††institutetext: School of physics, Institute for Research in Fundamental
Sciences (IPM)
P.O.Box 19395-5531, Tehran, Iran

# Neutral Exotica at FASER$\nu$ and SND@LHC

S. Ansarifard and Y. Farzan<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

The $(g-2)_{\mu}$ anomaly indicates that the second generation of leptons
should have new interactions beyond the standard model. The high flux of
$\nu_{\mu}$ and $\bar{\nu}_{\mu}$ at the forward experiments such as
FASER$\nu$ and SND@LHC makes them suitable setups to search for new
interactions of the second generation leptons. In this paper, we build a model
in which the second generation left-handed leptons couple to a new right-
handed neutrino, $N$ and a new Higgs doublet which also couples to the quarks.
The scattering of high energy $\nu_{\mu}$ off nuclei can produce $N$. We
investigate how forward experiments can test this model by looking for the $N$
production vertex followed by the displaced vertex of the $N$ decay.
Discovering even a single such event can be a harbinger to look for the
spectacular signals of the new Higgs doublet production at the LHC. We discuss
the possibility of explaining the $(g-2)_{\mu}$ anomaly by adding more
generations of $N$ which will lead to chain decays of $N$ and multiple leptons
with distinct signals both at forward experiments and at the CMS and ATLAS
detectors. Finally, we show that by adding a new light singlet scalar mixed
with the neutral component of the new Higgs doublet (i.e., 2HDM+$S$ model),
the statistics of the data sample can be dramatically increased.

##  1 Introduction

During the last 90 years, several breakthroughs in the field of particle
physics and high energies have been brought about by studying the properties
of neutrinos. This historical fact motivates scrutinizing neutrino interaction
at all possible energy intervals to look for signs of new non-standard
interactions. BOREXINO has made precise measurement of interaction of solar
neutrinos with energies of sub MeV to 14 MeV on the electron. The interaction
of neutrinos with energy range of few 10 GeV off nuclei has been studied with
great precision by experiments like NOMAD NOMAD:2001xxt . Finally, neutrino
telescopes such as ICECUBE, DEEPCORE, ANTARES, ARCA and ORCA detect the
scattering of higher energy atmospheric and cosmic neutrinos on nuclei but
these detectors cannot resolve the fine details of the scattering processes in
case an intermediate new particle is produced.

Neutrinos of all flavors with 100 GeV$-$1 TeV energies can be produced at the
Interaction Points of the LHC. The main detectors of the LHC, CMS and ATLAS,
are not designed to detect these neutrinos and miss them. Indeed, interactions
of colliding protons produce large flux of quarks along the beamline which
hadronize as pions, Kaons and charmed hadrons. The eventual decay of these
hadrons emit a large flux of neutrinos in the forward direction. During the
run III of the LHC in 2022-2024, two new detectors called FASER$\nu$
Abreu:2019yak  and SND@LHC SND  will detect these large fluxes of neutrinos
emitted in the forward direction. These detectors are designed to resolve even
the short track of the $\tau$-lepton produced by Charged Current (CC)
$\nu_{\tau}$ interaction. That is they have a superb spatial resolution which
makes them ideal for probing new feebly interacting particles that go through
chain decays.

Ref. Bakhti:2020vfq  demonstrates that FASER$\nu$ can explore a variety of
light dark matter models in which dark matter abundance is set via freeze-out
scenario with annihilation into pairs of intermediate light new neutral
particles. Ref. Falkowski:2021bkq  studies the capability of FASER$\nu$ to
probe effective beyond standard model couplings between neutrino and quarks
that may lead to anomalous Charged Current (CC) interaction of neutrinos:
$\nu+{\rm nulceus}\to l+X$. Ref. Ismail:2020yqc  makes a forecast of the
impact of the Neutral Current (NC) Non-Standard Interaction (NSI) of form
$[(V-A)(V\pm A)]$ on FASER$\nu$ data: $\nu+{\rm nucleous}\to\nu+X$. Ref.
Kling:2020iar  investigates the effects of light flavor gauge bosons at
FASER$\nu$ and SND@LHC. Ref. Bakhti:2020szu  as well Ref. Jho:2020jfz ;
Jodlowski:2020vhr  scrutinize $\nu+{\rm nucleus}\to N+X$ in which $N$ is a
heavy neutrino. In the prvious studies, the intermediate particle meditating
$\nu+{\rm nucleus}\to N+X$ is either a SM photon (interacting via dipole
interaction Jodlowski:2020vhr ; Ismail:2021dyp ) or a dark photon or
$Z^{\prime}$ Bakhti:2020szu ; Jho:2020jfz .

Large fluxes of $\nu_{\mu}$ and $\bar{\nu}_{\mu}$ at FASER$\nu$ and at SND@LHC
provide a unique opportunity to study possible interactions of the second
generation leptons with new particles. Such new interactions are also
motivated by the famous $(g-2)_{\mu}$ anomaly. In this paper, we build a model
in which the second generation left-handed leptons interact with a new right-
handed neutrino $N$ and a new scalar $SU(2)$ doublet that also couples to the
quarks. $N$ heavier than a few GeV is an uncharted territory which could not
be explored by previous neutrino scattering experiments but as we shall show
FASER$\nu$ and SND@LHC can explore $N$ with a mass up to $\sim 15$ GeV. We
outline the signals of the models at forward experiments and show that the
signals are background free. We propose methods to derive the mass and the
lifetime of $N$ from the data. We forecast the bounds that can be extracted
from the data of SND@LHC, FASER$\nu$ and its upgrade for HL-LHC on the
relevant effective couplings. We also argue that the model can be tested by
main detectors of the LHC. Indeed, a single signal event at these forward
experiment will be a great motivation to look for signatures of the model at
CMS and ATLAS.

We discuss the contribution to $(g-2)_{\mu}$ and find that to explain the
anomaly, the model should be extended to include more generations of right-
handed neutrinos and/or scalar doublets. Adding more scalar doublets may
increase the new effective coupling between $\bar{N}\nu$ and the quarks,
increasing the statistics of the signal at the forward experiments. Multiple
$N$ can have a more spectacular effect, leading to chain decays of right-
handed neutrinos at both forward experiments and at the main detectors of the
LHC (i.e., at CMS and ATLAS). In the process of chain decay, multiple charged
leptons can be emitted. Recent studies report such multiple lepton signal in
the LHC data vonBuddenbrock:2017gvy ; Fischer:2021sqw ; Hernandez:2019geu . As
discussed in Sabatta:2019nfg , the $(g-2)_{\mu}$ anomaly and these multilepton
excess at the LHC may be related. We have discussed the signatures of the
variant of the model with multiple $N$ at forward experiments and discuss how
such a variant can be distinguished from the minimal version of the model with
only a single $N$.

Finally, we add a lighter singlet scalar, $S$, mixed with the neutral
component of the scalar electroweak doublet. This makes the model a two Higgs
doublet plus $S$ model which is also motivated by some anomalies reported in
the LHC data vonBuddenbrock:2017gvy ; Fischer:2021sqw ; Hernandez:2019geu ;
Mathaha:2021buc ; Crivellin:2021ubm  (see, however, Fowlie:2021ldv ). We show
that with adding such a light scalar the statistics of the signal at forward
experiments can be dramatically increased.

This paper is organized as follows. In sect. 2, we describe the
characteristics of the forward experiments that are relevant for our analysis.
We then outline the concept of deriving information on the properties of new
neutral intermediate particles which might be produced and subsequently decay
inside the detector. In sect. 3, we propose the minimalistic version of our
model and describe its signals at forward experiments. We assess the
background for the signal. We compute the cross section of the $N$ production
by $\nu_{\mu}$ and $\bar{\nu}_{\mu}$ scattering off nucleons. We forecast the
bounds from forward experiments on the couplings in case of null signals. We
also compute maximum number of signal events, saturating the present bounds
and discuss how the parameters of the underlying model can be extracted in the
lucky situation of relatively large signal sample. In sect. 4, we compute the
contribution from the new interactions to $(g-2)_{\mu}$ and to the neutrino
masses. We then describe the potential signals at the main detectors of the
LHC, CMS and ATLAS. In sect. 5, we describe the signatures of a variant of the
model with multiple $N$ at forward experiments as well as at the main
detectors of the LHC. In sect. 6, we show that by adding a singlet scalar, the
$N$ production cross section can be significantly increased. In sect. 7, we
summarize the results and briefly discuss the implications for the detection
of atmospheric neutrinos by neutrino telescopes.

##  2 Forward experiments

In this section, we briefly review the setup and capabilities of SND@LHC,
FASER$\nu$ and its upgrades for high luminosity LHC. FASER$\nu$ and SND@LHC
will take data during the run III of the LHC. They will be both located at a
distance of 480 m from the ATLAS Interaction Point (IP) but at opposite sides
in the TI12 and TI18 tunnels Kling:2021gos . These detectors are designed to
detect neutrinos emitted from the IP in the forward direction. The main
sources of neutrinos are the decays of hadrons such as pions, Kaons, and
charmed hadrons produced at the IP with a momentum along the beamline.
Although the distance of both detectors from the IP are equal, the fluxes of
neutrinos at SND@LHC are predicted to be softer than those at the FASER$\nu$
because, while FASER$\nu$ is located exactly in the beamline direction (right
before the FASER detector), SND@LHC will be located slightly off-line. The
upgrade of FASER$\nu$ will be located at the proposed Forward Physics Facility
(FPF) at a distance of 620 m from IP. While during run III (2022-2024) 150
fb-1 integrated luminosity will be collected at ATLAS IP, during the high
luminosity upgrade the integrated luminosity will increase to 3000 fb-1. Ref.
Kling:2021gos  predicts fluxes of neutrinos in these three detectors.

The FASER$\nu$ and SND@LHC detectors are both made of Tungsten with masses of
1.2 tons Abreu:2019yak  and 850 kg SND , respectively. Indeed FASER$\nu$ is
made of 1000 layers of emulsion films interleaved with Tungsten plates of
thickness of 1 mm. The size of FASER$\nu$ is $25~{}{\rm cm}\times 25~{}{\rm
cm}\times 1.3~{}{\rm m}$ Abreu:2019yak . That of SND@LHC is $41.6~{}{\rm
cm}\times 38.7~{}{\rm cm}\times 32~{}{\rm cm}$. The upgrade of FASER$\nu$ will
be of size $50~{}{\rm cm}\times 50~{}{\rm cm}\times 5~{}{\rm m}$ and will
weight 10 tons felix_kling_2020_4059893 .

Notice that FASER$\nu$, being an emulsion detector, cannot record the timing
of events. The whole data taking period of run III will be divided into
periods with 10-50 fb-1 integrated luminosity after which the emulsion will be
processed Abreu:2019yak . As we will see in sect. 3, such division reduces
significantly the accidental background from standard model NC and CC events
to our signal.

The great advantage of these detectors is their spatial resolution which makes
them capable of resolving even the short $\tau$-tracks. For example,
FASER$\nu$ can resolve vertices with a precision of $\sigma_{pos}=0.4~{}\mu$m.
The angular resolution of tracks will be $\sqrt{2}\sigma_{pos}/L_{tr}$ where
$L_{tr}$ is the track size. As discussed in Bakhti:2020vfq , this makes
FASER$\nu$ ideal setup for studying the dark sector which go through chain
decays. The energy resolution is however modest and of order of 30 %
Abreu:2019yak .

Let us consider a new particle produced at FASER$\nu$ decaying to $n_{f}$
final charged particles. The mass of this particle can be extracted by
measuring the invariant mass of these final particles with a relative error of
$30\%\sqrt{n_{f}}$. If $N_{new}$ such events are observed, the relative
precision will be improved to $30\%\sqrt{n_{f}/N_{new}}$. The great precision
in the angular resolution can also help us to find out whether the decay
products of the new particle also include an invisible neutral particle or
not. This can be done by reconstructing transverse missing momentum relative
to the direction of the momentum of the intermediate new particle. Let us
consider a new particle (charged or neutral) whose production and decay
vertices are both inside the FASER$\nu$ detector and are reconstructed. Let us
denote the distance between these two vertices by $L_{new}$. The transverse
momentum of a possible invisible particle can be reconstructed by projecting
the momenta of the final charged particles onto the plane perpendicular to the
track of the invisible particles. If the final charged particles make an angle
smaller than $90^{\circ}$ with each other, the transverse momentum of the
missing light particle will be of order of $m_{N}/2$. If
$m_{N}/(2E_{new})>\sqrt{2}\sigma_{pos}/L_{new}$, the precision will be enough
to distinguish the emission of a neutral invisible particle in the final
state. By measuring $L_{new}$, the lifetime of the new particles can also be
extracted. The uncertainty in the deviation of the lifetime will be mostly
dominated by statistics: $\Delta\tau_{new}/\tau_{new}=2\log 2/\sqrt{N_{new}}$.

##  3 The minimal model and its prediction for forward experiments

In subsect. 3.1, we introduce the field content of the minimal model and its
Lagrangian. In subsect. 3.2, we compute the cross section of $\nu_{\mu}+{\rm
nucleon}\to N+X$ and predict the number of events at the forward experiments.
In subsect. 3.3, we study the decay modes of $N$. We calculate the average
energy of $N$ produced by $\nu_{\mu}$ with a given energy and use it to
estimate the decay length of $N$ at the forward experiments. In subsect. 3.4,
we describe the signals and estimate the potential backgrounds for it. We then
forecast the bounds that can be derived on the effective coupling by the
forward experiments if no signal is observed.

###  3.1 The model content and Lagrangian

Figure 1: Cross section of $\nu_{\mu}+{\rm nucleon}\to N+X$ versus the energy
of the initial neutrino for various $N$ mass. We have taken
$G_{u}=G_{d}=10^{-5}$ GeV-2.

In this subsection, we introduce the Lagrangian of the minimalistic version of
the model. We add a scalar doublet

$\Phi=\left[\begin{matrix}\Phi^{+}\cr\Phi^{0}\end{matrix}\right]$ (1)

and a right-handed singlet fermion, $N$ to the standard model. The right-
handed $N$ can be either Majorana or Dirac. We turn on the following Yukawa
interactions

$Y_{\alpha}\bar{N}\Phi^{T}cL_{\alpha}+Y_{d}\bar{d}\Phi^{\dagger}Q+Y_{u}\bar{u}\Phi^{T}cQ+{\rm
H.c.}$ (2)

where $L_{\alpha}$ is the left-handed lepton doublet of flavor $\alpha$ and
$Q$ is the first generation of left-handed quarks. As long as the masses of
$\Phi^{0}$ and $\Phi^{+}$ are heavier than $\sim 300$ GeV and
$Y_{u},Y_{d}\stackrel{{\scriptstyle<}}{{\sim}}0.3$, the present bounds from
the direct production of these particles at CMS and ATLAS can be avoided.
Moreover, as long as the splitting between the components is small, the bounds
from precision data can be satisfied Haller:2018nnx . However, $\Phi^{0}$ and
$\Phi^{+}$ will be within the reach of the high luminosity phase of the LHC or
even that of the run III. We shall discuss the possible signatures of direct
$\Phi$ production in sect. 4. The Yukawa coupling to the muon, $Y_{\mu}$ can
also lead to a contribution to $(g-2)_{\mu}$. As we shall see in sect. 4,
explaining the famous $(g-2)_{\mu}$ anomaly motivates large values of
$Y_{\mu}$, saturating the perturbativity condition, $Y_{\mu}\sim 3-4$. Ref.
Allwicher:2021rtd  shows that the models explaining the $(g-2)_{\mu}$ anomaly
with new Yukawa coupling generally need large Yukawa couplings. Here, our main
aim is not to explain the $(g-2)_{\mu}$ anomaly but we take $Y_{\mu}\sim O(1)$
to have a high rate of the $N$ production at FASER$\nu$ and SND@LHC detectors
due to $\nu_{\mu}$ interaction. Integrating out $\Phi^{0}$ and $\Phi^{+}$, we
obtain the following effective Lagrangian

$G_{u}\bar{N}_{R}\nu_{\mu}\bar{u}_{L}u_{R}+G_{d}\bar{N}_{R}\nu_{\mu}\bar{d}_{R}d_{L}+G_{L}\bar{N}_{R}\mu_{L}\bar{d}_{R}u_{L}+G_{R}\bar{N}_{R}\mu_{L}\bar{d}_{L}u_{R}+{\rm
H.c.}$ (3)

where

$G_{u}=\frac{Y_{\mu}Y_{u}^{*}}{m_{\Phi^{0}}^{2}},\
G_{d}=-\frac{Y_{\mu}Y_{d}}{m_{\Phi^{0}}^{2}},\
G_{L}=\frac{Y_{\mu}Y_{d}}{m_{\Phi^{+}}^{2}},\ {\rm and}\
G_{R}=\frac{Y_{\mu}Y_{u}^{*}}{m_{\Phi^{+}}^{2}}.$

Taking $Y_{u}\sim Y_{d}\sim 0.3$, $Y_{\mu}\sim 3$ and $m_{\Phi^{0}}\sim
m_{\Phi^{+}}\sim 300$ GeV, we find $G_{u}\sim G_{d}\sim G_{L}\sim G_{R}\sim
10^{-5}$ GeV-2.

Throughout this paper, we assume that only the second generation leptons
couple to $\Phi$ and $N$. We can also impose the following global $U(1)$
symmetry to set $Y_{e}=Y_{\tau}=0$:

$\Phi\to e^{i\alpha}\Phi,\ \ L_{\mu}\to e^{-i\alpha}L_{\mu},\ \ \mu_{R}\to
e^{-i\alpha}\mu_{R},\ \ N\to N,\ \ u\to e^{i\alpha}u,\ \ {\rm and}\ \ d\to
e^{-i\alpha}d,$ (4)

with the rest of SM fields including $Q$, the SM Higgs and the first and third
generation leptons being invariant. Notice that the $u$-quark and $d$-quark
masses as well as neutrino mixing break this $U(1)$ symmetry. As a result,
this symmetry explains the smallness of the masses of the first generation
quarks relative to the rest of quark masses, as a bonus. We shall discuss in
sect. 4 that this symmetry also prevents a large 1-loop contribution to the
neutrino mass in case $N$ is of Majorana type.

###  3.2 $N$ production by neutrino scattering on nuclei

Let us consider a neutrino of energy $E_{\nu}^{lab}$ colliding on a quark
which carries a fraction $x$ of the proton momentum. The $s$ Mandelstam
variable of the quark neutrino system will be

$s=x^{2}m_{p}^{2}+2xm_{p}E_{\nu}^{lab}$ (5)

where the first term is negligible. The energy of the neutrino and quark in
the center of mass is

$E_{\nu}=\left(\frac{xm_{p}E_{\nu}^{lab}}{2}.\right)^{1/2}.$ (6)

The cross section of scattering off a $u$ quark at the center of mass frame
with a scattering angle of $\theta$ can be written as

$\frac{d\sigma_{u}}{d\cos\theta}=\frac{G_{u}^{2}}{32\pi
v_{rel}}\left(1-\frac{m_{N}^{2}}{4E_{\nu}^{2}}\right)^{2}\left(\frac{m_{\Phi^{0}}^{2}}{t-m_{\Phi^{0}}^{2}}\right)^{2}\left(E_{\nu}^{2}(1-\cos\theta)^{2}+\frac{m_{N}^{2}}{4}(1-\cos^{2}\theta)\right)$
(7)

where $m_{N}$ is the mass of particle $N$. The Mandelstam variable $t$ can be
written as

$t=2E_{\nu}(E_{\nu}-\frac{m_{N}^{2}}{4E_{\nu}})(\cos\theta-1)\stackrel{{\scriptstyle<}}{{\sim}}O[(10~{}{\rm
GeV})^{2}]\ll m_{\Phi^{0}}^{2}$

so the second parenthesis in Eq. (7) can be approximated by 1. The cross
section of scattering off $d$-quark is given by a similar formula replacing
$G_{u}$ with $G_{d}$. Since the mediator is scalar, the cross sections of the
scattering off quark and antiquark are equal. Moreover, the cross section of
neutrino and antineutrino will be equal. We can therefore write the total
cross section of scattering on a nucleon ($i\in\\{{\rm proton,~{}neutron}\\})$
as

$\sigma_{tot}^{i}=\int_{-1}^{1}\int_{x_{min}}^{1}\left[\frac{d\sigma_{u}}{d\cos\theta}(f_{u}^{i}(x,t)+f_{\bar{u}}^{i}(x,t))+\frac{d\sigma_{d}}{d\cos\theta}(f_{d}^{i}(x,t)+f_{\bar{d}}^{i}(x,t))\right]dxd\cos\theta,$
(8)

where $x_{min}=m_{N}^{2}/(2m_{p}E_{\nu}^{lab})$. The nucleon parton
distribution functions, $f^{i}_{q}(x,t)$, are computed by LHAPDF6
buckley2015lhapdf6  using NNPDF3.1 ball2017parton . We have set
$G_{u}=G_{d}=10^{-5}$ GeV-2 throughout this paper. Notice that since we have
taken $G_{u}$ equal to $G_{d}$, the cross sections on u-quark and d-quark and
as a result on the proton and neutron will be equal. That is, from
$G_{u}=G_{d}$, we conclude $d\sigma_{u}/d\cos\theta=d\sigma_{d}/d\cos\theta$
and therefore $\sigma_{tot}^{n}=\sigma_{tot}^{p}$.

Figure 2: The number of $N$ and $\bar{N}$ particles produced by muon neutrino
and antineutrino fluxes at each energy bin at FASER$\nu$ during run III of the
LHC. We have taken $G_{u}=G_{d}=10^{-5}$ GeV-2. The fluxes of neutrinos and
antineutrinos are taken from Kling:2021gos .

The total cross section shown in Eq. (8) versus $E_{\nu}^{lab}$ is illustrated
in Fig. 1, taking $m_{N}=0.1,1,3,8,10$ GeV and $15$ GeV. As seen from the
figure, the curves converge for large $E_{\nu}^{lab}$. This is expected
because when the energy of the center of mass is much larger than the masses
of final particles, the dependence of the cross section on the masses becomes
weaker. Fig. 1 also shows that for $m_{N}<1$ GeV and $E_{\nu}^{lab}>100$ GeV,
the cross section is almost independent of $m_{N}$. This means that the major
contribution to the cross section comes from $x>{\rm few}\times 10^{-2}$. Even
in the limit $m_{N}\to 0$ and taking $G_{u}=G_{d}\sim G_{F}$, the cross
section in Eq. (8) is about one order of magnitude smaller than the cross
section of SM neutral current. This is partly due to the spinorial difference
of the amplitudes and partly due to the difference in the numerical factors in
the definitions of $G_{F}$ and $G_{u}$ or $G_{d}$ (cf., the definitions of
$G_{u}$ and $G_{d}$ in Eq. (3) with the Fermi interaction,
$(4G_{F}/\sqrt{2})(\bar{d}_{L}\gamma^{\mu}u_{L})(\bar{\nu}_{\mu}\gamma_{\mu}\mu_{L})$).

Table 1: The total predicted numbers of $N+\bar{N}$ to be detected by FASER$\nu$ and SND@LHC during run III of the LHC as well as by FASER$\nu 2$ during HL-LHC era, taking $G_{u}=G_{d}=10^{-5}$ GeV-2 and different values of $m_{N}$ as shown in the second row.  Number |  |  | $N+\stackrel{{\scriptstyle-}}{{N}}$ |  |  |
---|---|---|---|---|---|---
$m_{N}$ GeV | 0.1 | 1 | 3 | 8 | 10 | 15
SND@LHC | 19 | 18 | 13 | 5 | 3 | 1
FASER$\nu$ | 113 | 109 | 90 | 46 | 35 | 17
FASER$\nu 2$ | 7685 | 7394 | 6045 | 3019 | 2229 | 1015

The total number of $N$ and $\bar{N}$ produced at the detector will be

$\frac{M_{det}}{m_{p}}\int_{m_{N}^{2}/(2m_{p})}(F_{\nu_{\mu}}(E_{\nu}^{lab})+F_{\bar{\nu}_{\mu}}(E_{\nu}^{lab}))(r_{n}\sigma_{tot}^{n}+r_{p}\sigma_{tot}^{p})dE_{\nu}^{lab}$
(9)

where $M_{det}$ is the detector mass and $r_{n}$ and $r_{p}$ are fractions of
the neutron and proton in the nuclei of the detector, respectively:
$r_{p}=Z/A$ and $r_{n}=(A-Z)/A$. $F_{\nu_{\mu}}$ and $F_{\bar{\nu}_{\mu}}$
Kling:2021gos  are time integrated fluxes per unit area at the detector. Fig.
(2) illustrates the spectrum of $N+\bar{N}$ produced during the run III of the
LHC at FASER$\nu$. The total number of events at SND@LHC, FASER$\nu$ and
FASER$\nu 2$ for various values of $m_{N}$ are shown in table 1

###  3.3 $N$ decay

The produced $N$ travels a distance of $l\sim\Gamma_{N}^{-1}E_{N}/m_{N}$ where
$\Gamma_{N}$ is the total decay rate of $N$. Taking $N$ heavier than $\sim 3$
GeV, the following decay modes are available for the $N$ decay

$\Gamma(N\to\nu_{\mu}u\bar{u})=\frac{|G_{u}|^{2}}{|G_{d}|^{2}}\Gamma(N\to\nu_{\mu}d\bar{d})=\frac{|G_{u}|^{2}}{|G_{L}|^{2}+|G_{R}|^{2}}\Gamma(N\to\mu
u\bar{d})=\frac{G_{u}^{2}m_{N}^{5}}{1024\pi^{3}}$ (10)

where we have neglected the masses of the final particles. Notice that the
factor of three that usually appears in denominator from phase space
integration in the three body decay modes has been canceled out from Eq. (10)
by the factor of three in the numerator from summation on the color of the
final quarks. Moreover, we have assumed that $N$ is heavy enough so that the
meson resonances are not relevant for the $N$ decay. Of course, if $N$ is of
Majorana type, it can also decay into the charged conjugates of the above
final particles.

The average energy of $N$ produced by a neutrino of $E_{\nu}^{lab}$ can be
estimated as

$\langle
E_{N}^{lab}\rangle=\frac{\int_{-1}^{1}\int_{x_{min}}^{1}E_{N}^{lab}\left[\frac{d\sigma_{u}}{d\cos\theta}(f_{u}(x,t)+f_{\bar{u}}(x,t))+\frac{d\sigma_{d}}{d\cos\theta}(f_{d}(x,t)+f_{\bar{d}}(x,t))\right]dxd\cos\theta}{\sigma_{tot}},$
(11)

where $E_{N}^{lab}$ in the integrand can be written as

$E_{N}^{lab}=\frac{E_{\nu}^{lab}}{2}(1+\cos\theta)+\frac{m_{N}^{2}}{4xm_{p}}(1-\cos\theta).$

$\langle E_{N}^{lab}\rangle$ versus $E_{\nu}^{lab}$ is shown in Fig (3). As
expected $N$ carries a fraction of $O(0.3)$ of the energy of the initial
neutrino. For heavier $N$, this fraction is larger because the energy
available for the jets produced along with $N$ is lower.

Figure 3: Average energy of $N$ particles produced by incoming neutrinos with
energy of $E_{\nu}^{lab}$.

The momentum of $N$ produced by the scattering on a parton of associated
momentum fraction $x$ will make an angle of
$\sim\gamma^{-1}=(2xm_{p}/E_{\nu}^{lab})^{1/2}\sim 10^{-2}$ with the beamline.
It will decay after traveling a distance of

$l=\Gamma_{N}^{-1}\gamma_{N}=3~{}\mu{\rm m}\frac{\left(10^{-5}~{}{\rm
GeV}\right)^{2}}{|G_{u}|^{2}+|G_{d}|^{2}+|G_{L}|^{2}+|G_{R}|^{2}}\left(\frac{10~{}{\rm
GeV}}{m_{N}}\right)^{6}\left(\frac{E_{N}^{lab}}{200~{}{\rm GeV}}\right).$ (12)

###  3.4 Signal and background at forward experiments

The signature of the production of $N$ vertex is jets similar to the SM
neutral current neutrino interaction. As seen from Eq.(12), the dependence of
the $N$ decay vertex displacement on $m_{N}$ is very strong. For values of the
effective coupling saturating the upper limit, up to $m_{N}<14$ GeV, the
displacement can be large enough to disentangle thanks to the superb position
resolution of 0.4 $\mu$m of FASER$\nu$. Taking $m_{N}>2$ GeV, the displacement
will be smaller than 10 cm and therefore within the size of the detector.
Since $N$ is also relativistic, its decay products will be emitted within a
cone with an opening angle of $m_{N}/E_{N}\sim 10^{-3}-10^{-2}$. Again thanks
to the excellent angular resolution of FASER$\nu$, it can resolve the two jets
associated with $u\bar{u}$ (or $d\bar{d}$) as well as the two jets associated
with $u\bar{d}$ and the charged lepton. The topology of the events are
schematically shown in Fig. 4.

Figure 4: Schematic topology of the signal event. The green arrows show the
final particles from the $N$ decay. The solid green arrows show jets. The
dashed green arrow denotes lepton.

The model therefore predicts two signals:

  * •

A neutral-current-like event associated with the $N$ production plus another
vertex with two jets associated with $N\to\nu_{\mu}u\bar{u}$ or
$\nu_{\mu}d\bar{d}$. The second vertex lies within a cone with an apex at the
$N$ production vertex and an opening angle of $\gamma^{-1}\sim 10^{-2}$. The
angular separation of the two final jets from each other as well as from the
$N$ track (i.e., from the line connecting the two vertices) will be also of
order of $10^{-2}$. The sum of the momentum of the two jets projected onto the
plane perpendicular to the direction of the $N$ momentum (i.e., to the
direction of the line connecting the $N$ production and the $N$ decay
vertices) will be of order of a fraction of $m_{N}$ which should be
compensated with the transverse momentum of the final neutrino which escapes
detection. In almost half of the events, the transverse components of the jet
momenta make an angle smaller than $90^{\circ}$ with each other. For such
events, the transverse missing momentum will be relatively large and close to
$m_{N}/2$. As discussed in sect. 2, if the decay length of the intermediate
$N$ is larger than $\sqrt{2}\sigma_{pos}(2E_{N}/m_{N})\sim 20$ $\mu$m, the
missing transverse momentum can be reconstructed with enough precision to
verify the emission of an invisible particle along with the jets.

  * •

Again a neutral-current-like event associated with the $N$ production followed
by a vertex of two jets plus a muon track (associated with $N\to\mu
u\bar{d}$). The second vertex lies within a cone with an apex at the $N$
production vertex and an opening angle of $\gamma^{-1}\sim 10^{-2}$. In this
case there should not be any missing transverse momentum. By measurement of
the four-momenta of the decay products of $N$ and computing their invariant
mass, the mass of $N$ can be reconstructed. By measuring their total momenta
the energy of $N$ can be derived. Then, if the statistics is enough, the
average displacement of the vertices (average $l$) gives the lifetime of $N$.
The relative error in the extracting the mass and the lifetime of $N$ from
$N_{\mu u\bar{d}}$ number of such types of events will respectively be
$40\%/\sqrt{N_{\mu u\bar{d}}}$ and $2\log 2/\sqrt{N_{\mu u\bar{d}}}$ as shown
in Ref. Bakhti:2020vfq .

Let us now estimate the backgrounds for the signals, starting with the main
source of the background which is accidental alignment of two SM neutrino
scattering vertices. If two separate neutral current events accidentally
happen along the beam direction from each other in one data collecting period,
they can mimic the first signal described above when processing the emulsion.
Similarly, if a $\nu_{\mu}$ charged current event lies ahead of a neutral
current event, it can mimic the second signal described above. The volume of a
cone with a height of $l$ and an opening angle of $\gamma^{-1}$ is $\pi
l^{3}\gamma^{-2}/3$. The probability of one event to lie accidentally within a
cone with the apex at the other vertex is $p=(\pi l^{3}\gamma^{-2}/3)/({\rm
detector~{}size}).$ The numbers of fake signals from pile-up of the SM
vertices are therefore equal to $\mathcal{N}_{NC}^{2}\times p/2$ and
$\mathcal{N}_{NC}\times\mathcal{N}_{\mu}\times p$ where $\mathcal{N}_{NC}$ and
$\mathcal{N}_{\mu}$ are respectively the numbers of SM NC and $\nu_{\mu}$ CC
vertices passing the applied cuts in each data collecting period. Of course,
for heavier $N$, $l$ is smaller and the relevant probabilities decrease very
fast with a factor of $l^{3}\propto(2~{}{\rm GeV}/m_{N})^{18}$. Fig. 5 shows
the numbers of accidental backgrounds for $N\to\nu q\bar{q}$ (marked with NC)
as well as for $N\to\mu q^{\prime}\bar{q}$ (marked with CC) at FASER$\nu$ and
at FASER$\nu$2 versus the displacement of the second vertex, $l$. The second
vertex is supposed to be located within a cone with an apex at the first
vertex and an opening angle of $10^{-2}$, oriented in the forward direction as
depicted in Fig. 4. We have assumed data accumulation periods of 10, 20, 50
and 75 fb-1. To draw the figures we have used the results of Ismail:2020yqc ;
Kling:2021gos  to estimate $\mathcal{N}_{NC}$ and $\mathcal{N}_{CC}$. As seen
from the figure, with $l<$few cm, the accidental background will be
negligible. Notice that for the $m_{N}$ range of our interest, $l<$few cm
corresponds to $G>{\rm few}\times 10^{-7}$ GeV-2 which fortunately coincides
with the coupling range giving rise to fairly significant number of events at
the forward experiments.

Figure 5: Number of accidental backgrounds for the $N\to\nu q\bar{q}$ and
$N\to\mu q^{\prime}\bar{q}$ signals at FASER$\nu$ and FASER$\nu$2 for data
taking periods with collected data of 10, 20, 50 and 75 fb-1. Background
consists of a NC or CC vertex (respectively, marked as NC or CC) located
within a narrow cone of opening angle of $10^{-2}$ and a height of $l$ in
front of another neutral current vertex with a topology and orientation as
depicted in Fig. 4. The solid and dashed lines respectively correspond to
FASER$\nu$ and FASER$\nu$2\.

We now discuss other possible sources of the background. Neutral hadrons such
as neutrons produced in the neutral current events may mimic the signal
associated with $N$ decaying into $\nu q\bar{q}$. The number of such events is
not calculated but is expected to be much smaller than muon induced neutral
hadrons Ismail:2020yqc . Similarly to discriminating between NC events and
neutral hadron induced events Ismail:2020yqc , our signal can be distinguished
from this background by measuring the transverse momentum of visible final
tracks. The interaction length of neutral hadrons is of order of 10 cm.
Inserting a cut of few cm on the displacement of the second vertex as well as
using the criteria of selecting NC events (described in Ismail:2020yqc ) can
eliminate this background. Another potential source of background is displaced
vertex caused by the $\tau$ production by $\nu_{\tau}$ and its subsequent
decay at the detector. By measuring the total charge of the $N$ decay
products, $N$ can be discriminated against the $\tau$ lepton produced by the
charged current interaction of $\nu_{\tau}$. If the decay length of $N$ is of
order of 100-500 $\mu$m, it may be mistaken with neutral $D$-meson decaying
into $\mu$. However, $D$-mesons are mostly produced by charged current
interaction of $\nu_{\mu}$ off the strange sea quarks inside the proton and
neutron. That is the signature of $D$-meson at FASER$\nu$ is a charged current
vertex followed by a displaced vertex. Since in our model the first vertex is
neutral current, $D$-meson production will not constitute a background to our
signals. In principle, $D$-meson can also be produced via neutral current
interaction of neutrinos on intrinsic $c$ quark in the nucleons but the rate
will be too low to cause a significant background to our signal. We therefore
conclude that our signals with $l<$few cm are practically background-free. As
a result, detection of even a single event will be an indication of new
physics..

Fig. 6 shows the bound that can be set on $G_{u}=G_{d}$ versus $m_{N}$ by
FASER$\nu$ and SND@LHC during the run III of the LHC as well as by the upgrade
of FASER$\nu$ during the high luminosity run of the LHC. That is the curves
correspond to number of produced $N$ equal to 1. As seen from the figure in
case of null results, FASER$\nu$ can set a bound of $10^{-6}$ GeV-2 on
$G_{u}=G_{d}$ improving the theoretical limit by a factor of 10. Its upgrade
can improve by another order of magnitude. As shown in the figure, for
$m_{N}<15$ GeV, the dependence of the bound from forward experiments on
$m_{N}$ is mild. If $N$ is lighter than 3 GeV, it should have been already
discovered by lower energy neutrino scattering experiments such as NOMAD
NOMAD:2001xxt . This is yet another reason why we focus on $m_{N}>3$ GeV.

Figure 6: Forecasted upper bound on $G_{u}=G_{d}$ from SND@LHC, FASER$\nu$ and
its upgrade FASER$\nu$2 in case of null signal. We have assumed integrated
luminosity of 150 fb-1 and 3000 fb-1, respectively, for run III of the LHC
(SND@LHC and FASER$\nu$) and for HL-LHC (FASER$\nu$2).

##  4 Neutrino mass, $(g-2)_{\mu}$ and production of the heavy states at the
LHC

As mentioned before, the $Y_{\mu}$ coupling gives a contribution to
$(g-2)_{\mu}$ which can be written as Lavoura:2003xp ; Farzan:2009ji

$\Delta
a_{\mu}=\delta\left(\frac{g-2}{2}\right)=\frac{Y_{\mu}^{2}}{16\pi^{2}}\frac{m_{\mu}^{2}}{m_{\Phi^{+}}^{2}}K(m_{N}^{2}/m_{\Phi^{+}}^{2}),$

where

$K(t)=\frac{2t^{2}+5t-1}{12(t-1)^{3}}-\frac{t^{2}\log t}{2(t-1)^{4}}.$

For $m_{N}^{2}/m_{\Phi^{+}}^{2}\to 0$, $K\to 1/12$ so

$\Delta a_{\mu}=\delta\left(\frac{g-2}{2}\right)=5\times
10^{-10}\left(\frac{Y_{\mu}}{3}\right)^{2}\left(\frac{300~{}{\rm
GeV}}{m_{\Phi^{+}}^{2}}\right)^{2}.$

This contribution has the right sign to explain the anomaly but its magnitude
is too small to explain the deviation of the experimental result from the SM
prediction Muong-2:2021ojo . If we want to explain the $(g-2)_{\mu}$ anomaly,
we have to add more $\Phi$ or $N$ to the model. For example, four $\Phi$ with
couplings of $O(3)$ to the muon and $N$ can account for the deviation. In this
case, the effective couplings can increase by a factor of $\simeq 4$ leading
to a $O(16)$ fold increase in the statistics at forward experiments such as
FASER$\nu$. In case that model has more than one $N$ with masses smaller than
$O(10~{}{\rm GeV})$ and with couplings of $O(3)$ to muon and $\Phi$, the
signal of the model at forward experiments as well as at CMS and ATLAS will be
completely different. We shall discuss the phenomenology of such a variant of
the model with multiple $N$ in sect. 5. Let us now focus on the consequences
of the minimal version with single $N$ and $\Phi$.

In our model, the new scalar does not develop a VEV so neutrinos do not obtain
a Dirac mass term at the tree level. The neutrino mass can receive a
contribution at one loop level provided that (i) $N$ is a Majorana fermion and
(ii) there is a splitting between real (CP-even) and imaginary (CP-odd)
components of $\Phi^{0}$ which originates from a quartic coupling of form
$\lambda_{H\Phi}(H^{\dagger}\cdot\Phi)^{2}+H.c.$ Farzan:2009ji ; Boehm:2006mi
. Thus, if we take $N$ to be of Majorana type, the smallness of neutrino mass
constrains the splitting of the real and complex components of $\Phi^{0}$ or
equivalently $\lambda_{H\Phi}$. The smallness of $\lambda_{H\Phi}$ can be
explained by the global $U(1)$ symmetry introduced in Eq. (4). Such a symmetry
not only explains the smallness of $\lambda_{H\Phi}$ and the splitting but
also explains the smallness of the $u$ and $d$ masses in comparison to those
of higher generation quarks as mentioned before.

At the interaction points of the LHC, the components of $\Phi$ can be pair
produced via electroweak interaction with a cross section of $\sim 10$ fb
Farzan:2010fw . The $\Phi$ components can also be singly produced in
association of a recoiling gluon via the $Y_{u}$ and $Y_{d}$ couplings. Again
the cross section is expected to be of order of 10 fb as
$\alpha_{s}Y_{u}^{2},\alpha_{s}Y_{d}^{2}\sim e^{2}\alpha/\sin^{4}\theta_{W}$.
Since $Y_{\mu}\gg Y_{u},Y_{d},e/\sin\theta_{W}$, the dominant decay modes are
$\Phi^{0}\to N\bar{\nu}_{\mu}$ and $\Phi^{+}\to N\mu^{+}$. Subsequently, $N$
will decay to $\mu^{-}+$two jets or $\nu_{\mu}+$two jets. If $N$ is of
Majorana type, it can also decay into the charged conjugates of these final
states.

Let us shortly discuss two possible signals:

  * •

$\Phi^{+}$ in association of a gluon showing up as $\mu^{+}$ with displaced
vertex of $\mu^{-}+$two jets. If $N$ is of Majorana type, it can even decay
into $\mu^{+}+$two jets, providing a distinctive same sign signature.

  * •

$\Phi^{+}$ in association of a gluon showing up as $\mu^{+}$ with displaced
vertex of two jets+missing energy.

Similarly, $\Phi^{0}$ can be produced in association with the gluon. Moreover,
the pair productions
$\Phi^{+}\Phi^{-},\Phi^{+}\bar{\Phi}^{0},\Phi^{-}{\Phi}^{0},\Phi^{0}\bar{\Phi}^{0}$
and subsequent decays can take place. Exploring all the signals at CMS and
ATLAS and assessing their background is beyond the scope of the present paper.
However, this rich phenomenology with its distinctive signatures sounds very
promising. If FASER$\nu$ finds signals for this model, it will be a great
motivation for a dedicated search in the CMS and ATLAS data for the
distinctive predictions of the present model.

Notice that FASER$\nu$ cannot distinguish the nature of $N$ (i.e., Majorana vs
Dirac). The reason is that there is no way to know whether the detected
$\mu^{-}$ is initiated by $\nu_{\mu}$ via lepton number conserving processes
or by $\bar{\nu}_{\mu}$ via lepton number violating processes involving
Majorana $N$. However, at CMS and ATLAS, same sign muon signals coming from
$\Phi^{+}\to\mu^{+}N\to\mu^{+}\mu^{+}d\bar{u}$ and their charge conjugates
will testify for the lepton number violation and a Majorana type $N$.

##  5 Model with multiple sterile neutrinos

In this section, we shall discuss the phenomenological consequences of a
variant of the model introduced in sect. 4 with more than one right-handed
neutrinos with couplings of form

$Y_{\alpha i}\bar{N}_{i}\Phi^{T}cL_{\alpha}+{\rm H.c.}$ (13)

We shall take $Y_{\alpha i}\sim 3$. As we discussed in sect. 4, adding
multiple $N_{i}$ is motivated by the $(g-2)_{\mu}$ anomaly. We shall take all
$N_{i}$ heavier than $\sim 3$ GeV to avoid the bounds from early universe,
core collapse supernova and meson decay as well as from lower energy neutrino
scattering experiments such a NOMAD. Integrating out the heavy $\Phi$ states,
the coupling in Eq. (13) yields

$G_{ij}^{\mu}(\bar{N}_{i}\mu)(\bar{\mu}N_{j})+G_{ij}^{\nu}(\bar{N}_{i}\nu_{\mu})(\bar{\nu}_{\mu}N_{j})$
(14)

in which

$G_{ij}^{\mu}=\frac{Y_{\mu i}Y_{\mu j}^{*}}{m_{\Phi^{+}}^{2}}\ \ \ {\rm and}\
\ \ G_{ij}^{\nu}=\frac{Y_{\mu i}Y_{\mu j}^{*}}{m_{\Phi^{0}}^{2}}+{\rm H.c}.$
(15)

Taking $Y_{\mu i}\sim Y_{\mu j}\sim 3$ and $m_{\Phi^{+}}\sim m_{\Phi^{0}}\sim
300$ GeV, we find $G_{ij}^{\nu}\sim G_{ij}^{\mu}\sim 10^{-4}~{}{\rm
GeV}^{-2}$. All $N_{i}$ can be produced via the $\nu_{\mu}$ interaction in the
forward experiments as described in the previous section. They can also be
produced via the decay of the $\Phi$ components at the Interaction Point of
the LHC. The lightest $N_{i}$ decays into $\nu$+two jets or $\mu$+two jets as
described for the minimal model in the previous section. The heavier $N_{i}$
will however dominantly decay into lighter $N_{j}$ because $Y_{\mu i}\gg
Y_{u},Y_{d}$:

$N_{i}\to N_{j}\mu\bar{\mu}\ \ \ {\rm and}\ \ \ N_{i}\to
N_{j}\nu_{\mu}\bar{\nu}_{\mu}.$

For $m_{N_{j}}^{2}\ll m_{N_{i}}^{2}$, the decay length will be given by Eq.
(12), replacing $|G_{u}|^{2}+|G_{d}|^{2}+|G_{L}|^{2}+|G_{R}|^{2}$ with
$\sum_{j}(|G_{ij}^{\mu}|^{2}+|G_{ij}^{\nu}|^{2})/3$ where $j$ includes all
$N_{j}$ states lighter than $N_{i}$. The factor of 3 is due to the summation
on color in hadronic decay case. The factor
$\sum_{j}(|G_{ij}^{\mu}|^{2}+|G_{ij}^{\nu}|^{2})/3$ can be $\sim 30$ times
larger than $|G_{u}|^{2}+|G_{d}|^{2}+|G_{L}|^{2}+|G_{R}|^{2}$, making the
decay length 30 times smaller. The whole decay chain of $N_{i}$ will take
place inside the detector. As long as $N_{i}$ is lighter than $8-9$ GeV, the
displacement of the vertex can still be resolved at FASER$\nu$.

In case of $N_{i}\to\mu^{-}\mu^{+}N_{j}$, the di-muon can be resolved at
FASER$\nu$, providing a background-free signal even without resolving the
displacement of the vertex Bakhti:2020szu . For $N_{i}\to\nu\bar{\nu}N_{j}$,
the intermediate vertices in the decay chain of $N_{i}$ particles cannot
however be identified and located. The decay vertex of the final $N$ is
however guaranteed to be resolved because the final vertex involves two
detectable jets. If the lightest $N_{i}$ decays into muon+two jets, the total
momenta of final particles can be measured. If there is a missing transverse
momentum in the plane perpendicular to the line connecting the first neutral
current type vertex and the second $\mu+$two jets vertex, this would indicate
that the final $N$ was produced via decay chain in which $\nu\bar{\nu}$
pair(s) were emitted rather than directly by interaction of $\nu_{\mu}$ from
IP on the Tungsten at the detector.

##  6 Model with a light scalar singlet

Within the model described in the previous sections, the $N$ particles are
produced via $G_{u}$ and $G_{d}$ couplings which are suppressed by
$m_{\Phi^{0}}^{-2}$. If we build a model in which the neutral mediator is
lighter, the statistics can be higher. Let us consider a new singlet scalar
$S$ which has a mixing angle $\theta$ with $\Phi^{0}$. Such a mixing may
originate from trilinear coupling of form $AS^{\dagger}H^{\dagger}\cdot\Phi$.
To preserve the global symmetry introduced in Eq. (4), the relevant $U(1)$
charge of $S$ has to be equal to that of $\Phi$. The cross section of $N$
production shown in Eq. (7) will then be modified with replacement

$\left(\frac{m_{\Phi^{0}}^{2}}{t-m_{\Phi^{0}}^{2}}\right)^{2}\rightarrow\left(\frac{m_{\Phi^{0}}^{2}\cos^{2}\theta}{t-m_{\Phi^{0}}^{2}}+\frac{m_{\Phi^{0}}^{2}\sin^{2}\theta}{t-m_{S}^{2}}\right)^{2}.$
(16)

If $S$ is heavier than a few GeV, the bounds from meson decay can be avoided.
Notice that $m_{\Phi^{0}}^{2}\sin^{2}\alpha$ should not be much larger than
$m_{S}^{2}$; otherwise, the model will suffer from fine tuning. Taking
$|t|<m_{S}^{2}<m_{\Phi^{0}}^{2}$ and a sizable mixing, the replacement as in
Eq (16) enhances the $N$ production cross section and therefore the statistics
by a factor of

$4\times 10^{2}\left(\frac{30~{}{\rm
GeV}}{m_{S}}\right)^{2}\frac{\sin^{2}\theta}{0.2}.$

In sect. 4, we have observed that in the minimal version of the model, the
number of signal events cannot exceed $\sim 200$ for $m_{N}>3$ GeV. A number
of signal events above $O(4000)$ would indicate a light singlet scalar mixed
with $\Phi^{0}$. Such a scalar can be also produced at the Interaction Point
of the LHC via mixing with $\Phi^{0}$. The produced $S$ decays into $N$ and
$\nu_{\mu}$ with signatures that were already discussed in sect. 4.

Let us discuss the possibility of coherent enhancement of the cross section.
As argued before $|t|\sim(10~{}{\rm GeV})^{2}(1-\cos\theta)$. In order for the
amplitudes of the scatterings off various nucleons of Tungsten to sum up
coherently, $|t|$ should be of order of $(0.1~{}{\rm GeV})^{2}$ which
corresponds to $(1-\cos\theta)\stackrel{{\scriptstyle<}}{{\sim}}10^{-4}$. Due
to coherence, an enhancement of $A_{W}=183$ (corresponding to the mass number
of Tungsten) is expected. As long as $m_{S}\gg 0.1$ GeV, no enhancement in the
amplitude in this region is expected so the contribution of the coherent
scattering with $|t|<(0.1~{}{\rm GeV})^{2}$ to the total cross section of
scattering off Tungsten nucleus will be of order $A_{W}\Delta\cos\theta\sim
10^{-2}$ and therefore negligible.

##  7 Summary and discussion

We have proposed a model in which the left-handed doublet,
$L_{\mu}=(\nu_{\mu},\ \mu_{L})$ couples to new scalar doublet(s),
$\Phi=(\Phi^{+},\ \Phi^{0})$ and right-handed neutrino(s), $N$. If the
components of $\Phi$ are heavier than $\sim 300$ GeV and their coupling to
quarks is of order of $O(0.3)$ or smaller, they can escape the bounds from
direct production at the colliders as well as the bounds from precision data,
such as oblique parameters. Satisfying these bounds, the effective coupling
between $L_{\mu}$ and quarks after integrating out the heavy $\Phi$ components
can be as large as $10^{-5}$ GeV-2. In the minimal version of the model with
only a single $N$, the signatures of the model at forward experiments will be
a multiple jet vertex due to $N$ production (similar to that appearing due to
the neutral current interaction of neutrinos in the SM), followed by a
displaced vertex within a cone with an apex at the first vertex aligned in the
direction of the beam with a small opening of size $10^{-2}$. The topology of
the event is shown in Fig. 4. The decay of $N$ can produce either two jets
plus a charged muon $(u\bar{d}\mu)$ or two jets plus muon neutrino
($u\bar{u}\nu_{\mu}$ or $d\bar{d}\nu_{\mu}$). In the former case, all the
final particles are detectable so the mass of $N$ can be reconstructed at
forward experiments by measuring the four momenta of the final particles. In
the case of $N\to u\bar{u}\nu_{\mu},d\bar{d}\nu_{\mu}$, we have formulated the
condition under which the emission of $\nu_{\mu}$ can be observationally
distinguished by measurement of the transverse momenta of the final jets.

Notice that $\Phi$ and $N$ can also couple to the first and third generations
of left-handed leptons, $L_{e}$ and $L_{\tau}$. We have however focused on the
second generation as it is less constrained than the first generation.
Moreover, with a coupling to the second generation, the possibility of an
observable signal at forward experiments is higher thanks to the larger fluxes
of $\nu_{\mu}$ and $\bar{\nu}_{\mu}$ compared to those of $\nu_{e}$,
$\bar{\nu}_{e}$, $\nu_{\tau}$ and $\bar{\nu}_{\tau}$.

We have shown that the minimal model with only single $N$ and $\Phi$ can at
most account for 25 % of the $(g-2)_{\mu}$ anomaly but by adding more
generations of $\Phi$ and/or $N$, the anomaly can be completely explained. In
this case, the heavier $N$ will go through chain decays to the lightest $N$,
emitting either $\mu^{-}\mu^{+}$ pairs or $\nu_{\mu}\bar{\nu}_{\mu}$ pairs in
the process. The final $N$ will decay into jets plus a lepton, similarly to
the case of single $N$. The $\mu^{-}\mu^{+}$ pair emitted through chain decay
can be of course observed at forward experiments. We have shown that even in
case that the decays of intermediate $N$ particles produce only invisible
$\nu_{\mu}\bar{\nu}_{\mu}$, their production can be confirmed by measuring the
transverse momentum of the final particles.

We have argued that in both minimal version of the model with a single $N$ and
in its variants with multiple $N$, the predicted signals at FASER$\nu$ will be
background free. As a result, even a single event observed at forward
experiments can count as discovery. We have shown that in the case of null
results at FASER$\nu$ or SND@LHC, the bound on the relevant effective
couplings can be lowered down to $10^{-6}$ GeV-2. If $N$ is heavier than
$O(3)$ GeV, it could not be produced at lower energy scattering experiments
such as NOMAD NOMAD:2001xxt , CHORUS CHORUS:2007wlo , Miner$\nu$a
MINERvA:2021csy , CHARM II CHARMII:2008qag  and MicroBooNE MicroBooNE:2021fdt
. For the same reason, the future state-of-the-art DUNE experiment cannot test
the model, either. The energy of the neutrino beam at NuTeV experiment
Dore:2018ldz  was few 100 GeV so $N$ particles heavier than 3 GeV could be
produced at this experiment but the spatial resolution of NuTeV NuTeV:1999uck
was not fine enough to disentangle the second displaced vertex so the signals
would be mistaken for SM NC and CC vertices. IF FASER$\nu$ discovers signals
for $N$, the NuTeV data should be re-analyzed to correct parton distribution
function derived from it. Thus, the bound from FASER$\nu$ and SND@LHC will be
the strongest, only to be surpassed by the bounds to be provided by their own
upgrades. The upgrade of FASER$\nu$ for the high luminosity phase of the LHC
(FASER$\nu$2) can improve the bound down to $10^{-7}$ GeV-2.

If a discovery is made by forward experiments, it will be a strong motivation
for customized searches for the $\Phi^{+}$ and $\Phi^{0}$ production at the
CMS and ATLAS. These particles can be pair produced via electroweak
interactions or can be singly produced in association of a gluon via their
Yukawa couplings to the quarks at Interaction Point. They will then go through
decays as $\Phi^{+}\to N\mu^{+}\to{\rm two~{}jets}+\mu^{+}\mu^{-}$ or $\to{\rm
two~{}jets}+\mu^{+}\nu_{\mu}$ and $\Phi^{+}\to N\bar{\nu}_{\mu}\to{\rm
two~{}jets}+\mu^{-}\nu_{\mu}$ or $\to{\rm
two~{}jets}+\bar{\nu}_{\mu}\nu_{\mu}$. Moreover, if $N$ is a Majorana
particle, it can decay into $\mu^{+}$ instead of $\mu^{-}$, producing a same
sign muon signal. Thus, the Majorana nature of $N$ can be established by CMS
and ATLAS via detecting same sign muon signals.

If $N$ is of Majorana type, its couplings to $\nu_{\mu}$ can contribute to the
$\mu\mu$ component of neutrino mass matrix. Then, the smallness of neutrino
mass imposes a bound on the mass splittings between the CP-even and CP-odd
components of $\Phi^{0}$. We have devised a global U(1) symmetry explaining
this smallness. The same symmetry can also explain the smallness of the masses
of the first generation quarks as a bonus.

The $N$ particles can also be produced by high energy atmospheric neutrinos
scattering off nuclei inside the neutrino telescopes. The production of $N$
will lead to a cascade similar to those produced by the SM neutral current
interactions. Since the cross section of the new interactions is at most 10 %
of the cross section of the SM neutral current, the new physics can account
for less than $10\%$ of the cascade events registered by ICECUBE. The produced
$N$ will decay after traveling $\sim 2~{}{\rm m}(3~{}{\rm
GeV}/m_{N})^{6}(E_{N}/100~{}{\rm TeV})$. For heavy $N$, the decay length will
be too short to be resolved by ICECUBE. Moreover, the particles from the $N$
decay will be emitted almost parallel, making a small angle of
$m_{N}/(2E_{N})$ with each other. Since their total electric charge is zero,
it will be like propagation of a neutral particle in ice so the Cherenkov
emission from the particles of the $N$ decay may be too faint to be detected
at the neutrino telescopes.

###### Acknowledgements.

This project has received funding /support from the European Union′s Horizon
2020 research and innovation programme under the Marie Sklodowska -Curie grant
agreement No 860881-HIDDeN. YF has received financial support from Saramadan
under contract No. ISEF/M/400279 and No. ISEF/M/99169. SA is supported by a
grant from Basic Sciences Research Fund (No. BSRF-phys-399-01).

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