Displaced vertices as probes of sterile neutrino mixing at the LHC

We investigate the reach at the LHC to probe light sterile neutrinos with displaced vertices. We focus on sterile neutrinos $N$ with masses $m_{N} \sim $ (5-30) GeV, that are produced in rare decays of the Standard Model gauge bosons and decay inside the inner trackers of the LHC detectors. With a strategy that triggers on the prompt lepton accompanying the $N$ displaced vertex and considers charged tracks associated to it, we show that the 13 TeV LHC with $3000$/fb is able to probe active-sterile neutrino mixings down to $|V_{lN}|^2\approx 10^{-9}$, with $l=e,\mu$, which is an improvement of up to four orders of magnitude when comparing with current experimental limits from trileptons and proposed lepton-jets searches. In the case when $\tau$ mixing is present, mixing angles as low as $|V_{\tau N}|^2 \approx 10^{-8}$ can be accessed.


I. INTRODUCTION
Searches for new physics responsible for the lightness of neutrino masses [1] has been in the program of the LHC experiments for decades [2]. This new physics beyond the Standard Model (SM) may be explained by the so-called see-saw mechanism [3] that introduces the existence of heavy right-handed (sterile) neutrinos that mix with the neutrinos in the Standard Model [4,5]. For low enough mixing angles and sterile neutrino masses below the electroweak scale, the sterile neutrino N can be long-lived, and may decay with a characteristic displaced vertex (DV) signature inside particle detectors.
Experimental efforts to search for these states at hadron colliders normally focus on promptly decaying sterile neutrinos with masses m N ∼ O(100) GeV. Earlier searches by ATLAS [6] and CMS [7,8] have focus on the Majorana signature of same-sign dileptons and jets [9]. Only recently the CMS experiment has provided limits for N masses below 40 GeV in the search for three prompt charged leptons [10]. For current bounds on sterile neutrino mixing, see [11,12]. Attention to displaced vertex signatures, which can probe masses in the GeV range, is vastly growing. Recent phenomenological studies assessing the LHC sensitivity with displaced vertices to light sterile neutrinos in various models are studied in [13][14][15][16][17][18]. Despite the technical challenges in the reconstruction and modeling of the detector response to displaced vertices, this is an important signal of new physics as it is scarce in the Standard Model, and has to be explored further in order to ensure the successful exploration of new physics at the LHC, and across all mass ranges.
In this work we use a strategy motivated by the ATLAS multitrack displaced vertex search [19,20], recently validated in our previous work [21] in the context of a left-right symmetric model. Here we focus on a simplified model in which the Standard Model field content is extended by one heavy sterile neutrino, briefly described in Section II. We consider three cases for active-sterile neutrino mixing V lN , for each of the active flavours l = e, µ, τ .
The displaced vertex strategy implemented in each case is described in Section III. Discovery prospects and reach at the LHC are discussed in IV. We close the paper and summarize in Section V.

II. STERILE NEUTRINO SIMPLIFIED MODEL
We consider a simplified see-saw model of neutrinos based on the Standard Model gauge group, in which only one massive sterile neutrino N is present in the kinematic range of our interest. In this generic framework, N couples to the SM leptons via a small mixing in the electroweak currents. The charged and neutral current interactions of this model are described in [22].
We consider one neutrino flavour at a time, e, µ or τ , produced in W boson decays in association with the respective lepton flavour: W ± → N l ± . The N decays proceed via N → l ± qq, N → l ∓ l ± ν l and N → ν l qq. The N proper lifetime is given by [22] The relevant parameters are the sterile neutrino mass m N and active-sterile neutrino mixing |V lN | 2 , which we treat as independent. In general, small neutrino masses and mixing lead to macroscopic lifetimes. In principle one can have a large N lifetime by making the mixing very small (instead of the sterile neutrino mass). However, for smaller values of the mixing, the production rate of N also becomes smaller. We are interested in masses in the GeV range to access lifetimes of the order of picoseconds while scanning over mixings as low as |V lN | 2 = 10 −12 .
In Figure 1 we show different values of the proper lifetime in the (m N , |V lN | 2 ) plane.
We highlight the approximated region where vertices can efficiently be reconstructed inside the tracker region of ATLAS, for proper decay distances between 4 and 300 mm. We are interested in sterile neutrino mass between 5 GeV < m N < 30 GeV, which can be probed with a multitrack displaced vertex search at ATLAS or CMS.

III. SIMULATIONS AND SELECTION OF DISPLACED EVENTS
We generate a UFO [23] model with SARAH [24] and use SPheno [25,26] for the spectrum calculation of the sterile neutrino simplified model. We simulate events at √ s = 13 TeV for the process pp → W ± → N l ± . Generation is performed with Mad-Graph5 aMC@NLOv2.4.3 [27] at leading order. We normalize the corresponding value to match the experimental cross section in Ref [28]. The generated events are then interfaced active-sterile neutrino mixing angle. The shaded region represents roughly the region that can be accessed with current displaced vertex searches in the ATLAS inner tracker [19,20].
The promising decay channels for the sterile neutrino are semileptonically N → l ± qq and leptonically N → l ∓ l ± ν l . Decays via a neutral current such that N → ν l qq are also possible [22]. All these modes will lead to displaced vertices with charged tracks associated to them. For the case when only τ mixing is present, both semileptonic and leptonic decays lead to the presence of a τ lepton coming from the displaced vertex.
We propose a search inspired by the ATLAS multitrack displaced vertex analysis [19,20], which is sensitive to lifetimes of the order of picoseconds to about a nanosecond, so particle decays can be reconstructed with a displaced vertex signature inside the inner tracker. This strategy was developed in our recent work in Ref. [21], where we trigger on the prompt lepton coming from the W boson decay, impose cuts on the neutrino displaced vertex and its decay products, and apply vertex-level efficiencies (made public by ATLAS in [19]) to DVs that pass the required particle-level acceptance cuts. Since the parametrized selection efficiencies provided by ATLAS assumes all decay products are prompt from the DV, they 4 are not directly applicable to the case when there is a τ lepton coming from the displaced vertex 1 , so we consider decays to N → ν l qq only when τ mixing is present.
Prompt leptons are reconstructed considering the following: For electrons: We require an isolated electron within |η| < 2.5. We smear their momenta with a resolution of 2% at 10 GeV, falling linearly to 1% at 100 GeV, and then 1% flat.
For muons: We require an isolated muon within |η| < 2.5. We smear their momenta with a resolution between |η| of 2 and 0, linearly falling from 4% to 1.5%.
For taus: We implement a basic reconstruction following Ref. [32]. We start by reconstructing jets with FastJet 3.1.3 [33] using the anti−k t algorithm with distance parameter R = 0.4. Only jets with p T > 10 GeV and |η| < 2.5 are taken as seeds for τ reconstruction.
Charged constituents inside the jet must have p T > 1 GeV. If a truth τ candidate falls within a cone ∆R < 0.2 centered on the jet axis, it is selected.
The following selections are then imposed: 1. One prompt lepton (as reconstructed above) with p T > 25 GeV.
2. Decay position of the DV contained within transverse distance r DV < 300 mm, and |z DV | < 300 mm. The distance between the interaction point and the decay position must be bigger than > 4 mm. 1 The further displacement of τ 's inside the vertex will affect the vertex reconstruction efficiency from the subsequent displacement of taus (and from any heavy flavour quark in general). This was addressed for example in Ref. [31], when there are two b s coming from the displaced vertex. By allowing a bigger merging distance of 5 mm (instead of 1 mm) when forming a vertex, some efficiency is recovered. Since we do not implement a vertex reconstruction algorithm in this work, the loss in efficiency can not be estimated.
As pointed out in [21], with these selections we are still in a zero background region, where background comes mostly from instrumental sources.

IV. SENSITIVITY REACH
We analyze the region where a displaced search with the above selections can have sensitivity. The relevant parameters in the sterile neutrino model are the neutrino mass m N and active-sterile neutrino mixing |V lN | 2 .
We first chose a representative benchmark point with m N = 15 GeV, |V lN | 2 = 10 −8 and proper neutrino decay distance cτ N ≈ 50 mm to illustrate the effect of all analysis cuts from the previous section. We combine the second and third cuts in the "DV fiducial" entry in Table I. Events are generated at 13 TeV which corresponds to a production cross section of 0.09 fb. Cut-flows in the case of electron, muon and tau mixing are shown.
As already noted in the context of a left-right symmetric model in [21], for sterile neutrinos with masses below the electroweak scale this search strategy looses sensitivity, as lower masses will lead to softer decay products with limited amount of tracks available to make up a vertex. However, given the low background nature of this signature, the discovery of a displaced vertex signal in the sterile neutrino model is possible at the high-luminosity LHC.  (taken from Ref. [14] and [34], respectively). The DELPHI limit, taken from [2], is also shown.
A limit derived from 0νββ experiments is also shown (see text for more details). The 95% CL exclusion of the CMS 13 TeV trileptons search [10] is given by the filled grey region.

Figures 2, 3 and 4
show the estimated reach in the (m N , |V lN | 2 ) plane, in the case of electron, muon and tau mixing, respectively. We also show for comparison the projected sensitivities from the proposed SHiP [38] and MATHUSLA [37,39] experiments. For projected sensitivities for other proposals for heavy neutral leptons within m N < 5 GeV, see FASER [14,40,41] and CODEX-b [14,42].
In Figure 3, mixings as low as |V µN | 2 ≈ 2.2 × 10 −9 for 3000/fb and 13 TeV can be probed for m N = 20 GeV. We also show limits from proposals with lepton-jets searches, which is a complementary strategy. The curve labeled "lepton-jets-1" shows the exclusion form Ref. [35] at 13 TeV and 300/fb, where zero background is assumed. The curve labeled "lepton-jets-2" shows the 13 TeV limit in [36] and 300/fb, and considers additional background sources to the ones in [35]. An improvement of roughly two orders of magnitude in sensitivity is achieved with our strategy with 300/fb for masses 5 < m N < 25 GeV. The 95% CL exclusion of the CMS 13 TeV prompt trileptons search [10] is also shown, proving to be competitive with DELPHI [44].  [37]). The DELPHI limit, taken from [2], is also shown for comparison.
Z decays for comparison. For taus, a large luminosity sample will be needed for obtaining meaningful constraints. As the figure shows there is only a very narrow region testable with 300/fb. Mixings as low as |V τ N | 2 ≈ 8 × 10 −9 for m N = 16 GeV at 3000/fb can be probed with this strategy.

V. SUMMARY AND CONCLUSIONS
We study the potential of the LHC to probe light sterile neutrinos, and active-sterile neutrino mixing angles, with a displaced vertex strategy motivated by current multitrack DV searches at the 13 TeV LHC. We focus on a simplified model where the Standard Model is extended with one sterile neutrino N . Mixing with the three flavours e, µ and τ , are treated separately. In all cases, to our knowledge, we see that this strategy probes to be the most sensitive to date in the mass region of interest (5 GeV < m N < 30 GeV).
For e mixing, we show that this DV search is more sensitive than current neutrinoless double beta decay experiments. In the case of µ mixing, parts of the parameter space not accessible with other LHC searches (such as lepton-jets or trileptons) is possible. In both 9 cases, with 3000/fb, an improvement of up to four orders of magnitude (|V lN | 2 ≈ ×10 −9 for sterile neutrino masses between 5 and 20 GeV) in sensitivity is gained when comparing with the current experimental limits from trileptons searches at CMS [10].
Accessing τ mixing is less straightforward due to to the difficulty in reconstructing τ leptons, and also since the presence of a subsequently displaced τ coming from the displaced vertex affects the vertex reconstruction efficiency in a way that is not trivial to quantify.
We access τ mixing by considering sterile neutrino decays via a neutral current only, which still leads to a displaced vertex formed from hadronized tracks, to which publicly available DV efficiencies can be applied, thus avoiding the problematic τ in the DV. This may be an advantage of a multitrack based strategy, as opposed to tagging leptons coming from the DV, in constraining τ mixing.
We briefly comment on prospects for testing sterile neutrinos at LHCb. The authors in [45] discuss limits for a different model than the one presented in this work. They show that sterile neutrino masses around 9 GeV and mixings down to ≈ 10 −6 can be constrained in semileptonic sterile neutrino decays (µqq) at the 95% CL with current LHCb data [46].