#  $\nu_{\mu}$ and $\nu_{\tau}$ elastic scattering in Borexino

Kevin J. Kelly<EMAIL_ADDRESS>Department of Physics and Astronomy, Mitchell
Institute for Fundamental Physics and Astronomy, Texas A&M University, College
Station, Texas 77843, USA  Nityasa Mishra<EMAIL_ADDRESS>Department
of Physics and Astronomy, Mitchell Institute for Fundamental Physics and
Astronomy, Texas A&M University, College Station, Texas 77843, USA  Mudit Rai
<EMAIL_ADDRESS>Department of Physics and Astronomy, Mitchell Institute for
Fundamental Physics and Astronomy, Texas A&M University, College Station,
Texas 77843, USA  Louis E. Strigari<EMAIL_ADDRESS>Department of Physics
and Astronomy, Mitchell Institute for Fundamental Physics and Astronomy, Texas
A&M University, College Station, Texas 77843, USA

###### Abstract

We perform a detailed study of neutrino-electron elastic scattering using the
mono-energetic 7Be neutrinos in Borexino, with an emphasis on exploring the
differences between the contributions of $\nu_{e}$, $\nu_{\mu}$, and
$\nu_{\tau}$. We find that current data are capable of measuring these
components such that the contributions from $\nu_{\mu}$ and $\nu_{\tau}$
cannot be zero, although distinguishing between them is challenging – the
differences stemming from Standard Model radiative corrections are
insufficient without significantly more precise measurements. In studying
these components, we compare predicted neutrino-electron scattering event
rates within the Standard Model (accounting for neutrino oscillations), as
well as going beyond the Standard Model in two ways. We allow for non-unitary
evolution to modify neutrino oscillations, and find that with a larger
exposure (${\sim}30$x), Borexino may provide relevant information for
constraining non-unitarity, and that JUNO may be able to accomplish this with
its data collection of 7Be neutrinos. We also consider novel $\nu_{\mu}$\- and
$\nu_{\tau}$-electron scattering from a gauged $U(1)_{L_{\mu}-L_{\tau}}$
model, showing consistency with previous analyses of Borexino and this
scenario, but also demonstrating the impact of uncertainties on Standard Model
mixing parameters on these results.

††preprint: MI-HET-836

##  I Introduction

Neutrinos change flavor as they journey from the Sun to the Earth through a
combination of vacuum oscillations and matter-induced transformations,
described by the Large Mixing Angle (LMA)-MSW solution Wolfenstein:1977ue ;
Mikheyev:1985zog . For solar neutrino data, this process is well-approximated
by a two-flavor transformation model between electron neutrinos and a second
component which is combination of mu/tau neutrinos Haxton:2012wfz ;
Antonelli:2012qu .

Though the LMA-MSW model is effective at describing the combination of
experimental data sets Gann:2021ndb , as measurements continually improve and
next generation detectors come online JUNO:2023zty ; Capozzi:2018dat , a more
precise theoretical description is required. For example, the detection
channels used are unable to distinguish between these $\nu_{\mu}$ and
$\nu_{\tau}$ flavors. This is unfortunate, since identifying a $\nu_{\tau}$
component from the Sun would be particularly interesting since this mixing is
one of the only mechanisms to produce $\nu_{\tau}$ at the MeV scale.

Going beyond solar neutrinos, measuring the neutral current elastic scattering
of $\nu_{\mu}$ and $\nu_{\tau}$ has important, broader implications. At
energies characteristic of solar neutrinos, the $\nu_{\mu}$-electron elastic
scattering cross section has been measured at terrestrial experiments
Hasert:1973cr ; Ahrens:1983py ; Abe:1989qk ; Ahrens:1990fp ; Formaggio:2012cpf
, though there has been no direct measurement of the tau neutrino scattering
cross section. A measurement of this tau neutrino flux is important not only
to confirm the standard model of neutrino oscillations from the Sun, but also
to probe for possible effects from new physics. For example, non-standard
neutrino interactions (NSI) may be important in scattering at these energies
Dutta:2020che .

Recent analyses have discussed the prospects of separating the $\nu_{\mu}$ and
$\nu_{\tau}$ flavor components from the Sun, relying on detecting the presence
of radiative corrections to the cross section through future high precision
measurements of the solar neutrino flux. Ref. Mishra:2023jlq  considered the
prospects using CE$\nu$NS interactions at future large scale dark matter
detectors, while Ref. Brdar:2023ttb  considered the prospects for elastic
scattering at future large scale detectors such as DUNE, Hyper-K, or JUNO.
Both of these studies focused on the high-energy, 8B components of the solar
neutrino flux, which is the most accessible component given the thresholds of
the experiments.

Given the importance such a measurement, it is interesting to take a step back
and establish the sensitivity of current and past solar neutrino experiments
to $\nu_{\mu}$/$\nu_{\tau}$ components. This paper is focused on establishing
this sensitivity, and exploring implications for a precise measurements of the
$\nu_{\mu}$/$\nu_{\tau}$ scattering cross sections. We specifically focus on
the 7Be component of the flux as measured by Borexino Borexino:2017rsf , since
this component is measured with high statistics, and in a region of the
parameter space well-separated from experimental backgrounds. In addition,
since this is a mono-energetic flux, the neutrino survival probability is
straightforward to implement and use to establish sensitivity to the
scattering cross section. With similar motivation, previous studies have
considered the prospects for extracting new physics in the form of non-
standard neutrino interactions with Borexino Coloma:2022umy .

We consider mechanisms through which the elastic scattering may differ from
standard 3-flavor mixing, including models in which unitarity is violated, and
models with $L_{\mu}-L_{\tau}$ symmetry Datta:2018xty . We expand the concept
of the flavor triangle, which has been used to study 3-flavor mixing in
astrophysical neutrino sources IceCube:2015rro ; Tabrizi:2020vmo , to 3-flavor
mixing in solar neutrinos. Non-unitarity has been previously investigated in
the context of solar neutrinos and Borexino Moreno:2024pbw . We determine the
exposure at which Borexino would be sensitive to separation of $\nu_{\mu}$ and
$\nu_{\tau}$ fluxes, and explore the implications for both non-unitarity and
$L_{\mu}-L_{\tau}$ models.

This paper is organized as follows; Section II presents the theoretical models
that we consider. In Section III we present our analysis of the Borexino data,
and is Section IV we present the results of our analysis. In Section V we end
with discussion and implications of our results.

##  II Predicted Borexino Event-rate Observations

In this section we discuss the necessary pieces for event-rate calculations of
solar neutrinos scattering elastically off electrons in the Borexino detector
– specifically we discuss interaction cross sections in the SM and beyond in
Section II.1, and calculations of the expected flavor composition of the
neutrino flux at earth in Section II.2.

###  II.1 (B)SM Neutrino-electron Cross Sections

We are primarily interested in the elastic scattering of SM neutrinos with
electrons (EES) in the Borexino detector via the exchange of $Z$ (all flavors)
and $W$ (only $\nu_{e}$) bosons. We will also consider modifications to this
process due to a new, light mediator.

#### Tree-level EES

At tree level the cross-section for electron scattering with $\nu_{e}$ is
different from that with $\nu_{\mu}$ and $\nu_{\tau}$, due to the presence of
charged current interaction for electron flavor in addition to the neutral
current interaction, the latter of which is present for all the three flavors.
At tree level the differential cross-section is given by

$\frac{d\sigma_{e\alpha}}{dT}=\frac{m_{e}}{4\pi}\left[\left(c^{\alpha}_{L}\right)^{2}+\left(c_{R}\right)^{2}\left(1-\frac{T}{E_{\nu}}\right)^{2}-\left(c^{\alpha}_{L}\right)\left(c_{R}\right)\frac{m_{e}T}{E_{\nu}^{2}}\right]$
(1)

where $T$ and $E_{\nu}$ are the electron recoil energy and neutrino energy
respectively while $\alpha=e,\ \mu,\ \tau$ refers to the neutrino flavor. The
constants $c_{L}^{\alpha}$ and $c_{R}$ are

$\displaystyle
c_{L}^{\alpha}=2\sqrt{2}G_{F}\left(\sin^{2}\theta_{w}-\frac{1}{2}+\delta_{e\alpha}\right),\
\quad c_{R}=2\sqrt{2}G_{F}\sin^{2}\theta_{w}.$ (2)

The remaining constants are $\theta_{w}$ (the weak mixing angle) and $G_{F}$
(the Fermi constant). The Kronecker delta function $\delta_{e\alpha}$
appearing in $c_{L}^{\alpha}$ is due to the charged-current exchange available
for $\nu_{e}$ EES and causes the overall $\nu_{e}$ EES cross section to be
significantly larger (by nearly a factor of five) than that of
$\nu_{\mu,\tau}$.

#### Radiative corrections to EES

As seen above, for $\alpha=\mu$ or $\tau$, the differential cross section is
identical. However at one-loop order (e.g., via the exchange of a $W$ loop
causing a charged lepton to scatter off the target electron via the exchange
of a photon), leptonic mass effects exist causing $d\sigma_{e\mu}/dT\neq
d\sigma_{e\tau}/dT$. The differential cross sections may be expressed as
Tomalak:2019ibg

$\frac{d\sigma^{\rm
rad.}_{e\alpha}}{dT}=\left[1+\frac{\alpha}{\pi}\left(\delta_{v}+\delta_{s}+\delta_{I}+\delta_{II}\right)\right]\frac{d\sigma^{\rm
LO}_{e\alpha}}{dT}+\frac{d\sigma_{v}}{dT}+\frac{d\sigma^{\rm
dyn.}_{e\alpha}}{dT}+\frac{d\sigma^{\rm NF}}{dT}.$ (3)

Here, $d\sigma_{e\alpha}^{\rm LO}/dT$ is given by Eq. 1 with appropriate
redefinition of $c_{L,R}$ Tomalak:2019ibg . The quantities $\delta_{v}$ and
$\delta_{s}$ contain information about UV-finite and gauge-independent
virtual- and soft-photon corrections, respectively, and $\delta_{I}$ and
$\delta_{II}$ provide corrections due to bremsstrahlung. The additional terms
arise from QED vertex corrections ($d\sigma_{v}$), non-factorizable
contributions to the electron energy spectrum ($d\sigma^{\rm NF}$), and from
dynamical corrections due to closed fermion loops ($d\sigma_{e\alpha}^{\rm
dyn.}$) The exact expressions for the above-mentioned terms can be found in
Ref Tomalak:2019ibg .

Overall, the nontrivial flavor dependence (especially that which allows for
differentiation between $\nu_{\mu}$ and $\nu_{\tau}$ EES) is largely present
in $d\sigma_{e\alpha}^{\rm dyn.}$. In Fig. 1 we demonstrate these impacts for
7Be neutrinos with $E_{\nu}=0.862$ MeV as a function of the recoiling
electron’s kinetic energy. The two panels demonstrate the same information,
with the only difference being the y-axis range due to the relative strength
of $\nu_{e}$ EES (top panel) over that of $\nu_{\mu}$ and $\nu_{\tau}$ (bottom
panel). For each $\alpha=e,\ \mu,\ \tau$, we demonstrate the differential
cross section at tree level – solid lines, Eq. 1 – and including radiative
corrections – dashed, Eq. 3. Notably, including radiative corrections ends up
decreasing the $\nu_{e}$ EES cross section and increasing that of $\nu_{\mu}$
and $\nu_{\tau}$ Tomalak:2019ibg , while introducing some different
$T$-dependence in comparing $\nu_{\mu}$ (dashed orange) and $\nu_{\tau}$
(dashed green) rates. With precise enough measurements of the recoil energy
distribution of electrons, one can, in principle, differentiate between
$\nu_{\mu}$ and $\nu_{\tau}$ EES due to these effects. Previous results have
also studied these radiative corrections and their impact on coherent
scattering of neutrinos, e.g., Refs. Tomalak:2020zfh ; Brdar:2023ttb .

Figure 1: Differential cross-sections for neutrino-electron elastic scattering
for $\nu_{e}$ (top) and $\nu_{\mu,\tau}$ (bottom), where the difference in
panels is only the y-axis range. In each panel, we compare tree-level
differential cross sections (solid lines, identical for $\nu_{\mu}$ and
$\nu_{\tau}$) with those accounting for radiative corrections (dashed).

#### BSM Scattering with a light mediator

Neutrino-electron scattering can be modified, especially for low momentum
transfers, in the presence of new mediators that couple to both neutrinos and
to electrons – a feature present in many $U(1)$ extensions of the SM. One such
extension of theoretical interest (e.g. due to its potential for resolving the
$g-2$ anomaly of the muon) is $U(1)_{L_{\mu}-L_{\tau}}$, where the difference
between muon-lepton-number and tau-lepton-number is gauged and the new boson
associated with this, $Z^{\prime}$, gains a mass in the ${\sim}$MeV-GeV
regime. This mediator will couple (at tree level) to $\nu_{\mu}$ and
$\nu_{\tau}$ (but not $\nu_{e}$), and via loop-level effects Kamada:2015era ;
Bauer:2018onh ; Escudero:2019gzq , to electrons with an induced kinetic mixing
(we assume that the bare Lagrangian-level kinetic mixing between the
$Z^{\prime}$ and the SM is zero). When we consider this scenario, we will only
consider the modification of the neutrino-electron scattering in the detector
and not the potential modification of matter effects for neutrinos emerging
from the Sun.

In the presence of this new mediator with mass $m_{Z^{\prime}}$ and gauge
coupling $g_{\mu\tau}$, the coefficients in Eq. 1 for $\nu_{\mu,\tau}$
scattering are modified Altmannshofer:2019zhy ; Amaral:2020tga

$\displaystyle c_{L,R}\to
c_{L,R}\mp\frac{G_{F}}{\sqrt{2}}\frac{e^{2}g_{\mu\tau}^{2}}{6\pi^{2}}\log{\left(\frac{m_{\tau}^{2}}{m_{\mu}^{2}}\right)}\frac{v^{2}}{m_{Z^{\prime}}^{2}+2m_{e}T},$
(4)

where $v=246$ GeV is the Higgs vev, $e$ is the electric charge, and the $-$
($+$) sign corresponds to the modification for $\nu_{\mu}$ ($\nu_{\tau}$),
where this new interaction provides constructive (destructive) interference
with the SM. The impact on the $\nu_{e}e\rightarrow\nu_{e}e$ cross-section is
negligible and can be taken to be the same as the tree-level value.

###  II.2 Solar fluxes and oscillation probability

The sun produces electron neutrinos via nuclear fusion processes within its
core, predominantly in the “pp reaction” in which a proton (in the vicinity of
another proton) beta decays. In the subsequent reactions, a relatively smaller
fraction of electron neutrinos – the “7Be neutrinos” – are emitted as 7Be
nuclei are transformed to 7Li ones. These neutrinos are monoenergetic,
$E_{\nu}=0.862$ MeV, and their flux may be predicted given a solar model. In
the high-metallicity model of the sun Haxton:2012wfz , this prediction is
$5\times 10^{9}$ cm-2s-1, with a theoretical uncertainty of $6\%$. Other
decays in the sun produce large neutrino fluxes, e.g. 8B, however we focus in
this work on 7Be due to its high flux and the fact that it produces
monoenergetic neutrinos.

Due to their interactions with electrons in the Sun, the $\nu_{e}$ undergo
flavor-dependent oscillations into $\nu_{\mu,\tau}$ as they travel from the
core to the surface Wolfenstein:1977ue ; Mikheyev:1985zog . This oscillation
probability (due to the “slow” change of density from core to surface) may be
expressed as

$\displaystyle
P(\nu_{e}\rightarrow\nu_{\alpha})=\sum_{i=1}^{3}P\left(\nu_{e}\to\nu_{i}\right)\left\lvert
U_{\alpha i}\right\rvert^{2},$ (5)

where $P(\nu_{e}\rightarrow\nu_{i})$ ($i=1,\ 2,\ 3$) represents the adiabatic
transition probability of a (flavor-eigenstate) $\nu_{e}$ to a (mass-
eigenstate) $\nu_{i}$ during its propagation out of the Sun, and $\left\lvert
U_{\alpha i}\right\rvert^{2}$ represents the overlap between the mass-
eigenstate $\nu_{i}$ and the flavor-eigenstate $\nu_{\alpha}$ when the
neutrino interacts in the detector. The relation between the mass eigenstates
and the flavor eigenstates is

$\displaystyle|\nu_{i}\rangle=\sum_{\alpha}U_{\alpha i}|\nu_{\alpha}\rangle.$
(6)

The Hamiltonian that describes propagation in the Sun is made up of local
eigenvalues and can be written as

$\displaystyle H_{\rm fl.}={U}E_{\rm M}{U}^{\dagger}+V$ (7)

The energy eigenvalue matrix (diagonal in the mass basis) is $E_{\rm
M}=\frac{1}{2E_{\nu}}\textrm{diag}(m_{1}^{2},m_{2}^{2},m_{3}^{2})$. $V$ refers
to the (flavor-basis) matter potential, which is a function of electron number
density and nucleon number density in the matter, and is given by

$\displaystyle V=\mathrm{diag}(V_{\rm cc}+V_{\rm nc},V_{\rm nc},V_{\rm
nc}),\quad V_{\rm cc}=\sqrt{2}G_{F}n_{e},\quad V_{\rm
nc}=\frac{\sqrt{2}}{2}G_{F}n_{n}$ (8)

Consequently the $P(\nu_{e}\rightarrow\nu_{i})$ can be written as Blennow_2004
; Mishra:2023jlq

$\displaystyle P(\nu_{e}\rightarrow\nu_{i})$
$\displaystyle=\int_{0}^{R_{\odot}}\left\lvert\langle\nu_{i}|\nu_{e}(t)\rangle\right\rvert^{2}f(r)dr=\int_{0}^{R_{\odot}}\sum_{j=1}^{3}|\hat{U}_{ej}(\theta_{m})|^{2}P^{\rm
jump}_{ji}f(r)dr$ (9)

where $P^{\rm jump}_{ji}$ refers to the probability of jumping from one energy
eigenvector to the other. For adiabatic propagation, $P^{\rm
jump}_{ji}=\delta_{ji}$. The probability is averaged over the different
production zones within the Sun, weighted by the fraction of neutrinos
produced in each zone $f(r)$. The information on zonal neutrino production
fraction and flux information is taken from Ref. Bahcall_rep , and
$\hat{U}_{ej}(\theta_{m})$ is the eigenvector of Eq. 7.

In the following sections, we will be analyzing the Borexino experiment and
its measurement of 7Be neutrinos. We will generically be agnostic regarding
the relative rate of $\nu_{e}$, $\nu_{\mu}$, and $\nu_{\tau}$ interactions
comprising the neutrino-electron scattering for this fixed-energy portion of
the analysis. However, we will be interested in comparing the extracted rates
against those predicted by the flavor evolution described above. We will do so
assuming that the mixing of neutrinos is unitary or considering the case where
we relax that assumption. The following subsections explore those two
situations in turn.

#### Oscillations with Unitary Mixing

Assuming only three neutrinos exist, we can use the PDG parameterization
ParticleDataGroup:2022pth  to describe the $3\times 3$ unitary mixing matrix
$U_{\alpha i}$ – we assume that the three mixing angles and one CP-violating
phase are determined to the level summarized by the most recent NuFit results
nu_fit ; nu_fit_web .

The survival probability of electron neutrinos from the Sun depends
predominantly on the $U_{ei}$ elements, or, in terms of mixing angles,
$\theta_{12}$ and $\theta_{13}$. However, the appearance probabilities into
$\nu_{\mu,\tau}$ exhibit dependence on $\theta_{23}$ and $\cos\delta_{\rm
CP}$. Since these two are less well-measured, there is more uncertainty on the
expected fluxes of $\nu_{\mu}$ and $\nu_{\tau}$ arriving at the Earth.

If we consider the best-fit mixing angle values nu_fit  as well as assuming
three-flavor neutrino evolution through the Sun Mishra:2023jlq ; Bahcall_rep ,
we arrive at the expectation for 7Be neutrinos that

$\displaystyle P(\nu_{e}\to\nu_{e})^{\rm BF}$ $\displaystyle=0.528,$ (10)
$\displaystyle P(\nu_{e}\to\nu_{\mu})^{\rm BF}$ $\displaystyle=0.204,$ (11)
$\displaystyle P(\nu_{e}\to\nu_{\tau})^{\rm BF}$ $\displaystyle=0.267.$ (12)

Given the current uncertainty on oscillation parameters nu_fit_web , we
determine that the predicted $P(\nu_{e}\to\nu_{e})$ has a $5\%$ relative
uncertainty. We will return to the range of allowed
$P(\nu_{e}\to\nu_{\mu,\tau})$ in later discussion.

#### Non-Unitary Mixing

Considering the possible existence of one or more heavier “sterile” neutrino
states Blennow:2016jkn ; Escrihuela:2016ube , the unitary mixing matrix may
expand to an $n\times n$ matrix ($n$ being the total number of neutrino
species). This introduces non-unitary effects when examining the mixing of the
three active neutrino species, leading to a non-unitary nature within the
$3\times 3$ framework.

If the mixing among the three light neutrinos is not unitary Antusch:2006vwa ;
Parke:2015goa ; Ellis:2020hus , then the flavor evolution of neutrinos through
the Sun, as well as the probability of interacting as a certain neutrino
flavor eigenstate in an Earth-based detector, are modified. This could
generically lead to an overall reduction in the expected rate of solar
neutrino events at Borexino as well as in deviations from the expected
relative flavor compositions in this event rate. Generally, we may
parameterize the nonunitary analogue of the mixing matrix, called $N$, as

$\displaystyle N$ $\displaystyle=\alpha U=\begin{bmatrix}\alpha_{11}&0&0\\\
\alpha_{21}&\alpha_{22}&0\\\
\alpha_{31}&\alpha_{32}&\alpha_{33}\end{bmatrix}U^{3\times 3}$ (13)

where $U^{3\times 3}$ is the standard, unitary leptonic mixing matrix and
unitarity is restored if $\alpha_{ij}\to\delta_{ij}$. The $\alpha_{ij}$ are
constrained by numerous observations to be close to $\delta_{ij}$, but the
specific value of these constraints depends on the assumed source of unitarity
violation. This includes, for instance, whether a heavy neutrino inducing this
violation exists below or above the electroweak scale – we refer the
interested reader to Refs. Escrihuela:2016ube ; Blennow:2016jkn  for in-depth
discussion on these effects. When making predictions of the expected 7Be event
rate due to non-unitarity, we will adopt the “neutrino-only” constraints from
Escrihuela:2016ube  which allow for ${\sim}$percent level deviations at the
$3\sigma$ CL (with the exception of $\alpha_{33}$ which may be as small as
approximately $0.76$ and $|\alpha_{31}|$ which may be as large as
approximately $0.10$).

Importantly, any non-unitarity impacts the Hamiltonian of flavor-evolution
through the Sun: Eq. 7 becomes

$\displaystyle H_{\rm fl.}\longrightarrow{N}E_{\rm
M}{N}^{\dagger}+(N{N}^{\dagger})V(N{N}^{\dagger})$

This then impacts Eq. 9 – the eigenvectors of this Hamiltonian enter into the
expected $\nu_{i}$ production. The final effect is in modifying the
probability that a $\nu_{i}$ interacts “as” a $\nu_{\alpha}$ in the detector –
in the unitary scenario this probability is $\left\lvert U_{\alpha
i}\right\rvert^{2}$, as in Eq. 5. Without unitarity, this overlap is in
principle different and is given by $\left\lvert N_{\alpha
i}\right\rvert^{2}$.

Figure 2: Expected event rate in Borexino for neutrino-electron scattering as
a function of $N_{h}$, the number of hits. Data points, with error bars, are
shown as black points. The highlighted blue region, labeled the “7Be Window,”
is where we analyze the data. Several flux components are highlighted,
including the three different flavor contributions (as well as the total) of
7Be neutrinos, as well as different background and other solar neutrino
components.

##  III Borexino Data Analysis

In this section we describe our analysis of the Borexino data, with a
particular emphasis on the 7Be solar neutrino component. Borexino is a 278-ton
ultra-pure organic liquid scintillator located at LNGS in Italy. In its most
recent Phase-III data set, Borexino performs a multi-component spectral
analysis of the solar neutrino flux scattering off electrons in the detector,
obtaining flux measurements of the $pp$, $pep$, CNO, 8B, and 7Be solar
neutrino components. The 7Be flux is the best-measured, in part because it
occupies a region of relatively low background with respect to the recoiling
electron energy. Borexino Phase-II data have constrained the 7Be flux at the
$2.7\%$ level, in comparison with, for instance, of $pp$ neutrinos ($7.9\%$)
and 8B ones ($8\%$).

Because of the low backgrounds and the mono-energetic flux, 7Be neutrinos
allow us to precisely discern the flavor composition of the neutrino signal by
removing the dependence on neutrino energy. In order to isolate the 7Be
signal, we focus on the energy window from 250 $N_{h}$ to 320 $N_{h}$, where
$N_{h}$ corresponds to the number of hits on the detector photomultipliers,
and serves as an estimator of the recoil energy of the electron scattered by
the incoming neutrino Coloma:2022umy . By confining our analysis to this
specific range, we minimize the contributions of various backgrounds as well
as neutrinos from other solar-neutrino production mechanisms (CNO, $pep$,
$pp$, etc.).

Since 7Be neutrinos are mono-energetic($E_{0}$), the number of events for any
flavor $\alpha$ can be written as:

$N_{\alpha
i}=\xi\int_{N_{h}^{i}}^{N_{h}^{i+1}}\int\int_{E_{\nu}=E_{0}}\frac{d\phi_{e\alpha}(E_{\nu})}{dE_{\nu}}\frac{d\sigma_{e\alpha}(E_{\nu},T)}{dT}\frac{d\mathcal{R}(T,N_{h})}{dT}dE_{\nu}dTdN_{h}=\phi
P_{e\alpha}\sigma_{e\alpha}$ (14)

here $\xi$ is the exposure for Borexino Phase III (running for 1431.6 days,
71.3 ton fiducial volume). This quantity also contains information for
detector efficiency. The term $\frac{d\mathcal{R}(T,N_{h})}{dT}$ is the
resolution function and $\sigma$ is the EES cross section taken from Section
II – we refer the reader to Refs. Gonzalez_Garcia_2024 ; Coloma:2022umy  for
more details on the resolution function, $\xi$ and $N_{h}(T)$. Unless
otherwise stated, we include radiative corrections when considering the EES
cross section. For 7Be, $d\phi_{e\alpha}/dE_{\nu}$ is a delta function and so
this term integrates trivially, giving $N_{\alpha i}\propto\phi P_{e\alpha}$.
Fig. 2 presents our reproduction of the Borexino expected/observed event rate
with different contributions labeled. We have also separately identified the
contributions to the 7Be event rate by neutrino flavor, assuming the best-fit
probabilities discussed above.

We construct a $\chi^{2}$ test statistic that incorporates the experimental
data $N_{i}^{\rm obs.}$ in the $i$th bin, as well as the total expected
background ($B_{i}$) and signal ($S_{i}$) rates in that bin. For simplicity,
we assume that our signal consists only of the 7Be contributions and that the
background rates include the following: 210Bi, 85Kr, 210Po, Borexino’s
“external” backgrounds, as well as the solar-neutrino contributions from CNO
and pep neutrinos. We also include the bin-by-bin uncertainty from Borexino,
$\sigma_{i}$, giving our test statistic

$\displaystyle\chi^{2}=\sum_{i}\frac{\left(N_{i}^{\rm
obs.}-\left(B_{i}+S_{i}\right)\right)^{2}}{\sigma_{i}^{2}}+\chi^{2}_{\rm
ext.},$ (15)

where $\chi^{2}_{\rm ext.}$ allows for external information to be included in
the form of a prior.

Since we are interested in the sensitivity of Borexino to the individual
flavor components of the 7Be neutrinos, we parameterize $S_{i}$ as

$\displaystyle S_{i}\equiv n\sum_{\alpha=e,\mu\tau}S_{\alpha i},\quad
S_{\alpha i}\equiv\frac{f_{\alpha}}{P_{e\alpha}^{\rm BF}}N_{\alpha i}^{\rm
BF},$ (16)

where $P_{e\alpha}^{\rm BF}$ are the best-fit expected oscillation
probabilities cf Eqs. 10, 11 and 12 and $N_{\alpha i}^{\rm BF}$ is the
expected event rate in bin $i$ for flavor $\alpha$ assuming the best-fit
oscillation probability in Eq. 14. The overall normalization parameter $n$
that rescales all of the 7Be components is to account for uncertainty in the
overall flux of the neutrinos from the Sun. Generically, this allows for four
free parameters – $\left\\{n,\ f_{e},\ f_{\mu},\ f_{\tau}\right\\}$ – however,
considering all four to vary simultaneously will result in perfect
degeneracies among them making it impossible to constrain each independently.
In the following subsection, we will detail the choices we make regarding
which parameter(s) to consider and what external information to include in
$\chi^{2}_{\rm ext.}$. We will also discuss how our measurements of these four
parameters allow for direct comparisons of expected, allowed values of the
oscillation probabilities $P_{e\alpha}$.

In general, we will perform our analyses keeping the background rates $B_{i}$
fixed. In all of the cases that we present, we have also considered allowing
one background at a time to vary with a Gaussian prior on its normalization at
the $10\%$ level. We find that this does not significantly impact the results
of our analysis and so we opt to keep the normalizations fixed in what we
display.

##  IV Analysis Choices & Results

In this section we detail the analysis choices we make when studying Borexino
and its sensitivity to the 7Be flux. We perform a number of different
analyses, beginning with a simplified two-flavor approach, followed by two
distinct three-flavor studies. Finally, we explore the Borexino sensitivity to
new gauge bosons ($L_{\mu}-L_{\tau}$) and the impact of other underlying model
parameters to this sensitivity.

###  IV.1 Two-flavor Analysis

We first consider an effective two-flavor analysis in which we divide the
signal into electron-flavor contributions ($\nu_{e}$) and “others”
($\nu_{x}$). Instead of considering $S_{\mu i}$ and $S_{\tau i}$ separately as
in Eq. 16, we define an effective $S_{xi}$ using

$\displaystyle S_{xi}\equiv\frac{f_{x}}{P_{e\mu}^{\rm BF}+P_{e\tau}^{\rm
BF}}\left(N_{\mu i}^{\rm BF}+N_{\tau i}^{\rm BF}\right)$ (17)

If we fix the normalization such that $n=1$, then the $f_{\alpha}$ parameters
map on directly to the oscillation probabilities $P_{e\alpha}$. In that case,
we have a number of theoretical expectations against which we can compare.
First, unitary mixing requires $f_{e}+f_{x}=1$. If we consider knowledge of
three-neutrino mixing and constraints from other solar neutrino experiments,
then there is a further restriction on the range of the allowed $f_{e}$ within
this range. However, if we allow for non-unitary mixing, more of the $f_{e}$
vs. $f_{x}$ parameter space is allowed. We will consider two scenarios with
$n=1$ fixed – one in which $f_{e}$ is allowed to vary freely and one in which
it is constrained, adding a prior $\chi^{2}_{\rm ext.}=(f_{e}-P_{ee}^{\rm
BF.})^{2}/\sigma_{e}^{2}$, with $\sigma_{e}=P_{ee}^{\rm BF.}\times 5\%$.

Figure 3: Measurements of Borexino Phase-III at $1\sigma$ (dashed) and
$2\sigma$ (solid) confidence for a number of analysis scenarios. Left panel:
the two-flavor analysis of Section IV.1. We consider three different
scenarios, all with the normalization $n$ fixed – one in which $f_{e}$ is
fully unconstrained (light blue), one in which a $5\%$ prior is included (dark
blue), and one in which a further $30$x factor is included on the exposure
(dark gray). Best-fit points are marked with an ’x.’ In the left panel, we
compare with expected regions of parameter space when considering unitarity
(orange), the MSW solution (brown), and allowing for present-day uncertainty
of non-unitarity (green). Right panel: the three-flavor analysis of Section
IV.2, where $n$ is fixed and $f_{e}$ is profiled subject to a $5\%$
uncertainty. The dark blue (light blue) contours display measurements with
current ($30$x exposure) data, compared against predicted points in parameter
space for unitary (orange) or non-unitary (green) mixing. See text for further
detail.

Our results are shown in the left panel of Fig. 3 for the case where $f_{e}$
is left unconstrained (light blue) vs. when a $5\%$ prior is included (dark
blue). For both cases, we show the best-fit point (as an ‘x’) along with
$1\sigma$ (dashed) and $2\sigma$ (solid) CL contours. We also display a
hypothetical measurement that could be performed with $30$ times the exposure
of Borexino (dark gray), assuming that uncertainties are dominantly
statistical so that the test-statistic $\chi^{2}$ scales linearly with this
factor of 30. In comparison, we show (orange) the line in parameter space for
which these measurements are consistent with unitarity, $f_{e}+f_{x}=1$, with
the MSW solution for higher-energy solar neutrinos (brown), as well as points
in parameter space allowed when current constraints on non-unitary mixing are
considered (green). While some of these points exist outside of our current
$1/2\sigma$ CL contours, we do not purport to improve on existing non-
unitarity constraints, as these measurement contours can expand significantly
if we allow the normalization $n$ to vary, even at the $6\%$ level.

In the left panel of Fig. 3, we note that, without any external information on
$f_{e}$, the Borexino data are consistent with $f_{x}=0$ (i.e., no $\nu_{\mu}$
or $\nu_{\tau}$ contribution) at $1\sigma$ CL – this is driven both by (a) the
relatively similar shape between $\nu_{e}$ and $\nu_{\mu,\tau}$ contributions
to the Borexino event rate, as in Fig. 2, as well as (b) the fact that the
$\nu_{e}$ EES cross section is significantly larger than that of
$\nu_{\mu,\tau}$. However, once the $5\%$ prior is included (dark blue
contours), we can exclude $f_{x}=0$ at greater than $1\sigma$ CL and the best-
fit point moves very close to the region predicted by unitarity and the MSW
solution. Finally, we note that this analysis includes the presence of
radiative corrections in the $\nu_{\alpha}$ EES cross sections. This decision
does not impact the measurements of $f_{e}$ significantly, nor the uncertainty
on the measurement of $f_{x}$; however, it does shift the best-fit value of
$f_{x}$ substantially – this is summarized in Table 1

Parameter | Measurement without Radiative Corrections | Measurement with Radiative Corrections
---|---|---
$f_{e}$ | $0.529\pm 0.0762\ (0.1245)$ | $0.529\pm 0.0762\ (0.1246)$
$f_{x}$ | $0.3889\pm 0.3545\ (0.583)$ | $0.4529\pm 0.341\ (0.561)$
Table 1: Change in extracted measurements of $f_{e}$ vs. $f_{x}$ (in the
analysis of Section IV.1) when radiative corrections are or are not included
in the calculation of the EES cross sections. In the measurement
uncertainties, the first number (second number in parentheses) corresponds to
the $1\sigma$ ($2\sigma$) CL uncertainty.

###  IV.2 Three-flavor Analysis

We also consider a more complete three-flavor analysis, incorporating
contributions from all three neutrino flavors: $\nu_{e}$, $\nu_{\mu}$, and
$\nu_{\tau}$. We allow $f_{e}$, $f_{\mu}$, and $f_{\tau}$ to vary, fixing the
normalization $n$. In that case, $f_{\alpha}$ correspond directly to inferred
measurements of $P_{e\alpha}$. Since $P_{e\alpha}$ may be effectively
determined by other solar neutrino experiments, we will include a $5\%$ prior
on $f_{e}$ as in the above via $\chi^{2}_{\rm ext.}$.

The resulting measurements of $f_{\mu}$ and $f_{\tau}$, after profiling over
$f_{e}$, are shown in the right panel of Fig. 3 – we show this assuming
current experimental data (dark blue) and with a hypothetical factor of 30
larger exposure as in the above (light blue). As with the above, we show
$1\sigma$ (dashed) and $2\sigma$ (solid) CL contours in both cases.
Unsurprisingly, we find that there is not significant power in distinguishing
the $\nu_{\mu}$ vs. $\nu_{\tau}$ components given current experimental data,
or even with this significantly larger exposure considered. The allowed
regions of parameter space assuming unitary mixing (orange, with the variance
of $\theta_{23}$ and $\delta_{\rm CP}$ driving the motion) and non-unitary
mixing (green) are superimposed here – we see that especially when considering
current data, these hypotheses are very well consistent with our measurements.
We also observe that the point $(f_{\mu},\ f_{\tau})=(0,\ 0)$ is disfavored at
over $1\sigma$ CL (consistent with how in the two-flavor case, we saw
$f_{x}=0$ was disfavored when a $5\%$ prior on $f_{e}$ was included) – here we
can quantify that it is disfavored relative to the best-fit-point by
$\Delta\chi^{2}=4.09$. However, if we repeat this same analysis using only
tree-level calculations for the $\nu_{\alpha}$ EES cross sections, we see that
this point is only disfavored at the level of $\Delta\chi^{2}=2.76$ –
including radiative corrections in our $\nu_{\mu}$ and $\nu_{\tau}$ EES cross
sections makes these relatively small contributions easier to detect.

###  IV.3 Flavor Triangle Measurements

In a similar fashion to our three-flavor analysis of Section IV.2, we can
choose to scan over a different set of parameters – $n$ as well as two of the
three $f_{\alpha}$, assuming that the third is constrained by
$\sum_{\alpha}f_{\alpha}=1$. In doing so, we are assuming that the flux
normalization can vary (subject to a $6\%$ prior given theoretical uncertainty
Haxton:2012wfz ) and that all of the events present are due to electron
neutrinos, or their oscillation into $\nu_{\mu}$ or $\nu_{\tau}$. By making
this choice, we may present our measurement in terms of a ternary flavor
triangle, as done e.g. with high-energy astrophysical neutrinos with IceCube
IceCube:2015rro ; Arguelles:2015dca ; Bustamante:2016ciw ; Song:2020nfh  or
with diffuse supernova neutrinos Tabrizi:2020vmo .

Figure 4: Flavor triangle showing the measurement capability of Borexino
Phase-III at $1\sigma$ (dashed) and $2\sigma$ (solid), assuming present data
(dark blue) as well as a hypothetical $30$x exposure (light blue). In both
cases, a $6\%$ prior is included on the normalization $n$ which is profiled,
and the sum of the probabilities $\sum_{\alpha}P_{e\alpha}$ is fixed to be
one. Colored regions indicate the portions of parameter space expected under
unitary (orange) and non-unitary (green) mixing, subject to present-day
uncertainties.

The resulting measurements presented as a flavor triangle are shown in Fig. 4,
where each point on the figure automatically guarantees
$\sum_{\alpha}f_{\alpha}=1$. Since we also allow $n$ to vary, subject to a
$6\%$ Gaussian prior, we choose to interpret the $f_{\alpha}$ as probabilities
$P_{e\alpha}$. In this figure, the dark blue contours (dashed: $1\sigma$ CL,
solid: $2\sigma$ CL) correspond to our current capabilities using 7Be
neutrinos at Borexino, whereas the light blue contours demonstrate the
possibility of a factor of 30 larger exposure as considered in the previous
two subsections. Even though the only prior information we consider is this
$6\%$ uncertainty on $N$, we find that we can reasonably constrain $f_{e}$ to
be between ${\sim}[0.4,\ 0.7]$ at $2\sigma$ CL. We contrast these measurements
against the predicted relative event rates/oscillation probabilities when we
consider unitary, three-neutrino mixing (orange), as well as going beyond to
allow for non-unitary mixing (green). Since $n$ is allowed to vary (in
contrast to the left panel of Fig. 3 in our two-flavor analysis), we see that
these non-unitary points are all within the current measurement’s $1\sigma$ CL
contours. However, we see that with this hypothetical factor of 30 larger
exposure, we begin to probe this region more significantly.

In an ideal world, such a measurement would also allow for distinction between
$P_{e\mu}$ and $P_{e\tau}$, but with the current tools we have, the data are
consistent with either of these being zero – the test-statistic is very flat
along the diagonal direction connecting between $P_{e\mu}=0$ and
$P_{e\tau}=0$. Even with a substantially larger exposure (beyond the $30$x we
have presented here), this separation is very challenging. We find that with
an increase in the exposure by a factor of approximately $3000$, this
differentiation would be possible. Given this, we conclude that other
mechanisms are likely necessary to detect the difference between $\nu_{\mu}$
and $\nu_{\tau}$ EES processes if working within the standard model, where
their differences are relatively small and only separated due to radiative
corrections.

###  IV.4 $L_{\mu}-L_{\tau}$ Analysis

Finally, when considering a light gauge boson coupled to $L_{\mu}-L_{\tau}$,
as in Eq. 4, we consider the same $S_{\alpha i}$ but now allowing $N_{\mu i}$
and $N_{\tau i}$ to depend on the new gauge boson’s mass $m_{Z^{{}^{\prime}}}$
and coupling $g_{\mu\tau}$. This enters via the differential cross section in
Eq. 14. In the most general sense, we are sensitive to five free parameters:
these two new-physics ones, and three among
$\left\\{n,f_{e},f_{\mu},f_{\tau}\right\\}$. In order to compare against
previous literature results, we will consider a number of different analysis
choices regarding sensitivity to
$\left\\{m_{Z^{{}^{\prime}}},g_{\mu\tau}\right\\}$. First, we will fix
$\left\\{n,f_{e},f_{\mu},f_{\tau}\right\\}$ to their best-fit values – this
will serve as a direct comparison to other analyses of Borexino and
$L_{\mu}-L_{\tau}$ gauge bosons, however we expect that it will yield the most
aggressive constraints. Then, we will consider three different sets of prior
knowledge in the form of $\chi^{2}_{\rm ext.}$, in all cases constraining
$\sum_{\alpha}f_{\alpha}=1$:

  * •

Include $6\%$ normalization uncertainty on $n$ Haxton:2012wfz , allowing
$f_{e}$ and $f_{\mu}$ to vary freely between $[0,1]$.

  * •

Include $5\%$ relative uncertainty on $f_{e}$ (from theoretical knowledge of
oscillation parameters impacting $P_{ee}$) but letting $n$ vary freely between
$[0,2]$ and $f_{\mu}$ to vary freely as well.

  * •

Combining the two above cases, $6\%$ uncertainty on $n$ and $5\%$ on $f_{e}$.

Figure 5:  Constraints on the $L_{\mu}-L_{\tau}$ model from the Borexino data.
The dark blue is the $2\sigma$ contour for Borexino Phase III data using
$L_{\mu}-L_{\tau}$ model, keeping the probabilities and flux normalization
fixed to the best-fit value. The light blue and orange lines are $2\sigma$
contours calculated by scanning over $M_{Z^{\prime}}$ and $g_{Z^{\prime}}$,
$n$, $p_{e}$ and $p_{\mu}$, with having a 5$\%$ Gaussian prior on $p_{e}$ and
a 6$\%$ on $n$ respectively. The green line is the previous estimation based
on Phase I and Phase II data. Altmannshofer:2019zhy

The resulting $2\sigma$ constraints we derive in these different situations
are shown in Fig. 5, in comparison with results analyzing previous Borexino
data Altmannshofer:2019zhy  (green). We see significant (factors of ${\sim}2$
in the size of $g_{\mu\tau}$) differences in the derived constraints depending
on what prior information we include. If we fix the expected normalization as
well as the oscillation probabilities (in the form of $P_{e\alpha}$), we
arrive at a result (dark blue) that is consistent with the previous results of
Ref. Altmannshofer:2019zhy , given the increase in exposure in this data
release. However, if we allow the normalization $n$ to vary freely as well as
the oscillation probabilities (subject to $\sum_{\alpha}P_{e\alpha}=1$), we
find that the Borexino data cannot constrain this model at all. This is
because, without any external prior information, the data are consistent with
the entirety of the 7Be events coming from $\nu_{e}$ EES and
$P_{e\mu}=P_{e\tau}=0$ (as we saw in Section IV.1). If there are no
$\nu_{\mu,\tau}$, then we cannot test the $U(1)_{L_{\mu}-L_{\tau}}$ model.
However, things improve when prior information is included. For instance, if
we still allow the normalization to vary freely but constrain $P_{ee}$ at the
$5\%$ level, we can constrain this model as shown by the light blue line – we
consider this to be a very conservative constraint. If instead of including
prior information on $P_{ee}$, we do so on the normalization $n$ at the $6\%$
level, we arrive at the orange curve, approximately a factor of $2$ weaker
than the everything-fixed result in dark blue. Finally, we have also
considered including priors on both $P_{ee}$ and $n$ – that result is
degenerate with the orange curve seen here. Overall we find that while
Borexino is capable of using its 7Be neutrino events to constrain this new-
physics scenario, some precaution is warranted given the uncertain flavor
composition of these low-energy neutrinos.

##  V Discussion & Conclusions

In this paper we have performed a full 3-flavor analysis of Borexino Phase-III
data, focusing on the 7Be solar neutrino signal. We have focused on 7Be
because it is a mono-energetic line and is confined to a region of relatively
low backgrounds. We considered the impact of radiative corrections in the
calculation of $\nu_{\mu}$ and $\nu_{\tau}$ elastic scattering with electrons,
allowing us to gauge the current sensitivity of high-precision solar neutrino
data to differentiate between these two processes. While the current results
are not able to distinguish $\nu_{\mu}$ vs. $\nu_{\tau}$ elastic scattering,
we find that a significantly larger exposure, approximately 3000 times that of
Borexino, would be able to perform this exciting differentiation. We have also
found that accounting for radiative corrections in our calculations makes the
presence of $\nu_{\mu}$ and $\nu_{\tau}$ fluxes easier to detect than if they
are ignored in such an analysis.

Throughout our work, we have considered both a simplified, two-flavor approach
(combining the $\nu_{\mu}$ and $\nu_{\tau}$ components to compare them against
the $\nu_{e}$ one) as well as considering these three contributions
independently. In all of these analyses, we have compared our resulting
measurements of these components against expectations, both within the
standard, three-neutrino (unitary evolution) framework, as well as going
beyond. We have demonstrated, for instance, that with $\mathcal{O}(30)$ times
larger exposure, an experiment such as Borexino is sensitive to new levels of
non-unitarity present in the leptonic sector. In constructing these analyses,
we have also presented for the first time a flavor triangle in terms of the
flavor components as they arrive at the Earth for low-energy, solar neutrinos
coming from 7Be decays. Comparisons of these flavor triangles for other low-
energy components will yield fruitful information as measurements improve in
the coming era.

Finally, we have also considered the possibility of beyond-the-standard-model
contributions to low-energy neutrino-electron scattering, with an emphasis on
models that primarily impact $\nu_{\mu}$ and $\nu_{\tau}$ scattering. The
demonstrative model that we have focused on is the $U(1)$ extension of the SM
in which $L_{\mu}-L_{\tau}$ is gauged, allowing for new interactions between
$\nu_{\mu,\tau}$ and electrons to arise from a light, MeV-scale mediator. We
have performed comparisons against existing constraints using Borexino’s 7Be
neutrino sample on this model, and we demonstrate consistency, however we
preach caution due to the impact of uncertainties (e.g., on the flavor
composition of the 7Be neutrinos) in arriving at such constraints.

While bounding non-unitary models may not be possible for Borexino, future
solar neutrino experiments should be able to improve bounds. In particular,
the JUNO experiment JUNO:2023zty  will be sensitive to 7Be solar neutrinos,
and it may be possible to perform a similar analysis on the JUNO signals and
backgrounds as performed in this paper. As a crude estimate, with ${\sim}10$
years of data taking, for an ideal background scenario at JUNO, the flux
uncertainty will be reduced by over an order of magnitude, which would probe
the non-unitarity regime discussed in our analysis. Additional possibilities
exist in studying 8B neutrinos at DUNE or lower-energy $pp$ neutrinos at
future Xenon-/Argon-based dark matter direct detection experiments. However,
these signals do not have the advantage exploited throughout this work from
the monoenergetic nature of the 7Be neutrinos. Nevertheless, future analyses
of this nature as we progress into the next generation of neutrino and dark
matter experiments will naturally extend our understanding of solar neutrinos
and their flavor compositions.

## Acknowledgements

The authors are all supported by the DOE Grant No. DE-SC0010813. We are very
grateful to O. Tomalak and R. Plestid for several discussions on radiative
corrections for EES.

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