# Minimal dirac neutrino mass models from \(\hbox {U}(1)_{\mathrm{R}}\) gauge symmetry and left–right asymmetry at colliders

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## Abstract

In this work, we propose minimal realizations for generating Dirac neutrino masses in the context of a right-handed abelian gauge extension of the Standard Model. Utilizing only \(U(1)_R\) symmetry, we address and analyze the possibilities of Dirac neutrino mass generation via (a) *tree-level seesaw* and (b) *radiative correction at the one-loop level*. One of the presented radiative models implements the attractive *scotogenic* model that links neutrino mass with Dark Matter (DM), where the stability of the DM is guaranteed from a residual discrete symmetry emerging from \(U(1)_R\). Since only the right-handed fermions carry non-zero charges under the \(U(1)_R\), this framework leads to sizable and distinctive Left–Right asymmetry as well as Forward–Backward asymmetry discriminating from \(U(1)_{B-L}\) models and can be tested at the colliders. We analyze the current experimental bounds and present the discovery reach limits for the new heavy gauge boson \(Z^{\prime }\) at the LHC and ILC. Furthermore, we also study the associated charged lepton flavor violating processes, dark matter phenomenology and cosmological constraints of these models.

## 1 Introduction

Neutrino oscillation data [1, 2, 3, 4] indicates that at-least two neutrinos have tiny masses. The origin of the neutrino mass is one of the unsolved mysteries in Particle Physics. The minimal way to obtain the non-zero neutrino masses is to introduce three right-handed neutrinos that are singlets under the Standard Model (SM). Consequently, Dirac neutrino mass term at the tree-level is allowed and has the form: \({\mathcal {L}}_Y\supset y_{\nu }{\overline{L}}_L{\widetilde{H}}\nu _R\). However, this leads to unnaturally small Yukawa couplings for neutrinos (\(y_{\nu } \le 10^{-11}\)). There have been many proposals to naturally induce neutrino mass mostly by using the seesaw mechanism [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16] or via radiative mechanism [17, 18, 19, 20]. Most of the models of neutrino mass generation assume that the neutrinos are Majorana^{1} type in nature. Whether neutrinos are Dirac or Majorana type particles is still an open question. This issue can be resolved by neutrinoless double beta decay experiments [23, 24, 25, 26]. However, up-to-now there is no concluding evidence from these experiments.

Recently, there has been a growing interest in models where neutrinos are assumed to be Dirac particles. Many of these models use ad hoc discrete symmetries [27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37] to forbid the aforementioned unnaturally small tree-level Yukawa term as well as Majorana mass terms. However, it is more appealing to forbid all these unwanted terms utilizing simple gauge extension of the SM instead of imposing discrete or continuous global symmetries. This choice is motivated by the fact that contrary to gauge symmetries, global symmetries are known not to be respected by the gravitational interactions [38, 39, 40, 41, 42].

In this work, we extend the SM with \(U(1)_R\) gauge symmetry, under which only the SM right-handed fermions are charged and the left-handed fermions transform trivially. This realization is very simple in nature and has several compelling features to be discussed in great details. Introducing only the three right-handed neutrinos all the gauge anomalies can be canceled and \(U(1)_{R}\) symmetry can be utilized to forbid all the unwanted terms to build desired models of Dirac neutrino mass. Within this framework, by employing the \(U(1)_R\) symmetry we construct a tree-level Dirac seesaw model [43] and two models where neutrino mass appears at the one-loop level. One of these loop models presented in this work is the most minimal model of radiative Dirac neutrino mass [44] and the second model uses the scotogenic mechanism [45] that links two seemingly uncorrelated phenomena: neutrino mass with Dark Matter (DM). As we will discuss, the stability of the DM in the latter scenario is a consequence of a residual \({\mathcal {Z}}_2\) discrete symmetry that emerges from the spontaneous breaking of the \(U(1)_R\) gauge symmetry.

Among other simple possibilities, one can also extend the SM with \(U(1)_{B-L}\) gauge symmetry [46, 47, 48, 49] for generating the Dirac neutrino mass [44, 50, 51, 52, 53, 54, 55]. Both of the two possibilities are attractive and can be regarded as the minimal gauge extensions of the SM. However, the phenomenology of \(U(1)_R\) model is very distinctive compared to the \(U(1)_{B-L}\) case. In the literature, gauged \(U(1)_{B-L}\) symmetry has been extensively studied whereas gauged \(U(1)_R\) extension has received very little attention.

Unlike the \(U(1)_{B-L}\) case, in our set-up, the SM Higgs doublet is charged under this \(U(1)_R\) symmetry to allow the desired Yukawa interactions to generate mass for the charged fermions, this leads to interactions with the new gauge boson that is absent in \(U(1)_{B-L}\) model. The running of the Higgs quartic coupling gets modified due to having such interactions with the new gauge boson \(Z^{\prime }\) that can make the Higgs vacuum stable [56]. Due to the same reason, the SM Higgs phenomenology also gets altered [57].

We show by detail analysis that despite their abelian nature, \(U(1)_{R}\) and \(U(1)_{B-L}\) have distinguishable phenomenology. The primary reason that leads to different features is: \(U(1)_R\) gauge boson couples only to the right-handed chiral fermions, whereas \(U(1)_{B-L}\) is chirality-universal. As a consequence, \(U(1)_R\) model leads to large left–right (LR) asymmetry and also forward–backward (FB) asymmetry that can be tested in the current and future colliders that make use of the polarized initial states, such as in ILC. We also comment on the differences of our \(U(1)_R\) scenario with the other \(U(1)_R\) models existing in the literature. Slightly different features emerge as a result of different charge assignment of the right-handed neutrinos in our set-up for the realization of Dirac neutrino mass. In the existing \(U(1)_R\) models, flavor universal charge assignment for the right-handed neutrinos are considered and neutrinos are assumed to be Majorana particles. Whereas, in our set-up, neutrinos are Dirac particles that demands non-universal charge assignment for the right-handed neutrinos under \(U(1)_R\). Neutrinos being Dirac in nature also leads to null neutrinoless double beta decay signal.

The originality of this work is, by employing only the gauged \(U(1)_R\) symmetry, we construct Dirac neutrino masses at the tree-level and one-loop level (with or without DM) which has not been done before and, by a detailed study of the phenomenology associated to the new heavy gauge boson, we show that \(U(1)_R\) model is very promising to be discovered in the future colliders. Due to the presence of the TeV or sub-TeV scale BSM particles, these models can give rise to sizable rate for the charged lepton flavor violating processes which we also analyze. On top of that, we bring both the dark matter and the neutrino mass generation issues under one umbrella without imposing any additional symmetry and, work out the associated dark matter phenomenology. We also discuss the cosmological consequences due to the presence of the light right-handed neutrinos in our framework.

The paper is organized as follows. In Sect. 2, we discuss the framework where SM is extended by an abelian gauge symmetry \(U(1)_R\). In Sect. 3, we present the minimal Dirac neutrino mass models in details, along with the particle spectrum and charge assignments. In Sect. 4, we discuss the running of the \(U(1)_R\) coupling. Charged lepton flavor violating processes are analyzed in Sect. 5. We have also done the associated dark matter phenomenology in Sect. 6 for the scotogenic model. Furthermore, we analyze the collider implications in Sect. 7. In Sect. 8, we study the constraints from cosmological measurement and finally, we conclude in Sect. 9.

## 2 Framework

*k*. Hence, all the charges are determined once \(R_H\) is fixed, which can take any value. The anomaly is canceled by the presence of the right-handed neutrinos that in general can carry non-universal charge under \(U(1)_R\). Under the symmetry of the theory, the quantum numbers of all the particles are shown in Table 1.

Quantum numbers of the fermions and the SM Higgs doublet

Multiplets | \(SU(3)_C\times SU(2)_L\times U(1)_Y\times U(1)_{R}\) |
---|---|

Quarks | \({Q_L}_i (3,2,\frac{1}{6},{0})\) |

\({u_R}_i (3,1,\frac{2}{3},{R_H})\) | |

\({d_R}_i (3,1,-\frac{1}{3},\{{-R_H}\})\) | |

Leptons | \({L_L}_i (1,2,-\frac{1}{2},{0})\) |

\({\ell _R}_i (1,1,-1,{-R_H})\) | |

\({\nu _R}_i (1,1,0,\{{R_{\nu _1},R_{\nu _2},R_{\nu _3}}\})\) | |

Higgs | \(H (1,2,\frac{1}{2},{R_H})\) |

*O*(10) TeV scale or above. We assume that \(U(1)_R\) gets broken spontaneously by the VEV of a SM singlet \(\chi (1,1,0,R_{\chi })\) that must carry non-zero charge (\(R_{\chi }\ne 0\)) under \(U(1)_R\). As a result of this symmetry breaking, the imaginary part of \(\chi \) will be eaten up by the corresponding gauge boson \(X_{\mu }\) to become massive. Since EW symmetry also needs to break down around the

*O*(100) GeV scale, one can compute the masses of the gauge bosons from the covariant derivatives associated with the SM Higgs

*H*and the SM singlet scalar \(\chi \):

*Z*-boson and a heavy \(Z^{\prime }\)-boson (\(M_{Z}<M_{Z^{\prime }}\)). The corresponding masses are given by:

*Z*-boson gets modified as a consequence of \(U(1)_R\) gauge extension. Precision measurement of the SM

*Z*-boson puts bound on the scale of the new physics. From the experimental measurements, the bound on the lower limit of the new physics scale can be found by imposing the constraint \(\Delta M_{Z} \le 2.1\) MeV [63]. For our case, this bound can be translated into:

*H*and the SM singlet scalar \(\chi \) that breaks \(U(1)_R\) will be used in Sects. 3 and 7).

Couplings of the fermions with the new gauge boson. Here we use the notation: \(c_{2w}=\cos (2\theta _w)\). \({\mathcal {N}}_{L,R}\) is any vector-like fermion singlet under the SM and carries \(R_{{\mathcal {N}}}\) charge under \(U(1)_R\). If a model does not contain vector-like fermions, we set \(R_{{\mathcal {N}}}=0\)

Fermion, \(\psi \) | Coupling, \(g_{\psi }\) |
---|---|

Quarks | \(g_{u_L}=-\frac{1}{6}\frac{g}{c_w}(1+2c_{2w})s_X\) |

\(g_{d_L}=\frac{1}{6}\frac{g}{c_w}(2+c_{2w})s_X\) | |

\(g_{u_R}=\frac{2}{3}\frac{g}{c_w}s^2_ws_X+ g_Rc_XR_H\) | |

\(g_{d_R}=-\frac{1}{3}\frac{g}{c_w}s^2_ws_X- g_Rc_XR_H\) | |

Leptons | \(g_{\nu _L}=-\frac{1}{2}\frac{g}{c_w}s_X\) |

\(g_{\ell _L}=\frac{1}{2}\frac{g}{c_w}c_{2w}s_X\) | |

\(g_{\ell _R}=-\frac{g}{c_w}s^2_ws_X- g_Rc_XR_H\) | |

\(g_{\nu _{R_i}}=g_Rc_XR_{\nu _i}\) | |

Vector-like fermions | \(g_{{\mathcal {N}}}=g_Rc_XR_{{\mathcal {N}}}\) |

Based on the framework introduced in this section, we construct various minimal models of Dirac neutrino masses in Sect. 3 and study various phenomenology in the subsequent sections.

## 3 Dirac neutrino mass models

By adopting the set-up as discussed above in this section, we construct models of Dirac neutrino masses. Within this set-up, if the solution \(R_{\nu _i}=R_H\) is chosen which is allowed by the anomaly cancellation conditions, then tree-level Dirac mass term \(y_{\nu }v_H {\overline{\nu }}_L\nu _R\) is allowed and observed oscillation data requires tiny Yukawa couplings of order \(y_{\nu }\sim 10^{-11}\). This is expected not to be a natural scenario, hence due to aesthetic reason we generate naturally small Dirac neutrino mass by exploiting the already existing symmetries in the theory. This requires the implementation of the flavor non-universal solution of the anomaly cancellation conditions, in such a scenario \(U(1)_R\) symmetry plays the vital role in forbidding the direct Dirac mass term and also all Majorana mass terms for the neutrinos.

In this section, we explore three different models within our framework where neutrinos receive naturally small Dirac mass either at the tree-level or at the one-loop level. Furthermore, we also show that the stability of DM can be assured by a residual discrete symmetry resulting from the spontaneous symmetry breaking of \(U(1)_R\). In the literature, utilizing \(U(1)_R\) symmetry, two-loop Majorana neutrino mass is constructed with the imposition of an additional \({\mathcal {Z}}_2\) symmetry in [58, 59] and three types of seesaw cases are discussed, standard type-I seesaw in [60], type-II seesaw in [61] and inverse seesaw model in [62]. In constructing the inverse seesaw model, in addition to \(U(1)_R\), additional flavor dependent U(1) symmetries are also imposed in [62]. In all these models, neutrinos are assumed to be Majorana particles which is not the case in our scenario.

### 3.1 Tree-level dirac seesaw

^{2}For the realization of this scenario, we introduce three generations of vector-like fermions that are singlets under the SM: \({\mathcal {N}}_{L,R}(1,1,0,R_{{\mathcal {N}}})\). In this model, the quantum numbers of the multiplets are shown in Table 3 and the corresponding Feynman diagram for neutrino mass generation is shown in Fig. 1. This choice of the particle content allows one to write the following Yukawa coupling terms relevant for neutrino mass generation:

Quantum numbers of the fermions and the scalars in Dirac seesaw model

Multiplets | \(SU(3)_C\times SU(2)_L\times U(1)_Y\times U(1)_{R}\) |
---|---|

Leptons | \({L_L}_i (1,2,-\frac{1}{2},{0})\) |

\({\ell _R}_i (1,1,-1,{-1})\) | |

\({\nu _R}_i (1,1,0,\{{-5, 4, 4}\})\) | |

Scalars | \(H (1,2,\frac{1}{2},{1})\) |

\(\chi (1,1,0,{3})\) | |

Vector-like fermion | \({\mathcal {N}}_{L,R} (1,1,0,{1})\) |

In this scenario two chiral massless states appear, one of them is \(\nu _{R_1}\), which is a consequence of its charge being different from the other two generations. In principle, all three generations of neutrinos can be given Dirac mass if the model is extended by a second SM singlet \(\chi ^{\prime }(1,1,0,-6)\). When this field acquires an induced VEV all neutrinos become massive. This new SM singlet scalar, if introduced, gets an induced VEV from a cubic coupling of the form: \(\mu \chi ^2 \chi ^{\prime } +h.c.\). Alternatively, without specifying the ultraviolet completion of the model, a small Dirac neutrino mass for the massless chiral states can be generated via the dimension-5 operator \(\overline{{\mathcal {N}}}_L\nu _R\langle \chi \rangle \langle \chi \rangle /\Lambda \) once \(U(1)_R\) is broken spontaneously.

### 3.2 Simplest one-loop implementation

Quantum numbers of the fermions and the scalars in radiative Dirac model

Multiplets | \(SU(3)_C\times SU(2)_L\times U(1)_Y\times U(1)_{R}\) |
---|---|

Leptons | \({L_L}_i (1,2,-\frac{1}{2},{0})\) |

\({\ell _R}_i (1,1,-1,{-1})\) | |

\({\nu _R}_i (1,1,0,\{{-5, 4, 4}\})\) | |

Scalars | \(H (1,2,\frac{1}{2},{1})\) |

\(\chi (1,1,0,{3})\) | |

\(S^+_1 (1,1,1,{0})\) | |

\(S^+_2 (1,1,1,{-3})\) |

This is the most minimal radiative Dirac neutrino mass mechanism which was constructed by employing a \({\mathcal {Z}}_2\) symmetry in [64] and just recently in [44, 52] by utilizing \(U(1)_{B-L}\) symmetry. As a result of the anti-symmetric property of the Yukawa couplings \(y^{S_1}\), one pair of chiral states remains massless to all orders, higher dimensional operators cannot induce mass to all the neutrinos. As already pointed out, neutrino oscillation data is not in conflict with one massless state.

### 3.3 Scotogenic dirac neutrino mass

Quantum numbers of the fermions and the scalars in scotogenic Dirac neutrino mass model

Multiplets | \(SU(3)_C\times SU(2)_L\times U(1)_Y\times U(1)_{R}\) |
---|---|

Leptons | \({L_L}_i (1,2,-\frac{1}{2},{0})\) |

\({\ell _R}_i (1,1,-1,{-1})\) | |

\({\nu _R}_i (1,1,0,\{{-5, 4, 4}\})\) | |

Scalars | \(H (1,2,\frac{1}{2},{1})\) |

\(\chi (1,1,0,{3})\) | |

\(S (1,1,0,{-\frac{7}{2}})\) | |

\(\eta (1,2,\frac{1}{2},{\frac{1}{2}})\) | |

Vector-like fermion | \({\mathcal {N}}_{L,R} (1,1,0,{\frac{1}{2}})\) |

*S*and the second Higgs doublet \(\eta \) do not acquire any VEV and the loop-diagram is completed by making use of the quartic coupling \(V\supset \lambda _D\eta ^{\dagger }H\chi S+h.c.\). Here for simplicity, we assume that the SM Higgs does not mix with the other CP-even states, consequently, the mixing between \(S^0\) and \(\eta ^0\) originates from the quartic coupling \(\lambda _{D}\) (and similarly for the CP-odd states). Then the neutrino mass matrix is given by:

Since \(\nu _{R_1}\) carries a charge of \(-5\), a pair of chiral states associated with this state remains massless. However, in this scotogenic version, unlike the simplest one-loop model presented in the previous sub-section, all the neutrinos can be given mass by extending the model further. Here just for completeness, we discuss a straightforward extension, even though this is not required since one massless neutrino is not in conflict with the experimental data. If the model defined by Table 5 is extended by two SM singlets \(\chi ^{\prime }(1,1,0,-6)\) and a \(S^{\prime }(1,1,0,\frac{11}{2})\), all the neutrinos will get non-zero mass. The VEV of the field \(\chi ^{\prime }\) can be induced by the allowed cubic term of the form \(\mu \chi ^2\chi ^{\prime }+h.c.\) whereas, \(S^{\prime }\) does not get any induced VEV.

Here we comment on the DM candidate present in this model. As aforementioned, we do not introduce new symmetries by hand to stabilize the DM. In search of finding the unbroken symmetry, first, we rescale all the \(U(1)_R\) charges of the particles in the theory given in Table 5 including the quark fields in such a way that the magnitude of the minimum charge is unity. From this rescaling, it is obvious that when the \(U(1)_R\) symmetry is broken spontaneously by the VEV of the \(\chi \) field that carries six units of rescaled charge leads to: \(U(1)_R\rightarrow {\mathcal {Z}}_6\). However, since the SM Higgs doublet carries a charge of two units under this surviving \({\mathcal {Z}}_6\) symmetry, its VEV further breaks this symmetry down to: \({\mathcal {Z}}_6\rightarrow {\mathcal {Z}}_2\). This unbroken discrete \({\mathcal {Z}}_2\) symmetry can stabilize the DM particle in our theory. Under this residual symmetry, all the SM particles are even, whereas only the scalars \(S, \eta \) and vector-like fermions \({\mathcal {N}}_{L,R}\) are odd and can be the DM candidate. Phenomenology associated with the DM matter in this scotogenic model will be discussed in Sect. 6.

## 4 Running of the \(U(1)_R\) gauge coupling

Utilizing the basic set-up defined in Sect. 2, we have constructed three different models in Sect. 3, which correspond to three different coefficients \(b_R=\{179/3, 56, 731/12\}\) for the Dirac seesaw, simplest one-loop, and Scotogenic models respectively. For demonstration purpose, we choose \(\mu _0=10\) TeV and show the scale \(\Lambda _{Landau}\) as a function of gauge coupling in Fig. 4 for the three different models discussed in this work. As expected, the higher the value of \(g_R\), smaller the \(\Lambda _{Landau}\) gets.

## 5 Lepton flavor violation

^{3}

In Fig. 6, we have shown the contour plots for branching ratio predictions for the cLFV processes: \(\mu \rightarrow e + \gamma \) (top left), \(\tau \rightarrow e + \gamma \) (top right) and \(\tau \rightarrow \mu + \gamma \) (bottom) as a function of mass \((m_{H_1})\) and Yukawa \(\left| y^{S_1}_{i \alpha } y_{i \beta }^{S_1*}\right| \) plane in simplest one-loop Dirac neutrino mass model. Red solid lines indicate the current bounds on branching ratios: 4.2 \(\times 10^{-13}\) [68] for the \(\mu \rightarrow e + \gamma \) (top left) process, 3.3 \(\times 10^{-8}\) [69] for the \(\tau \rightarrow e + \gamma \) (top right) process and 4.4 \(\times 10^{-8}\) [69] for the \(\tau \rightarrow \mu + \gamma \) (top right) process. Red dashed lines indicate the future projected bounds on the branching ratios: 6 \(\times 10^{-14}\) [70] for the \(\mu \rightarrow e + \gamma \) (top left), 3 \(\times 10^{-9}\) [71] for the \(\tau \rightarrow e + \gamma \) (top right) and 3 \(\times 10^{-9}\) [71] for the \(\tau \rightarrow \mu + \gamma \) (top right) processes respectively. For simplicity, we choose \(m_{H_2}=m_{H_1}+100\) GeV. As we can see from the Fig. 6, \(\mu \rightarrow e + \gamma \) is the most constraining cLFV process in this model. Since this could lead to sizable rates, it can be tested in the upcoming experiments.

## 6 Dark matter phenomenology

*S*in Refs. [75, 76]. In the following analysis, we consider \({\mathcal {N}}_1\) to be the lightest among all of these particles, hence serves as a good candidate for DM (for simplicity we will drop the subscript from \({\mathcal {N}}_1\) in the following). We aim to study the DM phenomenology associated with the vector-like Dirac fermion \(\mathcal {N_{L,R}}\) here. Due to Dirac nature of the dark matter, the phenomenology associated with it is very different from the Majorana fermionic dark matter scenario [77].

*t*- channel scalar (\(S, \eta _0,\eta ^+\)) exchanges. The representative Feynman diagrams for the annihilation of DM particle are shown in Fig. 8. It is important to mention that for the Majorana fermionic dark matter case, the annihilation rate is

*p*-wave (\(\sim v^2\)) suppressed since the vector coupling to a self-conjugate particle vanishes, on the contrary, the annihilation rate is not suppressed for the Dirac scenario (

*s*-wave). The non-relativistic form for this annihilation cross-section can be found here [80]. In Fig. 9, we analyze the dark matter relic abundance as a function of dark matter mass \(m_{DM}\) for various gauge couplings \(g_R\) (left) and \(Z'\) boson masses (right). Horizontal red and blue lines represent WMAP [78] relic density constraint \(0.094 \le \Omega _{\mathrm{DM}} h^2 \le 0.128\) and the PLANCK constraint \(0.112 \le \Omega _{\mathrm{DM}} h^2 \le 0.128\) [79] respectively. For simplicity, we set \(m_{Z'}=10\) TeV (left) and provide the relic abundance prediction for two different values of gauge coupling (\(g_R= 0.1\) and 0.277). For the right plot in Fig. 9, DM relic abundance is analyzed for two different values of the \(Z'\) masses \(m_{Z'}=10\) and 20 TeV setting \(g_R= 0.1\). As expected, we can satisfy the WMAP [78] relic density constraint \(0.094 \le \Omega _{\mathrm{DM}} h^2 \le 0.128\) and the PLANCK constraint \(0.112 \le \Omega _{\mathrm{DM}} h^2 \le 0.128\) [79] for most of the parameter space in our model as long as \(m_{DM}\) is not too far away from \(m_{Z'}/2\) mass. Throughout our DM analysis, we make sure that we are consistent with the SM

*Z*- boson mass correction constraint while choosing specific \(g_R\) and \(m_{Z'}\) values.

In addition to the relic density, we also take into account the constraints from DM direct detection experiments. In case of Majorana fermionic dark matter, at the tree-level, the spin-independent DM-nucleon scattering cross-section vanishes. However, at the loop-level, the spin-independent operators can be generated and hence it is considerably suppressed. The dominant direct detection signal remains the spin-dependent DM-nucleon scattering cross-section which for the Majorana fermionic dark matter is four times that for the Dirac-fermionic dark matter case. In general, the \(Z'\) interactions induce both spin-independent (SI) and spin-dependent (SD) scattering with nuclei. The representative Feynman diagram for the DM-nucleon scattering is shown in Fig. 10. Particularly, in the scotogenic Dirac neutrino mass model, DM can interact with nucleon through *t*-channel \(Z'\) exchange. Hence, large coherent spin-independent scattering may occur since both dark matter and the valence quarks of nucleons possess vector interactions with \(Z'\) and this process is severely constrained by present direct detection experiment bounds. The DM-nucleon scattering cross-section is estimated in Ref. [80]. In Fig. 11, we analyze the spin-independent dark matter-nucleon scattering cross-section, \(\sigma \) (in pb) as a function of the dark matter mass \(m_{DM}\) with different gauge coupling \(g_R=0.2, 0.277\). For this plot, we set \(m_{Z'}=10\) TeV. Yellow, blue and green color solid lines represent current direct detection cross-section limits from LUX-2017 [81], XENON1T [82] and PandaX-II (2017) [83] experiments respectively. As can be seen from Fig. 11, we can satisfy all the present direct detection experiment bounds as long as we are consistent with the other severe bounds on mass \(m_{Z'}\) and \(g_R\) arising from colliders to be discussed in the next section.

## 7 Collider implications

Models with extra \(U(1)_R\) implies a new \(Z^{\prime }\) neutral boson, which contains a plethora of phenomenological implications at colliders. Here we mainly focus on the phenomenology of the heavy gauge boson \(Z^{\prime }\) emerging from \(U(1)_R\).

### 7.1 Constraint on heavy gauge boson \(Z^{\prime }\) from LEP

*f*indicates all the fermions in the model and \(\eta \) takes care of the chirality structure coefficients. The exchange of the new \(Z^{\prime }\) boson state emerging from \(U(1)_R\) can be stated in a similar way:

### 7.2 Heavy gauge boson \(Z^{\prime }\) at the LHC

^{4}The present lack of any signal for di-lepton resonances at the LHC dictates the stringent bound on the \(Z^{\prime }\) mass and \(U(1)_R\) coupling constant \(g_R\) in our model as the production cross-section solely depends on these two free parameters. Throughout our analysis, we consider that the mixing \(Z-Z^{\prime }\) angle is not very sensitive (\(s_X=0\)). In order to obtain the constraints on these parameter space, we use the dedicated search for new resonant high-mass phenomena in di-electron and di-muon final states using 36.1 \(\hbox {fb}^{-1}\) of proton-proton collision data, collected at \(\sqrt{s} = 13\) TeV by the ATLAS collaboration [85]. The searches for high mass phenomena in di-jet final states [86] will also impose bound on the model parameter space, but it is somewhat weaker than the di-lepton searches due to large QCD background. For our analysis, we implement our models in FeynRules_v2.0 package [87] and simulate the events for the process \(pp \rightarrow Z^{\prime } \rightarrow e^+ e^- ({\mu }^+ {\mu }^-) \) with MadGraph5_aMC@NLO_v3_0_1 code [88]. Then, using parton distribution function (PDF) NNPDF23_lo_as_0130 [89], the cross-section and cut efficiencies are estimated. Since no significant deviation from the SM prediction is observed in experimental searches [85] for high-mass phenomena in di-lepton final states, the upper limit on the cross-section is derived from the experimental analyses [85] using \(\sigma \times \) BR \(= N_{rec}/(A \times \epsilon \times \int L dt)\), where \(N_{rec}\) is the number of reconstructed heavy \(Z^{\prime }\) candidate, \(\sigma \) is the resonant production cross-section of the heavy \(Z^{\prime }\), BR is the branching ratio of \(Z^{\prime }\) decaying into di-lepton final states , \(A\times \epsilon \) is the acceptance times efficiency of the cuts for the analysis. In Fig. 12, we have shown the upper limits on the cross-section at 95\(\%\) C.L. for the process \(pp \rightarrow Z^{\prime } \rightarrow l^+ l^- \) as a function of the di-lepton invariant mass using ATLAS results [85] at \(\sqrt{s} = 13\) TeV with \(36.1 \hbox {fb}^{-1}\) integrated luminosity. Red solid, dashed and dotted lines in Fig. 12 indicate the model predicted cross-section for three different values of \(U(1)_R\) gauge coupling constant \(g_R=0.5, ~0.3, ~0.1\) respectively. We find that \(Z^{\prime }\) mass should be heavier

^{5}than 4.4, 3.9 and 2.9 TeV for three different values of \(U(1)_R\) gauge coupling constant \(g_R=0.5, ~0.3\) and 0.1.

In Fig. 13, we have shown all the current experimental bounds in \(M_{Z^{\prime }} - g_R\) plane. Red meshed zone is excluded from the current experimental di-lepton searches [85]. The cyan meshed zone is forbidden from the LEP constraint [84] and the blue meshed zone is excluded from the limit on SM Z boson mass correction: \(\frac{1}{3}M_{Z^{\prime }}/g_R > 12.082\) TeV as aforementioned. We can see from Fig. 13 that the most stringent bound in \(M_{Z^{\prime }} - g_R \) plane is coming from direct \(Z^{\prime }\) searches at the LHC. After imposing all the current experimental bounds, we analyze the future discovery prospect of this heavy gauge boson \(Z^{\prime }\) within the allowed parameter space in \(M_{Z^{\prime }} - g_R \) plane looking at the prompt di-lepton resonance signature at the LHC. We find that a wider region of parameter space in \(M_{Z^{\prime }} - g_R\) plane can be tested at the future collider experiment. Black, green, purple and brown dashed lines represent the projected discovery reach at \(5 \sigma \) significance at 13 TeV LHC for 100 \(\hbox {fb}^{-1}\), 300 \(\hbox {fb}^{-1}\), 500 \(\hbox {fb}^{-1}\) and 1 \(\hbox {ab}^{-1}\) luminosities. On the top of that, the right-handed chirality structure of \(U(1)_R\) can be investigated at the LHC by measuring Forward-Backward (FB) and top polarization asymmetries in \(Z^{\prime } \rightarrow t{\bar{t}}\) mode [92] and which can discriminate our \(U(1)_R\) \(Z^{\prime }\) interaction from the other \(Z^{\prime }\) interactions in \(U(1)_{B-L}\) model. The investigation of other exotic decay modes \(( {\mathcal {N}}{\mathcal {N}}, \mathrm ~ \chi \chi , \mathrm ~ S_2^+ S_2^- )\) of heavy \(Z^{\prime }\) is beyond the scope of this article and shall be presented in a future work since these will lead to remarkable multi-lepton or displaced vertex signature [93, 94, 95, 96, 97, 98, 99] at the colliders.

### 7.3 Heavy gauge boson \(Z^{\prime }\) at the ILC

Due to the point-like structure of leptons and polarized initial and final state fermions, lepton colliders like ILC will provide much better precision of measurements. The purpose of the \(Z^{\prime }\) search at the ILC would be either to help identifying any \(Z^{\prime }\) discovered at the LHC or to extend the \(Z^{\prime }\) discovery reach (in an indirect fashion) following effective interaction. Even if the mass of the heavy gauge boson \(Z^{\prime }\) is too heavy to directly probe at the LHC, we will show that by measuring the process \(e^+e^- \rightarrow f^+ f^-\), the effective interaction dictated by Eq. 7.2 can be tested at the ILC. Furthermore, analysis with the polarized initial states at ILC can shed light on the chirality structure of the effective interaction and thus it can distinguish between the heavy gauge boson \(Z^{\prime }\) emerging from \(U(1)_R\) extended model and the \( Z^{\prime }\) from other *U*(1) extended model such as \(U(1)_{B-L}\). The process \(e^+e^- \rightarrow f^+ f^-\) typically exhibits asymmetries in the distributions of the final-state particles isolated by the angular- or polarization-dependence of the differential cross-section. These asymmetries can thus be utilized as a sensitive measurement of differences in interaction strength and to distinguish a small asymmetric signal at the lepton colliders. In the following, the asymmetries (Forward–Backward asymmetry, Left-Right asymmetry) related to this work will be described in great detail.

#### 7.3.1 Forward–backward asymmetry

The differential cross-section in Eq. 7.12 is asymmetric in polar angle, leading to a difference of cross-sections for \(Z^{\prime }\) decays between the forward and backward hemispheres. Earlier, LEP experiment [84] used Forward–backward asymmetries to measure the difference in the interaction strength of the *Z*-boson between left-handed and right-handed fermions, which gives a precision measurement of the weak mixing angle. Here we will show that our framework leads to sizable and distinctive Forward–Backward (FB) asymmetry discriminating from other models and which can be tested at the ILC, since only the right-handed fermions carry non-zero charges under the \(U(1)_R\). For earlier analysis of FB asymmetry in the context of other models as well as model-independent analysis see for example Refs. [60, 62, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110].

*Z*boson mass correction. Red dashed (solid) line represents \(\Delta A_{FB}\) for \(U(1)_R\) case for left (right) handed polarized cross-sections of the \(e^+ e^- \rightarrow \mu ^+ \mu ^-\) process, whereas blue dotted (dashed) line indicates \(\Delta A_{FB}\) for \(U(1)_{B-L}\) case for left (right) handed polarized cross-sections. From Fig. 14, we find that in case of \(U(1)_R\) model, it provides significant difference of \(\Delta A_{FB}\) for \(\sigma _R\) and \(\sigma _L\) due to the right-handed chirality structure of \( Z^{\prime }\) interaction from \(U(1)_R\), while in the case of \(U(1)_{B-L}\) model, it provides small difference. Hence by comparing the difference of \(\Delta A_{FB}\) for differently polarized cross-section \(\sigma _R\) and \(\sigma _L\) at the ILC, we can easily discriminate the \( Z^{\prime }\) interaction from \(U(1)_R\) and \(U(1)_{B-L}\) model. As we can see from Fig. 14 that there are significant region for \(M_{Z^{\prime }}/3 g_R> 12.082\) TeV which can give more than \(2\sigma \) sensitivity for FB asymmetry by looking at \(e^+ e^- \rightarrow \mu ^+ \mu ^-\) process at the ILC. We can also expect much higher sensitivity while combining different final fermionic states such as other leptonic modes (\(e^+ e^-, {\tau }^+{\tau }\)) as well as hadronic modes

*jj*. Moreover, the sensitivity to \(Z^{\prime }\) interactions can be enhanced by analyzing the scattering angular distribution in details, although it is beyond the scope of our paper.

#### 7.3.2 Left–right asymmetry

*Z*boson mass correction. Red (blue) solid line represents \(\Delta A_{LR}\) for \(U(1)_R\) (\(U(1)_{B-L}\)) case. From Fig. 15, we find that in case of \(U(1)_R\) model, it provides remarkably large LR asymmetry \(\Delta A_{LR}\) due to the right-handed chirality structure of \( Z^{\prime }\) interaction from \(U(1)_R\), while in case of \(U(1)_{B-L}\) model, it gives a smaller contribution. Hence by comparing the difference of \(\Delta A_{LR}\) at the ILC, we can easily discriminate the \( Z^{\prime }\) interaction from \(U(1)_R\) and \(U(1)_{B-L}\) model. As we can see from Fig. 15 that there is a significant region for \(M_{Z^{\prime }}/3 g_R> 12.082\) TeV which can give more than \(3\sigma \) sensitivity for LR asymmetry by looking at \(e^+ e^- \rightarrow \mu ^+ \mu ^-\) process at the ILC. Even if, we can achieve \(5 \sigma \) sensitivity for a larger parameter space in our framework if integrated luminosity of ILC is upgraded to 5 \(\hbox {ab}^{-1}\). Although, measurement of both the FB and LR asymmetries at the ILC can discriminate \(Z^{\prime }\) interaction for \(U(1)_R\) model from other

*U*(1) extended models such as \(U(1)_{B-L}\) model, it is needless to mention that the LR asymmetry provides much better sensitivity than the FB asymmetry in our case. In Fig. 16, we have shown the survived parameter space in \(M_{Z^{\prime }} - g_R\) plane satisfying all existing bounds and which can be probed at the ILC in future by looking at LR asymmetry strength. Green and yellow shaded zones correspond to sensitivity confidence levels 1\(\sigma \) and 2\(\sigma \) by measuring LR asymmetry for \(U(1)_R\) extended model at the ILC. For higher \(Z^{\prime }\) mass (above \(\sim \) 10 TeV), it is too heavy to directly produce and probe at the LHC looking at prompt di-lepton signature. On the other hand, ILC can probe the heavy \(Z^{\prime }\) effective interaction and LR asymmetry can pin down/distinguish our \(U(1)_R\) model from other existing

*U*(1) extended model for a large region of the parameter space. Thus, \(Z^{\prime }\) search at the ILC would help to identify the origin of \(Z^{\prime }\) boson as well as to extend the \(Z^{\prime }\) discovery reach following effective interaction.

## 8 Constraint from cosmology

*g*(

*T*) is the relativistic degrees of freedom at temperature T, with the well-known quantities \(g(T_{dec}^{\nu _L})=43/4\) and \(T_{dec}^{\nu _L}=2.3\) MeV [114]. For the following computation, we take the temperature-dependent degrees of freedom from the data listed in Table S2 of Ref. [115], and by utilizing the cubic spline interpolation method, we present

*g*as a function of

*T*in Fig. 17 (left plot).

*f*respectively.

By plugging Eqs. (8.3)–(8.5) in Eq. (8.2) and then solving numerically, we present our result of \(\Delta N_{eff}\) as a function of \(M_{Z^{\prime }}/g_R\) in Fig. 17 (right plot). From this figure, one sees that cosmology provides strong bound on the mass of the new gauge boson based on the associated decoupling temperature of the right-handed neutrinos. The blue curve corresponds to the contribution of all the three right-handed neutrinos and the red dashed line represents the current experimental upper bound on the deviation of \(\Delta N_{eff}\). This bound puts the restriction \(M_{Z^{\prime }}/g_R \gtrsim 26.5\) TeV, which is quite stronger than the LEP bound \(M_{Z^{\prime }}/g_R \gtrsim 3.59\) TeV, however, lies within the constraint provided by the SM *Z*-boson mass correction \(M_{Z^{\prime }}/g_R \gtrsim 36.2\) TeV. The framework presented in this work puts larger bound on the mass of the new gauge boson from cosmology due to large charge assignment of the right-handed neutrinos compared to the conventional \(U(1)_{B-L}\) models with universal charge, \(M_{Z^{\prime }}/g_{B-L} \gtrsim 14\) TeV [119, 120].

## 9 Conclusions

We believe that the scale of new physics is not far from the EW scale and a simple extension of the SM should be able to address a few of the unsolved problems of the SM. Adopting this belief, in this work, we have explored the possibility of one of the most minimal gauge extensions of the SM which is \(U(1)_R\) that is responsible for generating Dirac neutrino mass and may also stabilize the DM particle. Cancellations of the gauge anomalies are guaranteed by the presence of the right-handed neutrinos that pair up with the left-handed partners to form Dirac neutrinos. Furthermore, this \(U(1)_R\) symmetry is sufficient to forbid all the unwanted terms for constructing naturally light Dirac neutrino mass models without imposing any additional symmetries by hand. The chiral non-universal structure of our framework induces asymmetries, such as forward–backward asymmetry and especially left–right asymmetry that are very distinct compared to any other *U*(1) models. By performing detailed phenomenological studies of the associated gauge boson, we have derived the constraints on the \(U(1)_R\) model parameter space and analyzed the prospect of its testability at the collider such as at LHC and ILC. We have shown that a heavy \(Z^{\prime }\) (emerging from \(U(1)_R\)), even if its mass is substantially higher than the center of mass energy available at the ILC, would manifest itself at tree-level by its propagator effects producing sizable contributions to the LR asymmetry or FB asymmetry. This can be taken as an initial guide to explore the \(U(1)_R\) model at colliders. These models can lead to large lepton flavor violating observables which we have studied and they could give a complementary test for these models. In this work, we have also analyzed the possibility of having a viable Dirac fermionic DM candidate stabilized by the residual discrete symmetry originating from \(U(1)_R\), which connects to SM via \(Z'\) portal coupling in a framework that also cater for neutrino mass generation. The DM phenomenology is shown to be crucially dictated by the interaction of \({\mathcal {N}}\) with \(Z'\). Furthermore, we have inspected the constraints coming from the cosmological measurements and compared this result with the different collider bounds. For a comparison, here we provide a benchmark point by fixing the gauge coupling \(g_R=0.056\). With this, the current lower bound on the \(Z^{\prime }\) mass is \(M_{Z^{\prime }}> 4.25\) TeV from 13 TeV LHC data with \(36.1 fb^{-1}\) luminosity, and the future projection reach limit translates into \(M_{Z^{\prime }}> 4.67\) TeV with \(100 fb^{-1}\) luminosity. Whereas for the same value of the gauge coupling, the ILC has the discovery reach of 4.63 TeV at the \(2\sigma \) confidence level looking at the left–right asymmetry. The corresponding bounds from LEP, *Z*-boson mass correction and from cosmology are \(M_{Z^{\prime }}> 0.2, 2, 1.49\) TeV respectively, which are somewhat weaker compared to LHC and ILC bounds. To summarize, the presented Dirac neutrino mass models are well motivated and have rich phenomenology.

## Footnotes

## Notes

### Acknowledgements

We thank K. S. Babu, Bhupal Dev and S. Nandi for useful discussions. The work of SJ and VPK was in part supported by US Department of Energy Grant Number DE-SC 0016013. The work of SJ was also supported in part by the Neutrino Theory Network Program. SJ thanks the Theoretical Physics Department at Washington University in St. Louis for warm hospitality during the completion of this work.

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