Multi-lepton signatures of vector-like leptons with flavor

We investigate collider signatures of standard model extensions featuring vector-like leptons and a flavorful scalar sector. Such a framework arises naturally within asymptotically safe model building, which tames the UV behavior of the standard model towards the Planck scale and beyond. We focus on values of Yukawa couplings and masses which allow to explain the present data on the muon and electron anomalous magnetic moments. Using a CMS search based on $77.4 \, \rm{fb}^{-1}$ at the $\sqrt{s}=13$ TeV LHC we find that flavorful vector-like leptons are excluded for masses below around $300$ GeV if they are singlets under $SU(2)_L$, and around $800$ GeV if they are doublets. Exploiting the flavor-violating-like decays of the scalars, we design novel null test observables based on opposite sign opposite flavor invariant masses. These multi-lepton distributions allow to signal new physics and to extract mass hierarchies in reach of near-future searches at the LHC and the HL-LHC.

use L = (ν, L ) T and E = R , respectively, and denote the Higgs doublet by H. All SM leptons and VLLs carry a lepton flavor index i = 1, 2, 3, which is often suppressed to avoid clutter. Both models also contain complex scalars S ij , with two flavor indices i, j = 1, 2, 3, and which are singlets under the SM gauge interactions. In the interaction basis, the models' BSM Yukawa sectors read where the contraction of gauge indices is assumed. Here we followed [9] and identified SU (3)-flavor symmetries of the leptons with ones of the VLLs. This identification has important consequences for phenomenology: Each lepton flavor is conserved and leptons couple universally within (1), and the BSM Yukawas y, κ, κ become single couplings, instead of being tensors. While y is key in variants of the asymptotically safe framework [4,7], in models like (1) with mixed SM-BSM Yukawas its presence is not required to achieve a controlled UV-behavior [9]. As in addition the phenomenological implications of y are less relevant we do not consider it in the numerical analysis.
After spontaneous symmetry breaking, the vector-like fermions and the leptons mix, see [9,10] for details. To be specific, we denote the lightest three mass eigenstates by leptons and the others by VLLs, and continue to use the notation as introduced above. Z → data [14] constrains the mixing angles θ of left-handed (right-handed) leptons in the singlet (doublet) model as θ κv h / √ 2M F < O(10 −2 ). Here, v h 246 GeV is the Higgs vacuum expectation value (vev), and we denote by M F the common mass of all flavor and SU (2) L -components of the VLLs. We learn that θ, κ 1, which allows for a small angle approximation. At first order in κ and κ , the interactions in the mass basis in the singlet model read where A µ denotes the photon, h corresponds to the physical Higgs boson with M h = 125 GeV and e, g, θ w are the electromagnetic coupling, the SU (2) L coupling and the weak mixing angle, respectively. The remaining couplings fulfill For the doublet model, we find with couplings The vertex νγ µ ψ − L W + µ arises only at higher order, see [9] for details, and can be safely neglected for the purpose of this analysis.
The coupling κ is needed to account for the electron AMM anomaly ∆a e [12,13], however, can be chosen with some freedom since parameters of the scalar sector also play a role here, see [9,10] for details. For simplicity, we fix κ/κ = 10 −2 , consistent with Z-decay constraints and both AMMs.
Let us briefly comment on the scalar sector of the BSM framework [9,10,26]. The presence of H and the flavorful scalar matrix field S ij allows for a substantial scalar potential with in total three quartic couplings plus a Higgs portal one δS † SH † H. In addition to successful electroweak symmetry breaking the diagonal entries of S can acquire a non-trivial vev, v s . Interestingly, two different configurations exist: A universal ground state in which diagonal entries have the same vev, and one in which the vev points in a single flavor direction, breaking universality spontaneously.
Both the portal δ and v s are instrumental in achieving the chirally enhanced 1-loop contributions explaining AMMs. However, as discussed in the next section, the impact of δ, v s on the present collider study is negligible, and we do not consider them in this work.
To summarize, in the following we perform a collider analysis in the two models, one with three flavors of singlet VLLs (2), and one with three flavors of doublet VLLs (4), featuring nine flavored scalar singlets and with parameters M S , M F , κ/κ = 10 −2 , κ = κ (M S , M F ) .
Using (6) together with the muon AMM data to express κ in terms of M S and M F renders the numerical predictions for production and decay of the BSM sector in terms of the latter two masses.
We discuss more general settings in Sec. VIII The reduction of the models' BSM parameter space (i.e. BSM masses, Yukawas, and quartic couplings) onto the set (7)  for the remaining BSM Yukawas and quartics, using the methods of [9,10]. This completes the discussion of our setup.

III. LHC PRODUCTION AND DECAY
At the LHC, VLLs can be produced in pairs (upper plots) or singly (lower plots) in quark fusion through electroweak interactions shown in Fig. 1. Pair production occurs through s-channel photon or Z (Fig.1a), and in the doublet model additionally through s-channel W exchange (Fig.1b). Single production is induced by the Yukawa portal coupling κ through fermion mixing and Z, W -exchange ( Fig.1c, d). All three flavors are produced universally. Additional contributions to VLL production arise through s-channel Higgs and BSM scalars S ii induced by Higgs-scalar mixing (not shown). Due to both quark-Yukawa and parton-luminosity suppression these contributions to matrix elements are suppressed by at least two orders of magnitude with respect to electroweak contributions and thus negligible. Further production channels through Yukawa interactions open up at lepton colliders, briefly discussed in [9].
In Fig. 2 we show pair-and single-production cross sections for a single species ψ i -with lepton flavor index i fixed -at the LHC with √ s = 13 TeV as a function of M F for M S = 500 GeV and the procedure in (7). In both the singlet (left) and doublet model (right) the pair-production cross section is roughly three orders of magnitude larger than the single production cross section. This is due to the fact that single production is only induced by mixing between SM leptons and VLLs, while dominant pair production of VLLs at the LHC occurs through electroweak gauge interactions.
κ , the larger of the BSM Yukawa couplings, is irrelevant also for single production, but turns out to be important for BSM sector decays.
In the singlet model, the decay rates of the possible decay channels of the VLLs are where j dominates if kinematically allowed, as seen in Fig. 3 (left). Quantitatively, in the large-M F limit, the decays through the S ij dominate over Higgs-mediated decays (decays through weak bosons) for κ κ/ GeV and the procedure described in (7).
In the doublet model, we obtain the decay rates The corresponding branching ratios of the ψ − and the ψ 0 decays are shown in Fig. 3 (right).
As in the singlet model, the decays to BSM scalars dominate for large κ if allowed by the mass hierarchy of the BSM sector. As already stated in the previous section we assume that the ψ − and ψ 0 are degenerate in mass; we therefore neglect small isospin splitting induced by electromagnetic interaction ∆m = M ψ −1 − M ψ 0 = g 2 /(4π) sin 2 θ W M Z /2 0.4 GeV, that also allows for rare inter-   multiplet decays ψ − → ψ 0 W − * . The smallness of the splitting prohibits that for instance searches in R-parity violating SUSY models into four light leptons [27] apply to the VLL models.

Doublet
The decays of VLLs to S ij plus lepton are a singular feature of the models, which distinguishes them from other theories with VLLs, such as [23,28]. Moreover, the S ij can decay through fermion mixing to lepton final states, in which they can be searched for. Specifically, the singlet model features the cascade decays Similarly, in the doublet model the decays of the charged and neutral VLLs proceed as These processes preserve flavor; however, the scalar decay yields a dilepton pair with different-flavor charged leptons for i = j, which looks as if lepton flavor has been violated and cleanly signals new physics. While the scalars may also decay to dibosons through triangle loops, or to two VLLs through the coupling y if M S < 2M F , see (1) and [9] for details, here we assume that these rates are negligible.
In this work we are interested in final states with at least four light leptons (4L), where a light lepton is an electron or a muon, as in [24]. When the ψ i are pair-produced and decay through Eqs. (10) and (11), only certain flavor final states of each single decay can contribute to a 4L final state. These are given in Tab. I. Notice that the decay chains (10), (11) allow to observe resonance State Decay modes

IV. EVENT SIMULATION
In this section we describe the procedure used to generate a sample of events with 4L final states at the LHC. We employ FeynRules [29] to compute the Feynman rules at leading order (LO) for the models in Eqs. (2) and (4). The particles and Feynman rules are then implemented into UFO models [30]. These UFO models are interfaced to the Monte Carlo generator Mad- For the event generation we adapt settings similar to the CMS study [24]. We focus on the final states with at least four light leptons, that is, muons and electrons and require the missing transverse momentum, p miss T , to be smaller than 50 GeV. This cut serves to resemble the signal region considered by CMS and to suppress contributions from neutrinos in the decay of the electroweak bosons. Electrons and muons are required to have a minimal transverse momentum of p T ≥ 20 GeV.
Similarly to the CMS analysis, we neglect all events with a light-lepton invariant mass, m , smaller . We also consider ttZ, triboson and ZZ production as these processes contribute to the SM background for the distributions studied in [24]. The ZZ production includes contributions from virtual photons via pp → γ * γ * , γ * Z. ZZj final states are included via multijet merging in PYTHIA8 [34]. We also take into account in the cross section gluon-fusion contributions gg → ZZ, where the lowest order is induced at 1-loop. SM background processes are computed at LO within MadGraph5_aMC@NLO using the same set of PDF sets. Higher order corrections to SM production cross sections are taken from literature [35][36][37][38][39][40][41][42] and are taken into account by applying k factors to the LO distributions. To perform a simulation of the detector response we shower and hadronize the events with PYTHIA8 and use DELPHES3 [43] for the fast detector simulation, yielding events at particle level. Jets are clustered with the anti-k t algorithm [44] with a radius parameter R = 0.5 applying the FastJet package [45]. All criteria for the analysis are taken from the CMS default card for simplicity. In Tab. II we summarize the values for the parameters and signal selection cuts used in the event generation and detector simulation.

V. CONSTRAINTS FROM CMS DATA
Here we confront the models (2), (4) to the CMS search [24] using the 4L final state, which is expected to be the channel most sensitive to contributions stemming from three generations of VLLs. In Sec. V A, we study decay chains into 4L final states and their multiplicities. In Sec. V B we compare the distributions of the scalar sum of transverse momenta of the four light leptons (e, µ) with the largest transverse momenta, L T , with CMS data to obtain constraints on BSM masses.

A. 4L multiplicities
4L final states stem from both single and pair production of VLLs. Due to the flavor structure of the BSM sector, the following decay chains include 4L final states in the singlet model: where i , j , k are flavor indices and q i = u, d, c, s, b. We also indicate the values that the lepton flavor indices can take, and between parentheses the number of 4L final states of each chain after summing over all indices. Note that, for the first decay chain in (12) In the doublet model, the negatively charged state ψ − decays into 4L final states as in (12) with the exception of the decays with 8-fold multiplicity. These correspond to W -mediated decays jν j , which are subleading in the doublet model. Additionally, when the ψ 0 are produced, 4L final states arise through The first decay chain in Eq. (12), involving a six charged-lepton final state, is the only one where production of the third generation ψ 3 can give rise to a 4L final state. In all other cases, ψ 3 production yields at most three light leptons in the final state, since a τ + τ − pair is always produced due to flavor conservation. In Fig. 4 we give examples of Feynman diagrams for the different decay chains, with jets (a) or without them (b), and from single production (c).   (2) and (4), while the blue curves correspond to third-generation VLL models as in [23]. The band widths include uncertainties discussed in Sec. IV. In

B.
L T distributions and CMS constraints CMS has searched for VLLs employing the scalar sum of the leading four light leptons' transverse momenta, L T , finding no significant discrepancies with the background [24]. To work out the implications of this analysis for the models (2) and (4) we compute the L T distributions for different values of M S and M F and fixed BSM Yukawas (7). After performing the detector simulation we compare the distributions to CMS data for 4L final states. We also compute L T distributions for the dominant SM background processes of ZZ, triboson and ttZ production. We include the control region veto, two dilepton pairs with invariant masses 76 GeV < m 2 < 106 GeV, and set the bin width to 150 GeV as in [24].
Since our simulation of the SM background is performed at LO and only a fast detector simulation is publicly available, differences with the one by CMS are expected. In contrast to CMS, we can not perform a fit of the background distribution to a control region. This prohibits a quantitative reinterpretation of the data at precision level, which could be obtained from an actual experimental analysis only. Still, we find that our background simulation is in reasonable agreement with the shape and the bin content of the L T distribution. In view of the differences between our SM prediction and the one from CMS, and to make progress, in the following we refer to benchmarks as 'excluded' if the BSM distribution overshoots the CMS data in at least one of the bins by more than one sigma.
Our findings are summarized in In Fig. 7 we show the L T distributions for these benchmarks (long-dashed curves) to explicitly show that they pass 4L constraints, that is, are within the CMS plus 1σ range (hatched area).   We check for these conditions in the above order (a → b → c) and stop when one of the requirements is fulfilled. We define the observable m 2 _diff as invariant mass pairs that only fulfill condition c), where two particles of approximately equal invariant mass are found and at least one of them is computed from different-flavor leptons. All SM contributions to this observable are purely statistical, and therefore any significant excess away from SM resonances is an indication of new physics which can be explained by the VLL models of Eqs. (2) or (4).

B. m 3 and m 3 _diff
The m 3 and m 3 _diff observables are designed to reconstruct the invariant masses of the VLLs via their three-body decays. For each pair of two-particle invariant masses added to m 2 , we look for the additional lepton which stems from the decay of each ψ. We add to m 3 the pairs of three-particle invariant masses which fulfill one the following conditions: i) For two-particle invariant masses which reconstruct to a Z or Higgs (condition a in the previous section) each two-particle invariant mass is paired with an additional lepton present in the final state. The resulting three-particle invariant masses are added to m 3 if their difference is smaller than ∆M F , and no other combination presents a smaller difference.

C. Benchmark distributions for Run 2
We study the m 2 , m 2 _diff, m 3 , and m 3 _diff distributions for allowed benchmark values (yellow circles in Fig. 6) of the VLL mass M F and the BSM scalar mass M S for the full Run 2 data set. Results based on the algorithms described in Secs. VI A and VI B are shown in Fig. 8 and We learn the following generic features: i) The results are qualitatively similar for the singlet and doublet models, with a larger cross iii) Distributions can signal BSM physics also in the tails away from a narrow resonance peak, or if none is present; see for instance the black curves in Fig. 8 and Fig. 9. All benchmarks display detector simulation.
As we argued, the new observables have great sensitivity to flavorful BSM physics, and would benefit from higher luminosity. In the next section, we discuss perspectives for the HL-LHC.

VII. IMPLICATIONS FOR THE HL-LHC
As shown in Sec. VI C the discovery of a BSM sector consisting of VLLs and new scalars with a non-trivial flavor structure remains a challenging task at Run 2. Here we study the new observables In general, the coupling κ remains limited in magnitude from above by Z decays, inducing small but relevant effects in fermion mixing. On the other hand, κ is unconstrained by electroweak data.
As it is already rather sizable in the benchmark (7), we investigate the implications of a reduced κ .
The latter implies a suppression of ψ to S plus lepton decays. Since these modes are In the following we focus on the singlet model, since the allowed parameter space of BSM masses is larger, see Fig. 6. For simplicity we use κ = 1. We find that the benchmark M F = 800 GeV, M S = 500 GeV and κ = 1 is excluded by CMS data, even though the g −2 benchmark scenario with a larger coupling but the same BSM masses was found to fall within the allowed region (see Fig. 7).
Nevertheless, we observe that larger VLL masses M F ∼ 900 GeV are allowed for M S ∼ 500 GeV and κ = 1. Therefore, in this section we take M F = 900 GeV, M S = 500 GeV as one of our benchmarks.
We study as well the κ = 1 counterparts of the two remaining benchmarks considered in previous sections, which we find to be allowed.
The corresponding distributions of the new observables at the √ s = 13 TeV LHC and the full Run 2 data set are shown in Fig. 14. We observe that for m 2 and m 2 _diff (see Fig. 14 are substantially depleted with respect to m 3 , and higher luminosities would be beneficial.
In Fig. 15 we give the m 3 and m 3 _diff distributions after hadronization and detector simulation The m 2 and m 2 _diff spectra are very similar to the ones in Fig. 10 and therefore not shown.
The m 3 distributions show peaks for all three benchmarks, with O(1) events per bin for both M F = 300 GeV (blue) and M F = 900 GeV (black). For the latter, the m 3 _diff distribution allows for a null test, as the SM background is sufficiently suppressed while we find a peak with few events per bin in the M F = 900 GeV distribution.

IX. SUMMARY
We investigated opportunities at the LHC and the HL-LHC to search for flavorful vector-like leptons ψ i and new scalar singlets S ij . Such BSM sector (1) occurs in novel model building frameworks with favorable UV behavior [4][5][6], and particle physics phenomenology [7,10].
We considered two explicit BSM models of this kind, featuring three generations of either SU (2) L singlet or doublet VLLs, which can also accommodate present data of the muon and electron g − 2.
Key ingredients for flavor phenomenology are the mixed SM-BSM Yukawa couplings, the flavor matrix structure of the BSM scalars, the identification of lepton and VLL flavor, and fermion mixing after electroweak symmetry breaking. Although all BSM interactions (2) and (4)  For our study we implemented the models into UFO models using FeynRules. Predictions for observables including dominant SM background processes at pp-colliders are computed with MadGraph5_aMC@NLO together with PYTHIA8 and DELPHES3. We worked out constraints from a CMS search in final states with at least four light leptons (electrons, or muons) [24]. Results are summarized in Fig. 6, showing allowed regions (green and yellow circles) of VLL and scalar masses while accommodating g − 2 of the muon, (7). We find that, in general, regions around the M S = M F line are excluded by data due to the underlying enhancement of cross sections via on-shell S production. Lower limits for the VLL masses are around 300 GeV in the singlet model and 800 GeV in the doublet model. As such, our findings offer new constraints for the Planck safe models put forward in [10].
Predictions for the new observables m 2 , m 2 _diff, m 3 , and m 3 _diff after detector simulation are shown for several allowed benchmarks in Figs. 10 and 11 for the singlet and doublet model, respectively, for the full Run 2 data set with 150 fb −1 . The distributions exhibit a highly discriminating power on the BSM mass hierarchy, but suffer from marginal event rates and therefore would extremely benefit from higher luminosity. At the HL-LHC, for √ s = 14 TeV and a luminosity of 3000 fb −1 , we obtain O(10 2 ) events after detector simulation in some bins, see Figs. 16 and 17 for the singlet and doublet model, respectively. Hence, these new, optimized observables are very promising for higher luminosity runs at the LHC, to discover and discern hierarchies in flavorful models with multi-lepton final states.
Studying more general versions of our models in Sec. VIII we reduce κ , the key Yukawa for filling the "diff"-distributions. Results are shown in Fig. 15 for the invariant mass distributions after detector simulation. We again observe striking BSM signatures with diagnosing power. Let us also mention that the other colorless models with effectively κ = 0 put forward as asymptotically safe extensions of the SM [9] are not contributing significantly to the "diff"-observables, but could be probed using m 2 and m 3 or conventional VLL search strategies. as a large number of events migrate out of the peak bin but only very few migrate into it from neighboring bins. Bins with very few events show as well significant scaling factors, since small changes in the event count due to bin-to-bin migration can have a significant impact. In general, we find that the improved resolution at the HL-LHC results in a smaller suppression of the peaks due to detector effects compared to Run 2 CMS. This leads to larger scaling factors in the case of the HL-LHC, as seen in Tab. III.