Gauge Coupling Unification in Gauge-Higgs Grand Unification

We discuss renormalization group equations for gauge coupling constants in gauge-Higgs grand unification on five-dimensional Randall-Sundrum warped space. We show that all the four-dimensional Standard Model gauge coupling constants are asymptotically free and are effectively unified in $SO(11)$ gauge-Higgs grand unified theories on 5D Randall-Sundrum warped space.

show that in several SO (11) GHGUTs, the SM gauge couplings are asymptotically free at least at one-loop level and the three SM gauge coupling constants are almost the same values below the GUT scale M GU T = 1/L as long as M GU T = 1/L is much larger than its Kaluza-Klein (KK) mass scale m KK = πk/(e kL − 1) πke −kL , where k is the anti-de Sitter (AdS) curvature in 5D RS warped space. This paper is organized as follows. In Sec. 2, we discuss a RGE for a gauge coupling constant in 5D non-Abelian gauge theory. In Sec. 3, we discuss RGEs for the SM gauge coupling constants in the SO(11) GHGUT [34] and slightly modified ones. We find that the three SM gauge coupling constants are asymptotically free and they are unified in Sec. 3.1. Their several corrections are studied in Sec. 3.2. Section 4 is devoted to a summary and discussion.
2 RGEs for 4D gauge couplings on 5D RS warped space Let us first consider a non-Abelian gauge theory on 5D Randall-Sundrum (RS) warped spacetime.
We consider a model that contains bulk gauge and fermion fields. Its action is given by where L g.f. and L gh stands for gauge-fixing and ghost terms, respectively.

2)
where M = 1, 2, · · · , 5, T A are the generators of the Lie group G, its superscript A is the number of the generators of G, ξ is the gauge-fixing parameter, g is the gauge coupling constant. By using appropriate gauge-fixing and ghost terms discussed in e.g., Ref. [36], we get the KK mode expansion of the gauge field

5)
in a conformal coordinate z := e ky for |y| ≤ L, where k is the anti-de Sitter (AdS) curvature, L is the size of fifth dimension, f A n (z) and h A n (z) are described by using the Bessel functions. (See, e.g., Refs. [23,24].) Here we summarize some basic results for the RGEs for 4D gauge coupling constants. (See, e.g., [6].) We only consider the RGEs at the one-loop level, but we can find the RGEs at the two-loop level given in, e.g., Refs. [37][38][39]. The RGE for the gauge coupling constant is given by where β(g) is a β function for the gauge coupling constant. In general, a model contains real vector, Weyl fermion, and real scalar fields. The β function at one-loop level is given by where Vector, Weyl, and Real stand for real vector, Weyl fermion, and real scalar fields in terms of 4D theories, respectively. The vector bosons are gauge bosons, so 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 its branching rules, see Refs. [7,40].) It is convenient to use the β-function coefficient b By using α(µ) := g 2 (µ)/4π, we can rewrite the RGE in Eq. (2.7) as When b is a constant, we can solve it as Let us consider the RGE for 4D gauge coupling constant in 5D gauge theories given in Eq. (2.10) by using the β function coefficient given in Eq. (2.9), where it depends on its matter content at an energy scale µ. We take into account the contribution to the β function coefficient from not only zero modes but also KK modes below their masses less than renormalization scale µ, where since the contribution to the gauge coupling constant of the zero mode from each KK mode is almost the same as that from the zero mode, we neglect their difference between them. Under the approximation, once we know mass spectra in models, we can calculate the RGE for the gauge coupling constant at one-loop level. In general, it is difficult to write down exact mass spectra because it depends on orbifold boundary conditions and parameters of bulk and brane terms. For the zeroth approximation, the mass of zero modes is m = 0 and k-th KK modes is m = km KK . By using the mass spectra, the RGE of the gauge coupling constant can be divided into two regions: where b 0 is a β-function coefficient given from its zero modes, which can be calculated by using Eq. (2.9); ∆b KK is an additional β-function coefficient generated by a set of KK modes of all bulk fields, which can be also calculated by using Eq. (2.9). The β-function coefficient ∆b KK is because a 5D bulk gauge field is decomposed into 4D gauge and scalar fields and a 5D bulk fermion field is decomposed into 4D Dirac fermion fields. We solve the RGE in Eq. (2.12). The number of the set of KK modes for µ > m KK is approximately equal to the energy scale divided by the KK mass scale: (2.14) We integrate the RGE in Eq. (2.12) with respect to µ from M Z to µ (M Z < µ < m KK ): For µ > m KK , the gauge coupling constant is given by From Eq. (2.16), we find that for ∆b KK > 0, the gauge coupling constant diverges at a certain point while for ∆b KK < 0 and µ m KK , the gauge coupling constant reduces rapidly: Table 1: Summary for the adjoint representation of any Lie group G, where d(G) and C 2 (G) stand for the dimension and the quadratic Casimir invariant of the adjoint representation of G. See Refs. [6,7] in detail.
From Eq. (2.13) and the above discussion, we also find that the gauge coupling constant of a non-Abelian gauge field based on a simple Lie group G is asymptotically free when its matter content satisfies because of ∆b KK < 0. We can check which matter content can satisfy the condition in Eq. (2.19) for any classical and exceptional Lie group by using the quadratic Casimir invariant in Table 1 and the (second order) Dynkin index of irreducible representations of each simple Lie group G listed in Ref. [7]. Especially, by using Tables in Appendix A in Ref. [7], it is easy to check the cases for up to rank-15 simple Lie groups and D 16 = SO (32). Also, by using rank-n discussion, we can check it for any rank classical Lie group.

Gauge-Higgs grand unification
Let us consider the RGEs for gauge coupling constants in the SO(11) GHGUT shown in Table 2 and its slightly modified ones by using the results in the previous section. For the energy scale between M Z < µ < m KK , the RGEs for the SM gauge coupling constants at one-loop level are the same as the RGEs in the SM. To analyze this difference between the three SM gauge coupling constants, we introduce the following values:  where i, j = 3C, 2L, 1Y for the SM gauge coupling constants, α i (µ) = g 2 i /4π(i = 3C, 2L, 1Y ), α 3C (µ) is the SU (3) C gauge coupling constant, α 2L (µ) is the SU (2) L gauge coupling constant, and α 1Y (µ) is the U (1) Y gauge coupling constant, and we take the SU (5) normalization for U (1) Y . (i, j = 4C, 2L, 2R for the Pati-Salam gauge coupling constants). From Eqs. (3.1) and (3.2), we have the following relation: To discuss accuracy of unification, we introduce Ξ ij (µ) defined by  We check β function coefficients of the three SM gauge coupling constants by using the RGE in Eq. (2.9). The SM matter content or the zero mode matter content in the SO(11) GHGUTs is given in Table 3. By using the formula in Eq. (2.9) and the (second order) Dynkin indices listed in Refs. [6,7,40], we obtain the following well-known SM β-function coefficients: The RGE evolution for the SM gauge coupling constants in the SM is shown in Fig. 1, where we used the following input parameters for the three SM gauge coupling constants at µ = M Z = 91.1876 ± 0.0021 given in Ref. [41] , (3.8) where the relations between the EW gauge coupling constants α 2L (µ) and α 1Y (µ) and the electromagnetic (EM) gauge coupling constant α em (µ) and the Weinberg angle θ W (µ) are given by . (3.10) The experimental values of the EM gauge coupling constant and the Weinberg angle given in Ref. [41] are As well-known, GUTs based on the SU (5) gauge group and also other higher rank gauge group without intermediate scales predict the SM gauge coupling unification at the GUT scale M GU T . The relations between the SM gauge coupling constants α i (µ) are given by (3.13) They lead to (3.14) Obviously, sin 2 θ W (M GU T ) = sin 2 θ W (M Z ), so we have to take into account the effects for the RGEs for the SM gauge coupling constants between the EW scale and the GUT scale.
At present the value of α i (M Z ) has roughly 4-digit accuracy according to Ref. [41]. Thus, it is meaningless to discuss more than 4-digit accuracy for Ξ ij (µ), We regard ∀ |Ξ ij (µ)| < 10 −4 as an almost SM gauge coupling unification scale M GCU . From Fig. 1, in the SM, for any scale µ, ∀ |Ξ ij (µ)| cannot be less than 10 −4 , and then in the SM without any correction or only negligible ones, three gauge coupling constants are not unified. If there are intermediate symmetry breaking scales between an original GUT scale and the EW scale, then in general they contribute non-negligible effect for gauge coupling unification; it is discussed in e.g., 4D SO(10) GUTs [42][43][44][45][46] because one of examples is G GU T = SO(10) ⊃ G P S ⊃ G SM . The rank of the original GUT gauge group G GU T must be more than 4 because the rank of the SM gauge group G SM is 4. The rank of the SO(11) gauge group is 5, so we will discuss its intermediate scale effect in the SO(11) GHGUTs. Table 4: Summary for representations of the Lie group SO(11) satisfying a condition T (R) < (21/8)C 2 (SO(11) = 55) = 189/8, where SO(11) Irrep., d(G), T (R), and Type stand for the Dynkin label, the dimension, the Dynkin index, and the type of of the irreducible representations of SO (11), respectively. R and PR represent real and pseudo-real representations of SO (11).

Asymptotic freedom and gauge coupling unification
(See Ref. [7] in detail.) We check the asymptotic freedom condition given in Eq. (2.19) in SO (11) GHGUTs. To keep the success of the SO(11) gauge-Higgs grand unification in Ref. [34], such as automatic chiral anomaly cancellation for the gauge symmetries on the Planck and TeV branes, we use the same orbifold boundary conditions (BC); the orbifold BC on the Planck brane y = 0 breaks SO (11) to SO(10); the orbifold BC on the TeV brane y = L breaks SO (11) to SO(4) × SO (7) SU (2) × SU (2) × SO (7). The two orbifold BCs break SO(11) to the Pati-Salam gauge group G P S . The orbifold boundary conditions for the SO(11) vector representation 11 on the Planck and TeV branes are given by Also, by using the branching rules of the representations in Table 4 shown in Ref. [7], we find that the branching rules of pseudo-real representations of SO(11) lead to complex representations of its subgroup, while the branching rules of real representations of SO(11) lead to real representations of its subgroup. That is, we must use pseudo-real representations to realize a 4D chiral gauge theory. In Table 4, only the SO(11) spinor representation 32 is a pseudo-real representation of SO (11). (The SO(11) 320 representation is the second lowest dimensional pseudo-real representation listed in Ref. [7].) Also, the zero modes of each SO(11) spinor bulk fermion field are the five SM fermions plus one right-hand neutrino. Therefore, the matter content of SO(11) GHGUTs must contain at least three SO(11) spinor bulk fermion fields as the same as that in Ref. [34], so we subtract the contribution from the three SO(11) spinor bulk fermion fields. The asymptotic freedom condition is We consider which matter contents can satisfy the asymptotic freedom condition in Eq. (3.16). To maintain the number of chiral matter fields, if we introduce an SO(11) spinor bulk fermion field with a parity assignment, then we must also introduce another SO(11) spinor bulk fermion field with a the opposite parity assignment. From Table 4, the SO(11) 65 representation does not satisfy the condition. By using the condition in Eq. (3.16) and the Dynkin indices given in Table 2, we summarize the matter contents in Table 5 that satisfy three chiral generations of quarks and leptons and the asymptotic freedom condition in Eq. (3.16).
In the SO(11) GHGUT [34], a fermion number conservation lead to sufficient proton decay suppression [34]. When we impose the fermion number conservation, an SO(11) 55 bulk fermion with an orbifold BCs must have another SO(11) 55 bulk fermion with the opposite orbifold BCs; an SO(11) 11 bulk fermion with a orbifold BCs must have another SO(11) 11 bulk fermion with the opposite orbifold BCs. From Table 5, we cannot introduce any SO(11) 55 bulk fermion to n 55 n 32 n 11 keep the fermion number conservation without exotic fermion zero modes. The matter contents that satisfy three chiral generations of quarks and leptons, the asymptotic freedom condition in Eq. (3.16), and the fermion number conservation are shown in Table 6.
3 ≤ 10 −93+8n 10 6 Table 6: Matter contents that satisfy three chiral generations of quarks and leptons, the asymptotic freedom condition in Eq. (3.16), and the fermion number conservation. As in the previous section, we use approximate mass spectra of zero modes and k-th KK modes whose masses are m = 0 and m = km KK , respectively. We also use the gauge coupling constant in Eq. (2.16) for the three SM gauge group, where α −1 , b 0 , and ∆b KK should be replaced by α −1 i , b 0 i , and ∆b KK . α −1 i and b 0 i are dependent on the SM gauge group, while ∆b KK is independent from the SM gauge group. From Eq. (2.16), we find that the difference between the SO(11) GHGUTs and the SM is only its third term dependent on ∆b KK for µ > m KK . Also, the difference between α −1 i and α −1 j (i = j) in the SO(11) GHGUTs is the first and second terms in Eq. (2.16). Therefore, ∆ ij (µ) in the SO(11) GHGUTs are the same as those in the SM. (∆ ij (µ) in the SM are shown in the center figure in Fig. 1.) By using the asymptotic form of the gauge coupling constant given in Eq. (2.18), for µ m KK , Ξ ij (µ) can be written as Let us check what we can learn from Figs. 2, 3, and 4. From Fig. 2, we can clearly see that the three SM gauge coupling constants α i (i = 3C, 2L, 1Y ) are convergent into one and rapidly  Fig. 3, we find that for ∆b KK < 0, the SM gauge coupling constants decrease drastically above m KK and also they converge, where the convergent scales depend on ∆b KK ; for ∆b KK > 0, the SM gauge coupling constants increase drastically and also seem to converge above m KK , where our perturbative calculation is not reliable. From the right figure in Fig. 3, we find that regardless of KK mass scales m KK = 10 6 , 10 10 , 10 14 GeV, the SM gauge coupling constants decrease drastically above m KK and also they converge for ∆b KK < 0, where they diverge for ∆b KK > 0. From the right figure in Fig. 4, we find that for µ ∼ 10 10.5 GeV, Ξ ij (µ) ∼ 10 −4 , so we regard µ >  Fig. 4, and then M GCU starts around 10 11 GeV.
From the above discussion, we found that in the SO(11) GHGUTs the 4D SM gauge coupling constants are almost unified above M GCU regardless of the matter contents and their mass spectra and the SM gauge coupling constants are asymptotically free. We also found that M GCU depends on the matter contents given in Table 5.

Corrections for gauge coupling constants
We check whether the above analysis is valid even when we take into account several corrections. We divide our discussion into two cases, m KK < M P S M GU T = 1/L and m KK < M P S < M GU T = 1/L, where M P S is the symmetry breaking scale at which G P S gauge symmetry is broken in G SM gauge symmetry. This is because for m KK < M GU T = 1/L M P S , we use only the RGEs for the G SM gauge coupling constants, while for m KK < M P S < M GU T = 1/L, we have to use the RGEs for the G SM gauge coupling constants below M P S and the RGEs for the G P S gauge coupling constants above M P S . In the latter analysis, we have to take int account the matching conditions between the G P S gauge coupling constraints and the G SM gauge coupling constraints at the Pati-Salam scale M P S . (Note that for 4D non-SUSY SO(10) GUTs, this effect has been discussed in many articles, e.g., Refs. [42][43][44][45][46][47][48].)

m KK < M GU T M P S 1/L
Here we check whether the above analysis is valid even when we take into account mass spectra of bulk fields. Since mass spectra in the SO(11) GHGUTs depend on orbifold BCs and parameters of bulk and brane terms, it is almost impossible to use them in exact expression. Instead of them, we use approximate forms for flat space limit. We use the mass spectra of kth KK modes (k = 1, 2, · · · ) of bulk fields by their orbifold BCs for flat space limit: where N and D stand for Neumann and Dirichlet BCs, respectively. (X, Y ) (X, Y = N, D) stands for the orbifold BCs on the Planck and TeV branes, respectively. (This approximation is good for large k because the RS warped space is asymptotically flat space for short distance.) Only each field with (N, N ) contains a zero mode. For large k, a kth KK mass spectrum in RS warped space is approaching to that in flat space. For almost cases, the difference between warped and flat spaces leads to only tiny effect for RGEs because the contribution to the βfunction coefficient from each mode is logarithmic. In the following discussion, we use the above approximate mass spectra. By using the above approximation about mass spectra of the bulk fields, the RGE for the gauge coupling constant can be divided into three regions: where b 0 i is a β-function coefficients given from its zero modes, i.e., bulk fields with the orbifold BC (N, N ); δb KK i is an β-function coefficient by bulk fields with the orbifold BC (N, D) or (D, N ); ∆b KK is an additional β-function coefficient generated by a set of KK modes of all bulk fields, where b 0 i , δb KK i , and ∆b KK can be calculated by using Eq. (2.9). We solve the RGE in Eq. (3.20). As in Sec. 2, the number of the set of KK modes for µ > m KK is approximately equal to the energy scale divided by the KK mass scale k µ/m KK in Eq. (2.14). Under the approximation, we can solve the RGE, exactly, but that seems to be hard to see the contribution from mass splitting effects. We only write down rough approximate form for m KK ≥ µ, where for M Z < µ < m KK , (For the above expression, we ignored the contribution to α i (µ) from δb KK i between m KK /2 and m KK , and etc.) We find that the first and second terms in Eq. (3.21) are negligible compared with the third term for large µ.

Field
BC  Table 7: The orbifold BCs of the components of the SO(11) bulk gauge field A M = A µ ⊕ A y By using the above discussion, we calculate how much the mass splitting effect by the orbifold BCs contributes to the gauge coupling unification. First, we need to know the contribution for δb KK i from the SO(11) bulk gauge fields and the SO(11) 32 and 11 bulk fermion fields, but as long as the fermion number is preserved and their brane Dirac mass terms change the component fields with a Neumann BC to those with an effective Dirichlet BC on the Planck brane, they lead to the same contribution to all three gauge coupling constants: δb KK 1L = δb KK 2L = δb KK 3C . Therefore, we consider the contribution to δb KK i (i = 3C, 2L, 1Y ) from the SO(11) bulk gauge field. From Table 7, we get i term, and contributes to ∆ ij (µ). From the right figures µ − Ξ ij (µ) in Figs. 4 and 5, the convergence scale is changed, but this does not affect whether the SM gauge coupling constants converge or not. Therefore, we find that orbifold BCs or mass spectra affect the detail structure of gauge couplings described by ∆ ij (µ), but they do not affect the convergence of the SM gauge coupling constants described by Ξ ij (µ). We comment on the contribution to RGEs from the SO(10) spinor brane scalar field on the Planck brane in Table 2. Its non-vanishing VEV is responsible for breaking SO(10) to SU (5). There are twenty-one would-be NG modes. Nine modes are eaten by G P S /G SM gauge bosons, while twelve modes are uneaten because SO(10)/G P S gauge bosons absorb their corresponding 5th-dim. components of the 5D gauge field. The twelve modes become massive via their quantum correction, whose masses are expected to O(m KK ) or less depending on dynamics. They correspond to a complex scalar field with (3, 2) −1/6 under G SM . It is not any SU (5) multiplet, and it affects the gauge coupling unification. The contribution to the β-function coefficients of G SM is given by where this contribution vanishes effectively above the brane mass scale of φ 16 because the SO(10) full multiplet 16 contribute to the β-function coefficients of G SM . From Fig. 6   the formula in Eq. (2.9) and the Dynkin indices listed in Refs. [6,7,40], we obtain where i = 4C, 2L, 2R stand for SU (4) C , SU (2) L , SU (2) R , respectively.  We consider the contribution to δb KK i from the mass spectra of the SO(11) bulk gauge field. As we discussed before, the would-be NG bosons do not affect the RGEs for the SM gauge coupling constants. We can calculate δb KK i (i = 4C, 2L, 2R) by using the orbifold BCs of the SO(11) bulk gauge field shown in Table 9: We have to use the RGEs for three SM gauge coupling constants below M P S , while we have to use the RGEs for three Pati-Salam gauge coupling constants. To connect them, we use the following matching condition at the Pati-Salam scale M P S (m KK < M P S < M GU T ), where they are determined by the normalization conditions of the generators of G P S and G SM .

Summary and discussion
We discussed the RGEs for the 4D SM gauge coupling constants in the SO(11) gauge-Higgs grand unification scenario on the 5D RS warped spacetime. We found that the 4D SM gauge coupling constants are asymptotically free in the SO(11) GHGUTs with the matter contents shown in Tables 5 and 6, which satisfy ∆b KK < 0. We also discussed the SM gauge coupling unification. We showed that the three SM gauge coupling constants are effectively unified above the almost SM gauge coupling unification scale M GCU discussed in Sec. 3.1. We have not fixed the GUT or compactification scale M GU T = 1/L, but as long as M GU T = 1/L is larger than M GCU , there is no any inconsistency within at least the current experimental accuracy of the SM gauge coupling constants. In Sec. 3.2. we showed that the correction from the mass spectra of the SO(11) bulk gauge fields, the would-be NG boson, and the Pati-Salam scale does not affect the asymptotic freedom and gauge coupling unification of the SM gauge couplings, while they affect the detail structures of the RGE running. From the above, we find that the Weinberg angle at µ = M GU T , sin 2 θ W (M GU T ) = 3/8, is consistent with that at µ = M Z , sin 2 θ W (M Z ) 0.23. In this paper, we mainly considered the SO(11) GHGUTs, but our discussion can be applied for other GHGUTs. E.g., we have already found the asymptotic freedom condition for a gauge coupling constant in general GHGUTs based on any simple Lie group G in Eq. (2.19). It is very easy to list up the the matter contents that satisfy the asymptotic freedom condition by using Tables in Ref. [7].
We discussed the RGEs for the 4D SM gauge coupling constants in 5D RS warped spacetime by using the KK expansion. There is another approach about them by using AdS/CFT-like correspondence in Refs. [49][50][51][52][53].
aguchi for useful comments. This work was supported in part by Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research No. 23104009.