Pseudo-Goldstone Dark Matter in $SO(10)$

We propose a pseudo-Goldstone boson dark matter (pGDM) particle in $SO(10)$ grand unified theory (GUT). Due to its Goldstone nature, this pGDM evades the direct DM detection experiments which, otherwise, severely constrain the parameter space of DM models. In $SO(10)$, the pGDM is embedded as a linear combination of the Standard Model (SM) singlet scalars in ${\bf 16_H}$ and ${\bf 126_H}$ representations. We consider two scenarios for the intermediate route of $SO(10)$ symmetry breaking (SB) to the SM: $SU(5) \times U(1)_X$ and Pati-Salam the $SU(4)_c \times SU(2)_L \times SU(2)_R$ (4-2-2) gauge groups. The vacuum expectation value of ${\bf 126_H}$, which triggers the breaking of $U(1)_X$ and 4-2-2 symmetry in the two scenarios, respectively, determines the pGDM lifetime whose astrophysical lower bound provides one of the most stringent constraints. For the 4-2-2 route to $SO(10)$, the successful SM gauge coupling unification requires the 4-2-2 breaking scale to be ${\cal O} (10^{11})$ GeV, and most of the parameter space is excluded. For the $SU(5) \times U(1)_X$ route, on the other hand, the $U(1)_X$ breaking scale can be significantly higher, and a wide range of the parameter space is allowed. Furthermore, the proton lifetime in the $SU(5)$ case is predicted to be $4.53 \times 10^{34}$ years, which lies well within the sensitivity reach of the Hyper-Kamiokande experiment. We also examine the constraints on the model parameter space from the Large Hadron Collider and the indirect DM search by Fermi-LAT and MAGIC experiments.


I. INTRODUCTION
Astrophysical and cosmological observations [1] have provided compelling evidence for the existence of dark matter (DM), presumably a non-baryonic particle which constitutes about 85% of the observed matter energy density in the universe [2]. A (quasi-)stable, electrically neutral weakly interacting massive particle (WIMP), absent in the Standard Model (SM), is arguably one of the most attractive DM candidates. The interaction cross-section of WIMPs with the SM particles is typically subject to stringent bounds from the direct DM detection experiments for a wide range of the DM mass.
In Ref. [3] the authors have proposed a SM Higgs-portal DM scenario based on a softly broken global U (1) symmetry, which is called "pseudo-Goldstone DM (pGDM)". The model includes a SM singlet complex scalar field S with a U (1) global charge, whose vacuum expectation value (VEV) spontaneously breaks the U (1) symmetry, and a soft U (1) symmetry breaking term, where µ S is a real mass parameter. The resulting scalar potential is invariant under S → S † , so that the model has a Z 2 symmetry under which only the imaginary component of S (called χ) has an odd-parity, while the real component of S and all the SM fields have even-parity.
Hence, the pseudo-Goldstone scalar χ, which is the Goldstone boson associated with the spontaneous U (1) symmetry breaking in the limit of µ s → 0, is a stable DM candidate.
Thanks to its Goldstone nature, the coupling of the pGDM χ with the SM Higgs boson is proportional to its momentum. As a result, the scattering cross-section of the DM particle with a nucleon through the Higgs boson exchange vanishes in the non-relativistic limit, thus evading the direct DM detection constraints [3].
An ultraviolet (UV) completion of this pGDM scenario was proposed in Refs. [4,5]. The model is based on the gauged B − L (Baryon number minus Lepton number) extended SM [6][7][8][9][10][11][12][13], in which the accidental global B − L symmetry of the SM is gauged. In this model, the soft breaking term of Eq. (1), which is crucial to realize the pGDM scenario, is obtained and identifying Λ Φ A and Φ B with µ 2 S and S, respectively. This is a key idea for the UV completion of the pGDM scenario. It is important to note, however, that the B − L gauge interaction explicitly breaks the Z 2 symmetry responsible for the stability of the pGDM, and hence the pGDM can decay via the B − L gauge interaction. To satisfy the lifetime bound for a decaying DM, τ DM > 10 26 sec, from cosmic ray observations [14], the B − L symmetry breaking scale is required to be v BL 10 11 GeV [5]. In addition, since the Goldstone nature of the DM particle is lost due to the cubic scalar coupling of Eq. (2), the direct DM detection cross section does not exactly vanish [5].
In this paper we propose a UV completion of the pGDM scenario in an SO(10) grand unified theory (GUT), or in other words, an embedding of the gauge extension of pGDM in SO (10). The SO(10) grand unification is an interesting paradigm for the unification of strong, weak, and electromagnetic forces into a single force at a high energy scale. Each family of the SM quarks and leptons belongs in a single irreducible 16 representation of SO(10) along with a SM singlet fermion. Electric charge quantization is incorporated, and the SM singlet fermion in the 16 representation is identified with the right-handed neutrino, which paves a natural way for tiny neutrino masses generation by the seesaw mechanism [15][16][17][18][19]. We will show in the following how the pGDM scenario can be successfully implemented in the SO(10) framework.
The basic idea is as follows. We introduce a scalar field in the 16 representation (16 H ) 1 .
Along with a Higgs field in the 126 representation (126 H ), whose VEV generates Majorana mass terms for the right-handed neutrinos, consider the following SO(10) invariant term in the scalar potential: We represent the Higgs fields as follows: where v H,A,B are the VEVs of H, Φ A , and Φ B fields, resepctively, with v H = 246 GeV.
Requiring the scalar potential to exhibit a minimum around its VEV location leads to the following relations: After spontaneous symmetry breaking the Goldstone modes G ±,0 , are absorbed by the SM gauge bosons W ± and Z. The remaining mixing mass matrices are expressed as where the mass matrices for the CP -odd scalars (χ A , χ B ) and CP -even scalars (φ A , φ B , h) are given by In the CP -odd sector, one combination of χ A and χ B is the would-be Nambu-Goldstone mode (G) absorbed by the B − L gauge boson and the orthogonal combination is the pGDM where cos θ = 2v , and the mass of DM χ is given by For the CP -even sector, we consider the limiting case for the parameters in the mass matrix, Λv B , and λ HA , λ AB → 0. In this limit, φ A decouples from the low energy effective theory, and we only consider the sub-matrix for h and φ B . This sub-matrix is diagonalized for the mass eigenstate,h andφ B , defined as Their mass eigenvalues are given by where H , and the mixing angle θ H is given by Let us calculate the direct DM detection amplitude for the elastic scattering of DM with a nucleon. The scattering occurs at very low energies, so zero momentum transfer limit, t → 0 2 , is a good approximation to evaluate the scattering amplitude. In the limiting case with v A v B , the pGDM χ ≡ χ B to leading order in θ, such that the amplitude in the where Y hf f is the interaction of h with the SM fermions. Using Eq. (10), we obtain It is clear that the direct DM detection amplitude is negligibly small.

III. DM RELIC DENSITY AND INDIRECT DETECTION
The pGDM can interact with the SM particles through theh/φ B portal interaction as follows: with The couplings ofh/φ B to the SM fermions are as follows: 2 For t = 0 the maximum value of the DM direct detection cross section is approximately θ 2 H t 2 max × σ HP [4], where σ HP is the Higgs-portal DM cross section, θ H 0.1, |t max | = m 2 DM v 2 and v 10 −3 is the DM velocity. Hence, realistically we can expect the DM direct detection cross section to be closer to the neutrino floor. See, for example, [23]. The thermal relic abundance of 0.094 ≤ Ωh 2 ≤ 0.128 [2] is satisfied along the blue curves. The green shaded regions are excluded by the Higgs invisible decay constraint [24]. The brown and gray shaded segments along the each abundance curve is excluded after imposing Fermi-LAT + MAGIC bound obtained from annihilation of pGDM particles to bb and W + W − (see Fig. 2). For the We numerically evaluate the relic abundance of the DM using the software MicrOmegas [25]. The model is implemented in CalcHEP [26] by using LanHEP [27]. The DM relic abundance depends on the pGDM pair annihilation cross section, which is determined by 4 free parameters, m χ , θ H , v B and mB. In the following analysis, we fix sin θ H = 0.1 as our benchmark and impose the criterion that the DM relic abundance Ωh 2 = 0.120 ± 0.001 [2] measured by PLANCK collaboration at 3σ confidence level is satisfied.
A pair of DM particles annihilating into the SM particles which subsequently decay to produce gamma-rays can be searched by indirect DM detection experiments such as Fermi-LAT [28] and MAGIC [29]. These experiments have set an upper bound on the cross section of DM pair annihilation into bb and W + W − final states. If m χ < mh/2, the SM(-like) Higgs boson can decay into a pair of pGDM particles. The CMS result provides an upper bound on the branching ratio of the invisible Higgs boson decay, BR(h → χχ) ≤ 0.16 [24].
As previously mentioned, the pGDM gauge interaction explicitly violates the Z 2 symmetry, which would otherwise forbid the decay of the pGDM. Hence we also impose a lifetime bound on pGDM, τ DM > 10 26 sec [14], to further constrain the parameter space. In the following, we show the results of our analysis, including the lifetime constraints for both 4-2-2 and SU (5) × U (1) X scenarios, leaving out the details of the pGDM decay, which will be presented in the later sections.
Our overall conclusions is that the allowed region for the 4-2-2 scenario is significantly smaller than for the SU (5) × U (1) X case. This is because the VEV of 126 H (v A ) in the 4-2-2 case required by the successful SM gauge coupling unification is at the intermediate scale, 10 11 GeV, which is much smaller than the VEV scale required in the SU (5) × U (1) X case.
In particular, for the later case, the pGDM can be as heavy as m χ /GeV = 690, 715, 860 for mB/GeV = 80, 200, 500, respectively. In contrast, only the region close to the resonance points survive for the 4-2-2 case with the pGDM mass m χ O(10 2 ) GeV excluded. In parameter. The interesting feature of the U (1) X extended SM is that for x H = −4/5, the SM quarks and lepton are unified in 10 and 5 representations of SU (5) with U (1) X charges +1/5 and −3/5, respectively [32]. In the SO(10) embedding, the U (1) X charge is normalized to be Q X → 5/8 Q X [33].
The Higgs fields involved in breaking SO(10) down to the SM along with their decomposition under SU (5) × U (1) X are listed below: Here only the SU (5) × U (1) X multiplets depicted in bold develop nonzero VEVs to trigger the following symmetry breaking chain: Here, v GU T , v A,B , and v EW denote the VEVs of the corresponding scalars fields involved in the various stages of the symmetry breaking.
The Higgs potential relevant for the discussion of pGDM at low energies is expressed as Hence the low energy effective scalar potential matches the pGDM potential in Eq. (4).
It is known that the unification of the three SM gauge couplings can be achieved by introducing new vector-like quark/lepton pairs at certain mass scales [32,[34][35][36][37][38][39][40][41]. Let us introduce a new vector-like fermion pair in 16+16 representation under SO(10) and consider the following Lagrangian [33]: Here α i ≡ g 2 i /4π, g 2,3 respectively are the SU (3) c and SU (2) L SM gauge couplings, α 1 = g 2 1 /4π is related to the SM hypercharge gauge coupling by g 1 = 5/3 g Y , and we have set M D c = M Q for simplicity. In our RG analysis, we employ the low energy values of the SM gauge couplings at µ = m t = 172.44 GeV [42]: can be approximated as [43] where m p = 0.983 GeV is the proton mass. This is consistent with the current bound on proton lifetime obtained by the Super-Kamiokande experiment τ p (p → π 0 e + ) 1.6 × 10 34 yr [44]. Importantly, this is within the expected sensitivity reach of the future Hyper-Kamiokande experiment, τ p 1.3 × 10 35 yr [45]. Since the RG running of the three SM couplings only depends on two free parameters, there is a one-to-one correspondence between the masses M Q and M L to satisfy the unification of the couplings. As a result, we find that M Q 3.4 × 10 3 GeV, or equivalently M L 8.0 × 10 12 GeV, for the proton lifetime to be within the search reach of Hyper-Kamiokande experiment.
Next let us discuss the gauge interaction of the pGDM which explicitly breaks the Z 2parity and thereby enabling it to decay. For instance, the gauge interaction in the B − L extended model [5] is given by where In the SO(10) model, we can identify Z BL → Z X with Q A = − 5/2 and Q B = 5/8. For the interaction of the Z X gauge boson with the SM SU (2) L singlet quarks (q) and leptons ( ) we obtain where Q q = −3/ √ 40 and Q = +1/ √ 40. The Z X mass is given by where we have used v B v A in the last expression. Using Eqs. (27) and (28) for µ < v GU T . We impose the perturbative bound on 20λ 2 B /16π 2 < 1 at µ = v GU T , which is extrapolated to the upper bound on λ B (v B ). Thus, we find the lower bound on v B Here only the 4-2-2 multiplets depicted in bold develop a nonzero VEV, which results in the following symmetry breaking pattern: where v GU T , v A,B , and v EW denote the VEVs of the corresponding scalars fields involved at the various stages of the symmetry breaking.
In the fermion sector, the 10 H and 126 H Higgs field generate realistic fermion masses , while the SM singlet component in ( light-charged scalar. To illustrate the mass splitting, consider the following coupling: Their low energy values are fixed to match the measured values at µ = m t = 172.44 GeV [42]: For v A < µ < v GU T , our theory is based on the 4-2-2 gauge group. The relationships between the SM and the 4-2-2 gauge couplings at µ = v A are given by the tree-level matching conditions: Here given by With the initial values of the SM gauge couplings fixed at µ = m t , the RG equations for the SM and 4-2-2 gauge couplings can be solved analytically. After imposing the unification conditions v GU T and v A are uniquely determined. We find v A 3.
The relevant interactions for the pGDM decay, with κ = 3g 2 4 + 2g 2 R /2 = √ 2m Z R /v A , and the effective Z R charges for the SU (2) L singlet SM quarks (q) and SU (2) L doublet leptons ( ) given by Here δ is the mixing between the SU (4) c and SU (2) R components of Z R defined by tan δ = 3/2(g 4 /g R ). In Fig. 5  For the Φ B quartic coupling λ B we obtain the following RG equation Using the solution for the running of the 4-2-2 gauge couplings in Fig. 4   quartic inflation driven by a gauge singlet field with non-minimal coupling to gravity [75,76].
However, in our case with a 4-2-2 breaking scale significantly lower than 10 13 GeV, it would be more appropriate to use the 4-2-2 breaking scalar field as the inflaton. This scenario will be discussed in detail elsewhere [77]. Since there are only heavy GUT monpoles in the SU (5) × U (1) X scenario, the non-minimal inflation by using a SO(10) singlet field with Hubble parameter is of order 10 13 − 10 14 GeV [74] nicely works to resolve the monopole problem.

VII. CONCLUSION
The pGDM scenario is an interesting WIMP DM scenario which can avoid the very severe constraints from the direct DM search experiments. Because of the Goldstone nature of the pGDM, its coupling with the Higgs bosons vanishes in the limit of zero momentum transfer. In this paper, we have proposed an UV completion of the pGDM in the SO (10) GUT framework. For this completion, it is crucial to introduce the SO (10)

NOTE ADDED
During the write up of our manuscript we came across a recent paper [78] which discusses the same topic with SO(10) broken via the gauge symmetry SU (4) c × SU (2) L × SU 2) R .
Where there is overlap our results are in agreement. The SO(10) symmetry breaking via SU (5) × U (1) X has not been discussed in the above mentioned paper.