Heavy Majorana neutrino pair productions at the LHC in minimal U(1) extended Standard Model

In our recent paper [1], we explored a prospect of discovering the heavy Majorana right-handed neutrinos (RHNs) at the future LHC in the context of the minimal non-exotic U(1) extended Standard Model (SM), where a pair of RHNs are created via decay of resonantly produced massive U(1) gauge boson ($Z^{\prime}$). We pointed out that this model can yield a significant enhancement of the branching ratio of the $Z^\prime$ boson to a pair of RHNs, which is crucial for discovering the RHNs under the very severe LHC Run-2 constraint from the search for the $Z^\prime$ boson with dilepton final states. In this paper, we perform a general parameter scan to evaluate the maximum production rate of the same-sign dilepton final states (smoking gun signature of Majorana RHNs production) at the LHC, while reproducing the neutrino oscillation data. We also consider the minimal non-exotic U(1) model with an alternative charge assignment. In this case, we find a further enhancement of the branching ratio of the $Z^\prime$ boson to a pair of RHNs compared to the conventional case, which opens up a possibility of discovering the RHNs even before the $Z^\prime$ boson at the future LHC experiment.


I. INTRODUCTION
The experimental evidence of the neutrino oscillation [2] indicate that neutrinos have tiny but non-zero masses and flavor mixings. Since the neutrinos are massless in the Standard Model (SM), we need to extend the SM to incorporate the non-zero neutrino masses and flavor mixings. From the low energy effective theory viewpoint, we may introduce a dimension-5 operator [3] involving the Higgs and lepton doublets, which violates the lepton number by two units. After the electroweak symmetry breaking, neutrinos acquire tiny Majorana masses suppressed by the scale of the dimension-5 operator. For example, in a type-I seesaw [4][5][6][7][8], heavy Majorana right-handed neutrinos (RHNs), which are singlet under the SM gauge group, are introduced, and the dimension-5 operator is generated by integrating them out.
If the RHNs have masses around 1 TeV or smaller, they can be produced at the Large Hadron Collider (LHC). The same-sign dilepton in the final state, which indicates a violation of the lepton number, is a smoking-gun signature of RHN production. Since RHNs are singlet under the SM gauge group, they can be produced only through their mixings with the SM neutrinos.
To reproduce the observed light neutrino mass scale, m ν = O(0.1) eV, through the type-I seesaw mechanism with heavy neutrino masses at 1 TeV, a natural value of the light-heavy neutrino mixing parameter is estimated to be O(10 −6 ). With a general parametrization for the neutrino Dirac mass matrix [9], this mixing parameter can be much larger. However, it turns out to be still small 0.01 [10] in order to simultaneously satisfy a variety of experimental constraints, such as the neutrino oscillation data, the electroweak precision measurements and the lepton-flavor violating processes. Hence, the production rate of RHNs at the LHC is very suppressed.
In the simplest type-I seesaw scenario, the SM singlet RHNs are introduced only for the neutrino mass generation. The gauged B − L extended SM [11][12][13][14][15][16] may be a more compelling scenario, which incorporates the type-I seesaw mechanism. In this model, the global U(1) B−L (baryon number minus lepton number) symmetry in the SM is gauged and the RHNs play the essential role to cancel the gauge and mixed-gravitational anomalies. After the spontaneous breaking of the B − L symmetry, the RHNs acquire their Majorana masses, and the type-I seesaw mechanism is automatically implemented after the electroweak symmetry breaking.
This model provides a new mechanism for the production of RHNs at the LHC. Since the B − L gauge boson (Z ) couples with both the SM fermions and the RHNs, once the Z boson is resonantly produced at the LHC, its subsequent decay produces a pair of RHNs. Then, the RHNs decay into the SM particles through the light-heavy neutrino mixings: N → W ± ∓ , Zν , Zν , hν , and hν .
Recently, in the context of the gauged B − L models [17][18][19], the prospect of discovering the RHNs in the future LHC has been explored by simulation studies of a resonant Z boson production and its decay into a pair of RHNs. In Refs. [17,19], the authors have considered the trilepton final states, Z → N N → ± ∓ ∓ ν jj. For example, in Ref. [19] the signal-tobackground ratio of S/ √ B 10 has been obtained at the LHC with a 300 fb −1 luminosity, for the production cross section, σ(pp → Z → N N → ± ∓ ∓ ν jj) = 0.37 fb ( = e or µ), with the Z and the RHN masses fixed as m Z = 4 TeV and m N = 400 GeV, respectively. In Ref. [18], the authors have considered the final state with a same-sign dimuon and a boosted diboson, Z → N N → ± ± W ∓ W ∓ 1 . For fixed masses, m Z = 3 TeV and m N = m Z /4, they have obtained a cross section σ(pp → Z → N N → µ ± µ ± W ∓ W ∓ ) 0.1 fb for a 5σ discovery at the LHC with a 300 fb −1 luminosity.
Since the RHNs are produced from the Z boson decay, in exploring the future prospect of discovering the RHNs we need to consider the current LHC bound on the Z boson production, which is already very severe. 2 The primary mode for the Z boson search at the LHC is via the dilepton final states, pp → Z → + − ( = e or µ). The current upper bound on the Z boson production cross section times its branching ratio into a lepton pair (e + e − and µ + µ − combined) is given by σ(pp → Z → + − ) 0.2 fb, for m Z 3 TeV at the LHC Run-2 with 36.1 fb −1 luminosity [22]. Since the number of SM background events is very small for such a high Z boson mass region, we naively scale the current bound (at 95% confidence level) to a future bound as where L (in units of fb −1 ) is a luminosity at the future LHC. Here, we have assumed the worst case scenario, namely, there is still no indication of the Z boson production in the future 1 For previous studies of Z → N N → e ± µ ∓ W ± W ∓ , see, for example, Ref. [20]. 2 In Ref. [18], the authors have considered the U (1) (B−L)3 model [21], in which only the third generation fermions couple to the Z boson. Hence, the current LHC bound on the Z boson production is not applicable to the model, although their simulation results, which we employ in this paper, are model-independent.
σ(pp → Z → + − ) 2.4 × 10 −2 fb. Note that this value is much smaller than the RHN production cross section of O(0.1) fb obtained in the simulation studies. Taking into account the branching ratios N N → ± ∓ ∓ ν jj and N N → ± ± W ∓ W ∓ , the original production cross section σ(pp → Z → N N ) must be rather large. Therefore, an enhancement of the branching ratio BR(Z → N N ) over BR(Z → + − ) is crucial for the discovery of the RHNs in the future.
In the worst case scenario with the 300 fb −1 luminosity, we estimate an enhancement factor respectively. Even for the same sign dilepton final states, we have found that a huge enhancement factor is required. Note that we only have BR(Z →N N ) In this paper we consider a simple extension of the SM which can realize the branch- The model is based on the gauge group, In our recent paper [1], we considered the minimal U(1) X model with the conventional charge assignment and pointed out that the model can yield a significant enhancement of the branching ratio of Z boson to a pair of RHNs. We focused on the same-sign dimuon final state which is a smoking gun signature of Majorana RHNs production at the LHC. With such an enhancement and a realistic model-parameter choice to reproduce the neutrino oscillation data, we concluded that the possibility of discovering the RHNs in the future implies that the LHC experiments will discover the Z boson well before the RHNs. In this paper, we extend the analysis in our previous paper and perform a general parameter scan to evaluate the maximum production rate of the same-sign dilepton final state at the LHC, while reproducing the neutrino oscillation data. We also consider the alternative charge assignment and find a huge enhancement of the branching ratio of Z boson to a pair of RHNs compared to the conventional case. Performing a general parameter scan for this case, we find a possibility of discovering the RHNs even before the Z boson at the future LHC experiments.
The paper is organized as follows. In the next section, we present the minimal U(1) X model with a conventional charge assignment. After considering the production of the RHNs, we discuss the prospect of discovering the RHNs through their pair production from the decay of U(1) X gauge boson (Z ) at the future LHC experiments. In Sec. III, we present the minimal U(1) X model with an alternative charge assignment, and discuss the prospect of discovering the RHNs in this case. In Sec. IV, we consider the RHN decay process in details and employ a general parametrization for the neutrino Dirac mass matrix to reproduce the neutrino oscillation data. Performing general parameter scans, we evaluate the maximum branching ratio into the signal process, N N → ± ± W ∓ W ∓ , and discuss the prospect of discovering the RHN at the future LHC in the minimal U(1) X model with both the conventional and the alternative charge assignments. Sec. V is devoted to conclusions.

II. MINIMAL U(1) X MODEL
We first consider the minimal U(1) X extension of SM. 3 The model is based on the gauge the U(1) X is a generalization of U(1) B−L such that the U(1) X charges of particles are realized as a linear combination of the SM U(1) Y and U(1) B−L charges (the so-called non-exotic U(1) X model [23]). The structure of the theory is the same as the B − L model except for a U(1) X charge assignment. The particle content is listed in Table I. In addition to the SM particle content, this model includes three generations of RHNs (N i R ) required for gauge anomaly cancellations, a new Higgs field (Φ) which breaks the U(1) X gague symmetry, and a U(1) X gauge boson (Z ). The U(1) X charges are defined in terms of two real parameters x H and x Φ , which are the U(1) X charges associated with H and Φ, respectively. In this model, x Φ always appears as a product with the U(1) X gauge coupling and is not an independent free parameter, which we fix to be x Φ = 1 throughout this paper. Hence, U(1) X charges of the particles are defined by a single free parameter x H . Note that this model is identical to the minimal B − L model in the limit of x H = 0.
In the minimal U(1) X model, the Yukawa sector of the SM is extended to include where the first and second terms are the Dirac and Majorana Yukawa couplings. Here we use a diagonal basis for the Majorana Yukawa coupling without loss of generality. We assume a suitable Higgs potential for φ and H to develop their vacuum expectation values, v Φ and v h = 246 GeV, respectively. After the U(1) X and the electroweak symmetry breakings, U(1) X gauge boson mass, the Majorana masses for the RHNs, and neutrino Dirac masses are generated: where g X is the U(1) X gauge coupling, and we have used Let us now consider the RHN production via Z boson decay. The Z boson decay width into a pair of SM chiral fermions (f L ) is given by where N c = 1 (3) where, m N and Q N R are the mass and the U(1) X charge of the RHN, respectively.
In the left panel of Fig is nothing but the ratio of the N N and dilepton production cross section. Using Eqs. (5) and (6), this ratio is given by for only one generation of RHNs and charged leptons in the final states.
In the right panel of Fig. 1 (2)). Since the enhancement required for the trilepton final states is extremely large, we focus on the same sign dilepton and diboson final states in the rest of this section. 4 In the left panel of Fig. 1, we can see that the branching ratio to the dijet final states is also significantly enhanced. As we have commented in Ref. [1], the LHC constraint on the Z boson production cross section with the dilepton final states is still stronger than that with the dijet final states even with such an enhancement. Let us now consider an optimistic case and assume that the Z boson has been discovered at the LHC. In this case, we remove the constraint σ(pp → Z → + − ) 2.4 × 10 −2 fb.
According to [18], the cross section required for the 5σ discovery of the RHNs at the LHC with a 300 fb −1 luminosity is σ(pp → Z → N N → µ ± µ ± W ∓ W ∓ ) 0.1 fb. Although it is difficult for us to evaluate systematic errors, we here very naively require ad-hoc benchmark number of signal events to be 25 for the discovery of the Z boson production, since the number of SM background events for a high Z boson mass region (m Z 3 TeV) is very small. Hence, we estimate the luminosity (L) for 25 signal events of the Z boson production as follows: For a degenerate mass spectrum for the RHNs, , and we obtain is shown in the right panel of Fig. 1.  Hence, we obtain the luminosities L(fb −1 ) 102, 203 and 305 for one, two and three generations of degenerate RHNs, respectively. These luminosities will be reached in the near future.

III. ALTERNATIVE U(1) X MODEL
There is another way to assign the B − L charges for the three RHNs to achieve gauge anomaly cancellations. The B − L charge −4 is assigned to the first two generation of RHNs (N 1,2 ), while −5 for N 3 [31]. In addition to the SM particle content, the new particle content of this "alternative U(1) X model" is listed in We assume a suitable scalar potential for H, H E , Φ A , and Φ B , in which these scalars develop their vacuum expectation values as follows: where we require that v 2 h +ṽ 2 h = (246 GeV) 2 . Associated with the U(1) X symmetry breaking, the RHNs and the U(1) X gauge boson (Z ) acquire their masses as After the electroweak symmetry breaking, the neutrino Dirac masses, are generated, and hence the seesaw mechanism is automatically implemented.
Let us now consider the branching ratios for Z decay. Note that in the alternative U(1) X model, the charge assignment ensures the stability of N 3 R and it is naturally a dark matter (DM) candidate [33]. We may consider the scenario where the DM particle N 3 mainly communicates with the SM sector via Z boson exchange (Z portal DM). In this case, we expect that the relic abundance constraint leads to m 3 N m Z /2. In the following, we consider this case and the partial decay width of the Z into N 3 is neglected. The Z boson decay width formulas are given by Eqs.   Fig. 4 indicates that the RHNs will be discovered before the Z boson for −2.4 x H 0.

IV. REALISTIC HEAVY NEUTRINO BRANCHING RATIOS
In the above analysis and the simulation studies, BR(N → W µ) 0.5 is assumed. However, note that in a realistic scenario to reproduce the neutrino oscillation data, this branching ratio is smaller, which implies that more enhancement is required to obtain a sufficient number of signal events. In this section we consider the RHN decay processes in more details.
In the following analysis, we consider the case with degenerate RHNs, for simplicity. Using the Dirac and Majorana mass terms in Eqs. (3) or (10), the neutrino mass matrix is expressed as where In terms of the neutrino mass eigenstates, the charged current interaction is given by where α are the three generations of the charged SM leptons, and P L = (1 − γ 5 )/2. Similarly, the neutral current interaction is given by where θ W is the weak mixing angle.
From Eqs. (15) and (16), we parameterize the Dirac mass matrix as [9] where M N is a diagonal matrix for the mass eigenvalues of the RHNs and √ M N is defined as a matrix with each element of M N square rooted, O is a general orthogonal matrix, and the matrix √ D ν will be defined later. For the light neutrino mass spectrum, we consider both the normal hierarchy (NH), m 1 < m 2 < m 3 , and the inverted hierarchy (IH), m 3 < m 1 < m 2 .
Through its mixing with the SM leptons, a heavy neutrino mass eigenstate N i m (i = 1, 2, 3) decays into W , ν Z, and ν h with the corresponding partial decay widths: where  We first consider the minimal U(1) X model with three RHNs. In order to make our discussion simple, we assume the degeneracy of the heavy neutrinos in mass such as m N = m 1 N = m 2 N = m 3 N . Here, for simplicity, we fix the lightest neutrino mass eigenvalue as m lightest = 0.1× ∆m 2 12 , by which the elements of the matrix √ D ν ≡ diag( √ m 1 , √ m 2 , √ m 3 ) are uniquely fixed for both NH and IH cases. We parameterize the general orthogonal matrix O as where θ 1 , θ 2 , and θ 3 are complex numbers. With the inputs of the neutrino oscillation data and M N = m Z /4 with m Z = 3 TeV, we have performed a scan for the free parameters (θ 1 , θ 2 , θ 3 , ρ 1 , and ρ 2 ), and found the maximum values of the branching ratio as 0.337 and 0.157, for the NH and IH cases, respectively (see Fig. 5). In the analysis of Ref. [1], the orthogonal matrix in Eq. (22) is taken to be a unit matrix, and the branching ratios have been found to be 3 i=1 BR(N i m N i m → µ ± µ ± W ∓ W ∓ ) 0.210 and 0.154, for the NH and IH cases, respectively. Thus, a general parameter scan yields a larger branching ratios. The branching ratio for the NH case is almost twice as large, while the IH case is almost the same as before. If there is no indication of the Z boson production at the future LHC with a dilepton final state, we obtain an upper bound on the U (1) X gauge coupling for a fixed x H value and the Z boson mass. Using a narrow decay width approximation, the total production cross section of the Z boson is proportional to α X = g 2 X /(4π). We refer the results in Refs. [36,37] for the upper bound α X 0.01 7 for x H = −1.2 and m Z = 3 TeV from the ATLAS results with L = 36.1 fb −1 . The upper bound on α X scales as where L in units of fb −1 is a luminosity at the future LHC.  Let us now consider the alternative U(1) X model. Note that in this model only the first two generation RHNs are involved in the seesaw mechanism (the minimal seesaw [39,40]). In order to make our discussion simple, we assume the degeneracy of the heavy neutrinos in mass such as m N = m 1 N = m 2 N , and m 3 N m Z /2. The minimal seesaw scenario predicts one massless light neutrino eigenstate. In the NH case, the diagonal mass matrix is given by while in the IH case The matrices √ D ν for the NH and the IH are defined as respectively, and O is a general 2 × 2 orthogonal matrix given by where X and Y are real parameters. With the inputs of the neutrino oscillation data and M N = m Z /4 with m Z = 3 TeV, we have performed a scan for the free parameters (X, Y , and ρ 1 ), and found the maximum values of the branching ratio as 2 i=1 BR(N i m N i m → µ ± µ ± W ∓ W ∓ ) 0.148 and 0.0634, for the NH and IH cases, respectively (see Fig. 7). For both the NH and the IH, the maximum values for the branching ratios are obtained for ρ 1 π/2 and |Y | 2. The result becomes independent of Y for |Y | 2.
Using these realistic values for branching ratios to reproduce the neutrino oscillation data, we now re-evaluate the luminosity required for 25 signal events of the Z boson production.
For fixed values of m Z = 3 TeV and m N 1,2 = m Z /4, we show the luminosity as a function of Fig. 8. The dashed (dot-dashed) line corresponds to the NH (IH) case. Note that with a very large enhancement factor, the alternative U(1) X model allows us to discover the RHNs at the LHC well before the discovery of the Z boson. For example, for x H = −1.2, using the realistic branching ratios for the RHNs, 2 i=1 BR(N i m N i m → µ ± µ ± W ∓ W ∓ ) = 0.148 and 0.0634, the required luminosity is found to be L(fb −1 ) 1923 and 824 for the NH and the IH cases, respectively. For the B − L limit (x H = 0) case, we previously obtained L(fb −1 ) 325, for Hence for the NH (IH), Fig. 8 indicates that the RHNs will be discovered before the Z boson for −2.1 x H 0 (−1.7 x H −0.7).

V. CONCLUSIONS
We have investigated a prospect of discovering the RHNs in type-I seesaw at the LHC, which are pair produced from the decay of a resonantly produced Z boson. Recent simulation studies show that the discovery of the RHNs via Z → N N is promising at the future LHC with, for example, a 300 fb −1 luminosity. However, the production cross section of Z boson into dilepton final states (pp → Z → + − , where ± = e ± or µ ± ) is very severely constrained by the current LHC results. Imposing this constraint, we have found that a significant enhancement of the branching ratio BR(Z → N N ) over BR(Z → + − ) is crucial for the future discovery of RHNs.
For the minimal gauged U(1) X extension of the SM with the conventional and the alternative charge assignments, we have found that a significant enhancement, BR(Z → N N )/BR(Z → The branching ratio of BR(N → W µ) = 0.5 is commonly assumed in the simulation studies.
However, this branching ratio is not consistent with the neutrino oscillation data. Employing the general parameterization of the neutrino Dirac mass matrix to reproduce the neutrino oscillation data, we have performed a parameter scan to evaluate the maximal value for BR(N → W µ).
With the maximum enhancement factors and the maximum branching ratio, we have concluded for the minimal U(1) X model that a 5σ discovery of RHNs in the future according to the simulation studies implies that the Z boson must be discovered before the RHNs. In the alternative U(1) X model, we have obtained further enhancement of the signal cross section than the conventional case, and found a possibility of discovering the RHNs even before the Z boson at the future LHC experiment.

ACKNOWLEDGMENTS
The work of N.O. is supported in part by the United States Department of Energy (No. de-sc0013680).