Resonant production of $Z^{\prime}$ and signature of right-handed neutrinos within a 3-3-1 model

Recent simulation studies indicate that right-handed neutrinos may be discovered at future high luminosity LHC runs as long as a new neutral gauge boson, $Z^{\prime}$, have been discovered first. In this case, when $Z^{\prime}$ is resonantly produced at the LHC it, subsequently, decays into pair of right-handed neutrinos with mass up to TeV scale. Then the decay of these neutrinos leave as signature tri-lepton final states and/or same-sign di-muon and a boost di-boson. The prospect of discovering right-handed neutrinos through this mechanism has been developed within abelian gauge extension of the standard model. In this work we study the processes involved in this discovery mechanism within the framework of the 3-3-1 model.


I. INTRODUCTION
The future runs of the LHC at tens of TeVs and High luminosity lead to the tantalizing possibility of probing the TeV Type-I seesaw mechanism for small neutrino masses [1][2][3][4]. The signature of this mechanism is right-handed neutrino (RHN) with mass around TeV or less.
The challenge with the production of such RHNs depends on their mixing with the standard neutrinos which is very tiny, O(10 −6 ). The tiny mixing diminishes considerably their production at the LHC leaving us with very few events which difficult we have a good statistical significance.
To circumvent this problem, it was recently proposed that the discovery of RHNs may be conditioned to the existence of a new massive neutral gauge boson, Z , which interacts with all fermions, including the RHNs [5][6][7]. Then it is considered that Z is resonantly produced at the LHC through the process pp → Z , and that subsequently it decays in a pair of RHNs, Z → SS. Numerical simulations, having as benchmark model the B − L gauge model, have showed that under certain specific conditions, the signature of these neutrinos may be detected at the future Large hadron Collider (LHC) if they are kinematically accessible at the LHC.
The signature of RHN in this case is trilepton final state ± ∓ ∓ ν jj, or same-sign dimuon and a boosted diboson ∓ ∓ W ± W ± . The authors of Ref. [6] considered the latter channel and concluded that , for fixed masses, m Z = 3 TeV and m S = m Z /4, if we have σ(pp → Z → SS → ∓ ∓ W ± W ± ) ≈ 0.1, we may have a 5σ discovery at the LHC if we have a luminosity of 300fb −1 . For the trilepton final states, a signal-to-background ratio of S/ √ B 10 requires σ(pp → Z → S 3 S 3 → ± ∓ ± ν jj) 0.37fb for a luminosity of 300fb −1 with m Z = 4TeV and m S 3 = 400GeV, see Ref. [7].
In this paper we study the behavior of all processes involved in this mechanism within the [8,9]. The idea is to verify if the model fulfills the conditions that may lead to the discovery of RHNs. This model performs the type-I seesaw mechanism in a very peculiar way since the mass matrix of the RHNs and of the standard ones, both, involves the same Yukawa couplings [10][11][12]. This guarantee that the heaviest RHN has mass of 1TeV or less, which are accessible kinematically at the future LHC. Its particle content has also a new neutral gauge boson Z . The main aim is to check if around 10 −1 fb at the resonance of Z for the LHC running at 14 TeV with 300 fb −1 .
This work is organized as follows. In Sec. (II) we present the main aspect of the model, develop the seesaw mechanism and derive the neutral and charged current of the neutrinos with the gauge bosons of the model. In Sec. (III) we discuss resonant production of Z and its subsequent decay in pairs of RHNs and show in graphics the behavior of the cross sections involved in the mechanism with m Z and m S . Finally, in Sec. (IV), we present our conclusions.

II. THE MECHANISM
A. Revisiting the model In this section we revisit the TeV type I seesaw mechanism within the 331ν R [10]. Before, we present the main aspect of the model [8,9].
The leptonic content is arranged in triplet and singlet: where ∼ represent the way the triplet transform under the 3-3-1 symmetry. The index a(= 1, 2, 3) represents the three known generations of leptons. Notice that the third component of the lepton triplet is the right-handed neutrino.
In the quark sector, the first and second generations transform as anti-triplet, while the third generation transform as triplet under SU (3) L symmetry In addition to the left-handed field, we have right-handed fields as singlets under 3-3-1 symmetry with i = 1, 2.
The original scalar sector of the model involves three scalar triplets with η and χ transforming as (1, 3, −1/3) and ρ transforming as (1, 3, 2/3) under 3-3-1 symmetry. This content is sufficient to generate the correct masses for all massive fields [8,9,13], except neutrinos. We remember here that, in addition to the standard gauge bosons, Z µ , W ± µ and the photon, A µ , the model has five new massive gauge bosons

B. The TeV type-I seesaw mechanism
In order to generate neutrino masses, we add a scalar sextet to the original scalar content [9,11,12]: It was showed in [10] that with this scalar sextet TeV type-I seesaw mechanism is easily im- plemented. In what follow we revisit such result. The potential of the model with sextet is developed in [10] .
After the spontaneous breaking of the SU (3) C × SU (3) L × U (1) N to the standard symmetry, the scalar sextet splits into a triplet, a doublet and a singlet of scalar, where Y ∆ = −2, Y Φ = −1 and Y σ = 0 are the hyper-charges of the respective multiplets with The Yukawa interaction formed by the leptonic triplets and the scalar sextet is This interaction is factorized into the following ones with , the Yukawa interactions in Eq. (8) yields the following Dirac and Majorana mass terms for the neutrinos in basis where and with G being a symmetric matrix formed by the Yukawa couplings G ab . Throughout this paper, for simplicity, we neglect CP violation effects in the leptonic sector, which means that there are no phases in the above mass matrix and then all their elements are real. In what follow we The diagonalization of the mass matrix in Eq. (11), for the case v σ >> v Φ , leads to a mass relation among M D and M R known as the seesaw mechanism [14,15], where M ν is a mass matrix for the left-handed neutrinos and M R is the mass matrix for the right-handed neutrinos.
Having said that, in the more general case the mass matrix M D+M is diagonalized by the where M light is diagonal mass matrix and gives the masses of the standard neutrinos while M heavy is diagonal and gives the masses of the right-handed neutrinos. The rotating mix matrix, U , is given by where U P M N S and U R diagonalize M ν and M R , respectively. The rotating matrix U connects physical eigenstates with flavor ones. From now on we adopt the following notation: the light physical neutrinos will be call (N 1 , N 2 , N 3 ), while the heavy ones will be (S 1 , S 2 , S 3 ). The relations between mass and flavor neutrinos eigenstates are given by: where Now comes one of the main points of this TeV type-I seesaw mechanism. One should notice It is important to stress that σ 0 carries two units of lepton number. Hence, when σ 0 develops VEV, both the lepton number as well as the 3-3-1 symmetry are spontaneously broken. This means that the 3-3-1 symmetry breaking energy scale coincides with that of spontaneous breaking of the lepton number. The 3-3-1 symmetry breaking energy scale is expected to occur at TeV scale, as so will the spontaneous lepton number violation, i.e., v σ should belong to the TeV scale.
On the other hand, Φ does not carry lepton number. Its VEV generates, exclusively, Dirac mass terms for the neutrinos. The proposal of the seesaw mechanism is to yield neutrino masses at eV scale, which is accomplished if the ratio v 2 Φ vσ lies around eV. However, as we argued above v σ should be around TeV, then the eV neutrino requires v Φ around MeV.
Another interesting point is that the masses of the left-handed and right-handed neutrinos both share the same Yukawa coupling since that they originate from the same Yukawa interaction which is given in Eq. (8). This means that the Yukawa couplings G ab provides the pattern of the standard neutrinos as well as of the RHNs. This is a distinguishable fact of the model.
With the relation between the mass and flavor neutrino eigenstates, we are able to obtain the charged and neutral currents involving these neutrinos. Except by the photon, all the other gauge bosons of the model interact with the neutrinos where l L = (e L , µ L , τ L ) T and C W = cos θ W and S W = sin θ W with θ W being the Weinberg angle.
For the interactions between quarks and Z , see [13,[16][17][18] The According to Eq. (12) and (13), at least one RHN (S 1 ) may be light enough to have a long life and scape the LHC detectors and be considered as missing transverse energy. In this case it will contribute to the invisible Z decay, Z → S 1 S 1 , and we have to check if this is in agreement with the current experimental bound Γ exp inv = 503 ± 16 MeV [19]. As it is possible that at least one standard neutrino be massless, let us assume that one RHN, S 1 is massless, too.
According to the interactions above, the total width decay of the standard Z will include the contributions: Z → i,j (N i N i + N i S 1 + S 1 S 1 ) with i = 1, 2, 3. For the values of the parameters considered here, and using MadGraph5 [20], we obtain Γ inv = 507.367 MeV. Such prediction is in agreement with the experimental value for the invisible decay of Z standard gauge boson.

Finally, the interactions between gauge bosons and other fermions of the model is given in
Ref. [13]. Moreover, as the scalar sector of the model is very complex, and their interactions with neutrinos are sub-dominantly, once involve Yukawa couplings, then in our analysis we discard the interactions of the RHNs with the scalars of the model.

PAIR PRODUCTION OF RHNS AND THEIR SIGNATURE
The mass expression for the RHNs in the 331ν R model is M heavy = Gv σ . As the typical value of v σ is around TeV and the pattern of the matrix G is determined by the standard neutrino data, then assuming normal hierarchy it is reasonable to expect that G ij take values in the range 10 −3 − 10 −1 , which implies that at least one RHN may have mass of hundreds of GeVs. In the APENDIX we present a realistic scenario. For our study here we consider that the heaviest RNH neutrino (S 3 ) has mass of 200 GeV.
Concerning RHN with mass of hundreds of GeVs, recent simulation studies considered the prospect of its discovery at future LHC [5][6][7]21]. The framework used in the simulations was the B − L model [22][23][24][25][26]. The success of the simulations is conditioned to the discovery, first, of a new neutral gauge boson Z with mass of few TeVs associated to U (1) B−L . Then the idea is that when Z is resonantly produced through the process pp → Z , its subsequent decay in pair of RHNs, Z → S 3 S 3 , has as signature tri-lepton final states, S 3 S 3 → l ± l ± l ∓ νjj, or same-sign di-muon and a boost di-boson, S 3 S 3 → µ ± µ ± W ∓ W ∓ . As result, successful simulations require It is interesting to inspect, first, if the neutrino pair production cross section surpasses the charged lepton pair production one pp → Z → + − ( = e or µ). This is so because pp → Z → S 3 S 3 competes with di-lepton final states pp → Z → + − . Thus, an interesting model is one that gives σ(pp→Z →S 3 S 3 ) σ(pp→Z → + − ) > 1 in the resonance of Z . In order to check that, we implement the interactions of Eq. (17) in the MadGraph5 and obtain the values for the width decays provided by the 331ν R model.
Such widths depend on m Z and their behavior are displayed at Figure.(1). Perceive that the

4.7
(per generation). This is an encouraging result. Remember that the B − L model is the benchmark framework used in the simulations and it provides Γ(Z →N R N R ) Γ(Z → + − ) = 0.5. Alternative U (1) X models may increase considerably this rate [27][28][29] Before go further, we must examine the behavior of the original cross section σ(p p → Z → S 3 S 3 ) in order to check if it falls in the range of values that allows future prospect of discovering RHNs at the LHC at high luminosity. In Figure.(2), we present the behavior of this cross section with m S 3 . Note that it reproduces the RHN production cross section required for From the interactions in Eq. (17), and using Madgraph5, we estimate the following branching ratios: where = e and µ.
Let us first focus on that same-sign di-muon process. With these branching ratios in hand, we may analyze the behavior of the cross section σ(p p → Z → S 3 S 3 → ± ± W ∓ W ∓ ) with m S 3 for three values of m Z : 3 TeV, 4 TeV and 5 TeV. The results are displayed at Figure. (3). It was claimed in Ref. [6] that for m Z = 3 TeV and m S 3 = m Z /4, a cross section of 300 fb −1 . The results in Figure. (3) are in accordance with the simulation done in Ref. [6] For the case of tri-lepton final states, the behavior of the cross section with m S 3 is displayed at Figure. (4). It was claimed in Ref. [7] that for m Z = 4 TeV and m s 3 = 400 GeV, a cross section σ(pp → Z → S 3 S 3 → ± ∓ ∓ ν ) 0.37 fb is necessary for the LHC obtains a signalto-background ratio of S/ √ B 10 with 300 fb −1 . Although our results in Figure. (4) is a little smaller than the claimed by the simulation, however they agree in order of magnitude.
According to the Figures. (3) and (4), it seems that the process pp → Z → S 3 S 3 → ± ± W ∓ W ∓ appears to be the better channel to probe the RHNs within the 331ν R model.
We finish this section calculating the number of events predicted by the process pp →

IV. CONCLUSIONS
Right-handed neutrinos is an undoubted signature of type-I seesaw mechanism. Search for their signature is a very well motivated topic in particle physics. The benchmark framework of  this mechanism is the B − L model.
It is very hard to detect RHNs since it mix very weakly with the standard neutrinos.
However, it was recently perceived that the discovery of this particle may depend strongly of the discovery of a neutral gauge boson, Z . In this case RHNs production at the LHC may be considerably enhanced if its production occurs at the resonance of Z through the process p p → Z → S S. The main signature of these RHNs are trilepton final states,  provides the following masses for the standard neutrinos: m N 3 = 5.02 × 10 −2 eV, which yield the following mass differences (∆m 21 ) 2 = 7.50 × 10 −5 eV 2 (solars) (∆m 32 ) 2 = 2.524 × 10 −3 eV 2 (atmospheric), which explain solar and atmospheric neutrino oscillation experimental results [19].
This same set of Yukawa couplings define, also, the texture of the mass matrix of the heavy neutrinos given in Eq. (13). Diagonalizing that mass matrix we obtain: