Constraining the transitional magnetic dipole moment interaction between active and heavy neutrinos with the $Z'$ at FASER$\nu$

Abstract

We investigate the effects of the transitional magnetic dipole type interactions of the active-to-heavy-neutrino associated with a new neutral gauge boson $Z'$ on neutrino-nucleon scattering at the ForwArd Search ExpeRiment-$\nu$ (FASER$\nu$). We consider the neutral-current neutrino-nucleon scattering ($\nu A\rightarrow N A'$, where $A'$ denotes the disintegrated nuclei) as the production mechanism of the heavy neutrino N at FASER$\nu$ and estimate the sensitivity reach on the magnetic-type dipole coupling $\omega _{\nu _\alpha }$ for a range of heavy neutrino mass ($1\,\textrm{GeV}< \textrm{M}_{\textrm{N}} < 70 \mathrm{~GeV}$). 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.

1 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 [1, 2], 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 and y is the O(1) Yukawa coupling. There are other extensions [3, 4] 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 [5,6,7,8,9,10,11,12,13] 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 ms 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 the 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$\nu$ [5]. It is a $25\,\textrm{cm} \times 30\, \textrm{cm} \times 1.1\,\textrm{m}$ emulsion detector, consisting of 770 layers of emulsion films interleaved with 1-mm-thick tungsten plates with mass 1.2 tons [6]. 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) [5] and neutral-current (NC) interactions [14, 15]. The total cross-section $\sigma _{\nu A}$ (where A denotes 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 Literature [13, 16,17,18,19,20,21,22,23,24,25,26,27,28,29,30].

In this work, we consider a scenario of a heavy neutrino N with the presence of a light U(1) gauge boson $Z'$ [31,32,33]. The heavy neutrino N can couple to the active neutrinos via a transitional magnetic dipole type interaction of the $Z'$ boson: $\omega _{\nu _\alpha } {\overline{N}} \sigma ^{\mu \nu } \nu _\alpha \, Z'_{\mu \nu }$, where $Z'_{\mu \nu }$ is the field strength of the $Z'$ boson and $\omega _{\nu _\alpha }$ is the $Z'$-magnetic dipole moment. Such interaction enables sizable production of heavy neutrinos at FASER$\nu$ and also detectable decays via $N \rightarrow \nu _\alpha Z'$ followed by $Z'$ decays into the 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.¹ The advantage of including the $Z'$ interaction with the heavy neutrinos is to enable either prompt or detectable long decays within the FASER detector. Here the $Z'$ 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. Note that strictly speaking the transitional magnetic dipole interaction associated with $Z'$ is not exactly the same as the transitional magnetic dipole moment $\mu _\nu {\bar{N}} \sigma ^{\mu \nu } \nu F_{\mu \nu }$, where $F_{\mu \nu }$ is the electromagnetic field strength that has been measured in a number of experiments. In the following, we recast the current limits on $\mu _\nu$ into the $Z'$ magnetic dipole moment $\omega _{\nu }$.

We identify the events of heavy neutrinos through their decays into a pair of charged leptons, either in prompt or displaced decays, as well as into hadrons. The smaller the coupling $\omega _{\nu _\alpha }$ the longer the decay length will be. Thus, using the length of the whole FASER detector we can estimate the range of $\omega _{\nu _\alpha }$ 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. [16], in which the sterile neutrino is produced through the magnetic-dipole operator via the active neutrino up-scattering, mostly via the electron scattering channel $\nu e^- \rightarrow N e^-$. It was shown that [16] the High-Luminosity LHC can search for sterile neutrinos with an upgraded FASER$\nu$ experiment.

A study of the transitional magnetic dipole moment interactions between the active neutrinos and new sterile states at emulsion and liquid argon experiments was reported in [30], 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. Reference [30] obtained the 90% C.L. contour curve for the active-to-heavy-neutrino transitional magnetic dipole moment with heavy neutrino mass $M_N$ ranging from $10^{-3}~\textrm{GeV}~\textrm{to}~10~ \textrm{GeV}$ at the Forward Physics Facility detectors.

We summarize the differences from Ref. [30] as follows.

• 1. The transitional active-to-heavy neutrino dipole moment via photon $\mu _\nu {\bar{N}} \sigma ^{\mu \nu } \nu \, F_{\mu \nu }$ while we consider the interaction via a $Z'$: $\omega _\nu {\bar{N}} \sigma ^{\mu \nu } \nu \, Z'_{\mu \nu }$.

• 2. In Ref. [30], $\mu _\nu$ was constrained by the scattering with the electron: $\nu e^- \rightarrow N e^-$ while in our work $\omega _\nu$ is constrained by the scattering with nucleon: $\nu q \rightarrow N q'$. Since the number of protons and neutrons are numerous inside the tungsten nucleus, the corresponding scattering rate is larger.

• 3. Detection of the events is also different. In Ref. [30], the recoiled electron is detected and the decay length of N is classified into prompt, displaced, or unobserved clearly. On the other hand, most of the signal events in our work are prompt given the mass range of N that we consider.

• 4. The limits obtained in our work show improvements over those in Ref. [30].

The organization of the work is arranged as follows. We describe the effective interactions considered in the next section. In Sect. 3, we calculate the production rates of the heavy neutrino N via $\nu A$ scattering at FASER$\nu$. In Sect. 4, 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 Sect. 5.

2 The heavy neutrino model

The new matter content includes a heavy neutrino N and a U(1) gauge boson $Z'$. The effective Lagrangian described the interactions of N and $Z'$ with the standard model (SM) gauge bosons [34,35,36,37,38,39] and other particles [40,41,42,43,44] can be written as

$$\begin{aligned} {\mathcal {L}}_{\textrm{eff}}= & {} \sum _{\alpha =e,\mu ,\tau } \Biggr [ \omega _{\nu _\alpha } {\overline{N}}\sigma ^{\mu \nu }\nu _{{\alpha }} Z'_{\mu \nu }- \frac{g}{\sqrt{2}}V_{\alpha N}{\overline{N}} \gamma ^\mu P_L l_{\alpha } W_\mu ^+\nonumber \\{} & {} - \frac{g}{ \cos \theta _w} V_{\alpha N}{\overline{N}} \gamma ^\mu P_L\nu _{\alpha } Z_\mu +\mathrm{H.c.} \Biggr ] \nonumber \\{} & {} - \sum _{q,\nu ,l} \left[ g_{\textrm{q}}\bar{q}\gamma ^{\mu }q+ g_{\nu } {\bar{\nu }} \gamma ^{\mu } P_L \nu +g_{l} \bar{l} \gamma ^{\mu } l \right] Z'_{\mu } , \end{aligned}$$

(1)

where $g=e/\sin \theta _w$, $\theta _w$ is the Weinberg angle, $Z'_{\mu \nu }$ is the field strength of the $Z'$ 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 $\omega _{\nu _\alpha }$ denotes the coupling of the active-to-heavy-neutrino $Z'$ 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 $\omega _{\nu _\alpha }$ above the electroweak scale requires a Higgs insertion so that the $Z'$ transitional magnetic dipole interaction described in Eq. (1) originates from a dimension-6 operator. For example, an $SU(2)_L$ invariant operator

$$\begin{aligned} {{{\mathcal {L}}}} = \frac{e}{\Lambda ^2} \, Z'_{\mu \nu } \, {\overline{N}} \sigma ^{\mu \nu } {\tilde{\Phi }}^\dagger L + \mathrm{H.c.} {,} \end{aligned}$$

(2)

where L is the lepton doublet, ${\tilde{\Phi }} = i\tau ^2 \Phi ^*$ and $\Phi$ is the SM Higgs doublet. When the Higgs field takes on the vacuum expectation value, the operator becomes the $Z'$ transitional magnetic dipole moment interaction in Eq. (1). The $\omega _{\nu _\alpha }$’s are therefore envisioned to be of the order of $1/\Lambda ^2$ in form of $\omega _{\nu _\alpha } \sim \frac{e.vev}{\Lambda ^2}$. Note that there are existing bounds on the transitional magnetic moment dipole interaction $\mu _\nu {\overline{N}} \sigma ^{\mu \nu } \nu \, F_{\mu \nu }$, where $F_{\mu \nu }$ is the electromagnetic field strength, such as those from Borexino [45], XENON1T [46], LSND [47], MiniBooNE [47], CHARM-II [48], NOMAD [47], and LEP [47], for a wide mass range of N. For the mass range of our interests, the relevant ones come from NOMAD and LEP, namely, $\mu _{\nu _e}, \mu _{\nu _\tau } \lesssim 1.5 \times 10^{-7} \, \mu _\textrm{B}$, and $\mu _{\nu _\mu } \lesssim 5 \times 10^{-9} - 1.4 \times 10^{-7}\, \mu _\textrm{B}$ for $M_N = 1{-} 10$ GeV, where $\mu _\textrm{B}$ is the Bohr magneton. Note that the transitional magnetic dipole type interaction considered in this work is not exactly the same, because the $Z'_{\mu \nu }$ is associated with a new U(1) different from the $U(1)_{\textrm{EM}}$. Nevertheless, we can recast the current bounds on the transitional magnetic moment $\mu _{\nu _\alpha }$ [47] into the $Z'$ transitional magnetic-type moment $\omega _{\nu _\alpha }$ by

$$\begin{aligned} \omega _{\nu _\alpha }\simeq \frac{ (1\,\textrm{GeV})^2+M_{Z'}^2}{ (1\,\textrm{GeV})^2}\times \mu _{\nu _\alpha } \end{aligned}$$

(3)

where we assume the square of momentum transfer in the photon propagator is about $(1\, \textrm{GeV})^2$. The $Z'$ transitional magnetic-type moment $\omega _{\nu _\alpha }$ in Eq. (3), tends to transitional magnetic moment $\mu _{\nu _\alpha }$ when $M_{Z'}\rightarrow 0$. The upper bounds on $\omega _{\nu _\alpha }$ for $\alpha = e, \mu , \tau$ are listed in Table 1.

Table 1 The upper limits on $\omega _{\nu _\alpha }$ recast from the transition magnetic moment $\mu _{\nu _\alpha }$ from [47] using Eq. (3) for $M_{Z'} = 0.1, 1, 10$ GeV. Here Bohr magneton $\mu _\textrm{B}$ is given by $\mu _\textrm{B} = e \hbar /2m_e$, which is about $3 \times 10^{2}\;\textrm{GeV}^{-1}$ in natural units. The electron ($\mu _{\nu _e}$) and tau ($\mu _{\nu _\tau }$) transitional magnetic moments share the same numerical value. Consequently, the $Z'$ transitional magnetic moments associated with the aforementioned neutrino flavors ($\omega _{\nu _e}$ and $\omega _{\nu _\tau }$) also possess the same value, as indicated by Eq. (3)

The effective Lagrangian consists of two widely used beyond the SM models: (i) the renormalizable $Z'$ model with flavor-conserving quark, lepton and neutrino interactions [40,41,42,43,44] and (ii) an effective/simplified extension by introducing a right-handed (RH) neutrino, which is a singlet under the SM gauge symmetry [37,38,39], with a transitional magnetic dipole type interactions connecting the active and the heavy neutrino. Here we assume that $q = u, d$ have equal coupling strength $g_q$ with the $Z'$, $\nu = \nu _e,\nu _\mu , \nu _\tau$ have equal strength $g_\nu$ with the $Z'$, and equal coupling strength $g_l$ for $l=e,\mu ,\tau$.

Fig. 1

Production and decays of the heavy neutrino N at FASER$\nu$ detector. ($\nu q \rightarrow N q$ followed by $N \rightarrow \nu f \bar{f}'$, where $f,f'$ are SM fermions.)

3 Heavy neutrino production at FASER$\nu$

Fig. 2

Production cross sections of the heavy neutrino N at FASER$\nu$: $\sigma _{\nu _{\mu } A} = p\sigma _{\nu _{\mu } p}+n\sigma _{\nu _{\mu } n}$. Note that each $M_{Z'}$ curve is computed using different values of $\omega _{\nu _\mu }$ recast from $\mu _{\nu _\mu }$ bounds [$\omega _{\nu _\mu }(M_{Z'}=0.1 ~\textrm{GeV})=4.10\times ~10^{-9}\mu _\textrm{B}$, $\omega _{\nu _\mu }(M_{Z'}=1~\textrm{GeV})=8.12\times ~10^{-9}\mu _\textrm{B}$, and $\omega _{\nu _\mu }(M_{Z'}=10~\textrm{GeV})=4.10\times ~10^{-7}\mu _\textrm{B}$] shown in Table 1 and $g_q = 1$. That is why the cross-section curves do not follow any particular pattern in relation to the mass of $Z'$. 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 $E_\nu$ = 10,000 GeV (bottom-right), respectively. The blue, orange and green lines represent the production cross sections of the heavy neutrino N at FASER$\nu$ with $M_{Z'}= 0.1, 1$ and 10 GeV respectively. We have used NNPDF3.1NNLO PDF [49] for parton distribution functions

Fig. 3

Production cross sections of the heavy neutrino N at FASER$\nu$: $\sigma _{\nu _{\alpha } A} = p\sigma _{\nu _{\alpha } p}+n\sigma _{\nu _{\alpha } n}$, where $\alpha =e,\tau$. The production cross-section of Heavy Neutrino from the incoming electron and tau neutrino is the same. Note that each $M_{Z'}$ curve is computed using different values of $\omega _{\nu _{\alpha }}$ recast from $\mu _{\nu _\alpha }$ bounds [$\omega _{\nu _\alpha }(M_{Z'}=0.1 ~\textrm{GeV})=1.39\times ~10^{-7}\mu _\textrm{B}$, $\omega _{\nu _\alpha }(M_{Z'}=1~\textrm{GeV})=2.76\times ~10^{-7}\mu _\textrm{B}$, and $\omega _{\nu _\alpha }(M_{Z'}=10~\textrm{GeV})=1.39\times ~10^{-5}\mu _\textrm{B}$] shown in Table 1 and $g_q = 1$. That is why the cross-section curves do not follow any particular pattern in relation to the mass of $Z'$. 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 $E_\nu$ = 10,000 GeV (bottom-right), respectively. The blue, orange, and green lines represent the production cross sections of the heavy neutrino N at FASER$\nu$ with $M_{Z'}= 0.1$, 1 and 10 GeV respectively. We have used NNPDF3.1NNLO PDF [49] for parton distribution functions

In this section, we discuss heavy neutrino production at FASER$\nu$ detector from the neutrino-nucleon scattering $\nu A\rightarrow N A'$ mediated by a new neutral gauge boson $Z'$. 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 $Z'$ boson, which resembles the transitional magnetic dipole 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 [16, 18, 28].

We use ${\mathrm{MadGraph5aMC@NLO}}$ [50, 51] for the computation of fixed-target neutrino-nucleon scattering $\nu A\rightarrow N A'$ cross-sections. We build the model file for Eq. (1) using FeynRules [52]. For the production process, we consider 3 different neutral gauge boson masses, $M_{Z'}=0.1,1,10$ 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 $Z'$-quark coupling strength $g_q=1$ and the value of $\omega _{\mu _\alpha }$ is fixed by Eq. (3) using the current bounds of $\mu _{\nu _\alpha }\;(\mu _{\nu _\mu }\simeq 10^{-8}\mu _{B},\mu _{\nu _e}=\mu _{\nu _\tau }\simeq 10^{-7}\mu _{B})$ [47]. The production cross-section simply scales with $g_q^2$ and $\omega _{\nu _\alpha }^2$. Figures 2 and 3 shows the muon neutrino-nucleon and electron(tau) neutrino-nucleon fixed target cross-sections for different neutrino beam energies $E_\nu = 10{ -} 10{,}000$ GeV respectively. In each case, the cross-section decreases with an increment of $M_N$. Each $M_{Z'}$ curve is using different values of $\omega _{\nu _{\alpha }}$ recast from $\mu _{\nu _\alpha }$ bounds. In the calculation, we are using different values of $\omega _{\nu _\alpha }$ (see Table 1) for each $M_{Z'}$, that is why the cross-section curves are shown in Figs. 2 and 3 do not follow any particular pattern in relation to the mass of $Z'$. With the given value of $\omega _{\nu _\alpha }$ in Table 1, the cross-sections with $M_{Z'}=10$ GeV give the largest value, while the curves with $M_{Z'}= 1$ GeV give the smallest cross-section values and the curves with $M_{Z'}=0.1$ GeV have intermediate values for the whole heavy neutrino mass $M_N$ and the whole neutrino energy $E_\nu$. (Note that there is a cross-over in the cross-section curves for the incoming neutrino energy $E_\nu =10$ TeV with $M_{Z'}= 1, 0.1$ GeV. The values of $\omega _{\nu _\alpha }$ that we used result in different cross-sections for each neutrino flavor. With the given values of $\omega _{\nu _\alpha }$ used, the neutrino-nucleon scattering cross sections with the incoming electron neutrino $\nu _e$ and incoming tau neutrino $\nu _\tau$ are larger than that of the incoming muon neutrino $\nu _\mu$, even though the incoming muon neutrino flux is the highest among the neutrino fluxes. Production Cross-Section of different Heavy Neutrinos across the incoming Neutrino Energy Range is shown in Appendix A.

Here we have assumed the benchmark detector specification made of tungsten with dimensions $25\,\textrm{cm} \times 30\, \textrm{cm}\times 1.1\, \textrm{m}$ at the 14 TeV LHC with an integrated luminosity of $L = 150\,\textrm{fb}^{-1}$. We use the neutrino fluxes and energy spectra obtained in [5, 7] to study the neutrinos that pass through FASER$\nu$. It is well known that a huge number of hadrons, such as pions, kaons, and other hadrons, are produced along the beam direction. These hadrons will decay during the flight, thus producing a lot of neutrinos of all three flavors at very high energy up to a few TeV [7]. It was shown [11] that 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.

The flavors of the neutrinos $\nu _e$, $\nu _\mu$, and $\nu _\tau$ can be distinguished by identifying the $e,\mu$ and $\tau$ leptons produced in the interactions. Electrons are identified by detecting electromagnetic showers along a track. If a shower is found, the first film with activity will be checked to see if there is a single particle (an electron) or an $e^+ e^-$ pair (from the conversion of a $\gamma$ from a $\pi ^0$ decay). The separation of single particles from particle pairs will be performed based on the energy deposit measurements. Muons are identified by their track length in the detector. Since the detector has a total nuclear interaction length of $10.1\lambda _{int}$, all the hadrons from the neutrino interactions will interact, except for hadrons created in the far downstream part of the detector. Tau leptons are identified by detecting their short-lived decays [5].

Fig. 4

Total numbers of heavy neutrinos produced by scattering with the tungsten target at FASER$\nu$ via the processes $\nu _{\alpha } A \rightarrow N A'$ and ${\bar{\nu }}_{\alpha } A \rightarrow \bar{N} A'$ scattering $(\alpha = e, \mu ,\tau )$, computed with the values of $\omega _{\nu _\alpha }$[For muon neutrino ($\nu _\mu$): $\omega _{\nu _\mu }(M_{Z'}=0.1 ~\textrm{GeV})=4.10\times ~10^{-9}\mu _\textrm{B}$, $\omega _{\nu _\mu }(M_{Z'}=1~\textrm{GeV})=8.12\times ~10^{-9}\mu _\textrm{B}$, and $\omega _{\nu _\mu }(M_{Z'}=10~\textrm{GeV})=4.10\times ~10^{-7}\mu _\textrm{B}$; For electron ($\nu _e$) and tau ($\nu _\tau$) neutrinos: $\omega _{\nu _\alpha }(M_{Z'}=0.1 ~\textrm{GeV})=1.39\times ~10^{-7}\mu _\textrm{B}$, $\omega _{\nu _\alpha }(M_{Z'}=1~\textrm{GeV})=2.76\times ~10^{-7}\mu _\textrm{B}$, and $\omega _{\nu _\alpha }(M_{Z'}=10~\textrm{GeV})=1.39\times ~10^{-5}\mu _\textrm{B}$] shown in Table 1. Top-left: $\alpha = \mu$; Top-right: $\alpha = e$; Bottom-left: $\alpha = \tau$

Fig. 5

Schematic diagram of FASER$\nu$-FASER [6, 17,18,19,20,21, 24,25,26,27,28]

Fig. 6

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

Fig. 7

Background event rates, top left: muon events $\nu _{\mu }~A\rightarrow A'~\nu _{\mu }\mu ^+\mu ^-$ (solid and dashed line represent the neutrino and antineutrino events respectively). Top right: electron events $\nu _{e}~A\rightarrow A'~\nu _{e}e^+ e^-$. Bottom left: tau events, $\nu _{\tau }~A\rightarrow A'~\nu _{\tau }\tau ^+\tau ^-$

Fig. 8

Dominant branching fractions of the heavy neutrino N as a function of its mass $M_{N}$ for nonzero $V_{\mu N} =10^{-4}$, $g_f = 1$ for $f = q,\nu ,l$, and the value of $\omega _{\nu _\mu }$ [$\omega _{\nu _\mu }(M_{Z'}=0.1 ~\textrm{GeV})=4.10\times ~10^{-9}\mu _\textrm{B}$, $\omega _{\nu _\mu }(M_{Z'}=1~\textrm{GeV})=8.12\times ~10^{-9}\mu _\textrm{B}$, and $\omega _{\nu _\mu }(M_{Z'}=10~\textrm{GeV})=4.10\times ~10^{-7}\mu _\textrm{B}$] shown in Table 1, and for $M_{Z'}$ =0.1 (top-left), 1 (top-right), 10 (bottom-left) GeV

Fig. 9

Sensitivity reach to $\nu _\mu$-to-heavy-neutrino $Z'$ transitional magnetic dipole moment $\omega _{\nu _\mu }$ at FASER$\nu$ as a function of the heavy neutrino mass $M_N$ with different $Z'$ mass by considering only charged leptons in the final state. Note that we recast the current limits on the conventional transitional magnetic dipole coupling $\mu _{\nu _{\mu }}$ [47] from LEP 91 GeV and NOMAD to the $Z'$ transitional magnetic dipole moment $\omega _{\nu _{\mu }}$ using Eq. (3). Top-left: $M_{Z'} =~0.1\,\textrm{GeV}$; Top-right: $M_{Z'} =~1\,\textrm{GeV}$; bottom-left: $M_{Z'} =~10\,\textrm{GeV}$. The blue and orange curves represent the bound of $\omega _{\nu _\mu }$ with the number of background events equal to BG = 1 and 2, respectively

Fig. 10

Dominant branching fractions of the heavy neutrino N as a function of its mass $M_{N}$ for nonzero $V_{e N}=10^{-4}$, $g_f = 1$ for $f = q,\nu ,l$ and the value of $\omega _{\nu _e}$ [$\omega _{\nu _e}(M_{Z'}=0.1 ~\textrm{GeV})=1.39\times ~10^{-7}\mu _\textrm{B}$, $\omega _{\nu _e}(M_{Z'}=1~\textrm{GeV})=2.76\times ~10^{-7}\mu _\textrm{B}$, and $\omega _{\nu _e}(M_{Z'}=10~\textrm{GeV})=1.39\times ~10^{-5}\mu _\textrm{B}$] shown in Table 1, and for $M_{Z'}$ =0.1 (top-left), 1 (top-right), 10 (bottom-left) GeV

Fig. 11

Sensitivity reach to $\nu _e$-to-heavy-neutrino $Z'$ transitional magnetic dipole moment $\omega _{\nu _e}$ at FASER$\nu$ as a function of the heavy neutrino mass $M_N$ with different $Z'$ mass by considering only charged leptons in the final state. Note that we recast the current limits on the conventional transitional magnetic dipole coupling $\mu _{\nu _{e}}$ [47] from LEP 91 GeV to the $Z'$ transitional magnetic dipole moment $\omega _{\nu _{e}}$ using Eq. (3). Top-left: $M_{Z'} =~0.1\,\textrm{GeV}$; Top-right: $M_{Z'} =~1\,\textrm{GeV}$; bottom-left: $M_{Z'} =~10\,\textrm{GeV}$. BG = 1 and 2 represent the number of background events equal to 1 and 2, respectively. The blue and orange curves represent the bound of $\omega _{\nu _e}$ with the number of background events equal to 1 and 2, respectively

Fig. 12

Dominant branching fractions of the heavy neutrino N as a function of its mass $M_{N}$ for nonzero $V_{\tau N}= 10^{-4}$, $g_f = 1$ for $f = q,\nu ,l$ and the value of $\omega _{\nu _\tau }$ [$\omega _{\nu _\tau }(M_{Z'}=0.1 ~\textrm{GeV})=1.39\times ~10^{-7}\mu _\textrm{B}$, $\omega _{\nu _\tau }(M_{Z'}=1~\textrm{GeV})=2.76\times ~10^{-7}\mu _\textrm{B}$, and $\omega _{\nu _\tau }(M_{Z'}=10~\textrm{GeV})=1.39\times ~10^{-5}\mu _\textrm{B}$] shown in Table 1, and for $M_{Z'}$ = 0.1 (top-left), 1 (top-right), 10 (bottom-left) GeV

Fig. 13

Sensitivity reach to $\nu _\tau$-to-heavy-neutrino $Z'$ transitional magnetic dipole moment $\omega _{\nu _\tau }$ at FASER$\nu$ as a function of the heavy neutrino mass $M_N$ with different $Z'$ mass by considering only charged leptons in the final state. Note that we recast the current limits on the conventional transitional magnetic dipole coupling $\mu _{\nu _{\tau }}$ [47] from LEP 91 GeV to the $Z'$ transitional magnetic dipole moment $\omega _{\nu _{\tau }}$ using Eq. (3). Top-left: $M_{Z'} =~0.1\,\textrm{GeV}$; Top-right: $M_{Z'} =~1\,\textrm{GeV}$; bottom-left: $M_{Z'} =~10\,\textrm{GeV}$. BG = 1 and 2 represent the number of background events equal to 1 and 2, respectively. The blue and orange curves represent the bound of $\omega _{\nu _\tau }$ with the number of background events equal to 1 and 2, respectively

The expected number of heavy neutrino events from the three flavors of active neutrinos with three different $Z'$ masses in FASER$\nu$ is shown in Fig. 4. With the values of $\omega _{\nu _{\alpha }}$ shown in Table 1, the highest number of heavy neutrino events comes in with the electron neutrino $\nu _e$ scattering with tungsten nucleus ($\nu _{e} A \rightarrow N A'$), while the lowest one is from the muon neutrino $\nu _\mu$ scattering ($\nu _{\mu } A \rightarrow N A'$), and that from the tau neutrino $\nu _\tau$ scattering is in between. The heavy neutrino event rate follows the same trend of the cross-section curves with different values of $M_{Z'}$, heavy neutrino mass $M_N$, and neutrino flavor. The event rate with $M_{Z'}=10$ GeV is the highest in number for all incoming neutrino flavors, whereas the rate with $M_{Z'}=1$ GeV is the lowest and the event curves with $M_{Z'}=0.1$ GeV lies in between the two.

4 FASER$\nu$-FASER sensitivity towards active to heavy neutrino $Z'$ transitional magnetic dipole moment

Since the neutrino interaction is rather weak (in the SM sense), the heavy neutrinos N will not be entirely produced right after the incoming neutrinos hit the target at FASER$\nu$, instead, the N can be produced throughout the length of FASER$\nu$. The N so produced inside FASER$\nu$ can decay promptly or travel some distance before decay. The decay can be regarded as a long decay if we can reconstruct a displaced vertex of charged leptons or hadrons inside FASER$\nu$ or the decay volume of FASER. The construction of FASER$\nu$ allows a decay length longer than about $1\,\upmu \textrm{m}$ to be identified. Such a long decay, in principle, does not have any SM background given the precision of the decay vertex is high enough. This actually can happen if the coupling to $Z'$ is small enough that the decay length is longer than the resolution. This is an interesting possibility of exploring neutrino anomalous interactions, by which a new particle is produced at FASER$\nu$ and decays inside FASER$\nu$ or the decay volume of FASER. In our case, since both production and decay via the $Z'$ interactions, the $Z'$ transitional magnetic dipole moment has to be large enough for a meaningful production rate and thus the decay of N is mostly prompt. Only for a small mass range $M_N < 2$ GeV for $M_{Z'} =0.1,1$ GeV and $M_N < 10$ GeV for $M_{Z'} = 10$ GeV that the decay length of N is longer than $1\, \upmu \textrm{m}$.

The heavy neutrino produced at the FASER$\nu$ detector could decay into visible channels – charged leptons and quarks. The decay length is so short that we consider the whole range of FASER$\nu$ and FASER for the detection of the N. The decay probability from the location where N is produced at FASER$\nu$ to the very end of FASER is given by [18],

$$\begin{aligned} {\mathcal {P}}_{\textrm{detc}}=1-\exp \left( \frac{-\Delta }{\beta c\tau }\right) , \end{aligned}$$

(4)

where $\Delta$ is the total length of the detector $\Delta =L1+L2$. A schematic diagram of the detector is given in Fig. 5, where L1 and L2 are the length of the FASER$\nu$ detector and FASER decay volume, respectively. The dimension of FASER$\nu$ is $25(cm)\times 30(cm)\times 1.1(m)$.

The decays of the heavy neutrino N into SM particles are calculated using the ${\mathrm{MadGraph5aMC@NLO}}$ [50, 51, 53]. The angular distribution of the heavy neutrino produced at FASER$\nu$ is extracted from the output files with the extension “.lhe”.

We generate 10,000 events using ${\mathrm{MadGraph5aMC@NLO}}$ 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 to 70 GeV and the $Z'$ mass ($M_{Z'}=0.1$, 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 6 shows the angular distributions of the heavy neutrino versus the scattering angle $\theta$ for $E_\nu = 10 - 10,000$ 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 $M_{Z'}$ is not significant. More details about the angular distribution are given in Appendix B.

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

$$\begin{aligned} N^{\textrm{detc}}_{\alpha }= & {} N_{\alpha }^{\textrm{Prod}}(\nu _{\alpha } A \rightarrow N A')\times {\mathcal {P}}_{detc} \nonumber \\{} & {} \times \textrm{BR}(N\rightarrow \nu _{\alpha }~X~X) , \end{aligned}$$

(5)

where $N_{\alpha }^{\textrm{Prod}}(\nu _{\alpha } A \rightarrow N A')$ denotes the total number of heavy neutrino events produced at FASER$\nu$ from the $\nu _\alpha$ as shown in Fig. 4, ${\mathcal {P}}_{\textrm{detc}}$ is the probability of heavy neutrino decay within the decay volume of FASER, and $\textrm{BR}(N\rightarrow \nu _{\alpha }~X~X)$ represents the branching ratio of heavy neutrino N into $\nu _\alpha X X$. 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 signature is the decay of N into a pair of charged leptons (either $e^+ e^-$, $\mu ^+\mu ^-$, or $\tau ^+ \tau ^-$) or a pair of hadrons (from $q{\bar{q}}$) plus an active neutrino. If we can reconstruct a displaced vertex inside the decay volume of FASER, the background can be reduced substantially. However, in our case the magnetic dipole coupling has to be large enough for a sizable production rate and thus the decay of N is mostly prompt. We verified that in most of the mass range of $M_N$ and $M_{Z'}$ considered in this work, the decay length of N is shorter than about $1\,\upmu \textrm{m}$, except for a small mass range $M_N < 2$ GeV for $M_{Z'} =0.1,1$ GeV and $M_N < 10$ GeV for $M_{Z'} = 10$ GeV (see the decay length of N with $\omega _{\nu _e} = 1.4 \times 10^{-5}\,\mu _\textrm{B}$ as shown in Appendix C).

Here we consider the corresponding background for the decay of N into a pair of charged leptons. Such a final state in the SM can be produced by the process $\nu q \rightarrow \nu \mu ^+ \mu ^- q$, which is similar to the trident production in Literature [54,55,56,57,58,59,60,61,62,63]. We calculate the charged-lepton pair production rates, in particular, the number of events for the muon and electron pairs are only $10^{-1}$ while $\tau ^+\tau ^-$ channel has only $10^{-5}$ events (see Fig. 7).² This trident background is negligible. Besides this trident background, there are also other reducible backgrounds. The first is the charged leptons produced at the ATLAS IP or the charged leptons from the decay of hadrons produced at the ATLAS IP. These charged leptons are either deflected by the magnetic fields or absorbed by the rocks.³ The second source of charged leptons come from the charged-current interactions of $\nu e^- \rightarrow \nu e^-$ and $\nu N \rightarrow \ell ^- N'$ at FASER$\nu$. Note that these backgrounds only produce a single charged lepton while the signal of the N decay is a pair of charged leptons, which gives rise to a pair of oppositely charged tracks and thus can be easily identified. Given the uncertainties in the background estimation, we conservatively take the number of background events to be 1 (2). Thus, we present the sensitivity results in terms of 3-signal-event (5-signal-event) contour curves which correspond to 95% C.L. limits with the number of background events equal to 1 (2). We consider 3 benchmark models for the evaluation of the FASER and FASER$\nu$ sensitivity for the active-to-heavy-neutrino $Z'$ transitional magnetic moment. The three benchmark models are discussed below.

On the other hand, the SM event rates for a pair of hadrons have a large uncertainty and are much larger, such that the sensitivity obtained would not be as good as that using the charged-lepton channels. We will not consider the hadronic channel from now on.

4.1 Benchmark Model I (BM-I): heavy neutrino N and $Z'$ mixed with $\mu$ and $\nu _\mu$

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 $Z'$ boson $M_{Z'}$, the muon active-to-heavy-neutrino $Z'$ transitional magnetic dipole moment $\omega _{\nu _{\mu }}$, heavy neutrino mixing angle $V_{\mu N}$ and the $Z'$ 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 $Z'$ masses are shown in Fig. 8. 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., $N\rightarrow \nu _\mu \tau ^+ \tau ^-$ appears when $M_N > 2 m_\tau$. For $M_{Z'}<M_{N}$ the two-body decay of $N\rightarrow \nu _{\alpha } Z'$ dominates, followed by the $Z'$ boson decay into SM fermions $Z'\rightarrow f {\bar{f}}$ ($f = \nu , l, u, d$), according to the mass of $Z'$. The branching ratios of the heavy neutrino N through such a 2-body decay can be converted to the 3-body one using

$$\begin{aligned} \textrm{BR}(N \xrightarrow {Z'} \nu ~X~X )= \textrm{BR} (N\rightarrow \nu Z') \times \textrm{BR} (Z'\rightarrow X X). \end{aligned}$$

The sensitivity curves for the BM-I are shown in Fig. 9. The blue and orange curves represent the bound of $\omega _{\nu _\mu }$ with the number of background events equal to 1 and 2, respectively. Considering that the detector could probe the charged-lepton final state, the sensitivity reach on $\omega _{\nu _\mu }$ is the best at small $M_{Z'}\sim 0.1$ GeV and $M_{N} \lesssim 5$ GeV, and can go down to $\omega _{\nu _\mu } \sim 4\times 10^{-6} \, \textrm{GeV}^{-1}$ (with number of background events equal to 1), as shown in Fig. 9. Figure 9 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. 8. For $M_{Z'}=$ 0.1 GeV, $N\rightarrow \nu _\mu \nu {{\bar{\nu }}}$ is the dominant decay channel, while for other 2 cases $M_{Z'} = 1$ and 10 GeV $N \rightarrow \nu _\mu + \textrm{hadrons}$ becomes dominant. The larger $M_{Z'}$ and $M_N$, the weaker the limit on $\omega _{\nu _\mu }$ will be. Also, among all neutrino flavors, $\omega _{\nu _\mu }$ has the best sensitivity compared to other two benchmark models. We proceed to compare our sensitivity curves to the existing bounds (recast from $\mu _{\nu _\mu }$ using Eq. (3)). The red shaded region in Fig. 9 shows current constraints from LEP [47], based on the limits from monophoton searches. The purple region indicates the NOMAD constraint coming from a search for single photon production [47]. The following benchmark models are also compared with the LEP constraints. The bound of $\omega _{\nu _\mu }$ as the function of $M_{Z'}$ and $M_N$ is superior to the recast one from LEP (using Eq. 3). When comparing the sensitivity of $\omega _{\nu _\mu }$ at FASER$\nu$ to the recast one from NOMAD, NOMAD has better sensitivity than FASER$\nu$ in the $M_N= 1$ GeV to 3 GeV region, but, however, in the $M_N= 3$ GeV to 6 GeV region, FASER$\nu$ has better sensitivity. The sensitivity curve with the number of background events equal to 2 (orange curve) shows a slight reduction in that with the number of background events equal to 1 (blue curve). The other benchmark models exhibit a similar pattern.

The bump and the kink observed in the $M_{Z'}=1$ GeV curve between $M_N=30$ GeV to 50 GeV are primarily caused by the increase and decrease in hadron production. The hadronic channel affects the production of charged leptons (note that we are considering a pair of charged leptons in the final state). Similar pattern can be observed for the following benchmark models.

4.2 Benchmark Model-II (BM-II): heavy neutrino N and $Z'$ 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 $V_{e N}$ with the SM leptons, and has the non-zero active-to-heavy neutrino $Z'$ transitional magnetic dipole moment $\omega _{\nu _e}$.

The branching ratios of $N\rightarrow \nu _e~X~X$ are shown in Fig. 10 with three different $M_{Z'}$. When the mass of $M_{N}>M_{Z'}$, the two-body decay of heavy neutrino $N\rightarrow \nu Z'$ dominates, followed by the $Z' \rightarrow f {\bar{f}}$. For $M_{Z'}= 0.1$ GeV, $N\rightarrow \nu _e \nu {{\bar{\nu }}}$ is the dominant decay channel, while for the other 2 cases $M_{Z'} = 1$ and 10 GeV $N \rightarrow \nu _e + \textrm{hadrons}$ becomes dominant.

The sensitivity reach on $\mu _{\nu _{e}}$ versus $M_{N}$ with different $M_{Z'}$ is shown in Fig. 11. The blue and orange curves represent the bounds of $\omega _{\nu _e}$ when the number of background events is equal to 1 and 2, respectively. The sensitivity reach on $\mu _{\nu _{e}}$ is the best at small $M_{N} \lesssim 5$ GeV and $M_{Z'} \sim 0.1$ GeV and is around $\sim 10^{-5} \, \textrm{GeV}^{-1}$ (with the number of background events equal to 1). The $\omega _{\nu _\mu }$ sensitivity curves are better than those of $\omega _{\nu _e}$ curves. The bounds of $\omega _{\nu _e}$ derived from BM-II has better sensitivity compared to LEP.

4.3 Benchmark Model-III (BM-III): heavy neutrino N and $Z'$ 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 $Z'$ transitional magnetic dipole moment $\omega _{\nu _\tau }$. The branching ratios of $N\rightarrow \nu _\tau ~X~X$ are shown in Fig. 12. For $M_{Z'}=$ 0.1 GeV, $N\rightarrow \nu _\tau \nu {{\bar{\nu }}}$ is the dominant decay channel, while for other 2 cases $M_{Z'} = 1$ and 10 GeV $N \rightarrow \nu _\tau + \textrm{hadrons}$ becomes dominant. We can see similar pattern for the heavy neutrino decay in the BM-I and BM-II.

The sensitivity reach for the active-to-heavy-neutrino $Z'$ transitional magnetic dipole moment $\omega _{\nu _\tau }$ is depicted in Fig. 13. The blue and orange curves depict the limits of $\omega _{\nu _\tau }$ under different background event conditions, with the blue curve representing the bound for one background event, and the orange curve representing the bound for two background events. Among all three benchmark models, BM-III has the least sensitivity towards the $Z'$ transitional magnetic dipole moment, simply because of the small incoming tau-neutrino flux. The sensitivity reach on $\omega _{\nu _\tau }$ is the best at small $M_{N} \lesssim 5$ GeV and $M_{Z'} \sim 0.1$ GeV and is around $\sim 9\times 10^{-5} \, \textrm{GeV}^{-1}$ (with number of background events equal to 1). 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.

5 Conclusions

In this paper, we have studied the active-to-heavy-neutrino $Z'$ transitional magnetic dipole moment, which arises from the $Z'$ 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 $Z'$ 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 $\omega _{\nu _\alpha }$ 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 because of higher muon-neutrino flux. We obtained sensitivity curves by considering the final state with two-charged-leptons. While the FASER$\nu$ can achieve the best sensitivity at small $M_{Z'}$ and $M_{N}$ regime, the sensitivity limits get weaker with the increment of heavy neutrino and $Z'$ masses. The obtained limits presented in this study are based on the upper limits of conventional transitional magnetic dipole couplings. These limits provide valuable insights into the production cross-section of Heavy Neutrinos and contribute to our understanding of their interactions within the given framework.

It would be interesting to build a light $Z'$ model that can survive all existing constraints. An example is the Stueckelberg mechanism [64, 65] which can give a small mass to a gauge boson without symmetry breaking. A complete study is beyond the scope of the present work.

We have mentioned in the Introduction that there was a study of the magnetic dipole interactions between the active neutrinos and new sterile states [30], which considered the up-scattering of neutrinos on electrons. They obtained the 90% C.L. contour curve for the active-to-heavy-neutrino transitional magnetic dipole moment $\mu _{\nu _\alpha }$ with heavy neutrino mass $M_N$ ranging from $10^{-3}~\textrm{GeV}{-}10~ GeV$ at the Forward Physics Facility detectors. Dipole coupling strengths of a few $10^{-9}\mu _\textrm{B}$ ($\sim 10^{-6}\,\textrm{GeV}^{-1}$) for $\mu _{\nu _e}$, $\sim 10^{-9}\mu _\textrm{B}$ (a few $10^{-7}\,\textrm{GeV}^{-1}$) for $\mu _{\nu _\mu }$, and a few $10^{-8}\mu _\textrm{B}$ ($10^{-5}\,\textrm{GeV}^{-1}$) for $\mu _{\nu _\tau }$. According to [30], starting at $M_N\sim$ $10^{-1}$ GeV, the sensitivity weakens.

Appendices

Appendix A: Production cross-section of heavy neutrino across neutrino energy range

The production cross-section of three different Heavy Neutrinos ($M_N=1,3,5,10,30,50$ and $70~\textrm{GeV}$) with various combinations of $Z'$ as a function of incoming neutrino energy is depicted in Fig. 14.

Fig. 14

Production cross-section heavy neutrinos produced by scattering with the tungsten target at FASER$\nu$ $\sigma _{\nu _{\alpha } A} = p\sigma _{\nu _{\alpha } p}+n\sigma _{\nu _{\alpha } n}$ via the processes $\nu _{\alpha } A \rightarrow N A'$ scattering $(\alpha = e, \mu ,\tau )$, computed with the values of $\omega _{\nu _\alpha }$[For muon neutrino ($\nu _\mu$): $\omega _{\nu _\mu }(M_{Z'}=0.1 ~\textrm{GeV})=4.10\times ~10^{-9}\mu _\textrm{B}$, $\omega _{\nu _\mu }(M_{Z'}=1~\textrm{GeV})=8.12\times ~10^{-9}\mu _\textrm{B}$, and $\omega _{\nu _\mu }(M_{Z'}=10~\textrm{GeV})=4.10\times ~10^{-7}\mu _\textrm{B}$; For electron ($\nu _e$) and tau ($\nu _\tau$) neutrinos (The production cross-section of Heavy Neutrino from the incoming electron and tau neutrino is the same.): $\omega _{\nu _\alpha }(M_{Z'}=0.1 ~\textrm{GeV})=1.39\times ~10^{-7}\mu _\textrm{B}$, $\omega _{\nu _\alpha }(M_{Z'}=1~\textrm{GeV})=2.76\times ~10^{-7}\mu _\textrm{B}$, and $\omega _{\nu _\alpha }(M_{Z'}=10~\textrm{GeV})=1.39\times ~10^{-5}\mu _\textrm{B}$] shown in Table 1. Top: $\alpha = \mu$; Bottom: $\alpha = \tau , e$

The upper panel of Fig. 14 shows the production cross-section of Heavy Neutrinos from incoming muon neutrinos, while the lower panel represents the production cross-section of Heavy Neutrinos from incoming electron or tau neutrinos.

Appendix B: Angular distributions of the heavy neutrino

Angular distribution curves are obtained by using the ${\mathrm{MadGraph5aMC@NLO}}$ [50, 51, 53] and extracting the information from the output “.lhe” file. The angular distribution of the heavy neutral neutrino produced at FASER$\nu$ through the neutrino-nucleon scattering $\nu A\rightarrow N A'$ is shown in Fig. 15. Here we show the distributions with three $Z'$ masses $M_{Z'}$ = 0.1, 1 and 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 $M_{N}$.

Fig. 15

Angular distributions of the heavy neutrino N with different $M_{Z'}$ and different neutrino energy $E_{\nu }$

Appendix C: Heavy neutrino decay length

The decay length of the heavy neutrino N when it decays into an electron-neutrino $\nu _e$ and a pair of fermions ($N\rightarrow \nu _e f \bar{f'}$) is given in Fig. 16. The heavy neutrino decay N includes the SM mediators $W^\pm$,Z bosons and the extra $Z'$ boson. The decay of the heavy neutrino N into SM particles is calculated using the ${\mathrm{MadGraph5aMC@NLO}}$ [50, 51, 53].

The two-body partial decay widths of the heavy neutrino N and $Z'$ are given as follows.

$$\begin{aligned} \Gamma (N\rightarrow Z\nu _{\alpha })= & {} \frac{g^2V^2_{\alpha N} (M_N^2-M_Z^2)^2(M_N^2+2M_Z^2)}{128\pi C^2_w M^2_Z |M_N|^3}, \nonumber \\ \end{aligned}$$

(C1)

$$\begin{aligned} \Gamma (N\rightarrow W l)= & {} \frac{g^2 V^2_{\alpha N} }{64\pi M^2_W|M_N|^3 } \,\nonumber \\{} & {} \times \left[ (M_l^2-M_N^2)^2+(M_l^2+M_N^2) M^2_W-2M^4_W ) \right] \nonumber \\{} & {} \times \sqrt{(M^2_l-M_N^2)^2-2(M^2_l+M^2_N)M^2_W+M^4_W} , \nonumber \\\end{aligned}$$

(C2)

$$\begin{aligned} \Gamma (N\rightarrow Z'\nu _{\alpha })= & {} \frac{\omega ^2_{\nu _{\alpha }}(M_N^2-M_{Z'}^2)^2(2M_N^2+M^2_{Z'})}{16\pi |M_N|^3} , \nonumber \\\end{aligned}$$

(C3)

$$\begin{aligned} \Gamma (Z'\rightarrow l \bar{l})= & {} \frac{g^2_l(M^2_{Z'}-M_l^2)^2\sqrt{M_Z'^4-4M_l^2M_Z'^2}}{24\pi |M_{Z'}|^3} , \nonumber \\\end{aligned}$$

(C4)

$$\begin{aligned} \Gamma (Z'\rightarrow \nu {\bar{\nu }})= & {} \frac{ g^2_\nu M^4_{Z'}}{24\pi |M_{Z'}|^3}, \end{aligned}$$

(C5)

$$\begin{aligned} \Gamma (Z'\rightarrow q \bar{q})= & {} \frac{g_q^2(2M_q^2+M_{Z'}^2)\sqrt{(M_{Z'}^4-4M_q^2M_{Z'}^2)}}{4\pi |M_{Z'}|^3} \end{aligned}$$

(C6)

Fig. 16

The decay length of the heavy neutrino N decaying into an electron neutrino ($\nu _e$) and a pair of fermions $N\rightarrow \nu _e f \bar{f'}$, estimated with $V_{N e}=10^{-4}$, $g_f = 1$ for $f = q,\nu ,l$ and the value of $\omega _{\nu _e}$ [$\omega _{\nu _\textrm{e}}(M_{Z'}=0.1 ~\textrm{GeV})=1.39\times ~10^{-7}\mu _\textrm{B}$, $\omega _{\nu _e}(M_{Z'}=1~\textrm{GeV})=2.76\times ~10^{-7}\mu _\textrm{B}$, and $\omega _{\nu _e}(M_{Z'}=10~\textrm{GeV})=1.39\times ~10^{-5}\mu _\textrm{B}$] shown in Table 1