Sensitivity Reach on the Heavy Neutral Leptons and $\tau$-Neutrino Mixing $|U_{\tau N}|^2 $ at the HL-LHC

The model of heavy neutral leptons (HNLs) is one of the well-motivated models beyond the standard model (BSM) from both theoretical and phenomenological point of views. It is an indispensable ingredient to explain the puzzle of tiny neutrino masses and the origin of the matter-antimatter asymmetry in our Universe, based on the models in which the simplest Type-I seesaw mechanism can be embedded. The HNL with a mass up to the electroweak scale is an attractive scenario which can be readily tested in present or near-future experiments including the LHC. In this work, we study the decay rates of HNLs and find the sensitive parameter space of the mixing angles between the active neutrinos and HNLs. Since the mixing between $ \nu_{\tau} $ and HNL is not well established in literature compared with those of $\nu_e$ and $\nu_{\mu}$ for the HNL of mass in the electroweak scale, we focus on the channel $ pp\rightarrow W^{\pm(\ast)} + X\rightarrow \tau^{\pm} N + X$ to search for HNLs at the LHC 14 TeV. The targeted signature consists of three prompt charged leptons, which include at least two tau leptons. After the signal-background analysis, we further set sensitivity bounds on the mixing $ |U_{\tau N}|^2 $ with $ M_N $ at High-Luminosity LHC (HL-LHC). We predict the testable bounds from HL-LHC can be stronger than the previous LEP constraints and Electroweak Precision Data (EWPD), especially for $ M_N \lesssim $ 50 GeV can reach down to $ |U_{\tau N}|^2 \approx 2\times 10^{-6} $.


I. INTRODUCTION
Neutrino oscillation is one of the definite evidences of physics beyond the standard model (BSM), which implies that at least two of three active neutrinos are massive. However, there is no clear answer for the origin of neutrino-mass generation. Further, the matter-antimatter asymmetry in our Universe is another mystery that the SM cannot explain. To address these problems the conventional Type-I seesaw mechanism [1][2][3][4][5][6][7] with at least two superheavy right-handed neutrinos is one of the the simplest possibilities and widely discussed so far.
Thanks to the existence of heavy Majorana neutrinos, the observed tiny neutrino masses are naturally explained and their decays can be the source of the baryon asymmetry of the Universe (BAU) through a well-known mechanism called thermal leptogenesis [8].
Hence, if heavy Majorana neutrinos are discovered, it would be a clear signal of new physics without any doubts. Unfortunately, since the thermal leptogenesis requires the scale of the Majorana neutrinos to be superheavy, say more than 10 9 GeV [9], and the conventional Type-I seesaw can be perturbatively applied up to around the GUT scale, 10 15 GeV, we cannot directly produce and test such heavy particles in near-future terrestrial experiments.
However, this is not the end of the story because the allowed mass range for the heavy Majorana neutrinos can be very wide below the GUT scale. On the other hand, once the mass of the heavy Majorana neutrinos, which contribute to the seesaw mechanism, becomes below the pion mass in the minimal model, it would conflict with the constraints from the Big Bang Nucleosynthesis, since its lifetime becomes longer than 1 sec [10]. Therefore, the Type-I seesaw mechanism itself can be valid for the mass range of right-handed neutrinos between ∼ O(100 MeV) and the GUT scale.
Among a bunch of possibilities, the one with heavy Majorana neutrinos below the electroweak scale is an attractive scenario which can be readily tested in present or near future experiments. A model called the Neutrino Minimal Standard Model (νMSM) [11,12], in which the SM is extended only by introducing three heavy Majorana neutrinos, possesses two such neutrinos around the electroweak scale and one in the keV scale which also serves as a dark matter candidate. Since the neutrino Yukawa coupling of the keV-scale Majorana neutrino is so tiny compared with the other two that we can completely separate its physics from the others and simply focus on the dynamics of the other two heavier Majorana neutrinos, namely, the contribution from the keV-scale Majorana neutrino to the seesaw neutrino mass is small enough and the lightest active neutrino mass is suppressed enough compared with the solar neutrino mass scale. The other two Majorana neutrinos, which have the mass above the pion mass and below the EW scale, are responsible for the explanations of the observed atmospheric and solar neutrino mass scales and baryogenesis via neutrino oscillation [12,13].
Generically, the mass eigenstates of the heavy Majorana neutrinos are called heavy neutral leptons (HNLs) and labeled as N . The HNLs can be searched for at terrestrial experiments and, especially the testability at beam dump experiments where bunches of kaon and B mesons are produced when HNLs are lighter than the parent mesons (see e.g. [14][15][16][17]).
Furthermore, the HNLs can also be searched for at colliders like the LHC as well and searchable range of HNL mass becomes wider than the beam damp experiments (see e.q. [14,[18][19][20][21][22][23] and references therein). Actually, the lepton-number-violating (LNV) channels are the most specular signals and the definite discriminator of the models because the HNLs uniquely break lepton number which the SM always preserves. Not only for that but the lepton-number-conserving (LNC) channels can also provide strong hints for searching for the HNLs.
In this work, since the constraints for the mixing angles between ν τ and HNLs are not well established compared with those of ν e and ν µ for the electroweak scale HNLs, we focus on the channel pp → W ±( * ) + X → τ ± N + X to search for HNLs at the High-Luminosity LHC (HL-LHC). Our characteristic signature consists of three prompt charged leptons, where at least two tau leptons are included. With a detailed signal-background analysis we can set sensitivity bounds on the mixing angle |U τ N | 2 with M N at the HL-LHC. Especially, it can be improved almost one order of magnitude than the previous analyses when M N 50 GeV.
This is a significant improvement over previous studies.
where L SM is the SM Lagrangian based on SU (3) c × SU (2) L × U (1) Y gauge symmetry, the index α denotes the active flavors running for e, µ, and τ , and I is the HNL-flavor index running from 1 to 3. The fields , Φ, and ν R are the lepton doublet, the Higgs doublet, and the right-handed neutrino singlet, respectively. F αI 's are the neutrino Yukawa coupling constants and M I 's are the Majorana masses for the right-handed neutrinos.
After the Higgs field acquires the vacuum expectation value, there are two kinds of neutrino masses, namely, the Dirac neutrino masses defined as (M D ) αI ≡ F αI Φ and the Majorana neutrino masses, M I . In the mass basis of neutrinos, the tiny active neutrino masses can be explained by the hierarchical ratio between Dirac and Majorana masses as M 2 D /M I realized by the seesaw mechanism. In the mass basis, the HNLs are composed of mostly right-handed neutrinos but also small portion of left-handed neutrinos, thus, HNLs can have gauge interactions through the mixing denoted as U αI ≡ (M D ) αI /M I . Therefore, HNLs can be searched for at terrestrial experiments.
As discussed in a number works in literature (see e.g. [24] and references therein and also related papers) a certain amount of mass degeneracy between two HNLs is necessary for the success of baryogenesis. Then, we can simply rewrite the Majorana masses as M 2,3 = M N ±∆M/2 where M N is the common mass and ∆M denotes the slight mass difference. We do not stick ourselves to the valid parameter space for baryogenesis in the following studies, though. Between these two mass parameters, the common mass scale is more important than their slight difference for the purpose of HNLs searches since ∆M/M 1. Therefore, we can safely neglect the correction of ∆M and simply multiply a factor of 2 when we want to estimate physical observables, such as cross sections, for HNLs in the νMSM. In the following analyses and discussion, however, we focus on the case with one HNL just for simplicity and denote the mixing angle as U αN .
for the decay modes N → W ±( * ) l ∓ α , N → Z ( * ) ν α and N → ν α H of HNL in the low and medium mass regions.

B. Decay rates of the Heavy Neutral Leptons
Based on the mass range of HNLs, we can calculate its decay rate in three mass ranges: (1) low mass region (M N m W,Z ), (2) medium mass region (M N m t ) and (3) high mass region (M N m W,Z ). Here we only focus on the low and medium mass ranges in this study.
In the low and medium mass ranges of HNL, the major decay modes are N → W ±( * ) l ∓ α and N → Z ( * ) ν α , where W, Z bosons can be either on-shell or off-shell depending on M N .
Once HNL is heavier than the Higgs boson, the N → ν α H decay mode is also open. 1 All detailed formulas for these partial decay widths are collected in Appendix A. The branching ratios with the assumption |U eN | 2 = |U µN | 2 = |U τ N | 2 for the above decay modes of HNL in the above mass ranges are shown in Fig. 1 is dominant for the whole mass range, we focus on N → W ±( * ) l ∓ α in the following study. The dependence of the total decay rate Γ N on the square of mixing parameter U 2 αN 1 The partial decay width Γ(N → ν α H * ) is much smaller than the other two partial decay widths via the propagators of W or Z boson when M N < m H , so we can safely ignore this small contribution in our calculation. 2 Numerically, we take M N ≤ 25 GeV for the low mass range and 25 < M N ≤ 150 GeV for the medium mass range.

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10 -10 the decay length of HNL is quite small such that we can simply take the decay of HNL as prompt in most of the parameter space for each lepton flavor. In contrast, the low mass HNL with tiny U 2 αN can easily generate the displaced vertex signature after it has been produced at colliders [25,26], which is of immense interest in the upcoming LHC run.
The jump of the |U eN | 2 constraints from M N ≈ 50 to 100 GeV comes from the threshold of gauge boson masses m W,Z . In addition, we follow Eq. (2.18) in Ref. [14] for the constraint of 0νββ decay which is the strongest in Fig. 4.
We observe that the constraints on the mixing between ν τ and HNLs are relatively weaker the signatures consisting of three prompt charged leptons in the final state, of which at least two are tau leptons. We first study the kinematical behavior of the HNL in the production channel, pp → W ±( * ) + X → τ ± N + X, and then discuss the signatures for various final states from the HNL decays and discuss possible SM backgrounds. Finally, the details of simulations and event selections for both signals and SM backgrounds are displayed.
A. Kinematical behavior of the HNL in the production channel pp → W ±( * ) + X → τ ± N + X Based on the fact that the constraints on the mixing between ν τ and HNLs are relatively weaker than those of ν e and ν µ in various HNL mass ranges, we study the channel pp → W ±( * ) + X → τ ± N + X at the LHC 14 TeV to search for HNLs in this work. We first set U 2 eN = U 2 µN = 0 and only focus on the U 2 τ N dependence in the above production channel. The W boson propagator can be either on-shell or off-shell depending on the mass of HNLs.
We apply the Heavy Neutrino model file [40] from the model database of FeynRules [41] and use Madgraph5 aMC@NLO [42,43] to simulate this production channel at tree level Combined with the information from Figs. 2 and 3, there is still large allowed parameter space for prompt decays of HNLs in this production channel. Therefore, we focus on the case with prompt decays of HNLs first and leave the displaced vertex of HNLs aside in this paper.

B. Signature of the signals and possible SM backgrounds
We first divide the signal region to two parts: (1) on-shell W boson production region and (2) off-shell W boson production region. Different analysis strategies will be applied to each signal and SM backgrounds in these two regions. We focus on those final states with two τ leptons and one additional charged lepton in this work, and will explore the signature of two same-sign τ leptons with two jets as Ref. [44] in the future. Even though the mass peak of HNL could be fully reconstructed in the τ ± τ ± jj search channel, the severe QCD backgrounds will submerge such signal events. Conversely, the signature of two τ leptons with one additional charged lepton can effectively reduce those huge QCD backgrounds, but we need to carefully exploit kinematic properties of the final states to discriminate between the signal and SM backgrounds.
As shown before, there are still some possibilities to search for displaced τ leptons events from the low M N region with small mixing angles. This kind of signature has been studied in Ref. [45]. Therefore, we mainly focus on the prompt τ s in this work. On the other hand, τ leptons have both hadronic and leptonic decay modes. We choose hadronic τ lepton decays for all τ leptons in our study with the following two main reasons. First, hadronic τ lepton decays account for approximately 65% of all possible τ lepton decay modes. Therefore, we can save more τ lepton decay events from hadronic decay modes than leptonic decay modes.
Second, leptonic τ lepton decays can mimic the signals of only e's and µ's in the final state which cannot be distinguished at the LHC.
There are some irreducible and reducible SM backgrounds for the above three categories of signatures. We first consider the signal signature with two same-sign τ s, e/µ and mET , the backgrounds of which include 1. Irreducible SM backgrounds: 2. Reducible SM backgrounds: (1) EW processes : (2) tt associated processes: ttW ± /Z/H/γ * and tt + nj (n = 0-2).
Finally, the sources of SM backgrounds for the signal signature with three prompt τ s and mET are similar to those of two opposite-sign τ s, e/µ and mET . We will not repeatedly list them again.

C. Simulations and event selections
We use Madgraph5 aMC@NLO [42,43] to calculate the signal and background processes at leading order (LO) and generate MC events, perform parton showering and hadronization by Pythia8 [46], and employ the detection simulations by Delphes3 [47] with the ATLAS template. The NNPDF2.3LO PDF set was used and ME-PS matching with MLM prescription [48,49] was applied for all the signal and major SM backgrounds.
We include the emission of up to two additional partons for the signals with a matching scale set to be 30 GeV for M N 120 GeV and about one quarter of the M N for M N > 120 GeV. On the other hand, the matching scales for tt + nj and τ + τ − + nj (n = 0-2) are set to be 20 GeV and 30 GeV, respectively. All jets are reconstructed using the the anti-k T algorithm [50] in FastJets [51] with a radius parameter of R = 0.6. Furthermore, the electron, muon and tau lepton efficiencies in Delphes3 are modified to include the low P T regions inspired from the Ref. [52,53]. In order to study the Majorana nature of HNLs at the LHC, we classify our simulations and event selections in (1)  N (τ ± , l ∓ ) 2, 1, 5 < P l T < 40 GeV, |η l | < 2.5, 15 < P where l = e, µ. Since τ leptons and e/µ are relatively soft in this case compared with SM backgrounds, we reject those high P T regions to reduce background contributions inspired from the Ref. [52][53][54][55]. On the other hand, for M N > m W , we only choose the following conditions for them: where l = e, µ. Besides, the two same-sign τ candidates must be angularly separated enough by requiring ∆R τ ± τ ± > 0.4 in order to avoid overlapping. Other isolation criteria among e, µ, τ and jets are the same as the default settings of Delphes3. 3 In order to suppress the SM background contributions from both τ + τ − + nj and tt + nj (n = 0-2) with non-negligible jet fake to electron rate, we don't take into account of the signature with two opposite-sign τ s, e and mET in this study.
2. In order to reduce the τ lepton pair from the Drell-Yan process, we veto any oppositesign τ lepton for both the signal and backgrounds with 3. To suppress the contributions from backgrounds of tt associated processes, we reject the high missing transverse momentum P miss T events by requiring 4. To further reduce the contributions from backgrounds of tt associated processes, we apply the b-veto for both the signal and backgrounds with for M N < m W (M N > m W ). Moreover, for M N > m W , we further reduce background contributions by requiring the number of jets for both the signal and backgrounds with N (j) < 5 .
5. We require the minimum invariant mass for one of τ leptons and an extra e/µ to satisfy This τ lepton is most likely to be the second energetic one for small M N , but it becomes hard to be distinguished as M N increases. Here we use the transverse mass distribution for M T τ ± ∓ 1 P miss T to find the correct τ lepton from the HNL decay. We plot both M T τ ± 6. Finally, if M N < m W , the invariant mass of two same-sign τ leptons and an extra e/µ system is required to have 2. Two opposite-sign τ s, µ and mET In this scenario, we require two opposite-sign τ leptons and an extra µ in the final state with the following cut flow.
1. For M N < m W , we specifically take two soft opposite-sign τ leptons and an extra soft µ as the trigger in our events with the following conditions, On the other hand, for M N > m W , we choose instead the following conditions for them: Compared with Eq. (4), we require a stronger P µ T cut to further suppress soft radiation muons from τ τ + nj and tt + nj processes. Again, ∆R τ + τ − > 0.4 and other isolation criteria are set to avoid overlaps.
2. In order to reduce SM backgrounds with more than three τ leptons, we veto any samesign τ lepton for both signal and backgrounds with the same conditions as Eq. (5).
3. To further reduce the contributions from backgrounds of tt associated processes, we apply the following cuts for both signal and backgrounds: high P miss T rejection as Eq. (6), b-veto as Eq. (7). In addition, the cut N (j) < 5 is applied for M N > m W . 4. We require the minimum invariant mass for the τ leptons and an extra µ with opposite charges to satisfy Eq. (9). Compared with the case of same-sign τ s, it becomes more precise to pick up the correct τ lepton from the HNL decay.
5. Finally, if M N < m W , the invariant mass of two opposite-sign τ leptons and an extra µ system is required to have Two Same-Sign τ s Selection Flow    Table I and Table II, respectively. Here we set U 2 τ N = 10 −5 for all benchmark points. We list three major SM backgrounds in these two tables: W ± W ± W ∓ , W + W − Z/H/γ and tt + nj. The tt + nj is the dominant one among them before applying the selection cuts. On the other hand, the notation of Preselection includes Eqs. (3), (4) and (5) and Invariant Mass Selection includes Eqs. (9) and (10) Two Same-Sign τ s Selection Flow Table   Process σ Preselection P miss    Table I, Tables III and IV, respectively. Again, we set U 2 τ N = 10 −5 for all benchmark points. We list four major SM backgrounds in these two tables: W ± Z/H/γ, ZZ/γ, τ τ + nj and tt + nj. The τ τ + nj is the dominant one among them. The notation of Preselection includes Eqs. (11), (12) and (5)  For M N < m W , after passing all selection cuts, we can find the signal efficiencies around 0.5-1.4%, the efficiencies of W ± Z/H/γ and ZZ/γ are less than 0.1% and 0.2%, and that of  This is due to the different helicity structures between N → τ + l − α ν α and N → τ − l + α ν α that involve the W propagator with only the left-handed interaction, and causing the variation of P l T and P τ T distributions. In Fig 14, we also display these kinematical distributions for the signal M N = 50 GeV and three major SM backgrounds. We do not show kinematical distributions for τ τ + nj process because only very few events can pass the preselection Two Opposite-Sign τ s Selection Flow Table   Process σ Preselection P miss     criteria. As we expected, these selection criteria can also successfully distinguish most parts of the signal from SM backgrounds.
For M N > m W , after imposing all selection cuts, we can find the signal efficiencies around 0.7-4.3%, the efficiencies of W ± Z/H/γ and ZZ/γ are less than 2.7% and 3.4%, and those of τ τ + nj and tt + nj are even smaller, less than 1.6 × 10 −4 % and 6.0 × 10 −3 %, respectively.
Various kinematical distributions for the signal with M N = 85, 100, 125 and 150 GeV are shown in Fig. 15. Again, these distributions are similar to Fig. 11. In Fig. 16, we also display these kinematical distributions for the signal M N = 125 GeV and three major SM backgrounds. Again, kinematical distributions for τ τ + nj process are not shown in Fig. 16 for the same reason. It is clear that both M τ ± µ ∓ 1 and M T τ ± µ ∓     GeV and major SM backgrounds.
and DELPHI of Fig. 6 are added for comparison. We estimate the background uncertainties as √ B (we consider only the statistical one in this work) in the CLs method [56] where B is the total background event numbers. Also, the background-only hypothesis is assumed Likelihood. We observe that the sensitivity bounds from HL-LHC can be stronger than LEP and EWPD constraints in some parameter space, especially for two-same-sign τ selection which can reach down to |U τ N | 2 ≈ 2 × 10 −6 for M N 50 GeV, which is almost one order of magnitude better than the current constraint. These regions are close to the boundaries between the prompt and long-lived decays of HNLs at the LHC scale. Hence, our study in this paper can serve as a complementary sensitivity reach of Ref. [45] to make HNL searches in the channel pp → W ±( * ) + X → τ ± N + X more complete.  Tables I and III  For the low mass region (M N m W,Z ), we follow the calculations in Ref. [14,58,59] for the partial decay widths of N . Notice that we consider the inclusive approach, and take the parameter µ 0 ∼ m η = 957.78 ± 0.06 MeV for the mass threshold from which we start taking into account hadronic contributions via qq production.
For lepton and quark masses, we apply the values from PDG 2018 [39].
For the medium mass region (M N m t ), we take into account the both effects of onshell and off-shell W and Z bosons by including the width of these gauge bosons in the propagators. We follow the calculations in Ref. [14,60] for the the partial decay widths of N . Notice all the SM fermion masses of the final states have been neglected to simplify our calculations.
The functions F N is where P W comes from the propagator of the W boson with the form, where q 2 = M 2 N − 2M N E 1 and Γ W is the total decay width of W . We can simply obtain F N (M N , m Z , Γ Z ) by taking (m W , Γ W ) → (m Z , Γ Z ).
On the other hand, the function F S is given by and P Z comes from the propagator of the Z boson with the form, where q 2 3 = M 2 N − 2M N E 3 with E 3 = M N − E 1 − E 2 considering the decay of N at rest. Besides, we also take into account the N partial decay width to the Higgs boson and an active neutrino when N is heavier than the Higgs boson, Finally, we represent the total decay width of N as where we denoted the hadronic states H + = du, su, dc, sc, bu, bc and H 0 = qq. Then we further simplify Γ N as where a α (M N ) = 2 × Γ (lαH) + Γ (νH) + Γ (3ν) + β Γ (l β l β ν) + 2 × Γ (lαl β ν) + Γ (νH) , with α, β = e, µ, τ .