On the behaviour of composite resonances breaking lepton flavour universality

Within the context of composite Higgs models, recent hints on lepton-flavour non-universality in $B$ decays can be explained by a vector resonance $V$ with sizeable couplings to the Standard Model leptons ($\ell$). We argue that, in such a case, spin-$1/2$ leptonic resonances ($L$) are most probably light enough to open the decay mode $V \rightarrow L\ell$. This implies, in combination with the fact that couplings between composite resonances are much larger than those between composite and elementary fields, that this new decay can be important. In this paper, we explore under which conditions it dominates over other decay modes. Its discovery, however, requires a dedicated search strategy. Employing jet substructure techniques, we analyse the final state with largest branching ratio, namely $\mu^+\mu^- Z/h, Z/h \rightarrow$ jets. We show that (i) parameter space regions that were believed excluded by di-muon searches are still allowed, (ii) these regions can already be tested with the dedicated search we propose and (iii) $V$ masses as large as $\sim 3.5$ TeV can be probed at the LHC during the high-luminosity phase.

As we will argue, this solution implies that vectorlike partners (L) of the SM leptons ( ), can very well be lighter than V . In that case, the channel V → L opens and it can actually dominate the V decay width. It is certainly surprising that no single study has taken this effect into account. This has two major implications. (i) Contrary to the standard case studied so far, this setup survives all constraints from LHC data, including the strongest ones from di-muon searches. (ii) In already collected events containing two muons and a fat jet resulting from a Z or Higgs boson, new dedicated searches can reveal a clear peak in the invariant mass distribution of m µ1j1 = (p µ1 + p j1 ) 2 , with µ 1 and j 1 being the highest-p T muon and jet, respectively.

II. MODEL
The phenomenological Lagrangian describing the interactions between the spin-1 singlet V and the SM leptons is given by where m V is the mass of V and the ellipsis encode the kinetic term as well as other interactions not relevant for the subsequent discussion. We further define with and q SM leptons and quarks, respectively. Let us consider for simplicity λ ij ∼ δ 2 i δ 2 j and λ q ij ∼ δ 3 i δ 3 j , so that mainly the second-generation leptons and the thirdgeneration quarks couple to the vector resonance. Likewise, the LFU violating term arises in the physical basis after performing the CKM rotation in the down sector, and can therefore be estimated as Reproducing the LFU anomalies requires [24,25] for g V ∼ 1. Much larger values of g V are disfavoured by limits on neutrino trident production [26]. On the other hand, smaller values are disfavoured by measurements of ∆M s for values of m V in the natural region of CHMs, namely m V ∼ few TeV. We thus stick to this value henceforth, which is allowed even by the latest measurement of ∆M s [25].
The key point is that, if V is a resonance in a CHM, its couplings to the SM fermions originate from partial compositeness [27]. The more fundamental Lagrangian reads ∆L = 1 2 where / V = γ µ V µ , g e (g c ) is a weak (strong) coupling, / D is the SM covariant derivative, M L,Q , ∆ L,Q are dimensionful constants, Q = (T B) t and L = (E N ) t are composite SU (2) L fermion doublets and V and B are composite and elementary vectors, respectively. The physical vectors are admixtures of the latter with mixing angle θ. Likewise for the fermions. With a slight abuse of notation, we denote the physical fields with the same letters.
After rotating the heavy and the light degrees of freedom, the following relations hold: with m L,Q the physical masses before Electroweak Symmetry Breaking (EWSB) of the vector-like fermions and g the U (1) Y gauge coupling. In the expected limit g c g e , we find where g V L parametrises the strength of the V L interaction; see a pictorial representation in Fig. 1. Similar expressions hold in the quark sector. In this limit, cot θ is large, the mixing between V and the SM gauge bosons is small and V production in the s-channel at proton colliders is dominated by bottom quarks. Following Eq. 4, we obtain This implies that the degree of compositeness of the second-generation leptons is large, even larger than that of the left-handed quarks. On top of this, the top Yukawa y t is induced by proto-Yukawa interactions ∼ Y (T HQ + h.c.), with Y a dimensionless coupling, and T and Q composite resonances mixing with t R and q respectively. The phenomenology of V depends still on the dynamics of these particles. This can not be inferred from the anomaly data. Instead, we must rely on the possible structure of the CHM. Let us discuss different regimes and their implications. Let us assume that Q = Q and that T couples also to V with g c strength. Then y t ∼ Y sin φ q sin φ t , with φ t the degree of compositeness of t R . We can in turn distinguish two cases: (i) If Y 4 we get sin φ t ∼ 1. In this regime, t R is maximally composite, and V decays predominantly to tt.
(ii) If Y 4 then sin φ t can be significantly smaller than ∼ 1. In that case, the V decay width to SM particles is small. The dominant modes are V → T t and V → L . The former is more relevant for smaller values of Y , m Q and m T and for larger values of m L ; see Refs. [28][29][30][31] for dedicated analyses. The latter dominates otherwise. As an example, sin φ t ∼ 0.5 implies that g V L ∼ g V T t ∼ 2.5. In such case, for m V = 2 TeV, m L = 500 GeV and m T = 1.5 TeV we already get that Γ(V → L ) is around 1.25 times larger than Γ(V → T t).
Let us also notice that the aforementioned values for the masses fit well within the current experimental data on CHMs. Indeed, the Higgs in CHMs is an approximate Goldstone boson. Its mass is generated radiatively and grows with ∆ L,Q , because these are the main sources of explicit breaking of the shift symmetry. Thus, in order not to advocate a large cancellation between different breaking sources to keep the Higgs light, ∆ L must be small. Given the large value of sin φ , this implies a small m L ; see Eq. 6. In addition, there is no experimental reason for m L not to be in the sub-TeV region. In fact, due to the small EW pair-production cross section for heavy vector-like leptons, they can be as light as few hundreds GeVs [32]. Very dedicated searches will be needed to unravel larger masses even in the LHC High-Luminosity (HL) phase [33]. The preferred values of m T (and m Q ), namely 1 TeV [34][35][36][37][38][39][40], are instead in tension with current LHC data, at least in the minimal CHM [41]. Even masses as large as m T ∼ 1.3 TeV have been already ruled out in several scenarios [42].
If Q = Q and it does not couple to V , or if composite right-handed currents (such as T γ µ T ) couple less to the spin-1 resonance, then g V tt and g V T t can be arbitrarily small. In light of this observation, and given the discussion in point (ii) and the fact that tt and T t signatures have already been explored in the literature, we focus on this regime hereafter. We will show that V → L plays a dominant role in this case.
Finally, it is worth mentioning that other uncoloured composite vector resonances such as EW triplets, commonly present in CHMs too, couple directly to the Higgs and the longitudinal polarization of the gauge bosons [40]. Given that the latter are fully composite, non-singlet vectors decay mostly into them and not into pairs of heavy-light leptons [43,44]

III. COMPOSITE VECTOR PHENOMENOLOGY
The leading-order decay widths for V in the limit m V m , m q are We willl restrict ourselves to the regime m V < 2m L . Otherwise, the decay into two heavy fermions opens and V becomes typically too broad to be treated as a resonance [31]. We will consider a Benchmark Point (BP) defined by m V = 2 TeV, m L = 1.2 TeV and cot θ = 20. Departures from this assumption will be also discussed. We note that, while g V qq depends on m V and is weak, g V and g V L are approximately fixed and given by ∼ 1 and ∼ 2.5, respectively. They are all below the perturbative unitarity limit ∼ √ 4π. Likewise, Γ V /m V is never above 30 %. A perturbative approach to the collider phenomenology of V is therefore justified.
We do not aim to focus on any particular UV realization of the simplified Lagrangian in Eq. 5. Our aim is rather highlighting the implications of light lepton partners for the phenomenology of V . However, it must be noticed that composite muons give generally large corrections to the Zµ L µ L coupling. These can be avoided in left-right symmetric implementations of lepton compositeness [45]. This requires however the introduction of more degrees of freedom. For example, secondgeneration leptons in the minimal CHM [46] might mix with composite resonances transforming in the representation 10 of SO(5); see Ref. [13] 1 . The latter reduces to (2, 2) + (3, 1) + (1, 3) under the custodial symmetry group SO (4). Interestingly, the extra degrees of freedom, namely (1, 3) and (3, 1), do not affect the spin-1 vector decays for several reasons.
(i) In the regime we are interested in, pair-production of heavy leptons mediated by V is kinematically suppressed.
(ii) The custodial triplets do not mix with the SM fermions before EWSB. (Note also that the product of (2, 2) times any of the custodial triplets can not be a singlet, and hence the corresponding current does not couple to V .) Therefore, the extra new fermions can only be produced in association with SM fermions with Production cross section for E ± µ ∓ (thick solid blue) and µ + µ − (thin solid black) in the BP. The thin dashed black line represents the cross section for µ + µ − for mL > mV /2.
In dashed red, we see the current limits on this cross section from ATLAS [48].
a strength further suppressed by a factor of Y v/M , Y being the typical coupling between composite fermions. Provided this is not extremely large, the extra states can be ignored [47]. (Similar reasonings work for other representations.) In this regime the following relations hold with good accuracy: Hereafter, we assume this to be the case. The production cross section for pp → V → µ + µ − in the BP at the LHC with √ s = 13 TeV is depicted by the thin black solid line in Fig. 2. The thin black dashed line represents the would-be cross section in the absence of light L. The region above the thick red dashed curve is excluded according to the recent ATLAS analysis of Ref. [48]. Clearly, in the region where the heavylight topology is kinematically forbidden, di-muon constraints are extremely important, the limit on m V being close to 1.8 TeV. However, in the presence of light lepton partners, and given that the coupling between composite particles is larger than that between composite and elementary fields, the cross section pp → V → E ± µ ∓ dominates (thick blue solid line). The limit on m V imposed by di-muon searches gets then reduced by more than 500 GeV. Additionally, dijet searches as well as tt searches are less constraining. Other LHC searches are sensitive to the heavy-light channel. In particular, we considered searches for electroweakinos in multilepton events with large missing energy, such as that in Ref. [49]. The 13 fb −1 version of this analysis was first presented in Ref.
[50]. The latter is fully included in CheckMATE v2 [51]. Therefore, the reach of the former can be estimated by scaling the signal over the 95% CL limit by the luminosity ratio 36/15. We obtain that values of m V above 1 TeV are not constrained.
Other studies, such as searches for evidence of the type-III seesaw mechanism [52], focus on final states with more than two leptons, which are almost absent in our scenario. On balance, we find imperative to develop a dedicated search to unravel the origin of LFU in CHMs.

IV. NEW SEARCHES
Among the different heavy-light topologies, we focus on the channel pp → V → E ± µ ∓ , E ± → Z/h µ ± , with Z/h decaying hadronically 2 .
Signal events are generated using MadGraph v5 [53] and Pythia v6 [54], after including the relevant interactions in an UFO model [55] using Feynrules v2 [56]. For the subsequent analysis, we have used home-made routines based on Fasjet v3 [57] and ROOT v6 [58]. Muons are defined by p T > 20 GeV and |η| < 2.7. Jets are clustered according to the Cambridge-Aachen algorithm [59,60] with R = 1.2. Muons with p T > 50 GeV are removed from hadrons in the clustering process. The dominant background is given by µ + µ − + jets. We matched Monte Carlo background events with 1 and 2 jets using the MLM merging scheme [61] with a matching scale Q = 30 GeV. At the generator level, we also impose a cut on the p T of the muons, p µ T > 100 GeV. The matched cross section we obtain at LO at √ s = 13 TeV is ∼ 1.2 pb; we generated 10 million events. As basic cuts we require, first, the presence of exactly two opposite charged muons and at least one jet. The leading p T jet, j 1 , is required to have a significant mass drop [62], characterized by µ = 0.67, y cut = 0.3. This jet is further filtered [62,63] using a finer angular scale given by R filt = min{0.3, 0.5×R 12 }, with R 12 the angular separation of the two sub-jets obtained in the mass-drop procedure. This method impacts on the background by systematically moving m j1 to smaller values. Likewise, the h and Z boson mass peaks in the signal become significantly narrower. We also impose both muons to have a p T > 200 GeV. This stringent cut is motivated by the fact that muons originate from the decay of very heavy particles, while the background is mostly populated by soft leptons. More sophisticated jet substructure methods can improve on our result further [64][65][66][67].
Finally, we enforce the leading jet to have an invariant mass 80 GeV ≤ m j1 ≤ 130 GeV. A summary of the basic selection cuts is given in Tab. I. Their efficiency in the BP as well as in the background are also displayed. Interestingly, while a large fraction of the signal is kept, the background is reduced by more than two orders of magnitude.
After these basic cuts, m V can be reconstructed just as the invariant mass m rec V = m(µ 1 + µ 2 + j 1 ). The nor- malized distribution in the BP and in the background after the basic cuts is depicted in Fig. 3. Clearly, a cut on m rec V 1 TeV separates further the signal from the SM.
For large m L , the muon coming from E is typically that with largest p T , i.e. µ 1 . m L can then be obtained just as the invariant mass m rec L = m(µ 1 +j 1 ). The corresponding distribution in the signal as well as in the background after the basic cuts and after enforcing m rec V > 1 TeV is depicted in Fig. 4. We normalize to the expected number of events at L = 1 fb −1 . Remarkably, the signal is already comparable to the background in number of events, while concentrating at much larger values of m rec L . Given our ignorance on m V and m L , we set further cuts depending on these parameters. In particular, we require m rec Denoting by S and B the number of signal and background events, respectively, after all cuts, we estimate the significance as S = S √ S+B . In Fig. 5 we show, in green, the regions that can be excluded at the 95% CL (S = 2) as a function of m L and m V for different values of the collected luminosity. These are to be compared with the reach of di-muon searches, depicted in red. In the grey regions, the composite spin-1 resonance does not decay  sizeably into L . The reason is, that in the upper triangle L is heavier than (or very close in mass to) V . In the lower triangle, V decays mostly into pairs of heavy leptons.
Remarkably though, regions not yet tested by the LHC can start to be probed. Moreover, contrary to di-muon searches, our new analysis will also shed light on the high mass region.

V. CONCLUSIONS
If the origin of the apparent breaking of Lepton Flavour Universality (LFU) in B meson decays is due to a composite spin-1 resonance V , this has to couple to rather composite light leptons as well as quarks. We have shown that V should not be searched in di-muon final states, as it has been done so far, but rather into ditop or in final states containing both a heavy and a light fermions. Focusing on the case of composite muons, we have discussed when V decays mostly into a muon and a composite fermionic resonance, leading to a final state consisting of two muons and a boosted gauge or Higgs boson which express mainly as fat jets. Unravelling the physics responsible for breaking LFU in such a final state requires dedicated and tailored analyses to its kinematic features.
We have worked out one such analysis based on jet substructure techniques. Three main conclusions can be pointed out. (i) Parameter space regions that were thought excluded by searches for di-muon resonances, are still allowed. They are not even ruled out by other beyond the Standard Model analyses, including multilepton searches for electroweakinos or heavy leptons. (ii) Some of the allowed regions can already be probed at the 95% CL with our dedicated analysis. (iii) With more luminosity, e.g. 300 (3000) fb −1 , heavier resonances can be tested, e.g m V ∼ 2 (3) TeV.
Had we assumed λ 33 ∼ λ 22 in Eq. 3 3 , the cross section for our target final state would be reduced by an O(1) factor. The analogous topology with taus instead of muons would be also important. Not taking the latter into account, we estimate the reduced reach to this setup in the right panel of Fig. 5; see dashed green line. Finally, composite electrons give signatures very similar to the ones studied here, with electrons instead of muons in the final state. Our analysis can therefore be trivially extended to this latter case, for which we expect roughly the same sensitivity. On balance, we strongly encourage a re-analysis of current data.