Pinning down the leptophobic Z ′ in leptonic final states with Deep Learning

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I. INTRODUCTION
Many extensions of the Standard Model (SM), like the grand unified theories (GUTs) [1][2][3], contain an electrically neutral colour-singlet heavy gauge boson, Z ′ .To get such a particle in the spectrum in a bottom-up manner, one can simply add an extra local U(1) symmetry spontaneously broken by a complex scalar field with a TeV-scale vacuum expectation value (vev).The phenomenology of a TeV-scale Z ′ is well-explored in the literature (see, e.g., [4,5]).One of the current programs of the LHC is to look for the signature of Z ′ .However, so far, the focus has been only on the cases where it decays exclusively to the SM particles [6][7][8][9][10][11][12][13][14].The failure to find the Z ′ so far motivates us to look for other possibilities with nonstandard decay modes of Z ′ , i.e., where it can decay to beyond-the-SM (BSM) particles as well.
The strongest exclusion limits on the Z ′ parameter space usually come from the dilepton resonance searches [8,9].For instance, the current mass exclusion limit on a sequential Z ′ is about 5 TeV [8].The stringent dilepton exclusion limits can be evaded if the Z ′ is leptophobic, i.e., it does not couple (or couples very feebly) to the SM leptons.In the leptophobic case, the nonstandard decay modes of the Z ′ can become important as the branching ratios (BRs) to the new modes can be high.One interesting possibility is when the Z ′ decays to a pair of right-handed neutrinos (RHNs, N R ) [15][16][17][18][19][20].While appending an additional U(1) gauge symmetry to the SM group, one must ensure * tanumoy@iisertvm.ac.in † subhadip.mitra@iiit.ac.in ‡ cyrin.neeraj@research.iiit.ac.in § naveenreule20@iisertvm.ac.in ¶ kalp.shah@research.iiit.ac.in the cancellation of all gauge anomalies to maintain gauge invariance and renormalisability of the theory.To cancel the gauge anomalies, one can include RHNs in the particle spectrum [21] or rely on some special mechanism like the Green-Schwarz (GS) mechanism [22,23].The minimal GS mechanism does not require RHNs.However, the cancellation of pure gravity anomaly motivates the existence of RHNs.Therefore, many anomaly-free U(1) extensions of the SM contain RHNs since they can also help generate light neutrino masses through various seesaw mechanisms.
In our previous paper [24], we presented two theoretically well-motivated leptophobic Z ′ scenarios where the Z ′ dominantly decays to a pair of RHNs.Being leptophobic, these scenarios can easily bypass the dilepton exclusion limits.At the same time, a large BR of the Z ′ → N R N R decay and the subsequent decays of N R to W ℓ, Zν, and Hν final states open up many interesting leptonic final states (see, e.g., Fig. 1).The process pp → Z ′ → N R N R is important not only from the Z ′ -search point of view but also for the RHN searches.Since RHNs are SM singlets, producing them at the LHC is difficult as their production cross sections are small, suppressed by the light-heavy neutrino mixing angles.However, they can be copiously produced if they come from the decays of another BSM particle [25][26][27][28].
We considered the pp → Z ′ → N R N R channel in a leptophobic setup in Ref. [24].We studied the dilepton channel (which is easy to analyse with a cut-based analysis and shows a good sensitivity) to estimate the HL-LHC reach.Interestingly, we found that a large chunk of the parameter space beyond the reach of pp → Z ′ → j j can be probed through this channel at the HL-LHC.The decay of the RHN pair can lead to multilepton final states.The monolepton mode is complimentary but more challenging because of the difficulty in the background reduction.In this followup paper, we estimate the reach in the monolepton final arXiv:2307.01118v3[hep-ph] 4 Feb 2024 state with a deep neural network (DNN) model.
The RHNs we consider are around the TeV scale, much lighter than the standard type-I seesaw scale ∼ 10 14 GeV.To naturally realise the TeV-scale RHNs, we consider inverse-seesaw mechanism (ISM) for neutrino mass generation [29,30].As a result, unlike the Majorana-type RHNs in the type-I seesaw, the RHNs here are pseudo-Dirac in nature.If neutrino mass is generated using a type-I seesaw mechanism, because of the Majorana nature of the RHNs, the collider consequences will be different from the ISM.For example, from the pp → Z ′ → N R N R process, we get a same-sign dilepton signature, which is not present in the ISM framework.Earlier, in Refs.[31][32][33][34][35], different U(1) extensions using the ISM were considered in various contexts.Also, the future lepton colliders could be the best testing ground for the heavy neutrinos [36][37][38].
The rest of the paper is organised as follows.In Sec.II, we review the leptophobic models; in Sec.III, we present our analysis of the monolepton channel; in Sec.IV, we estimate the HL-LHC prospects of the monolepton channel.We also update our estimate of the dilepton prospects with the DNN in Sec.V. Finally, we conclude in Sec.VI.

II. LEPTOPHOBIC Z ′ MODELS
We look at the two examples of anomaly-free gauge extensions of the SM from Ref. [24], where a leptophobic Z ′ with a substantial Z ′ → N R N R branching is present.In one example, the RHNs cancel the mixed gauge-gravity anomaly while the GS mechanism cancels the rest.In the other example, the leptophobia of Z ′ arises in a GUT framework [39][40][41][42].Below, we briefly review the essential details of these two constructions and elaborate on some aspects for completeness.

A. Leptophobia with GS mechanism
In this model, the particle spectrum includes three RHNs (one for every generation) equally charged under the extra U(1) z .Even though the RHNs are not essential for gauge anomaly cancellation with the GS mechanism [23], the cancellation of the pure gravity anomaly motivates us to introduce them.The light-neutrino masses are generated through the ISM, which requires an extra chiral sterile fermion S L per generation.We show the U(1) z charges of various particles in Table I.The SM leptons are uncharged under U(1) z to make the Z ′ leptophobic.The nonzero U(1) z charges of the quarks ensure that the Z ′ can be produced at the LHC through quarkquark fusion.The U(1) z charge assignment is generationindependent, and we introduce two free parameters α and β to parametrise the right-handed up and down-type quark charges, respectively (α = β = −1 in Ref. [24]).The SM Higgs doublet is chargeless under U(1) z to minimise the mixing between Z and Z ′ .The scalar φ with unit U(1) z charge is the flavon field needed to generate the quark Yukawa interactions as shown in Eq. (5).The fermion masses arise after φ and H get vacuum expectation values.
The charge assignment in Table I leads to six anomalous triangle diagrams proportional to the traces of the product of the generators: Since gravity becomes prominent at high scales, it is natural to assume that the mixed gauge-gravity anomaly vanishes at low energy without the GS mechanism.Therefore, we set by hand.This gives us a relation between the U(1) z charges of the quarks and RHNs.
To cancel the other anomalies with the GS mechanism [43,44], we add new gauge-dependent terms in the Lagrangian with carefully chosen coefficients.The Peccei-Quinn (PQ) terms for the mixed gauge anomalies and the [U(1) z ] 3 anomaly can be written as Under the new U(1) z , the pseudoscalar (Goldstone) axion Θ transforms as Θ → Θ + Mg z θ z and the new gauge field transforms as Here, M is the scale at which U(1) z breaks through the Stückelberg mechanism, F z , F Y , F W , F S are the field strengths and g z , g ′ , g, g S are the coupling constants associated with the U(1) z , U(1) Y , SU(2) L , SU(3) c gauge groups, respectively.The generalised Chern-Simons (GCS) terms for other mixed anomalies are given as with and The coefficients C , D, E , and K can be solved in terms of the U(1) z charges of the fermions as to cancel the anomalies.The U(1) z invariant Yukawa interactions are written as where H = iσ 2 H * , and λ u and λ d are the Yukawa couplings.

Neutrino mass generation
The neutrino masses are generated by the ISM through the following higher-dimensional operators, We get the mass matrix for the neutrinos after the sequential breaking of the U(1) z and electroweak symmetries in the (ν c L N R S c L ) T basis as, where m D = λ ν v h , µ, and M R are 3 × 3 mass matrices in the generation space in general.The parameter µ is typically associated with lepton number violation and restores the lepton number symmetry in the µ → 0 limit.After diagonalising the matrix in Eq. ( 7), we get the active (light) neutrino mass matrix as [45], There is a double suppression of the light neutrino masses from the large M R (taken around the TeV scale) and the small µ.

B. Leptophobia in E 6 GUT models
A leptophobic Z ′ can be realised in some E 6 GUT models through gauge-boson kinetic mixing [39] even though the fermion couplings to the gauge bosons are not arbitrary free parameters.The U(1) z group in such models can come from a symmetry-breaking chain, like, The fundamental representation of E 6 is 27 dimensional, containing all the SM fermions along with two colourneutral isosinglets ν c and ν c , a colour-singlet isodoublet L = (N E) T and its conjugate L c , and a colour-triplet isosinglet D and its antiparticle D c .The RHN can be identified with either ν c or ν c .Under SO (10), the 27 dimensional representation breaks into a 16, a 10 and an 1.These further break into representations of SU (5).In the standard embedding (I in Table II), we put all the SM fermions and the RHN into the 16, and the new fermions in the 10 and the 1.There are five other ways one can embed these fermions, Particle leading to a total of six different embeddings [42].In Table II, we show the new charges of all the fermions for the six different embeddings.
From the chain in Eq. ( 9), the U(1) z charge Q z can be expressed as Q z = Q ψ cos θ − Q χ sin θ where θ is the E 6 mixing angle.However, there is no solution for θ where the lepton-Z ′ couplings vanish.Therefore, we look at the U(1) Y ↔ U(1) z gauge-kinetic mixing terms allowed by the gauge invariance, The kinetic mixing, parametrised by sin χ, can be removed by the following transformation [41]: However, this cancellation is valid only at a particular scale, implying that the mixing term can regenerate at higher orders.This makes the couplings energy-dependent.Hence, they must be evaluated at the TeV scale before relating to the experiments.After the above rotation, the couplings of the abelian gauge bosons with the fermions are given as [41], where δ = − gY sin χ/ gz .Now, the fermion couplings are functions of two free parameters θ and δ ; we can, in general, make the couplings of any two fields vanish.For example, for the standard embedding (Table II), leptophobia corresponds to θ = tan −1 3/5 and δ = −1/3 [42].Different combinations of θ and δ work in different embeddings, as shown in Table II.

III. COLLIDER ANALYSIS
For collider analysis, we use the following simplified Lagrangian, where z u,d are the effective U(1) z charges of up and down type quarks (for any quark q, z 2 q = z 2 q L + z 2 q R ).We show  TABLE IV.Projected number of events at the 14 TeV at different stages of analysis in the monolepton channel for the two benchmark points, BP1 ≡ (M Z ′ = 5.0, M N R = 2.0) TeV and BP2 ≡ (M Z ′ = 3.0, M N R = 0.5) TeV (see Fig. 2) with g z = 0.3.Here, S 1 and S 2 denote the number of signal events for BP1 and BP2, respectively.We obtain DNN(BP1) and DNN(BP2) after training the network on BP1 and BP2 training data, respectively.For the signal, we assume a K factor of 1.3 [46] and BR(Z  Invariant masses of the fatjet-lepton pairs.m j i j k Invariant masses of any two reconstructed AK-4 jets.III.Between the two colour-singlet isosinglets in Table II, we identify the one with higher absolute U(1) z charge with the RHN, i.e., z N = Max(z ν , z ν ).
We use FeynRules [47] to build the above model and obtain the Universal FeynRules Output (UFO).We simulate the hard scattering in MadGraph5 [48], showering and hadronisation in Pythia [49] and the LHC detector environment in Delphes3 [50].The events are generated at √ s = 14 TeV.Jets are clustered using the anti-k T algorithm [51] with R = 0.4 (we call them AK-4 jet).For the electron, the DeltaRMax parameter in the Delphes card (distance between the other isolated objects and the identified electron) for isolation has been modified to 0.2 from 0.5.

A. Event selection criteria: Monolepton channel
We show the signal topology in Fig. 1.In Ref. [24], we performed a cut-based analysis of the dilepton signature.
Here, in this paper, we mainly focus on the monolepton signature, which arises when one of the RHNs decays to a hadronically decaying W boson and a lepton (muon) while the other decays to a Z/H boson and a neutrino: As explained earlier, probing the monolepton channel is more challenging than the dilepton case as the backgrounds (see Table IV) are harder to tame with simple kinematic cuts.Hence, we employ a DNN model to enhance the signal sensitivity.We look at Z ′ masses between 3 − 7 TeV and RHNs ranging from 0.5 to 3.5 TeV.From the topology and the distributions of the final state, we design the following basic selection criteria for the signal events: , where H T is the scalar sum of the transverse momenta of all hadronic objects in an event.
(We consider the muon as it has the best detector sensitivity among the leptons.) C 3 : At least two AK-4 jets.We also demand that the leading jet has p T > 120 GeV and the subleading one has p T > 40 GeV.C 5 : The invariant mass between the reconstructed muon and missing transverse energy (/ E T ) must be greater than 140 GeV, i.e., M µ / E T > 140 GeV.
We pick the fatjet radius R = 0.7 after optimising to a boosted two-pronged jet.The constraints on fatjet mass (C 4 ) are chosen to include reconstructed W, Z and H fatjets. Since the missing energy and muon come from the decay of W -boson in all the backgrounds, we put a high cut on the invariant mass of reconstructed lepton and missing transverse energy pair in C 5 .

B. Background processes
The major backgrounds considered for the analysis are listed in Table IV, along with the cut efficiencies and the total number of events at the HL-LHC luminosity, 3 ab −1 .
We generate the background events with some generationlevel cuts to reduce the computation time.The W ± + jets process is the leading background before the cuts; the demand of a high-p T jet (C 3 ) and two fatjets (C 4 ) reduces this background significantly.The contributions from the W + W − + jets process drop drastically with the demand of two fatjets (C 4 ).The fatjet criterion (C 4 ) also cuts the t t + jets background as most of these events have just one fatjet.The hadronically-decaying top is sometimes reconstructed as the fatjet in a fraction of t t + jets events; the demand on the fatjet mass cuts down these events.After the event selection criteria are enforced, W + W − + jets and t t + jets become the leading backgrounds.

C. Kinematic picture and the Deep Learning model
We use the kinematic information of the identified objects, i.e., one muon, three AK-4 jets, and two fatjets, along with some derived quantities to create the input set of variables for the multivariate analysis-the complete list of variables is given in Table V where J is used to denote a fatjet whereas j is used to identify an AK-4 jet.
We show the distributions of the key features of the signal and the background for three benchmark parameters in Fig. 2. The H T (scalar sum of all hadronic objects) distributions show a clear separation between the signals and background due to the boosted nature of the final state; see Fig. 2(a).Moreover, for the signals, the transverse momentum distributions of the tagged muon, AK-4 jets, and the fatjet and the / E T distribution peak at considerably higher values than those of the background, see Figs. 2(b), 2(c), 2(d), 2(e).We also consider the nsubjettiness variables [52] which check for two-pronged (τ 2 /τ 1 ) and three-pronged (τ 3 /τ 2 ) nature of a fatjet for various values of β (β ∈ {1, 2, 3}).Since our fatjet parameters are tuned to identify a W -like (two-pronged) jet, the τ 2 /τ 1 ratios are the key variables.Of these, the τ 2 /τ 1 for β = 2 gives a clear separation between the signals and the background [Fig.2(f)].The τ 3 /τ 2 ratios do not offer clear separations, but, nevertheless, we include them due to the inclusive nature of the analysis cuts on the number of fatjets.
The variable m J i ℓ (invariant mass of a fatjet and the tagged muon) reconstructs the mass of the RHN very well-the trend can be clearly seen in Fig. 2(g).The ∆R J i / E T distributions in Fig. 2(h) can also distinguish the signals and the background well, as the final state objects come from the decay of an on-shell Z ′ .
To perform the classification task, we use a fully connected DNN model, the details of which are described in Appendix A. We pick the benchmark point (M Z ′ , M N R ) = (5.0,2.0) TeV to optimise the hyperparameters using a grid search.The performance metric is the signal significance (Z score), where N S and N B are the numbers of signal and background events the network allows at 3 ab −1 luminosity.

IV. MONOLEPTON CHANNEL PROSPECTS
Fig. 3 shows the HL-LHC discovery (5σ ) and exclusion (∼ 2σ ) reaches through the monolepton channel in a modelindependent manner.Assuming a fixed BR(Z ′ → N R N R ) = 50%, we mark the contours of the U(1) z gauge coupling, g z [Eq.( 2)], needed to achieve 5σ and 2σ significances on the M Z ′ − M N R plane.To calculate the number of events, N S and N B , we have used the final efficiencies of the DNN model at the working point with DNN response = 0.95.The g z contours are broadly insensitive to the mass of the RHN.
However, because of the high boosts, the event selection efficiency around M N R = 0.5 TeV (or less) is lower than those at larger values.Hence, one needs a larger g z to achieve a high signal significance.This can be seen from the slightly deformed contours between M N R = 1.0 TeV and M N R = 0.5 TeV.
In particular models, however, the Z ′ BR will not be fixed.For example, in Fig. 4, we show the 5σ and 2σ contours for the GS model and the six E 6 embeddings for g z = 0.72 (≈ g ew ); the corresponding U(1) z charges are found in Table III.As the M N R /M Z ′ ratio approaches the threshold value, 1/2, BR(Z ′ → N R N R ) drops.This is why the contours in Fig. 4 turn inwards with increasing M N R near the kinematic threshold (top-left edge).
In Fig. 5, we show the regions that will be beyond the reach of the dijet channel at the HL-LHC (approximately) but can be reached (≳ 2σ ) by the monolepton signal.To estimate the projected dijet reach, we have simply scaled down the current ATLAS dijet exclusion limits by the square root of the luminosity ratio ( 3000/140 ≈ 4.65), ignoring the difference between the centre-of-mass energies.

V. UPDATE: DILEPTON CHANNEL PROSPECTS
We update our earlier cut-based estimates of the dilepton prospects [24] with the current DNN model.The dileptonic signal comes from the following process, where the W bosons decay hadronically.See Ref. [24] for a discussion on the important background processes.The event-selection criteria remain similar to those in Sec.III A with some minor changes: in C 2 , we now demand two muons with p T > 120 GeV and the new fatjet window in C 4 is now [40,120] to only tag W -like fatjets.To suppress the huge Drell-Yan dilepton background, a Z-veto cut is necessary; we also change C 5 to demand that the invariant mass of the muon pair (m ℓℓ ) be greater than 140 GeV.We show the model-independent HL-LHC discovery (5σ ) and exclusion (∼ 2σ ) reaches through the dilepton channel in Fig. 6 and the 5σ and 2σ contours for the GS model and the six E 6 embeddings with g z = 0.72 in Fig. 7. Comparing Fig. 6 with Fig. 4 of Ref. [24], we see the DNN has enhanced the reaches (discovery and exclusion) significantly.

VI. CONCLUSIONS AND DISCUSSIONS
In this paper, we studied the prospects of a correlated search of a heavy neutral gauge boson Z ′ and the righthanded neutrino N R in leptophobic U(1) extensions.In particular, we analysed an experimentally unexplored channel pp → Z ′ → N R N R where the N R pair decays to a monomuon final state.While in Ref. [24] we only considered the GS model and the standard E 6 embedding, here, we study all possible embeddings of E 6 GUT models where a leptophobic Z ′ can dominantly decay to TeV-scale RHNs in large parts of the parameter space.Moreover, unlike the dilepton signature considered in our previous paper [24], probing the monolepton signature is highly challenging due to the huge background from the SM processes, which are hard to reduce with a cut-based analysis.Hence, in this paper, we used a DNN model to isolate the signal from the SM background, providing orders of magnitude gains in significance scores.With this, we found both mono and dilepton channels to be comparable and highly promising; for an order-electroweak coupling g z , the leptophobic Z ′ with mass up to 6 − 7 TeV can be discovered at the 14 TeV HL-LHC.Therefore, this follow-up paper improves upon our previous study on several counts: it covers more model possibilities with a more sophisticated analysis.It also covers the complimentary signatures-mono and dilepton channels-to present a comprehensive view of the prospects of this exciting but experimentally unexplored process.
Our findings point to a few directions for follow-up investigations.Studying the phenomenology of the Yukawa, scalar, and neutrino sectors of the GS and GUT models we consider will be interesting.It will also be interesting to explore the parameter space where the RHNs can produce displaced vertices (as we pointed out in Ref. [24]) or form N R -jets [53].On the other hand, our analysis is generic as it mainly relies on the kinematic features of the mono and dilepton final states produced from RHN pairs.Hence, the results-presented in Figs. 3 and 6 in a mostly modelindependent manner-can be mapped to other BSM scenarios not necessarily related to the leptophobic Z ′ .For example, we could consider (q Γ q)(N R Γ N R )-type operators (where ψ Γ ψ is a Dirac bilinear) in the N R -effective field theory framework [54][55][56][57][58] to produce RHN pairs at the LHC.Similarly, a pair of RHNs can also come from a leptoquark pair [28]: pp → ℓ q ℓ q → (N R j)(N R j).In such cases, where RHNs are resonantly produced in pairs, we can directly recast our results for comparable kinematics.For example, we can consider the simplest case of a charge-1/3 singlet scalar leptoquark.For M ℓ q = 1.5 TeV, M N R = 0.5 TeV (the kinematics is roughly comparable to the benchmark point BP2, defined in Fig. 2) and BR(ℓ q → N R j) = 100%, both the monolepton and the dilepton channels give Z > 5 Monolepton (1 muon) FIG. 4. HL-LHC reaches the GS model and the E 6 embeddings in the monolepton channel for g z = 0.72.The green regions (dotted contours) are discoverable (5σ ), and the red regions are excludable (2σ ).The U(1) z charges are found in Table III .(i.e., they are discoverable).The prospects are better than those reported in Ref. [28], which used no machine learning techniques.
Finally, some studies on the leptophobic Z ′ in supersymmetric models are available in the literature [59][60][61].Generally, in such models, the Z ′ can decay to a pair of charginos, producing a dilepton plus missing energy signal at the LHC [61]: Since there is no fatjet in the final states, our analysis is not directly applicable to this signal, but one can suitably modify the cuts to investigate it further.However, if a leptophobic Z ′ decays to a pair of neutralinos, those can decay to W bosons and charginos, producing the lepton(s) plus fatjets signature.In that case, the signal will resemble ours, allowing recast, even though no RHN is involved in the process.
of the particle contained in the 27 representation of E 6 for the six possible embeddings-the first one is called the standard embedding.Here, G SM ≡ SU(3) c × SU(2) L ×U(1) Y .The tan θ and δ values needed to get the leptophobic Z ′ boson in each embedding are shown in the bottom rows.

z u and z d for the six different E 6
embeddings and the GS model in Table

C 4 :
Exactly two fatjets, clustered using the anti-k T algorithm with R = 0.7 and p T > 200 GeV with mass M J ∈ [40, 165].

FIG. 5 .
FIG.5.Regions where the monolepton signal achieves at least 2σ significance at the HL-LHC but are beyond the projected reach in the dijet channel (see text).

10 FIG. 8 .
FIG.8.Schematic for the DNN model with four hidden layers.The number of nodes in a layer is shown on the top.

TABLE I .
Particle representations and quantum numbers in the GS anomaly-cancellation setup.SU

TABLE V .
Kinematic variables chosen to analyse the monolepton signature; τ's are the n-subjettiness ratios.