the first result
at 6.7$\sigma$, the updated significance level has now reached 11.6$\sigma$.
With this improvement, it is natural to expect a better sensitivity to any new
physics or improvement in the limits. To this aim, we analyze the new data to
constrain the neutrino electromagnetic properties if they contribute to the
CE$\nu$NS and derive constraints on neutrino magnetic moment, millicharge,
charge radius and neutrino anapole moment using the new COHERENT data.

In low energy scattering experiments with low target recoils, the sensitivity
to any new physics mainly depends on three factors: $(i)$ whether the new
physics interactions interfere with the SM interactions or not, $(ii)$ the
proportionality of the new physics coupling strength to the inverse power of
the target’s recoil energy, and $(iii)$ mass of the target particle. These
three factors are different for the four types of electromagnetic interactions
in the $\nu-$e elastic scattering and in the CE$\nu$NS processes, as we will
discuss later. For example, the neutrino millicharge is more sensitive to low
energy recoils [26, 27, 28] because it interferes with the SM interactions,
and its coupling is proportional to the inverse square of the target recoil
energy. We will discuss this aspect in more detail in section V. These
kinematical considerations are equally valid for the target recoils in the
dark matter scattering [29].

We organize the rest of the paper as follows. In the next section, we discuss
the basics of the differential cross-section of the CE$\nu$NS in the SM. Then,
in Sec. III, we discuss data analysis for the COHERENT setup with the new
data. Then, in Sec. IV, we introduce the electromagnetic properties of
neutrinos and derive constraints using the new COHERENT data. Next, in Sec. V,
we discuss in detail why millicharge neutrinos are kinematically more special
than the other electromagnetic properties of neutrinos. Finally, we provide
the conclusion of this work in sec VI.

##  II Coherent Elastic neutrino nucleus scattering

At the tree level in the SM, the differential cross-section of the neutrino
with flavor ‘$\alpha$’ scattering off the spin-0 nucleus of CsI with proton
number ‘$Z$’ and neutron number ‘$N$’ is given by [1, 2, 3, 4, 7, 8],

$\frac{d\sigma_{\alpha}}{dT}(E_{\nu},T)=\frac{G_{F}^{2}M}{\pi}\left[Zg_{p}^{V}+Ng_{n}^{V})\right]^{2}\left(1-\frac{T}{E_{\nu}}-\frac{MT}{2E_{\nu}^{2}}\right)F^{2}(q^{2})\,,$
(1)

where ‘$G_{F}$’ is the Fermi constant, ‘$E_{\nu}$’ is the energy of the
incoming neutrinos, ‘$T$’ is nuclear the recoil energy, $q^{2}=2MT$ is the
squared momentum transfer, and ‘$M$’ is the mass of the target nucleus. Here,
$g_{p}^{V}=(2g_{u}^{V}+g_{d}^{V})$ and $g_{n}^{V}=(g_{u}^{V}+2g_{d}^{V})$,
where $g_{u}^{V}$ and $g_{d}^{V}$ are the neutral current coupling constants
for the ‘up’ and ‘down’ quarks which, in terms of the weak mixing angle
‘$\theta_{W}$’ at tree level are given by

$\displaystyle g_{u}^{V}$ $\displaystyle=$
$\displaystyle\frac{1}{2}-\frac{4}{3}\sin^{2}\theta_{W}\,,$ (2) $\displaystyle
g_{d}^{V}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}+\frac{2}{3}\sin^{2}\theta_{W}\,.$ (3)

We will use $\rm sin^{2}\theta_{W}=0.23857\pm 0.00005$, the low energy
$(q\rightarrow 0)$ value evaluated in $\overline{\rm MS}$ scheme [30]. Using
the low energy value of $\rm sin^{2}\theta_{W}$ and including the small
radiative corrections [31, 30], we find the values of the coupling constants
$g_{u}^{V}=0.197,\ \ g_{d}^{V}=-0.353$ for electron neutrinos and
$g_{u}^{V}=0.191,\ \ g_{d}^{V}=-0.350$ for muon neutrinos. In Eq. (1),
$F(q^{2})$ is the nuclear form factor, and we use the Klein-Nystrand form [32]
as given in the following

$F(q^{2})=\frac{4\pi\rho_{0}}{Aq^{3}}[\sin(qR_{A})-qR_{A}\cos(qR_{A})]\left[\frac{1}{1+a^{2}q^{2}}\right],$
(4)

where $\rho_{0}$ is the normalized nuclear number density, $A$ is the atomic
number of CsI, $R_{A}=1.2A^{1/3}\,\mathrm{fm}$ is the nuclear radius, and
$a=0.7\,\mathrm{fm}$ is the range of the Yukawa potential.

##  III Data Analysis

The COHERENT detector receives a prompt signal from the mono-energetic (29.8
MeV) beam of muon-neutrinos $(\nu_{\mu})$ produced from the $\pi^{+}$ decay at
rest ($\pi^{+}\rightarrow\mu^{+}\nu_{\mu}$) at the Oak Ridge Spallation
Neutron Source. Subsequently, continuous fluxes of electron-neutrinos
($\nu_{e}$) and muon-anti-neutrinos ($\bar{\nu}_{\mu})$ with energy peaks,
respectively, around 35 Mev and 52.8 MeV produced in $\mu^{+}$ decays
($\mu^{+}\rightarrow\nu_{e}e^{+}\bar{\nu}_{\mu}$) with the characteristic time
scale of the muon lifetime, namely 2.2 $\mu$ sec, is received. The fluxes are
produced from $3.20\times 10^{23}$ protons on target from the liquid mercury.
The average production rate of the SNS neutrinos from the pion decay chain is
$r=0.0848$ neutrinos of each flavor per proton [25].

The detector, located at a distance $L=19.3$ m from the source, uses CsI[Na]
as a target, where the Na contributes small enough to be neglected. For such a
setup, the total number of events of the nuclear recoil in a given energy bin
‘$i$’ and neutrino flavor ‘$\alpha$’ reads

$N_{\alpha}^{i}=N\int_{T^{\prime\rm i}}^{T^{\prime\rm
i+1}}dT^{\prime}\int_{0}^{T^{\rm max}}dT\int_{E_{\nu}^{\rm min}}^{E_{\nu}^{\rm
max}}dE_{\nu}\frac{d\sigma_{\alpha}}{dT}(E_{\nu},T)\frac{d\phi_{\nu_{\alpha}}(E_{\nu})}{dE_{\nu}}\mathcal{E}(T^{\prime})G(T^{\prime},T),$
(5)

where $\mathcal{E}(T^{\prime})$ is the detection efficiency function,
$G(T^{\prime},T)$ is the gamma distribution function for the detector energy
resolution, $T$ and $T^{\prime}$ denotes the nuclear recoil energy and the
reconstructed recoil energy, respectively. Here,
$N=\left(2m_{\mathrm{det}}/M_{\rm CsI}\right)N_{A}$ is the total number of CsI
nucleons, $m_{\mathrm{det}}=14.57$ kg, $N_{A}$ is the Avogadro’s number,
$M_{\rm CsI}$ is the molar mass of CsI, $E_{\nu}^{\min}=\sqrt{MT/2}$, $M$ is
the mass of the target nucleus, $E_{\nu}^{\max}$ is the maximum neutrino
energy and $d\phi_{\nu_{\alpha}}(E_{\nu})/dE_{\nu}$ is the flux corresponding
to the flavor ‘$\alpha$’ [22].

The recent measurement of COHERENT [25] considers the recoiled energy-
dependent quenching factor, $f_{\rm q}(T^{\prime})$ and measures the energy
spectrum in terms of photo-electrons (p.e). Therefore, to calculate the total
number of events in a particular bin ‘i’ of photo-electrons, we use the
following relation between the recoil energy and the number of photo-electrons
($N_{\rm p.e}$)

$N_{\rm p.e.}=f_{q}(T^{\prime})\times T^{\prime}\times Y,$ (6)

where $\rm Y=13.35~{}\rm photons$/keV is the light yield and
$f_{q}(T^{\prime})$ is taken from [25].

To fit the data of the energy spectrum in Fig. 3 of ref. [25] with the model
including the SM and the new physics parameters, we use the following least-
square function

$\chi^{2}=\underset{i=2}{\overset{9}{\sum}}\left(\frac{N_{\rm obs}^{i}-N_{\rm
exp}^{i}(1+\alpha)-B^{i}(1+\beta)}{\sigma^{i}}\right)^{2}+\left(\frac{\alpha}{\sigma_{\alpha}}\right)^{2}+\left(\frac{\beta}{\sigma_{\beta}}\right)^{2}\,,$
(7)

where $N_{\rm obs}^{i}$ denotes the observed events in the $i$-th energy bin
and $\sigma^{i}$ is the relevant uncertainty. $N_{\rm exp}^{i}$ is the total
number of expected events, which is the sum of the three neutrino flavors as
given in eq. (5). $B^{i}$ is the sum of prior predicted beam-related neutron
and the neutrino-induced neutron backgrounds in a given energy bin. The first
and second penalty terms correspond to the systematic uncertainty of the
signal and backgrounds where ‘$\alpha$’ and ‘$\beta$’ are the corresponding
nuisance parameters. The uncertainty in the signal is $\sigma_{\alpha}$ =
0.127 and the uncertainty on the total background is $\sigma_{\beta}$ = 0.6.
The signal uncertainty includes a contribution from the neutrino flux,
quenching factor, efficiency, form factor and light yield. We took all
information from ref. [25]. One can expect stronger or even weaker constraints
with timing information since the only parameter affected by the timing
information is the total efficiency. However, since the time and recoil energy
is completely uncorrelated and thus efficiencies for the two cases are also
uncorrelated [25], our results are thus valid without incorporating the timing
information. In order to avoid the possible flavor dependence on the timing
information, we restrict our analysis to only flavor-conserving interactions.

##  IV Electromagnetic interactions of neutrinos in CE$\nu$NS

###  IV.1 Millicharge neutrinos

The electromagnetic contribution due to the electrically charged neutrinos,
parameterized in terms of $Q_{\alpha\alpha}$, to the SM weak interaction for
the coherent neutrino-nucleus ($\nu-N$) scattering is given by the
interactions

$\mathcal{L}_{\alpha}^{em}=-ie\left(Q_{\alpha\alpha}{\overline{\nu}_{\alpha}}\gamma_{\mu}{\nu}_{\alpha}+{\overline{N}}\gamma_{\mu}{N}\right)A^{\mu},$
(8)

where $A^{\mu}$ is mediating electromagnetic field and ‘$e$’ is the unit
electric charge. Since the electromagnetic interaction terms add coherently to
the vector part of the weak interaction, this modifies the weak mixing angle
$\theta_{W}$ in eq. (3) accordingly as

$\displaystyle\sin^{2}\theta_{W}\rightarrow\sin^{2}\theta_{W}\left(1-\frac{\pi\alpha_{em}}{\sqrt{2}\sin^{2}\theta_{W}G_{F}MT}Q_{\alpha\alpha}\right)$
(9)

where $\alpha_{em}$ is the fine structure constant.

We estimate the statistical significance of the millicharge neutrinos by
fitting the two parameters, $Q_{ee}$ and $Q_{\mu\mu}$. We consider two cases
while fitting parameters. First, we fit one parameter at a time and fix the
other to zero, and we show the result in the left-side plot of Fig. 1. Next,
we fit the two parameters together and show the results in the right-side plot
of Fig. 1. One can see from Fig. 1 that because of the interplay between the
SM and the neutrino millicharge contribution (interference effect) and the
dependence on the inverse square of the nuclear recoil energy in the cross-
section, the two-parameter fit prefers non-zero values at the best-fit
minimum. We obtain the following constraints from the one parameter fits at
90% C.L.,

$\displaystyle-0.55\times 10^{-7}<Q_{\mu\mu}/e<0.75\times 10^{-7}\,,$
$\displaystyle-1.10\times 10^{-7}<Q_{ee}/e<3.90\times 10^{-7}\,.$ (10)

Fig. 1: Neutrino millicharge (NMC) of the muon and electron flavors from the
new COHERENT data for one (left) and two (right) parameter fits. The red star
in the right-hand side plot corresponds to the best-fit value. See text for
discussion.

Stronger limits on millicharge neutrinos come from the observational studies
[33, 34, 35, 36, 37]. The strongest upper limit on the millicharge neutrino is
$Q_{\nu}\leq 2\times 10^{-15}e$ from the time arrival dispersion and the
energy spread of neutrinos from SN1987A [33]. The laboratory bounds from the
$\nu-e$ are also several orders of magnitude smaller in size than the bounds
of this study [38, 39, 40, 41, 42, 43, 44, 45, 46, 26]. For instance, the
TEXONO experiment derives the limit, $Q_{\nu}\leq 2.1\times 10^{-12}e$.
However, we can easily understand this difference from the kinematical
considerations, as we will discuss in Sec. V. However, the robustness of the
bounds depends on the experimental details. Exploring how the observational
constraints also depend on the kinematic details of the astrophysical
environments would be interesting.

###  IV.2 Neutrino magnetic moment

To understand the importance of interference and the low recoil energy
dependence of the millicharge neutrinos, we also compare the neutrino magnetic
moment to the same data. In general, for the Majorana ($M$) or Dirac ($D$)
neutrinos’ couplings to the electromagnetic field strength ($F^{\mu\nu}$), the
magnetic moments appear as [47, 48, 49, 50, 51]

${\cal L}^{M}=-\frac{1}{4}\bar{\nu}_{\alpha
L}^{c}\,\lambda_{\alpha\beta}^{M}\,\sigma_{\mu\nu}\,\nu_{\beta
L}\,F^{\mu\nu}~{}\mbox{ or }~{}{\cal L}^{D}=-\frac{1}{2}\bar{\nu}_{\alpha
R}\,\lambda_{\alpha\beta}^{D}\,\sigma_{\mu\nu}\,\nu_{\beta L}\,F^{\mu\nu}\,,$
(11)

where $\lambda^{X}=\mu^{X}-i\epsilon^{X}$, which is hermitian for the Dirac
neutrinos and antisymmetric for Majorana neutrinos. Only transition magnetic
moments are possible for Majorana neutrinos while the flavor diagonal is zero.
Because of the unknown final state neutrino flavor in a scattering process, in
practice, no distinction between Dirac and Majorana neutrinos is possible. For
simplicity, we only consider the flavor diagonal cases for the electron
($\mu_{\nu e}$) and muon neutrinos ($\mu_{\nu\mu}$). In the SM, a non-zero
neutrino magnetic moment can arise at the one-loop level, which quantifies as
follows, [47]

$\mu_{\alpha\beta}=\frac{3eG_{F}m_{\nu}}{8\sqrt{2}\pi^{2}}\sim 3\times
10^{-19}\mu_{B}\left(\frac{m_{\nu}}{1\rm eV}\right)$ (12)

As clear from eq. (11), the helicity of the final state neutrino changes in
interaction due to the magnetic moment. Therefore there is no interference
with the SM cross-section, and the corresponding contribution adds to the
standard model at the cross-section level. We add the following differential
cross-section [44] for the neutrino magnetic moment (MM) of neutrino
scattering off a spin-0 nucleus to the SM cross-section in eq. (1),

$\frac{d\sigma_{\alpha}^{MM}}{dT}(E_{\nu},T)=\left(\frac{\pi\alpha_{em}^{2}\,\mu_{\nu\alpha}^{2}}{m_{e}^{2}}\right)\text{
}\left(\frac{1}{T}-\frac{1}{E_{\nu}}+\frac{T}{4E_{\nu}^{2}}\right)Z^{2}F^{2}(q^{2}),$
(13)

where we take the magnetic moment ($\mu_{\nu\alpha}$) in units of Bohr’s
magneton ($\mu_{B}$), and $m_{e}$ is the electron mass. One can notice that
compared to the millicharge of neutrinos as given in eq. (9) in combination
with eq. (1), the neutrino magnetic moment has no interference with the SM,
and the dependence on the inverse power of the nuclear recoil is only linear
in the leading terms.

In this case, we consider two parameters $\mu_{\nu_{\mu}}$ and $\mu_{\nu_{e}}$
and fit them to the new COHERENT data [25] using eq. (7), first, with one
parameter at a time while keeping the other zero and then fitting two
parameters together. The results for both cases are shown respectively in the
left-side and right-side plots of Fig. 2. We obtain the following constraints
from the one-parameter fits at 90% C.L.,

$\displaystyle-0.04\times 10^{-8}<\mu_{\nu_{\mu}}/\mu_{B}<0.04\times
10^{-8}\,,$ $\displaystyle-0.40\times 10^{-8}<\mu_{\nu_{e}}/\mu_{B}<0.40\times
10^{-8}\,.$ (14)

Fig. 2: Neutrino magnetic moment (NMM) of the electron and muon neutrino from
the new COHERENT data for the one (left) and two (right) parameters fits. The
red star in the right-hand side plot corresponds to the best-fit value. See
text for discussion.

###  IV.3 Neutrino charge radius

Radiative corrections induce the neutrino charge radius for neutrinos in the
SM. In the general effective electromagnetic vertex of massive neutrinos,
$\bar{\nu}\Lambda_{\mu}\nu A^{\mu}$, the neutrino charge radius term is given
by [52, 53, 54],

$\Lambda_{\mu}(q)=\gamma_{\mu}F(q^{2})\simeq\gamma_{\mu}q^{2}\frac{\langle
r^{2}\rangle}{6}\,,$ (15)

where $q$ is the momentum transfer and $F(q^{2})$ is a form factor related to
the neutrino charge radius $\langle r_{\nu}^{2}\rangle$ via

$\langle
r_{\nu}^{2}\rangle=6\left.\frac{dF_{\nu}(q^{2})}{dq^{2}}\right|_{q^{2}=0}\,.$
(16)

The SM prediction for the charge radius of neutrino [52, 53, 54, 55, 56] is

$\langle r_{\nu_{\alpha}}^{2}\rangle_{\rm
SM}=-\frac{G_{F}}{2\sqrt{2}\pi^{2}}\left[3-2\ln\left(\frac{m_{\alpha}^{2}}{m_{W}^{2}}\right)\right],$
(17)

where $m_{\alpha}$ is the mass of the charged lepton associated with
$\nu_{\alpha}$ and $m_{W}$ is the mass of the $W^{\pm}$ boson. The numerical
values for the corresponding flavor of neutrinos are [52, 53, 54, 57, 55, 56]

$\displaystyle\langle r_{\nu_{e}}^{2}\rangle_{\rm SM}=-0.83\times 10^{-32}\ \
\rm cm^{2},$ $\displaystyle\langle r_{\nu_{\mu}}^{2}\rangle_{\rm
SM}=-0.48\times 10^{-32}\ \ \rm cm^{2},$ $\displaystyle\langle
r_{\nu_{\tau}}^{2}\rangle_{\rm SM}=-0.30\times 10^{-32}\ \ \rm cm^{2}.$ (18)

In the SM, only flavor diagonal charge radii exist, while in general,
transition charge radii are also possible [56]. We consider only the former
case whose contribution coherently adds to the SM cross-section. This
contribution adds to the coherent cross-section by making the following
replacement for the effective weak mixing angle in eq. (3),

$\displaystyle\sin^{2}\theta_{W}\rightarrow\sin^{2}\theta_{W}\left(1+\frac{\pi\alpha_{em}}{3\sqrt{2}\sin^{2}\theta_{W}G_{F}}\langle
r_{\nu_{\alpha}}^{2}\rangle\right)$ (19)

Notice that, unlike the millicharge neutrinos in eq. (9), the contribution to
the cross-section due to the neutrino charge radius does not directly depend
on the nuclear recoil energy and the target mass. Thus one cannot expect
enhanced sensitivity at low energy recoils. This was also noted before in refs
[26, 58]. We fit $\langle r_{\nu_{e}}^{2}\rangle$ and $\langle
r_{\nu_{\mu}}^{2}\rangle$ with the new COHERENT data first by taking one
parameter at a time and then the two parameters together. We show our results
in Fig. 3 and constraints from one parameter-at-a-time at 90% C.L. are in the
following.

$\displaystyle-0.60\times 10^{-30}<\langle r_{\nu_{\mu}}^{2}\rangle/{\rm
cm^{2}}<0.05\times 10^{-30}\,,$ $\displaystyle-0.67\times 10^{-30}<\langle
r_{\nu_{e}}^{2}\rangle/{\rm cm^{2}}<0.10\times 10^{-30}\,.$ (20)

###  IV.4 Neutrino anapole moment

If neutrino carries a non-zero charge radius, it can also have a non-zero
anapole moment [59, 60, 61, 62, 63, 64, 65, 55]. It determines the correlation
between the spin and charge distributions of neutrinos and has the same
dimensions as the charge radius. In the general vertex for electromagnetic
interactions, $\bar{\nu}\Lambda_{\mu}\nu A^{\mu}$, the anapole term is defined
by [61, 62, 65]

$\Lambda_{\mu}(q)=-\gamma_{\mu}\gamma_{5}F(q^{2})\simeq-\gamma_{\mu}\gamma_{5}q^{2}\textit{{a}}_{\nu}\,,$
(21)

where the form factor ‘$F(q^{2})$’ is related to the neutrino anapole moment
‘$\textit{{a}}_{\nu\alpha}$’ by the expression,

$\textit{{a}}_{\nu}=-\left.\frac{dF_{\nu}(q^{2})}{dq^{2}}\right|_{q^{2}=0}\,.$
(22)

By comparing with eq. (16), one can write the SM prediction for the neutrino
anapole moment in terms of the charge radius as [61, 62, 65, 55, 56]

$\textit{{a}}_{\nu_{\rm SM}}=-\frac{\langle r_{\nu}^{2}\rangle_{\rm SM}}{6},$
(23)

and numerical values accordingly are,

$\displaystyle\textit{{a}}_{\nu_{e\rm SM}}=4.98\times 10^{-32}\ \ \rm cm^{2},$
$\displaystyle\textit{{a}}_{\nu_{\mu\rm SM}}=2.88\times 10^{-32}\ \ \rm
cm^{2},$ $\displaystyle\textit{{a}}_{\nu_{\tau\rm SM}}=1.80\times 10^{-32}\ \
\rm cm^{2}.$ (24)

In the case of the CE$\nu$NS, one can add the contribution of the neutrino
anapole moment by replacing the effective weak mixing angle in eq. (3) as,

$\displaystyle\sin^{2}\theta_{W}\rightarrow\sin^{2}\theta_{W}\left(1-\frac{\pi\alpha_{em}}{18\sqrt{2}\sin^{2}\theta_{W}G_{F}}\textit{{a}}_{\nu_{\alpha}}\right)$
(25)

Again one can notice that, unlike the neutrino millicharges, the anapole
moment does not directly depend on the nuclear recoil energy and the target
mass, and one does not expect an enhanced sensitivity at low energy recoils.
We fit the parameters $\textit{{a}}_{\nu_{e\rm}}$ and
$\textit{{a}}_{\nu_{\mu}}$ with the new COHERENT data. First, we fit one
parameter at a time while fixing the other to zero and then fit the two
parameters together. We show our obtained results in Fig. 4, and constraints
from this analysis in the case of parameter-at-a-time at 90% C.L. are the
following.

$\displaystyle-0.30\times 10^{-30}<\textit{{a}}_{\nu_{\mu}}/{\rm
cm^{2}}<3.7\times 10^{-30}\,,$ $\displaystyle-0.60\times
10^{-30}<\textit{{a}}_{\nu_{e\rm}}/{\rm cm^{2}}<4.0\times 10^{-30}\,.$ (26)

Fig. 3: Neutrino charge radius (NCR) of the electron and muon neutrino from
the new COHERENT data for the one (left) and two (right) parameters fits. The
red star in the right-hand side plot corresponds to the best-fit value. See
text for discussion. Fig. 4: Neutrino anapole moment (NAM) of the electron and
muon neutrino from the new COHERENT data for the one (left) and two (right)
parameters fits. The red star in the right-hand side plot corresponds to the
best-fit value. See text for discussion.

##  V Theoretical and Experimental significance of millicharged neutrinos

The electric charge quantization comes from its empirical observation. The
standard model (SM) does not predict the origin of electric charge
quantization. However, several theories beyond the SM like those with magnetic
monopoles [66], grand unified theories [67, 68] and the extra dimension models
[69] do predict the charge quantization. In addition, there are several
extensions of the SM that predict new particles with fractional charges [70,
71, 72, 73, 74, 75]. The fractionally charged particles are also promising
candidates for dark matter [76, 77, 78, 79, 75, 80, 81, 82, 83, 84]. Even
charges of the SM particles can deviate from the integer multiple of ‘e/3’,
where ‘$e$’ is the magnitude of the unit electric charge. Neutrinos are the
favorite candidates for such particles, often called milli-charged neutrinos
[85, 86, 87, 88].

The SM is an anomaly-free gauge theory with several accidental global
symmetries. In the SM with one generation of fermions, neutrinos are neutral
because all the SM anomalies consistently cancel out. However, for the SM with
three generations, at least two of the three massless neutrinos could be
electrically charged. This dequantization leads to the emergence of three
gaugeable $U(1)$ symmetries, $L_{e}-L_{\mu}$, $L_{\mu}-L_{e}$ and
$L_{e}-L_{\tau}$. Only one of the three differences can be anomaly-free and
the corresponding difference in each case adds to the hypercharge of the SM.
Thus, it leads to the fractional charges of neutrinos and to the
dequantization of electric charge at least in the lepton sector [87, 88].

For massive neutrinos, the existence of fractional charges of neutrinos
depends on the nature of their masses. If neutrinos are Majorana fermions,
then new anomalies arise, but they cancel out, leaving neutrinos neutral [85,
86]. Consequently, the electric charge remains quantized in the minimally
extended SM. On the other hand, Dirac neutrinos with three right-handed
partners, singlets under $SU(3)_{c}\times SU(2)_{L}$, are electrically
charged. These small electric charges of Dirac neutrinos modify the charges of
charged leptons and quarks due to their hypercharges’ dependence on the right-
handed neutrinos’ hypercharges. It leads to the dequantization of electric
charges in general [87, 88]. This dequantization is related to the emergence
of anomaly-free gaugeable $B-L$ symmetry [85]. From the theoretical
considerations, there is no upper limit available on the millicharge
neutrinos. All the known limits are experimental [38, 39, 40, 41, 42, 43, 44,
45, 46, 26] or observational [33, 34, 35, 36, 37].

Neutrinos are electrically neutral in the SM. In several extensions of SM with
Dirac neutrinos, the electric charges of the neutrino can arbitrarily take any
values. Currently, all information about it comes from experiments. No
theoretically predicted value is available. More importantly, the limits
strongly depend on the process under consideration. Three critical factors
contribute to neutrino millicharges. These are the interference of their
amplitude with the standard model, inverse power dependence on the recoil
energy, and the size of the target mass. On the other hand, the other
electromagnetic properties partially depend on these factors. For instance,
there is the interference of the neutrino charge radius and anapole moment
with the SM, but there is no inverse recoil energy and target mass dependence.
Likewise, for the neutrino magnetic moment, there is no interference with SM
and no dependence on the target mass, while there is only one power of inverse
recoil energy dependence.

Fig. 5: Effects from interference and kinematics of the process.
$\mathbb{Left:}$ The ratio between the millicharge, magnetic moment, charge
radius, anapole moment cross-sections plus the SM cross-sections and the SM
cross-section as a function of the millicharge, magnetic moment, charge
radius, and anapole moment at fixed neutrino energy and nuclear recoil energy.
$\mathbb{Right:}$ The millicharge term in Eq. (9) as a function of $Q$ for the
target nuclear and electronic masses. See the text for the discussion.

We show the millicharge dependence on the interference and the inverse power
of the recoil energy for the relevant fixed neutrino energy (30 MeV) and
recoil energy (11 keV) in the left-hand side plots of Fig. 5. Here, the ratio
between the new physics plus the SM differential cross-section and the SM
alone cross-section for the four cases was taken as a function of the neutrino
millicharge, magnetic moment, charge radius, and anapole moment. In the region
of interest for CE$\nu$NS, millicharge interactions compete with the SM cross-
section up to $10^{-7}e$ and drop to zero $3.3\times 10^{-7}e$ at the start
and start growing afterward. In contrast, the magnetic moment starts deviation
from the SM at $10^{-9}\mu_{B}$, the charge radius starts deviation from the
SM at $10^{-31}cm^{2}$ and anapole moment starts deviation from the SM at
around $10^{-31}\rm cm^{2}$.

We plot the second term of eq. (9), writing as

$\displaystyle
f\left(Q\right)=\left(\frac{\pi\alpha_{em}}{\sqrt{2}\sin^{2}\theta_{W}G_{F}MT}Q_{\alpha\alpha}\right),$
(27)

to show the target mass dependence as a function of millicharge with target
mass dependence. The comparison between a CsI nuclear target and any
electronic target is shown in the right-hand side plot of Fig. 5. The
millicharge contribution is suppressed at $\mathcal{O}(10^{3})$ due to the
nuclear mass in comparison to electric recoils. This effect gets relatively
weaker at the cross-section level because the SM CE$\nu$NS cross-section is
directly proportional to the target mass times the sum of the squares of the
total number of protons and neutrons. However, the electronic targets still
dominate, and there are at least two orders of magnitude intrinsically
stronger constraints on millicharge neutrinos in this case than the CE$\nu$NS.
Notice that this property does not hold for the other three electromagnetic
properties because of no inverse dependence on the target mass. Thus, the
difference in the limits strongly depends on the difference between the
electronic and the nuclear masses, no matter how different the precision of
the two types of experiments is.

Flavor | $|\mu_{\nu}|[\times 10^{-11}\mu_{B}]$ | $Q_{v}\ [\times 10^{-13}e]$ |  $\left\langle r_{\nu}^{2}\right\rangle\ [\times 10^{-32}$cm${}^{2}]$ |  $a_{\nu}\ [\times 10^{-32}$cm${}^{2}]$
---|---|---|---|---
$\nu_{e}\ ($COH-2021$)$ | $\leq 400$ | $\ \ [-1.10,\ 3.90]\times 10^{6}\ $ | $\ [-67,10]$ | $[-60,400]$
$\nu_{\mu}($COH-2021$)$ | $\leq 40$ | $[-0.55,\ 0.75]\times 10^{6}\ $ | $[-60,5]$ | $[-30,370]$
$\nu_{\tau}($COH-2021$)$ | $-$ | $-$ | $-$ | $-$
$\nu_{e}\ ($COH-Prev$)$ | $\leq 860$ | $\ \ [-2.36,\ 8.38]\times 10^{6}\ $ | $\ [-55,13]$ | $[-78,330]$
$\nu_{\mu}($COH-Prev$)$ | $\leq 570$ | $[-1.18,\ 1.60]\times 10^{6}\ $ | $[-50,8]$ | $[-48,300]$
$\nu_{\tau}($COH-Prev$)$ | $-$ | $-$ | $-$ | $-$
$\nu_{e}\ ($Others$)$ | $\begin{array}[]{l}\leq 0.63\ \text{(XENONnT)}\\\ \leq 3.9\ \text{(Borexino)}\\\ \leq 110\ \text{(LAMPF)}\\\ \leq 11\ \text{(Super-K)}\\\ \leq 7.4\ \text{(TEXONO)}\\\ \leq 2.9\ \text{(GEMMA)}\end{array}$ | $\begin{array}[]{l}[-1.3,\ 4.7]\ \text{(XENONnT)}\\\ \leq 15\ \ \text{(Reactor)}\\\ \\\ \\\ \\\ \\\ \end{array}\ $ | $\ \ \begin{array}[]{l}[-45,\ 3.0]\ \text{(XENONnT)}\\\ [0.82,\ 1.27]\ \text{(Solar)}\\\ [-5.94,\ 8.28]\ \text{(LSND)}\\\ [-4.2,\ 6.6]\ \ \text{(TEXONO)}\\\ \\\ \\\ \end{array}$ | $\begin{array}[]{l}[-23,65]\\\ \text{(XENONnT)}\\\ \\\ \end{array}$
$\nu_{\mu}($Others$)$ | $\begin{array}[]{l}\leq 1.37\ \text{(XENONnT)}\\\ \leq 5.8\ \text{(Borexino)}\\\ \leq 68\ \text{(LSND)}\\\ \leq 74\ \text{(LAMPF)}\end{array}$ |  $[-8.9,\ 8.6]$ (XENONnT) | $\begin{array}[]{l}[-45,\ 52]\ \text{(XENONnT)}\\\ [-9,\ 31]\ \text{(Solar)}\\\ \leq 1.2\ \text{(CHARM-II)}\\\ [-4.2,\ 0.48]\ \text{(TEXONO)}\end{array}$ | $\begin{array}[]{l}[-95,89]\\\ \text{(XENONnT)}\\\ \\\ \end{array}$
$\nu_{\tau}($Others$)$ | $\begin{array}[]{l}\leq 1.24\ \text{(XENONnT)}\\\ \leq 5.8\ \text{(Borexino)}\\\ \leq 3.9\times 10^{4}\ \text{(DONUT)}\end{array}$ | $\begin{array}[]{l}[-7.9,\ 7.8]\ \text{(XENONnT)}\\\ \leq 10^{-8}\ \text{(Beam dump)}\\\ \\\ \end{array}$ | $\begin{array}[]{l}[-40,\ 45]\ \text{(XENONnT)}\\\ [-9,\ 31]\ \text{(Solar)}\\\ \\\ \end{array}$ | $\begin{array}[]{l}[-86,79]\\\ \text{(XENONnT)}\\\ \\\ \end{array}$
Table 1: 90% C.L. bounds on neutrino magnetic moment, neutrino millicharges,
charge radius and anapole moment from COHERENT new data (COH-2021), COHERENT
previous (COH-prev) data [27] and other laboratory experiments. Bounds for
XENONOnT were taken from ref. [89], Borexino from ref. [90], and solar from
ref. [58] while all other bounds for other experiments were taken from ref.
[91].

##  VI Summary and Conclusions

In this work, we have analyzed the new data from the COHERENT experiment to
derive limits on four types of neutrino electromagnetic interactions: the
neutrino millicharge, magnetic moment, charge radius, and neutrino anapole
moment. In the latest update of COHERENT experimental on the detection of
coherent neutrino-nucleus elastic scattering, the collaboration has reported
almost double statistics and improved the precision to twice their first
result. Furthermore, the statistical significance of the observed process has
now enhanced to 11.6$\sigma$. This improvement motivates the improved
constraints on any new physics. Finally, following our previous work, we have
analyzed the new data to constrain neutrino electromagnetic interactions.

We have derived one parameter at-a-time and two-parameter bounds for the two
types of neutrino flavors involved in the COHERENT experimental detection. In
general, there is an improvement in constraints by a factor of two than
previous constraints obtained from the COHERENT old data [44]. Also, the
constraints on the electromagnetic interactions of muon neutrino flavor are
stronger than those on the electron flavor because of the larger fluxes and,
thus, larger statistics of the muon neutrino flavors.

For comparison, we summarize limits from the COHERENT previous data, [5] and
neutrino-electron elastic scattering from several other experiments including
reactor, solar, and accelerator in Table 1. One can see that as expected the
bounds from the COHERENT new data have improved by a factor of two to the
previous bounds. On the other hand, as discussed before, the limits from
neutrino-electron scattering are stronger by several orders of magnitude in
most cases. The limits from the recent XENONnT experiment are currently the
strongest among all [89].

We have discussed the importance of millicharge neutrinos in detail.
Millicharge neutrinos are important from theoretical and observational points
of view. Their observation would be strong evidence for physics beyond the SM
and have significant astrophysical implications. It will support neutrinos’
Dirac nature and the electric charges’ dequantization in the standard model.
As shown in Fig. 1, there is a slight preference for the non-zero best value
of the neutrino millicharges. This effect is more prominent in the two-
parameter case of Fig. 1. This sensitivity can be understood by the unique
kinematic behavior of the millicharge neutrino interactions, as discussed in
detail. V. While the interference with the SM and unique dependence of the
neutrino millicharge interaction is obvious when we have shown its dependence
on the target mass by comparing the electronic recoil in fig. 5. We showed
that electronic targets could provide intrinsically stronger limits than
nuclear recoils, while their sizes depend on the specific interaction process.

In conclusion, we have obtained new constraints on all electromagnetic
interactions of neutrinos using the latest data from the COHERENT experiment.
The current and future experiments of the coherent elastic neutrino-nucleus
scattering experiments for both spallation neutron or reactor neutrino sources
would improve the sensitivity to electromagnetic interaction. In particular,
the neutrino millicharge and neutrino magnetic moment are more significant at
such lower recoils. Furthermore, a better understanding of neutrino
electromagnetic interactions will help understand other nonstandard neutrino
interactions and dark matter in direct detection experiments.

Note Added: When this work was under review, ref. [92] appeared on arXiv which
also discusses and constrains the neutrino electromagnetic properties using
the same new data set of COHERENT.

###### Acknowledgements.

The author thanks Kate Scholberg and Dan Pershey for providing helpful
information about the data and other details. I also thank Douglas McKay (KU)
for his valuable suggestions on the manuscript. The Alexander von Humboldt
Foundation financially supported this work.

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