$U(1)_\chi$, Seesaw Dark Matter, and Higgs Decay

It has recently been pointed out that the underlying symmetry of dark matter may well be $U(1)_\chi$ (coming from $SO(10) \to SU(5) \times U(1)_\chi$) with the dark parity of any given particle determined by $(-1)^{Q_\chi+2j}$, where $Q_\chi$ is its $U(1)_\chi$ charge and $j$ its spin angular momentum. Armed with this new insight, previous simple models of dark matter are reinterpreted, and a novel idea is proposed that light seesaw dark matter exists in analogy to light neutrinos and is produced by the rare decay of the standard-model Higgs boson.

To allow the quarks and leptons to acquire mass, the scalar 10 representation, which contains the necessary Higgs doublets, i.e. 10 = (5 * , −2) + (5, 2), with from (5 * , −2), (5, 2) respectively, is required, resulting in the allowed Yukawa couplings as desired. The nonzero vacuum expectation values φ 0 1,2 = v 1,2 also break electroweak SU (2) L × U (1) Y to electrodynamic U (1). Since the SM gauge bosons all have Q χ = 0, it is obvious (but not recognized for its importance until recently [1]) that all SM fermions have odd Q χ and all SM bosons have even Q χ . This means that each SM particle has even (−1) Qχ+2j where j is its spin angular momentum. It is thus a short step to realizing that any scalar with odd Q χ and any fermion with even Q χ would have odd (−1) Qχ+2j , making it a natural stabilizing symmetry for dark matter. Indeed, all previous simple models of dark matter based on an Z 2 discrete symmetry may be incorporated into such a framework.
The scalar 126 representation of SO(10) contains a singlet ζ ∼ (1, −10) under SU (5) × U (1) χ , which may be used to break U (1) χ at the TeV scale and would allow ν c (the righthanded neutrino) to obtain a large Majorana mass, thereby triggering the canonical seesaw mechanism for small Majorana neutrino masses. This is usually described as lepton number L breaking to lepton parity (−1) L [2], but here it is clear that it has to do with the breaking of gauge U (1) χ to (−1) Qχ .
In the minimal supersymmetric standard model (MSSM), R = (−1) 3B+L+2j is used to distinguish the SM particles from their superpartners, which belong thus to the dark sector if R is assumed conserved. Since R is identical to (−1) 3(B−L)+2j , it has long been recognized [3] that a theory with gauge B − L, broken by two units, would be a natural framework for dark matter. In particular, the decomposition shows that the fermionic 16 of SO(10) contains It should also be pointed out that in E 6 → SO(10) × U (1) ψ , the decomposition 27 = (16, −1) + (10, 2) + (1, −4) shows that Q ψ may be invoked as the underlying dark symmetry as well.
Reappraisal of Z 2 Dark Matter : It has been remarked that it is very easy to invent a model of dark matter. The first step is to introduce a new Z 2 symmetry under which all SM particles are even and a new neutral particle of your choice is odd. It should then have the appropriate mass and interaction to account for the relic abundance of dark matter in the Universe, but not excluded by direct or indirect search experiments.
The simplest such model [4] assumes a real scalar singlet, odd under Z 2 . It has been studied extensively [5] and is still a viable explanation of dark matter. In the framework of of 16 is such a particle. It is in fact the scalar analog of ν c . They have the same Q χ , but differ in spin. Hence one is dark matter and the other is not. In Ref. [2], they are both assigned odd lepton parity, which is now replaced by odd χ parity. If U (1) χ is indeed the origin of s, then it should be complex and it should have Z χ interactions. However, from the allowed ζ * ss trilinear scalar coupling, s splits into two real scalars with a large mass gap.
The lighter is dark matter and the heavier decays into the lighter plus a physical or virtual Z χ gauge boson. This would not affect the lighter scalar's suitability as dark matter, but would predict possible verifiable signatures involving Z χ . The present experimental bound on M Zχ is about 4.1 TeV from LHC (Large Hadron Collider) data [6,7], which may be improved [8] with further study.
Instead of choosing s ∼ (1, −5) from the 16 of SO(10), the scalar doublet (η 0 , η − ) ∼ (1, 2, −1/2, 3) may also be considered [9,10,11]. In that case, it is distinguished from This Z 2 discrete symmetry [12] allows η 0 to be dark matter [13], at least in principle. However, its interaction with quarks through the Z boson rules it out by direct-search experiments. In the SM, the allowed quartic coupling (Φ † η) 2 serves to split Re(η 0 ) from Im(η 0 ), and since Z only couples one to the other, the offending interaction with quarks is avoided kinematically in elastic nuclear recoil with a mass gap larger than a few hundred keV. This is known as the inert Higgs doublet model. In the case of Q χ , such a quartic coupling is forbidden, so if η originates from U (1) χ , other particles are needed for it to be dark matter. They turn out to be exactly s ∼ (1, −5) and ζ ∼ (1, −10), already discussed. The allowed couplings (η 0 φ 0 2 − η − φ + 2 )s, (φ 0 1 η 0 + φ + 1 η − )s combined with ζ * ss form the necessary effective quartic coupling as shown in Fig. 1. In this scenario, a linear combination of η 0 and s is dark matter.  (10). Since N has even Q χ , it is odd under (−1) Qχ+2j . However, it has no renormalizable interaction with the particles of the SM and thus not a good dark-matter candidate without some additional fundamental particle such as a singlet scalar [14,15] which has Q χ = 0, i.e. the scalar counterpart of N . A more interesting option is to combine N with the scalar doublet η discussed in the previous paragraph because there is now an allowed Yukawa coupling between the left-handed lepton doublet (ν, l) L with N through η, i.e. (η 0 ν L +η + l L )N L . This forms the basis of the scotogenic model [16] of radiative neutrino mass. Whereas the original model assumes the (Φ † η) 2 quartic scalar coupling, it must now be replaced by the effective operator of Fig. 1. The resulting diagram [17] for generating a radiative Majorana neutrino mass is then given by Fig. 2.
Whereas N could be dark matter, its only interaction with the particles of the SM is through the left-handed lepton doublet, and is known [18] to be restricted phenomenologically, thus limiting its viability as thermal dark matter. Hence a linear combination of s and η 0 is again the likely dark-matter candidate in this case. They both couple to Z χ but differently. Further study is then needed to reappraise this U (1) χ interpretation of the scotogenic model.
Once both s and N are present, the coupling s * N ν c is allowed. This has also recently been considered [19] with the assumption that it is very small so that a freeze-in mechanism applies to the decay of ν c to s and N . As for gauge U (1) χ anomaly cancellation, the fermion D c χ ∼ (1, 10) from the 126 * of SO(10) should be added. It may combine with D χ to form a Dirac fermion, as proposed recently [20]. Here D c χ is assumed to have an extra symmetry shared by the counterpart singlet N c ∼ (1, 0). This separate system is also assumed to be heavy and annihilate efficiently to SM particles through Z χ in the early Universe. Another possible but different connection between seesaw neutrino mass and dark matter has also been proposed [21], based on an imposed Z 4 discrete symmetry and a nonrenormalizable dimension-five coupling.
Consider now the interaction of D χ . It interacts mainly with Z χ . This is in analogy with ν which interacts mainly with Z and W ± . Just as ν decouples at a temperature of order 1 MeV, D χ would decouple at a temperature of order T ∼ 1 MeV(m Zχ /m Z ) 4/3 . There remains however a suppressed Yukawa coupling to Since m ζ R is heavy, the above interaction is only realized through HD χ D χ , coming from the mixing of the SM Higgs boson H with ζ, which is itself also suppressed, i.e. of order v/u.
With these two suppressions, the resulting interaction strength will be very small, as shown below. the other is heavier and is not relevant to the discussion below.

Higgs
The scalar interactions between the SM Higgs H and ζ is given by where Φ = (0, v + H/ √ 2) and ζ = u + ζ R / √ 2. The mass-squared matrix spanning (H, ζ R ) is then The H − ζ R mixing is then given by λ 3 v/λ 2 u. Hence the HD χ D χ coupling is The decay rate Γ H of H → D χ D χ is then where x = m Dχ /m H . If the reheating temperature of the Universe after inflation is below the decoupling temperature of D χ for thermal equilibrium and above m H , its only production mechanism is freeze-in through H decay before the latter decouples from the thermal bath.
The correct relic abundance is possible if f H is very small. Hence D χ could be FIMP (Feebly Interacting Massive Particle) dark matter [22], and for x << 1, the right number density is obtained for [23] f H ∼ 10 −12 x −1/2 .
As a numerical example which satisfies all the above conditions, let m Dχ = 5 GeV, then x = 0.04. Assuming λ 3 = 0.4, then f H = 5 × 10 −12 in Eq. (14) is obtained with m ζ R = 10 7 GeV. Assuming that this is also the value of m Zχ , then the decoupling temperature of D χ is about 5.2 TeV.
Since the U (1) χ breaking scale is about 10 7 GeV in this example of seesaw dark matter, the Z χ gauge boson is much too heavy to be discovered at the LHC. Furthermore, the interaction of D χ with quarks through Z χ is very much suppressed, so that it is not detectable in direct-search experiments.
Concluding Remarks : Using Q χ as a marker in SO(10) → SU (5) × U (1) χ so that (−1) Qχ+2j distinguishes dark matter from matter, previous simple models of dark matter are reap-praised. Furthermore, the notion is put forward that naturally light seesaw dark matter exists in parallel with naturally light seesaw neutrinos. In the latter, the left-handed doublet neutrino ν couples to a heavy singlet right-handed neutrino ν c through the SM Higgs doublet Φ, and ν c acquires a large Majorana mass through the singlet scalar ζ which also breaks U (1) χ and makes Z χ massive. As a result, ν gets a small seesaw mass. In the former,