aainstitutetext: Centre for Cosmology, Particle Physics and Phenomenology
(CP3), Université catholique de Louvain, Chemin du Cyclotron 2, B-1348
Louvain-la-Neuve, Belgiumbbinstitutetext: Department of Physics and McDonnell
Center for the Space Sciences, Washington University,
St. Louis, MO 63130, USAccinstitutetext: Institute of High Energy Physics,
Chinese Academy of Sciences, Beijing 100049, China

# Probing the $\nu_{R}$-philic $Z^{\prime}$ at DUNE near detectors

Garv Chauhan b P. S. Bhupal Dev c Xun-Jie Xu<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We consider a hidden $U(1)$ gauge symmetry under which only the right-handed
neutrinos ($\nu_{R}$) are charged. The corresponding gauge boson is referred
to as the $\nu_{R}$-philic $Z^{\prime}$. Despite the absence of direct gauge
couplings to ordinary matter at tree level, loop-induced couplings of the
$\nu_{R}$-philic $Z^{\prime}$ via left-right neutrino mixing can be
responsible for its experimental accessibility. An important feature of the
$\nu_{R}$-philic $Z^{\prime}$ is that its couplings to neutrinos are generally
much larger than its couplings to charged leptons and quarks, thus providing a
particularly interesting scenario for future neutrino experiments such as DUNE
to probe. We consider two approaches to probe the $\nu_{R}$-philic
$Z^{\prime}$ at DUNE near detectors via (i) searching for $Z^{\prime}$ decay
signals, and (ii) precision measurement of elastic neutrino-electron
scattering mediated by the $Z^{\prime}$ boson. We show that the former will
have sensitivity comparable to or better than previous beam dump experiments,
while the latter will improve current limits substantially for large neutrino
couplings.

##  1 Introduction

The discovery of neutrino oscillations has established non-zero neutrino
masses on a firm footing ParticleDataGroup:2020ssz . It also clearly points
towards the presence of Beyond the Standard Model (BSM) physics. There has
been monumental progress to understand the standard $3\times 3$ neutrino
mixing paradigm by measuring the mixing parameters with ever-increasing
accuracy but we are yet to understand the neutrino mass mechanism. There have
been several proposals in the literature (for reviews, see e.g. Refs.
Mohapatra:2006gs ; deGouvea:2016qpx ), which always come with new interactions
of neutrinos. Some of these ideas might be testable with the upcoming high-
intensity frontier experiments in the near future DUNE:2020fgq .

In this work, we consider a BSM scenario with a hidden $U(1)$ gauge symmetry
under which only the right-handed neutrinos ($\nu_{R}$) are charged. These
right-handed neutrinos interact with SM particles only through their mixing
with the active neutrinos. As a result, the new gauge boson $Z^{\prime}$
associated with the hidden $U(1)$ symmetry can interact with the SM particles
only through the left-right neutrino mixing and through $W^{\pm}/Z$-loop
induced couplings (see Fig. 1). A noteworthy feature of the $\nu_{R}$-philic
$Z^{\prime}$ is that its couplings to neutrinos (including the light ones via
mixing) are generically larger than its loop-induced couplings to electrons.
This is directly useful for probing this scenario at the intensity frontier
neutrino experiments like DUNE DUNE:2020ypp . Therefore, unlike most
$Z^{\prime}$ models for which there are already severe constraints from beam
dump experiments and $e^{+}e^{-}$ colliders, and hence, little space for DUNE
to probe, the $\nu_{R}$-philic $Z^{\prime}$ presents significantly better
prospect in neutrino experiments. In our case, due to stronger coupling to
neutrinos than quarks, $Z^{\prime}$ is primarily produced through charged pion
decays instead of neutral pion decays. This is in stark contrast to the
typical $Z^{\prime}$ models in literature, e.g. $B-L$ model, where neutral
pion decay is the dominant production mode for light $Z^{\prime}$
Berryman:2019dme ; Dev:2021qjj .

In this work, we study the detection prospects for the $\nu_{R}$-phillic
$Z^{\prime}$ model at the near detector complex of DUNE DUNE:2021tad  in two
different ways: (i) elastic $\nu$-$e$ scattering, and (ii) using DUNE as a
beam dump experiment. The process of elastic $\nu$-$e$ scattering has been
widely used to constrain new interactions of neutrinos in the literature
Bilmis:2015lja ; Lindner:2018kjo ; Bischer:2018zcz ; Ballett:2019xoj ;
Link:2019pbm ; Dev:2021xzd ; Chakraborty:2021apc , due to its rather small
theoretical uncertainties compared to the neutrino-nucleus scattering,
although the neutrino-electron scattering cross section is much smaller than
the neutrino-nucleus scattering cross section.111Since the successful
observation of coherent elastic neutrino-nucleus scattering (CE$\nu$NS)
COHERENT:2017ipa , neutrino-nucleus scattering has also been used to constrain
new physics, see e.g. Lindner:2016wff ; Dent:2017mpr ; Farzan:2018gtr ;
Abdullah:2018ykz ; Brdar:2018qqj ; Cadeddu:2018dux . Nevertheless, a large
neutrino flux at DUNE offers a significant statistical advantage to probe BSM
physics at the near detector. In addition to neutrino scattering, the
$Z^{\prime}$ can be directly produced from proton bremsstrahlung and decay of
charged/neutral mesons produced from proton beam striking the target. In this
way, DUNE can also be modeled as a beam dump experiment Berryman:2019dme .

A number of studies have shown that beam dump experiments in general have
great capabilities to constrain $Z^{\prime}$ with very weak couplings,
provided that its visible decay width is not too small Bjorken:2009mm ;
Andreas:2012mt ; Ilten:2018crw ; Bauer:2018onh ; Coy:2021wfs . For the
$\nu_{R}$-philic $Z^{\prime}$, the comparatively large couplings to neutrinos
would suppress the visible decay width but enhance the production rate. Both
aspects will be taken into account in this work.

The rest of paper is organized as follows: In Sec. 2, we discuss the framework
for the neutrinophilic scenario considered in this work. In Sec. 3, we study
prospects for this scenario at DUNE through neutrino-electron scattering. In
Sec. 4, we analyze the role of DUNE as a beam dump experiment in constraining
the neutrinophilic scenario. In Sec. 5, we discuss the implications of the
combined analysis from neutrino scattering and beam dump. Finally we conclude
in Sec. 6. Some details on the $Z^{\prime}$ production via proton
bremsstahlung are relegated to Appendix A.

##  2 Model Framework

We consider a $Z^{\prime}$ vector boson that is coupled to $n$ right-handed
neutrinos ($\nu_{R}$) as follows:222Following the same notations as in Ref.
Chauhan:2020mgv , we adopt two-component Weyl spinors for all chiral fermions.

${\cal L}\supset
g^{\prime}Z_{\mu}^{\prime}\sum_{i=1}^{n}\nu_{R,i}^{\dagger}\overline{\sigma}^{\mu}Q_{R,i}\nu_{R,i}\thinspace,$
(1)

where $g^{\prime}$ is the gauge coupling and $Q_{R,i}$ is the $U(1)^{\prime}$
charge of $\nu_{R,i}$. For brevity, we will rewrite Eq. (1) in the matrix
form:

${\cal L}\supset
g^{\prime}Z_{\mu}^{\prime}\nu_{R}^{\dagger}\overline{\sigma}^{\mu}Q_{R}\nu_{R}\thinspace,$
(2)

with $Q_{R}={\rm diag}(Q_{R,1},\ Q_{R,2},\ Q_{R,3},\cdots)$ and
$\nu_{R}=(\nu_{R,1},\ \nu_{R,2},\ \nu_{R,3},\cdots)^{T}$.

If $\nu_{R}$ are the only fermions charged under $U(1)^{\prime}$, the
cancellation of chiral anomalies requires $\sum_{i}Q_{R,i}^{3}=0$. Therefore,
one often introduces pairs of $\nu_{R}$ with opposite charges to attain the
cancellation Chang:2011kv ; Ma:2013yga ; Lindner:2013awa . Alternatively, one
may consider other charge assignment such as $Q_{R}=(1,\ 1,\ 1,\ -4,\ -4,\ 5)$
or $(3,\ 4,\ 5,\ -6)$ which also satisfies $\sum_{i}Q_{R,i}^{3}=0$. We refrain
from further discussions on the charge assignment as this can be rather model-
dependent while the results in this work can be simply rescaled by a factor of
${\cal O}(Q_{R})$.

The $\nu_{R}$ sector is connected to left-handed neutrinos $\nu_{L}$ via Dirac
mass terms. In addition, $\nu_{R}$ may have Majorana mass terms. Formulated in
the matrix form, the Dirac and Majorana mass terms are given by

${\cal
L}\supset\nu_{L}^{T}m_{D}\nu_{R}+\frac{1}{2}\nu_{R}^{T}m_{R}\nu_{R}+{\rm
h.c.}\thinspace,$ (3)

where $m_{D}$ is a $3\times n$ matrix and $m_{R}$ is a $n\times n$ matrix.

###  2.1 Loop-induced coupling

In the presence of the $Z^{\prime}$ couplings given by Eq. (2) and the
neutrino mass terms in Eq. (3), there are loop-induced couplings of
$Z^{\prime}$ to the SM charged leptons ($\ell$) and quarks ($u$ and $d$)
generated by the diagrams in Fig. 1. The loop-induced couplings are finite
(i.e. not UV divergent) when both light and heavy neutrino mass eigenstates
are included in the loop integral.

Figure 1: Loop-induced couplings of the $\nu_{R}$-philic $Z^{\prime}$ to
charged leptons ($\ell$) and quarks ($u$ and $d$). The $\times$ denotes the
Dirac mass insertion.

For illustration, let us first discuss the case with only one $\nu_{L}$ and
one $\nu_{R}$ so that $m_{D}$ and $m_{R}$ are $1\times 1$ matrices. In this
case, the loop-induced couplings read Chauhan:2020mgv :

${\cal L}_{{\rm
loop}}=\sum_{f}g_{f}f^{\dagger}\overline{\sigma}^{\mu}Z_{\mu}^{\prime}f\thinspace,$
(4)

where $f$ denotes SM chiral fermions and $g_{f}$ is determined as follows:

$g_{f}=\begin{cases}g_{f}^{(W)}+g_{f}^{(Z)}&f=\ell_{L}\\\ g_{f}^{(Z)}&{\rm
otherwise}\\\ \end{cases}\thinspace,$ (5)

with

$\displaystyle g_{f}^{(W)}$
$\displaystyle=-\frac{\sqrt{2}G_{F}m_{D}^{2}}{8\pi^{2}}g^{\prime}Q_{R}\thinspace,$
(6) $\displaystyle g_{f}^{(Z)}$
$\displaystyle=\frac{\sqrt{2}G_{F}m_{D}^{2}}{8\pi^{2}}g^{\prime}Q_{R}Q_{f}^{(Z)}\thinspace.$
(7)

Here the superscripts $(W)$ and $(Z)$ indicate contributions from the $W$ and
$Z$ diagrams in Fig. 1, respectively. The result of the $Z$ diagram depends on
the so-called $Z$ charge in the SM, defined as $Q_{f}^{(Z)}\equiv
I_{3}-Q_{{\rm em}}s_{W}^{2}$ with $I_{3}$ the isospin and $Q_{{\rm em}}$ the
electric charge (e.g. $Q_{\nu_{L}}^{(Z)}=1/2$,
$Q_{e_{L}}^{(Z)}=-1/2+s_{W}^{2}$, $Q_{e_{R}}^{(Z)}=s_{W}^{2}$), and
$s_{W}\equiv\sin\theta_{W}$, where $\theta_{W}$ is the weak mixing angle.

Eqs. (6) and (7) are derived under the approximation that $m_{D}/m_{R}\ll 1$.
The results are proportional to $m_{D}^{2}$ which can be understood from Fig.
1 since each diagram needs two mass insertions to connect $\nu_{L}$ and
$\nu_{R}$ lines. If $Z^{\prime}$ is replaced with a scalar, then the diagrams
would be proportional to $m_{R}^{-1}$, leading to more suppressed loop-induced
couplings Xu:2020qek .

For $3\nu_{L}+n\nu_{R}$, the loop-induced couplings are also finite and of the
same order of magnitude as Eqs. (6) and (7). However, with the most general
mass matrices, the loop-induced couplings cannot be written into compact and
simple forms—see Eqs. (3.8) and (3.9) in Ref. Chauhan:2020mgv . Besides, a
large number of free parameters in the flavor structure make it more difficult
to obtain simple correlations among the loop-induced couplings. For
simplicity, we concentrate on the case of $n=3$ and assume that $m_{D}$ and
$m_{R}$ can be simultaneously diagonalized as follows:333 That $m_{D}$ and
$m_{R}$ can be simultaneously diagonalized is a rather common feature in
flavor symmetry models—see e.g. Refs. Rodejohann:2015hka ; Rodejohann:2017lre
; Smirnov:2018luj .

$\displaystyle U_{L}^{T}m_{D}U_{R}$ $\displaystyle=m_{D}^{(d)}\thinspace,$ (8)
$\displaystyle U_{R}^{T}m_{R}U_{R}$ $\displaystyle=m_{R}^{(d)}\thinspace,$ (9)

where $U_{L}$ and $U_{R}$ are $3\times 3$ unitary matrices, $m_{D}^{(d)}$ and
$m_{R}^{(d)}$ are diagonal matrices.

The full neutrino mass matrix can be diagonalized by a $6\times 6$ unitary
matrix $U$:

$\displaystyle\left(\begin{array}[]{c}\nu_{L}\\\ \nu_{R}\end{array}\right)$
$\displaystyle=U\left(\begin{array}[]{c}\nu_{1,2,3}\\\
\nu_{4,5,6}\end{array}\right),$ (14) $\displaystyle
U^{T}\left(\begin{array}[]{cc}0&m_{D}\\\ m_{D}^{T}&m_{R}\end{array}\right)U$
$\displaystyle={\rm diag}(m_{1},\ \cdots,m_{6})\thinspace.$ (17)

Eqs. (8) and (9) allow us to write $U$ as

$U=\left(\begin{array}[]{cc}U_{L}\\\
&U_{R}\end{array}\right)U^{\prime}\thinspace,\ \
U^{\prime}=\left(\begin{array}[]{cc}-i\thinspace{\rm
diag}(c_{1},c_{2},c_{3})&{\rm diag}(s_{1},s_{2},s_{3})\\\ i\thinspace{\rm
diag}(s_{1},s_{2},s_{3})&{\rm diag}(c_{1},c_{2},c_{3})\end{array}\right),$
(18)

where $(s_{i},\ c_{i})=(\sin\theta_{i},\cos\theta_{i})$ and
$\theta_{i}=\arctan\sqrt{m_{i}/m_{i+3}}$.

With the above assumption, the loop-induced couplings previously computed in
Ref. Chauhan:2020mgv  can be reformulated into the following matrix form:

$\displaystyle g_{f}^{(W)}$
$\displaystyle=-\frac{\sqrt{2}G_{F}\left[m_{D}\hat{Q}_{R}m_{D}^{\dagger}\right]_{\alpha\beta}}{8\pi^{2}}g^{\prime}\thinspace,$
(19) $\displaystyle g_{f}^{(Z)}$
$\displaystyle=\frac{\sqrt{2}G_{F}\thinspace{\rm
tr}\left[m_{D}\hat{Q}_{R}m_{D}^{\dagger}\right]}{8\pi^{2}}g^{\prime}Q_{f}^{(Z)}\thinspace,$
(20)

where $\hat{Q}_{R}\equiv U_{R}{\rm
diag}(Q_{R,1},Q_{R,2},Q_{R,3})U_{R}^{\dagger}$ and $(\alpha,\ \beta)\in\\{e,\
\mu,\ \tau\\}$ are flavor indices. Eq. (19) implies that flavor-changing
couplings could be generated by the $W$ diagram. It can be flavor diagonal if
the charge assignments $(Q_{R,1},Q_{R,2},Q_{R,3})$ and $m_{D}^{(d)}$ satisfy

$\left[m_{D}^{(d)}\right]^{2}{\rm diag}(Q_{R,1},Q_{R,2},Q_{R,3})\propto
I_{{\rm 3\times 3}}\thinspace.$ (21)

In this case, we define

$\displaystyle\overline{m}_{D}^{2}$ $\displaystyle\equiv\frac{1}{3}{\rm
tr}\left\\{\left[m_{D}^{(d)}\right]^{2}{\rm
diag}(Q_{R,1},Q_{R,2},Q_{R,3})\right\\},$ (22) $\displaystyle\epsilon_{{\rm
loop}}$
$\displaystyle\equiv\frac{\sqrt{2}G_{F}\overline{m}_{D}^{2}}{8\pi^{2}}\thinspace,$
(23)

and rewrite Eqs. (19) and (20) as

$\displaystyle g_{f}^{(W)}$ $\displaystyle=-\epsilon_{{\rm loop}}\thinspace
g^{\prime}\thinspace,$ (24) $\displaystyle g_{f}^{(Z)}$
$\displaystyle=3\thinspace\epsilon_{{\rm loop}}\thinspace
g^{\prime}Q_{f}^{(Z)}\thinspace.$ (25)

###  2.2 Neutrino flavor states at low energies

Neutrinos in most experiments are produced via charged-current interactions.
Since $\nu_{L}$ are the eigenstates of such interactions, one might expect
that neutrinos at production are simply $\nu_{L}$. However, due to a small
fraction of heavy neutrino components in $\nu_{L}$, only a quantum
superposition of light mass eigenstates ($\nu_{1}$, $\nu_{2}$, $\nu_{3}$) can
be produced if the heavy mass eigenstates are heavier than the masses of the
particles responsible for neutrino production (typically $\pi^{\pm}$,
$K^{\pm},$ $\mu^{\pm}$, etc.). This implies that a neutrino state produced in
the laboratory is slightly different from $\nu_{L}$. To account for the
difference, we define neutrino flavor states at low energies as follows:

$\nu_{\alpha}^{{\rm lab}}=\sum_{i=1}^{3}\left(U_{L}\right)_{\alpha
i}\nu_{i}\thinspace,\ \alpha\in\\{e,\mu,\tau\\}\thinspace.$ (26)

Further, we introduce a set of orthogonal states $\nu_{\alpha}^{\perp{\rm
lab}}=\sum_{i=4}^{6}\left(U_{R}\right)_{\alpha i}\nu_{i}$ to complete the
basis so that $(\nu^{{\rm lab}},\ \nu^{\perp{\rm lab}})$ and $(\nu_{L},\
\nu_{R})$ are connected by a unitary transformation:

$\left(\begin{array}[]{c}\nu_{L}\\\
\nu_{R}\end{array}\right)=\left(\begin{array}[]{cc}-iC_{L}&S\\\
iS^{\dagger}&C_{R}\end{array}\right)\left(\begin{array}[]{c}\nu^{{\rm lab}}\\\
\nu^{\perp{\rm lab}}\end{array}\right),$ (27)

where

$\displaystyle C_{L,R}$ $\displaystyle=U_{L,R}{\rm
diag}(c_{1},c_{2},c_{3})U_{L,R}^{\dagger}\thinspace,$ (28) $\displaystyle S$
$\displaystyle=U_{L}{\rm diag}(s_{1},s_{2},s_{3})U_{R}^{\dagger}\thinspace.$
(29)

Since $c_{i}\approx 1$ and $s_{i}\sim m_{D}/m_{R}\ll 1$, the unitary matrix in
Eq. (27) performs only a small rotation from $(\nu^{{\rm lab}},\
\nu^{\perp{\rm lab}})$ to $(\nu_{L},\ \nu_{R})$. In the basis of $(\nu_{L},\
\nu_{R})$, the $Z^{\prime}$ couplings to the neutrino sector (including loop-
induced couplings to $\nu_{L}$) read:

${\cal L}\supset g^{\prime}\left(\nu_{L},\
\nu_{R}\right)\left(\begin{array}[]{cc}\frac{3}{2}\epsilon_{{\rm loop}}I_{{\rm
3\times 3}}\\\ &Q_{R}\end{array}\right)\left(\begin{array}[]{c}\nu_{L}\\\
\nu_{R}\end{array}\right)\thinspace.$ (30)

Applying the basis transformation in Eq. (27), we obtain the effective
couplings of $Z^{\prime}$ to $\nu^{{\rm lab}}$:

${\cal L}_{Z^{\prime},\nu^{{\rm
lab}}}=g^{\prime}Z_{\mu}^{\prime}\left(\nu^{{\rm
lab}}\right)^{\dagger}\overline{\sigma}^{\mu}\left[\frac{3}{2}\epsilon_{{\rm
loop}}C_{L}C_{L}^{\dagger}+SQ_{R}S^{\dagger}\right]\nu^{{\rm lab}}\thinspace.$
(31)

###  2.3 Effective $Z^{\prime}$ couplings

Let us summarize the effective couplings of $Z^{\prime}$ to normal matter and
neutrinos:

$\displaystyle{\cal L}$
$\displaystyle\supset\overline{\psi_{q}}\left[g_{qL}\gamma^{\mu}P_{L}+g_{qR}\gamma^{\mu}P_{R}\right]Z^{\prime}_{\mu}\psi_{q}$
$\displaystyle+\overline{\psi_{e}}\left[g_{eL}\gamma^{\mu}P_{L}+g_{eR}\gamma^{\mu}P_{R}\right]Z^{\prime}_{\mu}\psi_{e}+\overline{\psi_{\nu}}\left[g_{\nu}\gamma^{\mu}P_{L}\right]Z^{\prime}_{\mu}\psi_{\nu}\thinspace,$
(32)

where $q=u$ or $d$ and we use $\psi$ to denote four-component Dirac spinors.
For neutrinos, the left-handed part of $\psi_{\nu}$ is the neutrino state
produced in the laboratory, i.e. $P_{L}\psi_{\nu}=\nu^{{\rm lab}}$ where
$\nu^{{\rm lab}}$ has been introduced in Sec. 2.2. Combining the previous
results, we have the following couplings:

$\displaystyle g_{\nu}$
$\displaystyle=g^{\prime}\left[\frac{3}{2}\epsilon_{{\rm
loop}}C_{L}C_{L}^{\dagger}+SQ_{R}S^{\dagger}\right],$ (33)
$\displaystyle\left(g_{e_{L}},\ g_{e_{R}}\right)$
$\displaystyle=3\epsilon_{{\rm loop}}g^{\prime}\left(s_{W}^{2}-5/6,\
s_{W}^{2}\right),$ (34) $\displaystyle\left(g_{u_{L}},\ g_{u_{R}}\right)$
$\displaystyle=\epsilon_{{\rm loop}}g^{\prime}\left(1/2-2s_{W}^{2},\
-2s_{W}^{2}\right),$ (35) $\displaystyle\left(g_{d_{L}},\ g_{d_{R}}\right)$
$\displaystyle=\epsilon_{{\rm loop}}g^{\prime}\left(s_{W}^{2}-5/2,\
s_{W}^{2}\right).$ (36)

For later convenience, we also define

$g_{e}\equiv\sqrt{g_{e_{L}}^{2}+g_{e_{R}}^{2}}\thinspace,\
g_{q}\equiv\sqrt{g_{q_{L}}^{2}+g_{q_{R}}^{2}}\thinspace,$ (37)

and

$r\equiv\frac{g_{\nu}}{g_{e}}\thinspace.$ (38)

Note that all effective couplings are of the order of $\epsilon_{{\rm
loop}}g^{\prime}$ except for the mixing-induced part $SQ_{R}S^{\dagger}$ which
is of ${\cal O}(\sin^{2}\theta_{i})$ where the mixing angles $\theta_{i}$ are
defined below Eq. (18). According to Refs. deGouvea:2015euy ; Bolton:2019pcu ,
for heavy neutrino masses above the electroweak scale, the current constraints
on $\sin^{2}\theta_{i}$ typically vary from $10^{-3}$ for $10^{-2}$, depending
on lepton flavors. This allows for the scenario that the mixing-induced part
dominates over the loop-induced part ($\epsilon_{{\rm
loop}}C_{L}C_{L}^{\dagger}$) in $g_{\nu}$. More concretely, if we consider
$m_{R}\sim G_{F}^{-1/2}$, $\sin^{2}\theta_{i}\sim m_{D}^{2}/m_{R}^{2}$ should
be of the same order as $G_{F}m_{D}^{2}$, which is about a factor of
$8\pi^{2}\sim{\cal O}(10^{2})$ larger than the loop-induced part
$\epsilon_{{\rm loop}}C_{L}C_{L}^{\dagger}$. In this case, one expects that
$g_{\nu}\sim{\cal O}(10^{2})g_{e}$. For larger $m_{R}$ with fixed $m_{D}$, the
mixing-induced part decreases and eventually looses its dominance, leading to
$g_{\nu}\sim g_{e}$. Taking $m_{R}^{2}\in[1,\ 10^{2}]\thinspace G_{F}^{-1}$
and $\sin^{2}\theta_{i}=10^{-3}$ for example (neglecting the flavor
structure), we roughly have $m_{D}^{2}\in[10^{-3},\ 10^{-1}]\thinspace
G_{F}^{-1}$ and $\epsilon_{{\rm loop}}\in[1.8\times 10^{-5},\ 1.8\times
10^{-3}].$ Therefore, as a benchmark in this work, we can consider

$C_{L}C_{L}^{\dagger}\rightarrow 1\thinspace,\ SQ_{R}S^{\dagger}\rightarrow
10^{-3}\thinspace,\ \epsilon_{{\rm loop}}\in[10^{-5},\ 10^{-3}]\thinspace.$
(39)

##  3 DUNE as a neutrino scattering experiment

Having formulated the effective couplings of $Z^{\prime}$ to neutrinos and
electrons, we now study the sensitivity of the DUNE near detector to the
signal of elastic neutrino-electron scattering444Neutrino trident scattering
could also be used to probe $Z^{\prime}$. However, for our model the
sensitivity is weaker than elastic neutrino-electron scattering, as can be
expected from Fig. 7 of Ref. Ballett:2019xoj . caused by $Z^{\prime}$.

For a general $Z^{\prime}$ with the effective couplings given by Eq. (32), the
differential cross section of elastic neutrino-electron scattering including
both the SM and the new physics contributions reads Lindner:2018kjo ;
Link:2019pbm ; Marshall:2019vdy :

$\frac{d\sigma}{dT}=\frac{2m_{e}G_{F}^{2}}{\pi}\left[c_{L}^{2}+c_{R}^{2}\left(1-\frac{T}{E_{\nu}}\right)^{2}-c_{L}c_{R}\frac{m_{e}T}{E_{\nu}^{2}}\right],$
(40)

where

$\displaystyle c_{L}$ $\displaystyle=c_{L}^{({\rm
SM})}+\frac{g_{eL}g_{\nu}}{2\sqrt{2}G_{F}\left(2m_{e}T+m_{Z^{\prime}}^{2}\right)}\thinspace,\
\ c_{L}^{({\rm SM})}=-\frac{1}{2}+s_{W}^{2}+\delta_{\alpha e}\thinspace,$ (41)
$\displaystyle c_{R}$ $\displaystyle=c_{R}^{({\rm
SM})}+\frac{g_{eR}g_{\nu}}{2\sqrt{2}G_{F}\left(2m_{e}T+m_{Z^{\prime}}^{2}\right)}\thinspace,\
\ c_{R}^{({\rm SM})}=s_{W}^{2}\thinspace.$ (42)

Here $T$ denotes the recoil energy of the electron; $c_{L}$ and $c_{R}$ are
dimensionless quantities consisting of the SM and the new physics
contributions, as given by Eqs. (41) and (42). We have added $\delta_{\alpha
e}$ (where $\alpha$ denotes the the incoming neutrino flavor) in Eq. (41) to
take account of the SM charged current interaction: $\delta_{\alpha e}=1$ for
$\alpha=e$ and $\delta_{\alpha e}=0$ for $\alpha=\mu$ or $\tau$. Note that
$c_{R}^{({\rm SM})}$ is not affected by the presence of the charged current
contribution. For anti-neutrino scattering ($\overline{\nu}_{\alpha}+e$), Eq.
(40) should be modified with the interchange $c_{L}\leftrightarrow c_{R}$
while Eqs. (41) and (42) remain the same. In this work, we neglect flavor
transition in neutrino scattering since the contribution is suppressed due to
the loss of interference with the SM processes.

The event rate of elastic neutrino-electron scattering at the detector is
computed by Lindner:2018kjo ; Link:2019pbm :

$\frac{dN}{dT}=N_{e}\lambda_{{\rm POT}}\int\Phi(E_{\nu})\frac{d\sigma(T,\
E_{\nu})}{dT}\Theta(T_{\max}-T)dE_{\nu}\thinspace,$ (43)

where $N_{e}$ is the total electron number in the fiducial mass of the
detector; $\lambda_{{\rm POT}}$ is the number of protons-on-target (POT) at
the neutrino production facility; $\Phi$ is the neutrino flux per POT;
$\Theta$ is the Heaviside theta function with the maximal recoil energy
$T_{\max}$ determined by

$T_{\max}(E_{\nu})=\frac{2E_{\nu}^{2}}{m_{e}+2E_{\nu}}\thinspace.$ (44)

The event number in a given recoil energy bin $T\in[T_{i},T_{i}+\Delta T]$ is
computed as

$N_{i}=\int_{T_{i}}^{T_{i}+\Delta T}\frac{dN}{dT}dT\thinspace.$ (45)

The electron number $N_{e}$ can be computed as $N_{e}=18M_{{\rm Ar}}/m_{{\rm
Ar}}$ where the factor $18$ comes from the fact that each ${\rm Ar}$ atom has
18 electrons, $M_{{\rm Ar}}$ is the fiducial detector mass and $m_{{\rm
Ar}}=39.95u=37.21\ {\rm GeV}$ is the atomic mass of 40Ar. For the fiducial
mass, we consider only the liquid argon time-projection chamber (LArTPC) near
detector which has a fiducial mass of $M_{{\rm Ar}}=67.2$ ton DUNE:2020fgq .
For the number of POT, we take $\lambda_{{\rm POT}}=5\times 1.1\times 10^{21}$
for each of the neutrino and anti-neutrino modes, corresponding to the Long
Baseline Neutrino Facility (LBNF) at Fermilab operating at 1.2 MW intensity
for five years in each mode. The neutrino flux $\Phi$ can be obtained from the
latest DUNE Technical Design Report555See Fig. 5.4 in Ref. DUNE:2020lwj , Fig.
2.15 in Ref. DUNE:2021tad , and Fig. 2 in Ref. DUNE:2021cuw ..

With the above setup, we compute the event numbers666Our code is publicly
available at https://github.com/xunjiexu/Zprime-at-DUNE.  and perform the
sensitivity study with the following $\chi^{2}$ function previously adopted in
Ref. Lindner:2016wff :

$\chi^{2}=\left(\frac{a}{\sigma_{a}}\right)^{2}+\sum_{i}\left(\frac{(1+a)N_{i}-N_{i}^{0}}{\sigma_{i}}\right)^{2},$
(46)

where $a$ is a scale factor introduced to take into account the normalization
uncertainty (we assume that the dominant systematic uncertainty arises from
the normalization of event rates); $\sigma_{a}$ denotes the uncertainty of $a$
for which we set $\sigma_{a}=2\%$; the summation goes over all $T$ bins;
$N_{i}$ denotes the event number evaluated from Eq. (45) with $N_{i}^{0}$ the
corresponding SM value; and $\sigma_{i}\approx\sqrt{N_{i}}$ denotes the
statistical uncertainty on $N_{i}$. Here $a$ is treated as a nuisance
parameter which in the frequentist treatment ParticleDataGroup:2020ssz  is
marginalized by minimizing $\chi^{2}$ with respect to $a$. For Eq. (46), the
value of $a$ at the minimum, $a_{\min}$, can be computed analytically
Lindner:2016wff :

$a_{\min}=\frac{\sum_{i}(N_{i}^{0}-N_{i})N_{i}/\sigma_{i}^{2}}{\sigma_{a}^{-2}+\sum_{i}N_{i}^{2}/\sigma_{i}^{2}}\thinspace.$
(47)

We set the $T$ bins as follows: the interval $T\in[0,\ 8]\ {\rm GeV}$ is
divided evenly to 5 bins, and events above $8$ GeV are collected in a single
bin. Each bin has a sufficiently large event number so that the use of
$\sigma_{i}\approx\sqrt{N_{i}}$ is justified. The $\chi^{2}$ functions for
neutrino and anti-neutrino modes are computed separately and then combined.

Figure 2:  The events numbers for elastic neutrino-electron scattering at DUNE
near detector LArTPC. Depending on the angle $\beta$ defined in Eq. (48), a
generic $Z^{\prime}$ can lead to an excess or deficit in the event numbers.

Figure 3: Left: the sensitivity of DUNE as a neutrino scattering experiment to
a generic $Z^{\prime}$ for several selected values of $\beta$ defined in Eq.
(48). Right: the dependence of the results on $\beta$.

Fig. 2 shows the expected event numbers computed in the SM and in the new
physics scenario of a generic $Z^{\prime}$. We parametrize the ratio of
$g_{e_{L}}$ to $g_{e_{R}}$ as

$(g_{e_{L}},\ g_{e_{R}})=(\cos\beta,\ \sin\beta)g_{e}\thinspace.$ (48)

In our model, the couplings in Eq. (34) correspond to $\beta=159^{\circ}$.
This leads to excesses of event numbers in both neutrino and anti-neutrino
modes, as shown by the orange points in Fig. 2. For comparison, we also show
an example with $\beta=45^{\circ}$ (corresponding to
$g_{e_{L}}:g_{e_{R}}=1:1$) which could cause excesses in the anti-neutrino
mode but deficits in the neutrino mode. The two new physics examples assume
$m_{Z^{\prime}}=100$ MeV and $\sqrt{g_{e}g_{\nu}}=10^{-4}$.

In the left panel of Fig. 3, we present the 90% C.L. sensitivity reach of
DUNE-LArTPC for a few selected values of $\beta$. As one can see, the curves
nearly overlap for $m_{Z^{\prime}}\gtrsim 100$ MeV and show small differences
for $m_{Z^{\prime}}\lesssim 10$ MeV. To further check whether varying $\beta$
could cause significant changes, in the right panel we investigate the
dependence of the results on $\beta$ with three representative values of
$m_{Z^{\prime}}$ and find that the sensitivity reach varies at most by a
factor of $\sqrt{g_{e}g_{\nu}}/g_{0}\in[0.91,1.44]$ for $\beta\in[0,\ \pi]$,
where $g_{0}$ is defined as the value of $\sqrt{g_{e}g_{\nu}}$ at $\beta=0$.
For larger $m_{Z^{\prime}}$, the results are more insensitive to $\beta$. Fig.
3 can be considered as a model-independent analysis for $Z^{\prime}$ with
rather general couplings to left-/right-handed electrons and neutrinos.

##  4 DUNE as a beam dump experiment

At the neutrino production site of DUNE, $Z^{\prime}$ can be produced from the
proton beam striking the target. Due to its weak couplings to SM fermions, the
produced $Z^{\prime}$ boson can be long-lived and penetrate through the
shielding and earth between the near detector and the target. This possibility
allows us to consider DUNE as a beam dump experiment (like E137, E141, Orsay)
to probe $Z^{\prime}$ Berryman:2019dme ; Dev:2021qjj .

We consider the DUNE Multi-Purpose Detector (MPD) which is a High-Pressure Ar
gas TPC (HPgTPC)777Compared to LArTPC, HPgTPC has the advantage of lower
(roughly 50 times smaller) backgrounds. Hence it is more suitable for new
physics searches that do not require interactions with the content filled in
the detector. for the detection of $Z^{\prime}$ in this work. The DUNE MPD
will be located $579$ m away from the target, with a diameter of 5 m and a
length of 5 m Berryman:2019dme . Therefore, for $Z^{\prime}$ particles
produced at the target to cause observable signals at the DUNE MPD, the
transverse momentum, $p_{\perp}$, compared to the $Z^{\prime}$ momentum,
$p_{Z^{\prime}}$, needs to be sufficiently small:

$\frac{p_{\perp}}{p_{Z^{\prime}}}<\frac{R}{L_{1}}\approx 4.3\times
10^{-3}\equiv\theta_{{\rm max}}\thinspace,$ (49)

where $R=2.5$ m is the radius of the detector and $L_{1}=579$ m is the
distance of the detector to the target.

Figure 4: The number of $Z^{\prime}$ bosons produced via meson decays
($\pi^{0}\rightarrow Z^{\prime}\gamma$, $\eta\rightarrow Z^{\prime}\gamma$,
$\pi^{\pm}\rightarrow\ell^{\pm}\nu Z^{\prime}$,
$K^{\pm}\rightarrow\ell^{\pm}\nu Z^{\prime}$) and proton bremsstrahlung
($pp\rightarrow ppZ^{\prime}$) assuming $r=100$ and $1.47\times 10^{22}$ POT
from a 120 GeV proton beam.

There are several processes responsible for the production of $Z^{\prime}$ in
our model, namely meson decays ($\pi^{0}\rightarrow Z^{\prime}\gamma$,
$\eta\rightarrow Z^{\prime}\gamma$, $\pi^{\pm}\rightarrow\ell^{\pm}\nu
Z^{\prime}$, $K^{\pm}\rightarrow\ell^{\pm}\nu Z^{\prime}$) and proton
bremsstrahlung ($pp\rightarrow ppZ^{\prime}$). The latter can be computed
using an approximate formula in Ref. Blumlein:2013cua —see Appendix A for a
brief review. The former requires the production rates of relevant mesons,
which can be obtained from Monte-Carlo simulations of high-energy protons
scattering off nucleons. We adopt the production rates from Ref.
Berryman:2019dme  (neutral meson results from Fig. 4.1, charged meson results
from Fig. 5.2) and rescale them according to the effective couplings of
$Z^{\prime}$ in our model.

Fig. 4 shows the number of $Z^{\prime}$ bosons produced (denoted as $N_{{\rm
prod.}}$) via the aforementioned processes for $r=100$, within the small angle
determined by Eq. (49). Since in this case $Z^{\prime}$ is dominantly coupled
to neutrinos, in the low-mass regime the dominant production processes are
those with neutrino final states ($\pi^{\pm}\rightarrow\ell^{\pm}\nu
Z^{\prime}$, $K^{\pm}\rightarrow\ell^{\pm}\nu Z^{\prime}$). Note, however,
that these three-body decay processes are more suppressed in the phase space
than neutral meson decays ($\pi^{0}\rightarrow Z^{\prime}\gamma$,
$\eta\rightarrow Z^{\prime}\gamma$). Hence if all the couplings are of the
same order of magnitude, neutral meson decays would dominate over charged
meson decays Dev:2021qjj . When $m_{Z^{\prime}}$ is greater than the meson
masses, proton bremsstrahlung starts to dominate. In particular, the blue
curve in Fig. 4 peaks at $m_{Z^{\prime}}\approx 0.8$ GeV due to the resonance
caused by the $\rho$/$\omega$ meson deNiverville:2016rqh ; Faessler:2009tn .

The number of detectable events, $N_{{\rm det.}}$, can be computed by
Coy:2021wfs

$N_{{\rm det.}}=\int dp_{Z^{\prime}}\frac{dN_{{\rm
prod.}}(p_{Z^{\prime}})}{dp_{Z^{\prime}}}\thinspace P_{{\rm
decay}}(p_{Z^{\prime}})\thinspace{\rm BR}_{Z^{\prime}\rightarrow{\rm
vis.}}\thinspace,$ (50)

where ${\rm BR}_{Z^{\prime}\rightarrow{\rm vis.}}$ denotes the branching ratio
of $Z^{\prime}$ decaying to visible states and $P_{{\rm decay}}$ denotes the
probability that a single $Z^{\prime}$ particle travels from the target to the
detector and then decays in the detector. The momentum distribution of
$Z^{\prime}$ at the production, $dN_{{\rm prod.}}/dp_{Z^{\prime}}$, includes
contributions from meson decay processes and proton bremsstrahlung. The latter
is elaborated in Appendix A and the former can be obtained by combining the
kinematics of meson decays at rest and Lorentz boosts according to the meson
momentum distributions taken from Fig. 4 in Ref. Kopp:2006ky . The probability
$P_{{\rm decay}}$ is computed by Coy:2021wfs

$P_{{\rm
decay}}=e^{-L_{1}/L_{Z^{\prime}}}\left(1-e^{-L_{2}/L_{Z^{\prime}}}\right),$
(51)

with $L_{1}=579$ m, $L_{2}=5$ m, and $L_{Z^{\prime}}$ denotes the distance of
flight before $Z^{\prime}$ decay. The physical meaning of Eq. (51) is
manifest: it is the probability of $Z^{\prime}$ not decaying in $L_{1}$
multiplied by the probability of $Z^{\prime}$ decaying in $L_{2}$. The
distance of flight, $L_{Z^{\prime}}$, is computed by

$L_{Z^{\prime}}=\frac{\tau_{Z^{\prime}}v}{\sqrt{1-v^{2}}}\thinspace,$ (52)

where $1/\sqrt{1-v^{2}}$ is the Lorentz boost factor with the velocity
$v=p_{Z^{\prime}}/\sqrt{m_{Z^{\prime}}^{2}+p_{Z^{\prime}}^{2}}$. The lifetime
$\tau_{Z^{\prime}}$ takes into account all decay modes:

$\tau_{Z^{\prime}}^{-1}=\Gamma_{Z^{\prime}}=\Gamma_{Z^{\prime}\rightarrow{\rm
had.}}+\sum_{f\neq q}\Gamma_{Z^{\prime}\rightarrow f\overline{f}}\thinspace,$
(53)

where $\Gamma_{Z^{\prime}\rightarrow{\rm had.}}$ denotes the decay width of
$Z^{\prime}$ to hadronic states. The decay widths to other states
$\Gamma_{Z^{\prime}\rightarrow f\overline{f}}$ ($f\neq q$) are computed
analytically (see e.g. Ref. Blumlein:2013cua ):

$\Gamma_{Z^{\prime}\rightarrow
f\overline{f}}=\frac{g_{f}^{2}}{12\pi}m_{Z^{\prime}}\left(1+2\frac{m_{f}^{2}}{m_{Z^{\prime}}^{2}}\right)\sqrt{1-\frac{4m_{f}^{2}}{m_{Z^{\prime}}^{2}}}\thinspace.$
(54)

The hadronic decay width, $\Gamma_{Z^{\prime}\rightarrow{\rm had.}}$, can be
computed using experimental measurements of the hadron-to-muon cross section
ratio in $e^{+}e^{-}$ collisions. For $\Gamma_{Z^{\prime}\rightarrow{\rm
had.}}$, we adopt the same method previously used in Ref. Coy:2021wfs . The
visible decay width in Eq. (50) takes the form

${\rm BR}_{Z^{\prime}\rightarrow{\rm vis.}}\equiv
1-\frac{\Gamma_{Z^{\prime}\rightarrow{\rm\nu\overline{\nu}}}}{\Gamma_{Z^{\prime}}}\,.$
(55)

The effective coupling to neutrinos, $g_{\nu}$, plays a particularly important
role here because it reduces the visible decay width of $Z^{\prime}$. When
$g_{\nu}\gg g_{e}$ and $g_{q}$, we have ${\rm BR}_{Z^{\prime}\rightarrow{\rm
vis.}}\ll 1$ and hence a significantly suppressed event rate.

Figure 5: The sensitivity reach of DUNE MPD to the $\nu_{R}$-philic
$Z^{\prime}$ with loop-induced couplings. The results depend on the ratio $r$
defined in Eq. (38).

Using Eq. (50) with each ingredient explicated above, we compute the
sensitivity reach of DUNE MPD for our model. The results are presented in Fig.
5 with $r=g_{\nu}/g_{e}$ fixed at several benchmark values $(1,\ 5,\ 10,\
100)$. As expected, when $r$ increases, due to the aforementioned suppression
of the visible decay width, the DUNE MPD sensitivity becomes weaker.

##  5 Combined results and discussions

Figure 6: Combined results of the DUNE sensitivity to the $\nu_{R}$-philic
$Z^{\prime}$. All bounds depends on the ratio $r\equiv g_{\nu}/g_{e}=\\{1,\
5,\ 10,\ 10^{2}\\}$.

In this section, we compare the estimated sensitivities of DUNE-LArTPC (via
$\nu$-$e$ scattering) and DUNE-MPD (which could detect $Z^{\prime}$ decay)
with existing bounds. We focus on the mass range $1\ {\rm MeV}\leq
m_{Z^{\prime}}\leq 100\ {\rm GeV}$ where the DUNE experiment shows competitive
sensitivities.

For masses below 100 GeV, direct searches for $Z^{\prime}$ at BaBar, LHCb, and
NA48 put the strongest collider bounds. For masses ranging from MeV to GeV,
there are constraints from beam dump experiments such as E137, E141, and
Orsay. For these existing collider and beam dump bounds, we use the DARKCAST
package Ilten:2018crw  to recast the published bounds and obtain the shaded
exclusion regions in Fig. 6.

Since $Z^{\prime}$ is effectively coupled to charged leptons, the muon and
electron anomalous magnetic moments can be used to set constraints, which are
presented in Fig. 6, being labelled as $(g-2)_{\mu}$ and $(g-2)_{e}$,
respectively. These curves are obtained by setting $\Delta\alpha_{\mu}\leq
5.5\times 10^{-9}$ (5$\sigma$) Muong-2:2021ojo  and $\Delta\alpha_{e}\leq
13.8\times 10^{-13}$ (3$\sigma$) Morel:2020dww  where $\Delta\alpha_{\ell}$
($\ell=\mu$, $e$) is computed by Leveille:1977rc ; Coy:2021wfs

$\Delta\alpha_{\ell}=\frac{g_{\ell}^{2}}{4\pi^{2}}\left(\frac{m_{\ell}}{m_{Z^{\prime}}}\right)^{2}\int_{0}^{1}\frac{(1-x)x^{2}}{1-x+x^{2}(m_{\ell}/m_{Z^{\prime}})^{2}}dx\thinspace.$
(56)

There are also strong constraints on low-mass $Z^{\prime}$ from astrophysics
and cosmology. The most relevant one comes from the supernova event SN1987A.
Studies on SN1987A energy loss criteria have put restrictive constraints on
weakly coupled $Z^{\prime}$ below ${\cal O}(100)$ MeV Dent:2012mx ;
Rrapaj:2015wgs ; Chang:2016ntp . In Fig. 6, we adopt the combined SN1987A
bound from Ref. Knapen:2017xzo  for the $B-L$ gauge boson and rescale it
accordingly. Near the MeV mass scale or lower, there are constraints from
stellar cooling Redondo:2013lna  and the effective number of relativistic
degrees of freedom ($N_{\rm eff}$) Escudero:2019gzq . We do not show the
$N_{\rm eff}$ constraints as they are marginally relevant to the mass range
considered here.

From Fig. 6, we see that the ratio $r\equiv g_{\nu}/g_{e}$ is crucial when
comparing bounds from DUNE and the existing experiments. When $g_{\nu}$ and
$g_{e}$ are comparable ($r=1$), neutrino-electron scattering at DUNE-LArTPC
(dotted line) does not show a significant advantage over other experiments.
Nevertheless, there is a mass range from 10 MeV to 200 MeV for DUNE-LArTPC to
probe. When $r$ increases, both the DUNE-LArTPC sensitivity and collider
bounds become weaker, since they all depend on $g_{e}$ and/or $g_{q}$ which is
decreasing, given fixed mixing angles and hence fixed $g_{\nu}/g^{\prime}$.
However, as one can see, for large $r$, DUNE-LArTPC shows significantly better
sensitivity than other non-neutrino experiments because the former relies on
the product $g_{e}g_{\nu}$ while the latter typically relies on $g_{e}^{2}$ or
$g_{q}^{2}$. We emphasize here that the main feature of our framework is the
dominance of neutrino couplings in the $\nu_{R}$-philic $Z^{\prime}$ model,
which provides an excellent prospect for DUNE to probe.

As previously discussed in Sec. 4, DUNE-MPD can be used to probe $Z^{\prime}$
like a beam dump experiment. The sensitivity reach is shown as dashed curves
in Fig. 6. In this case, $g_{\nu}$ plays a positive role in production (it
leads to additional contributions of $Z^{\prime}$ production from meson decays
with neutrino final states), but a negative role in detection because it
increases the invisible decay width of $Z^{\prime}$. In particular, for $r\gg
1$, $Z^{\prime}$ dominantly decays to neutrinos. Overall, large $g_{\nu}$
reduces the sensitivity of DUNE-MPD. Therefore, when $r$ increases, DUNE-MPD
together with all other beam dump experiments quickly looses sensitivity to
the weak-coupling regime ($g^{\prime}\sim 10^{-4},\ 10^{-5}$). Consequently,
the DUNE-MPD sensitivity region shrinks quickly as we increase $r$ in Fig. 6.

##  6 Conclusion

Hidden $U(1)$ symmetries in the right-handed neutrino ($\nu_{R}$) sector are
theoretically well motivated and would give rise to an inherently dark gauge
boson which we refer to as the $\nu_{R}$-philic $Z^{\prime}$. Due to the loop-
suppressed couplings to normal matter and comparatively larger couplings to
neutrinos, neutrino experiments are the most suited to probe the
$\nu_{R}$-philic $Z^{\prime}$.

In this work we studied the sensitivity of the future DUNE experiment to the
$\nu_{R}$-philic $Z^{\prime}$. We considered two complementary near detectors,
DUNE-LArTPC and DUNE-MPD (HPgTPC), which could be sensitive to $Z^{\prime}$
signals via elastic $\nu$-$e$ scattering and via $Z^{\prime}$ decay
respectively. We stress here that the ratio of electron and neutrino
couplings, $r=g_{\nu}/g_{e}$, which is practically a free parameter in the
model, plays a crucial role. Larger neutrino couplings lead to higher elastic
$\nu$-$e$ scattering rates in DUNE-LArTPC but make $Z^{\prime}$ decay less
visible in DUNE-MPD due to the enhanced invisible decay width. The combined
results are shown in Fig. 6 and compared with existing bounds. For $r=1$ or
$5$, DUNE-MPD exhibits a significant advantage over other beam dump
experiments in the mass range $0.1{\rm GeV}\lesssim m_{Z^{\prime}}\lesssim
1{\rm GeV}$. On the other hand, for larger $r$ such as $r=10$ or $100$, DUNE-
LArTPC as a scattering experiment will be able to generate the leading
constraints, exceeding the collider bounds from BaBar, LHCb, etc. We conclude
that there is an excellent prospect of DUNE probing new physics hidden in the
sector of right-handed neutrinos.

###### Acknowledgements.

We thank Vedran Brdar, Matheus Hostert, Kevin Kelly and Doojin Kim for helpful
discussions and assistance. The work of B.D. is supported in part by the U.S.
Department of Energy under grant No. DE-SC0017987 and by a Fermilab Intensity
Frontier Fellowship. B.D. acknowledges the local hospitality at KITP where
part of this work was done. The research at KITP was supported in part by the
National Science Foundation under Grant No. NSF PHY-1748958. X.J.X is
supported in part by the National Natural Science Foundation of China under
grant No. 12141501.

##  Appendix A  The proton bremsstrahlung formula

For $Z^{\prime}$ emitted via proton bremsstrahlung ($pN\rightarrow
pNZ^{\prime}$), there is an approximate formula often used in the literature
Blumlein:2013cua ; deNiverville:2016rqh ; Berryman:2019dme ; SHiP:2020vbd
888We note that there is a discrepancy between the expressions of $w_{ba}$ in
Refs. Blumlein:2013cua ; SHiP:2020vbd  and Refs. deNiverville:2016rqh ;
Berryman:2019dme . We adopt the former since this is consistent with the
matrix element given by Eq. (6) in Ref. Blumlein:2013cua . :

$\frac{d^{2}N_{{\rm brem}}}{dzdp_{\perp}^{2}}=N_{{\rm
POT}}\frac{\sigma_{pN}(s^{\prime})}{\sigma_{pN}(s)}\left|F_{1,N}(m_{Z^{\prime}}^{2})\right|^{2}w_{ba}(z,\thinspace
p_{\perp}^{2})\thinspace,$ (57)

with

$\displaystyle w_{ba}(z,\thinspace p_{\perp}^{2})$
$\displaystyle=\frac{g_{p}^{2}}{8\pi^{2}H}\left[\frac{1+(1-z)^{2}}{z}-2z(1-z)\left(\frac{2m_{p}^{2}+m_{Z^{\prime}}^{2}}{H}-z^{2}\frac{2m_{p}^{4}}{H^{2}}\right)\right.$
$\displaystyle\left.\quad+2z(1-z)\left[1+(1-z)^{2}\right]\frac{m_{Z^{\prime}}^{2}m_{p}^{2}}{H^{2}}+2z(1-z)^{2}\frac{m_{Z^{\prime}}^{4}}{H^{2}}\right],$
(58) $\displaystyle H$ $\displaystyle\equiv
p_{\perp}^{2}+(1-z)m_{Z^{\prime}}^{2}+z^{2}m_{p}^{2}\thinspace.$ (59)

Here the kinematic variable $z$ is the ratio of the outgoing $Z^{\prime}$
momentum to the incoming proton momentum ($0<z<1$); $p_{\perp}$ is the
transverse component of the $Z^{\prime}$ momentum; $\sigma_{pN}$ is the
proton-nucleus cross section evaluated at $s=2m_{p}E_{p}$ and
$s^{\prime}\approx 2m_{p}(E_{p}-E_{Z^{\prime}})$;
$F_{1,N}(m_{Z^{\prime}}^{2})$ is the vector form factor of the nucleus
evaluated at $q^{2}=m_{Z^{\prime}}^{2}$ (timelike); and $g_{p}$ is the
effective coupling of $Z^{\prime}$ to the proton. Since the beam energy is
high, one can approximately view the target as free protons and neutrons at
rest. According to the simulations in Ref. Berryman:2019dme , the difference
between neutron and proton being the target particle is negligible at this
energy, which allows us to replace $\sigma_{pN}$ with the proton-proton cross
section, $\sigma_{pp}$, and replace $F_{1,N}$ with the corresponding form
factor of the proton, $F_{1,p}$. The cross section $\sigma_{pp}$ is computed
via an approximate formula from Ref. Blumlein:2013cua , and the form factor
$F_{1,p}$ is taken from Ref. deNiverville:2016rqh .

Due to the small angle in Eq. (49) which implies $p_{\perp}^{2}\ll H$, we can
integrate out $p_{\perp}^{2}$ in Eq. (57), which leads to

$\frac{dN_{{\rm brem}}}{dz}=N_{{\rm
POT}}\frac{\sigma_{pN}(s^{\prime})}{\sigma_{pN}(s)}\left|F_{1,N}(m_{Z^{\prime}}^{2})\right|^{2}w_{ba}(z)\thinspace,$
(60)

where

$w_{ba}(z)=\int_{0}^{E_{p}^{2}\theta_{{\rm max}}^{2}}w_{ba}(z,\thinspace
p_{\perp}^{2})dp_{\perp}^{2}=\frac{g_{p}^{2}\theta_{{\rm
max}}^{2}E_{p}^{2}}{8\pi^{2}}\frac{z\left(z^{2}-2z+2\right)}{\left[m_{p}^{2}z^{2}+m_{Z^{\prime}}^{2}(1-z)\right]}\thinspace.$
(61)

Note that, though $w_{ba}$ in Eq. (58) diverges at $z\rightarrow 0$, the
integrated form of $w_{ba}$ in Eq. (61) at $z\rightarrow 0$ has been
regularized by $m_{Z^{\prime}}^{2}$, i.e. $\lim_{z\rightarrow 0}w_{ba}(z)=0$
as long as $m_{Z^{\prime}}^{2}\neq 0$.

The proton bremsstrahlung formula presented above is valid only if the
incoming and outgoing protons as well as the $Z^{\prime}$ boson are ultra-
relativistic Blumlein:2013cua . Typically in the literature one imposes upper
and lower bounds on $z$, $z_{\min}<z<z_{\max}$. For instance, previous studies
on the SHiP experiment often take $(z_{\min},\ z_{\max})=(0.1,\ 0.9)$
deNiverville:2016rqh ; SHiP:2020vbd . For the DUNE setup, we set
$z_{\min}=5m_{Z^{\prime}}/E_{p}$ so that the outgoing $Z^{\prime}$ is
relativistic. If $z$ is very close to $1$, the outgoing proton would be non-
relativistic. Hence we set the upper bound $z_{\max}=1-5m_{p}/E_{p}$. The
factor $5$ here is chosen so that the relativistic approximation holds and
meanwhile the width of the integration interval ($z_{{\rm
max}}-z_{\min}\approx 1$) is not significantly reduced. We have checked that
varying this factor does not change the result significantly, provided that
the two conditions are satisfied.

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this prevents information linkage as these positional features take effect
after prediction. In addition, since temporal events repeat (in the same
order) at each epoch, we also speed up PINT’s training procedure by
precomputing and saving the positional features for each batch. To save up
space, we store the positional features as sparse matrices.

Table S3: Optimal hyperparameters for PINT. Dataset | $\beta/|\mathcal{N}|$ |  #Neighbors ($|\mathcal{N}|$) | #Layers
---|---|---|---
Reddit | $10^{-5}$ | $10$ | $2$
Wikipedia | $10^{-4}$ | $10$ | $2$
Twitter | $10^{-5}$ | $20$ | $2$
UCI | $10^{-5}$ | $10$ | $2$
Enron | $10^{-5}$ | $20$ | $2$
LastFM | $10^{-4}$ | $10$ | $2$

#### Hardware.

For all experiments, we use Tesla V100 GPU cards and consider a memory budget
of 32GB of RAM.

##  Appendix E Deletion and node-level events

Rossi et al. [27] propose handling edge deletions by simply updating memory
states of the edge’s endpoints, as if we were dealing with a usual edge
addition. However, it is not discussed whether the edge in question should be
excluded from the event list or if we should just add a novel event with edge
features that characterize deletion. If we choose the former, we may be unable
to recover the memory state of a node from its monotone TCT and the original
node features. Removing an edge from the event list also affects the
computation of node embeddings. Therefore, we advise practitioners to do the
latter when using PINT. It is worth mentioning that the vast majority of
models for temporal interaction prediction do not consider the possibility of
deletion events.

Regarding node-level events, PINT can accommodate node addition by simply
creating novel memory states. To deal with node feature updates, we can create
an edge event with both endpoints on that node, inducing a self-loop in the
dynamic graph. Also, we can combine (e.g., concatenate) the temporal features
in message-passage operations, similarly to the general formulation of the MP-
TGN framework [27]. Finally, we can deal with the removal of a node $v$ by
following our previous (edge deletion) procedure to delete all edges with
endpoints in $v$.

##  Appendix F Additional experiments

#### Time comparison.

Figure S7 compares the time per epoch for PINT and for the prior art (CAW,
TGN-Att, and TGAT) in the Enron and LastFM datasets. Following the trend in
Figure 7, Figure S7 further supports that PINT is generally slower than other
MP-TGNs but, after a few training epochs, is orders of magnitude faster than
CAW. In the case of Enron, the time CAW takes to complete an epoch is much
higher than the time we need to preprocess PINT’s positional features.

Figure S7: Time comparison: PINT versus TGNs (in log-scale) on Enron and
LastFM.

#### Experiments on node classification.

For completeness, we also evaluate PINT on node-level tasks (Wikipedia and
Reddit). We follow closely the experimental setup in Rossi et al. [27] and
compare against the baselines therein. Table S4 shows that PINT ranks first on
Reddit and second on Wikipedia. The values for PINT reflect the outcome of $5$
repetitions.

Table S4: Results for node classification (AUC). | Wikipedia | Reddit
---|---|---
CTDNE | 75.89 $\pm$ 0.5 | 59.43 $\pm$ 0.6
JODIE | 84.84 $\pm$ 1.2 | 61.83 $\pm$ 2.7
TGAT | 83.69 $\pm$ 0.7 | 65.56 $\pm$ 0.7
DyRep | 84.59 $\pm$ 2.2 | 62.91 $\pm$ 2.4
TGN-Att | 87.81 $\pm$ 0.3 | 67.06 $\pm$ 0.9
PINT | 87.59 $\pm$ 0.6 | 67.31 $\pm$ 0.2

plainnat references