Dirac Neutrino Mass Matrix and its Link to Freeze-in Dark Matter

Using a mechanism which allows naturally small Dirac neutrino masses and its linkage to a dark gauge $U(1)_D$ symmetry, a realistic Dirac neutrino mass matrix is derived from $S_3$. The dark sector naturally contains a fermion singlet having a small seesaw mass. It is thus a good candidate for freeze-in dark matter from the decay of the $U(1)_D$ Higgs boson.

Introduction : It has been shown recently [1] that naturally small Dirac neutrino masses may be linked to a dark U(1) D gauge symmetry. One specific model is studied here with the inclusion of an S 3 family symmetry, so that a realistic Dirac neutrino mass matrix is obtained. The dark sector consists of four singlet Majorana fermions. Its structure allows one to be the lightest from a seesaw mechanism akin to that used in canonical Majorana neutrino mass. It is thus very suitable as freeze-in dark matter which owes its relic abundance from the decay of the U(1) D Higgs boson.
The simple mechanism in question was first pointed out in 2001 [2]. Consider two Higgs dooublets Φ = (φ + , φ 0 ) and η = (η + , η 0 ), where η is distinguished from the standard-model (SM) Φ by a symmetry to be decided. Whereas Φ has the usual µ 2 < 0, the corresponding m 2 for η is positive and large. The aforesaid symmetry is assumed to be broken by the soft term µ ′ 2 Φ † η + H.c. The spontaneous breaking of the SU(2) L × U(1) Y gauge symmetry of the SM then results in the usual vacuum expectation φ 0 = v, but η 0 = v ′ is now given by −µ ′ 2 v/m 2 , which is suppressed by the small µ ′ 2 and large m 2 .
For neutrino mass, if ν R is chosen to transform in the same way as η, but not the other SM particles, then it pairs up with ν L to form a Dirac fermion with mass proportional to the small v ′ . If the symmetry chosen also forbids ν R to have a Majorana mass, then the neutrino is a Dirac fermion with a naturally small mass. This idea of achieving a small v ′ /v ratio is akin to that of the so-called Type II seesaw, as classified in Ref. [3] and explained in Ref. [4].
It is also easily generalized [5] and applicable to light quarks and charged leptons [6].
In this paper, following Ref. [1] which incorporates an anomaly-free U(1) D gauge symmetry to distinguish ν R from the other SM particles, a specific model of two massive Dirac neutrinos is proposed. With the implementation of an S 3 discrete family symmetry, a realistic Dirac neutrino mass matrix is obtained. The natural occurrence of light freeze-in dark matter is also discussed.
Concluding Remarks : The right-handed neutrino ν R has been proposed as the link [1] to dark matter by having it transform under a new U(1) D gauge symmetry. The small Dirac neutrino masses are enforced by a seesaw mechanism proposed in Ref. [2] using new Higgs doublets also transforming under U(1) D . With the help of the non-Abelian discrete symmetry S 3 , it is shown how a realistic neutrino mixing matrix may be obtained which is approximately cobimaximal [13].
After the spontaneous breaking of U(1) D , a dark parity remains for four Majorana fermions, the lightest of which has a seesaw mass. It is suitable as freeze-in dark matter with its relic abundance coming from the decay of the U(1) D Higgs boson.