Multilepton dark matter signals

The signatures of dark matter at the LHC commonly involve, in simplified scenarios, the production of a single particle plus large missing energy, from the undetected dark matter. However, in $Z'$-portal scenarios anomaly cancellation requires the presence of extra dark leptons in the dark sector. We investigate the signatures of the minimal scenarios of this kind, which involve cascade decays of the extra $Z'$ boson into the dark leptons, identifying a four-lepton signal as the most promising one. We estimate the sensitivity to this signal at the LHC, the high-luminosity LHC upgrade, a possible high-energy upgrade, as well as a future circular collider. For $Z'$ couplings compatible with current dijet constraints the multilepton signals can reach the $5\sigma$ level already at Run 2 of the LHC. At future colliders, couplings two orders of magnitude smaller than the electroweak coupling can be probed with $5\sigma$ sensitivity.


Introduction
One of the most attractive and popular frameworks for dark matter (DM) is the socalled Z ′ -portal [1][2][3][4][5][6][7][8][9][10][11][12][13][14], in which the DM particle, typically a fermion χ, singlet under the standard model (SM) gauge group, interacts with SM matter through the common interaction with a massive Z ′ boson associated to an extra gauge group, U(1) Y ′ . Usually, the most stringent bounds on this scenario arise from di-lepton production at the LHC [15,16] and DM direct-detection (DD) experiments [17]. This has led to consider leptophobic models, in which the only coupling of Z ′ in the SM sector is to quarks. Likewise, spinindependent DD cross-section is dramatically suppressed if the Z ′ coupling to the DM particle and/or to the quarks is axial [7,9,[18][19][20][21]. Although most of the analyses of these models have been done in the context of simplified dark matter models (SDMM), in which the DM particle, χ, and the mediator, Z ′ , are the only extra fields (see e.g. [22]), it has been recently stressed [3,10,20,23,24] that the ultraviolet (UV) completion of the model requires the presence of additional fields in the dark sector. Such UV completion is enormously simpler and more natural if the axial coupling of the Z ′ boson is to the DM particle, not to the quarks [24]. Actually, this is the only possibility if the Higgs sector contains less than three Higgs doublets. Then, leptophobia imposes that the U(1) Y ′ charge of the quarks must be universal, which means that this symmetry is identical to baryon number in the SM sector. Concerning the dark sector, besides the DM particle, i.e. the SM singlet χ, the minimal set of additional particles required to cancel all the anomalies consists of a SU(2) doublet, ψ, and a SU(2) singlet, η, both with non-vanishing U(1) Y and U(1) Y ′ charges. Moreover, there must be at least one extra scalar, S, whose vacuum expectation value (VEV) breaks the U(1) Y ′ group.
There are many possible assignments of the extra hypercharges in the dark sector consistent with anomaly cancellation, but only a few leading to axial Z ′ coupling of χ [8,20,23]. Among them, there is essentially only one in which a unique scalar gives mass not only to the Z ′ boson, but also to all particles in the dark sector, and avoids the presence of electrically charged stable particles [23].
The goal of this paper is to study the phenomenology of this model and explore its most distinctive signals at the LHC and future colliders. A preliminary analysis was performed in ref. [23], assuming that the extra dark particles, ψ, η, were very massive, so that they decouple, leading to an effective SDMM (with a fixed correlation between the Z ′ couplings to DM and SM). 1 However, since both the DM particle, χ, and the dark 'leptons', ψ, η, obtain their masses from the same S-VEV, it is natural to assume that these are of the same order. Actually, this is good news, as the obliged presence of the dark leptons offers a fortunate opportunity to test the scenario at the LHC through new and specific signals. As we will see, although the associated phenomenology has some similarities with that of supersymetric models, it also presents drastic differences, which motivate novel analyses of experimental beyond-the-SM signals.
As a matter of fact, the presence of the extra leptons not only affects the LHC phenomenology but may also modify the production of DM in the early universe. This happens in particular if the masses of any of these extra particles is close enough to the DM one to produce non-negligible co-annihilation effects. This enhances the region of the parameter space consistent with the DM relic density and, as we will see, improves the chances to detect the scenario at the LHC.
In section 2 we write down the model and the interactions of the dark leptons. The 1 For related work see [25][26][27]. constraints on the model parameters from dark matter relic density and direct detection are examined in section 3. With these constraints in mind, we address in section 4 the general features of the four-lepton signal we are interested in. Representative benchmark points are chosen in section 5, for which a detailed simulation is performed in section 6. The discussion of our results and possible implications for experimental searches are given in section 7.

The model 2.1 Matter content, Lagrangian and Spectrum
The simplest extension of the SM that accommodates a leptophobic Z ′ with axial coupling to DM has the following characteristics. The extra U(1) Y ′ gauge group is equivalent to baryon number in the SM sector (required by leptophobia). Regarding the dark sector, this consists of three (Dirac) fermions, χ, ψ, η, with the following SU(2) L ×U(1) Y ×U(1) Y ′ representations: where Y ′ q is the extra-hypercharge of the quarks, assumed positive, which if desired can be taken with the same normalisation as baryon number, i.e. Y ′ q = 1/3. All the previous fields are colour singlets, while in the SM sector only the quarks have non-vanishing Y ′ hypercharge. The specific charge-assignment (1) was first explicitly considered in [10]. The state χ is the one to naturally play the role of DM. Flipping the signs of the above ordinary hypercharges, i.e. Y ψ → −Y ψ , Y η → −Y η (independently) also leads to a consistent model, so there are in fact four minimal models with very similar characteristics; and we will focus in the one defined by the previous assignments. In addition, the dark sector must contain a scalar, S, whose VEV breaks the U(1) Y ′ group. 2 Requiring that the same VEV provides masses to the dark particles fixes the charges of S, 2 There might exist extra scalar states, but for the study of dark lepton signals from Z ′ boson decays performed in this paper it is enough to work in the simplest case with just one complex scalar singlet, S. For a discussion of Z ′ cascade decays into scalars in a model with two singlets see ref. [28].
Let us discuss now the most relevant pieces of the Lagrangian. The Yukawa-like terms involving the dark fermions read with φ the SM Higgs doublet, andφ = iσ 2 φ * in standard notation. We have assumed that the couplings y i , λ i are real. Let us mention that there are two additional terms, consistent with the gauge symmetry, that could be added to the previous Lagrangian, namely −λ LχL χ L S − λ RχR χ R S * plus their Hermitian conjugate. These terms induce a splitting of the two lightest degrees of freedom of the DM particle, thus spoiling its axial coupling to the Z ′ . Fortunately, they can be safely avoided by noticing that their absence is protected by a global 'dark lepton number' under which all dark fermions, χ, ψ, η, transform with the same charge. Let us also note that the mixing terms in the first line of (3) are crucial to enable the decay of the electrically-charged dark fermions, which otherwise would lead to cosmological disasters. 3 The relevant terms of the scalar Lagrangian involving the S−field read The mixing term is constrained by Higgs measurements [29] and does not play any relevant role in this analysis. The other two parameters can be traded by the S mass and VEV.
Let us now examine the spectrum of the model after symmetry breaking. When the scalars acquire a VEV, the mass terms for the dark leptons are We label the neutral mass eigenstates as N 1,2 and the charged ones as E 1,2 , with masses 3 Without this requirement there is another consistent assignment of ordinary hypercharges in eq. (1), [23]. Hence, the viability of this alternative model requires extra Higgs states in order to present analogous mixing terms.
The relation with weak eigenstates is The lightest neutral eigenstate N 1 is the dark matter candidate. Defining r = v/v s , the mixing angles for the neutral sector are given by tan 2θ N L = 2r , tan 2θ N R = 2r .
The mixing angles in the charged sector θ E L,R have analogous expressions with the replacements y 2 → y 1 , y 4 → y 3 , λ χ → λ η . The four masses m N 1,2 , m E 1,2 and four mixing angles θ N,E L,R are not independent parameters, and satisfy the relation that stems from the equality of the (2, 2) entries of the neutral and charged lepton mass matrices of eq. (6).
Typically the mixing angles are small (or close to π/2) if the Yukawa couplings (y 2 , y 4 for θ N L,R ; y 3 , y 3 for θ E L,R ) are small. As we will see, from DM direct-detection bounds, section 3, this is indeed the expected situation for the neutral angles, θ N L,R . Consequently, we expect the DM particle, N 1 , to be close to χ.
The expressions for the mass eigenvalues are cumbersome, but they get greatly simplified in the limit where left and right angles are equal, which occurs for Actually, this assumption has very mild implications on the collider phenomenology, the most important effect being the modification of angular distributions in decay chains.
With this simplification, one can obtain compact exact expressions for the masses, with and

Interactions in the mass basis
The interactions of the dark leptons with the various gauge bosons in the weak basis, {χ, ψ, η}, are either vectorial or axial, see eq. (1). In the mass eigenstate basis, {N 1,2 , E 1,2 }, they remain with this character provided the left-and right-handed mixing angles are equal. 4 In general, the interactions of dark leptons with the Z ′ boson can be written as with i, j = 1, 2. The mixing parameters for the left-handed neutral leptons are given by For the right-handed sector they have the same expressions but replacing θ N L by the corresponding angle θ N R . The mixing parameters for charged fields can be obtained simply by replacing the neutral mixing angles θ N L,R by θ E L,R . The interactions with the W boson read The left-handed mixing parameters are The expressions for right-handed mixings V R ij are the same as for V L ij above but replacing θ N,R L by θ N,E R . The interactions with the Z boson read where the left-handed mixing parameters are and the right-handed counterparts have similar expressions but replacing θ N,E L by θ N,E R . Photon interactions are flavour-diagonal, The interactions with the Higgs boson arise from the terms in the first line of (3). In the mass eigenstate basis, For convenience, the Yukawa couplings Y N,E ij can be parameterised in terms of masses and mixing angles. For the neutral sector they are For the charged leptons the Yukawa couplings Y E ij have similar expressions but replacing the masses and mixing angles by the corresponding ones in the charged sector.

Constraints from Dark matter
The thermal relic abundance of DM is determined by the efficiency of the processes that lead to its annihilation in the early universe. In the first place, there are the processes mediated by the Z ′ boson, in particular χχ → Z ′ → qq (recall here that the DM particle, N 1 , is close to a pure χ state). Besides, for heavy enough DM (which is not our case) there are processes χχ → Z ′ Z ′ with a χ in t−channel. In ref. [23] it was shown that, in order to reproduce the observed relic abundance abundance, the g Z ′ coupling involved in the previous processes must be fairly sizeable. This in turn leads to strong experimental bounds coming from di-jet production at LHC. Actually, there is a broad range of Z ′ masses, 500 GeV < ∼ m Z ′ < ∼ 3500 GeV, which is excluded on these grounds. If the scalar associated to the S field is sufficiently light, there are additional annihilation processes in play, which slightly reduces the required value of g Z ′ , leading to a (modest) enhancement of the allowed region. This situation makes challenging to probe the scenario at the LHC, since the resonant production of the new particles occurs essentially beyond the present energy limit.
On the other hand, the presence of the extra fields, ψ, η (or, more precisely, N 2 , E 1 , E 2 ) offers new possibilities to annihilate DM in the early universe, something not considered in ref. [23]. The most obvious one is the co-annihilation of the DM particle with one of these states. In this sense, the most convenient state to play this role is ψ, not only for the possibility of direct co-annihilations, but also because the direct interactions between χ and ψ in the Lagrangian (3) keep naturally the DM in thermal equilibrium with these extra degrees of freedom. 5 This additional source of annihilation relaxes the required value of g Z ′ in order to get the correct relic density. Actually, for m ψ close enough to m χ there is no even need of the Z ′ -mediated contribution to the annihilation. This means that the value of g Z ′ becomes in practice a free parameter, provided the gap between the two masses is the suitable one to produce the necessary amount of co-annihilation.
Co-annihilation processes are very sensitive to the mass gap between the DM particle and the co-annihilating one. For example, in our case, for m Z ′ = 2.5 TeV, m χ = 300 GeV, m ψ = 313.9 GeV, the observed relic density is entirely obtained thanks to co-annihilation processes, thus g Z ′ must be rather small to avoid an excess of annihilation. Decreasing m ψ further makes the co-annihilation too efficient, so that the relic density falls below the observed value. On the other hand, increasing m ψ , the efficiency of the co-annihilation drops quickly and the relic density becomes too large. This can be fixed by an appropriate increase of g Z ′ , and thus of the efficiency of the annihilation processes mediated by Z ′ . However, increasing m ψ in just 0.1 GeV requires to raise g Z ′ above perturbative levels. For LHC phenomenology this means that it is enough to set the value of m ψ at this narrow range and leave g Z ′ as a free-parameter. Notice also that the required value of m ψ is essentially independent of m Z ′ , since the annihilation of the ψ states mainly involve 5 We are referring here to co-annihilation in a generic sense, which includes not only co-annihilation stricto sensu, but also the transfer from the χ population to the ψ one (thanks to the thermal equilibrium), which is subsequently annihilated through much more efficient (weak-interaction) processes.

weak interactions.
A scenario of co-annihilation as the one depicted above requires a mass-ordering m χ < m ψ < m η , which implies in turn that |θ N L,R | ∈ [0, π/4], θ E L,R ∈ [π/4, 3π/4], where, for convenience, we have taken the definition ranges of the angles as −π/2 ≤ θ N L,R ≤ π/2, 0 ≤ θ E L,R ≤ π. In the limit y i → 0 the angles become θ N L,R = 0, θ E L,R = π/2. Note that the reason for the latter is simply that m ψ < m η in the charged mass matrix (6), while by definition Concerning the constraints from direct detection, the axial (vectorial) coupling of the Z ′ mediator to the DM particle (the quarks) leads to spin-dependent, velocity-suppressed DM-nucleon cross section, which is safe from present DD experimental bounds. However, the mixing of χ and ψ inside the DM particle, N 1 , leads to a non-vanishing N 1L N 1R H coupling, which is dangerous since it induces spin-independent cross section. The size of this coupling can be read from eqs. (21), (22). The corresponding bounds on y 2 , y 4 from DD exclusion limits are very strong. In particular, for y 2 = y 4 ≡ y (the case in which θ N L = θ N R ), the bound for the previous example is y 2 ≤ 4 × 10 −6 . Consequently, as mentioned in previous sections, from DD constraints one expects very small angles, |θ N L,R | 0.05, see eq. (8). In contrast, the size of the θ E L , θ E R angles is no restricted by DM phenomenology.

Features of the four-lepton signal
The fermionic decay modes of the Z ′ have partial widths If the mixing angles in the left and right-handed sector are equal, then Z F L ij = Z F R ij ≡ Z F ij and the latter equation simplifies to In the limit in which the Z ′ boson is much heavier than its decay products, and the Z ′ branching ratio to dark leptons is 9/11 ≃ 80%. For simplicity, we assume that the scalar singlet is heavier than M Z ′ /2, so that the Z ′ boson does not decay into scalar pairs.
The most promising signal for the kind of scenario analysed here is the production of four leptons in the final state through the process and the subsequent leptonic decays, N 2 → N 1 ℓ + ℓ − . This final state provides the best balance between signal branching ratio and SM background. We have also considered three-lepton signals, e.g. from Z ′ → N 2 N 2 when one of the charged leptons does not pass the minimum p T requirement. Unfortunately, these signals are swamped by the W Z background. We have also investigated five-lepton signals from Z ′ → E 2 E 2 → N 2 W N 2 W , with one W boson decaying hadronically and the other one leptonically. Despite the fivelepton signal is very clean, its branching ratio is too small to be competitive with the four-lepton one.
The features and visibility of this four-lepton signal essentially depend on four parameters, the Z ′ mass and coupling and the two neutral lepton masses, m N 1,2 , in a non-trivial and entangled way (the mixing angles may also affect the signal by modifying the branching ratios and angular distributions). In order to better understand the dependence, we study semi-analitically their influence in this section, previous to the simulation of selected benchmarks in section 5.
The decay N 2 → N 1 ℓ + ℓ − produces a lepton pair of invariant mass If m N 2 ∼ m N 1 , as required for the co-annihilation, the distinctive signature is a small invariant mass lepton pair. Therefore, the decay Z ′ → N 2 N 2 produces two same-flavour opposite-sign lepton pairs of small invariant mass. Moreover, most of the energy is taken by the N 1 N 1 pair. Let E * ℓ be the energy of either lepton in the N 2 rest frame, which has a maximum If which is a small fraction, and most of the energy is kept by N 1 . Because the N 1 are produced nearly at rest in the N 2 rest frame, in the laboratory frame the N 1 N 1 pair is approximately produced back-to-back, as the N 2 N 2 pair is. Therefore, their contribution to the missing energy cancels to a large extent.
Although the leptons are produced from the decay of a multi-TeV resonance, their transverse momentum is relatively small. As aforementioned, most of the energy is taken by the N 1 N 1 pair. The transverse momentum of the leptons p ℓ T has an upper bound

Benchmarks
This section is devoted to formulating benchmark points in the parameter space of the model that are consistent with all the phenomenological constraints (including those from DM) and are representative of the new phenomenology that emerges from this scenario.
Let us consider first the mixing angles θ N L,R , θ E L,R . Indeed, they are naturally small (θ N L,R ) or close to π/2 (θ E L,R ) due to to suppression factor r = v/v S in the expressions given around eq. (8). Still, they might show a substantial departure from those values. Note in particular that for θ N L,R the denominator in eq. (8) could be quite small since λ χ ≃ λ ψ in order to allow an efficient co-annihilation, typically λ ψ − λ ψ = O(10 −2 ). However, as discussed in section 3, to avoid problems with direct detection the Yukawa couplings y 2 , y 4 must be substantially smaller, O(10 −3 ), thus rendering the neutral angles, θ N L,R , very small. On the other hand, the precise values of y 2 , y 4 are irrelevant for most of the phenomenology, provided this bound is satisfied. We will take them so that θ N L = θ N R = 0.02 (more details below).
Concerning θ E L,R , although they are naturally close to π/2, they certainly could be quite different without conflicting any experimental data. This is illustrated in figure 3, where we have fixed all the parameters as in Benchmark 1 (31) below, except the y 1 , y 3 couplings, and hence θ E L,R . Scanning over y 1,3 with |y 1,3 | ≤ max(λ χ , λ ψ , λ η ) (in this case 0.51) gives the allowed blue region in the θ E L − θ E R plane. The area θ E L,R ∼ π/2 is the widest one and the departure from it is bounded, but still the possibility of sizeable mixings exists. However, such situation is inconvenient to test the model at the LHC. The reason is that sizeable cos θ E L,R would lead to m E 1 < m N 2 , as is illustrated by the expressions (11)- (13). If this mass gap is not tiny, the N 2 state would naturally decay as N 2 → E 1 W * (instead of N 1 Z * ), thus ruining the four-lepton signal. Consequently, in our benchmarks we will choose small y 1 , y 3 , so that θ E L = θ E R = π/2 − 0.02. Concerning the other parameters, we will fix g Z ′ at a weak-interaction size, g Z ′ Y ′ q = 0.2, and take two values for the mass of the extra gauge boson, namely m Z ′ = 2, 3 TeV. To be in the safe side we take a DM mass of 300 GeV, while the co-annihilating particle is 14 GeV heavier (obliged for a correct relic abundance). We have verified with CheckMATE [30][31][32] that the two benchmark points adopted for this study are not excluded by multi-lepton searches at LHC based on 36 fb −1 of data collected at √ s = 13 TeV. Finally, the mass of the heavier dark lepton, m E 2 has a sub-leading effect on the signals and we will fix it at 400 GeV. In summary our two benchmarks are Benchmark 1 Benchmark 2 M Z ′ = 3 TeV , and the same remaining parameters.
In both cases, as mentioned, we assume that the scalar S is heavy enough (i.e. m S > M Z ′ /2) to be ignored. If it were light it could be involved in additional decay-chains with dark leptons, a kind of signals that is out of the scope of this paper. Of course, the previous values are obtained with appropriate choices of the parameters in the initial Lagrangian (3). More precisely, for Benchmark 1: v s = 1111 GeV, λ χ = 0.38, λ ψ = 0.40, λ η = 0.51, while for Benchmark 2: v s = 1666 GeV, λ χ = 0.25, λ ψ = 0.27, λ η = 0.34, Strictly speaking, with the above parameters the masses m N 2 , m E 1 are not exactly degenerate and equal to m ψ , e.g. for Benchmark 1, using expressions (11)-(13), we get mass shifts ∆ N = 5.6 MeV, ∆ E = 34 MeV, which are negligible for LHC phenomenology.
In these scenarios we have the following decays of the heavy leptons: where f is any fermion except the top quark. These decays are mediated by an off-shell Z boson, and the final state with f = b receives a small contribution from Higgs boson exchange. The decays of interest, N 2 → N 1 ℓ + ℓ − have a branching ratio of 3.9% for ℓ = e, µ and 3.6% for ℓ = τ .
• E 1 → N 1 ff ′ , with ff ′ = dū, sc, ℓ − ν. The Z ′ → E 1 E 1 decay produces signals with zero, one or two soft leptons plus soft jets and small missing energy. Clearly, there is little hope for such signals.
• The heavier charged lepton can in principle decay The partial widths are proportional to (sin 2θ E ) 2 , (cos θ N × cos θ E ) 2 , (sin θ N × cos θ E ) 2 and (sin 2θ E × cos 2θ E ) 2 , respectively. Hence, for θ N ≃ 0 and θ E ≃ π/2, the E 2 → N 1 W decay is suppressed with respect to the others. The other three are sizeable if they are kinematically allowed, although E 2 → N 2 W is typically the dominant one. For our benchmarks it turns out that this is in fact the only kinematically allowed mode, so it has a nearly 100% branching ratio.
The decays to leptons of different flavour, i.e. Z ′ → N 1 N 2 , Z ′ → E 1 E 2 , are very suppressed in the scenarios with small mixings considered here. Note also that there is a contribution to four-lepton signals from Z ′ → E 2 E 2 , which we take into account in our simulations. We generate our signals (in the above benchmark scenarios) and the backgrounds using MadGraph5 [33]. For the signal processes the relevant Lagrangian is implemented in Feynrules [34] and interfaced to MadGraph5 using the universal Feynrules output [35]. Tau leptons are included in all processes. Hadronisation and parton showering are performed with Pythia 8 [36] and detector simulation using Delphes 3.4 [37] using the configuration for the ATLAS detector for LHC Runs 2 and 3. For HL-LHC and HE-LHC we use a card corresponding to the expected performance of the upgraded AT-LAS/CMS detectors [38], and for the FCC the corresponding card. In all cases, we relax the isolation requirement on charged leptons because in the signal the leptons from each N 2 decay are relatively close. This is an approximation, at the level of fast simulation, of the loose lepton isolation criteria that are applied in experimental searches with full control over the details of the lepton definition and reconstruction.
As pre-selection criteria, we require that events have exactly four leptons (electrons or muons), in two opposite-sign same-flavour pairs. We set a lower cut p ℓ T ≥ 10 GeV for all leptons and for LHC Runs 2 and 3 we select events that fulfill at least one of the following criteria from the Run 2 ATLAS trigger menu [39]: • one electron with p T ≥ 27 GeV; • one muon with p T ≥ 27 GeV; • two electrons with p T ≥ 18 GeV; • two muons with p T ≥ 15 GeV; • one muon with p T ≥ 23 GeV and another muon with p T ≥ 9 GeV; • one muon with p T ≥ 25 GeV and another muon with p T ≥ 8 GeV; • one electron with p T ≥ 17 GeV and two electrons with p T ≥ 9 GeV; • three muons with p T ≥ 6 GeV. The effect of the trigger on the signal efficiencies is minimal. For LHC upgrades the multilepton triggers are planned to even lower their thresholds; moreover, a low p T fourlepton trigger consumes very little bandwith (because the SM four-lepton background is quite small) and could easily be implemented. We therefore do not apply any trigger requirement for LHC upgrades and for the FCC.
The main irreducible backgrounds to our signals are four lepton production pp → 4ℓ, mediated by off-shell Z bosons and photons, Higgs production with decay H → ZZ * and five lepton production pp → 5ℓ + ν, also involving off-shell Z bosons and photons. Note that a much larger source of four leptons is for example tt production in the dilepton decay mode, with the two additional leptons from b quark decays. This and other backgrounds can be quite reduced by lepton isolation criteria and are an order of magnitude below the former irreducible backgrounds [40]. Since the tools to deal with this type of backgrounds are not available at the level of fast simulation and they are quite smaller than the irreducible ones, we do not include them in our calculations.
We point out that this signal shares some features with the exotic Higgs decays H → XX , X → ℓ + ℓ − , with X a new light boson or a pseudo-scalar, which is searched for at the LHC [40,41]. We show in figure 4 (left) the four-lepton invariant mass for the signals and the SM background, for a CM energy of 13 TeV and the two benchmark scenarios defined in Eqs. (31), (32). The four-lepton invariant mass is in the range not far from the Higgs mass and of course it does not display a peak. Notice the background peak at M Z , when two of the leptons are emitted in the radiative decay of an on-shell Z boson, and the smaller peak at M H , caused by H → ZZ * . On the right panel we show the missing energy distributions. The latter has some discrimination power between the signals and the backgound but for simplicity we do not use it as the improvement on the signal significance is small.
An excellent discrimination between the signals and the background is achieved by using the minimum sum of dilepton invariant masses Σm ℓℓ , defined as follows. Among the possible pairings of opposite-sign same-flavour pairs (ℓ + 1 ℓ − 1 ), (ℓ + 2 ℓ − 2 ) -there is only one pairing in e + e − µ + µ − events, but there are two if all the leptons have the same flavour -we select the one that minimises the sum of the two invariant masses m ℓ + This minimum is Σm ℓℓ . For the dominant process giving four leptons, Z ′ → N 2 N 2 , it turns out that Σm ℓℓ ≤ 2(m N 2 − m N 1 ), since there is at least one pairing, the one corresponding to leptons with the same mother particle, which fulfills such inequality, see eq. (27). Therefore we expect an accumulation of the four-lepton signal in that range of small Σm ℓℓ . This is fortunate since that is precisely the region where the background is less important. Actually, the only relevant background is four-lepton production, and the rest are two orders of magnitude below. (As mentioned, other backgrounds with leptons  Table 1: Cross sections (in fb) of the different signals and backgrounds at LHC and its upgrades at the pre-selection and final selection. The signal labels S1, S2 refer to benchmark scenarios 1 and 2, defined in Eqs (31,32). The background labels 4ℓ and 5ℓ correspond to four-and five-lepton production as described in the text.
from top / bottom quark decays are expected to be unimportant.) All this is illustrated in figure 5, which shows the distributions of signals and backgrounds for Run 2 (top, left), HL-LHC (top, right), HE-LHC (bottom, left) and FCC (bottom, right).
Two distinct regions for the signals can be distinguished. A region of small Σm ℓℓ arises from Z ′ → N 2 N 2 , with a small contribution from Z ′ → E 2 E 2 → N 2 W N 2 W , with hadronic W decay. As expected, this accumulation of signal occurs at Σm ℓℓ ≤ 2(m N 2 −m N 1 ), which equals 28 GeV in these examples. Besides this region, there are signal tails caused by Z ′ → E 2 E 2 → N 2 W N 2 W when one or both W bosons decay into electrons or muons. In this decay chain the e/µ resulting from W decay typically have larger p T than the ones from N 2 → N 1 ℓℓ and their pairing with other leptons does not result in small Σm ℓℓ . Concerning the background, the SM production of four leptons peaks at 2M Z , as expected, and it is three orders of magnitude smaller at the signal region of small Σm ℓℓ . This makes Run 2 Run 3 B 0.78 1.60 S (S1) 9.3 5.7σ 23.1 8.3σ S (S2) 1.9 1.3σ 5.1 2.7σ Table 2: Expected number of signal (S) and background (B) events, and statistical significance (n σ ) of the signal, for LHC Runs 2 and 3. The Z ′ coupling is set as g Z ′ Y ′ q = 0.2.
the Σm ℓℓ variable a very convenient one to bring to light a 'compressed' spectrum, as the one expected in co-annihilation regimes, provided the co-annihilating particle may decay with two leptons in the final state.
We require as selection criterium Σm ℓℓ ≤ 22 GeV in all cases. The breakdown of signal and background cross sections for the different processes considered is given in table 1. 6 The extra contribution to the four-lepton signal from Z ′ → E 2 E 2 amounts to 40%-50%; however, at the region of small Σm ℓℓ it is smaller, around 30%.
With the number of signal (S) and background (B) events obtained we compute the expected signal significances for Runs 2 and 3, using Poisson statistics. These numbers are collected in table 2. We do not include any systematic uncertainty, as the statistical one is clearly dominant for a background of less than two events. (For future colliders the background is larger than a handful of events but still one can use the sideband for a precise normalisation of the background; the Monte Carlo predictions for four-lepton production are reliable since it is an electroweak process.) For LHC upgrades it is not sensible to report the relative sensitivities in terms of signal significances n σ for a fixed coupling -in a scenario that would have been discovered with 5σ well before anyway. Instead, we give in table 3 the couplings for which the signals could be seen with 5σ significance for the two benchmarks. (Poisson statistics are still used for HL-LHC with a background of 12 events; for HE-LHC and FCC we use the Gaussian approximation.) Given the fact that this process has tiny background, the potential of future colliders is really impressive: couplings at the few percent level could be probed.
7 Summary and discussion WIMP models of thermal dark matter require an appropriate annihilation of the latter in the early universe, and thus interactions with the SM particles. The most obvious of such interactions are those mediated by a Higgs or a Z boson (Higgs and Z portals). However, this possibility is under strong experimental pressure, essentially from direct detection constraints. Then, one of the best motivated and popular scenarios of dark matter is when those interactions occur through a Z ′ boson (Z ′ portal). In order to avoid strong constraints from direct detection experiments and dilepton production at the LHC, it is highly convenient that the Z ′ couplings are both leptophobic and axial (to either quarks or dark matter). This framework has been much explored in the literature, but usually in the context of 'simplified dark matter models', where only the dark matter particle plus the mediator, Z ′ , are considered. This leads to very characteristic signals already searched for at the LHC, such as mono-Higgs [42,43], mono-top [44,45], mono-Z/W [46][47][48] and mono-jet [48,49] production. The common feature of these signals is the production of a SM particle together with large missing energy resulting from the undetected dark matter particle. Unfortunately, no positive signal has shown up in any of these experimental searches, up to date.
However, these simplified models are in fact not minimal, since they present various theoretical inconsistencies, in particular the lack of anomaly cancellation. The latter requires to extend the dark sector with at least two extra fermions, a SU(2) doublet and a SU(2) singlet, both with non-vanishing hypercharge and extra hypercharge (the one associated to the extra U(1) gauge group) [8,20,23]. The presence of these extra states affects both the phenomenology of dark matter both at the early universe, due to possibility of co-annihilations, and at the LHC, as novel dark matter signals may appear. The goal of this paper has been to explore this new phenomenology, with the focus on its possible detection at the LHC and future colliders.
We have studied a particularly clean signal consisting of four charged leptons, with (perhaps surprisingly) small missing energy, which arises from the cascade decay Z ′ → N 2 N 2 → N 1 ℓ + ℓ − N 1 ℓ + ℓ − . Its most salient feature is the presence of two opposite-sign same-flavour lepton pairs of low invariant mass. Because the main source of four leptons in the SM -barring other sources that produce them close to jets such as b quark decays -is on-shell ZZ production, the backgrounds for such signal are tiny.
Searches for this type of dark matter scenarios can be performed in four-lepton events by using the discriminant variable Σm ℓℓ , that is, the minimum sum of invariant masses of opposite-sign same-flavour pairs. The signals can be spotted as an excess at the low-Σm ℓℓ region. We have verified that such events can be triggered already at the LHC Run 2, and the expected backgrounds have quite different kinematical features, so that even if the signal has a small cross section, as it corresponds to the production of a TeV scale Z ′ , it could be seen for reasonable values of the model parameters.
Current searches for exotic Higgs decays H → XX → 4µ could also be extended to have sensitivity to the signals introduced here. For example, ref. [40] has a dedicated analysis for 1 GeV ≤ M X ≤ 15 GeV, considering the four-muon final state. However, the analysis focuses on a narrow four-lepton invariant mass window m 4ℓ ∈ [120, 130] GeV around the Higgs boson mass -adequate for a search of Higgs exotic decays -that unfortunately removes most of our signal, as it can readily be observed in figure 5. In addition to this cut, the event reconstruction is done by assuming the kinematics of the decay H → XX, and is sub-optimal for the signals addressed here.
Besides the production of an excess in the low-Σm ℓℓ region, which is the common feature, other characteristics of the signal depend on the model parameters. The full exploration of the relevant five-dimensional parameter space is cumbersome, but one can easily figure out, from the results in section 4, the behaviour for parameters other than those considered in the detailed simulation in section 6.
(i) Z ′ mass: for heavier Z ′ the cross section is obviously smaller; the leptons are produced with higher transverse momentum and therefore the efficiency for event selection is larger; the missing energy is also larger the heavier the Z ′ is.
(ii) N 1 , N 2 masses: the lepton p T are proportional to the relative mass splitting (m N 2 − m N 1 )/m N 2 , so a smaller mass difference or heavier N 2 makes the signals harder to see. It should be noticed here that the heavier N 1 , the smaller the relative splitting must be in order to enhance the co-annihilation effects. In addition, heavier N 1 , N 2 also implies smaller Z ′ → N 2 N 2 branching ratio.
(iii) E 2 mass: the signal receives a small contribution if E 2 is close to N 1 , N 2 , but otherwise the influence is moderate.
(iv) Coupling: the signal cross section scales with g 2 Z ′ but this coupling cannot be arbitrarily large, since one has the limit g Z ′ Y ′ q 0.3 from dijet resonance searches for the Z ′ masses considered.
As a final comment, let us remark that in this paper we have considered benchmark scenarios where the extra scalar(s) S necessary to provide the Z ′ and dark lepton masses do not play any role in Z ′ decays, by taking them heavy. Conversely, in ref. [28] the dark leptons where assumed heavy, to concentrate on the phenomenology of the Z ′ boson cascade decays into extra scalars. The perhaps more natural (and quite more complex) situation is to have scalars and dark leptons with mass of the same order, so that the new scalars can decay into dark leptons and vice versa. The analysis of this type of scenarios and their possible collider signals deserves further investigation.