Naturally Light Dirac Neutrinos from $SO(10) \times U(1)_\psi$

A new solution is presented where the right-handed neutrino $\nu_R$ in $SO(10)$ pairs up with $\nu_L$ to form a naturally light Dirac neutrino. It is based on the framework of $E_6 \to SO(10) \times U(1)_\psi$, then $SO(10) \to SU(5) \times U(1)_\chi$.

Introduction : Neutrinos (ν) are observed but not understood. They are not massless but very light [1]. They may be self-conjugate two-component spinors (Majorana) or fourcomponent spinors (Dirac) with right-handed components (ν R ) which have no electroweak interactions. Neutrinoless double beta decay experiments have so far no conclusive evidence for them to be Majorana, which is the prevalent theoretical thinking.
Why is the Dirac option disfavored? There are two well-known answers. (1) To have a Dirac neutrino, ν R must exist. However, under the standard-model (SM) gauge symmetry SU(3) C × SU(2) L × U(1) Y , it is a trivial singlet and not mandatory. If it is added anyway, then it is allowed a Majorana mass. Together with the Dirac mass linking ν L to ν R through the one SM Higgs doublet Φ = (φ + , φ 0 ), this forms the well-known 2 × 2 mass matrix With the usual reasonable assumption m D << m R , the famous seesaw mechanism yields a very small Majorana neutrino mass m ν ≃ m 2 D /m R . (2) To protect ν R from having a Majorana mass, a symmetry has to be imposed. The obvious one is lepton number. In that case, m R is forbidden, but m D is allowed. However there is no understanding why the Yukawa coupling which generates m D is so small, 10 −11 or less, since φ 0 = 174 GeV and m ν < 1.1 eV [2]. To have a compelling case for Dirac neutrinos, two conditions must be met.
• (A) The existence of ν R should not be ad hoc, but based on a well-motivated theoretical framework, within which it should not acquire a Majorana mass.
• (B) The smallness of the Dirac neutrino mass should be obtained naturally, without any extra symmetry.
In past studies [3], limited success has been achieved, but at the expense of extra symmetries, some of which are softly broken. Here a new solution based on E 6 → SO(10) × U(1) ψ , then SO(10) → SU(5) × U(1) χ is presented which satisfies for the first time both (A) and (B) in full measure.
Essential Existence of ν R : To justify the presence of ν R , it ought to transform under a symmetry related to those of the SM. It could be a global symmetry such as lepton number mentioned earlier. However, it would be more convincing and compelling if it were a gauge symmetry such as U(1) B−L [4] or SU(2) R × U(1) B−L [5]. A recently proposed alternative is a gauge U(1) D symmetry [6,7] not related to the SM but essential for dark matter, with ν R as the bridge between the two sectors.
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, thereby already not satisfying condition (A). To avoid this eventuality, there is a simple solution. 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 [8] for a general U(1) X symmetry and applied [9] to U(1) L for Dirac neutrinos. However, this mechanism does not by itself explain why the neutrino Higgs Yukawa couplings are so small.
The existence of ν R may also be justified in SU(6) as shown recently [10,11]. However, it is best known as the missing link which allows the 15 fermions (per family) of the SM to form a complete 16 representation of SO (10). Whereas the usual study of SO(10) proceeds from its left-right decomposition SU(3) C × SU(2) L × SU(2) R × U(1) B−L , here the SU(5) × U(1) χ alternative is considered: where It has been shown [12] that the U(1) χ charge may be used as a marker for dark matter, with dark parity given by (−1) Qχ+2j . Here it will be shown how a naturally small Dirac mass linking ν to ν c is obtained instead.
Naturally Small Dirac Neutrino Mass : To obtain a naturally small Dirac neutrino mass, the mechanism of Ref. [13] is the simplest solution. Let there be 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 soft dimension-two µ 2 Φ † η term, with m 2 Φ < 0 as usual, but m 2 η > 0 and large. Consequently, 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 [13], ν R also has a large Majorana mass, hence the ν L mass 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 mechanism has been used [6,7] in the framework of 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 connection between the SM and this new sector is a set of Higgs doublets which transform under both, so that ν R pairs up with ν L to form a Dirac neutrino. The particle content is chosen such that global lepton number is conserved as well as a dark parity or dark number.
A more compelling case [11] is to identify ν R as part of the fundamental representation
From the 16 F × 16 F × 10 S Yukawa term, it is seen that d c d and ee c couple to (5 * , −2), whereas u c u and νν c couple to (5,2). All would acquire masses if the scalar doublets However,Φ transform exactly as Φ 2,1 respectively. Hence both Φ 1,2 would couple to all quarks and leptons. This is the analog of the well-known property of the SU(2) L × SU(2) R scalar bidoublet, i.e.
which transforms exactly as η.
This results in as discussed earlier. This suppressed VEV is the source of the Dirac neutrino mass. As for the Φ 2 Φ 3 and Φ 2 Φ 5 terms, they are not allowed because the former does not obey SU (5) and the latter U(1) ψ . The only other allowed coupling is Φ 3 Φ 4 ζ 3 , which is expected because it comes from 351 × 351 × 351. Finally ζ 4 transforms as (1,  (10), and the (45, 2) of 120. The key is that whereas Φ 1,2 and Φ 3,4 are not split by themselves, the conjugate of Φ 5 lies in a different E 6 multiplet, so the would-be Φ 6 may be assumed much heavier and be ignored.
With the addition of the five singlet scalars of Table 1, it is shown how Φ 1,3,4 may develop VEV for the d c d, ee c , and u c u masses. The νν c masses come only from Φ 2 , which keeps a large positive mass-squared, so its VEV is induced and suppressed by the mechanism of Ref. [13].
The breaking of U(1) χ comes from ζ 4 which allows global lepton number to remain.
The fermions of this model are the same as in the SM, except that the neutrino is Dirac.
They form complete 16 representations of SO(10) which may have a remnant U(1) χ symmetry [12] from breaking to SU(5) × U(1) χ , instead through the usual left-right intermediate step.
There are possibly three scalar doublets (Φ 1,3,4 ) at the electroweak scale, together with a singlet ζ 4 at the U(1) χ breaking scale. The Z χ gauge should also appear at this scale, with known couplings to all SM particles. From present collider data, its mass is greater than a few TeV.