Seeking for sterile neutrinos with displaced leptons at the LHC

We study the signal of long-lived sterile neutrino at the LHC produced through the decay of the $W$ boson. It decays into charged lepton and jets. The characteristic signature is a hard prompt lepton and a lepton from the displaced decay of the sterile neutrino, which leads to a bundle of displaced tracks with large transverse impact parameter. Different from other studies, we neither reconstruct the displaced vertex nor place requirement on its invariant mass to maintain sensitivity for low sterile neutrino masses. Instead, we focus on the displaced track from the lepton. A difficulty for low mass sterile neutrino study is that the displaced lepton is usually \textit{non-isolated}. Therefore, leptons from heavy flavor quark is the major source of background. We closely follow a search for displaced electron plus muon search at CMS and study their control regions, which is related to our signal regions, in great detail to develop a robust estimation of the background for our signals. After further optimization on the signal limiting the number of jets, low $H_T$ and large lepton displacement $d_0$ to suppress SM background, we reach an exclusion sensitivity of about $10^{-8}$ ($10^{-5}$) for the mixing angle square at 10 (2) GeV sterile neutrino mass respectively. The strategy we propose can cover the light sterile masses complimentary to beam dump and forward detector experiments.

The origin of the neutrino masses is a puzzle of the Standard Model [1][2][3][4]. The parameters in the neutrino sector, such as the mass differences and mixing angles, have been measured with higher and higher precision [5]. At the same time, cosmological observations set an upper bound on the sum of neutrino masses to be smaller than 0.12 eV and the effective extra relativistic degrees of freedom to be N eff = 2.99 ± 0.17 from Planck 2018 data [6].
Many mechanisms have been introduced to incorporate the presence of the tiny neutrino masses. Among them, the seesaw type of solutions to the small neutrino mass is probably the most plausible. A heavier, often sterile, neutrino to realize the seesaw mechanism of a certain type would be a smoking gun signal for this class of models.
In this work, we focus on sterile neutrino production from W boson decay, which leads to a signature of a prompt lepton and a displaced but non-isolated lepton. A central challenge for new phenomenological studies on long-lived particles at the LHC is how to estimate the corresponding background. We overcome, at least partially, this challenge by extracting background behavior information from two very similar control regions measured and validated by an experimental search targeting different signatures at CMS [49]. We show the LHC sensitivity would be improved significantly in the regime of sterile neutrino mass around 1-20 GeV with a mixing angle squared between 10 −8 to 10 −3 .
The paper is organized as follows. In Sec.II, we provide a brief review of the seesaw models. In Sec.III, we discuss the properties of the sterile neutrino relevant for this study.
In Sec. IV, we present the proposed analysis with the corresponding background estimation and the resulting for model parameter coverage. Finally, we conclude and discuss future directions in Sec. V.

II. STERILE NEUTRINO MODELS
In this section, we briefly review a few classes of seesaw models to motivate the parameter space we focus on. We begin with the original seesaw models [50][51][52][53][54] with the interaction Lagrangian of the new sterile neutrino sector whereH = iσ 2 H * . The mass matrix in the flavor basis {ν L , N c } is where m D = λ ν v/ √ 2 with the vacuum expectation value of the Higgs field v = 246 GeV.
The mass of light and heavy neutrino are respectively, in the heavy Majorana mass limit. The mixing angle is sin θ = m D /m N , which yields a relation between the mixing angle, light (active) neutrino mass and heavy neutrino mass:  [55,56] or linear seesaw model [57][58][59]. These variations of seesaw models can be parameterized similarly, and they are more accessible (and testable) with a larger mixing angle while capable of reproducing the light neutrino mass. The phenomenology is controlled by the same set of phenomenologically relevant parameters sin θ and m N but with a different relation to Lagrangian parameters. In inverse seesaw, we have where µ is a free parameter with the relation Here m N becomes a Dirac mass term of new sterile neutrinos and µ is the Majorana mass of one of the new species.
For linear seesaw, the relation is where m ψ is an additional Dirac mass term small than m N , and its presence will violate the lepton number. With the additional parameters introduced, we can attribute neutrino mass to these new parameters. Thereafter, we only focus on sin θ and m N which are both free parameters.
In the later sections, we will constrain the mixing angle sin θ as a function of sterile neutrino mass m N with experimental studies. To simplify this discussion and focus on demonstrating our strategy, we will make the simple assumption that all three generations of the active neutrinos mix with N equally. In the context of more general seesaw models, making this assumption does not commit us to a particular neutrino mass hierarchy. The collider signal studied in this paper mainly rely on final states with electrons and muons, while the mixing with τ neutrino enters the decay width of the sterile neutrino if m N > m τ .
If we relax the assumption of universal mixing, the reach can be obtained from our result by rescaling relevant rates.
In the basic seesaw model, the mixing angle is not a free parameter for a particular sterile neutrino mass. It is fixed to be around which is very difficult to be probed at the colliders. Still, in the context of more extended models, larger mixing angle is allowed. It is these non-minimal models which would be the main target for LHC searches.

III. STERILE NEUTRINO PROPERTIES
In this section, we discuss the features and properties of sterile neutrino from the extended seesaw models, with mixing angle sin θ and mass m N as free parameters. Due to the mixing between SM neutrino ν and sterile neutrino N , the relevant interactions are where θ w is the weak mixing angle.
In a large region of parameter spaces of interests, the right-handed neutrino has a macroscopic lifetime with the details of partial decay widths following [35,60], see also [61]. The reference mixing angle squared 10 −12 is chosen for SM neutrino mass to 0.01 eV in the basic seesaw model.
For an intermediate value of the mixing angle squared 10 −6 , a 10 GeV sterile neutrino will still be long-lived at collider level with a proper lifetime of 12 mm. In Fig. 1, we show the partial widths, branching ratios and lifetime for heavy sterile neutrino N decay, with the assumption mixing angles between the three SM flavor neutrino are democratic sin θ νeN = sin θ νν N = sin θ ντ N .
At colliders, the production of the sterile neutrino is the same as SM neutrinos, with additional suppression from mixing angle and kinematics. The production through the onshell W boson is of particular interest here, given that the associated charge leptons can be used to trigger the signal processes. The long-lived sterile neutrino can then be analyzed without trigger penalties. The expected total number of sterile neutrino at the HL-LHC for one generation of democratic sterile neutrino is approximately where L = 3000 fb −1 is the integrated luminosity at the HL-LHC. show the partial decay width from mixing with ν e , ν µ , ν τ and total decay width respectively. In the middle panel, we show the branching ratios to leptonic channels ν , ννν and semi-leptonic channels jj and νjj. In the right panel, we show the lifetime as a function of m N with benchmark mixing angle sin θ = 10 −4 , 10 −5 and 10 −6 respectively.
It is worth noting that the existence of two other SM venues for heavy sterile neutrino production, namely the Z boson decay and the Higgs boson decay. These two channels will become competitive if displaced-track triggers becoming available [62], which can provide a similar amount of sterile neutrino in addition to the W boson decay. Furthermore, the Higgs channel has two exciting features, namely Higgs specific trigger options and large branching fractions. There are many sub-leading Higgs production channels which can be triggered on, especially the vector boson fusion channel and weak boson associated production channel.
Although the Higgs boson production rate is three to four orders of magnitude smaller than the W and Z boson, the branching fraction of the Higgs boson to ν + N can be five orders of magnitude larger, given its small total width. In this study, we do not include these channels and save these interesting new production modes for future studies in association with displaced triggers.
On the top panel of Fig. 2, we show the transverse momentum and pseudo-rapidity The process is generated by MadGraph5_aMC@NLO [63], and the parton shower is performed by is dictated by the maximum momentum it can obtain from W decay in the center of mass frame, (m 2 W + m 2 N ) / (2m W ). For the events with p N T larger than this value, the initial state radiation provides additional transverse momentum of the W boson system. In the bottom panel of Fig. 2, we show the p T distribution for both the prompt lepton from W decay and displaced lepton from N decay. The distribution of prompt lepton p T is similar to that of p N T , with the difference that the maximum momentum in center of mass frame be- The displaced lepton transverse momentum, p T , is dominantly distributed within [5,20] GeV, because the other two particles in the N decay take away approximately two thirds of the available energy.

IV. NEW SEARCHES STRATEGY FOR STERILE NEUTRINOS
There are many challenges to overcome to reach good sensitivities in the search for sterile neutrinos at the LHC. A major challenge is how to achieve a good signal selection efficiency with effective background suppression. Typical neutrino mass models point to tiny couplings between the sterile neutrino and the SM electroweak gauge bosons, leading to displaced decays of the sterile neutrinos. Therefore, a very effective strategy is necessary to pick out these displaced events at the LHC. Another aspect is triggering. The sterile neutrino can be produced in electroweak processes, which typically give rise to soft objects. At the same time, due to the low signal rates, the triggering needs to be as efficient as possible.
We consider the process of pp → W → N , with a subsequent displaced decay of the sterile neutrino N → jj. Since the prompt lepton is hard, one can in principal trigger the event using single lepton trigger. According to CMS Phase-2 upgrade of the level-1 trigger [66], the trigger thresholds (with track trigger) on a single lepton is p T > 27 (31) GeV for isolated (non-isolated) e, and 18 GeV for µ. As shown in the prompt lepton p T distribution in Fig. 2, the signal events will be reduced substantially by the single lepton trigger. Therefore, we consider double lepton trigger with the leading lepton p T > 19 GeV and the sub-leading lepton with p T > 10.5 GeV, also benefits from the track trigger [66]. In reality, the combination of triggers will be used in the experiment. Therefore, it is expected that the trigger efficiency will be better than using a single trigger category. Therefore, our approach on the triggering in this study is conservative.
For the off-line signal selection, the major advantage is the presence of a bundle of tracks with large (transverse) impact parameter, d 0 . We denote the decay position of sterile neutrino N as (x, y, z) with the production point being the origin, and the momentum of a daughter particle from N decay in x-y plane is denoted as (p x , p y ). We can define the transverse impact parameter d 0 as 1 We require the sterile neutrino N to decay within the detector region r < 0.5 m and GeV which is easy to satisfy for our signal, based on the p T distribution of displaced lepton in greater than 100 µm. The CR-III looks for a heavy flavor jet plus a displaced electron and CR-IV looks for a heavy flavor jet plus a displaced muon. CR-III is triggered by a singlet electron trigger (with electron p T > 20 GeV), while CR-IV is triggered by a muon plus jet trigger (with a muon p T > 10 GeV and jet p T > 12 GeV), both with low p T threshold for the leptons [68].
The detailed studies of these control regions in Ref. [49] confirm the following crucial fact about the displaced background distribution. First of all, the displaced leptons are dominantly from heavy flavor jets (b-jets) by using the "tag and probe" method where the jet recoil against the displaced lepton is tagged as a b-jet. The subleading background is from tt which is smaller by more than one order of magnitude, agreeing well with our simulation. Furthermore, for the displaced lepton, the (normalized) differential distribution as a function of the transverse impact parameter is shown to be the same for isolated and non-isolated leptons 2 .
In Fig. 2 of Ref. [49], the background events are dominated by heavy flavor. Summing up all the d 0 bins, the corresponding cross-sections are 16.6 pb for HF+e (CR-III) and 259 pb for HF+µ (CR-IV) respectively. We denote them as σ CMS HF+e and σ CMS HF+µ respectively. The HF+µ background is much larger than HF+e background because the muon plus jet trigger requires a softer lepton than the single electron trigger. We used these results to validate our own simulation. In particular, we generated bb events and require e or µ to show up in the event after hadron fragmentation using Pythia8. We apply the CR-III and CR-IV triggers separately and further require at least one b-tagged jet. For HF+e, the isolation requirement for the electron is ∆R e < 0.3 and the additional p T sum within the isolation cone should be less than 3.5% (6.5%) of the electron's p T in the barrel (endcap) region. For HF+µ, the isolation requirement for the muon is ∆R µ < 0.4 and the additional p T sum within the isolation cone should be less than 15% of the muon p T . We implement these requirements by modifying the Delphes3 [69]. We found the corresponding cross-sections to be 21.9 pb and 241.2 pb for HF+e and HF+µ backgrounds, which are consistent with our extraction from Ref. [49]. We denote them as σ icut bb(e) and σ icut bb(µ) which are the cross-sections after cuts with isolation requirement on leptons, denoted as "icut" in the superscript. Such  consistency provides confidence when we calculate the background to our new search 3 .
The displaced lepton in the signal is often non-isolated. Therefore, a major background comes from events with a displaced non-isolated lepton from heavy flavor jets and a prompt 3 We thank Bingxuan Liu on the analysis of Ref. [49] for clarifying the details of the control region in the analysis and confirming our scaling method.
lepton. Consequently, the leading backgrounds for our signal could be W + bb with W decaying leptonically, and tt with one of the top quarks decay leptonically. The number of the background can be calculated as where the "ncut" in the upper script means requiring jets with p j T > 20 GeV while having one non-isolated lepton in the final states. In the event generation, we require b quark GeV at parton level to ensure an energetic non-isolated lepton from its hadronic fragmentation. The lepton represent both e and µ. Further optimization selections are applied to both the signal and background, whose selection efficiencies are denoted as opt in Eq. 12. The cross-sections σ ncut tt→bb+ +X and σ ncut W +bb,W → ν are found to be 136 pb and 3.8 pb respectively, after applying the "ncut".
To further reduce the SM background, especially those from the tt process, additional cuts on the hadronic activities can help. In Fig. 4 We also include the efficiency information for the dilepton trigger requirement p 1 T > 19 GeV and p 2 T > 10.5 GeV. The effects of various selection cuts and the total efficiencies are given in Table. I.   After combining all the cuts and optimizations, the backgrounds from tt and W + bb are 0.017 pb and 0.022 pb respectively, which are comparable in size. Multiplying by the integrated luminosity 3000 fb −1 from HL-LHC, the corresponding number of SM backgrounds are about 51000 and 65100 respectively. We assume for the SM background, the d 0 distribution for the lepton does not depend strongly on the p T of the lepton and other optimization cuts on jets. Together with the fact that isolated and non-isolated lepton has the same distribution, we can use the normalized differential d 0 distribution from Ref. [49] to determine the background.
In Fig. 5, we plot the total number of SM background after applying d 0 > d 0 with all the cuts and optimizations applied 4 . The SM background decreases linearly with d 0 cut.
After comparing with the d 0 accumulative distribution for signal in Fig. 3, we adopt two benchmarks for the displacement, requiring d 0 > 0.2 cm and 2 cm, respectively. From With these, we derive the 95% C.L. sensitivity for sterile neutrino in red shaded region in  plane . The sensitivities for "ν e N " and "ν µ N " come from prompt lepton trigger with associated displaced vertex projection at HL-LHC [70] assuming zero background. The proposed sensitivity reach for MATHUSLA [45], FASER [44], DUNE [29] and SHiP [30,71] are also shown.
upper edge of the sensitivity region corresponds to the shorter lifetime. It is driven by the lower cut on the transverse impact parameter d 0 and we see clear advantage for smaller cut on d 0 , even though the number of SM background is larger. On the other hand, the lower edge of the red shaded region corresponds to the longer lifetime, which is insensitive to the d 0 cut.
In Fig. 6, we also show the projected sensitivity from a proposed displaced vertex search at the LHC in dot-dashed curves [70] and various fix target or satellite detector experiments [29,30,44,45,71] in dashed curves. It is clear that our more inclusive search allows us to explore parameter spaces of the sterile neutrino with lighter masses. Furthermore, when compared with other fixed target experiment and satellite detector experiments, our proposal covers new regions of sterile neutrinos whose lifetime is too short to have sufficient flux for these experiments. In the long run, if any discoveries were made, the regions overlapping with different searches and experiments will provide valuable information for the underlying new physics model.
In this paper, we study the signal of long-lived sterile neutrino N from W -boson decay, W → N , with the subsequent decay of N → jj. The characteristic feature is a hard prompt lepton with a displaced lepton with large transverse impact parameter d 0 . We neither reconstruct the displaced vertex nor cut on its invariant mass, therefore it can be sensitive for very low sterile neutrino mass. However, there is a crucial subtlety that with such small masses, the displaced lepton is usually non-isolated from the other two jets in the same N decay. To estimate the background, we have used the information from a search for displaced electron plus muon search at CMS [49] which studied relevant background in its control regions. It shows that for non-isolated lepton, those from heavy flavor quarks are the dominant SM background. Moreover, it demonstrates the important fact that the normalized d 0 differential distribution has the same shape for isolated and non-isolated leptons. Therefore, we can use their d 0 distribution for the non-isolated lepton from heavy flavor quark background. We recast their control region selection and found a good agreement with their observations. This ensures our background estimations are robust. After proposing the optimization cuts for the signal, we obtain the result for the sensitivity on parameters for the sterile neutrino. Our results can explore the region with light m N and relative large mixing angle, which is not covered by the displaced vertex searches at LHC, beam dump and far detectors experiments.
There are many new exciting opportunities at the LHC, such as displaced tracker triggers proposed to be implemented at low-level at CMS by Ref. [72]. This trigger can be handy as a general trigger for long-lived particles in a broad class of theories [73], especially for physics signals hard to be triggered on using traditional trigger. Although we have applied two lepton trigger in this study, adding the displaced tracker trigger into the trigger menu can effectively lower the p T requirement on the leptons. Moreover, the Phase-2 upgrade of the tracker system of CMS contains the Outer Tracker layer which is consisted of two closely spaced silicon sensors. It effectively doubles the hits of the track; therefore one can lower the requirement on the number of penetrating layers, which can increase the volume of the long-lived particle decay. It can significantly increase the sensitivity but needs further studies.
Note added. While we were finalizing this paper, Ref. [46], [47] and [48] appeared on the arXiv studying similar topics. The main focus of Ref. [46] is looking for sterile neutrino decay to lepton plus pions, using displaced vertex associated with low p T muon. And their mass coverage for sterile neutrino is from 5 GeV to 20 GeV. The second paper [47] studies long-lived sterile neutrino in displaced vertex searches and also using muon chambers to detect the muons from such vertex. It uses invariant mass cut and displacement cut to suppress the SM background. The third paper [48] requires displaced vertex reconstructed by two muons in the CMS muon detector.