Heavy Neutrinos at the FCC-hh in the $U(1)_{B-L}$ Model

We investigate the potential of the 100 TeV future circular collider (FCC-hh) to probe heavy neutrinos. We concentrate in particular on heavy neutrino production via a $U(1)_{B-L}$ $Z'$ gauge boson and contrast the resulting limits with that mediated by Standard Model weak currents. We consider heavy neutrino decays to semi-leptonic as well as fully leptonic final states, particularly with muon flavour, and we show the importance of considering searches both in prompt and displaced decays of the heavy neutrinos. For prompt final states, semi-leptonic modes are more promising due to smaller background and larger yields, and TeV-scale heavy neutrinos with active-sterile mixing compatible with light neutrino mass generation in a seesaw scenario can be probed for a 5 TeV $Z'$ and gauge coupling as low as $g_{B-L} = 10^{-2}$. Displaced vertex searches can extend this range to heavy neutrino masses as low as 10 GeV.


Introduction
The observation of small neutrino masses, which have no explanation within the Standard Model (SM) of particle physics, points towards new physics. Among possible scenarios which can explain neutrino masses, the U (1) B−L gauge model is one of the simplest anomaly free constructions [1,2]. It incorporates three right-handed (RH) neutrinos N i which acquire Majorana masses upon the spontaneous breaking of the U (1) B−L gauge symmetry. In turn, this induces light active neutrino masses via the seesaw mechanism. As opposed to the minimal seesaw scenario, the RH neutrinos are not completely sterile under the model's gauge symmetry. Instead, they are charged under U (1) B−L and they couple to the associated heavy gauge boson Z and B − L charged Higgs χ present in the model. This opens up a rich phenomenology as it not only provides the SM charged and neutral current portals for heavy neutrino production but also production via the Z and χ.
The B − L production mechanisms are not suppressed by the active-sterile mixing which is generically expected to be very small, |V lN | ∼ m ν /m N 10 −6 × (100 GeV/m N ), to accommodate the observed light neutrino masses m ν 0.1 eV. For such a small activesterile mixing, the heavy neutrinos, which can only decay through such suppressed channels, are long-lived with a proper decay length L 0 N ∼ 2.5 cm × (10 −6 /|V lN |) 2 (100 GeV/m N ) 5 for m N 100 GeV. This naturally leads to displaced vertex signatures at colliders for heavy neutrino masses m N ≈ 100 GeV.
Searches for RH neutrinos via SM mediated processes are being carried out at the LHC, including both prompt and displaced final states [3][4][5][6][7][8][9][10][11]. These searches currently put an upper limit of |V µN | 10 −3 on the active-sterile mixing with the muon and RH neutrino masses m N 100 GeV. The limits weaken rapidly for increasing heavy neutrino masses. This is expected because the SM W becomes increasingly off-shell and thus the production cross section is doubly suppressed, from the active-sterile mixing as well as the mass of the heavy RH neutrino. Further projections for RH neutrinos from the SM production, at the LHC , the proposed SHiP detector [39,40], LHeC [41] and the FCC [22,24,42,43] have been carried out. These studies rely on the minimal and ensured RH neutrino production mechanism, taking advantage of higher luminosity, larger detectors and increased center-of-mass energies. They remain fundamentally limited by the right-handed neutrino production cross-section suppressed by the active-sterile mixing. Together with the stringent cuts on the final state objects at FCC-hh, the studies show that probing the seesaw neutrino mass generation mechanism at future colliders via the minimal production channels will be very challenging.
This motivates the consideration of non-minimal models and probing neutrino mass generation mechanisms via exotic production modes. In this context, the B − L model offers an interesting avenue as it contains three additional heavy neutrino generation portals unsuppressed by the active-sterile mixing namely the heavy Z , heavy B − L Higgs and the SM Higgs. There are two main advantages of these additional modes, especially for Z mediated processes. Firstly, the right-handed neutrinos can have large p T depending on the Z mass which can be produced on-shell. Secondly, given that this on-shell Z can be heavier than the W boson, right-handed neutrino masses m N 100 GeV can be probed without penalising production cross sections as opposed to the SM case where this mass range necessitates off-shell W mediator and leads to an additional suppression in cross section. These considerations enable searching for right-handed neutrinos in regions of |V µN | and m N parameter space otherwise inaccessible. Analyses of B − L processes have been performed in the literature. This includes reinterpretations of existing Z production searches [44], analyses of the sensitivity at the lifetime frontier [45][46][47][48], explorations at the LHC [49][50][51][52][53], as well as prompt searches at the FCC-hh [54].
In this work, we look ahead and consider the potential of the FCC-hh detector at 100 TeV center-of-mass energy to probe right-handed neutrino production via the B − L Z . We contrast this mode with processes mediated by SM weak currents. In doing so, we not only exploit the gain in the center-of-mass energy and the higher luminosity (30 ab −1 ), but also the larger detector volume capable of capturing longer lived right-handed neutrinos. In doing such an exercise our aim is to demonstrate the complementarity between SM weak current and B − L mediated processes as well as to understand the prominent kinematic differences between FCC-hh and LHC, the two SM and B − L channels. We consequently illustrate regions of parameter space which can be probed by simple analyses while highlighting the necessity to develop more comprehensive analyses techniques.
The paper is organised as follows: We begin by reviewing the B − L gauge model and its immediate phenomenological consequences in Section 2. This is followed by a discussion of our analysis setup in Section 3. Using this as a basis, in Section 4 we first evaluate the sensitivity of FCC-hh for Z resonance production. We then discuss heavy neutrino production via SM W and B − L Z processes in Section 5. We show our sensitivity estimates in Section 6 and finally conclude in Section 7.
2 The B − L gauge model 2.1 Model setup and particle spectrum In addition to the particle content of the SM, the U (1) B−L model contains an Abelian gauge field B µ , a SM singlet scalar field χ and three RH neutrinos ν R,i . The gauge group is and −1, respectively. All SM particles carry their conventional B − L quantum number. The scalar potential includes all terms allowed by the symmetry, Here, χ is the vacuum expectation value (VEV) of the B−L Higgs χ. The mixing between the Higgs fields, induced by the term λ 3 in Eq. (2.1), can still be sizeable but is not relevant for our discussion. We denote the gauge coupling strength associated with the U (1) B−L symmetry by g B−L . In this paper, we neglect a kinetic mixing between the U (1) B−L and U (1) Y gauge bosons, i.e. we consider the minimal B − L gauge model. Such a mixing will be generated radiatively even if assumed zero at, e.g., the B − L breaking scale, . This is negligible for our analysis. We note that a finite kinetic mixing can be beneficial in searching for heavy neutrinos. Its main effect will be that the SM Z can decay to two heavy neutrinos, suppressed by , but not by the active-sterile neutrino mixing. The mass of the Z gauge boson is then simply given by m Z = g B−L χ . The model contains the additional Yukawa terms where L i are the SM lepton doublets,H = iσ 2 H * and a summation over the generation indices i, j = 1, 2, 3 is implied. The Yukawa matrices y ν and y M are a priori arbitrary. The RH neutrino masses are generated by breaking of the B − L symmetry, with the mass matrix given by M R = √ 2y M χ . The light neutrinos mix with the RH neutrinos via the Dirac mass matrix m GeV is the SM Higgs VEV. The combined 6 × 6 mass matrix in the (ν L , ν c R ) basis is then in block form In the seesaw limit we consider, ||M R || ||m D ||, the light and heavy neutrino mass matrices are m ν ∼ −m D · M −1 R · m T D and m N ∼ M R , respectively. The flavour (ν L,i , ν c R,i ) and mass (ν i , N i ) eigenstates of the light and heavy neutrinos are connected in block form as The mixing matrix V LL and the light neutrino masses are constrained by oscillation experiments to yield their observed values, i.e. the SM charged current lepton mixing V LL ≈ U PMNS (apart from small non-unitarity corrections and taking the basis in which the charged lepton mass matrix is diagonal). The approximately unitary matrix V RR describes the mixing among the RH neutrinos and the active-sterile mixing is V LR ≈ m D · M −1 R . We effectively consider the case of a single RH neutrino generation mixing with one SM neutrino at a time; specifically we focus on the mixing with the muon neutrino and hence take V LL , V RR ∼ 1 and V LR ∼ V µN with active-sterile mixing strength V µN suppressing the charged-current interaction of the muon with the RH neutrino. We take the RH neutrino mass m N and the mixing strength V µN as free model parameters. In order to generate a light neutrino mass m ν 0.1 eV via the seesaw mechanism with ||m D || ||M R ||, the mixing strength takes the generic value This is the generic expectation for the mixing strength of a single heavy Majorana neutrino generating a light neutrino mass of 0.1 eV. Both smaller and larger mixing is possible in more realistic and extended scenarios that incorporate all three generations of light neutrinos and at least two heavy neutrinos. Smaller mixing is possible by decoupling the heavy neutrino with other heavy neutrinos responsible for light neutrino mass generation.
Larger mixing, up to current experimental constraints, is possible if there is cancellation among contributions of different heavy neutrinos. This can be achieved with either three generations of heavy neutrinos (see, e.g., [55]) or extended scenarios with quasi-Dirac heavy neutrinos, usually referred to as inverse seesaw.

Production and decay of heavy neutrinos
Within the B − L model, heavy neutrinos can be produced at the FCC-hh via different mechanisms. The first and foremost are the SM mediators, i.e. the W and Z gauge bosons and the Higgs h 1 . In addition, there are the B − L mediators, namely the Z gauge boson and the heavy Higgs boson h 2 . Out of these mechanisms, the SM Higgs production pp → h 1 → N N will be suppressed by the Higgs mixing and the mass of the heavy neutrino with respect to the B − L breaking scale, m N / χ (m N < m h 1 /2) [29]. The alternative mode pp → h 1 → N ν is instead suppressed by the active-sterile neutrino mixing. The production via the heavy Higgs h 2 , pp → h 2 → N N , is possible as well but likewise suppressed by m N / χ , though higher neutrino masses can be probed. The mode pp → h 2 → N ν is suppressed by both the Higgs and the active-sterile neutrino mixing.
The best studied SM modes are via the W and Z gauge bosons, pp → W → N and pp → Z → N ν. They are generally suppressed by the active-sterile neutrino mixing and for m N > m W , m Z proceed through off-shell gauge bosons with further strong suppression for high N masses. The W mediated channel gives rise to a prompt lepton which helps in accepting events even if the heavy neutrino decays via a displaced vertex. We will thus focus on this mode.
Finally, the main production mode in our analysis is via the B − L Z gauge boson, pp → Z → N N . The production cross section σ(pp → Z ) at the LHC and FCC-hh is shown in Fig. 3 (left) in Section 4 to determine the reasonable range of the gauge coupling g B−L at the FCC-hh era. The partial decay width of Z → N N per generation of heavy neutrino is Considering the Z to be much heavier than the SM fermions into which it decays, the total decay width is with three generations of heavy neutrinos. In all mechanisms we must then consider the decays of the heavy neutrino. They proceed via the same mediators as discussed above. The h 1 , h 2 mediated decays are suppressed by the Yukawa coupling ∼ m N / χ and the heavy neutrino dominantly decays via SM W and Z, N → W ( * ) , Z ( * ) ν. We do not consider decays via Z such as N 2 → Z N 1 as we take into account only a single generation of heavy neutrino. In any case the gauge boson mediated decay widths are suppressed by the active-sterile mixing.
For heavy neutrino masses m N m W , m Z , decays will take place via off-shell W and Z. The branching ratios of such light heavy neutrinos are discussed in [29,56]. If the heavy neutrino mass is above this threshold, two body decays via on-shell W , Z occur. The heavy neutrino branching ratios in such cases are approximately independent of the mass and BR(N → W ) ≈ 70% while BR(N → Zν) ≈ 30%. Considering decays through a single SM lepton generation, the branching ratios are also independent of the active-sterile mixing.
With the SM W and Z decaying further, the N decays can thus result in N → + − ν corresponding to the W ( * ) and Z ( * ) mediated leptonic decays with visible final states. Finally, we also get N → jj which corresponds to the hadronic W , Z decays.
With the assumption of flavour-diagonal active-sterile neutrino mixing, we concentrate on one generation of heavy neutrinos, namely coupling to muons and muon neutrinos. Out of the several final states it can produce we will in particular consider N → µµν, N → µjj for neutrino masses lighter than 100 GeV and N → µW → µµν, N → µW → µjj for heavier neutrinos. We concentrate on muon flavour as muon performance in several parts of the detector is generally expected to be superior and we consider the µjj final state as it has a larger branching ratio.
While the branching ratios are largely independent of the heavy neutrino mass and the active-sterile neutrino mixing, the total decay width and thus the decay length crucially depend on it. For m N m Z , the proper decay length can be approximated as [56] For m N 100 GeV, decays are proportionally faster due to their on-shell nature. Approximately, the proper decay length in this regime is [56] L 0 (2.9) 3 Analysis setup

FCC-hh detector geometry
The future physics collider program has two major goals; first and foremost to measure the SM Higgs boson and SM electroweak sector properties as precisely as possible and a second to search for and potentially discover new physics which may be out of reach at the LHC. A two stage plan is proposed towards fulfilling these goals. The first stage is an electronpositron collider with a center-of-mass collision energy of 240 GeV called the FCC-ee 1 . The second will be a hadron-hadron collider with a center-of-mass collision energy of 100 TeV and an integrated luminosity of at least 10 times larger of the HL-LHC, i.e., 30 ab −1 [57]. In this paper, we focus on the reach of FCC-hh for a BSM particle, the heavy neutrino, therefore, we briefly discuss below the FCC-hh geometry.
Due to the high center-of-mass energy, the FCC-hh can in general produce boosted objects, therefore the FCC-hh detector is being designed to accept high pseudo-rapidities |η| 4. The detector subcomponents and geometry are shown in Fig. 1, taken from Ref. [57]. As detailed below we implement this geometry in our analysis in order to account for geometrical acceptance. Compared to the LHC detector, the FCC-detector is larger both in longitudinal and transverse directions in order to capture higher energy final states. The larger volume is of a particular importance in detecting long lived particles as larger lifetimes can be probed.
here L xy and L z are the transverse and perpendicular displacements respectively. We consider a decay prompt if L N ≤ 1 mm [57]. This is possible either when the mixing |V µN | is large or m N > 100 GeV, irrespective of the SM or B − L mediated production channels.

Signal final states
A number of different final states are possible depending on the production and decay of the heavy neutrino. We show the corresponding Feynman diagrams in Fig. 2, which includes the full production and decay chain for the RH neutrinos we consider in this work. The corresponding final states are summarised in Table 1. In general they contain a combination of muons, jets and missing energy. We separate the processes depending on the production mechanism i.e. SM or B − L, and also on the prompt or displaced category. Among the prompt production, five different final states are possible. Out of these final states, within this analysis we will not consider 3µ + 2j + E T final state. We foresee a signal -background discrimination on the basis of missing energy where available. When missing energy is not available, we will demonstrate the use of same sign (SS) or opposite sign (OS) muons in the final state. Later on we will explicitly demonstrate the effectiveness of these strategies to derive the final sensitivity. For the displaced final states, we will consider an inclusive analysis considering either one prompt lepton and one displaced vertex originating out of two displaced muons (for SM mediated heavy neutrino production), or one displaced vertex forming out of two displaced muons (for B−L mediated production).
In order to trigger on this signal, we use either a single prompt or displaced lepton trigger. In accordance with the FCC-hh Conceptual Design Report (CDR) [57], we require the leading muon p T greater than 150 GeV mimicking a trigger strategy. For a displaced muon trigger, no concrete numbers are as yet available, however we increase the p T cut by a factor 1.5 in order to keep in line with the experience of dealing with displaced versus prompt objects at the LHC [58]. Thus, for a displaced muon final state, we require a leading displaced muon with p T (µ 1 ) > 200 GeV.

Simulation details
To analyse the kinematics and later on sensitivity, we simulate the signal events using the following steps. We use the Universal FeynRules Output (UFO) [59] of B − L model developed in Ref. [29] in combination with the Monte Carlo event generator MadGraph5aMC@NLO -v2.6.7 [60] at parton level. The FeynRules [61,62] model file and UFO is publicly available from the FeynRules Model Database at [63]. For every signal sample, we generate 10 4 signal events. We then pass the generated parton level events on to PYTHIA v8.235 [64] which handles the initial and final state parton shower, hadronization, heavy hadron decays, etc. The clustering of the events is performed by FastJet v3.2.1 [65] 2 . Additionally, we use the NNPDF23_lo_as_0130_qed PDF set. Using the same setup we also produce 10 5 background events in each tt (leptonic), ZW W (leptonic), and µνZ (leptonic) final states. In particular, we consider only muon flavoured final states for backgrounds, which will be compatible with our signal final states.
Finally in order to minimise the Monte-Carlo efforts, heavy neutrinos are always decayed promptly. We simulate displacement via inverse sampling of the decay distribution. Specifically, the efficiency of our displaced vertex analysis relies on both the geometrical acceptance of the detector, geo , as well as the reconstruction effects, recon , and the total efficiency is DV = geo × recon . The geometrical acceptance of the detector includes the probability of the heavy neutrino travelling a distance L from the interaction point before decaying, given by the exponential density distribution where L N is the decay length of the boosted heavy neutrino in the laboratory frame. The efficiency geo is calculated by inverse sampling of the cumulative decay length distribution function, which in turn depends on the heavy neutrino mass and boost, and the activesterile mixing. The reconstruction efficiency is assumed recon = 100 % unless otherwise stated. Table 1: Different final states arising from prompt decays of heavy neutrinos for SM and B − L mediated production mechanism.

FCC-hh reach for heavy Z production
Before we proceed, we estimate the FCC and LHC potential to probe Z resonance mass. A detailed investigation of low mass Z production mechanisms and LHC limits was carried out in [69] and [70]. In this study, we are instead interested in high mass Z and the associated experimental sensitivity. To this end we exploit the existing CMS resonance search in dilepton final state at 140 fb −1 luminosity [66]. Comparable results can be obtained by recasting ATLAS search in the same final state [71]. We furthermore compliment these limits with the projected FCC-hh limits as detailed in [68]. We recast both these limits for the B − L coupling. The recast is performed as follows. We calculate the simulated cross section σ ref (pp → Z → ll) using the B − L model UFO with fixed g ref B−L without implementing any generator cuts, then we require the cross section σ(U.L.) to be equal to the upper limits from the Ref. [66], therefore (g U.L. ). In Fig. 3, we show the resulting Z production cross section (left panel) and the limits on the g B−L coupling (right panel) at HL-LHC (green) and FCC-hh (red), respectively. Comparing the Z production cross section (left panel) between HL-LHC and FCC-hh, it is clear that FCC-hh will gain significant cross section for the production of heavier Z , thus extending the reach for heavier resonances. This gain in cross section is complemented by a large luminosity of 30 ab −1 , and they are in turn reflected in the more stringent g B−L limits (right panel). Starting from about m Z = 4 TeV the FCC-hh will improve the g B−L limit by at least an order of magnitude. The lack of significant gain at low Z masses is due to the large dilepton background at the FCC-hh [68]. This background becomes much smaller at high mass, and correspondingly high mass resonances can be probed. For Z lighter than ∼ 4 TeV, the Z decays to heavy neutrinos can lead to displaced final states and may be beneficial for exploring lower Z mass due to smaller backgrounds. We will not explore this region any further however such a strategy might be worth exploring further.
It is thus clear that FCC-hh will in general be able to produce much heavier Z as compared to HL-LHC and correspondingly improve the limits on g B−L . This has important implications for the heavy neutrinos produced via Z . First and foremost, the heavy neutrino will in general be produced with a large boost which will produce more energetic decay products in the final state. Second, such large boosts will result in longer decay lengths of the heavy neutrinos producing more displaced objects in the detector.
We require σ(pp → Z ) > 10 −2 fb in order to achieve ∼ 10 signal events Z → N N final state at 30 ab −1 . Considering the FCC-hh's reach as shown in Fig. 3, we assume a benchmark point of m Z = 5 TeV and g B−L = 10 −2 in the rest of the paper.

Heavy neutrino production at the FCC-hh
Having understood the reach of the FCC-hh in terms of the Z mass and gauge coupling, we are now equipped to compute the heavy neutrino production cross section and decay modes. To this end, we explore the heavy neutrino production via the SM W, Z mediators and via the B −L Z . We furthermore also demonstrate effect of p T cut on leading lepton, in accordance with our trigger strategy. We simulate these processes as discussed in section 3.3.

Heavy neutrino production via SM W boson
The production of heavy neutrinos from SM W, Z bosons will always remain primary mechanism as it only relies on the active-sterile mixing naturally expected to generate neutrino masses. In this section we will illustrate the heavy neutrino production cross sections from both W and Z mediators, however for our final analysis we will consider only the W mediated process and use the µµν or µjj final state from the decay of the heavy neutrinos to assess the sensitivity. Among the lepton final states, only muon flavour is considered as it is cleaner and in general leads to higher detector efficiency compared to electrons or taus.
Benefiting from the 100 TeV collision energy, the cross section of the Drell-Yan process pp → W ± → µ ± ν will increase from 1.56 × 10 7 fb at the 13 TeV LHC to 1.05 × 10 8 fb at the 100 TeV FCC-hh.

Heavy neutrino production via Z boson
In the case of heavy neutrino production via a Z mediator, the production cross section is given by σ(pp → Z ) × BR(Z → N N ) [47], where the branching ratio is independent of the active-sterile mixing.
Assuming a 5 TeV Z gauge boson, with g B−L ∼ 10 −2 , which saturates the FCC-hh sensitivity and m N ∼ 10 GeV, the pp → Z → N N cross section reaches ∼ 10 −2 fb. This cross section will be approximately constant up to m N = 2.5 TeV, beyond which it will vanish due to phase space considerations.
In order to understand the complementarity of B − L heavy neutrino production with the SM mediated production processes, we compare the two cross sections before and after imposing the cut p T (µ 1 ) > 150 GeV, consistent with the trigger requirement. In Fig. 4 (left), we show the heavy neutrino production cross sections via SM W , Z, as well as B − L Z with N → µ + µ − ν for |V µN | = 10 −2 , g B−L = 10 −2 and m Z = 5 TeV (left panel). For the Z mediated process, only one of the two heavy neutrinos decays. We also show contours of 1 ab production cross section as a function of heavy neutrino mass and mixing angle (right panel).
Several features of these plots are to be noticed: Firstly, the production cross sections via SM mediators should be rescaled by the ratio of the square of the mixing angles for |V µ N | = 10 −2 . Such rescaling does not apply for the Z channel. Therefore, although for |V µ N | = 10 −2 , the SM mediated production cross section is much larger than the Z counterpart, for smaller mixing angles the situation will be reversed. The cross-over between Z and SM W mediated channels takes place for |V µN | ∼ 10 −5 without the p T cut, and for |V µN | ∼ 10 −3 after the cut. Secondly, the effect of the p T cut is much stronger on the SM mediated channel, while for the Z mediated process there is virtually no effect of the p T cut. This is because the heavy Z mass already leads to energetic displaced leptons in the final state, the lighter W and Z on the contrary lead to softer prompt leptons. As can be seen from the plot, the cross section via SM mediators can drop by three order magnitude after such a lepton p T cut. Finally, the SM Z mediated cross section is approximately one order magnitude smaller than the corresponding W mediated process, which justifies our decision to neglect the Z mediated processes.

Kinematics of the heavy neutrino and its final states
Having explained the heavy neutrino production cross sections in the previous section, we also discuss the kinematics of the final states by considering the two production channels of the heavy neutrinos either via a SM W or B − L Z . For the final states, we only take N → µ + µ − ν as an example. We compare the p T distributions of the resulting heavy neutrino, muons, and W ( * ) to illustrate advantages of each of the production modes.
For the SM mediated production, we compare the kinematics between LHC and FCC-hh center of mass energies. In Fig. 5 (top row), we plot the p T of the mediator W ( * ) (left panel) and N (right panel) in pp → W ( * ) → N µ at the 13 TeV LHC (blue dashed) and FCC-hh (red solid). As the W ( * ) boson is produced in s-channel, it has a small transverse momentum. In the left panel, with larger energy, the average of the p T (W ( * ) ) increases by a few GeV at the FCC-hh, while most of the W ( * ) bosons still possess vanishing p T . Correspondingly, as shown in Fig. 5 (right) the heavy neutrino p T does not increase significantly at the FCC-hh as compared to the LHC. The gain in sensitivity at the FCC-hh is therefore primarily due to the larger production cross section.
In Fig. 5 (bottom row), we plot the p T of the (prompt) muon from W ( * ) and from N decays. We fix the mass of the heavy neutrino to 10 GeV for concreteness. As the W ( * ) or the N do not gain significant energy at the FCC-hh as compared to the LHC, the resulting final states also do not gain any more p T , as reflected in the muon p T distributions shown. As we will see later, for this reason, FCC-hh does not lead to a large gain in sensitivity for SM mediated heavy neutrino production.
In contrast to the SM production, the heavy neutrinos in B − L production can have significant p T , as they are produced from a heavy resonance, i.e., a 5 TeV Z (as shown in Fig. 6 left). Therefore, in N → µµν decay, the final µ can have p T ∼ O(100) GeV (see Fig. 6 right). It thus becomes easier to pass the stringent p T requirements e.g. for trigger purposes. However, due to the presence of the heavy Z , it is also possible that the decay products of the heavy neutrino are collimated.
To illustrate this, in Fig. 7 (left), we plot the ∆R between the final state muons (and jet) for the leptonic N decay (N → µµν) for three heavy neutrino masses of 1 GeV (solid line), 10 GeV (dotted line) and 100 GeV (dashed line). We analyse here decays of only one of the heavy neutrinos. In principle, larger muon multiplicities can be obtained if decays of the second heavy neutrino are also considered. Such muons coming from two different heavy neutrinos will however not lead to a vertex and hence can be discarded. We therefore assume that only one heavy neutrino decays to illustrate potentially non-isolated muons. From these distributions it is clear that muons can be very collimated when the mass difference between Z and heavy neutrino is large. To demonstrate a similar situation in the semi-leptonic decay (N → µjj), in Fig. 7 (right) we plot the minimum ∆R between the muon and two of the jets emerging from N → µjj. In order to remove any soft hadronic activity in the event we consider jets with p T > 100 GeV. It can be seen that for heavy neutrinos lighter than 10 GeV, the ∆R is smaller than 0.2. This poses potential isolation problems in leptonic final states and can also lead to fat jets in hadronic final states.

Sensitivity
We now estimate the sensitivity of the FCC-hh with 30 ab −1 integrated luminosity to heavy neutrino production both via SM W and B − L Z mediators. For each of these production modes we consider two different heavy neutrino decay channels, namely µµν and µjj, which correspond to either the leptonic or hadronic W decay. For the SM W mediated mode, we always use the prompt lepton as a trigger. For the B − L Z mode, we use the prompt and the displaced lepton triggers. We furthermore consider any RH neutrino decaying within the detector volume as outlined in Section 3.1 and L N > 1 mm as displaced. Before we calculate the sensitivity, an estimation of the background is necessary. In general, this analysis can be carried out in several categories depending on charge and flavour composition of the signal processes (see [42] for such an exercise for SM mediated heavy neutrino production). We however restrict ourselves to muons in the final state and either consider same sign or opposite sign muons when a sufficient number of signal events are available. Unsurprisingly, the charge composition is relevant only when the signal originates from semi-leptonic decays of a heavy neutrino (µjj final state). As indicated before, the main signal and background discriminating variables we use are missing energy, number of jets, lepton charge composition and invariant mass of the system. These are summarised in Table 2. In Table 3, we show the main background processes and approximate number of expected events for a luminosity of 30 ab −1 after respective cuts.
We begin by discussing background composition and reduction methodologies for SM mediated heavy neutrino production and prompt decays. For heavy neutrino decays to muonic final states (µµµν), the background consists of µ ± νZ with leptonic Z decays. This background has approximately 10 5 events, therefore we require ∼ 10 3 signal events. It should be noted that we attempt no optimization in terms of final state flavour composition. For the hadronic final state, (µµjj), tt backgrounds are dominant. As the signal contains a fully visible final state, the resulting missing energy is small. Therefore, missing energy is a relevant discriminator here. For the opposite sign muons category requiring p T (µ 1 ) > 150 GeV, E T < 20 GeV and putting a b-jet veto on the leptonic tt decays leads to ∼ 10 4 events. Requiring same sign leptons (muons) in the final state leads to a more promising situation where backgrounds emerge from charge misidentification. Assuming an optimistic rate of 0.1% based on LHC detector performance [72] leads to ∼ 10 events [42].
For the prompt final states of the B − L mediated processes, two heavy neutrinos are produced. The signal final states we concentrate on are leptonic decays of both heavy neutrinos (4µ + MET) and semi-leptonic decays of both heavy neutrinos (2µ+4j) 3 . For 4µ + MET final state, triple boson background processes (ZW W ) have ∼ 10 3 events. For the 2µ + 4j final state, the tt mode in opposite sign muon and same sign muon final states are controlled by requiring low missing energy E T < 20 GeV and high invariant mass M (tt) > 4 TeV. For the opposite sign final states, there are still ∼ 10 3 background events after the cuts, while they become negligible after additionally requiring charge misidentification for the same sign final states.
Finally, for the displaced final states, in the case of SM W mediated production, we always use the available prompt lepton with a p T > 150 GeV to emulate the trigger and require either two displaced muons, in case of N → µµν decays or one displaced muon and one jet, for N → µjj decays. In addition, we require that the heavy neutrino decays within the detector volume, with geometry as described in Section 3.1. We consider the background to be negligible as the signal is sufficiently displaced [73][74][75], and only N signal > 3.09 is required from the Poisson distribution at 95% confidence level (C.L.).
The resulting sensitivity at 95% C.L. is shown in Fig. 8, for the µµν channel. For Z mediated production, we fix m Z = 5 TeV, while taking the B − L coupling at two representative values, g B−L = 10 −2 and 0.1, reflecting either a pessimistic view that the g B−L is taken near the projected sensitivity of the FCC-hh or an optimistic view with g B−L large but satisfying the current LHC limits.

SM Prompt
Background µ ± νZ 11.9 -3.55 × 10 5 Hadronic OS (µ ± µ ∓ jj) tt (leptonic decay) 1.84 -5.52× 10 4 Hadronic SS (µ ± µ ± jj) tt (leptonic decay) 1.84  Both neutrinos are assumed to decay via µµν in case of prompt B − L production and thus leading to a 4µ + E T final state. The production channels and decay modes are accordingly labelled, e.g., 'SM µµν DV' denotes the SM production mode with µµν final state, in a displaced vertex. Current best limits as collated in [76], which includes a displaced search at ATLAS [73] (small bump around m N 10 GeV), and projected sensitivities from the proposed SHiP [40] and FCC-ee [77] are also shown for comparison.
We start with analysing the SM production channels in displaced (blue shaded region) and prompt (red shaded region) signatures. The striking feature for this production channel is the limited FCC-hh sensitivity. As shown in Section 5.3, this is because the p T of the muons in the final state does not benefit from the larger collision energy. Instead, the stringent p T requirements, (p T (µ) > 28 GeV at the LHC [78] compared to p T (µ) > 150 GeV at the FCC-hh) limit the reach despite the increased cross section. Therefore, although the production cross section of N from this process can reach O(100) fb for active-sterile mixing |V µN | ∼ 10 −2 , the FCC-hh fails to probe a significant parameter space as muons in the final states do not have sufficient p T to pass the cuts. Therefore, we obtain a sensitivity |V µN | 10 −2 for the prompt final states of heavy neutrinos from the SM production at the FCC-hh, comparable to the reach of the LHC [4,11,79], as indicated by the 'Current' region. Although the cross section drops sharply as the W becomes off-shell when m N > m W , p T cuts become more efficient and the resulting sensitivity changes smoothly. The requirement of two same sign leptons helps with background control and in general leads to a sensitivity up to one order of magnitude stronger as compared to the opposite-sign signature. We therefore only show the sensitivity for the same-sign signature. For the displaced final states, as we look for heavy neutrinos The light blue band corresponds to the regime with light neutrino masses 0.01 eV < m ν < 0.3 eV via the seesaw mechanism. The shaded region labelled 'Current' is excluded by existing searches for heavy neutrinos [76], whereas 'SHiP' and 'FCC-ee' indicate the projected future sensitivity at the proposed SHiP [40] and FCC-ee [77], respectively. The proper heavy neutrino decay length L 0 N of 1 m is indicated by the corresponding curve.
with longer lifetime situated at lower |V µN |, the gain due to negligible background is however cancelled out by the reduced cross section, therefore the sensitivity to |V µN | becomes stronger by an order of magnitude only leading to |V µN | 10 −3 . The sensitivity vanishes for m N 10 GeV, as the displaced final states requires |V µN | ∼ 10 −4 , making the cross section insufficient to get any sensitivity in such a parameter space. Comparing to the projected limits from SHiP and FCC-hh, the FCC-hh is not competitive at lower m N , but can have and advantage at greater m N due to its larger collision energy.
The B − L heavy neutrino production complements the SM channel. In the prompt µµν final state, for g B−L = 10 −1 , the sensitivity extends up to m N = 2 TeV and activesterile mixing strengths |V µN | down to regimes where the heavy neutrino becomes long lived. Because both the neutrino production and decay branching ratios are independent  Figure 9: As Fig. 8, but for the semi-leptonic final state µjj.
of |V µN | (N does not decay to other flavours), only this promptness requirement limits the |V µN | reach. For the smaller value g B−L = 10 −2 , no sensitivity is obtained. For the displaced final state, the sensitivity in V µN follows heavy neutrino decay lengths between 1 mm L 10 m, extending the |V µN | reach by about three orders of magnitude. The break at m N ≈ 100 GeV arises due to the change from off-shell to on-shell decays. It is worth noting that combining prompt and displaced final states, the seesaw mechanism responsible for generating light neutrino masses can be tested for 20 GeV m N m Z /2, but displaced searches are also sensitive for g B−L = 10 −2 assuming them being free of background.
The corresponding sensitivity arising from prompt and displaced µjj final states is shown in Fig. 9. In the prompt B − L channel, both heavy neutrinos are assumed to decay as N → µjj thus leading to a 2µ + 4j final state. We show the prompt final state sensitivity again only for the opposite-sign channel. The most important feature is the higher sensitivity as compared to µµν final states in both displaced and prompt signatures because of the larger branching ratio of the semi-leptonic decay. For the SM µjj prompt final state, it is now possible to reach |V µN | ≈ 10 −3 for m N ≈ 30 GeV. Similar to the SM mediated case, an increased sensitivity in both prompt and displaced final states can be found for B − L heavy neutrino production as well. Otherwise, the overall sensitivity is similar to that of the fully leptonic mode but the prompt B − L µjj case is also sensitive for g B−L = 10 −2 up to masses of m N 2 TeV.
We have so far assumed recon = 100 % to detect and reconstruct the displaced ob-  jects in the detector. This is clearly an idealized assumption. In a realistic scenario, the efficiencies will be highest near the primary vertex and will degrade towards outer parts of the detector, see e.g. discussion in [73,80]. To illustrate the impact of less than perfect efficiencies, we show in Fig. 10 the sensitivity of the B − L µjj displaced final state as in Fig. 9 for an efficiency dependent on the lab frame decay length L N of the heavy neutrino, for L 0 < L N < L 1 , and eff = 0 otherwise. Here, L 0 = 0.025 m and L 1 = 5 m are the boundaries of the inner tracker of the FCC-hh as shown in Fig. 1. Hence, we only consider the inner tracker in detecting displaced objects. The efficiency in Eq. (6.1) peaks at eff (L N ) ≈ 10% for L N ≈ 0.5 m, i.e., we use a reduction of the order of that used in LHC analyses [73,80]. Such an order of magnitude reduction obviously worsens the FCChh reach but does not do so severely. As expected, the sensitivity is especially reduced in comparison with the eff = 100% case for smaller active-sterile mixing |V µN | as this corresponds to longer decay lengths. As shown in Fig. 7, the muons and jets in the final states tend to be very collimated, and we also illustrate the average ∆R(µj) between the muon and closest jet in Fig. 10. For ∆R(µj) 0.4, i.e., m N 300 GeV, different analyses strategies are beneficial due to the collimated muons and jets. This, in principle, applies to both the prompt and displaced B−L µjj signatures shown in Fig. 10. In the latter case, displaced fat jets can be considered which preserves sensitivity to small masses m N 10 GeV [81]. For the prompt signature, the interesting region with m N 100 GeV and |V µN | ≈ 10 −7 − 10 −6 , motivated by the light neutrino masses, corresponds to fairly large ∆R(µj) 0.1.

Conclusion
The absence of an explanation of neutrino masses within the SM demands new physics. Experimental confirmation for such a mechanism will have a profound impact on understanding the fundamental laws of nature and will help build the next standard model. One such scenario is the seesaw mechanism in which heavy right handed neutrinos are predicted. These can be produced at the LHC via Drell-Yan processes mediated by a SM W or a Z boson. The reach of colliders such as LHC or FCC is however limited as the heavy neutrino production is suppressed by the active-sterile mixing which is expected to be small to generate the correct light neutrino masses. This observation necessitates exploring non-minimal heavy neutrino production as a means to probe neutrino mass generation.
We have here determined the sensitivity of the FCC-hh towards heavy neutrino production within the B − L model, which is equipped with an additional B − L gauge boson Z . Using this gauge boson as a portal of heavy neutrino production, we explored the process pp → Z → N N and we have contrasted the resulting sensitivity against the assured but suppressed SM production channel pp → W → lN . We have concentrated on muon flavour focussing on two different neutrino decay modes, N → µµν and N → µjj. The FCC-hh with 30 ab −1 luminosity and 100 TeV center-of-mass energy can probe B − L Z gauge bosons with masses of the order 50 TeV. We use m Z = 5 TeV as benchmark with comparatively low values of the associated gauge coupling, namely g B−L = 10 −2 and 0.1. With such a U (1) B−L portal, the FCC-hh will be able to probe regions relevant for neutrino mass generation in the µµν and the µjj final state. Both prompt and displaced signals are relevant to cover a range of heavy neutrino mass 20 GeV m N 2 TeV with active-sterile mixing strength |V µN | ≈ 10 −7 − 10 −5 motivated by light neutrino masses, m ν ∼ |V µN | 2 m N ∼ 0.1 eV. In the given scenario considered, active-sterile mixing strengths as low as V µN ≈ 10 −12 can be probed. Such searches for displaced heavy neutrinos have smaller background and thus may even be the first direct signal of an exotic Z resonance, potentially shedding light on the R K anomaly [82].
The neutrino production processes mediated by the SM W and Z, while assured and independent of the U (1) B−L extension, are suppressed by this active-sterile mixing. Only dedicated facilities with high fluxes and the potential to probe long lived particles such as SHiP and FCC-ee have the ability to probe the required small active-sterile mixing strength but only for limited ranges of neutrino masses, namely m N ≈ 1 − 2 GeV and m N ≈ 50 GeV, respectively. In the same channel, the FCC-hh will only be able to probe active-sterile mixing strengths |V µN | ≈ 10 −3 − 10 −4 , to high for a successful generic seesaw mechanism. This sensitivity is also comparable to the HL-LHC; the gain in terms of luminosity and cross section are compensated by the harder p T requirements.
While the mechanism of neutrino mass generation may be adapted to incorporate heavy neutrinos with large active-sterile mixing, such as in inverse seesaw scenarios, the vanilla seesaw remains an attractive and suggestive proposition. The corresponding Yukawa couplings between the left-and right-handed neutrinos may be small, y ν ≈ 10 −6 for m N around the electroweak scale, but this is of the order of the Yukawa coupling of the electron. The minimal B −L model has the appeal of incorporating the origin of light neutrino masses by breaking lepton number spontaneously. This is still possible near the electroweak scale and, as we have shown, it can lead to striking prompt and displaced signatures. Heavy neutrino production via a Z mediator has the potential to probe regions of parameter space relevant for neutrino mass generation in multiple final states. A refinement of our analysis with a full detector simulation should be able to demonstrate the reach more accurately, however, our first exploration demonstrates the potential of the FCC to target heavy neutrinos and the origin of light neutrinos.