Pseudo-Nambu-Goldstone Dark Matter Model Inspired by Grand Unification

A pseudo-Nambu-Goldstone boson (pNGB) is an attractive candidate for dark matter (DM) due to the simple evasion of the current severe limits of DM direct detection experiments. One of the pNGB DM models has been proposed based on a {\it gauged} $U(1)_{B-L}$ symmetry. The pNGB has long enough lifetime to be a DM and thermal relic abundance of pNGB DM can be fit with the observed value against the constraints on the DM decays from the cosmic-ray observations. The pNGB DM model can be embedded into an $SO(10)$ pNGB DM model in the framework of an $SO(10)$ grand unified theory, whose $SO(10)$ is broken to the Pati-Salam gauge group at the unified scale, and further to the Standard Model gauge group at the intermediate scale. Unlike the previous pNGB DM model, the parameters such as the gauge coupling constants of $U(1)_{B-L}$, the kinetic mixing parameter of between $U(1)_Y$ and $U(1)_{B-L}$ are determined by solving the renormalization group equations for gauge coupling constants with appropriate matching conditions. From the constraints of the DM lifetime and gamma-ray observations, the pNGB DM mass must be less than $\mathcal{O}(100)$$\,$GeV. We find that the thermal relic abundance can be consistent with all the constraints when the DM mass is close to half of the CP even Higg masses.


Introduction
The existence of dark matter (DM) has been confirmed by several astronomical observations such as spiral galaxies [1,2], gravitational lensing [3], cosmic microwave background [4], and collision of bullet cluster [5]. There are no viable DM candidates in the Standard Model (SM), so the identification of DM plays an important role in particle physics as well as cosmology.
Due to the lack of understanding the nature of DM, there are a lot of DM candidates. One of the candidates is so-called Weakly Interacting Massive Particle (WIMP). To realize the the relic abundance of DM, the WIMP mass is expected to the range of O(10) GeV to O(100) TeV. Further, since the WIMPs have non-gravitational interaction, the direct and indirect detections are expected, but there are still no clear signals of WIMPs, which lead to the strong constraint for WIMP mass and interactions, especially from the direct detection.
In Refs. [19,20], a pNGB DM model is proposed based on G SM × U (1) B−L gauge groups, where G SM := SU (3) C × SU (2) L × U (1) Y . Two complex scalars with Q B−L = +1 and +2, denoted as S and Φ, and three right-handed neutrinos due to the gauge anomaly cancellation are introduced. The gauge symmetry is spontaneously broken via the nonvanishing vacuum expectation value (VEV) of the scalar fields S and Φ as below: (1.1) The results in the model are summarized below. The DM direct detection cross section is naturally suppressed as the same as other pNGB DM models. The pNGB can decay through the new high scale suppressed operators, but the pNGB has a lifetime long enough to be a DM in the wide range of the parameter space of the model. The thermal relic abundance of pNGB DM can be fit with the observed value against the constraints on the DM decays from the cosmic-ray observations. From other viewpoints, the charge quantization of U (1) Y , the gauge anomaly cancellation of G SM , and the almost SM gauge coupling unification even in non-supersymmetric SM seem to imply the existence of grand unification [22]. The unification scale is expected to be O(10 15 − 10 18 ) GeV, where the lower bound comes from the current non-observation of the nucleon decay [23] and the upper bound comes from the Planck scale. Also, the tiny neutrino masses from the neutrino oscillation data seem to suggest an intermediate scale O(10 10 − 10 14 ) GeV through a see-saw mechanism [24].
In this paper, we propose an SO(10) pNGB DM model in the framework of grand unified theories (GUTs). Each Weyl fermion in 16 of SO(10) contains one generation of quarks and leptons, which includes a right-handed neutrino [25]. The SM Higgs and two complex scalar fields S and Φ in Refs. [19,20] are assigned to a scalar field in 10, 16, and 126 of SO(10), respectively. There are several symmetry breaking patterns of SO (10) to G SM × U (1) B−L as below.  [27,28]. We mainly focus on the case of G I = G PS , but we also consider the possibility for such as G I = G LR , where the cases are not favored for a pNGB DM model under our assumption and experimental constraints. (For more information about GUT model building in general, see, e.g., Refs. [29,30]. ) We discuss the following three things. First, the value of the gauge kinetic mixing between U (1) Y and U (1) B−L is a free parameter in e.g., the non-GUT pNGB DM models [19,20], while that is determined mainly by the GUT gauge group in SO(10) models. Second, gauge coupling unification can be achieved due to the contribution from the additional scalar fields that contain a DM candidate. Then the intermediate scale M I , the unification scale M U , and the gauge coupling constant of U (1) B−L are fixed by using the renormalization group equations (RGEs) for gauge coupling constants. Third, the mass of the pNGB in the SO(10) pNGB DM model is limited to be O(10 − 100) GeV from experimental constraints.
The paper is organized as follows. In Sec. 2, we introduce the SO(10) pNGB DM model. In Sec. 3, we find gauge coupling unification determines mass scales and gauge coupling constants of the model. In Sec. 4, the constraints from experiments are discussed. Section 5 is devoted to summary and discussions.

The model
The model consists of an SO(10) gauge field A µ , fermions in 16 of SO(10), a real scalar field in 210 of SO (10), and complex scalar fields in 10, 16 and 126 of SO (10). The SO(10) gauge field contains G SM and U (1) B−L gauge fields. Each fermion in 16 of SO(10) corresponds to quarks and leptons. Scalar fields in 10, 16, and 126 of SO(10) include the Higgs H, S and Φ, respectively. A scalar field in 210 of SO(10) is responsible for breaking the SO(10) symmetry to G PS . The matter content in the SO(10) model is summarized in Table 1 The scalar potential V ({Φ x }) contains quadratic, cubic, and quartic coupling terms, where x = 10, 16, 126, 210. We consider the following symmetry breaking patterns of SO(10) broken to G PS at the unification scale M U by the nonvanishing vacuum expectation value (VEV) of the scalar field in 210 in SO(10), further to G SM at the intermediate scale M I by the VEV of the scalar field in 126 in SO(10), where the M U and M I will be determined by gauge coupling unification using the renormalization group equations (RGEs) for the gauge coupling constants in the next section.
We assume that the scalar fields H, S, and Φ develop the VEVs, which are parameterized by where h, s, and φ are CP-even modes, η s and η φ are CP-odd modes, and v, v s , and v φ are the VEVs of H, S, and Φ, respectively. The CP phase of the cubic term Φ * S 2 is eliminated by the field redefinition of Φ. In the limit µ c → 0, there are two independent global U (1) symmetries associated with the phase rotation of S and Φ. For µ c = 0, the U (1) symmetries are merged to the U (1) B−L (or U (1) X ) symmetry. Once U (1) B−L is broken, one of two CP-odd modes is absorbed by the U (1) B−L gauge field denoted as C µ , while the other appears as a physical pNGB whose mass is proportional to µ c . The scalar fields H, S, Φ have five modes; three of them are CP-even scalar modes and the other two are CP-odd modes. The mass matrix for the CP-even scalars in the (h, s, φ) basis is given by Since the matrix is real and symmetric, it can be diagonalized by a real orthogonal matrix. The gauge eigenstates (h, s, φ) are related with the mass eigenstates (h 1 , h 2 , h 3 ) as 2 When we take into account the nonvanishing VEV of Φ210, quadratic terms |H| 2 , |S| 2 , and |Φ| 2 and the cubic term Φ * S 2 also come from (Φ10Φ10)1(Φ210Φ210)1, (Φ16Φ * 16 )1(Φ210Φ210)1, (Φ 126 Φ * 126 )1(Φ210Φ210)1, Φ16Φ16Φ 126 Φ210, respectively. Therefore, each coefficient such as µc in Eq. (2.3) should be regarded as the total value including all the corresponding terms such as Φ16Φ16Φ 126 and Φ16Φ16Φ 126 Φ210.
where the approximate form of the real orthogonal matrix and its mixing angle are given by . (2.8) The masses of (h 1 , h 2 , h 3 ) are given by The mass eigenstate h 1 is identified as the SM-like Higgs boson with the mass m h 1 125 GeV, h 2 is a light CP-even scalar, and h 3 is a heavy CP-even scalar. The mass matrix of the CP-odd scalars in the gauge eigenstates (η s , η φ ) is given by The gauge eigenstates (η s , η φ ) are related with the mass eigenstates (χ,χ) as where the real orthogonal matrix is given by (2.14) By using the 2 × 2 real orthogonal matrix U o , the mass eigenvalues of (χ,χ) are given by Theχ is the NGB absorbed by the U (1) B−L gauge boson C µ , and χ is the pNGB identified as DM in the paper.

Gauge sector
The gauge kinetic term of the SO(10) can be canonically normalized at the unification scale M U as in Eq. (2.1). In general, the kinetic-mixing term of multiple U (1) symmetries are allowed for the case of at least two abelian groups because a field strength itself is gauge-invariant for abelian groups, while that is not gauge-invariant for non-abelian groups. So, in the energy scale . Therefore, it is expected that the kinetic mixing parameter between U (1) Y and U (1) B−L denoted as is non-zero at classical level.
To determine the value of the kinetic mixing parameter between U (1) Y and U (1) B−L , we focus on the kinetic terms of the gauge fields. First, from Eq. (2.1), the gauge kinetic term of SO(10) is given by Next, the gauge kinetic terms of G PS are given by where G µν , W a µν , and W a µν stand for the field strengths of SU (4) C , SU (2) L , and SU (2) R , respectively; the gauge kinetic terms and mass terms of SO(10)/G PS are omitted at M U . The gauge coupling constants are running from where G µν , B µν and C µν stand for the field strength of , respectively; the gauge kinetic terms and mass terms of where B µν and C µν stand for the field strength of U (1) Y and U (1) B−L , respectively; is the kinetic mixing parameter between U (1) Y and U (1) B−L . In the case, since the U (1) Y generator is given by the following linear combination of U (1) R and U (1) B−L Due to the orthogonality, the kinetic mixing parameter at µ = M I is given by The Lagrangian for the electro-magnetic neutral part of the SU (2) L × U (1) Y × U (1) B−L gauge fields including mass terms generated by the VEVs of the spontaneous SU (2) L × U (1) Y and U (1) B−L breaking scalar fields is given by where Z µ = cos θ W W 3 µ − sin θ W B µ is the usual Z boson, θ W is the Weinberg angle tan θ W := g 1 /g 2 ; g 1 and g 2 stand for the U (1) Y and SU (2) L coupling constants, respectively. The mass parameters are given by To discuss the physical implications of U (1) B−L gauge boson, we requires both diagonalizing the field strength terms and the mass terms. First, we diagonalize the kinetic term in Eq. (2.24) by using the following GL(2, R) transformation: whereB µ andĈ µ stand for the gauge fields of the U (1) Y and "U (1) B−L " in the physical basis. The transformation is exactly the same as that in Eq. (2.20). That is, "U (1) B−L " can be identified as U (1) X (⊂ SO(10)/SU (5)). Then, the gauge kinetic terms in Eq. (2.24) become Next, we consider the physical eigenstate via an O(3) rotation by diagonalizing the mass terms that arise after both U (1) B−L and SU (2) L × U (1) Y breaking. One mass eigenstate is massless corresponding to the photon A µ , while the other two denoted Z and Z receive masses. The mass terms of the neutral gauge boson in terms of (B µ , W 3 µ , C µ ) is given by (2.28) By using GL(2, R) transformation in Eq. (2.26), we change the basis whose kinetic term is diagonalized as below: (2.30) The above mass matrix is a real symmetric matrix. In fact, it can be diagonalized by using a real orthogonal matrix: where the mixing angle ζ is given by . (2.32) From the above, we find the masses of A µ , Z µ , and Z µ as where M 2 is given by In this section, we find that the gauge kinetic mixing in Refs. [19,20] is regarded as the mixing angle. In Appendix A, we will show this more explicitly.

Gauge coupling constants
To determine such as the U (1)  The RGE for the gauge coupling constant α i (µ) := g 2 i (µ)/4π at one-loop level is given in e.g., Refs. [29,30] by where i stands for a gauge group G; e.g., 4C stands for the gauge coupling constant of SU (4) C , and the beta function coefficient is given by where Vector, Weyl, and Real stand for real vector, Weyl fermion, and real scalar fields, respectively. Since the vector bosons are gauge bosons, they belong to the adjoint representation of the Lie group G: T (R V ) = C 2 (G). C 2 (G) is the quadratic Casimir invariant of the adjoint representation of G, and T (R i ) is a Dynkin index of the irreducible representation R i of G. Note that when the Lie group G is spontaneously broken into its Lie subgroup G , it is convenient to use the irreducible representations of G . (For the Dynkin index and the branching rules, see e.g., Refs. [30,45] or calculated by using appropriate computer programs such as Susyno [46], LieART [47,48], and GroupMath [49]. For the RGEs at the two-loop level, see, e.g., Refs. [50][51][52].) Let us consider the RGEs for gauge coupling constants in the pNGB DM model shown in Tables 2, 3, and 4. For the energy scale between M Z < µ < M I and M I < µ < M U , we use the RGEs for the gauge coupling constants of G SM and G PS , respectively. In the following calculation, we assume that there is only one intermediate scale M I and one unification scale M U , which should be recognized as effective scales.
We can obtain the beta function coefficients of the gauge coupling constants of G SM and G PS by using the generic RGE in Eq. (3.2) and the matter content of the model given in Tables 2,  3, and 4. The beta function coefficients of G SM in M Z < µ < M I are given by where i = 4C, 2L, 2R stand for SU (4) C , SU (2) L , SU (2) R , respectively. To distinguish the beta function coefficient of the SU (2) L in G SM and that in G PS , we use unprimed and primed, and the same notation is used below. To solve the above RGEs, we need to set the initial conditions at µ = M Z . The gauge coupling constants must satisfy the matching conditions between G SM and G PS at µ = M I and also the matching condition between G PS and SO(10) at µ = M U . They are listed below.
• The input parameters for the three SM gauge coupling constants at µ = M Z = 91.1876 ± 0.0021 GeV are given in Ref. [53]: • The matching conditions between G SM and G PS at µ = M I are given by where they are determined by the normalization conditions of the generators of G PS and G SM . (See e.g., Ref. [54] at one-loop level; Refs. [55,56] at two-loop level.) • The matching condition at the unification scale M U is given by where Since the standard normalization of U (1) B−L is not the same as that of "U (1) B−L "(⊂ SU (4) C /SU (3) C ), the modified normalization factor is used. The unified gauge coupling constants at µ = M U is given by 14) The energy dependence of the gauge coupling constants α i (µ) in the SO(10) pNGB model is plotted in Fig. 1. We comment on proton decay via a colored Higgs scalar or lepto-quark scalar denoted as S 1 in Ref. [59], which belongs to (3, 1, 1/3) under G SM . In the following, we omit Clebsch-Gordan coefficients for simplicity. When the lepto-quark scalar S 1 has di-quark and quarklepton couplings, there are proton decay modes such as p → e + π 0 , and the proton lifetime is roughly estimated as τ m 4 LQ /(|y| 2 |z| 2 m 5 p ), where m LQ is a lepto-quark mass, y and z represent generic values of relevant Yukawa coupling constants of the lepto-quark with the quark-lepton and quark-quark pairs, respectively. For example, for the lepto-quark with the intermediate scale mass m LQ = M I and the universal Yukawa coupling constants |y| = |z|, we obtain a constraint for the Yukawa coupling constants |y| = |z| 4.2 × 10 −6 from the current constraint τ (p → e + π 0 ) > 2.4 × 10 34 years at 90% CL. To apply this for the current model, for the scalar field S 1 in 10 of SO (10), which belongs to (6, 1, 1) under G PS , the mass of the lepto-quark scalar is the unification scale mass m LQ = M U and the Yukawa coupling constants are roughly expected as |y| = |z| |y (11) 10 |. The current constraint τ (p → e + π 0 ) > 2.4 × 10 34 years at 90% CL leads to |y (11) 10 | 0.68. To realize the mass of up quark, y 10 is roughly O(10 −5 ), so it is consistent with the current constraint, where the actual values of the Yukawa coupling constants depend on how to realized the observed quark and lepton masses. Next, for the scalar fields S 1(10,1,3) and S 1 (1,1,3) in 126 of SO (10), which belongs to (10, 1, 3) and (6, 1, 1) under G PS . The lepto-quark scalar S 1(10,1,3) and S 1(6,1,1) have the intermediate scale mass M I and the unification scale mass M U , respectively. For S 1 (10,1,3) , the Yukawa coupling couplings are given by |y| = 0 and |z| |y 126 |, so the proton decay mediated by S 1 (10,1,3) does not occur.
(For ε = 1, |y (11) 126 | 4.2 × 10 −6 .) Further, we comment on the relation between neutrino masses and the Yukawa coupling constants y 126 , so it is expected that the observed neutrino masses can be reproduced, but to perform it properly, we need to investigate how to reproduce the observed quark and charged lepton masses. We leave it for a future study.
Up to this point, we only consider the specific symmetry breaking pattern, SO(10) broken to G I = G PS at µ = M U in Eq.(1.2). We comment on other cases G I = G PS × D, G LR , G LR × D discussed in e.g., Refs. [41,55,56,60], where D stands for a discrete Z 2 left-right exchange symmetry [61,62]. (Note that the same analysis in SO(10) GUT models whose matter content is slightly different from the present model has been already discussed in e.g., Refs. [55,56] by using two-loop RGEs [63] and the corresponding matching condition [64,65].) To realize the appropriate symmetry breaking patterns, we need different SO(10) breaking Higgs fields; each G I = G PS , G PS × D, G LR , G LR × D is realized by the VEV of a scalar field in e.g., 210, 54, 45, 210 of SO(10), respectively.
The values of M I , M U , and α −1 U for several matter contents and symmetry breaking patterns are summarized in Table 5, which are estimated by using each analytical solution shown in Appendix B. Substituting the values of M U and α −1 U for the G PS × D and G LR × D cases into τ M 4 U /α 2 U m 5 p , rapid proton decay is expected. For the G LR case, the proton decay via lept-quark gauge bosons is consistent with the current experimental constraints, but the pNGB cannot be identified as DM because pNGB decays too rapidly or the observed relic abundance cannot be reproduced.

Long-lived pNGB as DM candidate
The DM lifetime should be longer than the age of the universe, 10 17 s at least. The bound on DM lifetime becomes stronger depending on DM decay channels due to the constraint of cosmicray observations. In particular, the bound from gamma-ray observations is strong as roughly τ χ 10 27 s for two body decays [66]. Since the DM lifetime is proportional to the power of the VEV v φ , it becomes longer for larger v φ . The evaluation of DM lifetime without GUT has been studied in Refs. [19,20], and it has turned out that the VEV should roughly be v φ 10 13 GeV in order to be consistent with the gamma-ray observations if three body decays χ → h i f f and Zf f can occur. Since in the current GUT pNGB model the kinetic mixing sin and the VEV v φ are fixed to be sin = − 2/5 and v φ 10 11 GeV by the requirement of the gauge coupling unification, the three body decays should kinematically be forbidden. Therefore we consider the mass region m χ O(100) GeV and estimate dominant four body decay channels. Before proceeding to four body decays, we comment on the two body decay channel χ → νν, which is possible even in the case m χ O(100) GeV. Similarly to the U (1) B−L model in the previous paper [19], this process occurs via the scalar mixing given by Eq. (2.14) and the mixing between the left-handed and right-handed neutrinos after the electroweak symmetry breaking. The decay width for this channel is calculated as where m ν i is the small neutrino mass eigenvalues. Eq. (4.1) roughly corresponds to the lifetime τ νν = O(10 34 ) s, which is too small to be observed in neutrino cosmic-rays [67,68] because of the suppression by the small neutrino mass squared m 2 ν i . Note that since the scale of the VEV in the GUT pNGB model is v φ 10 11 GeV which is much smaller than the previous analysis [19], the order of the lifetime for this channel is much shorter. However it is still too long to be detectable by experiments and observations.  The four body decay processes χ → f f f f mediated by h i , Z, Z can occur as shown in Fig. 2.
Note that if f and f are identical particles, additional diagrams exist due to interference. We numerically evaluated the decay width for all the four body decay processes using CalcHEP [69], and furthermore we took into account three body decay processes when these are kinematically possible. The results are shown in Fig. 3 in (m χ , v φ ) plane where the second Higgs mass is fixed to be m h 2 = 70 GeV (left) and 130 GeV (right). The orange region below the solid, dashed and dot-dashed lines are the region where the DM lifetime is shorter than the conservative bound τ χ = 10 27 s for the Higgs mixing angle sin θ = 10 −1 , 10 −2 , 10 −3 , respectively. 3 The horizontal black dotted line denotes v φ = M I = 10 11.10 GeV. The most part of the region in the plots is dominated by the four body decays except for the region m χ 60 GeV in the left panel where the three body decay χ → h 2 f f can open up. One can read off the upper bound of the DM mass m χ for a given mixing angle sin θ. Fig. 4 shows the parameter space in (m χ , m h 2 ) plane for the Higgs mixing angle sin θ = 10 −1 , 10 −2 and 10 −3 where v φ = M I . The region m χ m h 2 is strongly constrained by three body decay χ → h 2 f f while the other region is constrained by four body decays. In particular, if the second Higgs mass is degenerate with the SM-like Higgs boson (m h 1 m h 2 ), the four body decay width can be small and the constraint is weaken. This is because the effective coupling χ-f -f mediated by h 1 and h 2 becomes small when m h 1 m h 2 . Figure 5: Parameter space thermally reproducing the observed relic abundance consistent with some other observations. The red line represents the parameter space reproducing the correct thermal relic abundance Ω χ h 2 0.12. The orange and green region are excluded by gamma-ray observations coming from the DM decay and annihilations, respectively. The purple region are excluded by the constraints of the Higgs invisible decay h 1 → χχ and the Higgs signal strength. The gray region is perturbative unitarity bound λ S > 8π/3. Thermal relic abundance of DM is calculated using micrOMEGAs [70]. The results are shown in Fig. 5, where the other parameters are fixed to be m h 2 = 70 GeV, sin θ = 0.05 in the left panel and m h 2 = 130 GeV and sin θ = 0.05 the right panel. The red line denotes the parameter space which can reproduce the observed relic abundance of DM Ω χ h 2 0.12 [4]. The purple region is excluded by the constraints of the Higgs invisible decay and Higgs signal strength [71,72], and the gray region is excluded by the perturbative unitarity bound λ S < 8π/3 [73]. The green and orange region are ruled out by the constraints of the gamma-ray observations for DM annihilations [74] and four body decays [66], respectively. One can see that the thermal relic abundance can be consistent with all the constraints when the DM mass is rather close to the resonances m χ m h i /2. This is the characteristic due to the requirement from the gauge coupling unification in the current GUT pNGB model.
We comment on the allowed parameter space m χ m h i /2. For the second Higgs mass rather heavier than the SM-like Higgs mass, the constraint of the gamma-ray observations can be avoided only if the DM mass is light enough m χ 35 GeV as can be seen from Fig. 4. On the other hand, this mass region cannot be consistent with the thermal relic abundance of DM since it is far from the Higgs resonances. Therefore the mass region m h 2 m h 1 is completely excluded as long as thermal production mechanism of DM is assumed. For more precise calculations in the region m χ m h i /2, the effect of the early kinetic decoupling from the SM thermal bath should be taken into account [75,76]. If this effect is included, one can expect that the red line in Fig. 3 is shifted slightly upward.

Summary
In this paper, we proposed an SO (10)  The DM lifetime without GUT has analyzed in Refs. [19,20]. It suggests that the VEV should roughly be the VEV of Φ v φ 10 13 GeV in order to be consistent with the gamma-ray observations if three body decays χ → h i f f and Zf f are possible. In the current GUT pNGB model, the kinetic mixing and the VEV are fixed to be sin = − 2/5 and v φ 10 11 GeV, respectively. To satisfy the constraint from the gamma-ray observations, the pNGB DM mass must be m χ O(100) GeV to forbid the three body decays kinematically. In the mass region, the dominant contribution for DM decay channels comes from four body decay channels χ → ff f f . We find that the thermal relic abundance can be consistent with all the constraints when the DM mass is rather close to the resonances m χ m h i /2.

A.1 G PS → G SM
First, let us consider the following symmetry breaking pattern where E µ is the gauge field associated with U (1) B−L ⊂ SU (4) C and g B−L is the gauge coupling constant given by g B−L = and these gives the mass terms of the gauge fields where the mass matrices for the color charged vector bosons G 3,a µ and G 3,a µ are defined by The last term of Eq. (A.10) leads the mass mixing between U (1) B−L ⊂ SU (4) C and U (1) R ⊂ SU (2) R , and the massless direction becomes U (1) Y in the SM gauge group. From this term, the massive vector boson C µ and the orthogonal massless gauge boson B µ are introduced by where the mixing angle is defined by 13) and the mass of C µ becomes M 2 C = (g 2 R + 4g 2 B−L )(v 2 s /4 + v 2 φ ). In this basis, the Lagrangian is If the color charged vector bosons are dropped, the covariant derivative is rewritten by using these bosons as where the hypercharge is defined by 16) and the couplings are given by Correspondence between the pNGB model [19,20] and the SO(10) pNGB model We will discuss the kinetic mixing in the GUT model. First, from Eq. (A.12), B µ is written by using (W 3 µ , E µ ) as B µ = W 3 µ / cos + sin E µ / cos , and the field redefinition by cos leads the canonically normalized gauge kinetic terms. The massive direction of broken U (1) symmetry does not change in this rewriting. Then Let us introduce new fields after the rescaling by so that the massive direction does not change but the massless component is replaced. The relation between (W 3 µ , E µ ) and (B µ , C µ ) is given by The U (1) B−L × U (1) R gauge sector in the Lagrangian (A.14) is rewritten by using these fields as , and the covariant derivative is given by Gauged U (1) B−L model [19] pNGB in SO(10) GUT kinetic mixing gauge kinetic mixing of B µ and X µ : gauge kinetic mixing of B µ and C µ : = free parameter = mixing angle of (W 3 µ , E µ ) → (B µ , C µ ) in Eq. (A.14)  [19,20] and SO(10) GUT model. By this breaking pattern, the covariant derivative of G PS reduces to that of SU (3) C × SU (2) L × U (1) R3 × U (1) B−L as which is same to the last term of Eq. (A.10). In this breaking pattern, the charged gauge bosons become massive via the VEV of the adjoint Higgs fields. The mixing angle and correspondence between the mixing angle and kinetic mixing are same in the previous discussions.

B RGEs for gauge coupling constants
Here we analyze the RGEs for gauge coupling constants of G SM and G I = G PS , G LR , and SO (10) in the pNGB DM model. (For the RGE analysis, see e.g., Ref. [54].) The RGE for the gauge coupling constants given in Eq. (3.1) can be solve as when the beta function coefficients b i are constant in the energy range µ 0 < µ < µ 1 . In the following, we apply the solution for G SM , and G I = G PS , G LR cases.
In the following, we find the intermediate scale M I and M U can be described by using the gauge coupling constants of G SM at µ = M Z and the beta function coefficients of G SM and G I (= G PS , G LR ). Therefore, all the gauge coupling constants such as the unified gauge coupling constant α U can be analytically solved if they exist.

B.1 G I = G PS case
We list up the RGEs of G SM and G PS in M Z < µ < M I and M I < µ < M U , respectively, and the matching conditions at µ = M I , M U . where