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Introduction
The discovery of the Standard Model (SM) like Higgs boson at the Large Hadron Collider (LHC) experiment has proven that in the SM, masses of the fermions and gauge bosons are generated via the Brout-Englert-Higgs (BEH) mechanism.However, the origin of light neutrino masses and mixing still remains a key question, which can not be explained by the SM.A number of neutrino oscillation experiments have observed that, the solar and atmospheric neutrino mass splittings are ∆m 2  12 ∼ 10 −5 eV 2 and ∆m 2 13 ∼ 10 −3 eV 2 , and the mixing angles are θ 12 ∼ 32 • , θ 23 ∼ 45 • , and θ 13 ∼ 9 • [1].There have been a number of proposed beyond the Standard Model (BSM) extensions, which can explain the observed light neutrino masses and mixings and contain SM gauge singlet right-handed neutrinos (RHNs).The U (1) B−L model is such an extension, where the gauge sector is also extended by an additional U (1) B−L gauge group [2][3][4][5].The model contains three right handed neutrinos N i , which are SM gauge singlet, however carry non-trivial charges under the U (1) B−L gauge symmetry.Other than the three RHNs, the model also contains an extended gauge sector, and a complex scalar S. The SM neutrino mass in this model is generated from the lepton number violating (LNV) d = 5 seesaw operator LLHH/Λ [6,7] via Type-I seesaw mechanism [8][9][10][11][12].
Being SM gauge singlet RHNs interact with the SM particles only via their mixing with the active neutrinos, referred as active-sterile mixing, which is proportional to m ν /M N , where m ν and M N are the light neutrino and the RHN mass scales, respectively.Since m ν < eV, this mixing is small for RHN with mass few tens to hundreads of GeV, leading to an in-general suppressed production of a single RHN at a pp collider.This limitation can however be evaded in the gauged B − L model, since RHNs can be produced via additional production mechanism that involves unsuppressed interactions of RHN with BSM/SM particles [13,14].
Motivated by this, we study pair-production of RHNs in gauged B −L model, where the production primarily occurs via mediation of BSM gauge boson Z .Specifically, we focus on the low mass region of N satisfying M N < M W , for which N gives rise to distinctive model signatures.The produced N undergoes three body decays to few different final states, among which we consider N → µqq decay mode.For the considered mass range, two body decay of N is kinematically forbidden.Since the decay of N strongly depends on the active-sterile mixing, hence for mixing in agreement with eV light neutrino mass, the RHN undergoes displaced decays where its decay vertex is considerably displaced from the production vertex.Additionally, due to a large hierarchy between the masses of the Z and N , the decay product of N is collimated, resulting in a displaced fat-jet signature.We hence study distinctive long-lived signatures of N , that can be detected at the High-Luminosity upgrade of the LHC, and the future FCC-hh machine.For previous studies on displaced RHN decay, see [15][16][17][18][19][20][21].
In particular, we consider displaced decays of N in two different region of the detectors, a) in the inner detector (ID) and b) the farthest end of the detector, which is muon spectrometer (MS).To realise the former, N must have proper decay length ranging between ∼ mm to hundreds of mm, while for later, the decay length should be ∼ m.We note that the signal description between a) and b) differ widely.For the decay of N in ID, the signature contains two displaced fat-jets.For N decaying in the MS, the track properties in the ID, energy deposits in the calorimeter can not be used, which are used for the jet formation.Therefore, a jet description of the final state particles is inadequate in this case, and we instead perform a track-based analysis.We further extend the analysis by demanding displaced decay of at-least one RHN in the ID/MS.Assuming a background free environment we find that few tens of displaced fat-jet events can be observed at the HL-LHC for c.m.energy √ s = 14 TeV and luminosity L = 3 ab −1 .For FCC-hh that can operate with a much higher c.m.energy √ s = 100 TeV and luminosity L = 30 ab −1 , the maximum achievable events increases by order of magnitude.For the decay of N in the MS, the discovery prospect at the HL-LHC is rather low.This however improves significantly for FCC-hh.The discussion proceeds as follows: In Section.2, we present a brief review of the model, following which we present a discussion on existing constraints in Section.3. In Section.4, and Section.5, we discuss the displaced decays and pair-production of two N s.We present an extensive analysis for HL-LHC and FCC-hh in Section.6 and Section.7, respectively.
Finally, we present a summary in Section.8.

The B − L gauge model
We consider the gauge B − L model, which in addition to the particles of the SM sector also contains three right-handed neutrinos ν R , a BSM Higgs field S and a BSM gauge boson B µ .The gauge group is where L B−L is the B − L Lagrangian, and L SM is the Lagrangian for the SM sector.The B − L Lagrangian has the form with In the above, D µ represents the covariant derivative [22], where G α µ , W a µ , B µ are the SM gauge fields with associated couplings g s , g, g 1 and the respective generators are T α , T a , Y , respectively.The field denoted as B µ represents the gauge field for U (1) B−L gauge symmetry, g represents the respective gauge coupling and the B − L quantum number is denoted by Y B−L .In this paper, we neglect the mixing between U (1) B−L and U (1) Y to simplify the model, i.e. we consider the minimal gauged B − L model.This model is therefore valid in the limiting case of small mixing between B µ and B µ .The gauge sector of the model also includes the following kinetic term for the gauge field B µ , where In the above, ψ represents the SM fermion fields.These fields receive an additional term in their covariant derivatives, since they are non-trivially charged under the B − L gauge symmetry, with Y B−L = 1/3 and −1 for the quark and lepton fields, respectively.In the above a summation over the fermion species and generations is implied.
• Neutrino mass: In Eq. (2.2), the Yukawa matrix y ν represents the Dirac Yukawa coupling and y M is the Yukawa coupling connecting the RHNs with the complex scalar field S. The RHN mass is generated due to breaking of the B − L symmetry, with the mass matrix given by M R = y M S .The light neutrinos mix with the RHNs via the Dirac mass matrix m D = y ν v/ √ 2. The complete mass matrix in the (ν L , ν c R ) basis has the form where Here, v = φ 0 and v BL = S are the vacuum expectation values for electroweak and B − L symmetry breaking, respectively.In the seesaw limit, M R m D , the light and heavy neutrino masses are The flavour and mass eigenstates of the light and heavy neutrinos are connected as where ν m L and (N c R ) T represent the left-chiral mass basis for the light and heavy neutrinos, respectively.In our subsequent discussion, we represent the physical Majorana fields for the light and heavy neutrinos via ν m = ν m L + (ν m L ) c and N = N R + (N R ) c , respectively.In the above, we schematically write the 6-dimensional mixing matrix in terms of 3-dimensional blocks.Assuming the charged lepton mass matrix to be diagonal, the sub-block V ll can approximately be considered as the PMNS mixing matrix U PMNS .The other sub-block V lN represent the mixing between the light and RHN states and is referred as active-sterile mixing.In general, V lN is an arbitrary 3 × 3 matrix.However, to pursue a collider study on the proposed signature, it is sufficient for us to consider one generation of RHNs and only non-zero V µN , which we follow in the subsequent sections.
• Z Gauge boson mass: Due to the presence of an additional U (1) B−L gauge symmetry, the model contains BSM gauge bosons B µ .We refer to the massive state as Z .Similar to the RHNs, the additional neutral gauge boson mass M Z is generated via spontaneous breaking of B − L gauge symmetry.The mass of Z is related to the symmetry breaking scale v BL as where g is the associated B − L gauge coupling constant.
• SM Higgs and BSM Higgs: After spontaneous symmetry breaking (SSB), the SM Higgs doublet φ and BSM scalar S is given by with the dynamical states h 1 and h 2 .Owing to the non-zero λ 3 , h 1 and h 2 mix with each other which leads to the scalar mass matrix given by The basis states h 1 and h 2 can be rotated by suitable angle α to the new basis states H 1 and H 2 .The new basis states represents the physical basis states which are given by, where H 1 is the SM-like Higgs and H 2 is the mostly BSM Higgs.The mixing angle between the two states is (2.16) The mass square eigenvalues of H 1 and H 2 are given by, In what follows, we consider the scalar mixing to be negligible, for which the SM and BSM Higgs masses have the form, respectively Due to this choice of small mixing, any BSM Higgs production and its decay to two RHNs will be suppressed.The RHNs have charged current and neutral current interactions with the SM fields, with the scalars H 1 , H 2 and the gauge boson Z .The respective charged current interaction Lagrangian has the form, Limits in the M Z and g plane derived from ATLAS [23], and CMS [24] searches.The limit from LEP-II [25] has been shown by the dot-dashed red line.The green dashed and orange dashed lines represent the projection for HL-LHC [26] and FCC-hh [27], respectively.The black horizontal dotted line represents Γ/M Z = 0.6%.
where P L is the left-chirality projection operator P L = 1−γ 5 2 .Similarly, the NC interaction is given by ) In the above, i, j represent generation indices.The interaction with the SM and BSM Higgs have the form

Experimental constraints
The Z and N of the gauged B − L model are significantly constrained from the nonobservation of any direct signature at LEP, and LHC.In addition, eV light neutrino masses impose additional constraints on the model parameters.Below we briefly summarise the different existing constraints, • Heavy resonance search: the search for a massive resonance at LHC decaying to di-lepton/di-jet imposes tight limits on the respective production cross-section.The Feynman diagram representing pair-production of two RHNs via Z mediation and its displaced decays to a lqq final states.
13 TeV LHC search for a heavy resonance decaying into two leptons pp → Z → l + l − put a constraint M Z > 5.0 TeV at 90% C.L, assuming a 100% branching ratio of Z decaying into two leptons.For different branching ratios, the limit relaxes.In Fig. 1, we translate the di-lepton constraint from the LHC on M Z , and g plane.We adopt the following procedure in translating the bound.For the ATLAS search with L = 139 fb −1 , we consider σ th (pp → Z → l + l − ) < σ obs , where the observed limit corresponds to the stringent 95% C.L. limit from [23].The CMS limit is set on the relative cross section, σ rel = is the observed cross section in the m l + l − = 60 − 120 GeV window, which can be calculated as N obs /(Acc × Eff × L) 1 .In Fig. 1, we represent the ee + µµ combined limit by the purple line from the CMS search and the orange line represents the limit from the ATLAS search.As we consider the results for Γ/M Z = 0.6% for the CMS search, we show this by black horizontal dotted line.The dashed green line represents the future sensitivity of HL-LHC with 3000 fb −1 [26] and the dashed magenta line represents projection for FCC-hh [27] .We also note that the former search at LEP-II [25,[28][29][30] for a massive resonance constrains Z mass and gauge coupling, and thus the B − L breaking scale as v BL ≡ M Z /(2g ) ≥ 3.45 TeV.It has been shown by the dashed red line, which is considerably relaxed w.r.t. the present LHC limit.
• Search for RHN: Other than the constraint on Z , the RHN mass and mixings are also constrained, both from neutrino mass measurements as well as direct searches at the LHC.In the present work, we consider relatively low mass RH neutrinos, 10 GeV < M N < M W , which are mainly produced from the Z mediated channel.
light neutrino mass measurements.As we are working with a Type-I seesaw scenario with B − L gauge symmetry, the light neutrino mass m ν m 2 D /M R V 2 lN M N where the active-sterile mixing angle V lN m D /M R .This fixes the active-sterile mixing, The low mass RHN is further constrained by LHC searches for a heavy neutral lepton [31] via the decay mode pp → W → lN and the decay N → llν/ljj.The search for a displaced neutral lepton [31,32] in particular constrains active-sterile mixing as V µN < 10 −2 for RHN mass O(10) GeV.Other searches such as [33][34][35] mainly target heavier masses, where the RHN is not displaced, and hence not relevant for our case.
• LLP searches from SM Higgs decay: there are different exotic decays of the SM Higgs boson which give rise to distinctive signatures.The decay H 1 → N N followed by the decay of N into ljj/llν final states give rise to displaced lepton/displaced jet signature, as in the mass range of interest N has a large decay length.The decay is open kinematically, however it has a very suppressed branching ratio due to our choice of a small α.The partial decay width of this decay mode has the form , where For instance, the branching ratio BR(H 1 → N N ) becomes 0.005% for sin θ = 3×10 −3 and M N = 50 GeV.There are different CMS and ATLAS searches [36][37][38][39] to probe exotic decays of the SM Higgs into two LLP states, which can be compared with.This includes a search for exotic decays of the Higgs boson into LLP in the tracking system [36][37][38][39], or a search for LLP decaying in the ATLAS muon spectrometer [40][41][42], and in the CMS endcap muon detectors [43].Due to a very suppressed branching ratio, M N and V lN are unconstrained from these searches.

Pair production and decay of RHN at a pp machine
The RHN can be pair produced via the processes q q → Z → N N and q q → H 1 /H 2 → N N , among which the Higgs mediated channels give suppressed cross-sections, as we consider a small SM-BSM Higgs mixing sin α.Therefore, production in pair primarily occurs via pp → Z → N N channel.The cross-section depends strongly on the gauge coupling g , the mass of Z gauge boson M Z , its decay width Γ Z , as well as the mass of the RHN M N .We show the respective Feynman diagram for this process in Fig. 2.
In the left and right panel of Fig. 3, we show the pair production cross-section of RHN, σ(pp → Z → N N ) as a function of M Z at the HL-LHC and FCC-hh, respectively.This cross-section is calculated using MadGraph.For illustration, we assume two different values of couplings, g = 0.1, 0.003 and M N = 20 GeV.We show the LHC constraints (as shown in Fig. 1) by the black solid and dotted lines for g values 0.1 and 0.003, respectively.For instance, for the choice of gauge coupling g = 0.003, the heavy resonance search in l + l − decay channel rules out any value of M Z 0.7 TeV.For g = 0.1 the exclusion limit reaches higher value, M Z 4.5 TeV.Since in this figure we consider M Z satisfying M Z > 2M N , the on-shell production of Z and its decay will dominate the pair-production cross-section σ(pp → N N ).As can be seen from the figure, for the same gauge coupling g , the ratio of cross-section R = σ(pp → N N ) HL−LHC /σ(pp → N N ) FCC −hh vary over a wide range.For on-shell Z production with narrow width approximation Γ Z /M Z 10%, or less, this can be expressed as, With an increase in the c.m. energy from √ s = 14 TeV to 100 TeV, the ratio R increases by order of magnitude, and the increase is larger for higher masses.This occurs as the partonic c.m. energy is larger in a 100 TeV collider compared to the available partonic c.m. energy for 14 TeV.We also show the variation of cross-section for HL-LHC and FCC-hh w.r.t. the variation of both M N and M Z in Fig. 4. For this figure we consider the same values of gauge coupling that we use for Fig. 3 The shaded vertical region show the present constraint from LHC, both in the left and right panel.In the figure of left panel, the cross-section is significantly suppressed for M N > M Z /2, as can be seen from the figure.This happens as Z is not on-shell in this region.For Z with mass ∼ TeV and M N M Z , the cross-section can be in the sub-fb range.From the figure of right panel, this can be seen, that for higher Z mass the production cross section is almost independent of the mass of RHN for the variation of M N = 100 GeV to 2 TeV.
Decay of RHN: The RHN interacts with a lepton and a W gauge boson, which is governed by the active-sterile mixing V lN .It also interacts with light neutrino and a Z, H 1 , TeV.The vertically shaded region represents the LHC exclusion limit.
as well as, Z and BSM Higgs H 2 .For the mass range being considered in this work , the RHN N decays pre-dominantly via an off-shell W, Z, H 1 states into two SM fermions.The two body decay of RHN into a (lW, νZ, νH 1,2 ) state will only open if mass of N larger than the two body kinematic limit.The partial decay widths for N → lqq , νf f and ννν decay modes have the following expressions [44], In the above, I(x u , x d , x l ) = 12 The partial decay width for the other mode with a same α = β is given by, . The values of C f 1 and C f 2 are given in [44].
In addition to this there can also be two body decays N → l ± π ± , however, this is suppressed in our case, as we consider M N > 1 GeV ∼ the scale of non-perturbative QCD.Therefore, we do not consider this channel in our analysis.For M N = 10 − 70 GeV, the dominant decay mode is N → µjj with a branching ratio BR(N → µjj) 0.5.Branching ratios for other decay modes are BR(N → νjj) 0.2, BR(N → ν µ l + l − ) 0.10, BR(N → µeν e + µτ ν τ ) 0.15 and BR(N → ννν) 0.05.

RHN decay in inner tracker and muon spectrometer
We are interested in a signature which contains a pair of RHNs produced from Z decay and the decay of N to a N → lqq final state.To be specific, we consider the N → µqq final state and analyse the model signature.Signatures with an electron in the final state N → eqq can also potentially be explored by similar search strategies.Specifically, we consider two scenarios: (a) when the N decays within the inner detector (ID) of the HL-LHC/FCC-hh, and (b) in the muon spectrometer (MS).The search strategies for these two different decay modes differ widely.We will also consider a combination of these two, when one RHN decays in the ID and other in the MS.In order to analyse the model signatures, we simulate the events using the following steps.We use the FeynRules [45,46] model file and Universal FeynRules Output (UFO) [47] corresponding to Ref. [15], which we use in combination with the Monte Carlo event generator MadGraph5aMC@NLO -v2.6.7 [48] to generate events at the parton level.The FeynRules [45,46] model file and UFO is publicly available from the FeynRules Model Database at [49].For every signal sample, we generate 50000 signal events with MadGraph5aMC@NLO -v2.6.7,where we use NN23LO1 PDF set [50].We set max jet flavour at 5 to take account of the b quark contribution in the PDF.We then pass the generated parton level events on to PYTHIA v8.235 [51] which handles the initial and final state radiation, parton showering, hadronization, and heavy hadron decays.The clustering of the events and jet formation are performed by FastJet v3.2.1 [52].We consider a Cambridge-Aachen jet algorithm [53] for jet clustering with radius parameter R = 1.0.We analyse events at the generator level with PYTHIA v8.235.

Decay Probability of N
For M N < M W and for the viable range of active-sterile mixing that satisfies light neutrino mass constraint, the RHN N undergoes displaced decays with a displacement l ∼ mm or even longer.N decays via off-shell W, Z, H states with the decay length2  be seen, in the majority of events, decay length of N is more than a mm, thereby giving rise to a decay vertex considerably displaced from the production vertex of N .
As we have two RHNs from the channel pp → Z → N N , therefore the two RHNs can decay in different parts of the HL-LHC and FCC-hh detectors.The probability that both the RHNs decay within a distance interval (L 1 , L 2 ) from their production vertex is given by where cτ is proper decay length, θ represents the active-sterile mixing and b 1,2 = (βγ) 1,2 are the boost factors of the two RHNs.and is the probability distribution function of the boost.The probability that one RHN decay within interval (L 1 , L 2 ) and the other within interval (L 3 , L 4 ) is given by The decay probability of one RHN decaying within length L 1 − L 2 then becomes, where b is the boost factor of the respective RHN.We use the boost distributions of N from the HepMC file [54], which has been generated by showering the LHE events [55] from MadGraph.
To estimate the probability of decay of N in various parts of the HL-LHC and FCChh detectors, we adopt a simplistic approach.In  of the sub-detectors, where the decay of RHN leads to observable signatures.For Inner detector (ID) we demand RHN decays in between 2 mm and 300 mm.In this region, the vertex reconstruction efficiency is large [41].The decays of RHN in ID leads to multi-track displaced vertex.Similar to ID the decay of N in the MS creates a number of tracks.For this we consider that RHN decays within 4000-7000 mm where muon ROI trigger efficiency is higher [42].Hadronic decays of RHN in the outer edge of the ECal or in the HCal leads to a distinct signatures marked by significantly large energy deposit in HCal compared to that in the ECal [56].We however do not consider such signature in the present analysis, rather restrict to RHN decay in ID and in MS.For the FCC-hh detector coverage of various sub-detectors used in the analysis, we refer the readers FCC-CDR [57].
In Fig. 6, we show the probability of the RHNs decaying in different modules of the HL-LHC detector.The left panel represents the decay probability of a single N in the detector and in the right panel, we show the probability for both the N s to undergo displaced decays.The acronyms ID, CAL, MS, MET refer to the decay of RHN inside the inner detector, the calorimeter, the muon spectrometer, and outside of the CMS and ATLAS detector, respectively.From the figure, it is evident that the probability is maximal if the RHN decays inside the inner detector or outside of the detector, and it is minimal, if the decay happens in the muon spectrometer.The probability for other configurations, such as if the decay occurs in the calorimeter, or among the two N s one decays in the inner detector and another in calorimeter/muon spectrometer is somewhat in between.We show the respective probabilities for the FCC-hh in Fig. 7.
Fig. 6 and Fig. 7 illustrate useful information about the respective geometric probabilities of the RHN to undergo displaced decays.A naive comparison between the results derived for HL-LHC and FCC-hh shows that while the maximum decay probability of one/two N decaying within ID is very similar 95%, the probability for one N decaying in the ID and the other in MS is somewhat larger in FCC-hh than HL-LHC.A large probability along with a higher c.m. energy and a higher achievable integrated luminosity will result in a significantly large number of events, that can be observed at the FCC-hh.In the subsequent sections, we present a detailed discussion of the observable signal events at HL-LHC and FCC-hh, while taking into account both the kinematic and geometric cut efficiencies.

Signal description for N decaying in the ID
As we are focussing on a heavier Z with mass M Z ≥ 1 TeV, and lighter N with masses 10 − 70 GeV satisfying M N < M W , the N hence will be boosted, resulting in collimated decay products 3 .In Fig. 8, we show the ∆R = (∆η) 2 + (∆φ) 2 separation between the muon and the closest quark at the parton level, originating from the decay of a RHN.The ∆R separation between two closest quarks also shows similar features.As can be seen, in most of the events the muon-quark separation ∆R(µ, q) 0.4, where ∆R = 0.4 has often been utilised in experimental analyses as the isolation criterion between final states.Instead of abiding by the selection criterion with jet radius R = 0. Figure 8: ∆R separation between the µ and closest q at the partonic level.
∆R( , j), ∆R(j, j) > 0.4, we hence demand a large jet radius R = 1.0, referred to as fat-jet [58].Since we consider a large jet-radius and do not impose any specific isolation criterion, in the majority of the events, the lepton hence will be part of the fat-jet.Extending further, we also show the distributions of ∆R after jet clustering.In left panel of Fig. 9, we show the ∆R separation of the closest µ from the leading jet, for both √ s = 14 TeV and √ s = 100 TeV.It is evident from the figure that the ∆R separation between the µ and the j is ∆R(µ, j) 0.4, which mimics the partonic distribution, shown in Fig. 8.
As we have pointed out before, for the considered RHN state with mass M N < M W , the eV scale light neutrino mass demands a very suppressed active-sterile mixing V lN ∼ 10 −6 , thereby leading to a macroscopic decay length for N .Therefore, the final state originating from pp → Z → N N decay would be In the above J dis f at represents the fat-jet originating from N decay, which is also displaced.Since the fat-jet is formed by the decay products of N , which is considerably lighter than Z , satisfying M N M Z 1, the N and hence the leading and sub-leading jet have a high transverse momentum.We show the p T distribution of leading and sub-leading jets in the right panel of Fig. 9.As can be seen, for M Z = 1 TeV and M N = 20 GeV, the distribution peaks around p T ∼ M Z /2 = 500 GeV.For M Z = 8 TeV, the peaks appear at a higher value, p T ∼ M Z /2 = 4 TeV.This high transverse momentum of leading and sub-leading jets will be useful in designing the selection cuts.
We also note that the proposed signal can also be extended to include ejj final state.This however depends strongly on the active-sterile mixing V eN .With V eN V µN , the contribution from ejj channel will be small.Here, we adopt a simplistic approach and consider only V µN = 0, therefore do not consider ejj state in fat-jet description.
Figure 9: Left: ∆R separation between the µ and leading jet.Right: Transverse momentum distribution of the leading and sub-leading fat-jet.

Signal description for N decaying in the MS
If the RHN decays within the muon spectrometer, then one can not use the information on the number of tracks in the ID, p T of track, and other variables such as deposited energy in the calorimeter.Instead of performing an analysis based on observables of a fat-jet, we rather present a track based analysis.The final state originating from pp → Z → N N would be considered as where µ dis represents a muon, produced from the RHN displaced decays.In the above, X corresponds to all other final state particles, including hadrons, electrons, photons etc. which are generated due to RHN decays as well as due to showering.We represent all the charged particles generated from the decay of RHN as track 1 , track 2 , ..track n and Y represent any neutral particle in the final state that does not leave any footprint of a charged track in the muon spectrometer.
For these above mentioned model signatures, we first perform the analysis for HL-LHC with c.m. energy √ s = 14 TeV considering an integrated luminosity L = 3 ab −1 and later investigate the discovery prospect for FCC-hh with √ s = 100 TeV with L = 30 ab −1 .
To evaluate the number of events for the above mentioned two signal descriptions, we implement both the geometric and kinematic cuts, the details of which are provided in Section.6, and Section.7. The geometric cuts take into account the probability of RHN to decay within a specified region of HL-LHC/FCC-hh detector.For the geometric cut efficiency, we follow the prescription described in Section.5.1, and use Eq.(5.2), Eq. (5.3), and Eq.(5.4).As kinematic cuts, we use following few variables -p T of jet, p T of associated tracks, η of jet and tracks.Below, we briefly outline the adopted procedure, which we implement for the analysis: • We implement both kinematic and geometric cuts to evaluate the number of signal events that can be detected.Let g and k denote the geometric and kinematic cuts.A maximum of N events = 10 events can be observed in 2IDvx event category.The brown dashed line represents the FCC-ee projection [59,60], and the red shaded region represents the excluded region from [61,62].

Decay vertex in the inner-detector (IDvx):
We consider the following sets of cuts for the analysis of two displaced vertices in the ID, • C1: Both the RHN decay within ID L 1 = 2 mm and ID L 2 = 300 mm.
• C2: The leading and sub-leading fat-jet j 0,1 has to satisfy |η(j)| < 4.5 and p T (j) ≥ 150 GeV.The strong cut on jet-p T is motivated by the ATLAS analysis of boosted RHN [63] and other fat-jet analyses [64,65].
• C3: Additionally for each of the leading and sub-leading fat-jet, number of associated track with p T ≥ 1GeV, |η| < 2.5 should be n trk ≥ 4.This can be easily satisfied, as majority of events have a p T (track) > 1 GeV, as is evident from the left panel of Fig. 10.The number of tracks associated to the leading and sub-leading jet peaks around ∼ 7, hence the selection criterion on number of tracks is easily satisfied.These criteria are inspired from the ATLAS search for long-lived particles decaying into displaced hadronic jets in ID [41].
• C4: Finally we select the events if number of jets n j ≥ 2 and the above-mentioned cuts are satisfied for both the leading and sub-leading fat-jets.
As we outline in Section.5.1, we consider the boost distribution of the two N s from the events that satisfy the above mentioned kinematic cuts C2-C4 and evaluate the geometric cut-efficiency as P(L 1 , L 2 , √ s, M N , M Z , θ) using Eq.5.2, and Eq.5.3.In the left panel of Fig. 11, we show the contours for N events = 3 and 8 events in |V µN | and M N plane by blue solid lines.A maximum number of 10 2IDvx events can be observed with the luminosity L = 3 ab −1 .In the same plot, we also show the projection for FCC-ee in the channel e + e − → Z → νN by the brown dashed line, derived in [59,60].The region shaded in red is excluded from CMS and ATLAS searches in both prompt and displaced leptonic decay signatures of RHNs [60][61][62].We also show the active-sterile neutrino mixing required for light neutrino mass m ν in between 0.01 − 0.1 eV.We also estimate the number of events demanding decay of at least one N in the ID4 and show the result in the left panel of Fig. 12.For this, we consider only the probability of one RHN decay within 2 − 300 mm using Eq.5.4 and the kinematic cut-efficiencies as k ∼ 100%.We checked explicitly that more than 90% events satisfy the above mentioned kinematic cuts.With the requirement of at least one N decaying in the ID, the probability is slightly larger, which leads to maximum number of events as N events = 14 , that can be obtained with 3 ab −1 of data.

Decay vertex in the muon spectrometer (MSvx):
We consider that N has a long lifetime and it decays to µqq in the muon-spectrometer.We consider the following sets of selection cuts: • C1: We demand both the RHN decay occur within the muon-spectrometer between the length M S L 1 = 4000 mm, and M S L 2 = 7000 mm.The chosen length interval corresponds to the outer edge of the HCal and the middle station of muon chambers where the muon RoI trigger efficiency is higher [42].
• C3: Finally we select the events if two decay vertex and the associated tracks from the vertex satisfy the above selection criterion.We also demand Σ track P T (track) > 60 GeV, which along with the cut n trk ≥ 4 ensures a significant number of hits in the muon-spectrometer.
We evaluate the geometric cut-efficiencies following Eq.5.2 and Eq.5.3 using the event samples that satisfy selection cuts C2-C3.The kinematic and geometric cuts have been designed based on the ATLAS searche [42].In the right panel of Fig. 11, we show the number of events for the 2MSvx events.We also show the number of events demanding decay of at least one N in the muon spectrometer in the right panel of Fig. 12, where we consider only the geometric cut-efficiency.The notable difference between the 2IDvx events and 2MSvx events is that the number of events in the later case reduces significantly.This can be understood by referring Fig. 6, where we show the probability of two N decaying in different region of the HL-LHC detector.As can be seen that the probability of two N decaying in the muon spectrometer is much smaller P ∼ 1%, compared to the probability of two N decaying in the inner-detector, where probability is P > 99%.This order of magnitude difference in the geometric cut-efficiency is primarily responsible5 for the reduction in number of 2MSvx signal events.The prospect of detection of 1MSvx events is higher in HL-LHC compared to 2MSvx events, as can be seen by comparing right panel of Fig. 11 and Fig. 12.A maximum of N events = 3 events can be obtained with 3 ab −1 of data.We also show the events where one of two N s decays in the inner-detector and the other N decays in the muon spectrometer, which we refer as MSID event category.The geometric probability for this is somewhat similar to the respective probability of 2MSvx events, as can be seen from Fig. 6.We show the event contours in Fig. 16 assuming 100% kinematic cut-efficiency.

Projection for FCC-hh
We consider the same decay channel and signal description as we consider for HL-LHC and present results for M Z = 8 TeV, g = 0.1.The necessary details about the geometry of the inner detector, muon spectrometer that we consider for the analysis has been given in • C3: Additionally for each of the j 0,1 , number of associated tracks with p T ≥ 1GeV and |η| < 2.5 should satisfy n trk 4. We show the distribution of track-p T and number of tracks in Fig. 13 for M N = 20 GeV.For higher M N the number of tracks increases further.
• C4: Finally we select the events if number of jets n j ≥ 2 and the above-mentioned cuts are satisfied for both the leading and sub-leading fat-jets.Additionally, similar to the analysis of HL-LHC, we also present the results demanding displaced decay of at least one RHN 6 , for which we consider only the geometric cut efficiency.
In the left panel of Figure .14, we show the contours for 3 and 800 2IDvx events.We show the event contours corresponding to 1IDvx events in the left panel of Fig. 15.For both of these two scenarios, a huge number of events N events > 1000 can be observed at FCC-hh with L = 30 ab −1 .

Decay vertex in the muon spectrometer (MSvx):
If RHN decays to a µqq in the muon-spectrometer, we implement the following sets of selection cuts: • C1: We demand both the RHN decay occur within the muon spectrometer between the length M S L 1 = 6000 mm, and M S L 2 = 9000 mm.
• C2: For each of the tracks originating from RHN decay vertex we impose p T (track) > 1 GeV , |η(track)| < 2.7 and n trk ≥ 4. Additionally, Σ track P T (track) > 60 GeV.These cuts are similar to the cuts that we use for HL-LHC projection.active-sterile mixing |V µN | is required to obtain N event ∼ 3.00 events for the 2MSvx events, compared to 2IDvx events.This occurs due to a larger decay length of N for its decay into a muon spectrometer.Although the sensitivity reach for |V µN | for 2MSvx events have a significant overlap with the sensitivity reach for 2IDvx, however the former has added benefit as 2MSvx events in the muon spectrometer will serve as background free distinctive signal of the model.

Conclusion and outlook
The RHN N in the gauged B − L model can be pair-produced at a pp machine via the heavy Z mediated channel, and give distinctive signatures with displaced vertices.The pair-production of N is independent of active-sterile mixing, instead depends on the B − L gauge coupling g , mass of Z , and mass M N of N .Hence, even with an active-sterile mixing which is suppressed due to light neutrino mass constraint, fairly large cross-section σ ∼ fb and even higher can be achieved.We consider a relatively light N with M N < M W , that satisfies light neutrino mass constraint m ν < 0.1 eV.For our choice of M N , the constraint from eV light neutrino mass results in a suppressed active-sterile mixing |V N | ∼ 10 −6 .For the considered range of M N and small |V lN |, the RHN undergoes three body decays with the decay vertex considerably displaced from its production vertex.We find that, N undergoes a large displacement ∼ mm upto meter if M N varies in between 10-70 GeV.Additionally, due to M N M Z , the produced N is significantly boosted, resulting in collimated decay products.
We analyse the displaced decay signatures of such a light N that can be probed at the high-luminosity run of the LHC (HL-LHC) with c.m.energy √ s = 14 TeV and the future pp machine FCC-hh that can operate with c.m.energy √ s = 100 TeV.Specifically, we consider two scenarios, which are N decays a) within the inner-detector (ID) of the HL-LHC and FCC-hh detectors, and b) within the first few layers of the muon spectrometer (MS).For the former, N should have proper decay length ∼ mm to hundreds of mm, while for later, the decay length should be ∼ m.We emphasize that the signal description between a) and b) differ widely.For a detail analysis of the signature, we specifically consider N → µqq decay mode.For a), i.e., if the decay of N occurs in the ID, the collimated decay products of N result in a displaced fat-jet.For this case, we thus analyse model signature with two displaced fat-jets, and further extend the analysis to take into account displaced decay of at-least one N .We find that O(10) displaced fat-jet events can be observed at HL-LHC with 3 ab −1 of luminosity, and for FCC-hh, a significantly large number of events O(1000) can be observed with L = 30 ab −1 .For b), i.e., if N decays in the MS, fat-jet description can not be used, since energy deposit in the calorimeter, informations about the track in the ID etc are missing, which are being used for the formation of a jet.In this case, we instead perform an analysis that relies on the properties of tracks in the MS.We apply selection cuts on the number of tracks originated from RHN decay, p T and |η|, and summation of p T of the tracks associated with N .We find that for HL-LHC, even after using the full integrated luminosity, the sensitivity reach is significantly low.For FCC-hh, this improves by order of magnitude, as a large number of events O(100) can be observed with the full integrated luminosity.We also extend the analysis for the case, when one of the N decays in the ID and the other N decays in the MS, for which results are similar to b).
Our proposed signature can potentially be explored for other decay modes of N , including N → eqq , as well as N → νqq final states.The final state with electron depends on active sterile mixing |V eN |, and hence choosing a large V eN compared to V µN will make the contribution from N → eqq large.The N → νqq decay instead depends on all possible |V lN |.The final state in addition to displaced fat-jet or charged tracks, will also carry missing transverse energy.By applying a veto on MET will hence reduce the contamination from νqq decay mode.A detail analysis of the model signature including both eqq and νqq decays will be presented elsewhere.
where the scalar and fermionic fields S and ν Ri have B − L charges B − L = +2 and −1, respectively.The SM states have the conventional B − L charges.Here, the field S represents a complex scalar field, which acquires vacuum expectation value (vev) v BL = 0 and breaks the B − L gauge symmetry.The states ν Ri contribute to the light neutrino mass generation via the seesaw mechanism.The complete Lagrangian of the model has the form the field strength tensor of the B −L gauge group.The kinetic term for the RHN and the SM fermion fields are

Figure 3 :
Figure 3: Cross-section σ(pp → Z → N N ) as a function of M Z at the HL-LHC and FCChh, for M N = 20 GeV.Black solid (dashed) line represents LHC limits for g = 0.1 (3×10 −3 ) as shown in Fig. 1.

Figure 4 :
Figure 4: Cross-section σ(pp → Z → N N ) as a function of M N and M Z .Left: For g = 0.003 and c.m. energy √ s = 14 TeV.Right: For g = 0.1 and c.m. energy √ s = 100 , and N c = 3 is the color factor.The partial decay width for N → l − α ν β l + β with generation index α = β have the following expression,

τ N 25 . 5 ,Figure 5 :
Figure 5: Distribution of the decay length of N at the HL-LHC and at FCC-hh for the considered masses of Z and N .

Figure 6 :
Figure 6: Decay probability of one RHN (left) and two RHN (right) in different parts of the HL-LHC detector.For this figure, we consider M N = 20 GeV, M Z = 1 TeV.ID, MS, CAL and MET represent the decay of N in the inner detector, muon spectrometer, calorimeter and outside of the detector, respectively.In the right panel, IDMS (IDCAL) represents the joint probability of one N decaying in the ID and the other decaying in the MS (CAL).

Figure 7 :
Figure 7: As Fig. 6 but shows the decay probability for the FCC-hh detector with M N = 20 GeV and M Z = 8 TeV.

Figure 10 :
Figure 10: Left panel: p T of tracks associated to leading and sub-leading jets.Right panel: number of tracks associated to j 0,1 for HL-LHC.

Figure 11 :
Figure11: Contours of 2IDvx (left) and 2MSvx (right) events that can be achieved with L = 3 ab −1 .A maximum of N events = 10 events can be observed in 2IDvx event category.The brown dashed line represents the FCC-ee projection[59,60], and the red shaded region represents the excluded region from[61,62].

Figure 17 :
Figure 17: Contours for 3 events for various banchmark at FCC-hh

Table 1 :
Table. 1, we present the fiducial volume Coverage of geometric acceptance of various sub-detectors of the HL-LHC and FCC-hh.