# Dark Gauge *U*(1) symmetry for an alternative left–right model

## Abstract

An alternative left–right model of quarks and leptons, where the \(SU(2)_R\) lepton doublet \((\nu ,l)_R\) is replaced with \((n,l)_R\) so that \(n_R\) is not the Dirac mass partner of \(\nu _L\), has been known since 1987. Previous versions assumed a global \(U(1)_S\) symmetry to allow *n* to be identified as a dark-matter fermion. We propose here a gauge extension by the addition of extra fermions to render the model free of gauge anomalies, and just one singlet scalar to break \(U(1)_S\). This results in two layers of dark matter, one hidden behind the other.

## 1 Introduction

The alternative left–right model [1] of 1987 was inspired by the \(E_6\) decomposition to the standard \(SU(3)_C \times SU(2)_L \times U(1)_Y\) gauge symmetry through an \(SU(2)_R\), which does not have the conventional assignments of quarks and leptons. Instead of \((u,d)_R\) and \((\nu ,l)_R\) as doublets under \(SU(2)_R\), a new quark *h* and a new lepton *n* per family are added so that \((u,h)_R\) and \((n,e)_R\) are the \(SU(2)_R\) doublets, and \(h_L\), \(d_R\), \(n_L\), \(\nu _R\) are singlets.

This structure allows for the absence of tree-level flavor-changing neutral currents (unavoidable in the conventional model), as well as the existence of dark matter. The key new ingredient is a \(U(1)_S\) symmetry, which breaks together with \(SU(2)_R\), such that a residual global \(S'\) symmetry remains for the stabilization of dark matter. Previously [2, 3, 4], this \(U(1)_S\) was assumed to be global. We show in this paper how it may be promoted to a gauge symmetry. To accomplish this, new fermions are added to render the model free of gauge anomalies. The resulting theory has an automatic discrete \(Z_2\) symmetry which is unbroken as well as the global \(S'\), which is now broken to \(Z_3\). Hence dark matter has two components [5]. They are identified as one Dirac fermion (nontrivial under both \(Z_2\) and \(Z_3\)) and one complex scalar (nontrivial under \(Z_3\)).

In Sect. 2 we make a digression to the historical perspective which motivated this study. In Sect. 3 our model is described, with a complete list of its particle content. In Sect. 4 the gauge sector is shown in detail. In Sect. 5 the fermions are discussed with details of how they obtain masses. In Sect. 6 we deal with the scalars and show how the desirable pattern of symetry breaking is obtained. In Sect. 7 we discuss the present phenomenological constraints on the new \(Z'\) bosons and would-be dark-matter candidates. In Sect. 8 we show an example of two viable dark-matter candidates, both in terms of relic abundance and direct detection. In Sect. 9 we conclude.

## 2 Motivation and historical perspective

*SU*(2) doublets, the bidoublet

*u*mass matrix, and \(\delta _1\) contributes to the

*d*mass matrix. In other words, each quark sector gets its masses from two different Higgs particles. This means that flavor changing neutral currents (FCNC) are unavoidable at tree level through neutral Higgs exchange. This is a very strong constraint on the masses of these particles, of order 10–100 TeV. As such they are not likely to be observable at the Large Hadron Collider (LHC). On the general issue of FCNC, they are, of course, present in the SM, but only at the loop level, and they are known to be small and consistent with experimental data. In any extension of the SM, they may occur at tree level, and if so the scalar particles in question are required to be very heavy and out of reach of the LHC. It is thus a valid question to ask whether a model beyond the SM may be constructed with the absence of tree-level FCNC, so that it may have new scalars which are light enough to be discovered in addition to the SM Higgs boson of 125 GeV.

*u*quark mass matrix must be proportional to the

*d*quark mass matrix, which disagrees with data. The solution to this conundrum was pointed out 30 years ago [1]. It was discovered in the context of superstring-inspired \(E_6\) models, but applicable to the \(SU(2)_L \times SU(2)_R\) case [2, 3]. The idea is to add another quark

*h*to each family which has the same charge as

*d*, i.e. \(-1/3\). Both \(h_L\) and \(h_R\) are singlets in the SM, but they are distinguished from \(d_L\) and \(d_R\) in their \(SU(2)_R\) assignments, i.e.

*d*mass comes from an \(SU(2)_L\) Higgs doublet, the

*h*mass comes from an \(SU(2)_R\) Higgs doublet, and the

*u*mass comes from only \(\delta _1^0\) whereas \(\delta _2^0\) has no vacuum expectation value. Thus the model is guaranteed the absence of tree-level FCNC. It was realized a few years ago [2, 3] that this extra \(U(1)_S\) also serves the purpose of a dark symmetry, because even though it is broken, the combination \(T_{3R} + S\) or \(T_{3R} - S\) may remain unbroken and protects the condition \(\langle \delta _2^0 \rangle = 0\). In other words, the symmetry which allows us to solve the FCNC conundrum has now been connected to that of dark matter. Contrast this with most models of dark matter, where the existence of the dark symmetry is completely ad hoc, and unrelated to any other symmetry of the original model. This we believe is a good motivation for studying alternative left–right models. The logical next step is to ask the question whether it is possible for this \(U(1)_S\) to be gauged. What follows is a simple example of how it can be done and the resulting consequences.

## 3 Model

Particle content of proposed model of dark gauge *U*(1) symmetry

Particles | \(SU(3)_C\) | \(SU(2)_L\) | \(SU(2)_R\) | \(U(1)_X\) | \(U(1)_S\) |
---|---|---|---|---|---|

\((u,d)_L\) | 3 | 2 | 1 | 1/6 | 0 |

\((u,h)_R\) | 3 | 1 | 2 | 1/6 | \(-1/2\) |

\(d_R\) | 3 | 1 | 1 | \(-1/3\) | 0 |

\(h_L\) | 3 | 1 | 1 | \(-1/3\) | \(-1\) |

\((\nu ,l)_L\) | 1 | 2 | 1 | \(-1/2\) | 0 |

\((n,l)_R\) | 1 | 1 | 2 | \(-1/2\) | 1/2 |

\(\nu _R\) | 1 | 1 | 1 | 0 | 0 |

\(n_L\) | 1 | 1 | 1 | 0 | 1 |

\(\left( \phi _L^+,\phi _L^0\right) \) | 1 | 2 | 1 | 1 / 2 | 0 |

\(\left( \phi _R^+,\phi _R^0\right) \) | 1 | 1 | 2 | 1/2 | 1/2 |

\(\eta \) | 1 | 2 | 2 | 0 | \(-1/2\) |

\(\zeta \) | 1 | 1 | 1 | 0 | 1 |

\(\left( \psi _1^0,\psi _1^-\right) _R\) | 1 | 1 | 2 | \(-1/2\) | 2 |

\(\left( \psi _2^+,\psi _2^0\right) _R\) | 1 | 1 | 2 | 1 / 2 | 1 |

\(\chi _R^+\) | 1 | 1 | 1 | 1 | \(-3/2\) |

\(\chi _R^-\) | 1 | 1 | 1 | \(-1\) | \(-3/2\) |

\(\chi ^0_{1R}\) | 1 | 1 | 1 | 0 | \(-1/2\) |

\(\chi ^0_{2R}\) | 1 | 1 | 1 | 0 | \(-5/2\) |

\(\sigma \) | 1 | 1 | 1 | 0 | 3 |

*Z*. Some explicit Yukawa terms are

Particle content of proposed model under \((T_{3R} + S) \times Z_2\)

Particles | Gauge \(T_{3R} + S\) | Global \(S'\) | \(Z_3\) | \(Z_2\) |
---|---|---|---|---|

\(u,d,\nu ,l\) | 0 | 0 | 1 | + |

\((\phi _L^+,\phi _L^0), (\eta _2^+,\eta _2^0), \phi _R^0\) | 0 | 0 | 1 | + |

\(n, \phi _R^+, \zeta \) | 1 | 1 | \(\omega \) | + |

\(h, \left( \eta _1^0, \eta _1^-\right) \) | \(-1\) | \(-1\) | \(\omega ^2\) | + |

\(\psi ^+_{2R}, \chi ^+_R\) | \(3/2,-3/2\) | 0 | 1 | − |

\(\psi ^-_{1R}, \chi ^-_R\) | \(3/2,-3/2\) | 0 | 1 | − |

\(\psi ^0_{1R}, \psi ^0_{2R}\) | 5 / 2, 1 / 2 | \(1,-1\) | \(\omega ,\omega ^2\) | − |

\(\chi ^0_{1R},\chi ^0_{2R}\) | \(-1/2,-5/2\) | \(1,-1\) | \(\omega ,\omega ^2\) | − |

\(\sigma \) | 3 | 0 | 1 | + |

## 4 Gauge sector

*pp*collisions at the Large Hadron Collider (LHC).

## 5 Fermion sector

*h*and

*n*to decay into \(\zeta \).

## 6 Scalar sector

*V*satisfies the conditions

*H*, which may all be assumed to be small enough to avoid the constraints from dark-matter direct-search experiments. The addition of the scalar \(\zeta \) introduces two important new terms:

## 7 Present phenomenological constraints

*u*and

*d*quarks, and decay to charged leptons (\(e^-e^+\) and \(\mu ^-\mu ^+\)). As noted previously, in our chosen example, \(D_1\) is the lighter of the two. Hence current search limits for a \(Z'\) boson are applicable [7, 8]. The \(c_{u,d}\) coefficients used in the data analysis are

*B*is the branching fraction of \(Z'\) to \(e^-e^+\) and \(\mu ^-\mu ^+\). Assuming that \(D_1\) decays to all the particles listed in Table 2, except for the scalars which become the longitudinal components of the various gauge bosons, we find \(B = 1.2 \times 10^{-2}\). Based on the 2016 LHC 13 TeV data set from ATLAS [9], this translates to a bound of about 4 TeV on the \(D_1\) mass.

The would-be dark-matter candidate *n* is a Dirac fermion which couples to \(D_{1,2}\), which also couples to quarks. Hence severe limits exist on the masses of \(D_{1,2}\) from underground direct-search experiments as well. The annihilation cross section of *n* through \(D_{1,2}\) would then be too small, so that its relic abundance would be too big for it to be a dark-matter candidate. Its annihilation at rest through *s*-channel scalar exchange is *p*-wave suppressed and does not help, barring of course any accidental resonance enhancement. As for the *t*-channel diagrams, they also turn out to be too small. Suggestions of previous studies [2, 3] where *n* is chosen as dark matter are now ruled out.

## 8 Dark sector

Dark matter is envisioned to have two components. One is a Dirac fermion \(\chi _0\), which is a mixture of the four neutral fermions of odd \(Z_2\), and the other is a complex scalar boson which is mostly \(\zeta \), with the added assumption that \(m_{\chi _0}\) is significantly greater than \(m_\zeta \). The annihilation \(\chi _0 \bar{\chi }_0 \rightarrow \zeta \zeta ^*\) determines the relic abundance of \(\chi _0\), and the annihilation \(\zeta \zeta ^* \rightarrow H H\), where *H* is the standard-model Higgs boson, determines that of \(\zeta \). The direct \(\zeta \zeta ^* H\) coupling is assumed small to avoid the severe constraint in direct-search experiments.

*HH*be \(\lambda _0\), then the annihilation cross section of \(\zeta \zeta ^*\) to

*HH*times relative velocity is given by

*H*. They are suppressed by making the \(D_{1,2}\) masses heavy, and the

*H*couplings to \(\chi _0\) and \(\zeta \) small. In our example with \(m_\zeta = 150\) GeV, let us choose \(m_{\chi _0} = 500\) Gev and the relic abundances of both to be equal. From Fig. 1, these choices translate to \(\lambda _0 = 0.12\) and \(f_0 = 0.56\).

The \(\bar{\chi _0} \gamma _\mu \chi _0\) couplings to \(D_{1,2}\) depend on the \(2 \times 2\) mass matrix linking \((\chi _1,\psi _1)\) to \((\chi _2,\psi _2)\), which has two mixing angles and two mass eigenvalues, the smaller one being \(m_{\chi _0}\). By adjusting these parameters, it is possible to make the effective \(\chi _0\) interaction to any particular nucleus through \(D_{1,2}\) negligibly small. Hence there is no useful limit on the \(D_1\) mass in this case. Note that the amplitude cancellation here is through \(D_{1,2}\) and not necessarily through *u* and *d* quarks (which are not adjustable in this model), as would be necessary in models with only one vector mediator.

*H*mixing with \(\sigma _R\) and \(\phi _R^0\)). Let their effective interactions with quarks through

*H*exchange be given by

*H*.

*H*contribution to the \(\chi _0\) elastic cross section off nuclei, we replace \(m_\zeta \) with \(m_{\chi _0} = 500\) GeV in Eq. (51) and \(\lambda _{\zeta H}/2 m_\zeta \) with \(\epsilon /\sqrt{2}v_H\) in Eq. (52). Using the experimental data at 500 GeV, we obtain the bound.

## 9 Conclusion and outlook

In the context of the alternative left–right model, a new gauge \(U(1)_S\) symmetry has been proposed to stabilize dark matter. This is accomplished by the addition of a few new fermions to cancel all the gauge anomalies, as shown in Table 1. As a result of this particle content, an automatic unbroken \(Z_2\) symmetry exists on top of \(U(1)_S\), which is broken to a conserved residual \(Z_3\) symmetry. Thus dark matter has two components. One is the Dirac fermion \(\chi _0 \sim (\omega ,-)\) and the other the complex scalar \(\zeta \sim (\omega ,+)\) under \(Z_3 \times Z_2\). We have shown how they may account for the relic abundance of dark matter in the Universe, and satisfy present experimental search bounds.

Whereas we have no specific prediction for discovery in direct-search experiments, our model will be able to accommodate any positive result in the future, just like many other existing proposals. To single out our model, many additional details must also be confirmed. Foremost are the new gauge bosons \(D_{1,2}\). Whereas the LHC bound is about 4 TeV, the direct-search bound is much higher, provided that \(\zeta \) is a significant fraction of dark matter. If \(\chi _0\) dominates instead, the adjustment of free parameters of our model can lower this bound to below 4 TeV. In that case, future \(D_{1,2}\) observations are still possible at the LHC as more data become available.

Another is the exotic *h* quark which is easily produced if kinematically allowed. It would decay to *d* and \(\zeta \) through the direct \(\bar{d}_R h_L \zeta \) coupling of Eq. (29). Assuming that this branching fraction is 100%, the search at the LHC for 2 jets plus missing energy puts a limit on \(m_h\) of about 1.0 TeV, as reported by the CMS Collaboration [15] based on the \(\sqrt{s} = 13\) TeV data at the LHC with an integrated luminosity of 35.9 fb\(^{-1}\) for a single scalar quark.

If the \(\bar{d}_R h_L \zeta \) coupling is very small, then *h* may also decay significantly to *u* and a virtual \(W_R^-\), with \(W_R^-\) becoming \(\bar{n} l^-\), and \(\bar{n}\) becoming \(\bar{\nu } \zeta ^*\). This has no analog in the usual searches for supersymmetry or the fourth family because \(W_R\) is heavy (\(> 16\) TeV). To be specific, the final states of 2 jets plus \(l_1^- l_2^+\) plus missing energy should be searched for. As more data are accumulated at the LHC, such events may become observable.

## Notes

### Acknowledgements

This work was supported in part by the U. S. Department of Energy Grant No. DE-SC0008541.

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