Physics Letters B 785 (2018) 73–83

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Physics Letters B

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Exploring scalar and vector bileptons at the LHC in a 331 model ∗ Gennaro Corcella a, , Claudio Corianò b, Antonio Costantini b, Paul H. Frampton b a INFN, Laboratori Nazionali di Frascati, Via E. Fermi 40, 00044, Frascati RM, Italy b Dipartimento di Matematica e Fisica ‘Ennio De Giorgi’, Università del Salento and INFN, Sezione di Lecce, Via Arnesano, 73100 Lecce, Italy a r t i c l e i n f o a b s t r a c t

Article history: We present an analysis on the production of two same-sign pairs at the LHC, mediated by Received 14 June 2018 bileptons in the SU(3)c × SU(3)L × U (1)X theory, the so-called 331 model. Compared to other 331 Received in revised form 7 August 2018 scenarios, in this model the embedding of the hypercharge is obtained with the addition of 3 exotic Accepted 10 August 2018 ±± and doubly-charged vector gauge bosons with lepton numbers ±2in the spectrum (Y ), which Available online 16 August 2018 can mediate the production of four-lepton final states. Furthermore, a complete description of the model Editor: A. Ringwald requires the introduction of a Higgs scalar sector, which is a sextet of SU(3)L , necessary in order to ±± correctly account for the lepton masses. As a result of this, new doubly-charged scalar states H are part of the spectrum as well and can in principle compete with the vector bileptons in giving rise to four-lepton final states. We investigate both channels and study several observables at the LHC, for both signal and ZZ background. With respect to previous work on vector- and scalar-bilepton production, we use the most updated exclusion limits at the LHC and implement the 331 model in a full Monte Carlo simulation code, capable of being interfaced with any analysis framework. Our result is that the 331 signal can be discriminated from the background with a significance of 6–9 standard deviations, depending on the LHC luminosity, with the vector bileptons dominating over the scalar ones. © 2018 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3.

1. Introduction ternative paths of exploration. Such is the case of a version of the SU(3)c × SU(3)L × U (1)X model, also known as 331 model In the experimental searches for new physics of the fundamen- [1–3], where the requirement that the gauge couplings are real tal interactions at the LHC, the resolution of several open issues – provides a significant upper bound on the physical region in which which remain unanswered within the Standard Model (SM) – typ- its signal could be searched for. This property sets the vacuum ically call for larger gauge structures and a wider particle content. expectation values (vevs) of the Higgs bosons, which trigger the Such are the gauge-hierarchy problem in the Higgs sector or the breaking from the 331-symmetry scale down to the electroweak origin of light-neutrino masses, just to mention two of them, both one, around the TeV. The model that we consider allows for bilep- −− ++ requiring such extensions. Grand Unified Theories (GUTs) play cer- tons, i.e. gauge bosons (Y , Y ) of charge Q =±2 and lepton tainly an important role in this: unfortunately, Grand Unification number L =±2, and therefore we shall refer to it as the bilep- 12 15 needs an energy scale O(10 –10 ) GeV, which is far higher than ton model. In the family of 331 models, bileptons in the spectrum the electroweak one, currently probed at the LHC. Obviously, the are obtained only for special embeddings of the U (1)X symmetry process of identifying the main signatures of a certain symmetry- and of the charge (Q ) and hypercharge (Y ) generators in the local breaking pattern from the GUT scale down to the TeV scale is gauge structure. One additional feature of the model is that, un- far from being trivial, due to the enlarged symmetries and to a like the Standard Model or most chiral models formulated so far, parameter space of considerable size. There are, however, interest- extending the SM spectrum and symmetries, the number of (chi- ing scenarios where larger gauge symmetries can be discovered ral) fermion generations is underwritten by the cancellation of the or ruled out already at the TeV scale, which suggests some al- gauge anomalies. Gauge anomalies cancel among different fermion families, and select the number of generations to be 3: from this perspective, the model appears to be quite unique. Moreover, in Corresponding author. * the formulation of [1], which we shall adopt hereafter, the third E-mail addresses: [email protected] (G. Corcella), [email protected] (C. Corianò), [email protected] (A. Costantini), fermion family is treated asymmetrically with respect to the first [email protected] (P.H. Frampton). two families. https://doi.org/10.1016/j.physletb.2018.08.015 0370-2693/© 2018 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3. 74 G. Corcella et al. / Physics Letters B 785 (2018) 73–83

In a previous analysis [4]we have presented results for the pro- in a minimal 331 model were also studied in [12]. It was found ++ −− duction of pairs of vector bileptons (Y Y ) decaying into two that, since the coupling to is proportional to the lepton same-sign√ lepton pairs, in conjunction with two jets at the LHC, mass, such scalar bileptons mostly decay into τ -lepton pairs. Also, at s = 13 TeV and relying on a full Monte Carlo implementation according to [12], the rate into WW pairs is suppressed, being of the 331 model. Our study has been based on the selection of proportional to the vacuum expectation value of the Higgs giving a specific benchmark point in the parameter space, where the Y mass to the neutrinos, as well as decays into leptons of different bileptons have mass mY  875 GeV. We have shown that, by set- flavours, since the Yukawa couplings are diagonal. While the inves- ting appropriate cuts on the rapidities and transverse momenta of tigation in [11]was performed at leading order, by using the FORM the final-state leptons and jets, it is possible to suppress the Stan- package [13]to calculate the amplitudes, we shall undertake a full dard Model background. By including jets in the final state, one hadron-level investigation. We will implement the 331 model, in- obtains larger signal/background ratio, but nonetheless one has to cluding the sextet sector, into SARAH 4.9.3 [14], while the am- face the issue of jet reconstruction. plitudes for bilepton production at the LHC will be computed by In this paper, we wish to extend the investigation in [4]. In fact, the MadGraph code [15]; the simulation of parton showers and in [4]the explicit mass matrices of the scalars in a minimal ver- hadronization will be performed by using HERWIG 6 [16]. Also, as sion of the model, containing only 3 SU(3)L triplets, as well as the will be thoroughly debated later on, in our model doubly-charged minimization conditions of the potential, were presented. How- scalar bileptons decay in all lepton-flavour pairs with branching ra- + − ever, the model also allows for doubly-charged Higgs-like scalars tios 1/3, unlike Ref. [12], wherein the τ τ mode had the largest ++ −− (H , H ), which may give rise to multi-lepton final state, in the rate. ±± same manner as the vectors Y debated in [4]. We shall there- From the experimental viewpoint, to our knowledge, there has fore explore the production of same-sign lepton pairs at the LHC, been no actual search for vector bileptons at the LHC, whereas the mediated by both scalar and vector bileptons; unlike Ref. [4], we latest investigations on possible doubly-charged scalar Higgs boson shall veto final-state jets. production at the LHC were undertaken in [17] and [18]by ATLAS The production of doubly-charged vector bilepton pairs at the and CMS, respectively. In detail, the ATLAS analysis, performed at − LHC in jetless Drell–Yan processes was already investigated in 13 TeV and with 36 fb 1 of data, considered the so-called left-right [5–8]. In particular, the authors of [8]implemented the bilepton symmetric model (LRSM, see, e.g. Refs. [19,20]) and its numeri- model in a full Monte Carlo simulation framework and obtained ±± cal implementation in [21], where doubly-charged Higgs bosons √the exclusion limit mY > 850 GeV, by using the ATLAS rates at can couple to either left-handed or right-handed leptons. In this s = 7TeV [9]on expected and observed high-mass same-sign ±± − framework, assuming that the H bosons only decay into lepton lepton pairs and extending the results to 13 TeV and L = 50 fb 1. ±± pairs, exclusion limits were set in the range 770–870 GeV for H L However, the analysis in [8] assumes that the same-sign lepton- ±± −1 and 660–760 GeV for H . As for CMS, a luminosity of 12.9fb pair yields are independent of the bilepton spin: in fact, the AT- R was taken into account and limits between 800 and 820 were de- LAS analysis in [9]was performed for scalar doubly-charged Higgs termined, always under the assumption of a 100% branching ratio bosons, while the predictions in [8] concerned vector bileptons. into same-sign lepton pairs. It is precisely the goal of the present exploration understanding Our paper is organized as follows. In Section 2, we shall discuss whether, referring to the 331 realization in [1], one can sepa- the family embedding in the minimal 331 model, while Section 3 rate vector from scalar bileptons at the LHC, in different luminos- will be more specific on its scalar content, giving details on the ity regimes. A careful investigation on the production of doubly- triplet and sextet sectors, as well as on the lepton masses and charged particles at the LHC and the dependence of final-state physical Higgs bosons. Our phenomenological analysis will be pre- distributions on the spin was undertaken in [10]. The authors of sented in Section 4 and final comments and remarks will be given [10] considered an effective simplified model where the SM is in Section 5. extended by means of a SU(2)L group and the scalar, fermion and vector doubly-charged particles lie in the trivial, fundamental 2. Family embedding in the minimal 331 and adjoint representations of SU(2)L , respectively. By looking at transverse-momentum and angular distributions and varying the bilepton mass between 150 and 350 GeV, it was found that it could One of the main reasons for the appearance of exotic particles be possible to distinguish at the LHC the particle spin. in the spectrum of the 331 model is the specific embedding of Compared to our previous study, in the present paper we shall the hypercharge Y in the SU(3)L × U (1)X gauge symmetry. The explore a complete version of the model which includes both the embeddings of Y and of the charge operator Q em are obtained by SU(3)L triplet Higgses and the newly added scalar sextet sec- defining them as linear combinations of the diagonal generators of tor, which is necessary in order to account for the masses of SU(3)L . We recall that in the 331 case this is defined by the leptons. The inclusion of the scalar sextet opens up the de- ±± ± ± cay channels H → l l , which compete with the companion ±± ± ± Y3 = βT8 + X1 Y ¯ =−βT8 + X1 (1) Y → l l process. Many of the analytic expressions in the de- 3 scription of the sextet contributions, such as the rotation matrices ¯ for the 3 and the 3 representations of SU(3)L , respectively, with appearing in the extraction of the mass eigenstates of this sector, generators Ti = λi/2(i =1, ...8), corresponding to the Gell-Mann cannot be given in closed analytical form, since they would be too √1 matrices, and T8 = diag (1, 1, −2) . The charge operator is lengthy. As in [4]we shall choose a benchmark point of the model 2 3 and present our results numerically: in particular, as we are inter- given by ested in comparing vector- and scalar-bilepton signals, we shall set ++ ++ = + = − the doubly-charged Y and H masses to the same value. Q em,3 Y3 T3 Q em,3¯ Y3¯ T3 (2) Vector- and scalar-bilepton production at hadron colliders was also explored in [11], where the authors presented the total cross in the fundamental and anti-fundamental representations of = 1 − sections, expected number of events at the LHC, invariant-mass SU(3)L , respectively, where we have T3 diag 2 (1, 1, 0) . We and transverse-momentum spectra for a few values of the bilep- choose the SU(2)L × U (1)Y hypercharge assignments of the Stan- ton mass. The decay properties of doubly-charged Higgs bosons dard Model as Y (Q L ) = 1/6, Y (L) =−1/2, Y (uR ) = 2/3, Y (dR ) = G. Corcella et al. / Physics Letters B 785 (2018) 73–83 75

¯ −1/3 and Y (eR ) =−1. Denoting by q X the particle charges un- The lepton sector is assigned to the representation 3of the same der U (1)X , the breaking of the symmetry SU(3)L × U (1)X → gauge group. Conversely from the sector, there is a demo- SU(2)L × U (1)Y for the fundamental representation 3 reads cratic arrangement of the three lepton generations into triplets of SU(3)L , β β ⎛ ⎞ (3, q X ) → 2, √ + q X + 1, −√ + q X , (3) 2 3 3 eL = ⎝ ⎠ ∈ ¯ ¯ l νL , l (1, 3, 0), (16) while for the representation 3 c eR β β ¯  → − √ +  + + √ +  c = ∗ (3, q X ) 2, q X 1, q X . (4) with eR iσ2eR . In the following, we shall adopt for the leptons 2 3 3 i the notation la, where the subscripts (a, b or c) refer to the lep- The X-charge is fixed by the condition that the first two compo- ton generation (electrons, muons and taus), and the superscripts nents of the Q 1 and Q 2 triplets carry the same hypercharge as the (i, j, k = 1, 2, 3) are SU(3)L indices. For example, the generation a quark doublets Q L = (u, d)L of the Standard Model, yielding corresponds to electrons and the three elements of the triplet (16) 1 β are labelled as: q X = − √ . (5) 6 2 3 1 = 2 = 3 = c la eaL, la νaL, la eaR. (17) The U (1) charge of the triplet will then be Q (Q ) = diag(2/3, em √ em 1 For the hypercharge operator we have the decomposition under −1/3, 1/6 − 3β/2). Fermions with √exotic charges will be auto- SU(2)L × U (1)Y matically present in the case of β = 3, which is the parameter choice that we will consider hereafter in our analysis. The first two = − √β +  − √β +  √β +  Y3¯ (l) q X , q X , q X (18) families will then be assigned as 2 3 2 3 3 ⎛ ⎞ ⎛ ⎞ √  uL cL with q = 1/6 + β/(2 3) and Q (L) = diag(−1, 0, 1). Both left- = ⎝ ⎠ = ⎝ ⎠ ∈ − X em Q 1 dL , Q 2 sL , Q 1,2 (3, 3, 1/3) (6) and right-handed components of the SM leptons are fitted into the D L S L same SU(3)L multiplet. The scalars of the 331 model, responsible for the electroweak under SU(3)c × SU(3)L × U (1)X . The charge operator Q em on Q 1,2 symmetry breaking (EWSB), come in three triplets of SU(3)L : will then give ⎛ ⎞ ⎛ ⎞ ρ++ η+ Q (Q ) = diag(2/3, −1/3, −4/3), (7) + em 1,2 ρ = ⎝ ρ ⎠ ∈ (1, 3, 1), η = ⎝ η0 ⎠ ∈ (1, 3, 0), − with two exotic quarks D and S of charge −4/3. The third family ρ0 η ⎛ ⎞ (19) is instead assigned as 0 ⎛ ⎞ χ = ⎝ − ⎠ ∈ − bL χ χ (1, 3, 1). ⎝ ⎠ ¯ −− Q 3 = tL , Q 3 ∈ (3, 3, 2/3), (8) χ T L The breaking SU(3)L × U (1)X → U (1)em is obtained in two where the hypercharge content of the third exotic quark (T L ) is steps. The vacuum expectation value of the neutral component of ρ causes the breaking from SU(3) × U (1) to SU(2) × derived from the operator (Y3¯ ) L X L U (1)Y ; the usual spontaneous symmetry breaking mechanism β β β × = − √ +  − √ +  √ +  from SU(2)L U (1)Y to U (1)em is then obtained through the vevs Y3¯ (Q 3) diag q X , q X , q X , (9) 2 3 2 3 3 of the neutral components of η and χ . √  Before closing this section, we remind that in the models [1,2] with the condition q = 1 6 + 2 3 , giving X / β/( ) the coupling constants of SU(3) and U (1) , namely g and g , L X 3L X are related to the electroweak mixing angle θW in such a way that 1 1 5 2 Y ¯ (Q 3) = diag , , (10) g (μ) exhibits a Landau pole at a scale μ whenever sin θ (μ) = 3 6 6 3 X W 1/4[24].1 Therefore, the theory may lose its perturbativity even at and a scale about 3.5 TeV [25]. Nevertheless, the typical energy scale 1 2 5 of bilepton-pair production in [4] and in this paper is somewhat Q ¯ (Q 3) = diag − , , . (11) smaller, i.e. 2m ±±  1.75 TeV. Therefore, we shall assume that the em 3 3 3 3 Y Landau pole does not pose any threat to the perturbative analysis With these assignments, the charge of T L is Q em(T L ) = 5/3, allow- carried out in the present work. ing to distinguish between the third and the first two generations of quarks. Right-handed singlet quarks in the 331 model carry 3. The scalar sectors the usual SM charges (2/3 and −1/3for the u-type and d-type quarks), The model presented in the previous section exhibits the in- teresting feature of having both scalar and vector doubly-charged (dR , sR , bR ) ∈ (3, 1, −1/3) (12) bosons, which is a peculiarity of the minimal version of the 331 model. In fact it is possible to consider various versions of the (uR , cR ,tR ) ∈ (3, 1, 2/3), (13) SU(3)c × SU(3)L × U (1)X gauge symmetry,√ usually parametrized with the exception of the three right-handed exotics by β [22]. We discuss the case of β = 3, corresponding to the

(D R , S R ) ∈ (3, 1, −4/3) (14) 1 2 2 = The relation between the SU(3)L and U (1)X couplings reads: g X /g3L T R ∈ (3, 1, 5/3). (15) 2 2 sin θW /(1 − 4 sin θW ) [24,25]. 76 G. Corcella et al. / Physics Letters B 785 (2018) 73–83

= i · j ∗k ijk minimal version presented here, leading to vector boson with elec- Mab (la lb)η (23) tric charge equal to ±2. Doubly-charged states are interesting by themselves because they can have distinctive features in terms of is therefore antisymmetric under the exchange of the two flavours, implying that even Gab has to be antisymmetric. However, an allowed decay channels, for example the production of same-sign η antisymmetric G is not sufficient to provide mass to all lep- lepton pairs. In the context of the minimal 331 model there is an ab even more interesting possibility. In fact, one can test whether tons. η a same-sign lepton pair has been produced by either a scalar In fact, the diagonalization of G by means a unitary matrix η = † η or a vector boson. As we are going to explain, this feature will U , namely G U U , with G antisymmetric in flavour space, = also shed light on the presence of a higher representation of the implies that its 3 eigenvalues are given by (0, λ22, λ33), with =− SU(3)c × SU(3)L × U (1)X gauge group, namely the sextet. λ22 λ33, i.e. one eigenvalue is null and the other two are equal in magnitude. At the minimum of η, i.e. η = (0, vη, 0), one has: 3.1. The triplet sector η G Mab =−Tr( UMU†) ab (24) In the previous section we have seen that the EWSB mechanism 1 3 ∗ 1 3 ∗ = 2vηλ22 U2a l · l U + 2vηλ33 U3a l · l U , is realised in the 331 model by giving a vev to the neutral com- a b 2b a b 3b ponent of the triplets ρ, η and χ . The Yukawa interactions for SM 1 = 3 = c with la eaL and lb ebR. Introducing the linear combinations and exotic quarks are obtained by means of these scalar fields and ≡ 1 = are given by: E2L U2a la U2a eaL ∗ ∗ ∗ (25) ∗ ∗ ∗ 3 = c = ≡ c LYuk. = 1 + 2 + 3 U2b lb U2b ebR iσ2(U2b ebR) E2R , q,triplet yd Q 1η dR yd Q 2η sR yd Q 3χ bR + 1 ∗ ∗ + 2 ∗ ∗ + 3 ∗ then the antisymmetric contribution in flavour space becomes yu Q 1χ uR yu Q 2χ cR yu Q 3η tR (20)

1 ∗ ∗ 2 ∗ ∗ 3 ∗ Yuk c c + y Q 1 ρ D + y Q 2 ρ S + y Q 3 ρ T + h.c., L = − E R E R E R l, triplet 2vηλ22 E2L E2R E3L E3R , (26) i i i where yd, yu and yE are the Yukawa couplings for down-, up-type which is clearly insufficient to generate the lepton masses of three and exotic quarks, respectively. The masses of the exotic quarks are non-degenerate lepton families. We shall solve this problem by related to the vev of the neutral component of ρ = (0, 0, vρ ) via introducing a second invariant operator, with the inclusion of a the invariants sextet σ : ⎛ √ √ ⎞ ∗ ∗ ∗ ∗ ++ + Q D , Q S ∼ (3, 3, −1/3) × (1, 3¯, −1) × (3¯, 1, 4/3) σ σ / 2 σ 0/ 2 1 ρ R 1 ρ R 1 √ 1 √ ⎜ + − ⎟ ∗ ∼ ¯ × × ¯ − σ = ⎝ σ / 2 σ 0 σ / 2 ⎠ ∈ (1, 6, 0), (27) Q 3 ρT R (3, 3, 2/3) (1, 3, 1) (3, 1, 5/3), (21) 1 √ 1√ 2 − −− σ 0/ 2 σ / 2 σ responsible of the breaking SU(3)c × SU(3)L × U (1)X → SU(3)c × 2 2 × SU(2)L U (1)Y . It is clear that, being vρ vη,χ , the masses of the leading to the Yukawa term O i ∼ exotic quarks are (TeV) whenever the relation yE 1is satisfied. LYuk. = σ i · j ∗ l,sextet Gabla lbσi, j, (28) 3.2. The sextet sector which allows to build a singlet out of the representation 6of The need for introducing a sextet sector can be summarised i · j SU(3)L , contained in la lb, by combining it with the flavour- ∗ as follows. A typical Dirac mass term for the leptons in the SM symmetric σ , i.e. 6.¯ Notice that Gσ is symmetric in flavour ¯ ab is associated with the operator lL HeR , with lL = (veL, eL ) being space. ¯ the SU(2)L doublet, with the representation content (2, 1/2) × It is interesting to note that if one did not consider the sextet (2, 1/2) × (1, −1) (for l, H and eR , respectively) in SU(2)L × U (1)Y . it would not be possible for a doubly-charged scalar to decay into In the 331 the L and R components of the lepton (e) are in the same-sign leptons. In fact, if we leave aside the sextet contribu- same multiplet and therefore the identification of an SO(1, 3) × tion, the Yukawa for the leptons is related to the scalar triplet η SU(3) singlet needs two leptons in the same representation. It which does not possess any doubly-charged state. This means that L ±± → ± ± can be obtained (at least in part) with the operator revealing a possible decay H l l would be a distinctive sig- nature of the presence of the sextet representation in the context LYuk = η i αβ j ∗k ijk + of the 331 model. l, triplet Gab(laα lbβ )η h.c. = η i · j ∗k ijk + Gab la lb η h.c. (22) 3.3. Lepton mass matrices where the indices a and b run over the three generations of The lepton mass matrices are of course related to the Yukawa flavour, α and β are Weyl indices contracted in order to generate i · j ≡ i αβ j interactions by the Lagrangian an SO(1, 3) invariant (la lb laα lbβ ) from two Weyl fermions, and i, j, k = 1, 2, 3, are SU(3) indices. The use of η as a Higgs L LYuk. = LYuk. + LYuk. + h.c. (29) field is mandatory, since the components of the multiplet l j are l l,sextet l,triplet i j U (1)X singlets. The representation content of the operator l l ac- and are combinations of triplet and sextet contributions. The struc- ¯ ¯ a b cording to SU(3)L is given by 3 ×3 = 6 +3, with the 3extracted by ture of the mass matrix that emerges from the vevs of the neutral an anti-symmetrization over i and j via ijk. This allows to identify components of η and σ is thus given by: i j ∗k ijk l l η as an SU(3)L singlet. Considering that the two leptons √  a b η are anticommuting Weyl spinors, and that the αβ (Lorentz) and LYuk. = σ + · c l 2σ0Ga,b 2vηGab (eaL ebR) ijk (SU(3)L ) contractions introduce two sign flips under the a ↔ b   (30) + 0 σ T + exchange, the combination σ1 Gab νL iσ2νL h.c., G. Corcella et al. / Physics Letters B 785 (2018) 73–83 77

l which generates a Dirac mass matrix for the charged leptons Mab 3.4. Physical Higgs bosons νl and a Majorana mass matrix for neutrinos Mab : √ The inclusion of the sextet representation in the potential en- l = σ + η νl = 0 σ riches the phenomenology of the model and enlarges the num- Mab 2 σ0 Ga,b 2vη Gab , Mab σ1 Gab. (31) ber of physical states in the spectrum. In fact we now have, af- 0 0 ter electroweak symmetry breaking (EWSB) SU(3) × U (1) → In the expression above σ and σ1 are the vacuum expectation L X σ values of the neutral components of σ . For a vanishing G , as we SU(2)L × U (1)Y → U (1)em, five scalar Higgses, three pseudoscalar have already discussed, we will not be able to generate the lepton Higgses, four charged Higgses and three doubly-charged Higgses. masses consistently, nor any mass for the neutrinos, i.e. The (lepton-number conserving) potential of the model is given by [23] η l = νl = Mab 2vη Gab , M 0. (32) † † † † 2 † 2 V = m1 ρ ρ + m2 η η + m3 χ χ + λ1(ρ ρ) + λ2(η η) (35) On the contrary, in the limit Gη → 0, Eq. (31) becomes † 2 † † † † + λ3(χ χ) + λ12ρ ρη η + λ13ρ ρχ χ + † † + † † + † † + † † √ σ 0 λ23η ηχ χ ζ12ρ ηη ρ ζ13ρ χχ ρ ζ23η χχ η l = σ νl = √1 σ Mab 2 σ0 Gab , Ma,b Gab, (33) † † 2 † † 2 + m4 Tr(σ σ ) + λ4(Tr(σ σ )) + λ14ρ ρ Tr(σ σ ) + † † + † † + † † which has some interesting consequences. Since the Yukawa cou- λ24η η Tr(σ σ ) λ34χ χ Tr(σ σ ) λ44 Tr(σ σσ σ ) plings are the same for both leptons and neutrinos, we have to † † † † † † + ζ14ρ σσ ρ + ζ24η σσ η + ζ34χ σσ χ require σ 0  σ 0 , in order to obtain small neutrino masses. √ √ 1 ijk T † 0 + ( 2 fρηχ ρi η j χk + 2 fρσχρ σ χ For the goal of our analysis, we will assume that the vev of σ1 0 ≡ σ ijk ∗l ijk lmn vanishes, i.e. σ1 0. Clearly, if the matrix G is diagonal in + ξ14 ρ σliρ jηk + ξ24 ηiηlσ jmσkn flavour space, from Eq. (33)we will immediately conclude that ∗ + ijk l + h.c. the Yukawa coupling Gσ has to be chosen to be proportional to ξ34 χ σliχ jηk) the masses of the SM leptons. An interesting consequence of this The EWSB mechanism will cause a mixing among the Higgs ±± → ± ± σ is that the decay H l l , which is also proportional to G , fields; from Eq. (35)it is possible to obtain the explicit expres- and therefore to the lepton masses, will be enhanced for the heav- sions of the mass matrices of the scalar, pseudoscalar, charged and ier leptons, in particular for the , as thoroughly discussed in [12]. τ doubly-charged Higgses, by using standard procedures. In the bro- This is an almost unique situation which is not encountered in ken Higgs phase, the minimization conditions other models with doubly-charged scalars decaying into same-sign leptons [21]. ∂ V 0 = 0, φ =vφ,φ= ρ, η, χ, σ (36) However, for the sake of generality, in the following analysis we ∂ vφ will consider the most generic scenario where both contributions will define the tree-level vacuum. We remind that we are consid- Gσ and Gη are present. In this case the branching ratios of the ering massless neutrinos choosing the neutral field σ 0 to be inert. doubly-charged Higgs decaying into same-sign leptons do not have 1 The explicit expressions of the minimization conditions are then to be proportional to the masses of the lepton anymore. In partic- given by ular, after accounting for both Gσ and Gη, configurations wherein even scalar bileptons have the same rates into the three lepton 3 1 2 1 2 m1 vρ + λ1 v + λ12 vρ v − fρηχ vη vχ + λ13 vρ v (37) species are allowed, as occurs for vector bileptons. In the following, ρ 2 η 2 χ we shall hence concentrate our investigation on scenarios yielding 1 1 1 − √ + + 2 + 2 ξ14 vρ vη vσ fρσχ vχ vσ λ14 vρ vσ ζ14 vρ vσ 2 2 4 ±± → ± ±  ±± → ± ±  BR(Y l l ) BR(H l l ) 1/3 (34) = 0 for l = e, μ, τ . The condition in Eq. (34)is in fact particularly suit- 1 2 3 1 2 m2 vη + λ12 v vη + λ2 v − fρηχ vρ vχ + λ23 vη v (38) able to compare vector- and scalar-bilepton rates at the LHC and, 2 ρ η 2 χ for the time being, should be seen as part of our model. The as- 1 1 1 − √ 2 + √ 2 + 2 − 2 sumption of having equal branchings of 1/3 in the decay of the ξ14 vρ vσ vχ vσ λ24 vη vσ ξ24 vη vσ 2 2 2 2 2 scalar to all three lepton families allows to extend the universal- = ity of spin-1 bileptons also to the scalar sector, allowing to treat 0 the two states (scalar and vector) on a similar footing. This option 3 1 2 1 2 m3 vχ + λ3 v + λ13 v vχ − fρηχ vρ vη + λ23 v vχ (39) is clearly possible since we have 9 total parameters in the mass χ 2 ρ 2 η matrix which are constrained by 6 conditions. Three of them are 1 1 1 + √ + + 2 + 2 necessary in order to reproduce the lepton masses and the remain- ξ34 vη vχ vσ fρσχ vρ vσ λ34 vχ vσ ζ34 vχ vσ 2 2 4 ing three come from the requirement of having equal values of the = branching ratios of the scalars into the three lepton families. The 0 explicit expressions of the solutions of such conditions are very 1 2 3 1 3 m4 vσ + λ14 v vσ + λ44 v + λ4 v + fρσχ vρ vχ (40) involved and we have hence opted for a numerical scanning of 2 ρ σ 2 σ the mass matrices satisfying such requirements. This in general re- 1 1 1 1 − √ 2 + √ 2 + 2 + 2 quires that the ratio of the matrix elements of Gη over those of ξ14 vρ vη ξ34 vη vχ λ14 vρ vσ ζ14 vρ vσ −2 2 2 2 2 2 4 Gσ to be proportional to vσ /vη ∼ 10 . Of course, if possible ex- ± ± 1 1 1 perimental data deviated significantly from BR(l l ) = 1/3, then + 2 − 2 + 2 + 2 λ24 vη vσ ξ24 vη vσ λ34 vχ vσ ζ34 vχ vσ they would clearly favour a scalar bilepton, because lepton-flavour 2 2 4 universality is mandatory for vector bileptons. = 0 78 G. Corcella et al. / Physics Letters B 785 (2018) 73–83

Fig. 1. Typical contributions to events with two doubly-charged bosons in the final state and no extra jets. (a) and (b): contributions due to the mediation of a scalar (a) and a vector (b) boson. (c): t-channel exchange of exotic quarks Q .

1 These conditions are inserted into the tree-level mass matrices + = − + + − ∗ + − ∗ H W vηη vχ (χ ) vσ (σ2 ) , of the CP-even and CP-odd Higgs sectors, derived from Mij = NW   (46) ∂2 V /∂φ ∂φ , where V is the potential in Eq. (35): the explicit i j vev N = v2 + v2 + v2 ; expressions of the mass matrices are too cumbersome to be pre- W η χ σ sented here, although their calculation is rather straightforward. + = 1 + − − ∗ + + After a numerical diagonalization, we derive both the mass eigen- HY vρρ vη(η ) vσ σ1 , NY states and the Goldstone bosons.  (47) = 2 + 2 + 2 In this case we have 5 scalar Higgs bosons, one of them will NY vρ vη vσ . be the SM Higgs with mass about 125 GeV, along with 3 neutral pseudoscalar Higgs bosons, out of which 2 are the Goldstones of In particular, we are interested in the doubly-charged Higgses,  the Z and the Z massive vector bosons. In addition there are 6 where the number of physical states, after EWSB, is three, whereas charged Higgses, 2 of which are the charged Goldstones and 3 are we would have had only one physical doubly-charged Higgs if we doubly-charged Higgses, one of which is a Goldstone boson. The had not included the sextet. The physical doubly-charged Higgs Goldstones are exactly 8, as the massive vector bosons below the states are expressed in terms of the gauge eigenstates and the ele- electroweak scale. ments of the rotation matrix RC as Hereafter we shall give the schematic expression of the phys- ++ ++ −− ∗ ++ −− ∗ ical Higgs states, after EWSB, in terms of the gauge eigenstates, = 2C + 2C + 2C + 2C Hi Ri1 ρ Ri2 (χ ) Ri3 σ1 Ri4 (σ2 ) . (48) whose expressions contain only the vev of the various fields. In K ≡ K In particular, the structure of the corresponding Goldstone boson the following equations, Rij Rij(m1, m2, m3, λ1, λ2, ...) refers to the rotation matrix of each Higgs sector that depends on all the is  parameters of the potential in Eq. (35). Starting from the scalar √ ++ = 1 − ++ + −− ∗ − ++ (CP-even) Higgs bosons we have H0 vρρ vχ (χ ) 2vσ σ1 N√  (49) + −− ∗ = S 0 + S 0 + S 0 + S 0 + S 0 2vσ (σ2 ) , Hi Ri1Re ρ Ri2Re η Ri3Re χ Ri4Re σ Ri5Re σ1 , (41)  = 2 + 2 + 2 where N vρ vχ 4vσ is a normalization factor. expressed in terms of the rotation matrix of the scalar compo- S nents R . There are similar expressions for the pseudoscalars ±± ±± 3.5. Vertices for H and Y = P 0 + P 0 + P 0 + P 0 + P 0 Ai Ri1Im ρ Ri2Im η Ri3Im χ Ri4Im σ Ri5Im σ1 , In Fig. 1 we present the typical contributions to the partonic (42) ++ −− ±± cross section of the process pp → B B , where B denotes ±± in terms of the rotation matrix of the pseudoscalar compo- either a spin-0 or a spin-1 bilepton; each B decays into a same- nents RP . Here, however, we have two Goldstone bosons responsi- sign lepton pair. From Fig. 1, we learn that bilepton pairs can be ble for the generation of the masses of the neutral gauge bosons Z produced in Drell–Yan processes mediated by either a vector bo-  0 =  ··· and Z given by son (V γ , Z, Z ) or a scalar neutral Higgs (h1 h5); moreover,   their production can be mediated by the exchange of an exotic 1 1 = 0 − 0 + 0 quark Q in the t-channel as well. In principle, we may even have A0 vρIm ρ vηIm η vσ Im σ , N1 (43) BB production via an effective vertex in gluon-gluon fusion, but this contribution turned out to be negligible with respect to the N = v2 + v2 + v2 ; 1 ρ η σ subprocesses with initial-state quarks.    1 In the following, we wish to discuss the differences between 2 = − 0 + 0 = 2 + 2 A0 vρIm ρ vχ Im χ , N2 vρ vχ . (44) the couplings of scalar Higgses and vector bosons to scalar and N2 vector bileptons, as the production rates at the LHC crucially de- For the charged Higgs bosons the interaction eigenstates are ±± pend on such couplings. Considering first the case of vector Y , + + − ∗ + − ∗ + 0 1 ++ 2 −− 3 H = RC ρ + RC (η ) + RC η + RC (χ ) + RC σ the Lorentz structure of the V (p )Y (p )Y (p ) vertex is i i1 i2 i3 i4 i5 1 (45) + C − ∗ given in terms of the momenta by Ri6(σ2 ) C 1 2 3 = 2 − 1 + 3 − 2 + 1 − 3 with R being a rotation matrix of the charged sector. Even in this V (pμ, pν, pρ) gμν(pρ pρ) gνρ (pμ pμ) gμρ (pν pν). case we have two Goldstones because in the 331 model there are ± ± (50) the W and the Y gauge bosons. The explicit expressions of the  Goldstones are Characterizing the vector boson V as photon, Z or Z , we obtain: G. Corcella et al. / Physics Letters B 785 (2018) 73–83 79 ++ −− ++ −− γ Y Y − √i μ γα Y Y =−2ig2 sin θW V (pα , p , p ) −− g2γ P L μ ν μ ν dT¯ Y = 2 (56) i ++ −− 0 P ++ −− = − Z Y Y R Zα Yμ Yν g2(1 2cos2θW ) sin θW V (pα, pμ , pν ) 2 √i μ −− g2γ P L (51) DuY¯ = 2 (57)  0 P i  ++ −− R  ++ −− =− − 2 Z Y Y   Zα Yμ Yν g2 12 9sec θW V (pα , pμ , pν ), 2 ++ −− i 2 S S hi Y Y = g vρR + vχ R (58) 2 2 i1 i3 where θ is the Weinberg angle. W ++ −− =− In the case of the doubly-charged Higgs boson, the situation γ Y Y 2ig2 sin θW (59) ++ −− is slightly different: in fact, the interaction V 0 H H is gen- ++ −− i ZY Y = g2 (1 − 2cos2θW ) sec θW (60) erated after that the Higgses take a vev. The Lorentz structure of 2  the coupling will be of course proportional to the difference of the i  ++ −− =− − 2 1 2 = 1 − 2 Z Y Y g2 12 9sec θW . (61) momenta of the Higgs fields. Defining S pμ, pμ pμ pμ, we 2 have In the equations above, P L,R are the usual left- and right-handed = ∓ ++ −− projectors P L,R (1 γ5)/2. γα Hi H j     4. Phenomenological analysis at the LHC =− + 2 − 2C 2C + 2C 2C i sin θW g2 g1 cot θW 3 Ri1 R j1 Ri2 R j2   ++ −− In this section we wish to present a phenomenological analysis, H H + 2C 2C + 2C 2C i j aiming at exploring possible scalar- or vector-bilepton signals at 2g2 Ri3 R j3 Ri4 R j4 S pα , pα the LHC. ++ −− H H As in our previous work, we choose a specific benchmark point, =− i j 2ieδijS pα , pα (52) obtained after scanning the parameter space by employing the ++ −− SARAH 4.9.3 program and its UFO [28]interface. In doing so, we Zα Hi H j make use of the analytical expressions of the mass matrices and     i of the minimization conditions of the potential in Eqs. (37)–(40). 2 = sec θW cos 2θW g2 + g1 cot θW − 3 The mass eigenvalues are computed numerically, after varying all 2  the quartic couplings in Eq. (35) between −1 and 1 and the vac- − 2 − 2C 2C uum expectation value v , responsible of the first breaking of the g1 cot θW 3)Ri1 R j1 ρ    331 model, between 2 and 4TeV. Possible benchmark points are − + 2 − 2 2C 2C 2 g2 g1 cot θW 3 sin θW Ri2 R j2 then chosen in such a way that all particle masses are positive, the SM-like Higgs boson has mass at about 125 GeV and one is 2C 2C − g2 cos 2θW R R consistent with the current LHC exclusion limits on new physics i3 j3  ++ −− scenarios. Moreover, as discussed in [4], we require the couplings H H + 2 2C 2C i j 2g2 sin θW Ri4 R j4 S pα , pα (53) of the lightest neutral Higgs boson to the Standard Model fermions and bosons to be the same as in the Standard Model, within 10%  ++ −− accuracy. For the sake of comparison, we also choose the doubly- Zα Hi H j   charged vectors and scalars to have roughly the same mass and i sec2 θW 2 just above the present ATLAS and CMS exclusion limits [17,18]on =  3g1 cot θW − 3(cos 2θW − 1) 2 doubly-charged Higgs bosons. Furthermore, we are obviously inter- 2 12 − 9sec θW ested in enhancing possible 331-model signals. Limiting ourselves + − 2C 2C g2(2cos2θW 1) Ri1 R j1 to the Higgs and exotic sectors, the particle masses in our refer-  ence point are quoted in Table 1. 2 + 3g1 cot θW − 3(cos 2θW − 1) + 2g2(2cos2θW − 1) From Table 1, we learn that the 331 model, as expected, af-   ter including the sextet sector, yields 5 neutral scalar (h1, ...h5); ± ± ± × 2C 2C + − 2C 2C + 2C 2C 3 pseudoscalar (a1, a2 and a3) and 3 singly-charged (h , h , h ) Ri2 R j2 2g2(2cos2θW 1) Ri3 R j3 2Ri4 R j4 1 2 3 Higgs bosons: the lightest h1 is SM-like, whereas the others have −− ++ H Hi j mass between 1.8 and 6.5 TeV. In particular, h2 is roughly degen- × S pα , pα . (54) ± ± ± erate with a1 and h , while h4, h5, a2, a3, h and h have all 1 2 3 ±± mass about 6.5 TeV. As for doubly-charged particles, both Y ±± The interactions shown in Eq. (51) and Eq. (52)are clearly very and h have mass around 878 GeV, just above the current exclu- 1 ±± different, both in their Lorentz structures and in their dependence sion limit for doubly-charged scalars, while the other scalars h ±± 2 on the parameters of the model; therefore, different decay rates and h are in the 6.5 TeV range and the singly-charged vector 0 → ++ −− ± 3 V , hi B B are to be expected, according to whether B Y is roughly as heavy as the doubly-charged one. In our sce- is a scalar or a vector. It can be noticed that the expressions of nario, doubly-charged vectors and scalars decay only into lepton ++ −− the coupling in γ Y Y , i.e. 2g2 sin θW ≡ 2e, is apparently very pairs, with branching ratio 1/3 for each lepton family (ee, μμ or ++ −− different from the γ H H one, but one can show that, after ττ). The exotic quarks D, S and T in our reference point have in- simplifications, they turn out to be the same, as expected. The rel- stead mass between 1.65 and 1.70 TeV. In principle, such exotic evant vertices for vector bileptons are quarks can be produced in pairs at the LHC, with cross sections ⎧ in the range of 0.5–0.7 fb at 13 TeV and 0.8–1.1 fb at 14 TeV, and ⎨ − √i g γ μ P may deserve a complete phenomenological analysis, especially in ++ 2 L Y = 2 (55) the high-luminosity LHC phase. Nevertheless, in the present paper ⎩ √i μ g2γ P R we prefer to concentrate ourselves on the bilepton phenomenology 2 80 G. Corcella et al. / Physics Letters B 785 (2018) 73–83

Table 1 momentum cut.3 At 13 TeV LHC, after such cuts are applied, the Benchmark point for our collider study, consistent with the ∼ 125 GeV Higgs mass LO cross sections, computed by MadGraph, read and the present exclusion limits on BSM physics. Benchmark point σ (pp → YY → 4l)  4.3fb; σ (pp → HH → 4l)  0.3fb. (65) = = = mh1 126.3GeV mh2 1804.4GeV mh3 2474.0GeV = = Once again, the difference in the cross sections can be explained mh4 6499.8GeV mh5 6528.1GeV = = = ma1 1804.5GeV ma2 6496.0GeV ma3 6528.1GeV in terms of the spin of the intermediate bileptons. In the centre-of ± = ± = ± = mh 1804.5GeV mh 1873.4GeV mh 6498.1GeV mass frame, in fact, for scalar production, only the matrix element 1 2 3  m ±± = 878.3GeV m ±± = 6464.3GeV m ±± = 6527.7GeV h1 h2 h3 where the vector (γ , Z and Z ) has helicity zero with respect to ±± = ± =  = ++ −− ++ −− mY 878.3GeV mY 881.8GeV mZ 3247.6GeV the H H direction contributes; for decays into Y Y final = = = mD 1650.0GeV mS 1660.0GeV mT 1700.0GeV states also the ±1 helicity amplitudes are to be taken into account. ++ −− ++ −− For processes mediated by scalars (hi → H H /Y Y ), the ++ vector final states have still more helicity options since Y and −− and defer a thorough investigation on the production and decays Y can rearrange their helicities in a few different ways to of exotic quarks of charge 4/3 and 5/3 to future work [29].  achieve angular-momentum conservation, i.e. a total vanishing he- The Z boson deserves further comments. In [5]the relation licity in the centre-of-mass frame. We therefore confirm the find-  ings of Ref. [11], where a higher cross section for vector-bilepton 2 m ++ 3 − 12 sin θW production with respect to the scalars was obtained at 7 and Y   0.27 (62)  14 TeV. mZ 2cosθW  As for the background, final states with four charged leptons was determined between Z and vector-bilepton masses, and in may occur through intermediate Z-boson pairs: fact in Table 1 Eq. (62)is verified to a pretty good accuracy. More-  over, in our benchmark scenario the Z width is almost 700 GeV → → + − + −  pp ZZ (l l )(l l ). (66) and, as found out in [30] when exploring Z bosons in 331 mod-   els, our Z is leptophobic. Therefore, the searches for Z bosons After setting the same cuts as in (64), the LO cross section of the carried out so far by ATLAS [26] and CMS [27], which have set process (66)is given by exclusion limits around 4TeV on their mass, cannot be directly applied to our scenario, since such searches were mostly perfomed σ (pp → ZZ → 4l)  6.1fb. (67) for narrow resonances decaying into dilepton final states.2 In our  In principle, within the backgrounds, one should also consider SM reference point, the Z decays dominantly into qq¯ pairs, amounting + − Higgs-pair production (hh), with h → l l . However, because of to almost 70% of the total width, and has a significant branching ++ −− the tiny coupling of the Higgs boson to electrons and muons, such ratio into Y Y pairs, about 14%; its decay rate into doubly- ++ −− a background turns out to be negligible. charged scalars h h is instead rather small, roughly 1%. Such − 1 1 Assuming a luminosity of 300 fb 1, the number of same-sign a difference can be easily explained in terms of the particle spins:  ++ −− electron/muon pairs in processes mediated by YY, HH and ZZ the Z has spin 1 and therefore, in the decay into h h , only  1 1 are N(YY)  1302, N(HH)  120, N(ZZ)  1836. Defining the sig- the amplitude where the Z has zero helicity with respect to the ++ −− h h axis contributes. On the contrary, in a possible decay into nificance s to discriminate a signal S from a background B as 1 1  ++ −− vector states Z → Y Y , amplitudes with helicity 0 and ±1 ++ −− S with respect to the Y Y direction play a role. s =  , (68) + 2 In the following, we shall present results for the production of B σB two same-sign lepton pairs at the LHC, mediated by either vector σ being the systematic error on B, which we estimate as σ  or scalar bileptons in the 331 model: B B 0.1B, we find that the YY signal can be separate from the ZZ background with a significance s  6.9, while HH production is → ++ −− ++ −− → + + − − pp Y Y (H H ) (l l )(l l ), (63) overwhelmed by both Standard Model background (s = 0.6) and ±± possible vector-bilepton pairs (s = 0.9). where l = e, μ and, for simplicity, we have denoted by H the ±± At 14 TeV, the cross sections read: σ (YY)  6.0fb, σ (HH)  lightest doubly-charged Higgs boson h . The amplitude of pro- 1 0.4fb and σ (ZZ)  6.6fb, leading to N(YY)  17880, N(HH)  cess (63)is generated by the MadGraph code [15], matched with − √ 1260 and N(ZZ)  19740 events with 3000 fb 1 of data. There- HERWIG 6 for shower and hadronization [16]. We have set s = fore, in the high-luminosity phase of the LHC, one will be able to 13 TeV and chosen the NNPDFLO1 parton distributions [32], which discriminate vector-like bileptons from the background with a sig- are the default sets in MadGraph. nificance of about 9 standard deviations, while one is still unable As in Ref. [4], and along the lines of [17,18], we set the fol- to distinguish doubly-charged Higgses from YY (s = 0.70) or ZZ lowing acceptance cuts on the lepton transverse momentum (p ), T (s = 0.64) pairs. rapidity (η) and invariant opening angle (R): Besides total cross sections and significances, computed em- ploying the foreseen number of events, it is instructive studying | | pT ,l > 20 GeV, ηl < 2.5, Rll > 0.1. (64) some final-state observables, in order to understand how one can possibly detect (mostly vector-like) bileptons at LHC. In Fig. 2 we We point out that, since our signal originates from the decay of present the transverse momentum of the hardest and next-to- particles with mass almost 1TeV, the final-state electrons and hardest lepton (pT ,1 and pT ,2), the invariant opening angle be- muons will be pretty boosted, and therefore the actual values of tween them (R), the rapidity of the hardest lepton (η1), the the cuts in (64)are not really essential, especially the transverse- invariant mass (mll ) and the polar angle (θll ) between same-sign

 2 See, e.g., Ref. [31]on how the Z exclusion limits are modified in leptophobic 3 Our cuts are in fact a conservative choice of the so-called overlap-removal algo- models. rithm implemented by ATLAS to discriminate lepton and jet tracks [33]. G. Corcella et al. / Physics Letters B 785 (2018) 73–83 81

Fig. 2. Distributions of the transverse momentum of the hardest (a) and next-to-hardest lepton (b), same-sign lepton invariant mass (c), invariant opening angle between the two hardest leptons (d), rapidity of the leading lepton (e), polar angle between same-sign leptons (f). The solid blue histograms are the spectra yielded by vector bileptons, the red dots correspond to scalar doubly-charged Higgs bosons, the blue dashes to the ZZ Standard Model background. (For interpretation of the colours in the figure(s), the reader is referred to the web version of this article.) leptons. In any figure, the results corresponding to YY (black solid tonic spectra are always significantly below those yielded by ZZ histogram) HH (red dotted histogram) and ZZ production (blue background and/or YY-pair production. As for the transverse mo- dashed histogram) are displayed. Unlike Ref. [4], where all our menta (pT ,1 and pT ,2), the ZZ distributions are rather sharp and spectra were normalized to 1, in Fig. 2 all distributions are nor- peak at low pT , while those yielded by the HH and YY bilep- malized in such a way that the height of each bin, such as N(pT ), tons are much broader and peak about 1TeV (pT ,1) and at roughly yields the expected number of events for such√ values of pT , η, R, −1 700 (pT ,2, YY) and 800 GeV (pT ,2, HH). Such a result should θ and mll for a luminosity of 300 fb and s = 13 TeV. have been expected, since the Z decays into different-sign leptons, As one could foresee from the very cross section and signifi- ±± ±± cance evaluations, the general feature of such spectra is that the while Y and H into same-sign electrons and muons. As an- 331 signal can be discriminated from the ZZ background, while ticipated, for every value of pT , the HH spectrum is well below it is not possible to detect doubly-charged Higgs pairs as the lep- the YY one. 82 G. Corcella et al. / Physics Letters B 785 (2018) 73–83

Regarding the rapidity (η1,l) distribution of the leading lepton, of the helicity suppression, its cross section is too low for them to the 331-model spectra are narrower than the background and yield be substantially visible at the LHC and separable from the back- a larger event fraction around η1,l  0. Once again, since the ZZ ground and the vector-bilepton signal. background and the YY signal predict a number of events of a Our study therefore confirms that, as already anticipated in a similar order of magnitude, while those due to scalar pairs are previous analysis, the production of vector-bilepton pairs is the much lower, the rapidity spectrum can be useful to detect possi- striking feature of the 331 model and we believe that, given the ble vector bileptons, but not to separate them from doubly-charged large cross section and easy separation from the background, with Higgses. a significance between 6σ and 9σ , it should deserve a full ex- As for the polar angle between same-sign leptons θll , the perimental search. As for doubly-charged scalars, although the 331 ◦ YY-inherited spectrum is peaked around θll  1.2  70 , while the framework discussed in this work leads to a too low production background is much broader and maximum at about θ  0.7  cross section at the LHC, in order to achieve angular-momentum ◦ 40 ; the Higgs-like signal is instead negligible. conservation, we plan to explore how much this conclusion de- Concerning the same-sign lepton invariant mass mll , it is of pends on the actual setup for the family embedding and on the course easy to discriminate the 331 signals, peaking at mll  choice of the reference point, and whether there could be other 900 GeV, from the Z-pair background, which is instead a broad realizations of the model yielding a visible LHC rate even for scalar distribution, significant up to about 350 GeV and maximum around bileptons. Besides, it will be very interesting to study the phe- 70 GeV. As to the bileptons, both invariant-mass spectra are pretty nomenology of the exotic quarks predicted by our 331 model and ++ ++ narrow, which reflect the fact that Y and H have widths the LHC significance reach, especially in the high-luminosity and roughly equal to 7GeV and 400 MeV, respectively. high-energy phases. This is in progress as well. The distribution of the invariant opening angle R between the hardest and next-to-hardest leptons is rather broad for the Acknowledgement background, significant for 0 < R < 6, while YY pairs yield a dis-  tribution in the range 1 < R < 5, which, for R 3, even leads We acknowledge Antonio Sidoti for discussions on the cuts im- to more events that the background. The HH signal is possibly plemented in Eq. (64)and on the overlap-removal algorithm im- visible only for 2 < R < 4, but even in this range it is negligible plemented by the ATLAS Collaboration. This work is partially sup- with respect to the SM background and the YY signal. ported by INFN ‘Iniziative Specifiche’ QFT-HEP and ENP. Our conclusion is therefore that, as already argued in [4], the LHC will be sensitive to the spin-1 bileptons of the 331 model − References already at 13 TeV and 300 fb 1, and even more in the high- luminosity regime. If the LHC does not see any bilepton, it may [1] P.H. Frampton, Chiral dilepton model and the flavor question, Phys. Rev. Lett. mean either that bileptons do not exist or that they are scalars, 69 (1992) 2889. since we have shown that the production of doubly-charged Higgs [2] F. Pisano, V. Pleitez, An SU(3) × U(1) model of electroweak interactions, Phys. bosons in the 331 model is possibly overwhelmed by the SM back- Rev. D 46 (1992) 410. [3] M. Singer, J.W.F. Valle, J. Schechter, Canonical neutral current predictions from ground, as well as by vector bileptons. the weak electromagnetic group SU(3) × U (1), Phys. Rev. D 22 (1980) 738. 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