# Left-right model for dark matter

###### Abstract

We argue that dark matter can automatically arise from a gauge theory that possesses a non-minimal left-right gauge symmetry, , for and .

###### pacs:

12.60.-iIntroduction: The astrophysical and cosmological observations have implied that our universe is made of roundly 26% dark matter planck . If its present-day existence is a result of thermal relics of the early universe, it may be formed of weakly-interacting massive particles (WIMPs), which do not present in the standard model pdg . What is the nature of WIMPs? In this work, we show that the left-right models which generalize the electroweak symmetry to non-minimal left-right gauge groups can automatically provide the WIMPs from their gauge principles.

Proposal of the model: Consider the model based on the gauge group,

(1) |

which explicitly violates a symmetry between the left and right for . If as well as the left-right symmetry is imposed, it must be spontaneously broken, along with the gauge symmetry breaking, in order to recover the standard model and the new physics besides. In other words, it all marks a theory that induces the observed left-right asymmetries. Further, our following discussions apply for all cases for , and so forth, as well as for if the left-right symmetry is imposed or not, called non-minimal left-right gauge extensions.

Denote the Cartan generators for and as () and (), respectively. Further, we work in the basis of the generalized Gell-Mann matrices, i.e. , , and so forth (note that ). The electric charge operator is generally embedded as

(2) |

where

(3) |

Here, note that either for or for .

The fermion content which is anomaly free is arranged as

(4) |

(5) |

(6) |

The generation indices are and . The electric charge parameters satisfy and . If , and the fermion spectrum as given is completed. If , and a number of chiral fermion partners have not been listed, which transform trivially under the left-right gauge groups (as concretely introduced in the following cases). The coefficients, , can be determined through the parameters, , as

(7) |

where the initial conditions are and . We particularly provide , , and , which will appropriately be used for the realistic versions as shown below.

Suppose that the nontrivial fermion representations under the left-right gauge groups are only defining representations or the complex conjugates of defining representations, respectively, and that they are enlarged from those of the standard model. The (or ) anomaly cancelation requires the number of fermion -plets (or -plets) to be equal to that of fermion anti--plets (or anti--plets), since a representation and its conjugate have opposite anomalies. Hence, the number of fermion generations must be a multiple of 3—fundamental color number. The QCD asymptotic freedom demands that the number of fermion generations is smaller than or equal to for . Thus, the generation number is just three, as observed, and . That property for the standard model and the minimal left-right model disappears since the anomaly always vanishes for every fermion representation. Therefore, this approach provides a partial solution of the fermion flavor questions.

The gauge symmetry breaking and mass generation are done by several scalar multiplets. The first one is

(8) |

which couples to , , and . The second one is

(9) | |||||

which couples to . The multiplets, and , are minimally required if .

If , the third one includes scalar multiplets, (), where

(10) |

which couples to , , and . Here, the second factors are the singlet fermion partners as mentioned.

Otherwise, if , the third one includes scalar multiplets instead, also denoted by

(11) |

which couples to , , and , where the first factors are the singlet fermion partners as mentioned, and in this case .

We see that for or for always form a squared matrix of dimension, in which every diagonal element is electrically neutral as the case is. Additionally, there might be two extra scalar multiplets (i) that couples to , defined similarly to by replacing and (ii) that couples to as well as to ,

(12) |

Exceptionally, if either or , we do not necessarily introduce because can play its role instead. Note that provides Dirac masses for fermions, while provide Majorana masses for neutrinos. It is also shown that the gauge bosons can gain appropriate masses. The gauge symmetry of the model, , is broken down to , where is a discrete gauge symmetry.

Indeed, note that break the gauge symmetry down to , which leave conserved. This charge is only broken by . A transformation of takes the form, . is those ’s that conserve the vacuum,

(13) |

Since , we get , which yields for , and thus . Considering as well as multiplying the result by the spin parity, , we obtain a conserved operator, , the so-called W-parity (since the wrong particles shown hereafter transform nontrivially under it).

From the electric charge operator, the charges of the new fields that have equal two times their electric charges, respectively, which is a new observation of this work. It is valid for every new fermion as well as numerous scalar and gauge boson fields. The charge and W-parity for all the fields are listed in Table 1. Here, we have denoted , which is nontrivial for , with The particles that have nontrivial W-parity are named as W-particles since they possess wrong charges aforementioned, unlike the standard model particles. The particles that have , which include the standard model particles and others, are called as normal particles. Thus, the normal particles also include those that have a charge differing from the usual one (defined by the standard model) by even unit, such as . It is noteworthy that if the electric charge parameters get values like ordinary electric charges, i.e. , we have . In this case, W-parity behaves as R-parity, and the W-particles are just W-odd.

Field | Field | ||||
---|---|---|---|---|---|

1 | |||||

1 | |||||

1 | |||||

1 | 0 | 1 | |||

0 | 1 | ||||

0 | 1 | ||||

0 | 1 | 0 | 1 | ||

0 | 1 | ||||

0 | 1 | ||||

0 | 1 | ||||

0 | 1 | ||||

0 | 1 | ||||

0 | 1 | ||||

1 | 0 | 1 | |||

1 | 0 | 1 | |||

1 | |||||

The lightest W-particle (LWP) is stabilized due to W-parity conservation and kinematically suppressed. Further, there is no single W-field appearing in interactions since W-parity is conserved. If an interaction has of fields and of fields for some values of and , where and are integer. The W-parity conservation implies for some integer. This equation is valid for any parameter, which follows and . Thus, and always appear in pairs in such interactions. If there are interactions that include only of fields and of fields, it deduces , which is satisfied if , , and . and always appear in pairs in such interactions. If an interaction contains only of fields, we have , which implies . Thus, and its conjugate always appear in pairs in those interactions. We also see that the W-fields that have W-parity as , , or often enter the self-interactions of three W-particles, where the last two have W-parity of and/or .

If the LWP is electrically neutral, it may be a dark matter candidate. The models that provide dark matter candidates are listed in Table 2. We see that in the model, dark matter may be a neutral fermion, a neutral non-Hermitian gauge boson, or some neutral scalar. However, in the model, dark matter include only a neutral non-Hermitian gauge boson or some neutral scalar. Whilst, the model yields dark matter particle uniquely as a neutral scalar.

Model | Candidates |
---|---|

, , , , , , , possibly | |

, , , , , possibly | |

Realistic realizations: They include the models that correspond to , , ,, or in order to contain W-particles as well as satisfying QCD asymptotic freedom. The one-parameter models coupled to , and were firstly introduced in hd to solve the diphoton anomaly, which was an old story. Additionally, the implication for dark matter was firstly recognized in a particular setup for and dh1 . The two-parameter models are associated with . We have seven models for the three parameters (), where or equals 5.

Our proposal can naturally provide the small neutrino masses via seesaw mechanisms. It also supplies the tree-level flavor-changing neutral currents due to the non-universal couplings of the new neutral gauge bosons and scalars with ordinary quarks, which potentially explain the B physics anomalies (see dh1 and references therein). A further analysis yields that the normal particles have and as quantized as usual values, while those charges of W-particles arbitrarily depend on the parameters (see d for a similar treatment).

Conclusion: It has been shown that there is a bulk of wrong particles, which have related to the corresponding electric charge (most yield that equals two times the electric charge), responsible for dark matter. The candidate is the LWP which may include a neutral fermion, scalar, or vector, and is stabilized by W-parity.

## Acknowledgments

This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 103.01-2016.77.

## References

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