††institutetext: Max-Planck-Institut für Kernphysik,
Saupfercheckweg 1, 69117 Heidelberg, Germany

# Light vector bosons and the weak mixing angle in the light of new reactor-
based CE$\nu$NS experiments

Manfred Lindner  0000-0002-3704-6016 1 Thomas Rink111Corresponding author.
0000-0002-9293-1106 and Manibrata Sen  0000-0001-7948-4332 thomas.rink@mpi-
hd.mpg.de<EMAIL_ADDRESS>
###### Abstract

In this work, the sensitivity of future germanium-based reactor neutrino
experiments to the weak mixing angle $\sin^{2}\theta_{W}$, and to the presence
of new light vector bosons is investigated. By taking into account key
experimental features with their uncertainties and the application of a data-
driven and state-of-the-art reactor antineutrino spectrum, the impact of
detection threshold and experimental exposure is assessed in detail for an
experiment relying on germanium semiconductor detectors. With the established
analysis framework, the precision on the Weinberg angle, and capability of
probing the parameter space of a universally coupled mediator model, as well
as a U(1)B-L-symmetric model are quantified. Our investigation finds the next-
generation of germanium-based reactor neutrino experiments in good shape to
determine the Weinberg angle $\sin^{2}\theta_{W}$ with $<10$% precision using
the low-energetic neutrino channel of CE$\nu$NS. In addition, the current
limits on new light vector bosons determined by reactor experiments can be
lowered by about an order of magnitude via the combination of both CE$\nu$NS
and E$\nu$eS. Consequently, our findings provide strong phenomenological
support for future experimental endeavours close to a reactor site.

##  1 Introduction

With the first observations of coherent elastic neutrino-nucleus scattering
(CE$\nu$NS) Freedman:1973yd ; Tubbs:1975jx ; Freedman:1977xn  by the Coherent
collaboration COHERENT:2017ipa ; COHERENT:2020iec  the pathway towards
precision neutrino physics with small-scale detectors has been opened and
promises a bright future with many opportunities for physics within the
Standard Model (SM) and beyond (BSM). Thus, it is not surprising that already
these first measurements have been used for many SM and BSM investigations,
targeting, for example, the weak mixing angle at low momentum transfers
Cadeddu:2018izq ; Cadeddu:2020lky ; AristizabalSierra:2022axl ;
DeRomeri:2022twg , the neutron radius Cadeddu:2019eta ; Papoulias:2019lfi ;
Cadeddu:2021ijh ; DeRomeri:2022twg  in the context of the SM as well as
electromagnetic properties of the neutrino Papoulias:2017qdn ; Cadeddu:2018dux
; Khan:2019cvi , non-standard neutrino interactions (NSI) Lindner:2016wff ;
Papoulias:2017qdn ; Khan:2019cvi ; Coloma:2019mbs ; Denton:2022nol , light
mediators Dent:2016wcr ; Farzan:2018gtr ; Miranda:2020zji ; Cadeddu:2020nbr ;
Dutta:2022fdt , new or sterile fermions Brdar:2018qqj ; Chang:2020jwl ;
Miranda:2020syh ; Candela:2023rvt , dark matter Ge:2017mcq ; COHERENT:2019kwz
and even axion-like particles AristizabalSierra:2020rom  in the context of BSM
physics.

In the Coherent experiments, (anti-)neutrinos from pion-decays-at-rest
($\pi$DAR) were detected in multiple flavours: $\nu_{e}$, $\nu_{\mu}$,
$\bar{\nu}_{\mu}$. The relatively high neutrino energies up to $\sim 50$ MeV
and timing information about the incident neutrinos were among the key factors
which enabled a successful CE$\nu$NS detection. Additional measurements are
planned with further target materials such as argon, germanium (Ge), sodium
iodide and heavy water Akimov:2022oyb .

With the aim of detecting this process at lower (antineutrino) energies
($E_{\nu}\lesssim 10$ MeV), several experimental attempts have emerged close
to nuclear reactors: Connie CONNIE:2019swq , Conus CONUS:2020skt , Mi$\nu$er
Ang:2021mdp , Ncc-1701 Colaresi:2022obx , Neon Choi:2020gkm , $\nu$-cleus
Strauss:2017cuu , $\nu$GEN nGeN:2022uje , RED-100 Akimov:2022xvr , Ricochet
Ricochet:2021rjo , Texono TEXONO:2020vnv . These approaches try to exploit the
naturally high, single neutrino flavour emission of about $\sim
10^{20}\,\bar{\nu}_{e}$/GW/s close to a reactor core and probe CE$\nu$NS with
the full spectrum of modern detection technologies. Although the intrinsically
lower nuclear recoils from reactor antineutrinos render experimental
conditions even more demanding, a strong CE$\nu$NS signal from a reactor site
will provide a reference signal for future combined investigations of
CE$\nu$NS data sets. The lower momentum transfer in CE$\nu$NS of reactor
antineutrinos allows us to measure the weak mixing angle at even lower
energies Kumar:2013yoa ; ParticleDataGroup:2022pth ; AristizabalSierra:2022axl
, and study new light degrees of freedom Cerdeno:2016sfi ; Papoulias:2017qdn ;
Papoulias:2019txv ; Khan:2019cvi ; Cadeddu:2020nbr ; Miranda:2020tif ;
Miranda:2020zji ; Bertuzzo:2021opb ; DeRomeri:2022twg ; AtzoriCorona:2022moj ;
Li:2022jfl ; Coloma:2022avw . In this sense, both $\pi$DAR and reactor
approaches effectively complement each other.

The weak mixing angle $(\theta_{W})$, also known as the Weinberg angle, is an
important prediction of the SM. It relates the masses of the weak gauge bosons
and forms an integral component of the electroweak precision observable tests.
It can also be measured through a measurement of the SU$(2)_{\rm L}$ and the
U$(1)_{\rm Y}$ couplings. As a result, a precise measurement of $\theta_{W}$
is required to test the validity of the SM.

Naturally, by virtue of its definition, quantum corrections are relevant when
determining $\theta_{W}$, and hence its value depends on the scale at which
the measurements are made. This scale dependence has been confirmed by a
number of different experiments at different scales (see
ParticleDataGroup:2022pth  for a detailed review). The most precise
measurements of the weak mixing angle come from the LEP and the SLC -both
$e^{+}-e^{-}$ linear colliders- and use data collected at the $Z$-resonance to
report $\sin^{2}\theta_{W}=0.23153\pm 0.00016$ ALEPH:2005ab . Neutrino
experiments have also joined the quest, with the most precise measurement of
$\theta_{W}$ from neutrino scattering coming from the NuTEV experiment
NuTeV:2001whx . Using deep-inelastic scattering of neutrinos and
antineutrinos, NuTEV measured $\sin^{2}\theta_{W}=0.2407\pm 0.0016$ at an
average energy scale $\mu=4.6\,{\rm GeV}$ \- a result deviating from the LEP
measurements by $3.6\sigma$. This makes it crucial for other neutrino
scattering experiments to probe this discrepancy and check whether this is an
effect of unaccounted (nuclear) physics or a signal of BSM physics
deGouvea:2019wav . With the advent of CE$\nu$NS, it has become possible to
measure $\theta_{W}$ at lower energy scales Canas:2018rng ; Miranda:2020tif ;
Fernandez-Moroni:2020yyl ; DeRomeri:2022twg ; AristizabalSierra:2022axl . This
allows us to test the running of $\theta_{W}$ at lower energies, thereby
offering an important test of the quantum corrections affecting SM
measurements.

Clearly, the measurement of $\theta_{W}$ is dependent on the underlying
interaction channels for electrons and quarks. If hitherto unknown
interactions are present for the SM particles, our inference of $\theta_{W}$
can change. This is particularly interesting if the new physics consists of
light degrees of freedom since these can contribute to the kinematic regions
away from the $Z$ boson pole. In fact, current measurements of $\theta_{W}$
are not as precise at low momentum transfers as they are at the $Z$-pole. This
allows one to use CE$\nu$NS to test the interplay between $\theta_{W}$ and
various simple well-motivated extensions of the SM. For example, some of the
effects that could easily modify $\theta_{W}$ include the neutrino charge
radius, a modified neutrino vector coupling, and the introduction of light new
mediators, among others.

These are the major issues we want to address in this work using future
reactor antineutrino experiments. Further, Ge-based reactor experiments
currently lead the CE$\nu$NS investigations at reactor sites CONUS:2020skt ;
Colaresi:2022obx ; nGeN:2022uje , exhibit enormous scaling potential and
feature ongoing improvements in detection sensitivity Bonet:2023kob . With its
publications of several constraints on BSM physics CONUS:2021dwh ;
CONUS:2022qbb , we focus on the Conus experiment as a benchmark for the
investigations of this work. Using different realisations of the experiment,
we demonstrate the sensitivity to measuring the weak mixing angle within the
SM at low momentum transfers of $\mathcal{O}(10)\,{\rm MeV}$. We obtain that
the relative uncertainty of the mixing angle can improve to $\lesssim 10\%$
for feasible realisations of these experiments. Furthermore, we also test the
potential of these experiments to text simple U(1) extensions of the SM
involving a dark $Z^{\prime}$ gauge boson. We consider two scenarios : (1) a
universally coupled $Z^{\prime}$, and (2) $Z^{\prime}$ charged under a
U(1)B-L. We find that, depending on the experimental considerations, the
bounds on the couplings of these $Z^{\prime}$ can be stronger by an order of
magnitude than the current ones. This underlines the potential of high-purity
Ge detectors for CE$\nu$NS experiments.

This work is structured in the following way: Section 2 introduces the
interactions under study and discusses their individual characteristics. A
generic Ge-based reactor experiment is introduced in section 3, together with
the analysis framework we utilise for our investigations. The main results are
presented in terms of sensitivity estimates on the weak mixing angle as well
as on light vector bosons in section 4. Finally, we conclude with a summary
and an outlook in section 5.

##  2 Neutrino interactions in germanium detectors close to a nuclear reactor

In what follows we introduce the SM neutrino interaction channels relevant for
the analysis of this work, together with potential BSM modifications: elastic
neutrino-electron scattering (E$\nu$eS) and coherent elastic neutrino-nucleus
scattering (CE$\nu$NS). Both are, in principle, sensitive to the weak mixing
angle and new light vector bosons, for which we assume typical benchmark
models.

###  2.1 Elastic neutrino scatterings in the Standard Model

Antineutrinos emitted by a nuclear reactor can interact in two different ways
within a semiconductor detector, either via scattering off the electrons or
the nuclei of the underlying target material. Elastic scatterings of
(anti-)neutrinos off electrons are mediated by the exchange of electroweak
gauge bosons and can be expressed in the following compact differential cross
section Vogel:1989iv :

$\displaystyle\frac{\mathrm{d}\sigma}{\mathrm{d}T_{e}}(T_{e},E_{\nu})=\frac{G_{F}^{2}m_{e}}{2\pi}\Big{[}\left(g_{V}+g_{A}\right)^{2}+\left(g_{V}-g_{A}\right)^{2}\left(1-\frac{T_{e}}{E_{\nu}}\right)^{2}+\left(g_{A}^{2}-g_{V}^{2}\right)\frac{m_{e}T_{e}}{E_{\nu}^{2}}\Big{]}\,,$
(1)

where $T_{e}$ represents the electron recoil energy after the interaction,
$G_{F}$ the Fermi constant, $m_{e}$ the electron mass and $E_{\nu}$ the energy
of the incident (anti-)neutrino. The axial-vector and vector couplings
$g_{A,V}$ for the underlying currents depend on the neutrino flavour and are
defined as

$\displaystyle g_{V}$
$\displaystyle=\begin{cases}\frac{1}{2}+2\sin^{2}\theta_{W}\,,&\text{for
}\nu_{e}\,,\\\ -\frac{1}{2}+2\sin^{2}\theta_{W}\,,&\text{for
}\nu_{\mu},\nu_{\tau}\,,\end{cases}$ $\displaystyle
g_{A}=\begin{cases}\frac{1}{2}\,,&\text{for }\nu_{e}\,,\\\
-\frac{1}{2}\,,&\text{for }\nu_{\mu},\nu_{\tau}\,,\end{cases}$ (2)

while in the case of antineutrino scattering, the axial-vector couplings
change their sign, i.e. $g_{A}\rightarrow-g_{A}$.

For coherent elastic neutrino scattering off a nucleus, the neutrino energy
must not be too large such that the mediator’s wavelength extends over the
whole nucleus. This coherence condition is usually given by $qR\sim 1$, where
$q$ is the reaction’s momentum transfer and $R$ is the nucleus radius. This
can be translated into a condition on the energy of the incident neutrino: for
energies below $E_{\nu}\sim$19 MeV, which is the case for reactor
antineutrinos, coherent interactions of neutrinos scattering off a Ge nucleus
can be assumed. This corresponds to a maximum nuclear recoil energy of
$T_{A}\sim 11$ keV.

The differential CE$\nu$NS cross section is given by Freedman:1973yd

$\displaystyle\frac{d\sigma}{dT_{A}}(T_{A},E_{\nu})$
$\displaystyle=\frac{G_{F}^{2}m_{A}}{4\pi}Q_{W}^{2}\left(1-\frac{m_{A}T_{A}}{2E^{2}_{\nu}}\right)|F(T_{A})|^{2}\,,$
(3) $\displaystyle\text{with }Q_{W}$
$\displaystyle=\left[(1-4\sin^{2}\theta_{W})Z-N\right]\,.$

Here, $m_{A}$ is the nucleus mass and $Q_{W}$ is the weak nuclear charge,
which is defined by the number of protons $Z$ and neutrons $N$, respectively.
For the nuclear form factor, the Helm parameterisation is used

$\displaystyle
F(T_{A})=\frac{3j_{1}(q(T_{A})R_{1})}{q(T_{A})R_{1}}\exp\left[-\frac{(q(T_{A})s)^{2}}{2}\right]\,,$
(4)

with the spherical Bessel function $j_{1}$, the momentum transfer
$q^{2}=2m_{A}T_{A}$, the nuclear skin thickness $s\simeq 1$ fm,
$R_{1}=\sqrt{R^{2}-5s^{2}}$ and $R\simeq 1.2A^{1/3}$ fm Helm:1956zz ;
Engel:1991wq . Due to the smaller momentum transfer at a reactor site, the
impact of the nuclear form factor is negligible, i.e. $F\rightarrow 1$
CONUS:2021dwh . However, with increasing precision on the SM signal - as
investigated in this work - this quantity becomes more relevant.

The target recoil energies $T_{x}$ for $x=\\{e,A\\}$ depend on the scattering
angle $\theta$ (lab frame) and is given by

$\displaystyle
T_{x}=\frac{2m_{x}E^{2}_{\nu}\cos^{2}\theta}{(m_{x}+E_{\nu})^{2}-E^{2}_{\nu}\cos^{2}\theta}\xrightarrow{\theta\rightarrow
0}\frac{2E^{2}_{\nu}}{m_{x}+2E_{\nu}}\,,$ (5)

where the last step defines the maximal nuclear recoil $T_{x}^{\mathrm{max}}$.

While measuring E$\nu$eS is a rather straightforward procedure within a
semiconductor detector (recoiling electrons can directly be read out as charge
signals), the read-out of nuclear recoil energies left by CE$\nu$NS is more
involved for this detection technology that only detects charge/ionisation
signals. In the conversion process from nuclear to ionisation energy after a
CE$\nu$NS event, so-called signal quenching, roughly 80% of the initial energy
is lost in dissipative processes and, thus, not accessible. The ratio between
ionisation energy $E_{I}$ and nuclear recoil energy $T_{A}$ is called the
quenching factor and an energy-dependent quantity. Here, the conventional
Lindhard model is used to quantify signal quenching in Ge detectors
Lindhard:1961zz ; Bonhomme:2022lcz .222Currently, deviations from the Lindhard
descriptions have been reported at sub-keV energy Collar:2021fcl . Since
proposed physical explanations for an increased quenching effect have been
excluded AtzoriCorona:2023ais , the authors only apply the conventionally used
quenching model of Lindhard. However, we would like to emphasise the
importance of quenching for reliable CE$\nu$NS predictions and turn to the
experimental community to quickly solve this problem.

###  2.2 Modifications due to New Physics

The interaction channels presented above can be subject to modification of BSM
physics if new interactions with electrons and quarks are introduced. Here, we
assume the existence of an additional $Z$-like gauge boson with interactions
given by the following Lagrangian:

$\displaystyle\mathcal{L}_{Z^{\prime}}=Z^{\prime}_{\mu}\left(g^{\nu\mathrm{V}}_{Z^{\prime}}\bar{\nu}_{L}\gamma^{\mu}\nu_{L}+g^{e\mathrm{V}}_{Z^{\prime}}\bar{e}\gamma^{\mu}e+g^{q\mathrm{V}}_{Z^{\prime}}\bar{q}\gamma^{\mu}q\right)+\frac{1}{2}m^{2}_{Z^{\prime}}Z^{\prime}_{\mu}Z^{\prime\mu}\,,$
(6)

with the new vector couplings $g^{x\mathrm{V}}_{Z^{\prime}}$ for
$x\in\\{\nu,e,q\\}$, the boson mass $m_{Z^{\prime}}$ and $q\in\\{u,d\\}$ being
the first generation of quarks. For the time being, we leave aside any aspects
of model-building and only assume the existence of an appropriate mass
mechanism for the new light gauge boson. In this sense, we follow the
framework of simplified models usually used in collider studies
Salvioni:2009mt ; Abercrombie:2015wmb ; Cerdeno:2016sfi ; Arcadi:2017kky ;
Morgante:2018tiq . Its coherent interaction with nuclei contributes to another
weak nuclear charge that is given by

$\displaystyle
Q_{Z^{\prime}}=\left(2g^{u\mathrm{V}}_{Z^{\prime}}+g^{d\mathrm{V}}_{Z^{\prime}}\right)Z+\left(g^{u\mathrm{V}}_{Z^{\prime}}+2g^{d\mathrm{V}}_{Z^{\prime}}\right)N\,.$
(7)

As a consequence, the weak nuclear charge appearing in the conventional
CE$\nu$NS cross-section, cf. eq. (3), is modified by the presence of the new
vector couplings: $Q_{W}\rightarrow Q_{W}+Q_{Z^{\prime}}$.

The same applies to the $Z$-boson contribution of E$\nu$eS, but with the
difference that eq. (1) also contains a contribution due to the exchange of
$W$ boson. The modified E$\nu$eS cross section due to the exchange of an
additional $Z$-like boson is given by Cerdeno:2016sfi

$\displaystyle\left(\frac{d\sigma}{dT_{e}}\right)_{\mathrm{E\nu
eS\,+\,Z^{\prime}}}=\left(\frac{d\sigma}{dT_{e}}\right)_{\mathrm{E\nu
eS}}+\frac{\sqrt{2}G_{F}m_{e}g_{V}g_{Z^{\prime}}^{\nu\mathrm{V}}g_{Z^{\prime}}^{e\mathrm{V}}}{\pi(2m_{e}T_{e}+m_{Z^{\prime}}^{2})}+\frac{m_{e}(g_{Z^{\prime}}^{\nu\mathrm{V}}g_{Z^{\prime}}^{e\mathrm{V}})^{2}}{2\pi(2m_{e}T_{e}+m_{Z^{\prime}}^{2})^{2}}\,,$
(8)

with the usual SM vector coupling $g_{V}$ defined in eq. (2) as well as the
new $Z$ couplings $g^{\nu V}_{Z^{\prime}}$ and $g^{eV}_{Z^{\prime}}$ to
neutrinos and electrons, respectively.

In the end, we would like to mention that the explicit appearance of the
$Z^{\prime}$ propagator term in both, E$\nu$eS and CE$\nu$NS, cross sections
divides the model’s parameter space in two different kinematic regions: $q\ll
m_{Z^{\prime}}^{2}$ and $q\gg m_{Z^{\prime}}^{2}$. In the latter case, the
dependence on $m_{Z^{\prime}}$ is negligible and limits become independent of
the new boson mass. In the following, we introduce the models that are
investigated in the course of this work:

#### Universal coupling

A straightforward way to simplify the parameter space of the introduced U(1)
extension of the SM is to assume equal coupling to all (or at least the first
generation of) SM fermions, i.e. $g_{Z^{\prime}}\equiv
g^{\nu\mathrm{V}}_{Z^{\prime}}=g^{e\mathrm{V}}_{Z^{\prime}}=g^{u\mathrm{V}}_{Z^{\prime}}=g^{d\mathrm{V}}_{Z^{\prime}}$.
In doing so, the relevant parameter space is shrunk to only two parameters:
the universal coupling $g_{Z^{\prime}}$ and the boson mass $m_{Z^{\prime}}$.
Due to this charge assignment, the additional nuclear charge of eq. (7)
simplifies further $Q_{Z^{\prime}}\rightarrow
3\,g_{Z^{\prime}}\left(Z+N\right)$ such that the combined CE$\nu$NS cross
section is given by Cadeddu:2020nbr

$\displaystyle\frac{d\sigma}{dT_{A}}(T_{A},E_{\nu})$
$\displaystyle=\frac{G_{F}^{2}m_{A}}{\pi}\bigg{[}Q_{W}+\frac{3g_{Z^{\prime}}^{2}}{\sqrt{2}G_{F}}\frac{Z+N}{2m_{A}T_{A}+m_{Z^{\prime}}^{2}}\bigg{]}^{2}\left(1-\frac{m_{A}T_{A}}{2E^{2}_{\nu}}\right)|F(T_{A})|^{2}\,.$
(9)

The quadratic dependency on the weak nuclear charge in eq. (9) and the fact
that $Q_{W}<0$ for practical purposes, induces a degeneracy in the parameter
space. In case of $\big{[}Q_{W}+Q_{Z^{\prime}}]\simeq\pm Q_{W}$, the
experimental signature appear as SM-like. This leads to a situation in which
the light vector mediator contribution cannot be distinguished from SM
CE$\nu$NS and an exclusion demands targets with varying proton and neutron
numbers as well as different sensitivity to the nuclear recoil energies left
in the detector.

#### The benchmark model U(1)B-L

One of the simplest and most widely studied extensions of the SM is an
additional vector boson with coupling to a particle’s B$-$L charge. Besides a
quite natural definition of charge, i.e. $g_{Z^{\prime}}\equiv
g^{\nu\mathrm{V}}_{Z^{\prime}}=g^{e\mathrm{V}}_{Z^{\prime}}=-g^{u\mathrm{V}}_{Z^{\prime}}/3=-g^{d\mathrm{V}}_{Z^{\prime}}/3$,
this model arises from a variety of more complete SM extensions and is by
definition free of any gauge anomalies. Thus, it appears as a valid
representation of nature with an economic parameter space spanned only by the
boson mass $m_{Z^{\prime}}$ and the absolute coupling strength
$g_{Z^{\prime}}$. The resulting modified CE$\nu$NS cross-section is then given
by

$\displaystyle\frac{d\sigma}{dT_{A}}(T_{A},E_{\nu})$
$\displaystyle=\frac{G_{F}^{2}m_{A}}{\pi}\bigg{[}Q_{W}-\frac{g_{Z^{\prime}}^{2}}{\sqrt{2}G_{F}}\frac{Z+N}{2m_{A}T_{A}+m_{Z^{\prime}}^{2}}\bigg{]}^{2}\left(1-\frac{m_{A}T_{A}}{2E^{2}_{\nu}}\right)|F(T_{A})|^{2}\,.$
(10)

In this model, the parameter space is non-degenerate such that conclusive
exclusion limits can be deduced. Since the charge assignment in both models is
the same for neutrinos and electrons, both lead to the same modification of
E$\nu$eS. Therefore, we expect only additional events compared to the SM case
for E$\nu$eS.

Figure 1:  Detector (trigger) efficiencies applied throughout this work. The
blue curve - as achieved in ref. Bonhomme:2022lcz  \- is used for the
conservative and the expected threshold scenario, while the red curve
corresponds to our optimistic case. Vertical lines indicate the three
threshold benchmarks, cf. table 1.

det. thresholds [eV] | exp. exposure
---|---
180 (E1) | 5 kg$\cdot$yr
150 (E1) | 50 kg$\cdot$yr
100 (E2) | 500 kg$\cdot$yr

Table 1: Overview of assumed detector thresholds and experimental exposures.
The applied detector efficiencies are given in brackets and shown in figure 1.
All combinations of these values are considered in the analysis below.

##  3 Experimental set-up and investigation strategy

After clarifying the phenomenological foundations of this work, we define the
experimental framework of this analysis and the methodology of our sensitivity
estimation. Since Ge detectors are currently leading CE$\nu$NS investigations
at reactor sites CONUS:2020skt ; Colaresi:2022obx ; nGeN:2022uje , the main
focus of this work are future capabilities of Ge-based experiments.

###  3.1 Experimental set-up under study

As the strongest artificial (anti-)neutrino source on Earth, we select a
commercial nuclear power plant with a thermal power of 3.5 GW, a value in the
ballpark of currently running pressurised water reactors. A distance of $\sim
20$ m from the reactor core represents a good trade-off between potential
reactor-correlated background events and the expected neutrino flux
Hakenmuller:2019ecb . With these assumptions a total antineutrino flux of
$\phi\sim 1.5\cdot 10^{13}\,\bar{\nu}_{e}$/cm2/s is obtained. For the reactor
antineutrino emission spectrum, we use a recently proposed data-driven
approach DayaBay:2021dqj . For this, information from inverse beta decay (IBD)
measurements is used to construct an antineutrino emission spectrum tailored
to a given reactor fuel composition. We assume a typical configuration for a
pressurised water reactor, where the antineutrinos are mainly emitted in beta
decay chains of four isotopes. Thus, following fission fractions are used:
(235U, 238U, 239Pu, 241Pu) = (56.1, 7.6, 30.7, 5.6) %. The unfolded IBD
spectra of ref. DayaBay:2021dqj  (1.8-7 MeV), are combined with new
measurements of the spectrum’s high energy part (7-11 MeV) DayaBay:2022eyy
and low-energy information from updated summation methods (0-1.8 MeV)
Estienne:2019ujo .

As an experimental target, a Ge semiconductor detector with three benchmark
exposures is assumed: 5 kg$\cdot$yr, 50 kg$\cdot$yr and 500 kg$\cdot$yr. Since
commercial power plants exhibit short maintenance periods of $\sim 1$
month/year, experiments close to reactors are usually limited by their
background measurements, i.e. the reactor OFF periods. We take this into
account by using different run times for reactor ON and OFF periods and
optimistically assume $t_{\mathrm{OFF}}=0.5\cdot t_{\mathrm{OFF}}$. For the
detection threshold $E_{\mathrm{thr}}$, we follow current detector
developments and use again three different values in our investigation: 180 eV
(conservative), 150 eV (expected) and 100 eV (optimistic). An overview of the
assumed experimental parameters is given in table 1. To describe signal
quenching, i.e. the conversion of nuclear recoil energies to ionisation
signal, for CE$\nu$NS in our detectors, we follow recent measurements and
apply Lindhard’s model Lindhard:1961zz ; Bonhomme:2022lcz  down to detection
energies of 100 eV. In doing so, we remain conservative since increased signal
quenching for whatever reason also increases the expected signal counts and,
thus, the experimental reach. An energy-dependent experimental resolution is
incorporated via convolutions with a Gaussian whose width is also determined
by the chosen experimental detection threshold. In reality, a heuristic
relation between a detector’s energy threshold and a pulser’s full width at
half maximum (FWHM), i.e. FWHM of an artificial pulse at zero detection
energy, is observed: $E_{\mathrm{thr}}\sim 3\cdot$ FWHMpulser. Thus, in the
convolution that takes care of the detector resolution, the Gaussian width is
given by the following expression:

$\displaystyle\sigma(E)=\sqrt{\left(\frac{E_{\rm thr}}{3\cdot
2.355}\right)^{2}+\epsilon\cdot\mathcal{F}\cdot E}\,,$ (11)

with the assumed detection threshold, $E_{\rm thr}$, the energy needed to
create an energy-hole pair at 90K in a Ge semiconductor, $\epsilon=2.96$ eV,
and the Fano factor for Ge, given by $\mathcal{F}=0.11$. The first term
represents the pulser resolution (FWHM) at zero energy. To incorporate
realistic detector efficiencies, we apply the (trigger) efficiency curves
shown in figure 1.333The curves are basically Gaussian CDFs with parameters
chosen to fit realistic detection efficiencies at lowest energies
Bonhomme:2022lcz ; Bonet:2023kob . For the conservative and expected threshold
case, the blue curve is applied Bonhomme:2022lcz , whereas the optimistic case
uses a curve that exhibits 50% detection efficiency down to the threshold of
100 eV. The first is motivated by recent developments in Ge semiconductor
detectors Bonhomme:2022lcz , whereas the red curve has been manually chosen to
reach 50% efficiency down to $E_{I}=100$ eV. Although recent studies show that
additional background reduction techniques such as pulse shape discrimination
by (charge) signal rise times are in principle applicable at such low energies
Bonet:2023kob , their implementation with additional cuts will depend heavily
on the individual experimental configuration and is thus beyond the scope of
this work.

A crucial ingredient of any CE$\nu$NS measurement is to prove that reactor-
correlated background, especially neutrons, are under control
Hakenmuller:2019ecb , which we assume to be the case. The experimental
background level depends strongly on the individual experimental set-up, in
particular the applied detector and shield materials, and the overburden and
radioactivity on site. For more details on experimental background close to
reactors, we refer the reader to ref. Bonet:2021wjw . For this sensitivity
estimate, we simply assume flat and constant background levels, which is at
least justified far away from the detector’s electronic noise and for keV-
energies. For the energy region $1~{}\mathrm{keV}<E_{I}\leq
10~{}\mathrm{keV}$, mainly important for E$\nu$eS, we assume a background
level of $\sim 0.5$ cnts/kg/d/keV. For energies $E_{I}\leq 1~{}\mathrm{keV}$
that are relevant for CE$\nu$NS investigations, the obtained background levels
are usually higher and rise towards the detector threshold. Thus, we assign it
an overall background level of $\sim 10$ cnts/kg/d/keV, which was achieved by
Conus on their first experimental runs CONUS:2020skt . Further improvements in
these values might be expected in the future.

###  3.2 Sensitivity estimation

To determine the experimental sensitivity, we closely follow the method used
by the Conus collaboration and construct a likelihood function that fits
reactor ON and OFF periods simultaneously, while experimental uncertainties
are incorporated with Gaussian pull terms CONUS:2020skt ; CONUS:2021dwh .
Hence, the used (binned) likelihood function exhibits the form

$\displaystyle-2\log\mathcal{L}=-2\log\mathcal{L}_{\mathrm{ON}}-2\log\mathcal{L}_{\mathrm{OFF}}+\sum_{i}\frac{(\theta_{i}-\bar{\theta}_{i})^{2}}{\sigma_{\theta_{i}}}\,,$
(12)

where $\theta_{i}$ are the incorporated nuisance parameters,
$\bar{\theta}_{i}$ their best-fit values and $\sigma_{\theta_{i}}$ the
corresponding uncertainties. For this phenomenological analysis, we assume
experimental quantities like detection efficiency, the detector’s energy
calibration and resolution as well as active volume to be sufficiently known
such that corresponding uncertainties play a minor role and can be safely
ignored. The only experimental uncertainties incorporated in this analysis are
attributed to signal quenching (Lindhard model with $\sim 1$ % uncertainty in
its $k$ parameter, which reflects the quenching factor at $T_{A}\sim 1$ keV
Bonhomme:2022lcz ) and an overall reactor antineutrino flux normalisation (3%)
originating from uncertainties on its thermal power and the underlying fission
fractions. Furthermore, the background contribution in both energy regions is
assigned an unconstrained normalisation parameter $b$ in order to account for
potential fluctuations. Thus, the constructed analysis framework considers the
following quantities:

  * •

weak mixing angle $\sin^{2}\theta_{W}$, vector boson coupling $g_{Z^{\prime}}$
and mass $m_{Z^{\prime}}$

  * •

two background normalisation parameters: $b_{\mathrm{low}}$ ($E<1$ keV) and
$b_{\mathrm{high}}$ ($E>1$ keV)

  * •

two (pulled) nuisance parameters: Lindhard (quenching) parameter $k=0.162\pm
0.004$ and reactor antineutrino flux $\phi\sim(1.50\pm 0.04)\cdot
10^{13}\,\bar{\nu}_{e}$/cm2/s

For the investigation of light vector bosons, we use knowledge from previous
investigations of the Weinberg angle and include this via an additional
Gaussian pull-term. Here, we use a recent data-driven low-energy determination
through combined data sets of atomic-parity violation investigations on
caesium and Coherent CE$\nu$NS measurements, i.e.
$\sin^{2}\theta_{W}=0.2374\pm 0.0020$ AtzoriCorona:2023ktl .

In the following, we estimate the future experimental sensitivity with two
different methods: Asimov and Monte-Carlo (MC)-sampled data sets Cowan:2010js
; ParticleDataGroup:2022pth . We apply a (profile) likelihood ratio test and
deduce limits from a $\chi^{2}$-distributed test statistics.444For the
investigation of the weak mixing angle, we assume a two-sided statistics test,
since both downward and upward fluctuations might occur. The gauged U(1)B-L
model, only contributes additional signal events, thus, a one-sided statistics
test is applied. For the case of the universally coupled model, the degeneracy
of the parameter space also allows for a reduction of signal events for
CE$\nu$NS, while E$\nu$eS can still be enhanced. This degeneracy region is
indicated in the following results plots and limits on the smallest coupling
strength are determined via one-sided test statistics.  While the first method
directly yields exclusion limits that represent the full exploitation of
statistical information, the second approach gives a distribution of potential
limits, from which the median average is commonly defined as the expected
experimental sensitivity.

Figure 2: CE$\nu$NS signal expectation in dependence of target mass and
detection threshold for the chosen experimental site close to a reactor. The
optimistic detector efficiency of figure 1 has been chosen. The detector
resolution is dynamically set in dependence on the assumed threshold, cf. eq.
(11).

##  4 Results

Here, we highlight the effects of key experimental quantities like detection
threshold and experimental exposure, and present the results of our
sensitivity estimate on the weak mixing angle $\sin^{2}\theta_{W}$ and the
parameter space of two light vector boson models. The expected sensitivities
are determined for the combination of three different detector thresholds (100
eV, 150 eV, 180 eV) and three potential experimental exposures (5 kg$\cdot$yr,
50 kg$\cdot$yr, 500 kg$\cdot$yr), cf. table 1.

###  4.1 Signal expectations in future germanium-based reactor experiments

Before we present our results, we calculate the number of expected SM events
to quantify the statistical reach we could expect for the chosen experimental
configurations. The SM E$\nu$eS is rather constant up to detection energies of
100 keV with an event rate of $\sim 113$ cnts/kg/yr independent of the chosen
detection threshold $E_{\mathrm{thr}}$.

Its strong sensitivity reach stems from the fact that the SM contribution is
negligible compared to the assumed background contributions and that
$\mathcal{O}(1)$ couplings of BSM models already contribute sizable event
rates. Due to the kinematic term in eq. (3) and the effect of signal
quenching, CE$\nu$NS only extends up to energies of $\sim 750$ eV, however
with contributions that by far exceed the electron channel: (655, 988, 4481)
cnts/kg/yr for (180, 150, 100) eV.555A reactor antineutrino with $E_{\nu}$=11
MeV, leaves a maximal recoil of $T_{A}^{\mathrm{max}}\sim 3.6$ keV on a Ge
nucleus. A quenching factor of $\sim 0.2$ then leads to a maximal ionisation
signal of $E_{I}^{\rm max}\sim 750$ eV. The coherent nature of this
interaction channel enhances potential BSM contributions and renders it a
strong probe for new physics. To further underline its strong potential for
future BSM studies, we show the expected CE$\nu$NS event rate dependence on
the detector mass and threshold in figure 2. While experimental exposure (mass
$\cdot$ runtime) increases the event rate only linearly, improvements in the
detector performance in terms of lower detection threshold and detection
efficiency lead to a non-linear signal increase. Experimental development in
this direction will, therefore, be of great benefit to future investigations.
In actual experimental realisations, the expected signal might be further
enhanced by scaling detector mass and measurement time. Therefore, figure 2
highlights the importance of future detector development for individual
devices and quantifies the impact of further improvements in terms of the
expected CE$\nu$NS signal.

Figure 3: $\Delta\chi^{2}$ contours for the assumed experimental specifications (Asimov data sets). Background levels and detector thresholds are given in the boxes, while increasing exposures are shown in different colours. Assumptions on the background are similar to what has been achieved in the Conus experiment CONUS:2020skt ; CONUS:2022qbb . The corresponding limits at 90% C.L. are given by the intersection with the dashed line. Limits obtained from MC-sampled data are indicated by the dotted line. Exposure / $E_{\mathrm{thr}}$ | 180 eV | 150 eV | 100 eV
---|---|---|---
5 kg$\cdot$yr | (0.205, 0.273) | (0.211, 0.266) | (0.222, 0.254)
| (-14, 15) % | (-11, 12) % |  $\pm$7 %
50 kg$\cdot$yr | (0.218, 0.258) | (0.221, 0.254) | (0.225, 0.251)
| (-8, 9) % |  $\pm 7$ % | (-5, 6) %
500 kg$\cdot$yr | (0.224, 0.251) | (0.225, 0.251) | (0.225, 0.25)
|  $\pm 6$ % | (-5, 6) % | (-5, 6) %
Table 2: Error bands at 90% C.L. and relative uncertainties of the weak mixing
angle $\sin^{2}\theta_{W}$ for the different exposures and detection
thresholds under study.

###  4.2 Sensitivity to the weak mixing angle $\sin^{2}\theta_{W}$

Figure 4: Expected exclusion limits at 90 % C.L. on a universally coupled
light vector mediator for different exposures and detector thresholds.
Background levels and detector thresholds are given in the boxes, while the
exposures under study are illustrated with different colouring. Exemplary
limits from MC-sampled data represented by crosses to validate our results.
For comparison Conus limits from their first BSM investigation are indicated
in orange (E$\nu$eS-only) and purple (CE$\nu$NS-only) CONUS:2021dwh . The
degeneracy region for CE$\nu$NS, where the BSM model cannot be distinguished
from the pure SM interaction, is marked with grey-dotted lines.

Exposure / $E_{\mathrm{thr}}$ | 180 eV | 150 eV | 100 eV
---|---|---|---
5 kg$\cdot$yr |  $2.1\cdot 10^{-7}$ / $3.0\cdot 10^{-4}$ |  $2.0\cdot 10^{-7}$ / $2.7\cdot 10^{-4}$ |  $1.9\cdot 10^{-7}$ / $2.1\cdot 10^{-4}$
50 kg$\cdot$yr |  $1.4\cdot 10^{-7}$ / $2.3\cdot 10^{-4}$ |  $1.4\cdot 10^{-7}$ / $2.1\cdot 10^{-4}$ |  $1.3\cdot 10^{-7}$ / $1.9\cdot 10^{-4}$
500 kg$\cdot$yr |  $9.7\cdot 10^{-8}$ / $1.9\cdot 10^{-4}$ |  $9.4\cdot 10^{-8}$ / $1.9\cdot 10^{-4}$ |  $9.0\cdot 10^{-8}$ / $1.9\cdot 10^{-4}$

Table 3: Exemplary exclusion limits at 90% C.L. for a universally coupled
light vector boson. Values are given for regions where the experimental reach
is dominated by E$\nu$eS ($m_{Z^{\prime}}=1$ keV) and CE$\nu$NS
($m_{Z^{\prime}}=1$ GeV), respectively.

Since both CE$\nu$NS and E$\nu$eS cross sections exhibit a dependence on the
weak mixing angle, these interaction channels can in principle be utilised for
its investigation. However, limits expected from E$\nu$eS alone are not
competitive to existing bounds from electron scatterings. Moreover, a combined
analysis via both SM interaction channels is mainly driven by CE$\nu$NS. Thus,
we mainly focus on CE$\nu$NS and only present results obtained from energy
regions below 1 keV. The outcome of our sensitivity estimation is shown in
figure 3 in terms of $\Delta\chi^{2}$ contours for the detection thresholds
and experimental exposures under study.

Figure 5: Expected exclusion limits at 90 % C.L. on a light gauged B$-$L
vector mediator for different exposures and detector thresholds. Background
levels and detector thresholds are given in the boxes, while the exposures
under study are illustrated with different colouring. Exemplary limits from
MC-sampled data represented by crosses to validate our results. For comparison
Conus limits from their first BSM investigation are indicated in orange
(E$\nu$eS-only) and purple (CE$\nu$NS-only) CONUS:2021dwh . In contrast to the
previous case, the U(1)B-L does not exhibit any degeneracy region.

Exposure / $E_{\mathrm{thr}}$ | 180 eV | 150 eV | 100 eV
---|---|---|---
5 kg$\cdot$yr |  $2.1\cdot 10^{-7}$ / $5.3\cdot 10^{-4}$ |  $2.0\cdot 10^{-7}$ / $4.8\cdot 10^{-4}$ |  $1.9\cdot 10^{-7}$ / $3.6\cdot 10^{-4}$
50 kg$\cdot$yr |  $1.4\cdot 10^{-7}$ / $4.1\cdot 10^{-4}$ |  $1.4\cdot 10^{-7}$ / $3.7\cdot 10^{-4}$ |  $1.3\cdot 10^{-7}$ / $3.3\cdot 10^{-4}$
500 kg$\cdot$yr |  $9.6\cdot 10^{-8}$ / $3.4\cdot 10^{-4}$ |  $9.4\cdot 10^{-8}$ / $3.3\cdot 10^{-4}$ |  $8.9\cdot 10^{-8}$ / $3.2\cdot 10^{-4}$

Table 4: Exemplary exclusion limits at 90% C.L. for a light U(1)B-L vector
boson. Values are given for regions where the experimental reach is dominated
by E$\nu$eS ($m_{Z^{\prime}}=1$ keV) and CE$\nu$NS ($m_{Z^{\prime}}=1$ GeV),
respectively.

The corresponding error bands at 90% confidence level (C.L.) and relative
uncertainties are given in table 2. In general, results from Asimov data sets
and MC-sampled mock data agree well. As expected, the precision of the weak
mixing angle increases with lower detection thresholds and increased
experimental exposure. While future experiments will determine the weak mixing
angle at a level of $\lesssim\mathcal{O}(10)$ %, underlying systematic
uncertainties and the overall background level start to become more relevant
for larger exposures and lower detection thresholds. Hence, a detailed
investigation of systematic uncertainties and their subsequent reduction will
become crucial for future precision CE$\nu$NS experiments.

###  4.3 Sensitivity to new light vector mediators

Next, we present the expected sensitivity of future Ge-based reactor
experiments to BSM physics in terms of potentially existing new light
particles. In this investigation, we simultaneously study BSM physics
modifications to both CE$\nu$NS and E$\nu$eS as induced by the Lagrangian in
eq. (6). Thus, both channels are included in a combined likelihood function
that is fit to Asimov/MC-sampled mock data. Furthermore, the weak mixing angle
is allowed to vary within the incorporated uncertainties, cf. section 3, while
limits are determined for the vector coupling $g_{Z^{\prime}}$ for a fixed
boson mass $m_{Z^{\prime}}$. By repeating these fits, we can explore future
sensitivities to the parameter space that is spanned by the model parameters
by simultaneously allowing for potential changes in the weak mixing angle
within current experimental uncertainties. As a natural reference, we compare
the expected sensitivities to CE$\nu$NS and E$\nu$eS limits obtained by the
Conus experiments (after their conversion to our conventions) CONUS:2021dwh .

#### Universally coupled light vector boson

The expected limits for the case of a universally coupled light vector boson
are depicted in figure 4 with exemplary values given in table 3. The
degenerate region of parameter space, i.e. the parameter configuration that is
indistinguishable from the SM, is indicated by the grey bands. Due to a
lighter target mass, the obtained bounds in the keV-mass region are dominated
by the electron channel, whereas CE$\nu$NS sets exclusion limits for the
higher (GeV-)mass region. Comparing to the respective Conus limits, i.e.,
$5.2\cdot 10^{-7}$ for E$\nu$eS-only ($m_{Z^{\prime}}=1$ keV), and $1.8\cdot
10^{-3}$ for CE$\nu$NS-only ($m_{Z^{\prime}}=1$ GeV), future experimental
efforts promise an improvement of roughly one order of magnitude.

#### Light gauged B$-$L vector boson

Our results for the case of the U(1)B-L model are shown in figure 5 with
benchmark values given in table 4. Since the corresponding cross-section in
eq. (10) is well defined, the model’s parameter space can be directly
accessed. The different charge assignment compared to the previous model leads
to a modification of the cross section by a factor of $\sim 3$ and, thus,
slightly weaker bounds. Besides that, the expected sensitivities are similar
to the previous case. The corresponding benchmark limits from Conus are
$5.2\cdot 10^{-7}$ for E$\nu$eS-only ($m_{Z^{\prime}}=1$ keV) and $2.3\cdot
10^{-3}$ for CE$\nu$NS-only ($m_{Z^{\prime}}=1$ GeV), respectively.

Expected limits on the parameter space and corresponding benchmark points for
both light vector models in the case of an extended high energy region, i.e.
up to 100 keV, are given in appendix A.

Finally, we emphasise the complementarity between experiments using reactor
antineutrinos and those using (anti-)neutrinos from pion-decays-at-rest. The
huge antineutrino flux of nuclear reactors allows for strong restriction of
the models’ parameter spaces, especially via E$\nu$eS. The lower momentum
transfer of reactor antineutrino also renders CE$\nu$NS more sensitive to
lower mediator masses. On the other hand, higher (anti-)neutrino energies, as
provided by $\pi$DAR sources, allow the probe of heavier particles and give
them a higher sensitivity for the high mass region of the model studied, cf.
right plot in figure 6. Thus, by combining data from both sources, the
parameter space of new light vector particles can be efficiently probed with
CE$\nu$NS detectors.

##  5 Conclusions

In this paper, the potential of future germanium-based reactor experiments was
investigated, using the weak mixing angle $\sin^{2}\theta_{W}$ and new light
vector bosons as examples. In particular, the expected large signal of
CE$\nu$NS renders such experiments promising tools for future SM and BSM
investigations since it allows for “car-size” neutrino detectors close to the
strongest artificial neutrino sources on Earth. Although multiple
confirmations of successful CE$\nu$NS detection are still lacking at a reactor
site, this channel is currently probed with the full repertoire of modern
detection technologies. This study focused on the potential of germanium
semiconductor detectors as these are currently leading the field at reactor
sites. The enormous antineutrino flux further allows obtaining competitive
bounds using the electrons of the detector as target material via E$\nu$eS.

We investigated the expected sensitivity of potential set-ups for selected
detection thresholds and experimental exposure, cf. table 1, using the SM
interactions channels CE$\nu$ES and E$\nu$eS. For these selected
configurations, realistic detector efficiency and resolution are considered as
well as systematic uncertainties related to reactor antineutrinos and signal
quenching are taken into account. The signal expectations of CE$\nu$NS and
E$\nu$eS have been estimated and the impact of key experimental parameters
like detector mass and detection threshold are quantified using the expected
event rate of CE$\nu$NS as an example, cf. figure 2. In doing so, the
potential of future SM and BSM investigations has been highlighted.

Further, we obtained that the relative uncertainty of the weak mixing angle is
expected to improve to $\lesssim 10$ %, cf. figure 3 and table 2. Our results,
along with some of the current measurements, are summarised in the left-panel
of figure 6. We find that experiments like Conus\+ Ackermann:2024kxo  are
expected to have a better sensitivity to the weak mixing angle, at energies of
around 10 MeV, than the current measurements by the Coherent and the Dresden
experiments.

For the light vector models, bounds on the coupling strength $g_{Z^{\prime}}$
strengthen by roughly one order of magnitude, depending on the explicit
experimental configuration, cf. figure 4 and table 3 for the universally
coupled model as well as figure 5 and table 4 for the U(1)B-L model. As a
demonstration, we show the sensitivity of a Conus\+ resembling experiment to
the B$-$L model in the right panel of figure 6 for two different
configurations of the experimental parameters that seem realisable in the near
future. For comparison, the corresponding bounds from the Coherent
collaboration are also shown.

Figure 6: Summary of the investigations in this work. Left: Expected precision
to the weak mixing angle $\sin^{2}\theta_{W}$ in comparison to existing
measurements at different energy scales. Right: Expected sensitivity to the
parameter space of a gauged B$-$L vector boson in comparison to existing
limits from CE$\nu$NS experiments at a $\pi$DAR source and a nuclear reactor.

We found that limits on the E$\nu$eS ($m_{Z^{\prime}}=1$ keV) region are
stronger driven by increasing exposure due to the signal’s flatness. On the
contrary, limits in the CE$\nu$NS ($m_{Z^{\prime}}=1$ GeV) region benefit
stronger from an improved detection threshold. In general, the combination of
both SM channels in one analysis has proven to be a very efficient way of
accessing the available parameter space. Incorporating data from $\pi$DAR
CE$\nu$NS experiments will improve bounds for higher mediators masses. Thus,
exploiting this reactor-$\pi$DAR complementarity in future investigations is a
sufficient way of constraining the full parameter space.

This study further underlines the large potential of high-purity point-contact
germanium detectors, also in critical, non-laboratory environments. Ongoing
improvements in detector resolution, detection threshold and detection
efficiency in the sub-keV region will increase the expected CE$\nu$NS events
rates and consequently the reach to any potentially existing BSM physics
related to it. In addition, the scaling capability of this detection
technology further allows to linearly increase the signal rates. In that
context, background and environmental stability are key to guarantee this
initial potential.

With more small-scale technologies showing their capability of measuring
CE$\nu$NS both close to reactors and at $\pi$DAR sources, neutrino physics
enters a high-statistics phase when their full detector and scaling potential
is exploited. Consequently, SM and BSM reaches of CE$\nu$NS experiments will
start to be dominated by the systematic (experimental) uncertainties which
call for a stronger consideration in future sensitivity studies. Since the
current generation of dark matter direct detection experiments is already
beginning to probe the neutrino fog, e.g. $\sim 2$ cnts/(0.6 t$\cdot$yr) from
CE$\nu$NS of solar 8B neutrinos are expected in the Xenon1T experiment
XENON:2020gfr , the sun will become available as additional neutrino source in
future CE$\nu$NS studies. In summary, the prospects for future CE$\nu$NS
investigations are promising and there will be a huge playground for future
phenomenological studies, especially if experimenters and theorists work
closely together.

###### Acknowledgements.

The authors would like to thank the Conus collaboration for important insights
into current developments of germanium semiconductor detectors. Special thanks
are given to Aurélie Bonhomme, Janina Hakenmüller and Edgar Sanchez Garcia for
useful discussions.

##  Appendix A Limits on new light vector bosons with an extended high energy
range

In the following, we summarise results on the parameter space of the
investigated light vector models with an increased energy region. For all
plots and tables below, the upper end of energy region was increased from 10
keV to 100 keV. As a result, the effect of E$\nu$eS on the expected
sensitivity increases, and the limit contours and the given reference points
change slightly. Furthermore, individual model characteristics become more
apparent: A gauged B$-$L model only contributes additional events in both
CE$\nu$NS and E$\nu$sS, while there is some interplay between both channels in
the case of a universally coupled light mediators as CE$\nu$NS events are also
reduced in a certain region of parameter space.

###  A.1 Universally coupled light vector boson

Figure 7: Expected exclusion limits at 90 % C.L. on a universally coupled
light vector mediator for an extended energy region ($E_{I}\leq 100$ keV) and
different exposures and detector thresholds. Background levels and detector
thresholds are given in the boxes, while the exposures under study are
illustrated with different colouring. Exemplary limits from MC-sampled data
represented by crosses to validate our results. For comparison Conus limits
from their first BSM investigation are indicated in orange (E$\nu$eS-only) and
purple (CE$\nu$NS-only) CONUS:2021dwh . The degeneracy region for CE$\nu$NS,
where the BSM model cannot be distinguished from the pure SM interaction, is
marked with grey-dotted lines.

Exposure / $E_{\mathrm{thr}}$ | 180 eV | 150 eV | 100 eV
---|---|---|---
5 kg$\cdot$yr |  $8.0\cdot 10^{-8}$ / $3.0\cdot 10^{-4}$ |  $7.7\cdot 10^{-8}$ / $2.8\cdot 10^{-4}$ |  $7.2\cdot 10^{-8}$ / $2.2\cdot 10^{-4}$
50 kg$\cdot$yr |  $6.3\cdot 10^{-8}$ / $2.7\cdot 10^{-4}$ |  $6.2\cdot 10^{-8}$ / $2.5\cdot 10^{-4}$ |  $6.1\cdot 10^{-8}$ / $2.6\cdot 10^{-4}$
500 kg$\cdot$yr |  $5.7\cdot 10^{-8}$ / $2.9\cdot 10^{-4}$ |  $5.6\cdot 10^{-8}$ / $3.1\cdot 10^{-4}$ |  $5.5\cdot 10^{-8}$ / $3.5\cdot 10^{-4}$

Table 5: Exemplary exclusion limits at 90% C.L. for a universally coupled
light vector boson for an extended energy range ($E_{I}\leq 100$ keV). Values
are given for regions where the experimental reach is dominated by E$\nu$eS
($m_{Z^{\prime}}=1$ keV) and CE$\nu$NS ($m_{Z^{\prime}}=1$ GeV), respectively.

###  A.2 Light gauged B$-$L vector boson

Figure 8: Expected exclusion limits at 90 % C.L. on a light gauged B$-$L
vector mediator for an extended energy region ($E_{I}\leq 100$ keV) and
different exposures and detector thresholds. Background levels and detector
thresholds are given in the boxes, while the exposures under study are
illustrated with different colouring. Exemplary limits from MC-sampled data
represented by crosses to validate our results. For comparison Conus limits
from their first BSM investigation are indicated in orange (E$\nu$eS-only) and
purple (CE$\nu$NS-only) CONUS:2021dwh . In contrast to the previous case, the
U(1)B-L does not exhibit any degeneracy region.

Exposure / $E_{\mathrm{thr}}$ | 180 eV | 150 eV | 100 eV
---|---|---|---
5 kg$\cdot$yr |  $8.0\cdot 10^{-8}$ / $5.4\cdot 10^{-4}$ |  $7.7\cdot 10^{-8}$ / $4.8\cdot 10^{-4}$ |  $7.2\cdot 10^{-8}$ / $3.5\cdot 10^{-4}$
50 kg$\cdot$yr |  $6.3\cdot 10^{-8}$ / $3.9\cdot 10^{-4}$ |  $6.7\cdot 10^{-8}$ / $3.1\cdot 10^{-4}$ |  $6.1\cdot 10^{-8}$ / $2.1\cdot 10^{-4}$
500 kg$\cdot$yr |  $5.7\cdot 10^{-8}$ / $1.3\cdot 10^{-4}$ |  $5.6\cdot 10^{-8}$ / $1.0\cdot 10^{-4}$ |  $5.5\cdot 10^{-8}$ / $6.8\cdot 10^{-5}$

Table 6: Exemplary exclusion limits at 90% C.L. for a light U(1)B-L vector
boson for an extended energy range ($E_{I}\leq 100$ keV). Values are given for
regions where the experimental reach is dominated by E$\nu$eS
($m_{Z^{\prime}}=1$ keV) and CE$\nu$NS ($m_{Z^{\prime}}=1$ GeV), respectively.

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