Trimuon production at the LHC

No process without standard model (SM) background has been observed so far. As a tool in the study of a class of models containing doubly charged vector bileptons, we propose one such process that, violating lepton flavor number, has no contribution from the SM: pp → μ+μ+μ−e−. By carefully isolating the parameters to keep free, we are able to acquire a notion of how a possible PMNS-like matrix present in the relevant charged current parametrization could affect the observables. We found that an interesting section of the space of parameters can be explored at current LHC in the specified conditions.


I. INTRODUCTION
The Large Hadron Collider (LHC) has reached an excited stage of large amount of data collection. Even though conclusive hints for new physics have been evasive, current data and the data that is expected to be collected at the high luminosity (HL) stage opens the possibility of turning the LHC into a precision machine.
In this paper we explore the prospects for the LHC to discover doubly-charged vector bileptons with current luminosity. Most of the processes considered to find new physics have SM background. This is not the case for the process pp → µ + µ + e − e − (e + e + µ − µ − ) [1], or the trimuon events pp → µ + µ + µ − e − , both of which violate lepton flavor number, which is conserved to all orders in perturbation theory within the SM. Hence, having no background, this sort of processes may be the smoking gun of new physics and, specifically, of the discovery of doubly charged particles which are, in many models, the biggest candidates to trigger these processes.
Doubly charged particles appear in multiple scenarios beyond the standard model (BSM) with extended gauge groups; they may be scalars, fermions or vectors. See Ref. [2] and references therein. Among the possibilities, the more interesting one is the case of doubly charged vector bosons, because i) their couplings with leptons have almost the same intensity as that of the W ± of the SM; ii) this kind of particle is a very rare feature in models of new physics. They occur, for example, in the minimal 3-3-1 model (m331 for short) [3][4][5], and in SU (15) grand unification [6,7].
If these sort of particles do exist, then resonances in like-sign leptons' invariant mass could be observed at the LHC [8] in the (sub)process U ++ → + + . Interesting cases are when pp → e + e + µ − µ − [1] and when pp → µ + µ + µ − µ − [9,10]. This paper is organized as follows: In Sec. II, we write down the interactions relevant for our analysis. In * mario.barela@unesp.br † v.pleitez@unesp.br Sec. III, we describe the method and present our results. Our conclusions are presented in Sec. IV.

II. INTERACTIONS
The lepton-lepton-bilepton interaction that is relevant to the present analysis is the following: For general A and B matrices, this is a model independent parametrization for vector and axial interactions. We will focus here on a large class of models in which the charged leptons mass matrix is diagonalized by a biunitary transformation given byM = V † L M V R , defining L,R = V L,R L,R andM = diag(m e , m µ , m τ ), where the primed fermions are symmetry eigenstates and the unprimed ones are mass eigenstates. In this scenario, A is an antisymmetric matrix given by In a model independent way, the significant production mechanism of the bileptons is through Drell-Yan-like processes. We will need the bilepton-bilepton-Z interaction, generally given by where f (g, v) is a dimensionless function of the gauge coupling constants and of the VEVs of the model. Distinctive possible values for f (g, v) include i) f (g, v) = 2c W , which is the corresponding value of the SM W + W − Z vertex and ii) f (g, v) = −(1−4s 2 W )/c W , which is the vertex of the m331 model in a possible SM limit [11] if we use g 2 where g X , g are the gauge coupling constants of U (1) X and SU (3) L , respectively, and s W , c W are the sine and cosine of the the weak angle. Below we will consider only the latter case. Notice that in the m331 model the vector bilepton if Z-phobic.

arXiv:1912.05900v1 [hep-ph] 12 Dec 2019
The last required lagrangian is that of the bileptonbilepton-photon interaction, which is: here Q e (g) is the expression for the fundamental charge within the considered model, which, in general, is a function of the coupling constants. In our calculations, we will put Q e (g) = gs W , which is the corresponding electrical charge of the SM and also of the m331 when using, again, g 2

III. LHC PHENOMENOLOGY
As mentioned before, we will focus on the phenomenology of the vector bileptons at the LHC energy and luminosity. We note that in all the previous analysis only a diagonal version of Eq. (1) has been considered and, consequently, also the trimuon final state case was not specifically studied. This may be a too restrictive imposition since, taking the m331 model as an illustration, it is not possible to assume that the charged lepton mass matrix is in the diagonal basis. This is because to generate the correct mass of these particles it is necessary to have two contributions arising from different scalar multiplets: a triplet η and a sextet S. The mass matrix coming from the Yukawa interactions between the leptons and the triplet is antisymmetric while that from the sextet is symmetric. If we choose only the symmetric matrix (forgetting the sextet), the neutrino mass matrix becomes proportional to the charged leptons one, so that they are diagonalized by the same transformations, which, in turn, causes the resulting PMNS matrix to be unity.
In turn, this theoretically (probably) inescapable mixing causes the study of this processes to be very difficult in consequence of the number of free parameters. For this reason we perform a study of the trimuon end-state with more general non-diagonal mixing matrices, considering only the contribution of the bilepton U ±± together with the SM particles. Of course, we stress that in this case the unitarity of the model is not manifest, but we already know possible ultraviolet completions, say [3,4] or [7]. Eventually, directed studies of specific models should take all contributions into account.

A. The method
In order to obtain exclusion contours in the twodimensional parameter space M U × (V U ) eµ , where V U is the unitary matrix introduced below Eq. (1), we study the process pp → e − µ − µ + µ + (and its charge mirrored end state conjugate), which, obviously, has no SM background and hence may be easily distinguished if it does happen at all in the current experimental reach. However, further simplifications are needed if we are to be able to perform this study without arbitrarily fixing unknown parameters.
We start by recalling that we are considering only contributions containing the U ±± and the SM particles. Diagrams are shown in Fig. 1.
At this point we would be left with 5 free parameters: the boson's mass and the 3 angles and one phase that parametrize a 3 × 3 unitary matrix -all the matrix elements influence the results, if not explicitly in the vertex, through the particle's width. The next obvious choice of simplification is to make the matrix real, eliminating in this way the phase. At last, we decide to study the case of a symmetric V U , ending up with 3 parameters to deal with: the mass and the two real numbers that are the degrees of freedom of a 3 × 3 symmetric orthogonal matrix (actually, there is still one more discrete degree of freedom that label one of four different solutions of the orthogonality conditions for the 4 others matrix elements in terms of two specified ones, but this is not an issue). We stress that these assumptions, although not more strong than necessary for any similar analysis in the present phenomenologic context, are not the most general case or a prediction of a specific model.
A consequence of imposing a symmetry condition on V U is that there are no vectorial interactions between the bilepton and the leptons, since, again, A = V U − V T U . Our goal is to learn what is the behaviour of the signal upon the variation of the parameters m U ±± and (V U ) eµ . To do so, we chose the third free parameter from the analysis above to be (V U ) ee , which we fix at four different benchmark values: (V U ) ee = {0.001, 0.01, 0.1, 0.9}. We then perform one bidimensional scan for each value of (V U ) ee , for values between 100 GeV ≤ m U ±± ≤ 1200 GeV 1500 GeV > p T > 30 GeV, |η | < 2.5, ∆R > 0. 4 (4) at a center of mass energy √ s = 13 TeV and an integrated luminosity of L = 140 fb −1 . The resulting total cross section of each point is multiplied by two, to accommodate the charge reversed end-state pp > e + µ + µ − µ − which has identical numerical results in our case of (V U ) ij = (V U ) ji and is experimentally distinguishable from the original process in principle.

B. Results
The results are presented in figure 2 and 3. The y-axis is rescaled by a square-root function for better readability of the smaller values of (V U ) eµ . Since there is no background, we present directly contours of number of events instead of confidence level. The shown contours in Fig. 2 refer to the occurrence of 3 events, so that the region to the left of the curves may, in the respective case, be eliminated with a confidence level of 95% (in a Poisson statistic basis and without systematic uncertainties).
We observe that the contour is roughly identical for all (V U ) ee , which happens because the orthogonality constraints obligate the matrix elements to conspire in such a way that when, for fixed (V U ) eµ , (V U ) µµ increases, making the numerator of the cross section larger, the width decreases (roughly) exactly the right amount to make it stay the same. We see that the highest mass that may be eliminated in the specified conditions is of ∼1100 GeV, for (V U ) eµ ∼ 0.52.

IV. CONCLUSIONS
We see that at √ s = 13 TeV and L = 140 fb −1 , the trimuon end-state may be adequate to explore the possibility of doubly charged vectors with masses up to the TeV scale in favorable cases.
By adding other contributions due to the other particles of a given model, unless a fine tuned negative interference happens, our result that a vector bilepton can be observed at the LHC should not be affected. In fact, this seems to be the case since works that take into account, for instance, the leptoquarks of the m331 obtain upper bounds similar to those obtained in this paper, which suggests that the s-channel U ±± exchange is indeed the most important contribution -as some works explicitly state.
Of course, there might be processes that could, theoretically, impose lower limits that are higher than those obtained in this work for the mass of the doubly charged bileptons. One example we have noted is the purely leptonic µ → eēe decay, which, by an analytical calculation, we observe to be able to explore masses up to 5 TeV if the vector bilepton contribution to that process is the predominant [12]. However, in the mentioned reference a heavy sextet was assumed and the only degrees of freedom active at low energies were the three scalar triplets. In contrast, if the degrees of freedom of the scalar sextet are active at low energy light (neutral or doubly charged) scalar may induce large contributions to the µ → eēe decay, possibly relieving the lower limit for the mass of the boson U .
Concerning the trimuon events, we recall that many years ago this kind of process was apparently observed in several experiments using neutrino-nucleon scattering [13][14][15]. At that time it was difficult to accommo-date these events in electroweak models with a SU (2) ⊗ U (1) symmetry, but not in those with a SU (3) × U (1) one [16,17]. However, further experiment do not confirm the existence of these events with neutrino energies larger than 100 GeV [18]. If they do occur in nature, perhaps they could be observed at the LHC.
The objective of the present paper was to study the contribution of the vector bilepton alone to a hadronic process with lepton flavor violation, so that we could, also, make a more skeptical assessment of the unavoidable mixing matrix. Nevertheless, we emphasize that a given closed model that contain such bileptons could accommodate a great number of still free parameters, which makes a truly skeptical phenomenological analysis very complicated and that, eventually, a more detailed study, including more degrees of freedom of each said model, cannot be avoided.
We conclude that it is worth to continue studying such processes in a model dependent way to see, among other things, if, as we expect, there is no negative interference that can suppress the contribution of the vector bilepton U ±± and we urge LHC to search for this sort of resonance in the context of such models.