Linkage of Dirac Neutrinos to Dark U(1) Gauge Symmetry

It is shown how a mechanism which allows naturally small Dirac neutrino masses is linked to the existence of dark matter through an anomaly-free U(1) gauge symmetry of fermion singlets.

Introduction : There is a known mechanism since 2001 [1] for obtaining small Dirac fermion masses. It was originally used [1] in conjucntion with the seesaw mechanism for small Majorana neutrino masses, and later generalized in 2009 [2]. It has also been applied in 2016 [3] to light quark and lepton masses.
The idea is very simple. Start with the standard model (SM) of quarks and leptons with just one Higgs scalar doublet Φ = (φ + , φ 0 ). Add a second Higgs scalar doublet η = (η + , η 0 ) which is distinguished from Φ by a symmetry yet to be chosen. Depending on how quarks and leptons transform under this new symmetry, Φ and η may couple to different combinations of fermion doublets and singlets. These Yukawa couplings are dimension-four terms of the Lagrangian which must obey this new symmetry.
In the Higgs sector, this new symmetry is allowed to be broken softly or spontaneously, such that η 0 = v is naturally much smaller than φ 0 = v. The mechanism is an analog of the well-known Type II seesaw for neutrino mass. Consider for example the case where the new symmetry is global U(1) which is broken softly. Let where µ 2 is the soft symmetry breaking term, then v, v are determined by For m 2 1 < 0 but m 2 2 > 0 and |µ 2 | << m 2 2 , the solutions are implying thus |v | << |v|. In Ref. [1], the new symmetry is taken to be lepton number, under which η has L = −1 but ν R has L = 0. This choice forbidsν R (ν L φ 0 − l − L φ + ), but allows To explore further this mechanism, it is proposed that this new symmetry is gauged and that it enforces neutrinos to be Dirac fermions and requires the addition of neutral singlet fermions which become members of the dark sector, the lightest of which is the dark matter of the Universe.
Dark U(1) Gauge Symmetry : The minimal particle content of the SM has only ν L , not ν R , and only the Higgs doublet Φ. Hence neutrinos are massless. Knowing that they should be massive [4], the usuall remedy is to add ν R and to assume that it pairs up with ν L through φ 0 . However, since ν R is a particle outside the SM gauge framework, it has many possible different guises [5]. Here it will be assumed that it transforms under a new U (1) D gauge symmetry, whereas all SM particles do not. The linkage of ν R to the SM is achieved through a second Higgs doublet η which transforms under U (1) D in the same way as ν R . Using Eq. (4) with a very large m 2 , a sufficiently small v , say of the order 1 eV, may be obtained for a realistic Dirac neutrino mass.
To be a legitimate and viable theory of Dirac neutrinos in this framework, there are two important conditions yet to be discussed. First, the gauge U (1) D symmetry must not be broken in such a way that ν R gets a Majorana mass. Second, there must be additional fermions so that the theory is free of anomalies. The two conditions are also connected because the additional fermions themselves must also acquire mass through the scalars which break U (1) D .
Since only the new fermions transform under U (1) D , the two conditions for anomaly freedom are comprising of N singlets, with N to be determined. There are some simple solutions: .
In the next two sections, solutions (B) and (C) will be examined in more detail because they allow both Dirac neutrinos and an associated dark sector in a consistent framework.
As for the direct detection of ψ, it cannot proceed through Z D because the latter does The scalars required for this solution are a second doublet η ∼ −4, and two singlets χ 1 ∼ 2 and χ 2 ∼ 6. The (−4, −4) fermions are identified as ν R , so they obtain Dirac masses through η, again with small η 0 = v . The analog of Eq. (6) is Hence is again suppressed for large m 2 2 > 0.
Assuming that H is lighter than the dark-matter Majorana fermion ζ, the annihilation of ζζ → HH is shown in Fig. 2. The first diagram is also accompanied by its u−channel counterpart, which has the same amplitude in the limit that ζ is at rest. Let m ζ = f u √ 2 and x = m H /m ζ , then this cross section at rest multiplied by relative velocity is (14) As an example, let m ζ = 1 TeV and m H = 400 GeV, then the canonical value of 3 × 10 −26 cm 3 /s is obtained for f = 1.05. This implies u = 673 GeV. The limit on λ 13 /λ 3 is then 2.7 × 10 −3 from XENON data [6]. In this scenario, the Z D gauge boson has m 2 D = 8g 2 D (s 2 + 9c 2 )u 2 , so m D is of order a few TeV.  Type II) seesaw mechanism first pointed out in Ref. [1]. To implement this idea that Dirac neutrino mass is linked to a dark U (1) D gauge symmetry, a set of singlet fermions is required so that the theory is free of anomalies. The scalars which are used to break U (1) D must be such that all fermions acquire mass, and two residual symmetries must remain: one is the usual lepton number, the other is a stabilizing dark symmetry.
Three examples are presented. In (B), one singlet fermion is identified as ν R whereas the other six form three dark Dirac fermions. The stabilizing symmetry is global U(1). In (C), two singlet fermions are identified as ν R whereas the other four are Majorana fermions.
The dark symmetry is Z 2 parity. In (B), the lightest dark Dirac fermion ψ annihilates to the U (1) D gauge boson Z D to establish its relic abundance. In (C), it is the lightest dark Majorana fermion ζ annihilating to the U (1) D breaking scalar H. In both cases, directsearch constraints put an upper limit on h − H mixing of order 10 −4 . In (D), dark matter consists of two separate Dirac fermion components.