$U(1)_\chi$ and Seesaw Dirac Neutrinos

In the context of $SO(10) \to SU(5) \times U(1)_\chi$, it is shown how seesaw Dirac neutrinos may be obtained. In this framework, $U(1)$ lepton number is conserved, with which self-interaction dark matter with a light scalar dilepton mediator may be implemented. In addition, $U(1)$ baryon number may be broken to $(-1)^{3B}$, thereby generating a baryon asymmetry of the Universe. The axionic solution to the strong $CP$ problem may also be incorporated.

Introduction : In considering SO(10) grand unification, the common approach is to allow an intermediate step with left-right symmetry, i.e. SU (4) × SU (2) L × SU (2) R or SU (3) C × SU (2) L × SU (2) R × U (1) (B−L)/2 . This has the advantage of forcing into existence the righthanded SU (2) R lepton doublet (ν, l) R , so that ν R is the Dirac partner of the observed ν L which belongs to the SU (2) L lepton doublet (ν, l) L . At the same time, B − L becomes a gauge symmetry, and its breaking through an SU (2) R scalar triplet from the 126 of SO (10) also makes ν R massive, realizing thus the canonical seesaw mechanism for a naturally small Majorana ν L mass.
Another option [1,2] is to consider SO(10) → SU (5) × U (1) χ . This is seldom studied because SU (5) is a grand unified symmetry by itself, so U (1) χ is often thought to be unnecessary and uninteresting. Let the breaking of SU (5) to the standard-model (SM) gauge symmetry SU (3) C × SU (2) L × U (1) Y be at the scale M U . If U (1) χ survives down to a scale much below M U and not too far above the electroweak scale, there could be important consequences which have been largely overlooked. In fact, whereas the right-handed neutrino ν R is a singlet under SU (5), it has a nonzero charge under U (1) χ . The Higgs doublet which connects u L to u R also connects ν L to ν R . Hence a Dirac neutrino mass is again obtained and the seesaw mechanism operates as in the left-right case. On the other hand, the detailed phenomenology is very different. Whereas W ± R must exist at the left-right scale M R , it must be heavier than M U if SO(10) → SU (5) × U (1) χ . The Z χ gauge boson itself has well-defined couplings to the SM particles. Its existence is routinely searched for at the Large Hadron Collider (LHC), with the present mass limit [3,4] of about 4.1 TeV, which may be improved [5].
In this paper, new fermions and scalars transforming under U (1) χ are added to the SM to obtain a number desirable features. With the help of a softly broken Z 2 discrete symmetry, natrurally light seesaw Dirac neutrinos [6,7,8] may be obtained. The resulting Lagrangian conserves both B and L. Further addition of two scalars with L = −1, −2 enables the appearance of self-interacting leptonic dark matter [9]. The analog of leptogenesis (through a heavy singlet Majorana fermion which couples to leptons in the seesaw mechanism) is possible using a heavy singlet Majorana fermion which couples to a scalar diquark and an antiquark, thereby generating the baryon asymmetry of the Universe. A fermion color octet may also be introduced to support a Peccei-Quinn symmetry to obtain an invisible axion for solving the strong CP problem.
Seesaw Dirac Neutrinos : The spinorial 16 representation is again chosen for the three families of quarks and leptons and their decompositions shown in Table 1. The necessary Table 1: Fermion content of model.
Higgs scalars for fermion masses belong to the 10 representation, as shown in Table 2.  (10) representations from which they come. It should also be clear that incomplete SO(10) and SU (5) multiplets are considered here (which is the case for all realistic grand unified models). An important Z 2 discrete symmetry is imposed so that ν c , N, N c and η are odd, and the other fields are even. Since Φ † 1 transforms exactly like Φ 2 , the linear combination The Z 2 symmetry is respected by all dimension-four terms of the Lagrangian.
It will be broken softly by the dimension-three trilinear term µσΦ † η as well as spontaneously by η 0 = v 3 . The 4 × 4 neutrino mass matrix spanning (ν, ν c , N, N c ) is then given by where u = σ which breaks U (1) χ . The above mass matrix generates a seesaw Dirac neutrino Scalar Sector : The scalar potential consisting of Φ, η, and σ is given by The minimum of V satisfies the conditions Assuming that u >> v >> v 3 , the solutions to the above are The 3 × 3 mass-squared matrix spanning √ 2[Im(φ 0 ), Im(η 0 ), Im(σ)] is given by which has two zero eigenvalues and one massive eigenstate with The 3 × 3 mass-squared matrix spanning √ 2[Re(φ 0 ), Re(η 0 ), Re(σ)] is given by which is approximately diagonal with Note that the φ R − σ R mixing (with the assumption that φ R is much lighter than σ R ) is roughly λ Φσ vu/λ σ u 2 which is naturally suppressed by v/u. As for η R , its mass is dominated by −µvu/v 3 , and its mixing with φ R and σ R is suppressed by v 3 /v and v 3 /u respectively.
This justifies the diagonal approximation assumed here. is added, it may be assigned B and L numbers appropriately, according to its assumed interactions with the known quarks and leptons [10,11]. These assignments lie outside U (1) χ , hence Q χ is now not a marker of dark matter, as in previous studies [1,2].  With ζ as a scalar dilepton which couples only to the Dirac neutrinos, it is then a simple step to consider a scalar singlet ρ ∼ (1, −5) from the 16 of SO(10) with L = −1 so that it can be self-interacting dark-matter [12] with ζ as its light mediator, as proposed recently [9].
Since ζ decays only to two neutrinos, it does not disrupt the cosmic microwave background (CMB) from its enhanced production at late times due to the Sommerfeld effect. It removes an important objection [13,14] to models where the light mediator decays to electrons and photons, usually through Higgs mixing, which is forbidden here by L conservation.
Baryogenesis : Since lepton number is strictly conserved, the usual mechanism of generating the baryon asymmetry of the Universe through leptogenesis is not possible. However, the analog process of having a heavy Majorana fermion ψ decaying to B = ±1 final states [15] may be implemented with the addition of two scalar diquarks h 1,2 , as shown in Table 4.
The allowed couplings involving the new particles are   [16] and self-energy [17] diagrams which contribute to the CP asymmetry and thus the B asymmetry are depicted in Fig. 1.
They are completely analogous to those of leptogenesis where ψ 1,2 are replaced by ν c 1,2 , the asymmetry generated by the decay of ψ 1 assuming that ψ 2 is much heavier is given by where the ψ 1 decay rate is Γ 1 = |f 1 | 2 m ψ 1 /8π. Consider the parameter K = Γ 1 /H(T = m ψ 1 ), where the Hubble parameter is H = 1.66 √ g * (T 2 /M P l ), as a measure of the deviation from equilibrium. If K << 1, which means |f 1 | << 0.02, then the baryon asymmetry is of order /g * . Setting this to 10 −10 , and assuming m ψ 2 /m ψ i = 6, then |f 2 | = 10 −3 if the relative phase between f 1 and f 2 is of order 1.
Axionic dark matter : To obtain an axionic solution to the strong CP problem, a colored fermion is needed which has an anomalous Peccei-Quinn charge. Instead of the usual quark triplet, a fermion color octet, such as the gluino of supersymmetry, may be used [18]. In a nonsupersymmetric context, it may just be any fermion color octet [19]    It acquires a large mass through its coupling to a singlet scalar, the dynamical phase of which becomes the invisible axion and contributes to dark matter.