Search for heavy Majorana neutrinos in the $\tau$ final state at proton-electron colliders

We utilize the lepton number violation signal process $p\, e^- \to \tau^+ jjj$ to search for heavy Majorana neutrinos at future proton-electron colliders. The LHeC (FCC-eh) is considered to run with an electron beam energy of 60 GeV, a proton beam energy of 7 (50) TeV and an integrated luminosity of 1 (3) ab$^{-1}$, and the electron beam is considered to be unpolarized. We apply detector configurations and simulate signal and related standard model background events for both hadronic $\tau_h$ and leptonic $\tau_\ell$ final states, $\ell$ being a muon. After preselection, multivariate analyses are performed to reject the background. The strategy to reconstruct the heavy neutrino mass is developed and distributions of reconstructed mass are presented. Discovery sensitivities on parameter $|V_{\tau N}|^2 |V_{eN}|^2 / ( |V_{\tau N}|^2 + |V_{eN}|^2 )$ for the heavy neutrino mass between 10 and 3000 GeV are predicted. At the 2-$\sigma$ significance, the best discovery sensitivity is $\sim 1.2 \times10^{-5} \,\,(5.0 \times 10^{-6})$ at the LHeC (FCC-eh) when $m_N \sim 100$ GeV for the hadronic $\tau_h$ final state. Sensitivities for the leptonic $\tau_\ell$ final state are found to be similar to those for the hadronic $\tau_h$ final state for most of the parameter space investigated. We also derive the limits on mixing parameters from electroweak precision data (EWPD) and DELPHI experiment. Assuming $|V_{\tau N}|^2 = |V_{eN}|^2 = |V_{\ell N}|^2$, sensitivity bounds from the LHeC and FCC-eh experiments are found to be stronger than those from EWPD when $m_N \lesssim 900$ GeV, and also stronger than those from DELPHI when $m_N \gtrsim 70$ GeV. Constraints are also interpreted and compared in the $|V_{\tau N}|^2$ vs. $|V_{e N}|^2$ plane.


Introduction
The experiments of neutrino oscillation [1][2][3][4][5][6] show that at least two of three active neutrinos are massive.However, in the standard model (SM), neutrinos have only left-handed components and no right-handed components, so they cannot form a Dirac mass term, and the neutrino mass is strictly equal to zero.Therefore, the standard model needs to be slightly expanded.One important method is the seesaw mechanism [7][8][9][10][11][12][13][14][15][16][17], which introduces right-handed Majorana neutrinos N R and couple them with SM neutrinos to produce small neutrino masses M ν ′ ∼ M 2 D /M R , where M R is the Majorana mass of N R and M D is the Dirac mass and proportional to the Yukawa coupling between the standard model neutrino and N R .When M R ≫ M D , active neutrino acquires a small mass.
Search for the heavy neutrinos is crucial to verify the seesaw mechanism.Because the production cross section, decay width, and lifetime of N are determined by its mass m N and the parameter |V ℓN | 2 which is related to the matrix element describing the mixing of N with the SM neutrino of flavor ℓ = e, µ, τ , limits for such searches are usually given in the plane of the mixing parameter |V ℓN | 2 vs. m N .
Compared with plentiful studies focusing on the mixing parameters |V eN | 2 and |V µN | 2 , because of the challenges in detecting the final state taus, the mixing parameter |V τ N | 2 is more difficult to be probed, making it not well studied at current experiments.For heavy neutrinos with mass above 10 GeV, the main experimental constraints on the mixing parameter |V τ N | 2 are set by the DELPHI [46], and can be derived from the electroweak precision data (EWPD) [47][48][49][50][51][52][53] and rare decays of Z-boson [46].
Among them, in Ref. [45], some of our authors develop the search strategy for a heavy Majorana neutrino via the lepton number violation signal process pe − → µ + jjj at future electron-proton colliders.Assuming mixing parameters 2 and |V τ N | 2 = 0, the dominant SM background processes are considered, and discovery sensitivities on the mixing parameter |V ℓN | 2 are predicted for the heavy neutrino mass in the range of 10-3000 GeV.The results show that the sensitivities at electron-proton colliders are much stronger than the current experimental limits at the LHC for m N above 30 GeV.
In this consecutive study, we concentrate on the signal process pe − → τ + jjj with final state taus at the LHeC and FCC-eh.Similar to Ref. [45], the LHeC (FCC-eh) is supposed to run with an electron beam energy of 60 GeV, a proton beam energy of 7 (50) TeV and an integrated luminosity of 1 (3) ab −1 ; the electron beam is considered to be unpolarized; and to simplify the analyses, we consider the phenomenological simplified Type-I model and the scenario that only one generation of heavy neutrinos N is within the collider access.The N is assumed to mix with active neutrinos of tau and electron flavours, i.e.
Since taus are unstable, they decay either leptonically into muons and electrons, or hadronically into mesons, leading to final state leptons or tau-jets at colliders.Because leptonic and hadronic final states have different kinematics and background, to obtain and compare the sensitivities for both states, we perform their analyses individually and forecast sensitivities for heavy neutrinos in the range 10-3000 GeV at LHeC and FCC-eh.
The organization of our paper is as follows.In Sec. 2, we present the cross section of the signal process.In Sec. 3 and Sec. 4, we show the SM background processes and the analyses for the hadronic and leptonic final states, respectively.In Sec. 5, we present sensitivity curves in the mixing parameters vs. heavy neutrino mass plane at the LHeC and FCC-eh.The sensitivities obtained from this study are compared with those from current experiments in Sec. 6.We conclude in Sec. 7. Details of this study are listed in Appendices A−D.

The signal production process
As shown in Fig. 1, heavy Majorana neutrinos N can be produced via the t−channel exchange of W bosons at the pe colliders, and decay finally into τ + plus three jets.The lepton number of this process changes from +1 to −1, so it violates the conservation of lepton number.For the data simulation, we implement the Universal FeynRules Output model file [91,92] which extends the SM with additional heavy neutrinos interacting with active neutrinos into MadGraph5 [93] to generate the signal events.When generating collision events, the default "nn23lo1" parton distribution function of the proton is used.To maximize the event acceptance, following loose requirements are applied at the parton level in MadGraph (implemented in the run_card.datfile) for both the signal and background: the minimal value of transverse momentum for the jet (lepton) is set to be 5 (2) GeV; the pseudorapidity (η) range for both the lepton and jet is set to be |η| < 10; the minimal value of the solid angular distance, ∆R = ∆η 2 + ∆ϕ 2 , between final state objects is set to be 0.01.Similar to our previous work [19,94], the Pythia6 [95] program is modified to perform the parton showering and the hadronization for the pe colliders, while the configuration card files [96] for the LHeC and FCC-eh detectors are implemented to the Delphes program [97] to complete the detector simulation.For the jet reconstruction, jets are classified by using the FastJet package [98] with anti-k t algorithm and cone radius R = 0.4.For the final state tau reconstruction, the default rates in the Delphes configuration card files are used, where the identification efficiency for tau-tagging is set to be 40%, and the misidentification rate is set to be about 0.1%.For the signal data simulation, we fix the benchmark mixing parameter
To maintain consistency through this study, the production cross sections calculated by MadGraph5 are used to estimate the number of events for both the signal and the background processes.In Fig. 2, we plot the cross sections of the LNV signal pe − → τ + jjj via the Majorana heavy neutrino N as a function of heavy neutrino mass m N at the LHeC and FCC-eh.The signal productions are proportional to the parameter 2 which is fixed to be 5 × 10 −5 in Fig. 2. For large m N the cross sections for LHeC decrease much faster than those for FCC-eh.This behaviour can be understood from the parton distribution function of the proton, and is explained in details in Ref. [45].
3 Hadronic τ h final state

SM background processes
When tau decays hadronically, the final state contains one tau-jet τ h with positive charge, at least three regular jets and missing energy.There exist four types of background processes: pe − → τ + τ − e − jjj, τ + τ − ν e jjj, τ + ν τ e − jjj, and τ + ν τ ν e jjj, which we label as "B1-B4" in this article.They can have the similar final state as the signal when the final state τ − and e − are undetected.Besides, because the jet production is still substantial at pe colliders, the misidentified taus from jets needs to be considered.The multi-jet process "p e − → ν e jjjj with a j → τ + h " is the main background of such kind, and is labeled as "B5a".The Delphes configuration card files [96] are used for the LHeC and FCC-eh detector simulations in this study, and the misidentification rate for the tau-tagging is set to be about 0.1% in these files.Therefore, we use this value as benchmark to test the effect of misidentified tau-jets.
Table 1 shows the production cross sections of background processes at the LHeC and FCC-eh for the hadronic τ h final state.Among them, ν e jjjj has the largest cross sections without any requirements, followed by τ + τ − e − jjj and τ + ν τ e − jjj.For the remaining two background processes, due to their small cross sections, they do not play a dominant role.The number of simulated events for each background process is determined according to its importance in reducing the statistical uncertainty of the final sensitivities.At the LHeC (FCC-eh), we generate 2.1 (1.2) million τ + τ − e − jjj, 4.9 (3.0) million τ + τ − ν e jjj, 21 (19) million τ + ν τ e − jjj, 0.5 (0.5) million τ + ν τ ν e jjj, and 32 (25) million ν e jjjj events, respectively.

Data analysis
We apply following preselection to select the signal and reject the background events at the first stage.(i) Exactly one τ -jet with positive charge, i.e.N (τ + h ) = 1 and transverse momentum p T (τ h ) > 5 GeV; events with final state electron(s) or muon(s) are vetoed.(ii) All regular jets are sorted in descending order according to their transverse momenta and we require at least three jets, i.e.N (j) ≥ 3; for the p T thresholds of jets, when heavy neutrino mass is below 80 GeV, the p T of the first three leading jets are required to be greater than 10 GeV, while when the mass is above 80 GeV, we require the first two leading jets have p T greater than 20 GeV and the third one has p T greater than 10 GeV.(iii) To reject the background from misidentified taus, the solid angular distance ∆R between the regular jet and tau-jet is required to be larger than 0.8, i.e., ∆R(τ h , j) > 0.8.
, and for background processes after applying preselection (i)-(ii) for the hadronic τ h final state at the LHeC and FCC-eh.
To exploit the ∆R(τ h , j) observable, we calculate ∆R(τ h , j) values for all possible combinations of tau-and regular jets in the same event, and select the minimal value.Fig. 3 shows the distributions of such minimal ∆R(τ h , j) for the signal with benchmark m N = 120 GeV and , and for background processes after applying preselection (i)-(ii).One observes that the ν e jjjj process still dominates after applying preselection (i)-(ii), and can be rejected effectively by the ∆R(τ h , j) requirement.
In Table 2, we show the number of events for the signal with benchmark m N = 120 GeV and background processes after applying the preselection (i)-(iii) sequentially described above.One observes that, initially at the LHeC (FCC-eh), without any selection, B5a is a factor of 562 (417) larger than the sum of B1-B4, while the ratio is reduced to 21 (23) after preselection (i)-(ii) and becomes 14 (16) after preselection (iii), which means the preselection rejects B5a effectively.
Table 2: The number of events for the signal with benchmark m N = 120 GeV and , and for background processes after applying preselection (i)-(ii) sequentially for the hadronic τ h final state.The numbers correspond to the LHeC and FCC-eh with 1 and 3 ab −1 integrated luminosity, respectively.
To further reject the background, we input following nineteen observables into the Toolkit for Multivariate Analysis (TMVA) package [99] to perform the multivariate analysis (MVA).

C. Number of regular jets: N (j).
D. Magnitude and azimuthal angle of the missing transverse momentum: / E T , ϕ( / E T ).
E. The minimal value of solid angular distances between the final state tau-and regular jets: ∆R(τ h , j).
The Boosted Decision Tree (BDT) algorithm is adopted to perform the MVA and maximally reject the background.The default setting in the TMVA package is used, where the number of trees in the forest "NTrees" is set to be 850, the minimum percentage of training events required in a leaf node "MinNodeSize" is set to be 2.5%, the maximal depth of the decision tree allowed "MaxDepth" is set to be 3, the learning rate for AdaBoost algorithm "AdaBoostBeta" is set to be 0.5, the relative size of bagged event sample to original size of the data sample "BaggedSampleFraction" is set to be 0.5, and the number of grid points in variable range used in finding optimal cut in node splitting "nCuts" is set to be 20.In Fig. 4, we show BDT distributions for the total background and the benchmark signal with m N = 120 GeV at the LHeC and FCC-eh.Since the kinematics of signal 4 Leptonic τ µ final state

SM background processes
When tau decays leptonically, we select the final state with tau decaying into muons which we label as "τ µ " in this article.Thus, signature of final state has one positively charged muon, at least three regular jets plus missing energy.Besides the B1-B4 processes including taus, SM processes including muons can also contribute to the background in this scenario.Therefore, four additional background processes: µ + µ − e − jjj, µ + µ − ν e jjj, µ + ν µ e − jjj, and µ + ν µ ν e jjj are included, and labelled as "B5-B8" in this article.Table 3 shows the production cross sections of background processes at the LHeC and FCC-eh for the leptonic τ µ final state.

Data analysis
We apply following preselection to select the signal and reject the background events at the first stage.(i) Exactly one muon with positive charge, i.e.N (τ + µ ) = 1 and transverse momentum p T (τ µ ) > 5 GeV; events with final state electron(s) or tau-jet(s) are vetoed.(ii) All regular jets are sorted in descending order according to their transverse momenta  and we require at least three jets, i.e.N (j) ≥ 3; for the p T thresholds of jets, when heavy neutrino mass is below 80 GeV, the p T of the first three leading jets are required to be greater than 10 GeV, while when the mass is above 80 GeV, we require the first two leading jets have p T greater than 20 GeV and the third one has p T greater than 10 GeV.In Table 4, we show the number of events for the signal with benchmark m N = 120 GeV and dominant background processes after applying the preselection (i)-(ii) sequentially described above.Table 4: The number of events for the signal with benchmark m N = 120 GeV and , and for six dominant background processes after applying preselection (i)-(ii) sequentially for the leptonic τ µ final state.The numbers correspond to the LHeC and FCC-eh with 1 and 3 ab −1 integrated luminosity, respectively.
To further reject the background, we input following nineteen observables to perform the BDT-MVA.
C. Number of regular jets: N (j).

Results
In this section, based on our former analyses, we show the discovery sensitivities on the parameter for the heavy neutrino mass m N in the range of 10 to 3000 GeV.We also develop the strategy to reconstruct the heavy neutrino mass.Considering the heavy neutrino decays into one tau plus two jets for our signal, when m N is above W -boson mass, we check di-jet combinations for all jets with p T > 10 GeV, and recognize the combination with its invariant mass closest to W -boson mass as the di-jet (j + j) from the N decay.This di-jet is then combined with final state τ h ( τ µ ) and missing energy to reconstructed transverse mass M T of the whole (τ h +j +j + / E T ) ((τ µ +j +j + / E T )) system.Here, transverse mass , where E vis.T (⃗ p vis. T ) is the transverse energy (momentum) of the visible object or system, / E T ( ⃗ / p T ) is the missing transverse energy (momentum).E vis.T = (⃗ p vis. T ) 2 + (m vis. ) 2 , where m vis. is the invariant mass of the visible object or system./ E T = | ⃗ / p T |, assuming the invariant mass of the invisible object is zero.Fig 6 shows distributions of M T (τ h + j + j + / E T ) and M T (τ µ + j + j + / E T ) for the signal with benchmark m N = 120 GeV after preselection at the LHeC and FCC-eh for the hadronic τ h (left) and leptonic τ µ (right) final states, respectively.One observes that the transverse mass has sharp peak around m N , which means that it can be used to reconstruct the heavy neutrino mass.We note that above input observables listed in Sec.3.2 and Sec.4.2 including the four-momenta and angles are very basic and usually called low-level variables for the MVA.One can also construct some complicated observables and input such high-level variables to perform the MVA analysis.The BDT-MVA combines the information from all input observables with correlations.Because the low-level observables have already included the information of all final state objects, inputting the high-level observables will give similar BDT distributions and will not improve the final sensitivities a lot, c.f. Ref. [45].However, distributions of high-level observables are helpful for researchers to understand the kinematics of both signal and background.They are usually more distinct between the signal and background, and can be used to perform the cut-based analysis.Therefore, in Appendix C, we show distributions of some high-level observables after preselection for the signal and dominant background processes assuming m N = 120 GeV for the hadronic τ h and leptonic τ µ final states at the LHeC and FCC-eh, respectively.
After the preselection, the BDT selection is optimized according to the signal statistical significance calculated by Eq. (5.1) [100][101][102] for each mass case.
where N s (N b ) is the number of signal (total background) events after all selections.
In Table 5, we show selection efficiencies of preselection and BDT requirements.for both the signal with representative m N assumptions and background processes at the LHeC and FCC-eh for the hadronic τ h final state.Selection efficiencies for the leptonic τ µ final state are shown in Table 6.The total selection efficiency is the product of preselection and BDT selection efficiencies.The number of signal or background events after all selections can be calculated by multiplying the production cross section, collider luminosity and total selection efficiency.

LHeC H
as varying the heavy neutrino mass in the range of 10 to 3000 GeV for the hadronic τ h (H) and leptonic τ µ (L) final states at the LHeC and FCC-eh.Fig. 7 presents discovery sensitivities at the 2-and 5-σ significances on the parameter for the heavy neutrino mass in the range of 10 to 3000 GeV.Both sensitivities for the hadronic τ h and leptonic τ µ final states at the LHeC and FCC-eh are shown.At the LHeC, as m N changes from 10 GeV to 120 GeV, for the leptonic τ µ final state, the 2-σ discovery sensitivities on decrease from 8.4 × 10 −4 to 1.0 × 10 −5 , then begin to increase afterwards.The 2-σ discovery sensitivity at the FCC-eh has similar behavior as that at the LHeC, but its varying range is much smaller, i.e., between 1.3 × 10 −4 and 5.4 × 10 −6 .At both colliders, the 5-σ discovery sensitivities are slightly weaker than those for 2-σ.
For the hadronic τ h final state, at the LHeC, as m N changes from 10 GeV to 100 GeV, the 2-σ discovery sensitivities on , then begin to increase afterwards.The 2-σ discovery sensitivity at the FCC-eh also has similar behavior as that at the LHeC, but its varying range is smaller, i.e. between 1.9 × 10 −5 and 5.0 × 10 −6 .Compared with the leptonic τ µ final state, the hadronic τ h final state has bigger signal rate due to larger branching ratio for hadronic tau decay mode, but it suffers from the misidentified tau background which restricts its discovery sensitivities.Therefore, sensitivities for both final states are found to be similar for most of the parameter space investigated in this study.

Sensitivities compared with current experimental limits
In Fig. 8, assuming mixing parameters the 2-σ discovery sensitivities on |V ℓN | 2 are shown for both the hadronic τ h and leptonic τ µ final states at the LHeC and FCC-eh.With the same assumption, we also derive the limits on |V ℓN | 2 from Electroweak Precision Data (EWPD) [47][48][49][50][51]53] and DELPHI experiment [46], and show them in the same plot for the comparison.The details of derivation process are shown in Appendix A. We observe that sensitivity bounds from the LHeC and FCC-eh are stronger than those from EWPD when m N ≲ 900 GeV, and also stronger than those from DELPHI when m N ≳ 70 GeV.
[GeV] In Fig. 9, assuming representative heavy neutrino masses, we show the expected 2-σ discovery sensitivities at future LHeC and FCC-eh colliders in the |V τ N | 2 vs. |V eN | 2 plane, together with existed limits as a comparison.For any given mass, the expected 2-σ discovery sensitivity for a fixed number (as shown in Fig. 7), which behaves as a hyperbola curve in the |V τ N | 2 vs. |V eN | 2 plane.The top right parameter space of the curves is excluded.From Appendix A, yellow regions in all plots are allowed by the EWPD [47][48][49][50][51]53], while the cyan region in the top left plot is allowed by the DELPHI measurement [46].Constraints on |V eN | 2 from the LHC experiments [36,37,39] are shown as black dotted line, which appears only in the top left plot, because this constraint is too weak for large m N .From Appendix A, yellow regions in all plots are allowed by the EWPD [47][48][49][50][51]53], while the cyan region in the top left plot is allowed by the DELPHI measurement [46].Constraints on |V eN | 2 from the LHC experiments [36,37,39] are shown as black dotted line, which appears only in the top left plot, because these constraints are too weak for large m N .
In the top left plot, we choose m N = 70 GeV as a typical benchmark point when m N < m Z .In this case, results show that the future FCC-eh (both leptonic τ µ and hadronic τ h final states) and LHeC experiments (only hadronic τ h final state) are both possible to probe additional parameter space compared with the DELPHI experiment.In other three plots, we choose m N = 120, 300, and 900 GeV as benchmark points, and all of them locate in the range where m N > m Z .In these cases, the combination of measurements from EWPD and searches at pe collider constrain the parameter space synergistically, while the limits on |V eN | 2 from the LHC experiments are much weaker and not shown.Our results show that except at very large m N region, for example, m N ≳ 900 GeV, the expected sensitivity from LHeC experiment will be similar to that from EWPD, due to the quickly decreasing signal production cross section.For other mass ranges, future LHeC and FCC-eh experiments are expected to significantly enhance the discovery potential for a large portion of the |V τ N | 2 vs. |V eN | 2 plane, compared with the EWPD and LHC searches.

Conclusion and discussion
In this paper, we utilize the lepton number violation signal process of p e − → τ + jjj to search for heavy Majorana neutrinos at the future electron-proton colliders.We consider the LHeC (FCC-eh) running with an electron beam energy of 60 GeV, a proton beam energy of 7 (50) TeV and an integrated luminosity of 1 (3) ab −1 .The electron beam is considered to be unpolarized.To simplify the analyses, we consider the phenomenological simplified Type-I and assume that only one generation of heavy neutrinos N is within the collider access and mixes with active neutrinos of tau and electron flavours, i.e. |V τ N | 2 , |V eN | 2 ̸ = 0 and |V µN | 2 = 0. We perform analyses for both hadronic τ h and leptonic τ µ final states, and forecast discovery sensitivities for heavy neutrinos mass m N in the range 10-3000 GeV at LHeC and FCC-eh.
The cross sections of the LNV signal pe − → τ + jjj are presented in Fig. 2 as a function of m N at the LHeC and FCC-eh.We apply detector configurations and simulate signal and related SM background events for both hadronic τ h and leptonic τ µ final states.When tau decays hadronically (leptonically), we select the final state containing the tau-jet (muon).The preselections are applied to select the signal and reject the background events at the first stage.Then various observables are input, and the BDT algorithm is adopted to perform the multivariate analysis and maximally reject the background.The BDT distributions with different m N assumptions are presented and some high-level observables corresponding benchmark m N case are shown in appendices.
The strategy to reconstruct the heavy neutrino mass is developed and distributions of reconstructed mass are presented.After the preselection, the BDT selection is optimized to maximize the signal statistical significance.In Appendix D, we show selection efficiencies of preselection and BDT requirements.for both the signal with representative m N assumptions and background processes at the LHeC and FCC-eh.The discovery sensitivities on parameter for the heavy neutrino mass in the range of 10 to 3000 GeV are presented in Fig. 7.At the 2-σ significance, the best sensitivity is ∼ 1.2 × 10 −5 (5.0 × 10 −6 ) at the LHeC (FCC-eh) when m N ∼ 100 GeV for the hadronic τ h final state.The discovery sensitivities for the leptonic τ µ final state are found to be similar for most of the parameter space investigated than those for the hadronic τ h final state.
We derive the limits on the mixing parameters from EWPD and DELPHI experiment in Appendix A. Assuming pe collider searches are compared with those from the EWPD and DELPHI experiment in Fig. 8.We find that sensitivity bounds from the LHeC and FCC-eh are stronger than those from EWPD for almost all the mass range, and also stronger than those from DELPHI when m N ≳ 70 GeV.
To investigate the individual sensitivities on |V τ N | 2 , in Fig. 9, assuming representative m N , we show the expected 2-σ sensitivity curves from pe collider searches in the |V τ N | 2 vs. |V eN | 2 plane, together with existed limits from EWPD, DELPHI, and LHC experiments for comparison.Our results show that compared with current experimental limits, future pe experiments can probe large additional regions in the parameter space formed by |V τ N | 2 and |V eN | 2 , and thus significantly enhance the discovery potential for a large portion of the Compared our sensitivity results with those in Ref [45], at pe colliders, sensitivities for the τ + final state are weaker than the µ + final state.This is mainly because that (i) due to the challenges in detecting the final state taus, the signal rate is much lower for the τ + final state; (ii) the τ + final state also suffers from larger background.Besides, as pointed in Ref [45], because the total background cross section for the e + final state is about three times larger than that for the µ + final state, sensitivities for the e + final state is expected to be weaker than those for the µ + final state.However, because the p e − → e + jjj signal process can depend on the mixing parameter |V eN | 2 only, it is a unique channel to probe |V eN | 2 independent of other mixing parameters.In this sense, the detailed analyses of the e + final state are still meaningful, and we leave it for future studies.
A jet can be misidentified as a hadronic τ h in the detector, which can contribute to the background for the hadronic τ h final state.Because of the huge production cross section, even a small misidentification rate of jet to hadronic tau can still lead to large background.To estimate its effects, we include the multi-jet process "p e − → ν e jjjj" in the background.Our analysis indicates that the ∆R(τ, j) and BDT requirements can reject such background effectively.However, because this background is greatly affected by the detector's performance and the detectors of future pe colliders are still under development, it should be carefully considered for future studies.
A Constraints from Z → N ν, N ν rare decay channels and EWPD In this appendix, we show details about the constraints from Z-boson rare decay [46] and EWPD [47][48][49][50][51]53].In this paper, since we consider the signal process p e − → N (→ τ + W − ) j corresponding to both W -N -e and W -N -τ vertices, thus we assume mixing parameters |V eN | 2 and |V τ N | 2 are both nonzero.We also assume |V µN | 2 = 0 here for simplicity.

C Distributions of representative high-level observables
Here, / E T is the magnitude of the missing transverse momentum.p T and η are the transverse momentum and pseudorapidity of the final state object, respectively.∆ϕ(τ h /τ µ , / E T ) is the azimuthal angle difference between the final state τ h / τ µ and missing transverse momentum.M T (τ h /τ µ + / E T ) is the transverse mass of the system of final state τ h / τ µ and missing transverse momentum.The invariant mass of the di-jet M (j + j) denotes the reconstructed W -boson mass, and the transverse mass M T (τ h /τ µ +j +j + / E T ) measures the reconstructed N mass, which are explained in Sec.

Figure 1 :
Figure 1: The production process of the Lepton Number Violating (LNV) signal in the tau final state via the heavy Majorana neutrinos at pe colliders.
Figure 7: 2-and 5-σ discovery sensitivities on the parameter|V τ N | 2 |V eN | 2 / |V τ N | 2 + |V eN | 2as varying the heavy neutrino mass in the range of 10 to 3000 GeV for the hadronic τ h (H) and leptonic τ µ (L) final states at the LHeC and FCC-eh.

Figure 8 :
Figure 8: Assuming the mixing parameter |V ℓN | 2 = |V τ N | 2 = |V eN | 2 , the discovery sensitivities on |V ℓN | 2 at 2-σ significance as the heavy neutrino mass changes from 10 to 3000 GeV for Hadronic τ h (H) and Leptonic τ µ (L) final states at the LHeC and FCC-eh.The constraints from EWPD [47-51, 53] (green solid line) and DELPHI experiments [46] (purple solid line) are derived and displayed in the same plot for the comparison, see more details in Appendix A.

2 Figure 9 :
Figure9: Assuming m N = 70, 120, 300, 900 GeV, the 2-σ sensitivity curves for the Hadronic τ h (H, solid) and Leptonic τ µ (L, dashed) final states at the LHeC (red) and FCC-eh (blue) in the |V τ N | 2 vs. |V eN | 2 plane, where the top right parameter space of the curves are excluded.From Appendix A, yellow regions in all plots are allowed by the EWPD[47][48][49][50][51]53], while the cyan region in the top left plot is allowed by the DELPHI measurement[46].Constraints on |V eN | 2 from the LHC experiments[36,37,39] are shown as black dotted line, which appears only in the top left plot, because these constraints are too weak for large m N .

Table 1 :
The production cross sections of background processes at the LHeC and FCC-eh for the hadronic τ h final state.

Table 3 :
The production cross sections of background processes at the LHeC and FCC-eh for the leptonic τ µ final state.

Table 6 :
Similar as Table5, but for the leptonic τ µ final state.