$SO(10) \to SU(5) \times U(1)_\chi$ as the Origin of Dark Matter

In the decomposition of $SO(10)$ grand unification to $SU(5) \times U(1)_\chi$, two desirable features are obtained with the addition of one colored fermion octet $\Omega$, one electroweak fermion triplet $\Sigma$ and one complex scalar triplet $S$ to the particle content of the standard model with two Higgs doublets. They are (1) gauge coupling unification of $SU(3)_C \times SU(2)_L \times U(1)_Y$ to $SU(5)$, and (2) the automatic (predestined) emergence of dark matter, i.e. $\Omega$, $\Sigma$ and $S$, with dark parity given by $(-1)^{Q_\chi + 2j}$. It suggests that $U(1)_\chi$ may well be the underlying symmetry of the dark sector.

Introduction : The origin of dark matter and the symmetry which maintains it are important issues in particle and astroparticle physics. A prevalent supposition is supersymmetry which admits superparticles as belonging to the dark sector if R parity is imposed. Great hope was attached to the Large Hadron Collider (LHC) in discovering supersymmetry at the present 13 TeV total center-of-mass energy, but no sign of such has yet been reported. Is there another underlying framework for dark matter? More importantly, does this underlying framework provide as well the dark symmetry required [1,2], instead of having it imposed as in supersymmetry? The answer, as suggested in this paper, is U (1) χ which is the possible residual symmetry in the breaking of SO (10) to SU (5).
In most studies of SO(10) grand unification, the breaking to a left-right extension of the standard model is considered. In that case, since U (1) B−L is an Abelian gauge factor in the decomposition SO(10) → SU (3) C × SU (2) L × SU (2) R × U (1) B−L with the electric charge Q = I 3L + I 3R + (B − L)/2, the parity (−1) 3(B−L)+2j may be used to distinguish matter from dark matter and coincides with the R parity of supersymmetry. Many studies [3,4,5,6,7,8,9,10,11,12] have been made using this obvious connection. On the other hand, the (trivial) breaking of SO (10) to SU (5) is usually considered to be uninteresting, because it reduces to studying SU (5) grand unification by itself. The possible residual U (1) χ symmetry is treated as an unimportant peripheral issue, allowing LHC data to put a limit on the Z χ boson mass of about 4.1 TeV [13,14].
Whereas examples of automatic (predestined) dark matter are possible in the standard model (SM) itself [1] or some of its simple gauge extensions [2], their origin is unexplained.
In the context of U (1) χ , because all SM fermions have odd charge Q χ and all SM bosons have even Q χ , the dark sector in this framework consists simply of all fermions with even Q χ and scalars with odd Q χ . The stability symmetry of dark matter is thus revealed to be (−1) Qχ+2j . Note the important fact that Q χ is not part of the electric charge, whereas B − L is. This means that Z χ is independent of the photon as well as the Z boson, whereas the B − L gauge boson would not be. Note that each complete fermion multiplet of SU (5), i.e. 5 * or 10, has its own unique Q χ , i.e. 3 or −1, whereas the complete fermion multiplet 16 of SO(10) has different B − L for its various components, separated by its SU (3) C × SU (2) L × SU (2) R content. Consequently, Q χ is more desirable than B − L as a marker of dark matter. There is also an important difference in their corresponding phenomenology.
R appear in the latter scenario but not in the former.
To find a marker for SO(10) multiplets, consider the decomposition In that case, Q ψ takes the role of Q χ , and Z ψ is independent of the three neutral gauge and Σ ∼ (1, 3, 0, 0) are added together with the scalar S ∼ (1, 3, 0, −5) at the TeV scale, then SU (5) gauge unification is achieved with M U ∼ 10 16 GeV. This is a new realization where the particles added to those of the SM are all in the dark sector. It suggests that matter and dark matter are related to each other [15]. It is also qualitatively different from previous studies requiring SO (10) gauge unification. In particular, the scalar triplet S from the 144 of SO(10) is unique to this proposal.
Hence the allowed Yukawa couplings are as desired. Note that U (1) χ is broken by 2 units as φ This suggests strongly that a dark sector exists where Q χ is even for fermions and odd for scalars so they all have odd R χ . The next step is to identify possible candidates which will also enforce gauge SU (5) unification, thus justifying the role of U (1) χ as the residual symmetry from the breaking of SO(10) to SU (5).
The particle content so far consists of all the SM fermions and gauge bosons together with two Higgs doublets and one singlet. There is also the Z χ gauge boson at the TeV scale.
Experimentally, the three gauge couplings corresponding to SU (3) C × SU (2) L × U (1) Y are measured, but it is well-known that they do not extrapolate to a single value at a possible unification scale, based on this particle content. On the other hand, this may be achieved simply with the addition of two fermion and one scalar multiplets, all belonging to the dark sector at the TeV scale.
Consider the one-loop renormalization-group equations governing the evolution of gauge couplings with mass scale: where α i = g 2 i /4π and the numbers b i are determined by the particle content of the model between M 1 and M 2 . In the SM with one Higgs scalar doublet, these are given by where N F = 3 is the number of quark and lepton families and b Y has been normalized by which contains (24, −5) and thus (1, 3, 0, −5). Note that 16 * × 10 = 16 + 144.
As a result, Ω contributes ∆b C = (2/3)3 = 2, Σ contributes ∆b L = (2/3)2 = 4/3, and S contributes ∆b L = (1/3)2 = 2/3. Note that Ω, Σ, and S all have odd R χ . Assuming unification at M U and normalizing α Y by 5/3, the evolution equations are then given by where α C , α L , α Y are evaluated at M Z . Their central values are [17] The idea that octets and triplets are important in SU (5) gauge unification is not new [18,19,20,21]. However, the role of U (1) χ was not recognized. Otherwise, the choice here follows closely that of Ref. [21]. Note that the chosen particle content is free of gauge anomalies even with the inclusion of U (1) χ . Eliminating α U and using Eq.

= ln
Subtracting the two equations to eliminate M U , the condition which breaks U (1) χ . Consider first the colored fermion octet Ω. It is just like the gluino of supersymmetry, except that it is stable here because there are no scalar quarks. However, because it has strong interactions, bound states do exist [23,24,25] from the exchange of gluons. These gluinonia would then decay into quark pairs. At the LHC, they may be searched for, as shown in Ref. [25]. Since M Ω is predicted here to be relatively light, this signature is a possible verification of its existence.
If M S > M Σ , then S decays to νΣ through the U (1) χ allowed f S S * ν c Σ Yukawa coupling and the neutrino ν − ν c mixing ∼ m ν /M R , with a decay rate In this scenario, Σ 0 is stable. Its relevance as dark matter has been studied in detail [22].
In particular, the radiative splitting of Σ ± with Σ 0 is known [26] to be positive from gauge boson interactions, but is limited [1] to 167 MeV. This means that whereas Σ 0 is the dark matter today, its relic abundance is determined in the early Universe with the coexistence of Σ ± , i.e. all annihilation channels involving Σ 0 , Σ ± have to be considered. This was done in Ref. [22] and 2.28 < M Σ < 2.42 TeV (26) was obtained. Similar results are obtained in supersymmetry with a pure wino as dark matter [27]. The difference is that the wino has many other interactions which are absent in the case of Σ. As for direct detection, Σ 0 does not couple to the Z χ or Z or any scalar at tree level, but may interact with quarks in one and two loops. However, these effects are known to be small [28], with the spin-independent cross section below 2 × 10 −47 cm 2 .
If M S < M Σ , then S 0 is dark matter. Since it is a scalar, it has possible quartic interactions with the two Higgs doublets and one singlet. Any such interaction must be suppressed to avoid the constraint from direct-search experiments because all R χ even scalars couple to quarks. This means that the annihilation cross section of S 0 to scalars would be too small, so its relic abundance should again be determined by gauge interactions, as in the case of Σ.
Phenomenology of the U (1) χ Sector : The contribution of the SM particles to b χ is where the factor 1/40 has been inserted to normalize Q χ . The addition of a second Higgs doublet at M φ contributes ∆b χ = 1/15, and those of ν c and the scalar singlet ζ ∼ (1, 1, 0, −10) contribute ∆b χ = 25/12, whereas S contributes ∆b χ = 5/8. Hence given by Using M Zχ = 4.1 TeV from the LHC lower bound, the Z − Z χ mixing is then at most which is consistent with precision electroweak measurements [17]. In fact, ζ always appears together with (ζ ) * . In other words, because of the chosen particle content, ζ does not interact singly with any combination of the available fields. After spontaneous symmetry breaking, the resulting Higgs scalar H χ = √ 2(Re(ζ ) − v χ ) behaves as a particle with even R χ even though its original Q χ is odd.

Two Variations with Dirac Neutrinos
In They imply that ζ has L = −2 and S has L = −1. In other words, the proposal of Ref. [29] of leptonic dark matter S 0 with scalar dilepton mediator ζ is realized, where ζ decays only to two neutrinos. Assumming ζ to be light (10 to 100 MeV), the self-interacting dark matter S 0 may then explain [30] the central flatness of the density profile of dwarf galaxies [31].
On the other hand, ζ has a large production cross section through Sommerfeld enhancement at late times. Its decay to electrons and photons would disrupt [32] the cosmic microwave background (CMB) and be ruled out [33] by the precision observation data now available [34].
Here ζ decays only to neutrinos, thus solving this important problem. Note that if S is not a triplet but a singlet, then the Yukawa terms ζ SSS and ζ ζS would be allowed [29], in which case U (1) L breaks to Z 3 and S * transforms as ζ, so it cannot be dark matter. As it is, the fact that S ∼ (1, 3, 0, −5) allows it to be stable dark matter, as well as a contributer to the gauge unification of SU (5)  To verify this proposal that U (1) χ is the origin of dark matter, the production of Z χ would be a necessary first step. However, its mass is not precisely predicted, only that it should be at the TeV scale. At present, the LHC bound [13,14] is about 4.1 TeV. However, a more detailed study [35] shows that it can be improved. Other new particles to look for are the bound states of Ω, i.e. gluinonia, which are possible [25] with higher luminosity at the LHC.
It should also be pointed out that if supersymmetry is imposed on SO(10) → SU (5) × U (1) χ , then the origin of R parity is again traced to Q χ . In other words, there is no need to impose it to distinguish the would-be identical Higgs and lepton superfields in the minimal supersymmetric standard model, because they now have different Q χ . Hence it could turn out that supersymmetry is there after all, but to explain dark matter, U (1) χ is still the key.
In that case, gauge coupling unification comes about from the presence of the gluino (acting as Ω), the wino (acting as Σ), the bino, and two higgsino doublets (replacing S), as well as complete multiplets of squarks and sleptons.
As remarked earlier, there are three equivalent markers of dark matter: (−1) 3(B−L)+2j , (−1) Qχ+2j , and (−1) Q ψ +2j . They are all even for the known SM particles and odd for would-be particles of the dark sector. The choice of the latter is somewhat arbitrary and many studies have been made. In Ref. [3], (−1) 3(B−L) was considered in the context of supersymmetry and SO(10) → SU (3) C × SU (2) L × SU (2) R × U (1) B−L was advocated but not SO(10) → SU (5) × U (1) χ , although SU (5) by itself was discussed and discarded. Subsequent work all follow this lead. In Refs. [4,5], supersymmetric SO(10) was considered with conserved (−1) 3(B−L) . The dark sector consists of all superpartners of the SM particles and the various Higgs multiplets used to break the symmetry at various scales. In Ref. [6], the scalar singlet and elctroweak doublet contained in the 16 of SO(10) are considered as dark matter in a nonsupersymmetric context. In Ref. [7], SO(10) → SU (5) × (−1) Qχ was considered. This work is closest to the present proposal, but there Z χ is superheavy and unobservable, whereas here the SM particles have Q χ charges and couple to Z χ at the TeV scale. Also, the chosen dark sector is completely different. In Ref. [8], nonsupersymmetric SO(10) was considered using (−1) 3(B−L) as a marker, but U (1) χ was not mentioned and Z χ is explicitly absent. However, this model is close to the present proposal in its dark-matter content, i.e. the electroweak fermion triplet Σ. In Refs. [9,10], nonsupersymmetric SO (10) was considered with various breaking scales. There is again no Z χ and the scenarios for dark matter are different. In Ref. [11], nonsupersymmetric SO(10) → SU (5) × U (1) χ was mentioned but it breaks to the SM at the unification scale. Hence Z χ is again superheavy. The dark sector mimics those of supersymmetry, i.e. the gauginos and the higgsinos. In Ref. [12], nonsupersymmetric SO(10) was considered, where fermions in a vectorlike 10 L + 10 R of SO(10) belong to the dark sector. These are analogs of the higgsinos in supersymmetry, i.e.
the fermionic partners of the scalar 10 of SO(10) which contains the Higgs bidoublet.
In summary, these references are related but also very different from the scenario discussed in this paper, which is indeed new and not contained in any previous study of this specific topic.