# Constraining the Active-to-Heavy-Neutrino transitional magnetic moments
associated with the $Z^{\prime}$ interactions at FASER$\nu$.

Kingman Cheunga,b,c, C.J. Ousepha,b a Department of Physics, National Tsing
Hua University, Hsinchu 30013, Taiwan
b Center for Theory and Computation, National Tsing Hua University, Hsinchu
30013, Taiwan
c Division of Quantum Phases and Devices, School of Physics, Konkuk
University, Seoul 143-701, Republic of Korea

(August 29, 2024)

###### Abstract

We investigate the effects of the transitional magnetic dipole moment of the
active-to-heavy-neutrino associated with a new neutral gauge boson
$Z^{\prime}$ on neutrino-nucleon scattering at the ForwArd Search
ExpeRiment-$\nu$ (FASER$\nu$). We consider the neutral-current neutrino-
nucleon scattering ($\nu A\rightarrow NA$) as the production mechanism of the
heavy neutrino $N$ at FASER$\nu$ and estimate the sensitivity reach on the
magnetic moment coupling $\mu_{\nu_{\alpha}}$ for a range of heavy neutrino
mass ($1\,{\rm GeV}<M_{N}<70{\rm GeV}$). In this study, we consider three
benchmark models, in which the heavy neutrino is coupled to $L_{e}$, $L_{\mu}$
or $L_{\tau}$ doublet, respectively.

##  I Introduction

Heavy neutrinos beyond the current known species of active neutrinos are
predicted in a number of grand unified theories (GUT), such as GUT based on
$SO(10)$, and are motivated as the natural explanation why the active
neutrinos are so light compared with other matter fermions. One of the natural
models is the seesaw model Yanagida:1979as ; Mohapatra:1979ia , in which the
heavy neutrinos are right-handed neutrinos of mass $O(10^{11-14})$ GeV, that
explains the order of active mass to be $y(vev)^{2}/M_{N}$, where the $vev\sim
10^{2}$ GeV is the electroweak vacuum expectation value. There are other
extensions Dev:2009aw ; Deppisch:2004fa  of the ideas of the seesaw mechanism
that predict intermediate mass neutrinos and can also be as light as a few
GeV’s. Here in this work we focus on the heavy neutrinos of mass in the range
of $1-70$ GeV.

FASER FASER:2019dxq ; Kling:2021gos ; FASER:2019aik ; FASER:2021ljd ;
FASER:2021cpr ; FASER:2020gpr ; FASER:2018bac ; Feng:2022inv  is one of the
neutrino-nucleon scattering experiments being approved and coming up very soon
in the next run of the LHC. It is located approximately 480 meters down the
beam direction from the ATLAS detector interaction point (IP). It is well
known that a large number of hadrons are produced along the beam direction,
such as pions, kaons, and other hadrons. During the flight, these hadrons will
decay thus producing neutrinos of all three flavors at very high energies up
to a few TeV. The neutrinos so produced can pass through hundred meters of
rock and concrete between the ATLAS IP and the FASER detector, whereas most
other particles will be either deflected or absorbed before reaching FASER.
FASER is unique in terms of the energy range of the neutrinos as beams
participating in $\nu-A$ scattering. It ranges from a hundred GeV to a few TeV
with the energy spectrum peaked at a few hundreds GeV. Other similar setups,
such as DUNE near detector, have lower energy ranges. It is an important
avenue of the energy frontier for neutrino physics, which is an area often
considered to be on the intensity frontier.

The FASER detector will be complemented with a new component called
FASER$\nu$FASER:2019dxq . It is a 25cm × 25cm × 1.35m emulsion detector,
consisting of 1000 layers of emulsion films interleaved with 1-mm-thick
tungsten plates with mass 1.2 tons. The main goal of FASER$\nu$ is to
distinguish various flavors of neutrinos. The FASER$\nu$ has the ability to
detect both charged-current(CC) FASER:2019dxq  and neutral-current (NC)
interactions Ismail:2020yqc ; Ansarifard:2021elw . The total cross-section
$\sigma_{\nu A}$ (where $A$ denote the target nucleus) of the neutrino
scattering at the FASER$\nu$ detector can be expressed as $\sigma_{\nu
A}=p\sigma_{\nu p}+n\sigma_{\nu n}$, where $\sigma_{\nu p}$ and $\sigma_{\nu
n}$ are the neutrino-proton and neutrino-neutron scattering cross sections,
and $p$ and $n$ are the number of protons and neutrons in the tungsten atom,
respectively. Various new physics scenarios at FASER and FASER$\nu$ detector
were reported in literatureJodlowski:2020vhr ; Feng:2022inv ; Feng:2017vli ;
Kling:2018wct ; Feng:2018pew ; Deppisch:2019kvs ; Cheung:2021tmx ;
Bakhti:2020szu ; Jho:2020jfz ; Okada:2020cue ; Bahraminasr:2020ssz ;
Kelly:2020pcy ; Falkowski:2021bkq ; FASER:2018eoc ; Cottin:2021lzz ;
Ismail:2021dyp .

In this work, we consider a scenario of a heavy neutrino $N$ with the presence
of a light $U(1)$ gauge boson $Z^{\prime}$Das:2022rbl ; Das:2019fee ;
Chiang:2019ajm . The heavy neutrino $N$ can couple to the active neutrinos via
a transitional magnetic dipole moment type interaction of the $Z^{\prime}$
boson: $\overline{N}\sigma^{\mu\nu}\nu_{\alpha}\,Z^{\prime}_{\mu\nu}$, where
$Z^{\prime}_{\mu\nu}$ is the field strength of the $Z^{\prime}$ boson. Such an
interaction enables sizable production of the heavy neutrinos at FASER$\nu$
and also detectable decays via $N\to\nu_{\alpha}Z^{\prime}$ followed by
$Z^{\prime}$ decays into standard model (SM) fermions. In literature, the
works that only considered the $N$ decays via the SM $W$ and $Z$ often
obtained very long decays certainly outside the detectors. The advantage of
including the $Z^{\prime}$ interaction with the heavy neutrinos is to enable
either prompt or detectable long decays within the FASER detector. Here the
$Z^{\prime}$ boson also couples to the SM fermions $q,\nu,l$. For completeness
we also include the couplings of $N$ to the SM charged leptons and neutrinos
via the $W$ and $Z$ bosons, respectively, although their contributions to
production and decays of $N$ are very small.

We identify the events of heavy neutrinos through its decays into a pair of
charged leptons, either in prompt or displaced decays, as well as into
hadrons. The smaller the coupling $\mu_{\nu_{\alpha}}$ the longer the decay
length will be. Thus, using the length of the whole FASER detector we can
estimate the range of $\mu_{\nu_{\alpha}}$ that FASER and FASER$\nu$ can
probe. The final result would be the sensitivity that FASER$\nu$ can reach in
this scenario.

Note that another scenario of sterile neutrino was considered in Ref.
Jodlowski:2020vhr , in which the sterile neutrino is produced through the
magnetic-dipole operator via the active neutrino up-scattering, mostly via the
electron scattering channel $\nu e^{-}\to Ne^{-}$. It was shown that
Jodlowski:2020vhr  the High-Luminosity LHC can search for sterile neutrinos
with an upgraded FASER$\nu$ experiment.

The organization of the work is arranged as follows. We describe the effective
interactions considered in the next section. In Sec. III, we calculate the
production rates of the heavy neutrino $N$ via $\nu A$ scattering at
FASER$\nu$. In Sec. IV, we calculate the number of events for the decays of
the heavy neutrino and then estimate the sensitivity reach at FASER$\nu$. We
conclude in Sec. V.

##  II The Heavy Neutrino Model

The new matter content includes a heavy neutrino $N$ and a $U(1)$ gauge boson
$Z^{\prime}$. The effective Lagrangian describing the interactions of $N$ and
$Z^{\prime}$ with the standard model (SM) gauge bosons Drewes:2015iva ;
Shrock:1978ft ; FMMF:1994yvb ; Pascoli:2018heg ; Alva:2014gxa ; Atre:2009rg
and other particles He:1991qd ; Hewett:1988xc ; Leike:1998wr ;
Langacker:2008yv ; Lee:2008zzl can be written as

$\displaystyle\mathcal{L}_{\rm eff}$ $\displaystyle=$
$\displaystyle\sum_{\alpha=e,\mu,\tau}\Biggr{[}\mu_{\nu_{\alpha}}\overline{N}\sigma^{\mu\nu}\nu_{\rm\alpha}Z^{\prime}_{\mu\nu}-\frac{g}{\sqrt{2}}V_{\alpha
N}\overline{N}\gamma^{\mu}P_{L}l_{\alpha}W_{\mu}^{+}-\frac{g}{\cos\theta_{w}}V_{\alpha
N}\overline{N}\gamma^{\mu}P_{L}\nu_{\alpha}Z_{\mu}+\rm H.c.\Biggr{]}$ (1)
$\displaystyle-$ $\displaystyle\sum_{q,\nu,l}\left[g_{\rm
q}\bar{q}\gamma^{\mu}q+g_{\nu}\bar{\nu}\gamma^{\mu}P_{L}\nu+g_{l}\bar{l}\gamma^{\mu}l\right]Z^{\prime}_{\mu}\;,$

where $g=e/\sin\theta_{w}$, $\theta_{w}$ is the Weinberg angle,
$Z^{\prime}_{\mu\nu}$ is the field strength of the $Z^{\prime}$ boson, the sum
over $q=u,d$, $\nu=\nu_{e},\nu_{\mu},\nu_{\tau}$ and $l=e,\mu,\tau$, and the
mixing parameter $V_{\alpha N}$ denotes the mixing between $\nu_{\alpha}$ and
$N$. Here $\mu_{\nu_{\alpha}}$ denotes the coupling of the active-to-heavy-
neutrino transitional magnetic dipole moment, for which the first term in the
Lagrangian Eq. (1) is valid up to a cut-off energy scale $\Lambda$. An
interpretation of $\mu_{\nu_{\alpha}}$ above the electroweak scale requires a
Higgs insertion so that the transitional magnetic dipole interaction described
in Eq. (1) originates from a dimension-6 operator. The $\mu_{\nu_{\alpha}}$’s
are therefore envisioned to be of the order of $1/\Lambda^{2}$ in form of
$\mu_{\nu_{\alpha}}\sim\frac{e.vev}{\Lambda^{2}}$.

The effective Lagrangian consists of two widely used beyond the SM models: (i)
the renormalizable $Z^{\prime}$ model with flavor-conserving quark, lepton and
neutrino interactions He:1991qd ; Hewett:1988xc ; Leike:1998wr ;
Langacker:2008yv ; Lee:2008zzl  and (ii) an effective/simplified extension by
introducing a right-handed (RH) neutrino, which is a singlet under the SM
gauge symmetryPascoli:2018heg ; Alva:2014gxa ; Atre:2009rg , with a
transitional dipole moment connecting the active and the heavy neutrino.
Without loss of generality, we assume that $q=u,d$ have equal coupling
strength $g_{q}$ with the $Z^{\prime}$, $\nu=\nu_{e},\nu_{\mu},\nu_{\tau}$
have equal strength $g_{\nu}$ with the $Z^{\prime}$, and equal coupling
strength $g_{l}$ for $l=e,\mu,\tau$.

Figure 1:  Production and decays of the heavy neutrino $N$ at FASER$\nu$.
($\nu q\rightarrow Nq$ followed by $N\rightarrow\nu f\bar{f}^{\prime}$, where
$f,f^{\prime}$ are SM fermions.)

##  III Heavy Neutrino production at FASER$\nu$

Figure 2:  Production cross sections of the heavy neutrino $N$ at FASER$\nu$:
$\sigma_{\nu A}=p\sigma_{\nu p}+n\sigma_{\nu n}$, computed with
$\Lambda=1000\leavevmode\nobreak\ \rm GeV$, and $g_{q}=1$. Here $A$ is the
tungsten nucleus with $p$ protons and $n$ neutrons. The energy $E_{\nu}$ of
the incoming neutrino beam is $E_{\nu}$ = 10 GeV (top left), $E_{\nu}$ = 100
GeV (top-right), $E_{\nu}$ = 1000 GeV (bottom-left), and $E_{\nu}$ = 10000
GeV. (bottom-right), respectively.

In this section, we discuss heavy neutrino production at FASER$\nu$ detector
from the neutrino-nucleon scattering $\nu A\rightarrow NA$ mediated by a new
neutral gauge boson $Z^{\prime}$. The heavy neutrino is produced at FASER$\nu$
and it will decay into SM particles ($N\rightarrow\nu l^{+}l^{-}$,
$N\rightarrow\nu q\bar{q}$, $N\rightarrow\nu\nu\bar{\nu}$). As shown in Eq.
(1) the heavy neutrino $N$ has a dipole-like vertex with the active neutrino
and the new gauge $Z^{\prime}$ boson, which resembles the transitional
magnetic moment. The Feynman diagram of the production and decay process of
heavy neutrino $N$ is shown in Fig. 1. Discovery potential for heavy neutral
leptons at FASER through the SM gauge boson decays was discussed in
Jodlowski:2020vhr ; FASER:2018eoc ; Kling:2018wct .

We use $\rm MadGraph5aMC@NLO$ Alwall:2011uj ; Alwall:2014hca  for the
computation of fixed-target neutrino-nucleon scattering $\nu A\rightarrow NA$
cross-sections. We build the model file for Eq. (1) using FeynRules
Alloul:2013bka . For the production process we consider 3 different neutral
gauge boson masses, $M_{Z^{\prime}}=0.1,1,10\leavevmode\nobreak\ \rm GeV$, and
the heavy neutrino mass from 1 to 70 GeV ($M_{N}<M_{w}$). FASER$\nu$ is
providing a unique region of neutrino energy 1 GeV – 10 TeV for heavy neutrino
production. Note that we fixed the value of tree level $Z^{\prime}$-quark
coupling strength $g_{q}=1$ and the cut-off energy scale value $\Lambda=1000$
GeV. The production cross section simply scales with $g_{q}^{2}$ and
$1/\Lambda^{4}$. Figure 2 shows the neutrino-nucleon fixed target cross-
sections for different neutrino beam energies $E_{\nu}=10-10000$ GeV. In each
case, the cross-section decreases with increment of both $M_{Z^{\prime}}$ and
$M_{N}$. Here we have assumed the benchmark detector specification made of
tungsten with dimensions $25\,{\rm cm}\times 25\,{\rm cm}\times 1\,{\rm m}$ at
the 14 TeV LHC with an integrated luminosity of $L=150\,{\rm fb}^{-1}$. We use
the neutrino fluxes and energy spectra obtained in FASER:2019dxq ;
Kling:2021gos  to study the neutrinos that pass through FASER$\nu$. We find
that the muon neutrinos are mostly produced from charged-pion decays, electron
neutrinos from hyperon, kaon and $D$-meson decays, and tau neutrinos from
$D_{s}$ meson decays. With average energies ranging from 600 GeV to 1 TeV, the
spectra of the three neutrino flavors cover a broad energy range.

Figure 3:  Total number of heavy neutrinos produced by scattering with the
tungsten target at FASER$\nu$ with $\nu_{\mu}A\rightarrow NA$ scattering (top-
left), with the $\nu_{e}A\rightarrow NA$ scattering (top-right), and with
$\nu_{\tau}A\rightarrow NA$ scattering (bottom-left).

The expected number of heavy neutrino events from the three flavors of active
neutrinos with three different $Z^{\prime}$ masses in FASER$\nu$ is shown in
Fig. 3. The highest number of heavy neutrino events come in with the muon
neutrino $\nu_{\mu}$ scattering with tungsten nucleus($\nu_{\mu}A\rightarrow
NA$), while the lowest one is from the tau neutrino $\nu_{\tau}$ scattering
($\nu_{\tau}A\rightarrow NA$), and those from the electron neutrino $\nu_{e}$
scattering is in between. Total number of events follow the same behavior as
the scattering cross-section. We can summarize Fig. 3 as follows,
$N_{\nu_{\tau}}(M_{Z^{\prime}},M_{N},\mu_{\nu_{\tau}},g_{q})<N_{\nu_{e}}(M_{Z^{\prime}},M_{N},\mu_{\nu_{e}},g_{q})<N_{\nu_{\mu}}(M_{Z^{\prime}},M_{N},\mu_{\nu_{\mu}},g_{q})$.

Figure 4:  Schematic diagram of FASER$\nu$-FASER Feng:2017vli ; Kling:2018wct
; Feng:2018pew ; Deppisch:2019kvs ; Cheung:2021tmx ; Okada:2020cue ;
Bahraminasr:2020ssz ; Kelly:2020pcy ; Falkowski:2021bkq ; FASER:2018eoc

##  IV FASER$\nu$-FASER Sensitivity towards active to heavy Neutrino
transitional magnetic moments

The heavy neutrino produced at the FASER$\nu$ detector could decay into
visible channels – charged leptons and quarks. The probability of heavy
neutrino decay within the detector volume is given byKling:2018wct ,

$\mathcal{P}_{\rm detc}=1-\exp(\frac{-d}{\beta c\tau})\;,$ (2)

where $d$ is defined as $d\equiv{\rm min}(\Delta,\frac{R}{\sin\theta})$,
$\Delta$ is the length of the detector, $R$ is the radius of the detector and
$\theta$ is the angle between the outgoing heavy neutrino $N$ along with
horizontal beam axis. A schematic diagram of the detector is given in Fig. 4.
The decays of the heavy neutrino $N$ into SM particles are calculated using
the $\rm MadGraph5aMC@NLO$ Alwall:2011uj ; Alwall:2014hca ; Alwall:2014bza .
The angular distribution of the heavy neutrino produced at FASER$\nu$ is
extracted from the output files with the extension $\rm``.lhe"$.

We generate 10,000 events using $\rm MadGraph5aMC@NLO$ for the computation of
the angular distribution of the heavy neutrino with respect to the horizontal
axis. We consider different sets of the heavy neutrino mass ranging from 1 GeV
to 70 GeV and the $Z^{\prime}$ mass ($M_{Z^{\prime}}$=0.1, 1, 10 GeV) with
neutrino energy $E_{\nu}$ ranging from 10 GeV to 10 TeV. We determine the
fraction of events at a particular scattering angle from the angular
distribution of heavy neutrino $N$.

Figure 5:  Angular distribution of the heavy neutrino of mass $M_{N}$ = 1 GeV
with different sets of neutrino beam energies. The mediator $Z^{\prime}$ mass
is set at $M_{Z^{\prime}}$ = 0.1 GeV

Figure 5 shows the angular distributions of the heavy neutrino versus the
scattering angle $\theta$ for $E_{\nu}=10-10000$ GeV. As expected the high
energy neutrino so produced at the FASER$\nu$ travels very close to the beam
axis. The higher the incoming neutrino beam energy and the heavier the $N$,
the closer to the beam axis the heavy neutrino $N$ travels (i.e. the smaller
the scattering angle). The variation with different $M_{Z^{\prime}}$ is not
significant. More details about the angular distribution are given in appendix
A.

The total number of the heavy neutrino decays detected within the FASER$\nu$
detector is given by

$N^{\rm detc}_{\alpha}=N_{\alpha}^{\rm Prod}(\nu_{\alpha}A\rightarrow
NA)\times\mathcal{P}_{detc}\times{\rm
BR}(N\rightarrow\nu_{\alpha}\leavevmode\nobreak\ X\leavevmode\nobreak\ X)\;,$
(3)

where $N_{\alpha}^{\rm Prod}(\nu_{\alpha}A\rightarrow NA)$ denotes the the
total number of heavy neutrino events produced at FASER$\nu$ from the
$\nu_{\alpha}$ as shown in Fig 3, $\rm\mathcal{P}_{detc}$ is the probability
of heavy neutrino decay within the detector volume Eq.(2), and ${\rm
BR}(N\rightarrow\nu_{\alpha}\leavevmode\nobreak\ X\leavevmode\nobreak\ X)$
represents the branching ratio of heavy neutrino $N$ into $\nu_{\alpha}XX$.
There are two possible detecting channels in the $N$ decays: (i)
$N\rightarrow\nu l^{+}l^{-}$ (a pair of charged leptons), and (ii)
$N\rightarrow\nu q\bar{q}$ (a pair of quarks appearing as hadrons).

The final-state charged leptons and quarks from heavy neutrino $N$ decays are
detectable in the $\rm FASER\nu$ and FASER detector. We also assume that
backgrounds can be reduced to negligible levels for both cases of prompt and
displaced decays of the heavy neutrino. Thus, we present the sensitivity
results in terms of 3-signal-event contour curves which correspond to 95% C.L.
limits with zero background Tian:2022rsi . We consider 3 benchmark models for
the evaluation of the FASER and FASER$\nu$ sensitivity for the active-to-
heavy-neutrino transitional magnetic moment. The three benchmark models are
discussed below.

###  IV.1 Benchmark Model I (BM-I): Heavy Neutrino $N$ and $Z^{\prime}$ mixed
with $\mu$ and $\nu_{\mu}$

Figure 6:  Dominant branching fractions of the heavy neutrino $N$ as a
function of its mass $M_{N}$ for nonzero $V_{\mu N}$ and $\mu_{\nu_{\mu}}$
with $M_{Z^{\prime}}$ =0.1 (top-left), 1 (top-right), 10 (bottom-left) GeV.

We now focus on the heavy neutrino $N$ that couples to only one of the SM
lepton doublets, $L_{\mu}$. This model depends on 7 parameters: the mass of
heavy neutrino $M_{N}$, the mass of $Z^{\prime}$ boson $M_{Z^{\prime}}$, the
muon active-to-heavy-neutrino transitional magnetic moment $\mu_{\nu_{\mu}}$,
heavy neutrino mixing angle $V_{\mu N}$ and the $Z^{\prime}$ coupling with
quarks $g_{q}$, with charged leptons $g_{l}$ and with neutrinos $g_{\nu}$. The
rest of parameters are set at zero.

The branching ratios for the BM-I with three different $Z^{\prime}$ masses are
shown in Fig. 6. The branching ratios vary with $M_{N}$ and once the mass
$M_{N}$ is above a particular threshold, the corresponding new channel
appears, e.g., $N\to\nu_{\mu}\tau^{+}\tau^{-}$ appears when $M_{N}>2m_{\tau}$.
For $M_{Z^{\prime}}<M_{N}$ the two-body decay of
$N\rightarrow\nu_{\alpha}Z^{\prime}$ dominates, followed by the $Z^{\prime}$
boson decay into SM fermions $Z^{\prime}\rightarrow f\bar{f}$ ($f=\nu,l,u,d$),
according to the mass of $Z^{\prime}$. The branching ratios of the heavy
neutrino $N$ through such a 2-body decay can be converted to the 3-body one
using

${\rm BR}(N\xrightarrow{Z^{\prime}}\nu\leavevmode\nobreak\
X\leavevmode\nobreak\ X)={\rm BR}(N\rightarrow\nu Z^{\prime})\times{\rm
BR}(Z^{\prime}\rightarrow XX).$

Figure 7:  Sensitivity reach to $\nu_{\mu}$-to-heavy-neutrino transitional
magnetic moment $\mu_{\nu_{\mu}}$ at FASER$\nu$ as a function of the heavy
neutrino mass $M_{N}$ with different $Z^{\prime}$ mass. Right: considering
quarks and charged leptons in the the final state; left: considering only
charged leptons in the final state.

The sensitivity curves for the BM-I are shown in Fig. 7. The right panel
considers that the detector could probe both hadron and charged-lepton final
states, while the left panel only considers the charged-lepton final state. By
comparing the two plots we can see that the right panel shows better
sensitivity. In both panels, the sensitivity reach on $\mu_{\nu_{\mu}}$ is the
best at very small $M_{Z^{\prime}}$ and $M_{N}$ region. For
$M_{Z^{\prime}}=0.1\rm\leavevmode\nobreak\ GeV$ and
$M_{N}=1\rm\leavevmode\nobreak\ GeV$ the sensitivity reach of
$\mu_{\nu_{\mu}}$ is about $\sim 5\times 10^{-6}\,{\rm GeV}^{-1}$ (see the
right panel of Fig 7), and $\sim 10^{-5}\,{\rm GeV}^{-1}$ (left panel of Fig.
7). The left panel of Fig. 7 depends on $N\rightarrow\nu_{\mu}l^{+}l^{-}$,
from which we can see that the change in the curve pattern can be read from
the branching ratio curves in Fig. 6. For $M_{Z^{\prime}}=$ 0.1 GeV,
$N\rightarrow\nu_{\mu}\nu\bar{\nu}$ is the dominant decay channel, while for
other 2 cases $M_{Z^{\prime}}=1$ and 10 GeV $N\rightarrow\nu_{\mu}+\rm
hadrons$ becomes dominant. The higher $M_{Z^{\prime}}$ and $M_{N}$, the weaker
the limit on $\mu_{\nu_{\mu}}$ will be. Also, among all neutrino flavors,
$\mu_{\nu_{\mu}}$ has the best sensitivity compared to other two benchmark
models.

###  IV.2 Benchmark Model-II (BM-II): Heavy Neutrino $N$ and $Z^{\prime}$
mixed with $e$ and $\nu_{e}$

In this BM-II, the heavy neutrino coupled with the $L_{e}$ doublet. Similar to
BM-I, this BM-II also depends on 7 parameters. Here the heavy neutrino has the
non-zero mixing angle $V_{eN}$ with the SM leptons, and has the non-zero
active-to-heavy neutrino transitional magnetic moment $\mu_{\nu_{e}}$.

Figure 8:  Dominant branching fractions of the heavy neutrino $N$ as a
function of its mass $M_{N}$ for nonzero $V_{eN}$ and $\mu_{\nu_{e}}$ with
$M_{Z^{\prime}}$ =0.1 (top-left), 1 (top-right), 10 (bottom-left) GeV.

The branching ratios of $N\rightarrow\nu_{e}\leavevmode\nobreak\
X\leavevmode\nobreak\ X$ are shown in Fig. 8 with three different
$M_{Z^{\prime}}$. When the mass of $M_{N}>M_{Z^{\prime}}$, the two-body decay
of heavy neutrino $N\rightarrow\nu Z^{\prime}$ dominates, followed by the
$Z^{\prime}\to f\bar{f}$.

Figure 9:  Sensitivity reach to the $\nu_{e}$-to-heavy-neutrino transitional
magnetic moment $\mu_{\nu_{e}}$ at FASER$\nu$ as a function of the heavy
neutrino mass $M_{N}$ with different $Z^{\prime}$ mass. Right: considering
both quarks and charged leptons in the final state; left: considering only
charged leptons in the final state.

The sensitivity reach on $\mu_{\nu_{e}}$ versus $M_{N}$ with different
$M_{Z^{\prime}}$ is shown in Fig. 9. The sensitivity reach on $\mu_{\nu_{e}}$
is the best at very small $M_{N}$ and $M_{Z^{\prime}}$ and is around $\sim
2\times 10^{-5}\,{\rm GeV}^{-1}$ (right panel of Fig. 9) and $\sim 5\times
10^{-5}\,{\rm GeV}^{-1}$ (left panel of Fig. 9). The right and left panels of
Fig. 9 follow the same notation as Fig. 7 of BM-I. The $\mu_{\nu_{\mu}}$
sensitivity curves are better than those of $\mu_{\nu_{e}}$ curves.

###  IV.3 Benchmark Model-III (BM-III): Heavy Neutrino $N$ and $Z^{\prime}$
mixed with $\tau$ and $\nu_{\tau}$

In this model heavy neutrino $N$ only mixes with the $\tau$ lepton doublet
through a nonzero mixing angle $V_{\tau N}$ and a non-zero transitional
magnetic moment $\mu_{\nu_{\tau}}$. The branching ratios of
$N\rightarrow\nu_{\tau}\leavevmode\nobreak\ X\leavevmode\nobreak\ X$ are shown
in Fig. 10. For $M_{Z^{\prime}}=$ 0.1 GeV,
$N\rightarrow\nu_{\tau}\nu\bar{\nu}$ is the dominant decay channel, while for
other 2 cases $M_{Z^{\prime}}=1$ and 10 GeV $N\rightarrow\nu_{\tau}+\rm
hadrons$ becomes dominant. We can see similar pattern for the heavy neutrino
decay in the BM-I and BM-II.

Figure 10:  Dominant branching fractions of the heavy neutrino $N$ as a
function of its mass $M_{N}$ for nonzero $V_{\tau N}$ and $\mu_{\nu_{\tau}}$
with $M_{Z^{\prime}}$ = 0.1 (top-left), 1 (top-right), 10 (bottom-left) GeV.

The sensitivity reach for the active-to-heavy-neutrino transitional magnetic
moment $\mu_{\nu_{\tau}}$ is depicted in Fig. 11. Among all three benchmark
models, BM-III has the least sensitivity towards the transitional magnetic
moment, simply because of the small incoming tau-neutrino flux. The left and
right panels of Fig. 11 follow the same convention as those of BM-I and BM-II.
The sensitivity reach on $\mu_{\nu_{\tau}}$ is the best at very small $M_{N}$
and $M_{Z^{\prime}}$ and is around $\sim 2\times 10^{-4}\,{\rm GeV}^{-1}$
(right panel of figure 11) and $\sim 3\times 10^{-4}\,{\rm GeV}^{-1}$ (left
panel of figure 11). Among all three benchmark models, BM-I has the best
sensitivity reach compared to the other two. It is due to the difference in
the flux of each neutrino flavor.

Figure 11:  Sensitivity reach to the $\nu_{\tau}$-to-heavy-neutrino
transitional magnetic moment $\mu_{\nu_{\tau}}$ at FASER$\nu$ as a function of
the heavy neutrino mass $M_{N}$ with different $Z^{\prime}$ mass. Right:
considering quarks and charged leptons in the final state; left: considering
only charged leptons in the final state.

##  V Conclusions

In this paper, we have studied the active-to-heavy-neutrino transitional
magnetic moment which arises from the $Z^{\prime}$ boson interactions between
the heavy and active neutrinos at FASER$\nu$. We consider the neutrino-nucleon
scattering as the production process of the heavy neutrino, and we see that if
the heavy neutrino can decay into charged leptons or hadrons, it can be
identified with negligible backgrounds inside the FASER$\nu$ detector. We
investigated the advantage of FASER$\nu$ in a wide mass range search for the
heavy neutrino $N$ and $Z^{\prime}$ to determine the flavor dependence of the
coupling between the active neutrinos and the heavy neutrino, for which we
found that FASER$\nu$ is sensitive to $\mu_{\nu_{\alpha}}$ because of large
incoming neutrino flux.

In this study we have considered three benchmark models in which the heavy
neutrino couples to $L_{\mu}$, $L_{e}$ or $L_{\tau}$ SM doublet. Among all the
three models, the BM-I in which the heavy neutrino couples to $L_{\mu}$ has
the best sensitivity reach compared to the other two. We obtained sensitivity
curves by considering two different final states: (i) charged leptons and (ii)
charged leptons and hadrons. The second final state provides better
sensitivity reach on $\mu_{\nu_{\alpha}}$. While the FASER$\nu$ can achieve
the best sensitivity at small $M_{Z^{\prime}}$ and $M_{N}$ regime, the
sensitivity limits get weaker with the increment of heavy neutrino and
$Z^{\prime}$ masses. We did not include antineutrinos in our study; if we had
included antineutrinos, the sensitivity curves could be improved by a factor
of 2.

A study of the magnetic dipole interactions between the active neutrinos and
new sterile states at emulsion and liquid argon experiments was reported in
Ismail:2021dyp , which considered the up-scattering of neutrinos on electrons
produces an electron-recoil signature that can probe new regions of parameter
space at the High Luminosity LHC (HL-LHC), in particular for liquid argon
detectors due to low momentum thresholds. They also considered the decay of
the sterile neutrino through the dipole operator, which leads to a photon that
could be displaced from the production vertex. Bohr magneton $\mu_{B}$ is
given by $\mu_{B}=e\hbar/2m_{e}$, which is about $3\times 10^{2}\;{\rm
GeV}^{-1}$ in natural units. Reference Ismail:2021dyp  obtained the 90% C.L.
contour curve for the active-to-heavy-neutrino transitional magnetic moment
with heavy neutrino mass $M_{N}$ ranging from $10^{-3}\leavevmode\nobreak\ \rm
GeV-10\leavevmode\nobreak\ GeV$, the Forward Physics Facility (FPF) detectors
can reach down to dipole coupling strengths of a few $10^{-9}\mu_{B}$ ($\sim
10^{-6}\rm\leavevmode\nobreak\ GeV^{-1}$) for $\mu_{\nu_{e}}$, $\sim
10^{-9}\mu_{B}$(a few $10^{-7}\rm\leavevmode\nobreak\ GeV^{-1}$) for
$\mu_{\nu_{\mu}}$, and a few $10^{-8}\mu_{B}$($10^{-5}\rm\leavevmode\nobreak\
GeV^{-1}$) for $\mu_{\nu_{\tau}}$. According to Ismail:2021dyp , starting at
$M_{N}\sim$ $10^{-1}$ GeV, the sensitivity weakens.

## Acknowledgements

The work was supported in parts by the MoST of Taiwan with the grant
no.110-110-2112-M-007-017-MY3.

##  Appendix A Angular distributions of the heavy neutrino

Angular distribution curves are obtained by using the $\rm MadGraph5aMC@NLO$
Alwall:2011uj ; Alwall:2014hca ; Alwall:2014bza  and extracting the
information from the output $\rm``.lhe"$ file. The angular distribution of the
heavy neutral neutrino produced at FASER$\nu$ through the neutrino-nucleon
scattering $\nu A\rightarrow NA$ is shown in Fig. 12. Here we show the
distributions with three $Z^{\prime}$ masses $M_{Z^{\prime}}$ = 0.1 ,1 10 GeV
and with different sets of neutrino energies from 10 GeV to 10 TeV. The
distribution curves are approaching to ‘zero’ by the increment of incoming
active neutrino energy $E_{\nu}$, and the heavy neutrino Mass $M_{N}$.

Figure 12:  Angular distributions of the heavy neutrino $N$ with different
$M_{Z^{\prime}}$ and different neutrino energy $E_{\nu}$.

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