Dark Matter Spin Characterisation in Mono-$Z$ Channels

The $B-L$ Supersymmetric Standard Model (BLSSM) is an ideal testing ground of the spin nature of Dark Matter (DM) as it offers amongst its candidates both a spin-1/2 (the lightest neutralino) and spin-0 (the lightest right-handed sneutrino) state. We show that the mono-$Z$ channel can be used at the Large Hadron Collider (LHC) to diagnose whether a DM signal is characterised within the BLSSM by a fermionic or (pseudo)scalar DM particle. Sensitivity to either hypothesis can be obtained after only 100 fb$^{-1}$ of luminosity following Runs 2 and 3 of the LHC.


I. INTRODUCTION
DM is one of the firm evidences of physics Beyond the Standard Model (BSM). Searches for DM at the LHC through Missing Transverse Energy (MET or / E T ) and probing a single particle, like mono-jet, -photon, -Z and -Higgs, are one of the most promising methods for establishing DM existence directly in an experiment. However, the nature of DM remains as one of the foremost open questions in particle physics, especially whether the DM is a fermionic or bosonic particle.
Fermionic DM is predicted by several BSMs, like the Minimal Supersymmetric Standard Model (MSSM), in which the lightest neutralino (a fermionic superpartner of the neutral scalar and gauge bosons of the SM) is a quite popular example of weak scale DM. Scalar DM has been analysed in models with extra inert singlet or doublet Higgs bosons. Here, we will perform a comparative study for the two types of DM, predicted by the same model, the BLSSM, in different regions of parameter space.
The BLSSM is a natural extension of the MSSM with an extra U (1) B−L . It accounts for non-vanishing neutrino masses through a low scale seesaw mechanism, which can be an inverse seesaw (see Ref. [1] for a review). In this scenario, it is quite possible to have the lightest neutralino or the lightest right-handed sneutrino as the Lightest Supersymmetric Particle (LSP), so that any of these can be a stable DM candidate [2]. A detailed analysis of BLSSM DM candidates has been performed in [2,3] (see also [4]). Therein, it was shown that, for a wide region of parameter space, the lightest right-handed sneutrino, with mass of order O(100) GeV, can be a viable DM candidate that satisfies the limits of relic abundance and also the scattering cross sections with nuclei. The chances of the lightest neutralino being the actual DM state are much less in comparison, however, in some regions of the parameter space, it is still possible to have it as the origin of DM, in particular, in the form of the lightest B − L neutralino. Further, in Ref. [3], it was shown that the Fermi Large Area Telescope (FermiLAT) can be sensitive to the DM spin (and eventually distinguish between the sneutrino and neutralino hypotheses) in the study of high-energy γ-ray spectra emitted from DM (co)annihilation into W ± boson pairs (in turn emitting photons).
Furthermore, we studied several single-particle signatures of the BLSSM DM at the LHC, i.e., mono-jet,photon, -Z and -Higgs signals, induced by new channels mediated by the heavy Z (in the few TeV range) pertaining to the (broken) U (1) B−L group [5,6]. The salient feature of this BLSSM specific channel is that the final state mono-probe carries a very large MET. Hence, it is a clean signal, almost free from SM background. It was argued that, with luminosities of order 100 fb −1 , mono-jet events associated with BLSSM DM can be accessible at the LHC while mono-photon, -Z and -Higgs signals can be used as diagnostic tools of the underlying scenario.
In this paper, we expand on all these results, by showing that DM spin can be accessed at the LHC in the mono-Z channel. We prove this result by showing that the angular distributions of the final state lepton emerging from a subsequent Z decay, for both neutralino and right-handed sneutrino DM, are significantly different from each other. This is in contrast to the result that these distributions are identical in mono-jet, -photon and -Higgs (owing to the fact that jets and γ's do not couple directly to DM while Higgs radiation is isotropic), thus being insensitive to the DM spin.
This paper is organised as follows. In Sect. II we briefly highlight the possibility of having both (pseudo)scalar and fermionic DM in the BLSSM with an inverse seesaw mechanism. Sect. III is dedicated to the mono-Z analysis in these two DM scenarios. In Sect. IV we discuss the impact of the DM spin on the angular distributions of the corresponding final leptons. Our conclusions and final remarks are given in Sect. V.

II. SCALAR VERSUS FERMIONIC DM
The BLSSM is based on the gauge group SU boson Z and three chiral singlet superfieldsν i with B−L charge = −1 are introduced for the consistency of the model. Finally, three chiral singlet superfieldsŜ 1 with B − L charge = +2 and three chiral singlet superfieldŝ S 2 with B − L charge = −2 are considered to implement the inverse seesaw mechanism [8]. The superpotential is given by The neutralinos,χ 0 i (i = 1, . . . , 7), are the physical (mass) superpositions of the three fermionic partners of the neutral gauge bosons, called gauginos, of the neutral MSSM Higgs bosons (H 0 1 andH 0 2 ), called Higgsinos, and of the B − L scalar bosons (η 1 andη 2 ). In this regard, the lightest neutralino, in the basis ψ 0 = {B,W 3 ,H 0 1 ,H 0 2 ,B ,η 1 ,η 2 }, decomposes as The lightest sneutrinoν 1 (either a CP-even state,ν R 1 , or a CP-odd one,ν I 1 ) can be expressed in terms ofν + L ,ν + R andS + 2 (e.g., in case of it being CP-even) as , 0, 0}, which confirms that the lightest sneutrino is mainly right-handed (i.e., a combination ofν + R andS + 2 ). It is worth mentioning that, due to the U (1) Y and U (1) B−L gauge kinetic mixing, the mass of the extra neutral gauge boson, Z , is given by whereg is the gauge kinetic mixing coupling. Also, the mixing angle between Z and Z , which is experimentally limited to < ∼ O(10 −3 ), is given by The relevant interactions of the lightest neutralino and lightest right-handed sneutrino with the Z and Z bosons are given by where ∆Vnm = V * in V1n − V * im V1m. Fig. 1 shows the total cross section for pp → Z → Z(→ l + l − ) + 2ν1 (l = e, µ), based on the diagrams (top panels) in Fig. 2 (summed and squared, thereby capturing the relative interference too), for different masses of the Z andν1 after satisfying all Higgs data constraints by using HiggsBounds [9] and HiggsSignals [10]. The scanned points have been generated over the following intervals of the BLSSM fundamental parameters: TeV and 3 TeV ≤ v 2 ≤ 5 TeV plus, to ensure that the lightestν1 is the LSP, we kept M1 = M2 = M3 = 6 TeV. A benchmark point will be chosen from the scanned ones to perform a detailed Monte Carlo analysis. As the latter will be based around Z production and decay, we also have made sure that, on the one hand, the scan points do not fall out of the LEP (indirect) constraints and, on the other hand, the ensuing Z will not have been discovered via LHC (direct) searches in Drell-Yan (DY) mode already. We meet these conditions by adjusting the parameters of the chosen point as follows: M Z = 2.9 TeV, Mν 1 90 GeV, g B−L = 0.5 and g = −0.25.
TeV mapped over the Z andν1 masses for the BLSSM with an inverse seesaw mechanism.

III. MONO-Z ANALYSIS
In the following, we will develop an analysis aimed at extracting information about the lightest right-handed sneutrino of the BLSSM as the DM candidate through a dedicated mono-Z search using a Machine Learning (ML) algorithm called Boosted Decision Tree (BDT) [11,12]. The key to this approach is to rely on a mono-jet evidence of DM in a kinematic regime compatible with Z production and decay 1 , so that, under a model dependent assumption (i.e., assuming the BLSSM), one can extract mono-Z signatures leading to the identification of the DM properties, chiefly, of its spin. In fact, an intriguing feature of the mono-Z analysis is the possible spin characterisation of DM. Spin determination methods rely heavily on the final state spins and the chiral structure of the couplings. The 2-body decays of neutralinos to a massive Z boson and a DM neutralino produce a Z boson in three helicity states, ±1 (transverse) and 0 (longitudinal). Reconstructing the three polarisation states through the angular distributions of the Z boson leptonic decays through χ 0 i →χ 0 1 Z(→ l + l − ) in the rest frame of the decaying Z boson leads to a clear characterisation of the spin state of the Z boson. The angular distribution of the transverse states are ∝ (1 ± cos 2 θ) while the angular distribution of the longitudinal state is ∝ sin 2 θ, where θ is the angle between the lepton momentum direction and the Z boson one in the latter rest frame. The decay width of the neutralinoχ 0 i to Transversely (T ) and Longitudinally (L) polarised Z bosons is given by [16]. It is worth mentioning that the decay width of the longitudinal component of a Z boson is suppressed with respect to its transverse ones [17].
The 2-body decays of heavier sneutrinos to a massive Z boson and sneutrino DM,νi →ν1Z(→ l + l − ), produce a Z boson in a zero-helicity (longitudinal) state only. This is because the helicity has to be conserved in the S-matrix and the fact thatνi andν1 are (pseudo)scalars forces the produced Z boson to have a unique state (cf Fig. 1 in [17]). Fig. 3 shows the angular distribution of the final state lepton l forχ 0 i →χ 0 1 Z transitions in red and that forνi →ν1Z limit [14,15]. This means that Z and h SM propagators are offshell, unlike the Z one. Further, the Z couplings to sneutrinos aremuch stronger than those of the Z and h SM . Finally, we will enforce a stiff MET cut to enhance the Z component of the signal. ones in blue. It is also worth noting that, in Refs. [18][19][20], a similar approach based on angular distributions of leptonic Z boson decays emerging fromχ 0 1 → ZG transitions, withG being light gravitino, was considered to distinguish between a Higgsino-and gaugino-like neutralino in a model with Gauge-Mediated Supersymmetry Breaking (GMSB).

IV. RESULTS
Given the Feynman diagrams underpinning mono-Z production in the BLSSM case for sneutrino DM (see Fig. 2, top panels), the Z boson decaying leptonically can be reconstructed as such by constraining the emerging electron and muon pairs to reproduce MZ within experimental di-lepton mass resolution (we will not include Z → jet decays in the   FIG. 4: Transverse momentum of the leading jet (left) and of the di-lepton final state (right), with S1 the signal process with Z ISR (Fig. 2 left) and S2 the signal process with Z FSR (Fig. 2 right). signal definition). The dominant irreducible background is ZZ → l + l −ν ν and the other large noise in this category is W + W − → l + νl −ν . As we reconstruct the Z boson (specifically, by selecting the lepton pair that gives the closest value to the measured mass of the Z boson), the reducible backgrounds must contain Z → l + l − . Given the hadronic environment of the LHC, additional jet activity is possible. Hence, the final list of backgrounds in this category is as follows: Z + jets, ZZ → l + l − + jets and ZW → l + l − + jets. In addi-tion, there are other reducible di-lepton backgrounds with jets that we have dealt with: tt → l + νbl −νb , as well as W ± + jets, which is reduced by an MET cut. The last quantitatively important background, purely leptonic, is ZW → l + l − lν, with one electron misidentified as jet. As preselection cuts we require pT (l) > 10 GeV, pT (j) > 20 GeV, |η(l/j)| < 2.5 (where j represents any jet and l any lepton) and / E T > 50 GeV. Tab. I shows the signal and background composition, wherein we emphasise the dominance of S2 over S1 owing to theνi multiplicity in the former, while the latter only sees the involvement ofν1. Moreover, we stress that, while the signal is mediated by a heavy gauge boson, Z , that leads to large MET, the whole background is not, thus we will eventually force the / E T > 100 GeV condition into the BDT. Process pp → tt 597  Fig. 2, top panels) and the dominant background processes considered in our analysis. The samples have been produced with the following cuts: pT (l) > 10 GeV, pT (j) > 20 GeV and / E T > 50 GeV.
Upon enforcing all kinematic conditions above, relevant distributions are given as an input to our BDT in order to perform a Multi-Variate Analysis (MVA) [21]. An important feature of the MVA is that it can rank the input variables according to their ability to separate between signal and background events. For illustrative purposes, we show the first two variables ranked by the BDT for the signals S1 and S2 as well as all backgrounds separately in Fig. 4. Herein, a peculiar feature is the fact that the signal is mediated by a heavy Z , so that this causes the ISR jet in S2 to recoil against a very massive object. Such kinematics pushes the transverse momentum distribution of the leading jet to be peaked around half of Z mass. This does not occur for S1, though, owing to the presence of also the ISR Z. For the di-lepton transverse momentum, both signals have a much stiffer spectrum than any of the backgrounds, again, owing to Z balancing the heavy Z (in S1) or else being ejected by the decay of the latter at large pT (in S2).
The discriminating power of the BDT relies on the fact that the signals and backgrounds may be characterised by different features that can be encoded into several distributions. For completeness, we sketch the first 9 most important variables, as ranked by the BDT, in Fig. 5 (wherein backgrounds are shown cumulatively). Further, Tab. II shows the BDT ranking of all input variables according to the their power in separating the signal and background events. Our ML approach is then based on a set of BDTs where each tree yields a binary output depending on whether an event is classified as signal-or background-like during the training session. The most important feature of the MVA algorithm is its possibility to combine the various discriminating kinematic distributions into one main discriminator, the BDT response, and thus dealing with only one variable to maximise the signal rate over the background one. The BDT response ranges between −1 and +1 corresponding to pure background and pure signal, respectively.
After the aforementioned kinematic cuts (preselection), the total number of events for the signal is 656 while for the background is 2.3 × 10 7 , both of which are passed to the MVA   Fig. 6 (left) with signal events in blue and background ones in red. Enhancing the BDT cut efficiency is done by maximising the function S/ √ S + B, where S is the total signal rate and B is the background one at the given luminosity. Hence, for the optimal value of the BDT cut set at 0.48, the remaining signal events (222) and background ones (285) yield a significance of 9.8σ. This corresponds to a signal extraction efficiency of 34% and a background rejection efficiency of 1.2 × 10 −5 . Fig. 6 (right) shows the signal efficiency in blue and the background rejection efficiency in red versus the BDT cut with the corresponding significance in green.
Finally, notice that the analysis has been performed at a Center-of-Mass (CM) energy of 14 TeV and integrated luminosity of 100 fb −1 . For the simulation of the signal and background event samples, we have used MadGraph5 (v2.4.3) [22]. Parton shower and hadronisation have been carried out by PYTHIA6 [23,24] while a fast detector simulation by Delphes [25] was used.

V. CONCLUSIONS
We have shown that a ML based approach, as opposed to a standard cut-flow one, is well suited to extract a mono-Z(→ l + l − ) signal of the BLSSM at the LHC, with 14 TeV and 100 fb −1 of energy and luminosity, respectively. The latter is emerging from a heavy Z boson decaying into sneutrinos, the lightest of which is the DM state of this scenario, eventually yielding a di-lepton plus MET signature with additional jet activity. Furthermore, the ability of the Z boson to couple directly to the DM state enables one to access the spin properties of the latter, specifically, by studying the angular behaviour of either lepton relative to the Z boson direction in its rest frame. We have illustrated this phenomenology using a single benchmark point in the BLSSM, compliant with current experimental limits. We defer to a future publication the illustration of such an approach applied to the entire BLSSM parameter space [26].