Testing neutrino dipole portal by long-lived particle detectors at the LHC

Abstract

We discuss the potential of using detectors aimed for searching long-lived particles (LLP) at the high-luminosity LHC run, to probe the neutrino dipole models. This is achieved by taking the heavy neutral leptons (HNL) of the models as candidates of the LLPs. Taking into account the dipole couplings to the weak bosons, $d_{W,Z}$, which control the production of the HNLs at the LHC, we discuss the reach on the electromagnetic dipole couplings, $d_\gamma$, by searching for a single high-energy photon at LLP detectors. Four typical scenarios are considered in this paper, scenario A, B with $d_{W}=0$ or $d_{Z}=0$, and scenario C, D with $d_{W,Z}\gg d_\gamma$. We show the sensitivity on $d_\gamma$, can be fairly different depending on the relations between the $d_{W,Z}$ and $d_\gamma$. And the LLP detectors can potentially extend the sensitivity on dipole couplings during the high-luminosity runs of the LHC in certain scenarios.

1 Introduction

The discovery of tiny neutrino masses, with non-explanation within the Standard Model (SM) of the particle physics, is regarded as one of the most direct evidence points towards new physics beyond the SM. In efforts to explain the neutrino masses, additional right-handed neutrinos, also referred as the heavy neutral leptons (HNL) N are widely considered [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24]. They are singlets under the SM gauge groups. However, the HNLs can still interact with SM leptons L and Higgs field H via a Yukawa interaction, $\mathcal {L} \supset NHL$, which accounts for the generation of the tiny Dirac neutrino masses.

The experimental searches for such HNLs have received a lot attention, see Ref. [25] for a recent review. Among them, the production of the HNL from the Yukawa interaction, so called the neutrino portal is widely considered. New interactions to the N can lead to novel signatures and features in their production and decay. For example, HNLs with gauge interactions are studied in Refs. [7,8,9,10,11, 26]. In this work, we focus on another case, where the HNLs couple to the SM via the so-called diople portal, $\mathcal {L} \supset d \bar{\nu }_L \sigma _{\mu \nu } F^{\mu \nu } N$, where $F^{\mu \nu }$ stands for the electromagnetic field strength tensor, d is the strength of magnetic dipole, and $\nu _L$ is the SM neutrino [27]. This case is interesting, especially if the neutrino portal is subdominant.

The dipole portal models have been investigated at different existing experiments in various literature [27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55]. Reference [27] summarises the limits on the neutrino magnetic dipole based at colliders, beam-dump and neutrino experiments, astrophysics, cosmology, dark matter searches as well as future projection at the proposed SHiP detector. Reference [34] revisits the limits at a neutrino or dark matter experiment by the detection of an upscattering event mediated via a transition magnetic moment. Reference [35] discussed the possibility at experiments aiming for solar neutrinos. Ultrahigh energy neutrino telescopes can also be used to probe the dipole models, sensitive to very massive HNLs [44]. Meanwhile, projections at other proposed future experiments are also investigated for Forward LHC Detectors [36, 38], Icecube [31], SuperCDMS [33], DUNE [37], CE$\nu$NS and E$\nu$ES [39], as well as electron colliders [52,53,54].

In most of scenarios considered by the existing literature, the dipole models can be simplified, only including the coupling $d_\gamma$between the sterile, active neutrinos and electromagnetic field strength tensor, as the energy scale is below the electroweak (EW) scale. Nonetheless, if the energy possessed by the HNLs is comparable or even higher than the electroweak scale, e.g. HNLs produced at colliders, the SM gauge invariant dipole couplings $d_W$ and $d_Z$ should also be considered.

In this work, we investigate the possibility where the HNLs are produced at the Large Hadron Collider (LHC), and detected at the detectors aiming for searching long-lived particles (LLP), including FASER [56], MoEDAL-MAPP [57, 58] and FACET [59] at the high luminosity runs of the LHC (HL-LHC). The beam-dump experiments can also be sensitive to the case where the HNLs are LLPs. Comparing to the existing studies using beam-dump experiments, owing to the high energy scale at the LHC, the SM gauge invariant dipole couplings can play a crucial role. As we will shown in the rest of the paper, depending on the SM gauge invariant dipole couplings, better sensitivity on the electromagnetic dipole couplings than the current limits can be yielded using LLP detectors.

We orangise the paper in the following order. In Sect. 2, we briefly introduce the neutrino dipole portal model. The LLP detectors at the LHC is discussed at Sect. 3, followed by the investigation of their sensitivity for the dipole portal model at Sect. 4. And we conclude this paper in Sect. 5.

2 Neutrino dipole portal model

The effective Lagrangian of the neutrino dipole $\mathcal {L} \supset d \bar{\nu }_L \sigma _{\mu \nu } F^{\mu \nu } N$ is only applicable at low energies. The Lagrangian of the neutrino dipole, which respect the full gauge symmetries of the SM can be written as [27]

$$\begin{aligned} \mathcal {L} \supset \bar{L}(d_{\mathcal{W}}^k \mathcal{W}_{\mu \nu }^a \tau ^a + d_B^k B^{\mu \nu }) \tilde{H}\sigma _{\mu \nu } N+\mathrm {H.c.}, \end{aligned}$$

(2.1)

$\tilde{H}=i\sigma _2H^*$ and $\tau ^a=\sigma ^a/2$, where $\sigma ^a$ is the Pauli matrix. In this form, it can describe the new physics beyond the EW scale.

After spontaneous symmetry breaking, the Lagrangian becomes

$$\begin{aligned} \mathcal {L} \supset d_W^k(\bar{\ell ^k} W^-_{\mu \nu } \sigma _{\mu \nu } N) + \bar{\nu }_L^k(d_\gamma ^k F_{\mu \nu }-d_Z^k Z_{\mu \nu }) \sigma _{\mu \nu } N +\mathrm {H.c.} \end{aligned}$$

(2.2)

Hence, the right-handed neutrinos N couple to SM photon, Z and W bosons via the dipole couplings $d_\gamma ^k$,$d_Z^k$, and $d_W^k$ respectively.

For a given lepton flavor k, the dipole couplings $d_\gamma ^k$,$d_Z^k$, and $d_W^k$ in the broken phase are linearly dependent by only two parameters $d_\mathcal{W}$ and $d_B$ in the unbroken phase, such that¹

$$\begin{aligned} d_\gamma= & {} \frac{v}{\sqrt{2}}\left( d_B\cos \theta _{w}+\frac{d_{\mathcal{W}}}{2}\sin \theta _w\right) , \nonumber \\ d_Z= & {} \frac{v}{\sqrt{2}}\left( \frac{d_{\mathcal{W}}}{2}\cos \theta _{w}-d_B\sin \theta _w\right) , \nonumber \\ d_W= & {} \frac{v}{\sqrt{2}}\frac{d_{\mathcal{W}}}{2}\sqrt{2}. \end{aligned}$$

(2.3)

By further assuming $d_{\mathcal{W}}=a\times d_B$, we have

$$\begin{aligned} d_Z= & {} \frac{d_\gamma (a\cos \theta _w-2\sin \theta _w)}{2\cos \theta _w+a\sin \theta _w}, \nonumber \\ d_W= & {} \frac{\sqrt{2}a d_\gamma }{2\cos \theta _w+a\sin \theta _w}. \end{aligned}$$

(2.4)

The above expressions are only true if the effective field theory (EFT) was valid at the LHC. The dipole couplings $d_{\mathcal{W}, B}$ are dim-6 operators, while $d_{\gamma , Z, W}$ are generated after spontaneous symmetry breaking, so are dim-5 operators. The EFT should be valid with the largest $d_\gamma \sim \frac{v}{\Lambda ^{2}} \sim \frac{100~ \text {GeV}}{\Lambda ^{2}}$ [27, 60], and $d_\gamma \sim \frac{1~ \text {GeV}}{\Lambda ^{2}}$ in the perturbative limit. In our following calculation, since the production of N mainly comes from on-shell decay of the W/Z at the LHC, the EFT is valid as long as the cutoff scale $\Lambda \gtrsim M_{W/Z}$ which indicates that the $d_\gamma$ can be as large as $\mathcal {O} (10^{-(3-4)})$.

The dipole couplings can be connected to the generation of the neutrino masses via loop diagrams, if a Majorana mass term $\mathfrak {m}_N$ exists. However, in this paper, we consider the HNL as purely Dirac fermion, or quasi-Dirac with a small Majorana-type mass splitting satisfying $\mathfrak {m}_N\ll m_N$ [27]. Large dipole couplings can still be compatible to the observed tiny neutrino masses, since they are decoupled, therefore as free parameters.

Thus, we have three independent free parameters in our model

$$\begin{aligned} (m_N, d_\gamma , a), \end{aligned}$$

(2.5)

where $m_N$ is the mass of the HNL.

Fig. 1

The Feynman diagrams of the production of the right-handed neutrinos N at the LHC

3 Signals of the HNLs at the LHC

3.1 Production and decay of the HNL

We consider the HNL at the LHC produced by the decay of the gauge bosons, i.e. $pp \rightarrow W^{\pm } \rightarrow N l^{\pm }$, and $pp \rightarrow Z, \gamma \rightarrow N \nu$, as shown in Fig. 1.²$^,$³ The production of the HNL via gauge boson decays can also be triggered by the active-sterile neutrino mixings. Nevertheless, if the neutrino masses were generated via type-I seesaw, the active-sterile neutrino mixings should be tiny, thus this contribution can be negligible, The production cross section depends on the couplings of N to the gauge bosons, $d_W$, $d_Z$ and $d_\gamma$ as well as $m_N$, therefore by $(m_N, d_\gamma , a)$. The N subsequently decays via the same couplings, with the decay width

$$\begin{aligned}{} & {} \Gamma _{N \rightarrow \nu \gamma } = \frac{|d_\gamma |^2 m_N^3}{4 \pi }, \end{aligned}$$

(3.1)

$$\begin{aligned}{} & {} \Gamma _{N \rightarrow \nu Z} = \frac{|d_Z|^2 (m_N^2-M_Z^2)^2 (2m_N^2+M_Z^2)}{8 \pi m_N^3} \Theta (m_N > M_Z),\nonumber \\ \end{aligned}$$

(3.2)

$$\begin{aligned}{} & {} \Gamma _{N \rightarrow W \ell } = \frac{|d_W|^2}{8 \pi m_N^3} \sqrt{(m_N^2-(M_W-m_\ell )^2(m_N^2+(M_W-m_\ell )^2))} \nonumber \\{} & {} \qquad \times (2 m_\ell ^2(2m_\ell ^2-4m_N^2-M_W^2)\nonumber \\{} & {} \qquad +(m_N^2-M_W^2)(2m_N^2+M_W^2))\Theta (m_N > M_W+m_\ell ). \end{aligned}$$

(3.3)

N can also decay via off-shell W and Z [61, 62],

$$\begin{aligned} \Gamma _{N \rightarrow \text {2body} }\propto & {} |d_{W,Z}|^2 \frac{G_F^2 m_N^3 f_M^2}{10 \pi }, \end{aligned}$$

(3.4)

$$\begin{aligned} \Gamma _{N \rightarrow \text {3body} }\propto & {} |d_{W,Z}|^2 \frac{G_F^2 m_N^5}{100 \pi ^3}, \end{aligned}$$

(3.5)

where $G_F$ and $f_M$ are Fermi constant and meson decay width, respectively. As we focus on the N which can lead to LLP signals at the LHC, for most of the parameter space with $m_N \lesssim 2$ GeV, we only have appreciable $\Gamma _{N \rightarrow \nu \gamma }$, hence $\textrm{Br}(N \rightarrow \nu \gamma )\simeq \text {100 \%}$ and $\Gamma (N) \propto |d_\gamma |^2$.

Having understood the expressions of the production and decay of the N, Monte-Carlo simulations are performed to analyse the kinematics. We use the Universal FeynRules Output (UFO) [63, 64] of the neutrino dipole model developed in Ref. [52], which is fed to the event generator MadGraph5aMC@NLO -v2.6.7 [65] to generate events at parton level. Shower, hadronization, etc are handled by PYTHIA v8.306 [66]. Detector level simulation and the clustering of the events by later purpose is performed by Delphes v3.5.0 [67] and FastJet v3.2.1 [68], respectively.

Fig. 2

Left: the cross section of the processes $pp \rightarrow W^{\pm } \rightarrow N l^{\pm }$ (solid), and $pp \rightarrow Z,\gamma \rightarrow N \nu$ (dashed) at the 13 TeV LHC as a function of a, when $d_\gamma = 10^{-5}$ and $m_N = 0.1$ GeV. Right: same but as a function of $m_N$ when $d_\gamma = 10^{-5}$ for Scenario A ($a=0$), B ($a=2 \tan \theta _w$), C ($a=-3$), and D ($a=-3.73$)

The cross sections of the processes $pp \rightarrow W^{\pm } \rightarrow N l^{\pm }$ (blue line), and $pp \rightarrow Z/\gamma \rightarrow N \nu$ (orange line) at the 13 TeV LHC as a function of a when $d_\gamma = 10^{-5}$ and $m_N = 0.1$ GeV, are shown in Fig. 2 left. It is clear that the cross sections depend strongly on a. For the W mediated processes, they are only controlled by $d_W$, which has a singularity with $a = -2 \cot \theta _w \approx -3.73$. Whereas their cross section becomes vanishing when a approaches zero leading to $d_W\sim 0$. The $Z,\gamma$ mediated processes have shown similar behavior, only they get minimum cross section where $a = 2 \tan \theta _w$ with $d_Z =$0. The minimum is non-vanishing since the $\gamma$ mediated processes still exist.

To this end, we select four typical scenarios to reflect the dependence on the high energy couplings $d_W$ and $d_Z$, where $a=0$ for Scenario A, and $a=2 \tan \theta _w, -3$ and $-$3.73 for Scenario B, C and D, respectively, as summarised in Table 1. We further show the dependence on the HNL mass $m_N$ for the two scenarios in Fig. 2 right with $d_\gamma = 10^{-5}$. For Scenario A, W mediated processes vanish, while $Z,\gamma$ mediated processes can still get a constant value about 30 fb when $m_N < M_W$, and drop off gradually to below 1 fb when $m_N$ approaches 100 GeV. Things becomes different when look at Scenario B, now the Z mediated processes vanishes, the $N \nu$ final states can still be produced via $\gamma$ with only $\sim 10$ fb cross section. The W mediated processes have similar cross section comparing to the Z ones for Scenario A. As for the Scenario C and D, now W mediated processes get larger cross section than $Z/\gamma$, reaches $\mathcal {O}(10^{4,5} )$ fb, while dropping sharply to below 1 fb when $m_N$ approaches 100 GeV. And the $Z,\gamma$ mediated processes have similar behavior.

Table 1 The four scenarios we taken in this paper

In Fig. 3, we present the radiative decay branching ratio $\textrm{Br}(N \rightarrow \nu \gamma )$ as a function of $m_N$ for Scenarios A and D. We only show these two scenarios, since Scenario B and C are similar to A and D, respectively. It can be found that in Scenario A there always be $\textrm{Br}(N \rightarrow \nu \gamma )\simeq 1$ until $m_N > M_Z$ in which the decay channel into on-shell Z boson $N\rightarrow Z\nu$ opens. Whereas in Scenario D, the radiative decay branching ratio starts to decrease rapidly from $m_N\gtrsim 10$ GeV, since the decays via an off-shell W, Z become sizeable. Due to the large ratio of $d_{W,Z}/d_\gamma$ for Scenario D, $\textrm{Br}(N \rightarrow \nu \gamma )$ is vanishing once $m_N > M_{W}$, opposite to Scenario A where it is still appreciable. And decays into on-shell W, Z become the dominant channels. We show the proper decay length of HNL, $L_N^0$ in ($m_N$,$d_\gamma$) plane. Current limits from Refs. [27, 36] are overlaid for Scenario A, while the limits for Scenario D will be shown later. From the figure, we obtain a useful analytical approximation of $L_N^0$ for $m_N \ll M_W$ no matter what value of a,

$$\begin{aligned} L_N^0 \approx 2.5~\text {cm} \times \left( \frac{d_\gamma }{10^{-5}}\right) ^{-2} \times \left( \frac{m_N}{0.1~\text {GeV}}\right) ^{-3}. \end{aligned}$$

(3.6)

It is evident to find that under current limits, the HNLs can have decay length of $\mathcal {O}$(m), which means they can be regarded as candidates of LLPs. The difference between the two scenarios in decay length do not enter into the parameter space interesting for LLPs consideration where $m_N < 10$ GeV, as shown that the decay length are only different between Scenario A and B when $L_N^0 \lesssim 10^{-6}$ m.

Fig. 3

Left: $\textrm{Br}(N \rightarrow X Y)$ as a function of $m_N$ for Scenarios A and D. Right: proper decay length of the HNL in ($m_N$,$d_\gamma$) plane. The solid (dashed) lines correspond to Scenario A (D). Current limits from Refs. [27, 36] are overlaid for comparison, only for Scenario A

This is important for the following analyses of the LLP signals. To generate macroscopic decay length of one particle for it to become a LLP, feeble interactions are required. If the LLPs are produced and decayed via the same interactions, this will leads to insignificant signal events in most cases. Nevertheless, in the model we consider, the production is controlled by $d_{Z,W,\gamma }$ or $(a, d_{\gamma })$, whereas the decay does not depend on a in our consideration of LLP signals. This means that without making the N not long-lived anymore, the production rates of N at LHC can be larger depending on the value of a in our model.

3.2 Analyses for the long-lived particle detectors at the LHC

Bear that in mind, we proceed the detailed analyses for LLP signals in this section. Although there exists quite a lot searches for LLPs at the multi-purpose detectors at the LHC, i.e. ATLAS, CMS and LHCb, no signatures of LLPs are found so far [69].

Benefited from their large distance to the interaction point (IP) and shields to stop the SM final states, specialized detectors aimed at probing LLPs might lead to more positive prospect of the discovery of the LLPs. Among them, the FASER and MoEDAL-MAPP detectors are already installed and operated since Run-3 of the LHC. The FASER detector is located about 480 ms away from the IP of the ATLAS experiment, at a very forward direction. The MoEDAL-MAPP (MAPP) detector is a new subdetector of the MoEDAL experiment, which is located about 50-100 ms away from the IP of the LHCb. In the meantime, other designs of LLP detectors such as AL3X [70], ANUBIS [71], CODEX-b [72], FACET [59] and MATHUSLA [73] detectors are also proposed. A short review for all of these detectors can be found in Ref. [25]. Considering the proposed detectors, we focus on the ones which can reconstruct the photon signals, including FASER, MAPP and FACET[4]. We take FACET to compare with FASER, since they are both at the forward direction. We focus on the phase two designs of the FASER (FASER-2) and MAPP (MAPP-2) detectors at the HL-LHC, since they have larger geometrical coverage and luminosity, providing optimistic reach of the LLP signals. FACET are also considered to be operated at the HL-LHC. We summarise the geometrical coverage and luminosity for the detectors we considered in Table 2.⁴

Table 2 The geometrical coverage and luminosity corresponding for FASER-2 [56], MAPP-2 [57, 58], and FACET [59]

The expected number of the observed events at these LLP detectors can be expressed as

$$\begin{aligned} N_{\text {signal}}/\mathcal {L} \approx \sigma (pp \rightarrow W/Z, \gamma \rightarrow N \ell /\nu ) \times \epsilon _{\text {kin}} \times \epsilon _{\text {geo}},\nonumber \\ \end{aligned}$$

(3.7)

here $\mathcal {L}$ is the integrated luminosity. $\epsilon _{\text {kin, geo}}$ are the efficiencies due to the trigger requirement, and geometrical acceptance, respectively. A kinematic threshold, $E_{vis} > 100$ GeV is put for FASER-2, following Ref. [36].

At FASER-2, for such high energies, the background can be suppressed. The main background for this single high-energy photon can be induced by the neutrino and muon. The neutrino-induced background can be cut away by the use of a dedicated preshower detector. While the muon-induced background can be vetoed by detecting the accompanying time-coincident muon [36, 74]. It still remains to be difficult to estimate the number of residual background events in a reliable way, and it has beyond the scope of our current study. Therefore, we only show the results with fixed number of signal events for each detectors. $N_{\text {signal}}=3,~30$ is going to be shown for FASER-2, as the background has been discussed in detailed. The information of the background at FACET and MAPP-2 is not yet provided yet in literature.

The geometrical acceptance is estimated as follows. In principle, $\epsilon _{\text {geo}}$ is related to the probability of the HNL to decay inside the detector volume, which is a function of the momentum p, angle to the beam line $\theta$, and lab frame decay length $L_N^\text {lab}$, such as [56]

$$\begin{aligned} \mathcal {P}(p,\theta )= & {} \left( e^{-(L-\Delta )/L_N^\text {lab} }- e^{-L/L_N^\text {lab}} \right) \Theta (R-\tan \theta L) \nonumber \\\approx & {} \frac{\Delta }{d} e^{-L/L_N^\text {lab}} \Theta (R- \theta L) \, \end{aligned}$$

(3.8)

where $\Theta$ is the Heaviside step function, L, R, and $\Delta$ are the distance to the IP, radius in the xoy plane and length of the detector. $L_N^\text {lab}=c\tau \beta \gamma = c \tau p/m$ is the lab frame decay length of the LLP, where $c \tau$ is the proper decay length. However, Eq. 3.8 requires L and R, being constants for different $\theta$, so it is only applicable for detectors like FASER-2 and FACET placed at a very forward direction and symmetric around the beam line. For MAPP-2, which have more complicated shape, we apply Monte-Carlo methods by inverse sampling of the cumulative distribution function according to the lifetime of the HNL.

Fig. 4

In Scenario A, the distribution of the p and $\theta$ (left) for $10^5$ events, as well as $L_N^\text {lab}$ and $\theta$ (right) for $10^6$ events of the HNLs from $pp \rightarrow W/Z,\gamma \rightarrow N \ell /\nu$ process. The approximate coverage of the FASER-2 (red), MAPP-2 (blue), and FACET (green) detectors is overlaid for comparison. The colours represent the weight of each bin, which is normalised to one. We fix $m_N =$ 0.1 GeV, and $d_\gamma = 10^{-5}$

Fig. 5

The same but for Scenario D

To roughly illustrate how the probability varies for different detectors, we show the distribution of the momentum p, angle to the beam line $\theta$, and lab frame decay length $L_N^\text {lab}$ for the HNLs in Figs. 4 and 5, at one benchmark where $m_N =$ 0.1 GeV and $d_\gamma = 10^{-5}$ for Scenario A and D, respectively. Again, Scenario B and C are similar to A and D, therefore not shown. The approximate coverage of the FASER-2 (red), MAPP-2 (blue), and FACET (green) detectors is overlaid for comparison. Nonetheless, the coverage on the $\phi$ (xoy plane) is not been shown, thus the resulting geometrical acceptance should be smaller comparing to the ones estimated from the figure.

In Fig. 4 left, we show the distribution of p and $\theta$ of the HNLs for $10^5$ events in Scenario A. As shown in Eq. 3.6, the proper decay length $L_N^0$ is about $2.5~ \text {cm}$ for this benchmark. The lab frame decay length equals to $L_N^0 \times p/m_N$, therefore each detector requires the p to be inside certain range to make the HNLs likely to decay within its volume. Nevertheless, the HNLs can still decay inside the detector volume for other values of $L_N^\text {lab}$, since their decay follow exponential distribution, but the probability is rather low. Both the Z and $\gamma$ mediated processes contribute to the distribution for Scenario A. The distribution from Z mediated process peaks around the line where $p_T = M_{Z}/2$, since the transverse momentum of the N is approximately half the mass of the mother particle Z for a 1$\rightarrow$2 process, when $m_N \ll M_{Z}$. However, for $\gamma$ mediated process, the distribution peaks where $p_T = p_T(\gamma )/2$, which can come from the remnant of the mesons masses, therefore covers a broader parameter space, especially for low $\theta$ region. Among these detectors, MAPP-2 located the closet to the peak of the Z mediated distribution. Whereas FACET and FASER-2 are located too far away from the Z peak, but benefited from the coverage of low $\theta$ of the $\gamma$ mediated distribution, therefore can still obtain appreciable acceptance.

The effects of the trigger can be seen in Fig. 4 left,, i.e. $p > 200$ GeV from  $E_{vis}>$ 100 GeV, as $E_{vis}\approx p/2$ since both photon and neutrino are almost massless. At this benchmark, we can see that this trigger does not result in any difference, since the requirement for the HNLs to decay inside detector volume already ask them to be energetic enough. Especially, $p \sim 2$ TeV is needed for FASER-2. However, when discuss other parameters, the proper decay length can be larger, so lower Lorentz factor subsequently lower p of the HNLs are required. Since $L_N^0 \propto d_\gamma ^{-2} \times m_N^{-3}$, so the momentum required $p \propto d_\gamma ^{2} \times m_N^{3}$. For instance, when $m_N =$ 0.1 GeV, if $d_\gamma = 10^{-6}$ instead of $10^{-5}$, FASER-2 now requires $p \sim 20$ GeV, which makes the $p>$ 200 GeV trigger effective to cut almost all the events. Generally speaking, trigger effects for the kinematical efficiencies $\epsilon _{\text {eff}}$ make the lowest $d_\gamma$ the detectors can reach larger, i.e. worse sensitivity. For a $p>$ $p_\text {low}$ trigger, the lowest $d_\gamma$ becomes $\sqrt{p_\text {low}}$ times larger, and about one magnitude for the $p>$ 200 GeV trigger.

In Fig. 4 right, we show the distribution of $L_N^\text {lab}$ and $\theta$ of the HNL for Scenario A. This figure is quite similar to the left one, only the x axis is scaled with a factor of 0.25 m $\times$ GeV$^{-1}$, and the $L_N^\text {lab}$ contains exponential distribution since each N decays exponentially. For each HNL, we simulate 10 events for the exponential distribution, so the statistics is higher, reaching $10^6$ events. Due to the exponential distribution, the distribution is modified, the parameter space far away from the peak now gets the tail from the exponential distribution. For example, FASER-2 now locates inside the bins with weight about $10^{-2}$, which is larger from Fig. 4 left. It severs as a more direct view of the geometrical acceptance of these detectors.

Comparing the distribution between Scenario A and D with Figs. 4 and 5, both the distribution of the momentum p, angle to the beam line $\theta$, and lab frame decay length $L_N^\text {lab}$ has shown appreciable difference. The contribution from $\gamma$ mediated process is insignificant in Scenario B since its cross section are much lower than the ones mediated by W and Z, therefore the distribution only surround where $p_T = M_{W,Z}/2$. For Scenario B, as shown in Fig. 5, now FASER-2 and FACET locate too far away from the peak, only get the tail of the exponential distribution. On the other hand, MAPP-2 are closer to the peak, thus still covers similar weight of events as in Scenario A.

Fig. 6

The geometrical efficiencies of the aforementioned detectors for Scenario A (left) and D (right). The $\epsilon _{\text {geo}}$ required to make $N_{\text {signal}}=$ 3 for Scenario A and D is demonstrated as the dashed (dotted) black lines at 3000 (300) fb$^{-1}$ luminosity. We fix $m_N =$ 0.1 GeV

We refer to Fig. 6 for the detailed geometrical acceptance $\epsilon _{\text {geo}}$ of each detector at the same benchmark for Scenarios A and D. When $d_\gamma = 10^{-5}$ and $m_N =$ 0.1 GeV, the geometrical acceptance $\epsilon _{\text {geo}}$ is about $10^{-4}$ for MAPP-2, $10^{-3~(-5)}$ for FACET, and $10^{-4~(-7)}$ for FASER-2, in Scenario A (D). This is smaller as than it shown up in Figs. 4 and 5 right, and FASER-2, MAPP-2 as well as FACET only gets small fraction of bins in Fig. 4 right. The difference between Scenario A and D, is from the different contribution of the $\gamma$ mediated process. The $\gamma$ mediated process can lead to appreciable distribution of HNLs for low $\theta$ as shown in Fig. 4 left, therefore FASER-2 and FACET get larger acceptance in Scenario A where the contribution of this process is significant. The number of signal events $N_{\text {signal}}$ can be obtained from Eq. 3.7. $\sigma (pp \rightarrow W/ Z, \gamma \rightarrow N \ell /\nu )$ is about $(d_{\gamma }/10^{-5})^2 \times 10^{1(6)}$ fb, when $a = 0~(-3.73)$ for Scenario A (D) and $m_N = 0.1$ GeV from Fig. 2 left.

At the HL-LHC, with 3000 (300) fb$^{-1}$ integrated luminosity for the IP of FACET and FASER-2 (MAPP-2), the $\epsilon _{\text {geo}}$ required to make $N_{\text {signal}}=$ 3 for Scenario A and D are demonstrated as the dashed black lines. Below the lines, the detectors suffer in low geometrical acceptance, leading to low signal events and vice versa. The range of $d_\gamma$ to make $N_{\text {signal}} >$ 3 can be estimated from the intersection points of the $\epsilon _{\text {geo}}$ curves of the detectors and the $N_{\text {signal}}=$ 3 lines. For Scenario A, when $m_N =$ 0.1 GeV, we get $d_\gamma \gtrsim 10^{-5~(-6)}$ for FASER-2 and MAPP-2 (FACET) detectors. For Scenario D, we have $d_\gamma \gtrsim 10^{-6}$ in order to make $N_{\text {signal}}>$ 3 for FASER-2, MAPP-2 and FACET.

4 Results

Now we show the sensitivity at the HL-LHC. According to the Lagrangian in Eq. 2.2, $d_\gamma$ can vary for different lepton flavours k, where $k = e, \mu , \tau$. Several existing limits depends on the lepton flavours, and we lack of the limits for the $\tau$. Therefore, for each scenarios, we show two different figures, one for the case when $d_\gamma$ is universal, another one when $d_\gamma$ corresponds to $\tau$ flavour. Only the sensitivity at FASER-2 is shown here. FACET and MAPP-2 might also be potentially sensitive to the monophoton signature, while the detailed analyses to accounting the background and reconstruction efficiency are not provided yet in the literature, we only estimate the number of signal events of them in Appendix A. The current limits are taken from Refs. [27, 36] considering the CHARM-II [75], LSND [76], MineBooNE [77], NOMAD [78,79,80], LEP  [81, 82], ATLAS and CMS at the LHC [83, 84] ⁵ and Supernova SN 1987 [87,88,89] experiments.

Fig. 7

Number of signal events of the LLP detectors including the FASER-2 (red) at the HL-LHC, in the ($m_N$, $d_\gamma$) plane for the Scenario A (top) and B (bottom). The red solid curve represents $N_{\text {signal}} = 3, 30$ at FASER-2 from bottom to up. Current limits taken from Refs. [27, 36] are overlaid for comparison. Left: for the universal coupling case. Right: for the case where the dipole portal couples to $\tau$ only

In general, the LLP and other detectors at colliders are complementary to each other, as the LLP detectors probe where the N is light, and CMS, ATLAS and LEP the opposite. The results for Scenario A is demonstrated in Fig. 7. For the universal coupling case as displayed in Fig. 7 left, the curves for FASER-2 roughly tracks the curves where $L_N^{0} \sim \mathcal {O}(\text {m})$ as shown in Fig. 3, until the coupling $d_\gamma \lesssim 10^{-5}$, becoming too small to yield sufficient cross section for $m_N \gtrsim 10^{-1}$ GeV. FASER-2 can get where $d_\gamma \approx 10^{-5}$. The reason is already explained in Fig. 6. In Fig. 7 left, the results are shown in comparison with the current limits for the universal coupling case. The coverage of the FASER-2 detectors in $m_N$ is within the ones of the CHARM experiment and neutrino scattering experiments, LSND [76] and MiniBooNE [77]. Due to the enormous number of events using by these experiments, they have very high precision, therefore reaching lower $d_\gamma$ comparing to the FASER-2 detectors. Anyway, our efforts are not in vain, when we consider the case where the dipole portal couples to $\tau$ only in Fig. 7 right. Now only the limits from the LEP, ATLAS and SN 1987 are effective, excluding $d_\gamma \gtrsim 10^{-4}$. Therefore, our results from the FASER-2 detectors are proved to be fairly useful, since they exceed the current limits by roughly one magnitude, when $m_N \lesssim 0.1$ GeV.

Now we move to the Scenario B, comparing to A, the FASER-2 has similar sensitivity, as their production cross section and decays branching ratio alike. Since $d_Z=0$ instead of $d_W=0$, therefore the current limits from ATLAS and LEP via Z decays are no longer valid. The searches for W mediated processes at the CMS applies, if the couplings are not $\tau$ only, since the searches aimed at light lepton final states. The searches for mono-photon signatures at the LEP are still applicable with much weaker limits. Thus, now the FASER-2 can give about two magnitude better sensitivity in the $\tau$ couplings only case.

Fig. 8

Same as Fig. 7, but for Scenario C (top) and D (bottom). The original limits from LEP, CMS and ATLAS are scaled, and hence shown prominently

As for Scenario C and D, the high scale couplings $d_{W,Z}$ are effective. Since these couplings are about much larger than $d_\gamma$ as indicated in Table 1, the cross section of N production at LHC is more than $10^{2,4}$ times larger the one in Scenario A. The larger cross section subsequently results in better reach at $d_\gamma$. From Fig. 8, the lowest $d_\gamma$ can be probed is $10^{-5.5~(-6)}$ for FASER-2 in Scenario C (D). Additionally, we redraw the current limits at high energy environment via analyses for prompt final states. We adopt ATLAS and CMS analyses, as well as the LEP analysis. Now these analyses benefited from the enlarged cross section as well, reaches to $d_\gamma \approx 10^{-(4.5-5.5)}$ in Scenario C, and $10^{-(5.5-6.5)}$ in Scenario D, only when $m_N \sim 0.1-90$ GeV. This is because these analyses is only sensitive to the HNL with $L_N^{\text {lab}} \lesssim 1$ m [53, 85, 86], and $\textrm{Br}(N \rightarrow \nu \gamma )$ drops sharply once $m_N > M_{W,Z}$ as shown in Fig. 3 left.

We compare them with the current limits, finding that the FASER-2 detector still can not compete with the low energy neutrino scattering and the CHARM experiments in the universal coupling case. When look at the case where the dipole portal couples to $\tau$ only in Fig. 8 right, the low energy neutrino scattering and the CHARM as well as CMS experiments do not apply, as it is sensitive to the $e, \mu$ final states only. In Scenario C, now the FASER-2 yield similar sensitivity to the ones from LEP, and better than the ones from ATLAS. In Scenario D, it seems LEP and ATLAS fully take the advantage of large $d_{W,Z}$, leading to roughly half magnitude better limits.

5 Conclusion

In pursuit of the explanation for the observed neutrino masses, many models assuming the existence of the HNLs are brought up. Among them, we focus on the neutrino dipole models within a dimension-6 EFT framework. This model contains high scale operators containing the couplings $d_{W,Z}$, which control the production of the HNLs at a high energy environment, e.g. the LHC.

The current constraints are stringent on such models, with the upper limits $d_\gamma \sim 10^{-6}$ for $m_N < 1$ GeV, have already brought us to where the HNLs are long-lived. Although this case is already considered in Ref. [36], which employ the FASER-2 detector to search for the HNLs produced secondarily in neutrino interactions at the FASER$\nu$, and can probe lower $d_\gamma$ due to the large number of HNL produced from the neutrino interactions in the tungsten layers. The dependence on the high scale operators $d_{W,Z}$ is however not considered. In this paper, we discuss the effects of different relations between $d_{W,Z}$, and the low scale coupling $d_\gamma$, then estimate the sensitivity of the LLP detector, FASER-2, with the HNL produced primarily.

The LLP detectors, located far away from the IP of the LHC, can be sensitive to new particles which are light and weak coupled to the SM, leading to long decay length. Although weak couplings can lead to low statistics, this is overcome since the high scale couplings can produce large number of the HNLs, no matter the low scale decay coupling is.

We choose four scenarios for comparison to show the dependence on the relations between $d_{W,Z}$ and $d_\gamma$. In Scenarios A and B with either $d_{W}=0$ or $d_{Z=0}$ and $d_{Z/W}$ is comparable to $d_\gamma$, the production rates are mainly controlled by the $d_\gamma$, while Scenario C and D dominantly controlled by $d_{W,Z}$ since $d_{W}$ and $d_{Z}$ are far larger than $d_\gamma$. For the former scenarios A and B, we show that the FASER-2 detectors can reach $d_\gamma \approx 10^{-5}$ when $m_N \lesssim 0.1$ GeV. Although this parameter space is already ruled out by neutrino scattering experiments, e.g. MiniBooNE and LSND, as well as the CHARM experiment, for $d_\gamma$ corresponds to the $e, \mu$ flavours or if it is universal, it is about one or two magnitude lower than the current limits including the ones at LEP, CMS and ATLAS, when the dipole only couples to $\tau$. For the latter scenarios C and D, since the production is enhanced by the choices of $d_{W,Z}$, the FASER-2 detectors can now reach $d_\gamma \approx 10^{-6}$. However, since the productions at LEP, CMS and ATLAS are directly connected to the $d_{W,Z}$, now the limits from them is comparable to the FASER-2 in Scenario C, and better for half magnitude in Scenario D.

We also shown the projected number of signal events for the proposed MAPP-2 and FACET detectors in App. A, which can potentially yield better sensitivity if the background can be controlled, and we leave the dedicated analyses for future study.

Appendix A: Projected sensitivity of other future LHC experiments

The FACET and MAPP-2 can potentially probe the dipole couplings as well. Since the detailed discussion of the background is not provided yet in literature, we only show the fixed number of signal events, and employ the same kinematic trigger as FASER-2. $N_{\text {signal}}=3,~1000$ is taken for MAPP-2 and FACET, to roughly indicate the potential reach of the model, if the number of background events can be controlled in a certain amount.

Their results are shown in Figs. 9 and 10. In Scenarios A and B, FACET can yield roughly half magnitude better sensitivity than the FASER-2, and even more than the MAPP-2, leading to fairly positive prospect especially in the $\tau$ couplings only case, if the background can be controlled. When it comes to the Scenario C and D, MAPP and FACET now have very similar sensitivity, and can potentially exceed all the current limits by about half magnitude when $m_N \sim 1$ GeV, if the background can be controlled. In a pessimistic view, if the background events can not be controlled, positive prospect can still be achieved even if we ask for $10^3$ signal events, which corresponds to $10^6$ number of background events, the sensitivity is still comparable to the ones from FASER-2.

Comparing the LLP detectors, those located at the forward direction, including FASER-2 and FACET, have shown drastic different geometrical efficiencies in these two scenarios, due to the different contribution from the $\gamma$ mediated processes. For them, the meson mediated processes might proved to be more powerful, and we leave them in the upcoming work.

Fig. 9

Number of signal events of the LLP detectors including the FASER-2 (red), MAPP-2 (blue) and FACET (green) at the HL-LHC, in the ($m_N$, $d_\gamma$) plane for the Scenario A (top) and B (bottom). The dotted and dashed curves corresponds to $N_{\text {signal}} = 3$, 1000 at MAPP (blue) and FACET (green). Current limits taken from Refs. [27, 36] are overlaid for comparison. Left: for the universal coupling case. Right: for the case where the dipole portal couples to $\tau$ only

Fig. 10

Same as Fig. 9, but for Scenario C (top) and D (bottom)