Naturally Light Dirac Neutrinos from SU(6)

A known mechanism for obtaining naturally light Dirac neutrinos is implemented in the context of $SU(6) \to SU(5) \times U(1)_N$.

Introduction : Whether or not neutrinos are Majorana or Dirac is a fundamental issue in particle physics. Experimentally, there is still no evidence for one or the other, although it is known that at least two neutrinos must have masses, because of neutrino oscillations [1].
Theoretically, the standard model (SM) requires only neutrinos in the left-handed SU (2) L × U (1) Y doublets (ν, l) L . The singlet ν R is not necessary because it is trivial under the SM gauge group SU (3) C × SU (2) L × U (1) Y . To have a Dirac neutrino, ν R must exist. To justify its existence, a gauge extension of the SM is often considered, either [3], which may be incorporated into the grand unified structure of SO (10). The breaking of U (1) B−L [and SU (2) R ] is usually assumed without hesitation to allow ν R to obtain a large Majorana mass, so that ν L gets a tiny seesaw Majorana mass, as is well-known.
There is actually another option. This breaking does not have to be ∆L = 2. If it is ∆L = 3 for example, then neutrinos are Dirac. This was first pointed out [4] for a general U (1) X symmetry and applied [5] to U (1) L for Dirac neutrinos. However, this mechanism does not by itself explain why the neutrino Higgs Yukawa couplings are so small.
To overcome this problem, the mechanism of Ref. [6] is the simplest solution. The idea is to have at least two Higgs doublets, say Φ = (φ + , φ 0 ) and η = (η + , η 0 ) which are distinguished by some symmetry, so thatν R ν L couples to η 0 , but not φ 0 . This symmetry is then broken by the µ 2 Φ † η term, under the condition that m 2 Φ < 0 and m 2 η > 0 and large. In that case, the vacuum expectation value η 0 is given by −µ 2 φ 0 /m 2 η , which is naturally small, implying thus a very small Dirac neutrino mass. In the original application [6], ν R is also allowed a large Majorana mass, hence the mass of ν L is doubly suppressed. In that case, m η could well be of order 1 TeV. On the other hand, if the symmetry and the particle content are such that ν R is prevented from having a Majorana mass, then a much larger m η works just as well for a tiny Dirac neutrino mass.
Recently this idea has been applied [7,8] using a gauge U (1) D symmetry under which the SM particles do not transform, but ν R and other fermion singlets do. The U (1) D symmetry is broken by singlet scalars which transform only under U (1) D . The bridge between the SM and this new sector is a set of Higgs doublets which transform under both. The particle content is chosen such that global lepton number is conserved as well as a dark parity or dark number.
In this paper, instead of the ad hoc U (1) D symmetry, ν R is identified as part of the fundamental representation of SU (6) which breaks to SU (5) × U (1) N . Following a recent analysis [9], it is shown how naturally small Dirac neutrino masses occur in this context.

Description of Model
: Starting with the well-known SU (5) model [10] of grand unification, an extension to SU (6) is straightforward [11,12]. Instead of having the anomaly-free combination of 5 * and 10 under SU (5) for each family, there should be now two 6 * = (5 * , −1)+ (1,5) and one 15 = (10, 2) Table 1. Note that all are left-handed. The added U (1) X is a global symmetry imposed on the dimensionfour couplings of the resulting Lagrangian, but softly broken by bilinear and trilinear scalar terms. The scalars which break the SU (6) symmetry and allow these fermions to acquire masses are listed in Table 2. The 35 S breaks SU (6) at the garnd unification scale u 1 to   together with a dark parity Z D 2 which may be identified as (−1) 3B+L+2j , where j is the intrinsic spin of the particle, as shown in Table 3.
The lightest mass eigenstate is possible dark matter. However as shown in Ref. [9], because of its SU (6) antecedent, N may decay through a superheavy gauge boson in analogy to proton decay. Note also that v 3 is very small. If u 3 is absent, then N gets a small seesaw mass proportional to v 1 v 3 /u 2 . This makes it a candidate for very light freeze-in dark matter, as described in Ref. [8]. In such a scenario, the 21 * scalar is not required.
Concluding Remarks : It has been shown that a framework exists for naturally small Dirac neutrino masses in the context of SU (6) → SU (5) × U (1) N , where the right-handed neutrino singlet ν c is embedded as shown in Eq. (1). With the implementation of a softly broken global U (1) X symmetry as given in Tables 1 and 2, the residual symmetries of baryon number B and lepton number L are preserved as given in Table 3, with dark parity identified as The reason for a conserved lepton number parallels that of Ref. [4], i.e. the interplay of U (1) X and the chosen SU (6) representations makes it impossible for ν c to be a Majorana fermion.
The dark sector consists of D leptoquarks and L = 0 fermions N and the vectorlike doublet (E 0 , E − ). Whereas N mixes with E 0 andĒ 0 , the former may be almost a mass eigensate and considered as dark matter. It is presumably of order a few TeV, but if the 21 * scalar is removed from Table 2, then u 3 = 0 and Eq. (2) yields a very small mass for N which then becomes freeze-in dark matter.
Suppose the U (1) N breaking scale is much higher than a few TeV, then only the SM particles are observable, with the important difference that neutrinos are Dirac, and there are two Higgs doublets. The other possible addition is the Majorana fermion N as freeze-in dark matter. If u 3 = 0, then its mass is proportional to v 1 v 3 /u 2 as remarked earlier, which is perhaps too small because v 3 is responsible for the Dirac neutrino mass and u 2 is now assumed to be very large. However, the 21 * scalar may be retained, and u 3 = 0 rendered small (but not too small) by the same mechanism [6] which makes v 3 small, i.e. of the form −µ 23 u 2 2 /m 2 3 from the term 6 S × 6 S × 21 * S , where m 3 is the mass of the scalar 21 * .