An SO(10) \times SO(10)' model for common origin of neutrino masses, ordinary and dark matter-antimatter asymmetries

We propose an SO(10) \times SO(10)' model to simultaneously realize a seesaw for Dirac neutrino masses and a leptogenesis for ordinary and dark matter-antimatter asymmetries. A (16\times \overline{16}')_H scalar crossing the SO(10) and SO(10)' sectors plays an essential role in this seesaw-leptogenesis scenario. As a result of lepton number conservation, the lightest dark nucleon as the dark matter particle should have a determined mass around 15 GeV to explain the comparable fractions of ordinary and dark matter in the present universe. The (16\times \overline{16}')_H scalar also mediates a U(1)_{em} \times U(1)'_{em} kinetic mixing after the ordinary and dark left-right symmetry breaking so that we can expect a dark nucleon scattering in direct detection experiments and/or a dark nucleon decay in indirect detection experiments. If a proper mirror symmetry is imposed, our Dirac seesaw will not require more unknown parameters than the canonical Majorana seesaw.

On the other hand, the dark and ordinary matter contribute comparable energy densities in the present universe [47]. This coincidence can be understood in a nature way if the dark matter relic density is a dark matter-antimatter asymmetry  and has a common origin with the ordinary matter-antimatter asymmetry. The mirror world based on the gauge groups [SU (3) is a very attractive asymmetric dark matter scenario [33,. The mirror models can contain a tiny U(1) Y × U(1) ′ Y kinetic mixing input by hand to open a window for dark matter direct detections.
In this paper we shall propose an SO(10) × SO(10) ′ model with a (16 × 16 ′ ) H scalar to simultaneously realize a seesaw for Dirac neutrino masses and a leptogenesis for ordinary and dark matter-antimatter asymmetries. After the ordinary and dark left-right symmetry breaking, the (16 × 16 ′ ) H scalar can acquire an induced vacuum expectation value. The ordinary right-handed neutrinos and the dark left-handed neutrinos then can form three heavy Dirac fermions to highly suppress the masses between the ordinary left-handed neutrinos and the dark right-handed neutrinos. Meanwhile, these heavy Dirac fermions can decay to generate a lepton asymmetry in the ordinary leptons and an opposite lepton asymmetry in the dark leptons. The SU(2) L and SU(2) ′ R sphaleron processes respectively can transfer such lepton asymmetries to an ordinary baryon asymmetry and a dark baryon asymmetry. With calculable lepton-to-baryon conversations in the ordinary and dark sectors, the lightest dark nucleon as the dark matter particle should have a predictive mass about 15 GeV to explain the ordinary and dark matter in the present universe as the ordinary proton has a known -1 -

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mass about 1 GeV. Benefited from the U(1) em × U (1) ′ em kinetic mixing mediated by the (16 × 16 ′ ) H scalar, the dark proton as the dark matter particle can scatter off the ordinary nucleons at a testable level while the dark proton/neutron as the dark matter particle can decay to produce the ordinary fermion pairs. Furthermore, a softly broken mirror symmetry can be imposed to simplify the parameter choice.

Fields and symmetry breaking
In the ordinary SO(10) sector, we have the fermions and scalars including Φ(1, 2, 2, 0) ∈ 10 H and/or others , where the brackets following the fields describe the transformations under the SU(3) c × SU(2) L × SU(2) R × U(1) B−L gauge groups. Accordingly, the fermions and scalars in the dark SO(10) ′ sector contain Φ ′ (1, 2, 2, 0) ∈ 10 ′ H and/or others , where the brackets give the SU There is also a (16 × 16 ′ ) H scalar crossing the SO(10) and SO(10) ′ sectors, For simplicity, we shall not consider the details of the SO(10) and SO(10) ′ symmetry breaking. Instead, we shall demonstrate at the left-right level. The ordinary and dark leftright symmetries are expected to have the breaking patterns as below, We further impose a U(1) G global symmetry under which (χ * L , χ R ) and (χ ′ * R , χ ′ L ) carry a same charge. This means the following cubic terms should be absent from the scalar potential. Therefore the neutral components of the scalars χ L , ∆ L,R , χ ′ R , ∆ ′ R,L will not acquire any induced vacuum expectation values. Accordingly, we can give a nonzero Σ RL ′ ≤ χ R,L ′ and a zero Σ LR ′ from the scalar interactions as below,

Dirac neutrinos and lepton asymmetries
We write down the Yukawa couplings relevant for the fermion mass generation, When the left-right symmetries are broken down to the electroweak symmetries, we can derive Figure 1. The heavy masses between the ordinary right-handed neutrinos ν R and the dark lefthanded neutrinos ν ′ L are responsible for suppressing the masses between the ordinary left-handed neutrinos ν L and the dark right-handed neutrinos ν ′ R .
Here the Higgs scalars φ and φ ′ with the vacuum expectation values, are responsible for spontaneously breaking the ordinary and dark electroweak symmetries. According to the symmetry breaking pattern (2.4), the fermion masses thus should be Note the dark charged leptons should be the so-called pseudo-Dirac particles for v ′ em ≪ v ′ . As for the ordinary and dark neutrinos, their mass matrix can be block-diagonalized if the off-diagonal blocks are much lighter than the diagonal block, Clearly, the ordinary left-handed neutrinos and the dark right-handed neutrinos can form the extremely light Dirac neutrinos as their masses are highly suppressed by the masses between the ordinary right-handed neutrinos and the dark left-handed neutrinos. This Dirac seesaw is definitely a variation of the canonical Majorana seesaw, see figure 1. For the following discussions we can conveniently define the mass eigenstates by a proper phase rotation, Figure 2. The lepton number conserving decays of the heavy Dirac fermions N i = ν Ri + ν ′ Li into the ordinary leptons l L as well as into the dark leptons l ′ R . The CP conjugation is not shown for simplicity.
As long as the CP is not conserved, the heavy Dirac fermions composed of the ordinary right-handed neutrinos and the dark left-handed neutrinos can have the lepton-numberconserving decays to generate a lepton asymmetryη L stored in the ordinary leptons and an opposite lepton asymmetryη ′ L stored in the dark leptons, Here ε N i is the CP asymmetry defined as below, . We can calculate the decay width at tree level, and the CP asymmetry at one-loop level, The relevant diagrams are shown in figure 2.

Dark matter mass
In the absence of other baryon asymmetries, the produced ordinary lepton asymmetryη L is equivalent to an ordinary B − L asymmetry η B−L = −η L while the dark lepton asymmetrȳ η ′ L is equivalent to a dark B − L asymmetry η ′ B−L = −η ′ L . The ordinary SU(2) L sphaleron processes and the dark SU(2) ′ R sphaleron processes then will partially transfer the ordinary and dark B − L asymmetries to an ordinary baryon asymmetry η B and a dark baryon asymmetry η ′ B , respectively, Note when computing the dark lepton-to-baryon conversation factor C ′ we should take the [SU(2) ′ R ]-triplet scalar ∆ ′ R into account since this scalar drives the dark electromagnetic symmetry breaking much below the dark electroweak scale.
After the dark electromagnetic symmetry breaking, the dark charged leptons acquire a lepton-number-violating Majorana mass term so that the final dark charged lepton asymmetry cannot survive at all [144]. The lightest dark charged lepton denoted as the dark electron will leave a thermally produced relic density, Here α ′ is the dark fine-structure constant. It is easy to check the dark electron will only give a negligible relic density if its mass is at the GeV scale. Furthermore, we will show later the dark photon can efficiently decay into the ordinary fermion pairs. Therefore, if the lightest dark nucleon N ′ is expected to serve as the dark matter particle, its mass should be determined by GeV

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In the above calculation we have simplified the left-right level interactions as Due to the U(1) em × U(1) ′ em kinetic mixing, the physically dark photon will couple to not only the dark charged fermions but also the ordinary charged fermions although the physically ordinary photon doesn't couple to the dark charged fermions, where the physical photons have been defined by [145] A Once the kinematics is allowed, the dark photon can efficiently decay into the ordinary charged fermion pairs, with the dark photon mass m 2Â ′ = 16πα ′ v ′2 em and the ordinary electric charges Q e,µ,τ = −1, Q d,s,b = − 1 3 and Q u,c,t = + 2 3 . The dark photon can mediate an elastic scattering of the dark nucleons off the ordinary nucleons. If the dark proton is the dark matter particle, its scattering will have a spinindependent cross section, (5.7) Such dark matter scattering can be verified in the direct detection experiments [146]. If the dark neutron is the dark matter particle, its scattering off the ordinary nucleons will be further suppressed by its dark magnetic moment [136]. In the present SO(10) × SO(10) ′ framework, we can expect a dark nucleon decay according to the ordinary proton decay. It should be noted the dark leptoquark scalars Ω ′ R,L can be allowed much lighter than the ordinary ones Ω L,R . This means the dark nucleon decay can be fast enough to open a window for the indirect detection experiments although the ordinary proton decay is extremely slow. For example, we can have the dark matter decay chains p ′ → π ′0 e ′+ (or n ′ → π ′0ν′ R ), π ′0 → γ ′ γ ′ , γ ′ → e + e − , uū, dd, µ + µ − , . . .. Clearly, if the dark photon mass is about 1-2 MeV, the dark matter should mostly decay into the positron/electron pairs. The dark electromagnetic interactions will lead to a dark matter self-interaction. For example, if the dark proton is the dark matter particle, we can have the self-interacting cross section as below, In the case the dark neutron is the dark matter particle, its self-interaction should be determined by a dark magnetic moment and hence should be further suppressed. The dark strong interactions will also result in the dark matter self-interaction. We have known the scattering of the ordinary neutrons off the ordinary protons should have a cross section σ np ∼ 10 −24 cm 2 . The isospin symmetry then can give σ pp ≃ σ nn ≃ σ np . We hence can estimate the cross sections of the dark nucleons' self-interactions to be with Λ QCD and Λ QCD ′ being the ordinary and dark hadronic scales. It is easy to see that the self-interactions (5.8) and (5.9) can be consistent with the constraints from simulations and observations [147][148][149][150],

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We then can make use of the beta functions of the ordinary and dark QCDs to determine Since the dark hadronic scale is lighter than the dark quark masses, we can simply ignore the dark QCD contributions to the masses of the dark baryons and mesons such as In this case, the dark proton is the lightest dark nucleon and hence is the dark matter particle. Another interesting consequence of the above parameter choice is that the Dirac seesaw (3.5) now can be given by which doesn't contain unknown parameters compared with the canonical Majorana seesaw.

Summary
In this paper we have proposed an SO(10) × SO(10) ′ model to simultaneously explain the smallness of the Dirac neutrino masses and the coincidence between the ordinary and dark matter. Specifically we introduced a (16×16 ′ ) H scalar crossing the ordinary SO(10) sector and the dark SO(10) ′ sector. This (16 × 16 ′ ) H scalar can acquire an induced vacuum expectation value after the 16 H and 16 ′ H scalars drive the spontaneous breaking of the ordinary and dark left-right symmetries. Consequently the ordinary right-handed neutrinos and the dark lefthanded neutrinos can form the heavy Dirac fermions to highly suppress the masses between the ordinary left-handed neutrinos and the dark right-handed neutrinos. The decays of such heavy Dirac fermions can generate an ordinary lepton asymmetry and an opposite dark lepton asymmetry. We hence can obtain an ordinary baryon asymmetry and a dark baryon asymmetry due to the SU(2) L and SU(2) ′ R sphaleron processes. By taking into account the difference between the ordinary and dark lepton-to-baryon conversations, we can expect the lightest dark nucleon as the dark matter particle to have a determined mass around 15 GeV. Furthermore, the (16 × 16 ′ ) H scalar can mediate a small U(1) em × U(1) ′ em kinetic mixing after the ordinary and dark left-right symmetry breaking. Therefore, the dark proton as the dark matter particle can be verified by the direct and indirect detection experiments. Alternatively, if the dark neutron is the dark matter particle, it can be only found by the indirect detection experiments. Our model can accommodate a softly broken mirror symmetry to simplify the parameters.