# Phenomenology of the gauge symmetry for right-handed fermions

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## Abstract

In this paper we investigate the phenomenology of the U(1) gauge symmetry for right-handed fermions, where three right-handed neutrinos are introduced for anomalies cancellations. Constraints on the new gauge boson \(Z_{\mathbf {R}}\) from \(Z-Z^\prime \) mixing as well as the upper bound of \(Z^\prime \) production cross section in di-lepton channel at the LHC are presented. We further study the neutrino mass and the phenomenology of \(Z_{\mathbf {R}}\)-portal dark matter in this model. The lightest right-handed neutrino can be the cold dark matter candidate stabilized by a \(Z_2\) flavor symmetry. Our study shows that active neutrino masses can be generated via the modified type-II seesaw mechanism; right-handed neutrino is available dark matter candidate for its mass being very heavy, or for its mass at near the resonant regime of the SM Higgs and(or) the new bosons; constraint from the dilepton search at the LHC is stronger than that from the \(Z-Z^\prime \) mixing only for \(g_\mathbf {R}<0.121\), where \(g_\mathbf {R}\) is the new gauge coupling.

## 1 Introduction

In this paper phenomenology relevant to the \(U(1)_{\mathbf {R}}\) is investigated. We first focus on constraint on the model from \(Z-Z^\prime \) mixing since there is tree-level mixing between *Z* and \(Z^\prime \) in the local \(U(1)_{\mathbf {R}}\) model. Mixing angles of the \(Z_\mathbf {R}\) with *Z* and \(\gamma \) as well as the mass spectrum of gauge bosons are calculated. It shows that the \(Z-Z^\prime \) mixing puts a lower bound on \(v_\Phi /v\), which is the function of \(g_{\mathbf {R}}\), where *v* is the vacuum expectation value (VEV) of the SM Higgs, \(v_\Phi \) is the VEV of the scalar singlet \(\Phi \) that breaks the \(U(1)_{\mathbf {R}}\) spontaneously, \(g_{\mathbf {R}} \) is the gauge coupling of \(U(1)_{\mathbf {R}}\), while the precisely measured *Z* boson mass puts a stronger constraint on \(v_\Phi /v\): \(v_\Phi /v>73.32\).

Then we study how to naturally realize neutrino masses in the \(U(1)_{\mathbf {R}}\) model. The discovery of the neutrino oscillations has confirmed that neutrinos are massive and lepton flavors are mixed, which provides the first evidence for new physics beyond the SM. Canonical seesaw mechanisms [18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30] provide a natural way in understanding the tiny but non-zero neutrino masses. In the \(U(1)_{\mathbf {R}}\) model the mass matrix of right-handed Majorana neutrinos originates from their Yukawa couplings with \(\Phi \), and their masses are thus at the TeV-scale, such that it predicts a tiny Yukawa coupling of right-handed neutrinos with left-handed lepton doublets if active neutrino masses are generated from the type-I seesaw mechanism. We show that neutrino masses can be generated from the modified type-II seesaw mechanism, where the scalar triplet carries no \(U(1)_{\mathbf {R}}\) charge and its coupling with other scalars breaks the \(\mathbf {B-L}\) explicitly.

After that we focus on the phenomenology of \(Z_\mathbf {R}\)-portal dark matter. The fact that about \(26.8\%\) of the universe is made of dark matter with relic abundance \(\Omega h^2\) = 0.1189, has been firmly established, while the nature of the dark matter is still unclear. Imposing the \(Z_2\)-symmetry on right-handed neutrinos only, the lightest right-handed neutrino \(\mathbf {N}\) can be cold dark matter candidate. We study constraint on the model from the observed dark matter relic density, the exclusion limits of the spin-independent direct detection cross section mediated by \(\Phi \) and *H*, as well as the spin-dependent direct detection cross section mediated by the \(Z_\mathbf {R}\). It shows that right-handed neutrino dark matter is available only for its mass at near the resonance of SM Higgs, new scalar singlet and the \(Z_\mathbf {R}\), or for its mass being very heavy.

Finally we investigate collider signatures of the \(Z_\mathbf {R}\) at the LHC.Comparing its production cross section at the LHC the with upper limits given by the ATLAS, we get the lower limit on the \(Z_{\mathbf {R}}\) mass, which is the function of \(g_\mathbf {R}\). It shows that the constraint from the dilepton search at the LHC is stronger than that from the \(Z-Z_\mathbf {R}\) mixing only for \(g_\mathbf {R}<0.121\).

The paper is organized as follows: In Sect. 2 we introduce the model in detail. In Sect. 3 we study constraint on the model from \(Z-Z^\prime \) mixing. Sections. 4 and 5 are focused on the neutrino mass and the dark matter phenomenology, respectively. We study the collider signature of \(Z_{\mathbf {R}}\) in Sect. 6. The last part is concluding remarks.

## 2 The model

Quantum numbers of various fields under the local \(U(1)_{\mathbf {R}}\), \(\ell _L\) and \(Q_L\) are left-handed fermion doublets, \(\mathbf {N}_R\) is right-handed neutrinos, \(\Phi \) is the electroweak singlet scalar

Fields | \(\ell \) | \(Q_L\) | \(\mathbf {N}_R\) | \(E_R\) | \(U_R\) | \(D_R\) | | \(\Phi \) |
---|---|---|---|---|---|---|---|---|

\(U(1)_\mathbf {R}\) | 0 | 0 | \(\beta \) | \(-\beta \) | \(\beta \) | \(-\beta \) | \(\beta \) | \(-2\beta \) |

\(Z_2\) | + | + | − | + | + | + | + | + |

The anomaly cancellation conditions of the \(U(1)_{\mathbf {R}}\)

Anomalies | Anomaly free conditions |
---|---|

\(SU(3)_C^2 U(1)_{\mathbf {R}}^{}\) | \(- 2 (\beta ) - 2 (-\beta ) =0 \) |

\(U(1)_Y^2 U(1)_{\mathbf {R}}^{} \) | \(-\left[ 3 \left( {2\over 3}\right) ^2 \beta +3 \left( {1 \over 3} \right) ^2 (-\beta )+ (-1)^2 (-\beta )\right] = 0\) |

\(U(1)_{\mathbf {R}}^2 U(1)_Y\) | \(-\beta ^2 \left[ 3 \times {2 \over 3} - 3 \times {1 \over 3} -1 \right] =0 \) |

\(U(1)_{\mathbf {R}}\) | \( -\left[ \beta +(- \beta )\right] - 3 [ \beta + (-\beta ) ]=0 \) |

\(U(1)_{\mathbf {R}}^3\) | \(-[\beta ^3 + (-\beta )^3] - 3 [\beta ^3 + (-\beta )^3] =0\) |

*v*the VEV of

*H*and \(\Phi \equiv (\phi +iG^0_s +v_\Phi )/\sqrt{2}\) with \(v_\Phi \) the VEV of \(\Phi \), \(Y_U,~Y_D\) and \(Y_E\) are \(3\times 3\) Yukawa matrices. The Yukawa interaction of right-handed neutrinos with left-handed lepton doublets is forbidden by the \(Z_2^{}\) discrete flavor symmetry and the lightest \(\mathbf {N}_R^{}\) is the cold dark matter candidate. Imposing the minimization conditions, one has

*v*and \(v_\Phi \), while all other parameters can be reconstructed by them. The mixing angle \(\alpha \) is constrained by the data from Higgs measurements at the LHC. Universal Higgs fits [38] to the data of ATLAS and CMS collaborations were performed in Refs. [9, 10, 39], and one has \(\cos \alpha >0.865\) at the 95% confidence level. The constraint from electroweak precision observables is usually weaker than that from universal Higgs fit [40]. For the beta functions of \(\lambda _i\) and \(g_{\mathbf {R}}\) as well as their impacts on the vacuum stability of the SM Higgs, we refer the reader to Ref. [17] for detail, where the stability of the SM vacuum can be enhanced by the new gauge coupling.

## 3 Vector boson Masses and Mixings

*Z*and \(Z_\mathbf {R}\) at the tree-level.

^{1}The mass matrix of neutral vector bosons in the basis \((B_\mu ^{},~W_\mu ^3,~B_\mu ^\prime )\) is given by:

*Z*and \(Z_{\mathbf {R}} \) are

Due to the \(Z-Z_\mathbf {R}\) mixing, \(Z_\mathbf {R}\) may decay into charged gauge boson pairs \(W^-W^+\), which can be used to place constraint on the \(Z-Z_\mathbf {R}\) mixing using diboson production at the LHC. It shows that the \(Z-Z_\mathbf {R}\) mixing should be less than \(0.7\sim 2\times 10^{-3}\) [41] derived from the data recoded by ATLAS and CMS collaborations at \(\sqrt{s}=13~\text {TeV}\) with integrated luminosities of \(13.2~\mathrm{fb^{-1}}\) and \(35.9~\mathrm{fb}^{-1}\), respectively. In Fig. 1, we show contours of the \(\theta _{13}\) (left-panel) and \(\theta _{23}^{}\) (right-panel) in the \(g_{\mathbf {R}}-v_\Phi /v\) plane. We take \(\alpha (M_Z)^{-1}=127.918\), \(\sin ^2 \theta _W (M_Z^{})=0.23122\) and \(M_Z=91.1876~\mathrm{GeV}\) [42], which are used to get values of *g* and \(g^\prime \) respectively. One can see from the right-panel of Fig. 1 that, the scale of the \(U(1)_{\mathbf {R}}\) breaking should be at least one order higher than the electroweak scale. According to Eq. (7), the Z-boson mass is slightly changed in the \(U(1)_\mathbf {R}^{}\) model. There is thus constraint from the precision measurement of the *Z* boson mass. We show in the Fig. 2 contours of \(\Delta M_Z^{} \), namely \(M_Z^{}-M_Z^\mathrm{observed}\), in the \(g_\mathbf {R}^{} -v_\Phi /v\) plane. It shows that \(\Delta M_Z^{} \) is insensitive to the \(g_{\mathbf {R}}\) when it is larger than 0.04. Using the ambiguity of the Z-boson mass given by the PDG [42], which is \(\Delta M_Z^{}<0.0021~\text {GeV}\), one has \(v_\Phi /v>73.32\) (\(v_\Phi >18~\text {TeV}\)).

Before proceeding, we discuss how to distinguish the \(U(1)_{\mathbf {R}}\) from the \(U(1)_{\mathbf {B-L}}\), which share many similarities. From previous study, it is clear that precision measurement of the *Z* boson mass at the CEPC and (or) ILC can test the \(U(1)_{\mathbf {R}}\), so here we mainly discuss how to distinguish them in colliders. First, \(Z_\mathbf {R}^{}\) couples to the SM Higgs, while there is no such interaction for the \(Z^\prime \) of \(U(1)_{\mathbf {B-L}}\). The new interaction takes the form: \( 1/2 g_R^2 Z_{\mathbf {R} \mu }^{} Z_\mathbf {R}^\mu (h^2 + 2 h v)\), where the effect of \(Z-Z_\mathbf {R}\) mixing is neglected. As a result, one can discriminate \(U(1)_{\mathbf {R}}\) from \(U(1)_\mathbf {B-L}\) via the process \(\bar{q} q \rightarrow Z_\mathbf {R}^{} + h\), with \(Z_\mathbf {R}\) decaying into di-lepton and *h* decaying into \(b\bar{b}\), in future 100 TeV proton-proton collider. Second, \(Z_\mathbf {R}\) only couple to right-handed fermions, while \(Z^\prime \) of \(U(1)_{\mathbf {B-L}}\) couples to both left-handed and right-handed fermions. One can write down partially-polarized differential cross section for the process \(e^-e^+ \rightarrow Z_\mathbf {R} (Z^\prime ) \rightarrow f \bar{f} \) at the CEPC and (or) ILC, with concrete helicity of initial electron(positron) and summed over the helicity of final states. It can then be applied to study the sensitivity to new gauge bosons through the measurement of a forward-backward asymmetry defined in Ref. [66] at lepton colliders. A detailed analysis will be out of the reach of this paper.

## 4 Neutrino masses

## 5 Dark matter

^{2}We evaluate the relic abundance of the dark matter and study its implications in dark matter direct detections in this section. The dark matter \(\mathbf {N}\) mainly couple to the \(Z_\mathbf {R}\), \(\hat{s}\) and \(\hat{h}\), with Lagrangian:

*h*and

*s*respectively. The interaction of \(\mathbf {N}\) with

*Z*can be neglected due to the tiny mixing angle \(\theta _{23}\).

^{3}

*n*is governed by the thermal average of reduced annihilation cross sections \(\langle \sigma v\rangle \), which can be approximated with the non-relativistic expansion: \(\langle \sigma v\rangle = a + b \langle v^2\rangle \). Contributions of various channels can be written as

*f*.

## 6 Collider signatures

*f*is (not) identical particle. \(Z_\mathbf {R}\) can also decay into diboson pair (\(W^+W^-\)) due to its mixing with

*Z*, which is suppressed by \(\theta _{23}^2\) and will be neglected in the following analysis. We show in the Fig. 5 the branching ratio of \(Z_\mathbf {R}\) decaying into various final states as the function of \(m_\mathbf {N}\) by setting \(m_{Z_\mathbf {R}}=3.6~\text {TeV}\), where the solid, dashed and dotted lines correspond to the decay channel of \(\overline{\mathbf {N}}\mathbf {N}\), \(\bar{e} e\) and \(\bar{t} t\) respectively. It shows that the branching ratio of dilepton with discrete flavor is about \(4.5\%\).

*Z*boson mass only for \(g_\mathbf {R}<0.121\) (\(M_{Z_R}<4.37~\text {TeV}\)).

## 7 Conclusion

Although the \(U(1)_{\mathbf {R}}\) extension to the SM shares the same merit as the \(U(1)_{\mathbf {B-L}}\) extension of the SM on anomalies cancellations, its phenomenology was not investigated in detail in any reference except its effect in the vacuum stability of the SM Higgs. In this paper we constrained the parameter space of the model using the updated results of the \(Z-Z^\prime \) mixing as well as the search of new resonance in dilepton channel at the LHC. Our investigation shows that the constraint from the dilepton search at the LHC is stronger than that from the \(Z-Z_\mathbf {R}\) mixing derived from diboson search at the LHC and the precision measurement of *Z* boson mass only for \(g_\mathbf {R}<0.121\). We further studied the phenomenology of the \(Z_\mathbf {R}\)-portal dark matter and the possibility of generating active neutrino masses in the same model. It shows that the right-handed neutrino dark matter is self-consistent for its mass at near the resonant regime of \(\hat{h}\), \(\hat{s}\) and \(Z_\mathbf {R}\) or for its mass being very heavy; the Majorana masses of active neutrinos can be generated from the modified type-II seesaw mechanism. It should be mentioned that the collider signature of \(\hat{s}\) is Di-Higgs in various channels. We refer the reader to Ref. [65] for the Di-Higgs searches at various colliders for detail. \(\hat{s}\) may also be searched at the LHC via the \(pp\rightarrow \hat{s}\rightarrow Z_\mathbf {R}Z_\mathbf {R}\rightarrow \ell _\alpha ^+ \ell _\alpha ^- \ell _\beta ^+\ell ^-_\beta \) process if \(m_{\hat{s}} > 2 m_{Z_\mathbf {R}}\).

**[Note added]:** When this paper was being finalized, the paper [66] appeared, which partially overlaps with this one in discussing constraint from the precision measurement of Z-boson mass, They use approximated formulae when do this analysis, while we present both full analytical and numerical results in this paper. Their study is largely complementary to ours.

## Footnotes

- 1.
There might be kinematic mixing term \(\varepsilon B^{\mu \nu }_{} B^\prime _{\mu \nu }\) in the Lagrangian, which may exist in any U(1) extension of the SM and may lead to the mixing between neutral gauge bosons. Here we assume \(\varepsilon \rightarrow 0\), because the tree-level mixing is the key feature that discriminate our model from other U(1)s. Furthermore, \(Z-Z^\prime \) mixing may arise at loop-level, which is subdominant and is thus neglected. For expressions of loop induced \(Z-Z^\prime \) mixing, we refer the reader to the appendix of our earlier paper Ref. [52] for detail.

- 2.
- 3.
As was discussed in Sect. 3 the mixing between \(Z_\mathbf {R}^{}\) and the SM

*Z*bosons is constrained to be less than \(\mathcal{O} (1) \times 10^{-3}\). As a result the process mediated by the*Z*-boson can be safely neglected in calculating dark matter relic density and constraints of direct detection, because its effects are at least two orders smaller than those of \(Z_\mathbf {R}^{}\) for its mass at the TeV scale.

## Notes

### Acknowledgements

We thank to Huai-ke Guo for his help on numerical calculations and to Xiaohui Liu for helpful conversation. This work was supported by the National Natural Science Foundation of China under Grant No. 11775025 and the Fundamental Research Funds for the Central Universities.

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