Search for like-sign dileptons plus two jets signal in the framework of the manifest left-right symmetric model

The left-right symmetric model may present evidence of new physics at the era of the Large Hadron Collider (LHC). We use its framework to investigate the lepton number violating process pp → e±e±jj + X at the 14 TeV LHC. We show that for an integrated luminosity of 300 fb−1, the discovery contour of the right-handed boson WR and the right-handed electron neutrino Ne as a result of the pp → WR±\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {W}_R^{\pm } $$\end{document} → e±N → e±e±jj process can be expanded upon considering an additional channel mediated by the right-handed doubly charged Higgs δR±±\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\delta}_R^{\pm \pm } $$\end{document}, i.e. pp → WR±\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {W}_R^{\pm } $$\end{document} → WR∓∗δR±±∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {W}_R^{\mp \ast }{\delta}_R^{\pm \pm \ast } $$\end{document} → e±e±jj.


Abstract:
The left-right symmetric model may present evidence of new physics at the era of the Large Hadron Collider (LHC). We use its framework to investigate the lepton number violating process pp → e ± e ± jj + X at the 14 TeV LHC. We show that for an integrated luminosity of 300 fb −1 , the discovery contour of the right-handed boson W R and the righthanded electron neutrino N e as a result of the pp → W ± R → e ± N → e ± e ± jj process can be expanded upon considering an additional channel mediated by the right-handed doubly charged Higgs δ ±± R , i.e. pp → W ± R → W ∓ * R δ ±± *

Introduction
The shortcomings of the Standard Model (SM) are the motivation to search for signals of new physics beyond it. This is the main goal of the Large Hadron Collider (LHC). The SM is incapable of explaining a number of fundamental issues, such as the hierarchy problem (resulting from the large difference between the weak force and the gravitational force), the dark matter problem and the number of families in the quark and lepton sector. It is, therefore, reasonable to assume that new physics beyond the SM will be discovered in the coming years. Among the possible attractive platforms for new physics are left-right symmetric models (LRSM) [1][2][3][4][5][6][7]. The LRSM addresses two specific deficiencies in the SM: (i) Parity violation in the weak interactions, and (ii) non-zero neutrino masses implied by the experimental evidence of neutrino oscillation [8]. In particular, the left-right symmetry which underlies LRSM restores Parity symmetry at energies appreciably higher than the electroweak scale, resulting in the addition of three new gauge boson fields, W R1,2,3 . Furthermore, in LRSM the neutrinos are massive and their nature (i.e., whether they are of Majorana or Dirac type) depends on the details of the LRSM.
Early constructions of the LRSM comprise a Higgs sector with a Higgs bidoublet and two Higgs doublets [1][2][3]. In such a setup, the neutrinos are of Dirac type and no natural explanation for their small masses is provided. A later version, the manifest (or quasimanifest -see below) LRSM, incorporates a Higgs bidoublet and two Higgs triplets, and JHEP01(2021)031 implies the existence of Majorana type neutrinos [4][5][6][7]. This later version provides a natural setup for the smallness of neutrino masses, relating their mass scale to the large left-right symmetry breaking scale through the see-saw mechanism [9][10][11].
In this work we examine a process which, if detected at the LHC, will be able to provide direct evidence to the correctness of the LRSM with regard to two primary points: the left-right symmetry breaking scale and the see-saw mechanism. The process, which is lepton number violating (LNV), is pp → l ± l ± + 2j + X. (1.1) This process has a unique signal of two same sign leptons (like-sign dileptons -LSD) and two jets (LSD + 2 jets), with no missing energy. It has two classes of channels: the first class consists of channels which comprise the heavy right-handed (RH) boson W R and the RH Majorana neutrino N [12,13], 1 and the second class consists of channels which comprise W R and the RH doubly charged Higgs δ ±± R [14][15][16]. The signal of the process, if observed, may allow one to 1. detect the RH gauge boson W R , 2. trace the see-saw mechanism and the Majorana nature of the neutrino by detecting a heavy RH neutrino -which is associated to the spontaneous left-right symmetry breaking scale by its heavy mass, 3. establish the existence of the charged Higgs bosons and further confirm the Higgs triplet nature.
In this work we investigate the above process at the (14 TeV) LHC within the framework of the manifest LRSM. Former studies of the LSD + 2 jets signal at the LHC (within the framework of the LRSM) focused on a Drell-Yan production of a RH neutrino and a lepton via the W boson, followed by the neutrino decay to a second same sign lepton and two jets [17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33]: We extend these studies by including all the possible diagrams and searching for the dominant contributions among the possible LRSM amplitudes of the process. We find that an additional channel, mediated by the RH gauge boson and the doubly charged Higgs, can significantly contribute to the LSD + 2 jets signal when considering the reach of the LHC at √ s = 14 TeV. The work is organized as follows. In section 2 we describe the manifest LRSM Lagrangian structure. In section 3 we briefly mention constraints on the parameter space which are relevant to the study of the LSD + 2 jets phenomenology. In section 4 we investigate the process. We find and evaluate the cross sections of its dominant channels,

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and reconstruct the heavy gauge boson W R , the RH electron neutrino N e and the doubly charged Higgs δ ±± R from the relevant kinematic variables of signal and background events. We then plot the discovery potential (associated with the process) of these particles at the LHC ( √ s = 14 TeV). We summarize in section 5.

General model description
The LRSM is based on the gauge group All the fermion fields in the model are assigned in doublets, including the RH fermions which transform as doublets under the new symmetry of the model, SU(2) R . In addition to the fermion fields, seven gauge fields, W L,R and B (corresponding to the groups SU(2) L,R and U(1) B−L respectively) and eight gluon fields G a (a = 1 . . . 8) are introduced in order to obtain gauge invariance. The appropriate coupling constants of the G a µ , W L,R µ and the B µ fields are g s , g L,R and g = g B−L , respectively. The scalar content of the model includes three Higgs multiplets: a bidoublet (denoted as φ), a RH and a left-handed (LH) triplet (denoted as ∆ L and ∆ R , respectively). The covariant derivatives of the multiplets are conventionally given in the adjoint representation, so that the triplets are also converted to the adjoint representation. Thus, the 2 × 2 bidoublet-equivalent field matrices ∆ L,R = 1 √ 2 σ · ∆ L,R are introduced (the three-vector σ contains the Pauli matrices as components). The model field content is given in table 1.
The requirement that the Lagrangian is invariant under the left-right symmetry (ψ represents the L and the Q fermion doublets, see table 1) As spontaneous symmetry breaking (SSB) occurs, the above Higgs multiplets break the left-right symmetry into the U(1) Q observed symmetry, where the electromagnetic charge Q is defined by the modified Gell-Mann-Nishijima formula [34] At the first stage of the SSB the RH Higgs triplet, ∆ R , acquires a VEV v R : This stage occurs at an energy scale which is much larger than the electroweak scale. At the second stage the bidoublet acquires a VEV

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Model fields Content to bilinear fermion field products which form singlets under SU(2) L × SU(2) R × U(1) B−L : Yukawa matrices in flavor space (which have to be hermitian due to the left-right symmetry). 2 The gauge boson masses arise from the Higgs term in the Lagrangian, which consists of the Higgs kinetic terms and the potential of the Higgs multiplets: After SSB, the charged and neutral gauge boson masses are generated through the above Higgs kinetic terms and are given by where g ≡ g L = g R denote the SU(2) gauge couplings and θ w is the weak mixing angle, defined in the SM as e = g L sin θ w (e being the electromagnetic coupling). 3

LRSM constraints
We proceed with briefly summarizing the constraints on the relevant LRSM parameter space. We start with the quark masses, which are formed after SSB and are given by the Yukawa mass terms (see eq. (2.7)): The observation that there is a quark mass scale in which the top quark mass is much larger than the bottom quark mass implies, assuming the absence of fine tuning, that k 1 2 The manifest/quasi-manifest LRSM is realized upon assuming the left-right symmetry together with the lack of explicit CP violation in the Higgs potential. These assumptions imply, up to a sign, an identity between the corresponding elements of the left and the right CKM matrices (quasi-manifest LRSM). In this work we take the right CKM to be fully identical to the left CKM (manifest LRSM). 3 We neglect here a small mixing angle and use the chiral eigenstates WL,R instead of the mass eigenstates W1,2, respectively. The rotation (mixing angle) between the former and the latter respective states, ξ, is bound from above due to the Schwarz inequality [35]: This small mixing is, as stated above, neglected here.

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and k 2 are not of the same order (and neither are h Q andh Q ). In this work we therefore use k 1 k 2 , and in particular the natural setting k 2 /k 1 ∼ m b /m t (see also [36]). Additional constraints exist on the masses of the RH gauge boson W R and the heavy bidoublet Higgs bosons, and originate from flavor changing neutral current (FCNC) effects: d,s , K mixings, as well as b → sγ. These mixings receive significant contributions from the above mentioned particles and provide lower bounds on their masses: M W R ≥ 2.9 TeV, and H 0 1 , A 0 1 10 TeV [37][38][39][40][41][42][43][44][45][46][47]. A more stringent constraint on the mass of W R can be derived from calculating the radiative Higgs decay process H → γγ [48]. This constraint, which depends on the doubly charged Higgs mass (chosen to be 0.8 TeV in this work), provides a lower bound of M W R 3.8 TeV. 4 Direct lower bounds on W R mass are comparable with the theoretical bounds [54][55][56][57], whereas direct lower bounds on (heavy) neutral and singly charged Higgs bosons vary between ∼ 150 GeV − 1.6 TeV, depending on the search mode for the neutral or singly charged Higgs sectors [58].
The direct bounds on the RH and LH doubly charged Higgs boson masses were obtained by searching for pair-produced "left-handed" states δ ±± L and "right-handed" states δ ±± [59]. The lower bound on the δ ±± L mass ranges from 770 GeV 6 The lower bound on the mass of δ ±± R varies from 660 GeV to 760 GeV, where again l = e ± , µ ± and B(δ ±± where it varies between ∼ 500 GeV − 600 GeV. One particular constraint applies on the upper limit of the mass ratio between N and W R . Theoretical considerations based on stability and perturbativity of the effective potential (see [48] and also [60] for earlier results) suggest that it is possible for the heavy neutrinos in the LRSM to be heavier than W R depending on the measure of perturbativity within the LRSM parameter space. A ratio of can be allowed without ruining neither stability nor perturbativity. 7 4 Extensions of the minimal LRSM which ameliorate indirect constraints are not ruled out. There are already a number of simple changes to the manifest LRSM which relax these constraints and allow coexistence of LHC detectable WR alongside lower limits on at least part of the Higgs sector (e.g. by adding extra SU(2)R quark doublet or Higgs multiplets to the model, by using a higher dimensional operator, or by differentiating the left and right couplings and mixing matrices -see [49][50][51][52][53]). These adjustments may add new processes which lead to the LSD+2 signal. We won't consider these cases here. 5 The terms "left-handed" and "right-handed" describe the chirality (left or right) of the weak isospin T3 coupled to the doubly charged Higgs: does not contribute to the LSD + 2 jets signal due to vanishing of relevant couplingssee text. 7 The measure of perurtbativity is the ratio between the self-generated 1-loop and the corresponding tree-level parameter [48]: α Eq. (3.2) corresponds to a maximal 100% perurbativity of α3.

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Some constraints are required by unitarity. The optical theorem suggests the following bounds on the Higgs potential parameters: ρ 1,2 < 2π and α 3 , ρ 3 < 8π. In addition, the α 3 parameter should satisfy the above mentioned FCNC constraints on the masses of W R and the heavy bidoublet Higgs bosons. In terms of the particle mass terms, the constraint is This leads to an interplay between the α 3 parameter and the Higgs right triplet VEV v R : The lower bound on α 3 is thus directly related to the lower bound on the squared heavy neutral Higgs masses, and inversely related to the squared mass of W R . Choosing the Higgs bidoublet VEVs so that k 2 /k 1 ∼ m b /m t (see discussion below eq. (3.1)) and setting α 3 = 4.8 (∼ 18% perturbativity, see footnote 7) yields M W R 3 TeV, which is in agreement with [37][38][39][40][41][42][43][44][45][46][47]. 8 This setting is used in section 4 to explore the LSD + 2 jets signal at the LHC (see also A).
It is worthwhile to explore the allowed parameter space constituting the mass term of the RH doubly charged Higgs δ ±± R (which contributes to the LSD + 2 jets signal, as shown below). Incorporating the above constraints into a contour plot of the δ ±± R mass term, which is given by ). For clarity, the perturbativity measure of α 3 is denoted (see footnote 7). Further constraints are related to the Yukawa matrix h M which governs the couplings of the doubly charged Higgs to leptons (see below). The constraints on its elements originate from different low energy processes, such as µ →ēee, Bhabha scattering, extra coupling to (g − 2) µ , muonium (µ + e − ) transformation to anti-muonium and µ → eγ decay [66][67][68][69].
Finally, relevant constraints also arise from the neutrinoless double β decay process (0νββ) [6,[70][71][72][73][74][75][76]. Its non-observation sets a higher bound on the ratio between the mixing elements of the heavy Majorana neutrinos (N e , N µ , N τ ) and their masses [77,78]: where K L is a mixing matrix in the LH lepton charged current (see below). 9 Experimental limits are also extracted in present experiments by converting the sensitivity to the 0νββ process into a 0νββ decay strength parameter which has the dimension of mass. This parameter is denoted as the effective Majorana mass m ββ ≡ | ν=νe,νµ,ντ

The LSD + 2 jets signal
The relevant pp → l ± l ± jj diagrams can be grouped according to two characteristics. The first is the kinematics structure of each diagram, which in this case can be either an s-channel or a t-channel. An additional distinctive feature is the presence of either a JHEP01(2021)031   mediating Majorana neutrino or a mediating doubly charged Higgs boson in the diagram. 11 This is illustrated in the subfigures of figure 2 (s-channel diagrams) and figure 3 (t-channel diagrams). We first note that for the tested parameters 12 the interference terms and tchannel diagrams were found to have a negligible contribution to the squared amplitude matrix of the pp → l ± l ± jj process. This contribution accumulates to a fraction of O(10 −2 ) of the dominant, diagonal s-channel contributions. We therefore neglect it and focus on the latter. Within the s-channel group there are two dominant diagrams, one from each diagram type (neutrino mediated and doubly charged Higgs mediated), with significant contributions to the signal. 11 We will therefore refer to each diagram as either neutrino mediated or doubly charged Higgs mediated. 12 The relevant parameters are set subject to the constraints detailed in section 3 (see A). In addition, the RH neutrino masses are chosen to be degenerate and constrained to MN /MW R ≤ 7.3 (see text). The mass of the RH doubly charged Higgs δ ++ R is set to 800 GeV.

Doubly charged Higgs mediated diagrams
Almost all of the contributions from the s-channel doubly charged Higgs mediated diagrams (see figure 2b) can be safely neglected at the 14 TeV LHC with an integrated luminosity of 300 fb −1 . The only significant contribution from this group to the squared amplitude matrix belongs to the diagonal element of the (s-channel) process pp → W + R → W − * R δ ++ * R → e + e + jj. 14 We first discuss its features, and later continue and shortly discuss the suppressing factors due to which the other processes shown in figure 2b were found to have negligible contributions to the signal.

Significant contribution to the cross section: pp
The vertex factor of the W R production is governed by the covariant derivative in the fermion-gauge interaction term of the Lagrangian. Since the manifest LRSM consists of identical left and right CKM matrices as well as equal left and right gauge couplings (g := g L = g R ), this vertex factor has identical size to its opposite parity SM analogue.
vertex, which originates from the Higgs kinetic term (D µ ∆ R ) † D µ ∆ R , and is given by the Lagrangian term The vertex factor (with v R of TeV scale) is significant. The δ ++ R propagates and decays into two positrons via an interaction which is governed by the Lagrangian term (see eq. (2.7) for the general Yukawa interaction) where the Yukawa matrix h M is given by

3)
K R is a 6 × 3 mixing matrix for RH leptons 15 and M ν diag is the 6 × 6 neutrino diagonal mass matrix. We chose, as mentioned above, degenerate heavy neutrino masses (namely M N ). As a result we obtain for the positron-positron case This Yukawa coupling, in light of eq. (3.2), can be significant. 13 While the manifest LRSM contains eight singly charged Higgs gauge eigenstates (δ ± L,R , φ ± 1,2 ), in fact the only mass eigenstates which participate in the processes leading to the signal are H ± 2 (see B and discussion below).
14 For the sake of simplicity and without loss of generality, we will deal in this subsection and the next one with diagrams leading to two positively charged leptons, although our discussion also applies to the signreversal diagrams. Moreover, although the expected cross section is similar for the three lepton generation pairs (as opposed to the background in which it is unnecessarily the same), we choose to deal with the first generation only in order to correspond with previous works related to this signal (see refs. [17,18] The electroweak coupling factor of the vertex does not counter-balance the negligibility of the diagrams, which results from the following: (i) the gauge eigenstate δ ± R consists of a highly suppressed fraction of the physical (massive) Higgs eigenstate H ± 2 : (as mentioned above, natural setting is used: k 2 /k 1 ∼ m b /m t , and the coefficient goes as ∼ k 1 / √ 2v R , (< 1/30)), (ii) the mass of H ± 2 is constrained by a lower limit. This is due to its dominant mass term (of v R scale) which is identical to the dominant term of the lower bounded heavy neutral Higgs particles H 0 1 and A 0 1 (see eq. (3.4)). Therefore, at the mass region in which the dominant channels can be discovered 17 -H ± 2 is significantly heavier than W R and its propagator factor is relatively suppressed, (iii) the coupling between H ± 2 and the proton quarks is proportional to the quarks masses, and thus very small (this applies only to the second process).
In general, the triple-Higgs vertices involving one doubly charged Higgs and two singly charged Higgs bosons arise from the ρ, α and β terms in the scalar potential: These interaction terms are, at most, highly suppressed. We first note that the β i parameters of the manifest/quasi-manifest LRSM must vanish in order to reduce the mass scale of the non-SM gauge bosons without the need to fine-tune (and thus be able to theoretically allow possible observation at the LHC, see [9][10][11]). As for the ρ i and α i terms in the potential, their contribution to the LSD+2 signal is negligible, the reason being is that all the 16 Replacing WR with the SM gauge boson interacting right-handedly results in a vertex with a small mixing angle factor. This factor, as explained in footnote 3, is ignored. 17 The two dominant channels, i.e. the doubly charged Higgs mediated channel (discussed above) and the neutrino mediated channel (discussed below).

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relevant triple field products include the singly charged δ ± L,R states. These states, in turn, comprise, respectively, the H ± 1 mass eigenstates, which are not part of any bidoublet eigenstate and therefore do not participate in the Yukawa coupling to the quarks, and a highly suppressed fraction of the heavy and tiny Yukawa coupled H ± 2 Higgs mass eigenstates (see discussion above and B).
As this vertex factor is proportional to v L , it vanishes in the framework of the manifest / quasi-manifest LRSM due to phenomenological considerations (see [9][10][11]). 18 In this case one cannot replace W L with a singly charged Higgs since, although a term W + L δ + L δ −− L appears in the Lagrangian, in the manifest LRSM the δ ± L states are decoupled from the quarks (see above).

Majorana neutrino mediated diagrams
We now consider the (Majorana) neutrino mediated group of processes which lead to the LSD + 2 jets signal (see figure 2a). As discussed above, the processes we consider are characterized by an s-channel production of a (Majorana) neutrino and a positron/electron through an exchange of a gauge or a Higgs boson. The formed neutrino then decays into two jets plus a positron/electron -with an equal probability due to the Majorana nature of the neutrino (we are only concerned with signals which consist of two electrons or two positrons). As in the case of the doubly charged Higgs mediated diagrams, also many of the neutrino mediated diagrams can be safely neglected within the present LHC reach. We find that the significant contribution to the squared amplitude matrix arrives from the diagonal element which comprises the squared amplitude of the diagram pp → W + R → e + N e → e + e + jj (as was first found by Keung and Senjanović, see [12]). We will shortly discuss this channel and the alternative suppressed channels in this diagram-group.

Significant contribution to the cross section: pp → W +
R → e + N e → e + e + jj. The produced W R decays into a positron 19 plus a heavy or a light electron-neutrino via theN e W + R e or theν e W + R e vertex, respectively (where N e (ν e ) is the heavy (light) electronneutrino element in a six-vector, N , constructed from three light and three heavy neutrino (Majorana) mass eigenstates). These vertices are governed by the lepton RH charged current terms:

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(where K L,R are 6 × 3 left and right mixing matrices, respectively, in the lepton sector 20 (see also eq. (4.3))). The decay of W R through a RH current with a heavy neutrino is not suppressed for the chosen K R mixing matrix used here. 21 The probability of the alternative RH decay, through a light neutrino, is smaller by an order of M ν /M N (the quotient of the light and heavy neutrino masses). 22 The heavy neutrino (a Majorana particle) decays in equal rates into a same sign positron (plus two jets) and an opposite sign electron (plus two jets). 23 Negligible contributions to the cross section: pp → W + L → e + (N, ν) e → e + e + jj. The produced W L decays into a heavy or a light neutrino via theN e W + L e orν e W + L e vertex, respectively. The Lagrangian terms controlling these vertices are the chiral mirror of the terms related to the decay of W R . They are given by These two options are suppressed. The heavy neutrino as part of a LH charged current is suppressed due to the seesaw mechanism (see also footnote 22), and disfavored with a probability of O(M ν /M N ) in comparison to the light neutrino. But despite its dominance, the decay into the light neutrino also leads to a suppressed LSD production, the reason being is as follows. In terms of its helicity, a Majorana neutrino behaves at a weak current vertex as if it were a Dirac particle [87]. Thus, since the light neutrino is relativistic (but not massless), it is dominantly emitted alongside the positron at theν e W + L e vertex with a negative helicity, and has a highly suppressed probability (of O((M/E) 2 )) to be emitted in the "wrong" (positive) helicity. Now, as only the "wrong" helicity state can be absorbed without suppression at the next vertex (ν e W + L e, which emits a second positron at a LH charged current and therefore interacts significantly only for an incoming antineutrino with a positive helicity), this channel is therefore also highly suppressed. 24 pp → H + 2 → e + (N, ν) e → e + e + jj. The H + 2 production and propagation parts of the diagram render it negligible. The H + 2 production originate from the Yukawa couplings between the quarks and the bidoublet singly charged fields. These couplings are proportional to the masses of the light quarks within the proton (and are negligible in comparison 20 The mixing matrices KL,R contain both heavy-light mixing and mixing among generations. The intergeneration mixing elements in KR which connect heavy neutrinos and charged leptons (the non-diagonal elements in the block) are negelected here for the sake of simplicity -see also footnote 15. 22 This is well demonstrated by the (type I) seesaw mechanism, where a diagonalization of a neutrino mass matrix composed of the masses related to the possible types of field billinear products gives eigenvalues of a light and a heavy neutrino masses, the geometric mean of which is the electroweak scale Dirac mass. 23 There are two non-negligible decay branches of the heavy electron neutrino Ne which lead to a second same sign lepton plus two jets: via RH and LH currents. While the LH current will be unfavored by a factor of Mν /MN because of the above mentioned seesaw mechanism, it is also enhanced due to the on-shell formation of WL. Both of these decay branches are part of the pp → W + R → e + Ne → e + e + jj channel. 24 While replacing the WL at the second vertex with WR eliminates this source of suppression, it in turn leads to a suppression due to the seesaw mechanism which favors a heavy neutrino in the RH current. to the weak coupling g involved in the W L,R production). Moreover, the H + 2 mass has a lower bound of approximately 10 TeV (see discussion above) -substantially heavier than an LHC detectable W R .

The cross section contributions from the two dominant diagrams
The cross-section arising from the two dominant diagonal terms of the squared amplitude matrix is plotted in figure 4. In the figure, and throughout this work, we have worked with degenerate heavy neutrinos and with positrons and electrons as l ± l ± . While the neutrino mediated channel is dominant at the lower M Ne /M W R range, it is inversely related to this ratio due to decreasing available phase space in the decay W ± R → N e e ± and due to the neutrino propagator factor which reduces the cross section for M Ne > M W R . Conversely, the doubly charged Higgs mediated channel is enhanced as M Ne /M W R increases towards unity since the W R decay width decreases. As M Ne passes M W R the decreasing of the W R decay width is stopped, and at that point the increasing decay width of δ ++ R , 25 which reduces its propagator factor, starts to slowly reduce the channel cross-section as well. 25 The Yukawa coupling of the δ ++ R ll vertex depends on MN /MW R , as discussed above -see eq. (4.4).

Background analysis and sensitivity estimates
The SM processes which were considered as a potential background to the LSD + 2 jets signal are the ones leading to signatures with two electrons/positrons and at least two hadronic jets. The background processes which were considered are pp → ZZ, ZW ± , tt → W ± W ∓ bb, W ± W ± W ∓ → e ± e ± + jets + . . . (4.11) These processes do not violate lepton number and therefore contain (in addition to the signal ingredients) also opposite sign leptons and/or neutrinos in the final state. For generating signal and background events we used the CalcHEP [88] software with cteq6l1 parton distribution functions and an implementation of the manifest LRSM [89]. We used PYTHIA [90,91] for the showering and hadronization routines. The K-factor for the signal was calculated using FEWZ (for the mass range of M W R ∼ 3−6 TeV) [92]. 26 For the detector simulation we used DELPHES [94] with ATLAS card (i.e. ATLAS detector specifications), and selected events with two isolated positrons/electrons and at least two jets in the final state. Processing the generated events was performed using the MadGraph [95] interface. In order to reduce the background without considerably affecting the signal, the following cuts were applied on the selected events of the signal and the background: • Each of the two jets with the highest 27 E T is required to have E T > 100 GeV, • The invariant mass of the ee system is required to be larger than 200 GeV, • The ee system is required to consist of either two positrons or two electrons.
The overall efficiency of the detector and the signal event selection is given in table 2 for the two dominant channels. The relatively low efficiency for M Ne /M W R = 0.1 in the N e mediated channel is due to highly boosted N e , leading to difficulties in separating its decay products in the detector. This feature does not occur in the δ ±± R channel, where the kinematics of the signal particles is independent of N e .
The signal and background events which survive the event selection and subsequently pass the above cuts are tagged as S and B, respectively, and are used as a common event 26 The K-factor is 1.3 for the signal and 1.4 for the background processes (see also [93]). 27 We assume that the two leading jets come from Ne.

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pool. In order to reconstruct the relevant LRSM particles, the pool events are probed for the following invariant mass systems: e ± 1 e ± 2 j 1 j 2 (which corresponds to reconstructing the W R mass), e ± j 1 j 2 (the N e mass reconstruction) and e ± 1 e ± 2 (the δ ±± R mass reconstruction). The reconstructed LRSM masses associated with three chosen mass coordinates are shown in figure 5. In the figure we plot the number of events as a function of the reconstructed masses of W ± R (top panel), N e 28 (middle panel) and δ ±± R (bottom panel). It is evident from the plots that the background becomes low before the signal peak for each of the three particle masses, which come out clear and distinct within the reach of the LHC.
In order to map the mass region with sensitivity to the signal we proceed as follows: in every mass coordinate tested for the signal we use the available detectable systems (i.e. e ± 1 e ± 1 j 1 j 2 , e ± j 1 j 2 and e ± 1 e ± 2 ) to compare the signal and the background events which pass the above event selection and cuts. We apply the two following selection rules on the available S and B counts for each system (in a selected mass window, see ref. [17]): 1. The presence of at least 10 signal events, and 2. The signal must exceed five statistical fluctuations of the background ( S √ B ≥ 5).
We require at least one of the above systems to pass both of these selection rules to allow discovery. For a given mass setting, passing the discovery criterion is related to the joint contribution of the two channels. In addition, we check whether the discovery can be attributed to any of the two channels independently. We continue and map the region boundaries of the independent and the joint (combined) channels with corresponding discovery contours in the (M W R , M Ne ) plane (with a chosen M δ ±± R = 0.8 TeV). The constructed discovery contours are shown in figure 6. They measure the detector sensitivity to the two channels, both separately and jointly (combined).
After data-taking at an integrated luminosity of 300 fb −1 , the discovery limits reach a maximal M W R of 5.3 TeV and a maximal M Ne of 3.6 TeV for the N e mediated channel alone.
Upon adding the contribution of the δ ±± R mediated channel, the maximal M W R limit associated with the N e mediated channel is pushed out marginally by 30 GeV at M Ne 1.6 TeV. For N e masses below 1.6 TeV the M Ne discovery limit is pushed out by less than 30 GeV. For N e masses above 1.6 TeV the upper M Ne discovery limit 29 for W R masses going down to 4.4 TeV rises by up to ∼ 100 GeV. As the mass of W R gets nearer (and then below) 4 TeV the significance of the δ ±± R channel markedly grows 30 and the upper N e mass discovery limit starts increasing rapidly as M W R declines. This rapid increase reaches an upper discovery limit of M Ne ∼ 33 TeV for M W R = 3 TeV, beyond the allowed theoretical limit (see eq. (3.2)).
As a more general (and clarifying) perspective to the above discussion it is beneficial to demonstrate the ultimate contribution of the Higgs mediated channel to the signal in terms of the number of detected events as a function of W R and δ ±± R masses in a two-dimensional 28 Since we were unable to deductively identify the lepton which originates from the Ne decay, both combinations of e ± j1j2 had to be examined (resulting in a wider mass spectra). 29 As shown in figure 6, for each given WR mass there is a lower and a higher Ne mass discovery limit. 30 The number of events from the Ne mediated channel becomes negligible as MN e approaches and passes MW R for a 14 TeV LHC and an integrated luminosity of 300 fb −1 .  . Discovery potential for the two dominant channels of the pp → W R → e ± e ± jj process at the 14 TeV LHC for an integrated luminosity of 300 fb −1 . The coordinates denoted by correspond to the three mass settings of figure 5. The red line boundary of the excluded area (i.e. the mass region where no excess of like-sign dilepton events in comparison to the SM was found) is based on the works in refs. [54][55][56][57]. The lower and the upper part of the y-axis have different (linear) scales in order to highlight the independent and the combined sensitivity regions associated with the two channels. coordinate occurs for M Ne = M W R (yet it decreases slowly and retains much of its relative strength throughout considerably heavier N e mass range, as shown in figure 4.). 31 The figure demonstrates, particularly for the lower mass areas (and at this chosen luminosity), that the Higgs mediated channel can be discovered either independently or jointly with the neutrino mediated channel.

Summary
The importance of the LRSM lies mainly in the fact that it restores parity symmetry at higher energy scales and provides a natural setup for the observed neutrino oscillations phenomena, based on the see-saw mechanism. One particular process within the framework of the LRSM which is able to supply direct evidence to both the left-right symmetry breaking scale and the correctness of the see-saw mechanism is pp → e ± e ± jj+X, generating a signal of two same sign leptons (electrons/positrons in this work) plus two jets. We evaluated the cross sections of the dominant process channels leading to this signal at the 14 TeV LHC, with an integrated luminosity of 300 fb −1 -for a chosen set of parameters. We estimated the mass regions in which the LHC is sensitive to the two dominant process channels, and showed that the contribution of the doubly charged Higgs mediated channel can significantly expand the discovery potential which arises from the neutrino mediated channel alone. 31 We remind that the masses of the three heavy neutrinos are restricted to be equal in this work.

A Parameter settings used
The manifest LRSM model file used in this work is described in ref. [89], and is based on the model as described in ref. [11]. The parameter settings used are as follows: • Higgs VEVs (in GeV)

B Higgs physical eigenstates
The Higgs multiplets consist of 20 degrees of freedom, i.e. 20 real fields. Obtaining the fields eigensystem is done by diagonalizing the squared-mass matrix: The eigenstates consist of 34 32 The lepton mixing parameters KL,R are 6 × 3 matrices in the lepton sector which connect the charged leptons to the six Majorana neutrinos. 33 The PMNS matrix is the lepton sector analogue to the CKM quark matrix (see [86]). 34 We use the unitary gauge.  3. Four singly charged scalar eigenstates H ± 1 , H ± 2 , G ± L , G ± R (G ± L and G ± R are Goldstone bosons), 4. Two doubly charged scalar eigenstates H ±± L , H ±± R . The corresponding eigenvalues/masses are given, e.g., in the third ref. in [9][10][11]. The nonphysical Higgs fields may be written in terms of the above eigenstates as follows (the φ 0 1 , φ 0 2 and δ 0 R states are given in the approximation v R k + , where k ± ≡ k 2 1 ± k 2 2 ): Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.