Coupling Sum Rules and Oblique Corrections in Gauge-Higgs Unification

In GUT inspired $SO(5) \times U(1)_X \times SU(3)_C$ gauge-Higgs unification (GHU) in the Randall-Sundrum warped spacetime, the $W$ and $Z$ couplings of all 4D fermion modes become nontrivial. The $W$ and $Z$ couplings of zero-mode quarks and leptons slightly deviate from those in the SM, and the couplings take the matrix form in the space of Kaluza-Klein (KK) states. In particular, the 4D couplings and mass spectra in the KK states depend on the Aharonov-Bohm phase $ \theta_H$ in the fifth dimension. Nevertheless there emerge three astonishing sum rules among those coupling matrices, which guarantees the finiteness of certain combinations of corrections to vacuum polarization tensors. We confirm by numerical evaluation that the equality in the sum rules holds with 5 to 7 digits accuracy. Based on the sum rules we propose improved oblique parameters in GHU. Oblique corrections due to fermion 1-loop diagrams are found to be small.


Introduction
Although the standard model (SM), SU (3) C × SU (2) L × U (1) Y gauge theory, has been successful in describing almost all of the phenomena at low energies, it has a severe gauge hierarchy problem when embedded in a larger theory such as grand unification.One possible answer to this problem is the gauge-Higgs unification (GHU) scenario in which gauge symmetry is dynamically broken by an Aharonov-Bohm (AB) phase, θ H , in the fifth dimension.The 4D Higgs boson is identified with a 4D fluctuation mode of θ H . [1][2][3][4][5][6][7][8][9][10][11][12][13][14] Many GHU models have been proposed, among which SO(5) × U (1) X × SU (3) C GHU models in the Randall-Sundrum (RS) warped space [15] turn out promising candidates for describing physics beyond the SM.The SO(5) × U (1) X × SU (3) C gauge symmetry naturally incorporates the custodial symmetry in the Higgs boson sector.[9] The orbifold boundary condition breaks SO(5) to SO(4) ≃ SU (2) L × SU (2) R .The SU (2) R × U (1) X symmetry is spontaneously broken to U (1) Y by a brane scalar on the UV brane in the RS space.The resultant SU (2) L × U (1) Y SM symmetry is dynamically broken to U (1) EM by the Hosotani mechanism.The 4D Higgs boson is a zero mode of the fifth-dimensional component of gauge fields in SO(5)/SO(4) which generates an AB phase in the fifth dimension.The finite Higgs boson mass m H ∼ 125. 1 GeV is generated at the quantum level by the dynamics of the AB phase θ H .
The GHU B-model is successful in many respects.It reproduces the quark, lepton, gauge and Higgs boson spectra (except for the small mass of the up quark), and yields nearly the same gauge couplings of the SM particles.It can incorporate the Cabbibo-Kobayashi-Maskawa (CKM) matrix structure in the W couplings with Flavour-Changing-Neutral-Currents (FCNC) naturally suppressed.[19] Many of the physical quantities at low energies are described mainly by the AB phase θ H , being mostly independent of the parameters in the dark fermion sector.[20] Both A-and B-models predict Z ′ particles as KK excited states of photon, Z boson and Z R boson.Z ′ couplings of quarks and leptons exhibit large parity violation which can be explored and tested at 250 GeV e − e + International Linear Collider (ILC).[21] By examining the dependence of event numbers on the polarization of incident electron and positron beams the A-and B-models can be clearly distinguished.[22][23][24][25][26] Effects of Z ′ bosons can be seen in single Higgs boson production processes as well.[27] It has been established that useful and convenient quantities to investigate new physics beyond the SM are the oblique parameters S, T and U of Peskin-Takeuchi, which represent corrections to vacuum polarization tensors of W , Z and photon.[28][29][30] In the early study of gauge theory in the RS warped space it was argued that SO(5) × U (1) X GHU models may yield appreciable corrections to S and T .[31] To evaluate oblique corrections at the one loop level in GHU, one has to know mass spectrum and gauge couplings of all KK modes.In Ref. [31] S and T were expressed in terms of truncated propagators of gauge bosons and fermions, by adopting a perturbative expansion in θ H .
In this paper we use exactly determined mass spectra of fermion KK states at general θ H at the tree level, and evaluate W and Z couplings of the fermion KK states by making use of exactly determined wave functions of both gauge bosons and fermions in the SO(5)× U (1) X space.It has been known that the W and Z couplings of quarks and leptons are nearly the same as in the SM.The W and Z couplings of the KK modes of quark and lepton multiplets become highly nontrivial, however.They are not diagonal in the KK states at θ H ̸ = 0, taking the matrix form with nontrivial off-diagonal elements.Further the wave functions of the W and Z bosons have substantial components not only in the SU (2) L × U (1) Y space but also in the entire SO(5) × U (1) X space, which necessitates refinement of the definition of oblique parameters.
One loop corrections to the vacuum polarization tensors of W , Z and photon contain divergences.In the 4D SU (2) L × U (1) Y gauge theory certain combinations of those vacuum polarization tensors, represented by S, T and U , are finite.In the GUT inspired SO(5) × U (1) X × SU (3) C GHU various combinations of the KK states of fermions run along the loops in the propagators of gauge bosons.It will be shown that there appear sum rules among the W and Z coupling matrices.We shall confirm those sum rules by numerically evaluating the mass spectra of the KK modes and their W and Z coupling matrices.The sum rules are found to hold with 5 to 7 digits accuracy.With these sum rules certain combinations of the 1-loop corrections to the vacuum polarization tensors become finite, which leads to improved oblique parameters.It will be seen that corrections to the improved oblique parameters are small.The total corrections are found to be S ∼ 0.01, T ∼ 0.12 and U ∼ 0.00004 for θ H = 0.1 and the KK mass scale m KK = 13 TeV, which is consistent with the current experimental data.[32] Although the gauge couplings in GHU in the RS space have highly nontrivial matrix structure, they satisfy remarkable identities.The identities in the sum rules discussed in the present paper are associated with two-point functions of gauge fields.It has been known that the exact identities hold in the combinations of the couplings appearing in three-point functions of gauge fields in orbifold gauge theory.[33,34] Triangle loop diagrams generally give rise to chiral anomalies in 4D gauge theory.In 5D orbifold gauge theory anomaly coefficients for the three legs of various 4D KK modes of gauge fields vary with the AB phase in the fifth dimension.This phenomenon is called the anomaly flow by an AB phase.Along triangle loops all possible KK modes of fermions run.The sum of all those loop contributions leads to the total anomaly coefficient which is expressed in terms of the values of the wave functions of gauge fields at the UV and IR branes in the RS space and orbifold boundary conditions of the fermions.In other words there hold sum rules for gauge coupling matrices of the third order.
In Section 2 the GUT inspired SO(5) × U (1) X × SU (3) C GHU is described.We explain how to determine the mass spectrum and wave functions of gauge bosons and quark-lepton multiplets.In Section 3 the W and Z couplings of all fermion modes are determined.In Section 4 fermion one-loop corrections to the vacuum polarization tensors of the W boson, Z boson and photon are evaluated.In Section 5 we show that there appear three sum rules among the W and Z coupling matrices of quarks, leptons and their KK modes.The sum rules are confirmed by numerical evaluation.Based on the coupling sum rules the improved oblique parameters are introduced in Section 6.The finite corrections to the S, T and U parameters are evaluated, and are found to be small.Section 7 is devoted to a summary and discussions.In Appendix A basis functions used to express wave functions of gauge bosons and fermions are summarized.In Appendix B wave functions of KK modes of down-type quarks and neutrinos are given.
The action of the GUT inspired GHU has been given in Ref. [18].Let Ψ J collectively denote all fermion fields in the bulk.Then the action of the fermions in the bulk becomes where Ψ = iΨ † γ 0 and (2.8) The dimensionless parameter c in D(c) is called the bulk mass parameter, which controls the wave functions of the zero modes of the fermions.m α D and m β V are pseudo-Dirac mass terms.The action for the Majorana fermion field ( χα ) is where M αβ represents Majorana masses.In the present paper we take M αβ = M α δ αβ for simplicity.
In the electroweak sector there are two 5D gauge couplings, g A and g B , corresponding to the gauge groups SO(5) and U (1) X , respectively.The 5D gauge coupling g 5D Y of U (1) Y is given by (2.11) The 4D SU (2) L and U (1) Y gauge coupling constants are given by (2.12) The bare weak mixing angle θ 0 W is given by (2.13) As is seen below, the mixing angle determined from the ratio of m W to m Z slightly differs from the one in (2.13) even at the tree level in GHU in the RS space; m W /m Z | tree ̸ = cos θ 0 W .The 4D Higgs boson field is a part of A the tensor representation is expanded as is expanded as . (2.16) 4D neutral Higgs field H(x) is the fluctuation mode of the AB phase θ H .

Spectrum and wave functions of gauge fields
When the VEV |⟨ Φ⟩| = w is sufficiently large, w ≫ m KK , the spectra of the W and Z towers, {m where the functions C(z; λ) and S(z; λ) are given in Appendix A. Note (1 + s 2 ϕ ) −1 = cos 2 θ 0 W .The lowest modes are W = W (0) and Z = Z (0) .For z L ≫ 1 their masses at the tree level are approximately given by In this paper m Z = m tree Z = 91.1876GeV is taken as one of the input parameters.As typical values we take m KK = 13 TeV, θ H = 0.1, sin 2 θ 0 W = 0.230634 and α EM (m Z ) = 1/128, which implies that z L = 3.86953 × 10 11 and kL = 26.6816.The precise values determined from (2.17) give m Z (0) cos θ 0 W /m W (0) = 1.00002.Each mode of the gauge boson tower has components in the SO(5) × U (1) X space.Let us decompose the SO(5) generators for the Z tower component, and for the photon tower component.Here where the normalization factors {r in each mode.Ŝ(z, λ)is given in (A.1).The photon (γ (0) ) coupling is e Q EM where e = g w sin θ 0 W and

Spectrum and wave functions of fermion fields
The spectra {m n = kλ n } of the KK towers of up-type quarks and charged leptons are determined by the zeros of where S L/R (z; λ, c) is defined in (A. e , e ′ ) are SU (2) R doublets.5D (u, u ′ )(x, z) fields are expanded as where, in terms of functions defined in (A.3), . (2.26) The normalization factor in each mode is determined by the condition Note that with the use of (2.24) one can express . (2.28) At θ H = 0, λ u (0) = 0 so that the zero mode u (0) has purely chiral structure; (u L ) becomes an SU (2) L doublet, whereas u Wave functions of {e (n) } have the same structure as those of {u (n) }.Formulas for {e (n) } are obtained from (2.25)-(2.28)by replacing u (n) and c u by e (n) and c e .
For mass spectra and wave functions of down-type quark and neutrino multiplets the SO(5) singlet fields Ψ (3,1) ± and Majorana brane fermions χ intertwine.In general the coupling constants κ's in the brane interactions given in Eq. (2.10) are not diagonal in the generation space.The L 1 term in (2.10), with ⟨ Φ⟩ ̸ = 0, leads to the Kobayashi-Maskawa mixing matrix in the quark sector.Further complex κ αβ 's give rise to CP -violation phases.
In the present paper we analyze the case in which the brane interactions given in Eq. (2.10) are diagonal in the generation space.
In the first generation the d and d ′ components in Ψ (3,4) and Ψ (3,1) ± ≡ D ± intertwine with each other.The mass spectrum {m n = kλ n } is determined by the zeros of where The resolution of this problem is left for future investigation.In this paper we take (m u , m d ) = (20, 2.9) MeV at the m Z scale.This does not affect gauge couplings and KK spectra of (u, d) multiplets in the discussions below as m u , m d ≪ m KK ∼ 13 TeV.
In the first generation there are two types of series, {d (n) ; n ≥ 0} and {D (n) ; n ≥ 1}. 3 For θ H = 0.1 and m KK = 13 TeV, the mass spectra of the KK excited states (2.30) Wave functions are normalized by in each mode.The explicit forms of the wave functions are given in (B.1) and (B.2).
In the neutrino sector the ν and ν ′ components in Ψ (1,4) and the brane Majorana fermion χ mix with each other.χ(x) = χc (x) has a Majorana mass M .The brane interaction L 3 in (2.10) generates a mixing brane mass term (m Because of the Majorana mass term eigen modes in the neutrino sector have both left-and right-handed components.The mass spectra {m ν ±(n) = kλ ν ±(n) } are determined by where S L L/R = S L/R (1; λ, c e ) etc. in the first generation.For c e < − 1 2 and M > 0 the gauge-Higgs seesaw mechanism, similar to the inverse seesaw mechanism, is at work in the K + ν series to generate a small neutrino mass.[36] (

2.33)
There arises no light mode in the K − ν series.For the KK excited modes the two series are nearly degenerate; λ ν −(n) ≃ λ ν +(n) for n ≥ 1.We note that with c e > 1 2 the mass of the lightest mode becomes ∼ m 2 e M z 2ce+1 L /(2c e + 1)m 2 B so that unnecessarily large m B is required to reproduce a small neutrino mass.Further c e > 1 2 yields an additional exotic mode with a mass of O(10 GeV).We adopt c e , c µ , c τ < − 1 2 .The Majorana field χ is decomposed as The fields are expanded as Wave functions are normalized as in each mode.The explicit forms of the wave functions are given in (B.3).

W and Z couplings
The γ, W and Z couplings of the fermion fields are contained in the part of the action where ΨJ = z −2 Ψ J .By inserting the KK expansions of the gauge and fermion fields into (3.1),γ (n) , W (n) , and Z (n) couplings among the fermion KK modes are evaluated.The photon γ = γ (0) couplings are universal.They are diagonal in the KK space, and are given by e Q EM = g w sin θ 0 couplings in the first generation of the quark multiplets are given by

3). The values in the SM correspond to ĝWud
L,00 = 1 and ĝWud R,00 = 0.In the RS space off-diagonal components of ĝWud L/R,nm are non-vanishing.For θ H = 0.1 and m KK = 13 TeV, for instance, the coupling matrices are given by ĝWud where 10 −7 , for instance, implies O(10 −7 ).Notice that the couplings for KK excited states are nearly vector-like.However, they have very small axial-vector components; ) is O(10 −3 ) or less.As is shown below, those small numbers must be properly taken into account to establish the coupling sum rules.The ĝWuD L/R,nm couplings are very small; max |ĝ W uD L/R,nm | ∼ 2 × 10 −6 .In the lepton sector there are two types of the neutrino towers, {ν +(n) } (ν e1 series) and {ν −(n) } (ν e2 series), both of which have W couplings.In the first generation ĝWν e1 e L,nm where The values in the SM correspond to ĝWν e1 e L,00   The W -couplings of the zero modes, namely the couplings of quarks and leptons in three generations, are summarized in Table 2. Except for (t, b) the W couplings are universal to high accuracy; ĝW L ∼ 0.997645 ≡ ĝW,GHU and ĝW R ∼ 0. The observed lepton coupling should be identified as g obs w = g w ĝW,GHU .
The Z = Z (0) couplings are evaluated similarly.The couplings in the up-type quark sector are given in the form As is suggested from the structure of the wave functions in (2.22), it is convenient to decompose the Z couplings into the U (1) EM part and the rest.We write 22) the couplings are given by ĝZu,su2 where T 3 u = 1 2 and Q u = 2 3 .One finds that ĝZu,su2 The couplings in the space of KK excited states, namely n, m ≥ 1 elements of ĝZu nm , are nearly vector-like.The axial-vector components are small; ĝZu In the down-type quark sector there are {d (n) } and {D (n) } series.The couplings are written as With the decomposition ĝZdd L/R,nm = ĝZdd,su2 L/R,nm − sin 2 θ 0 W ĝZdd,EM L/R,nm etc, the couplings are given by ĝZdd,su2 Here Also ĝZdD L/R,nm = (ĝ ZDd L/R,mn ) * = O(10 −6 ) or less.The Z couplings of charged lepton multiplets have the same structure as in the up-type quark sector.The couplings of the electron multiplet are where ĝZe L/R,nm are given by the expressions obtained by replacing, in (3.10) and (3.(3.17) The axial-vector components in the space of KK excited states are small; ĝZe A,nm (n, m ≥ 1) are O(10 −3 ) or less.
As shown in (3.10) and (3.11), the Z-coupling is decomposed into the su2 part and EM part.The su2 part consists of three components; the  The W W Z coupling is evaluated similarly.Triple couplings are written as The SM value is g SM W W Z = g w cos θ SM W .In the current model one finds . (3.25) For θ H = 0.1 and m KK = 13 TeV (3.26) The deviation in the W † W Z = W †(0) W (0) Z (0) coupling is extremely tiny.
We note that where K div is defined in Eq. (4.4).It will be seen convenient to take µ = m KK in (4.7) to evaluate finite parts of oblique corrections in the current model.
In Section 3 we have obtained W and Z couplings of fermions.We adopt the convention for vector and axial-vector couplings given by g . To simplify expressions we introduce ĝV,nℓ ĝV ′ ,ℓn + ĝA,nℓ ĝA ′ ,ℓn Then, for the W W vacuum polarization, contributions from the (u, d) multiplets are given by where N C = 3.Note that ĝW † ud V /A,ℓn = (ĝ W ud V /A,nℓ ) * etc. Contributions from the (ν e , e) multiplets are For the ZZ vacuum polarization, contributions from the (u, d) multiplets are Here we have set ĝZdD V /A,n0 = ĝZDd V /A,0n = ĝZDD V /A,n0 = ĝZDD V /A,0n = 0. Contributions from the (ν e , e) multiplets are (4.12) The photon couplings are universal.They are diagonal and vectorlike.Noting that m), one finds for the γγ vacuum polarization that where For the Zγ vacuum polarization one finds that Note that Π γγ (0) = Π Zγ (0) = 0 as a consequence of the Ward-Takahashi identity in Expressions for Π(p 2 ) for the second and third generations are obtained similarly.

Coupling sum rules
Each Π(p 2 ) in the previous section contains divergent terms proportional to Ê.In the SM some of them are absorbed by renormalization constants, and specific combinations of the Π(p 2 )'s, namely the S, T and U combinations, remain finite.[28][29][30] In GHU all KK modes of fermions contribute to Π(p 2 ), and their couplings ĝW V /A and ĝZ V /A are highly nontrivial.The couplings ĝW V /A and ĝZ V /A take the matrix form with nonvanishing offdiagonal elements.Further even in the subspace of the KK excited states the axial vector couplings are nonvanishing.
In this section we show that there exist three identities among W and Z coupling matrices in each fermion doublet-multiplet, which are associated with the divergent terms in Π W W (p 2 ), Π ZZ (p 2 ) and Π Zγ (p 2 ).We define the W 3 coupling matrix ĝW 3 V , say for the (u, d) multiplet, by We stress that ĝW 3 u V and ĝW 3 dd V slightly differ from ĝZu,su2 We define Here Tr in Q 2 u Tr I implies the trace over the u (n) states.Then the divergent parts of Π ud (p 2 ) are expressed as and we define Tr ĝWνeae The divergent parts of Π νee (p 2 ) are given by the expressions in (5.3)where N C = 1 and the superscript 'ud' is replaced by 'ν e e'.Note that these coefficients depend on the fermion doublet; Although all of the coupling matrices ĝV/A are rather nontrivial as shown in Section 3, there appear astonishing relations among A 0 , A 1 , B, D 0 and D 1 .We are going to establish, by numerical evaluation from the coupling matrices, the following coupling sum rules to high accuracy, where − ĝZdd,su2 L,00 , h νee = ĝZν e11 L,00 − ĝZe,su2 L,00 . (5.7) Similar relations hold for the second and third generations.For the (t, b) doublet, we use . Numerical values of ĝZ,su2 L,00 and ĝW L,00 for θ H = 0.1, m KK = 13 TeV and M = 10 3 TeV are summarized in Table 4.The factors h are close to, but not exactly 1.
. so that the relations in (5.6) are satisfied for each doublet.In the current GHU model the relations are highly nontrivial.We have included contributions coming from the KK modes n = 0 to n = 12.The mass spectrum of the KK states and the 13-by-13 coupling matrices are determined with double precision.To confirm the accuracy of the coupling sum rules we introduce Table 4: The couplings ĝZ,su2 L,00 and ĝW L,00 for θ H = 0.1, m KK = 13 TeV and M = 10 3 TeV. (5.8) Obtained results for A 0 , A 1 , ∆ S , ∆ T and ∆ U are summarized in Table 5.
It is seen that the coupling sum rules (5.6) are valid with 5 to 7 digits accuracy, at least where left-handed (right-handed) quarks and leptons are SU (2) eff doublets (singlets), and L,00 for β = (β u , β d ).The factor h β is not equal to 1 in GHU even at the tree level.It is very close to ĝWβ L,00 .The relevant quantity for the forward-backward asymmetry in e − e + → f f at the Z pole, for instance, is sin 2 θ 0 W /h β which is about sin 2 θ SM W .The h factor in (5.9) effectively appears in the relation A 0 = hB, which affects the definition of the S parameter in GHU as discussed below.

Improved oblique parameters
The oblique parameters S, T and U of Peskin-Takeuchi are useful to investigate new physics beyond the SM.These parameters are expressed in terms of the vacuum polarization tensors of W , Z and photon.Certain combinations of those vacuum polarization tensors are finite, and are expected to represent important parts of the corrections to physical quantities.
In GHU some improvement is necessary.In the most general situation the S, T and U parameters should be defined as certain combinations of the vacuum polarization tensors of all SO(5)×U (1) X gauge fields including the KK excited modes.Only the combinations which are finite at the quantum level could serve as quantities measuring corrections to physical quantities.In this section we examine the finite corrections to the S, T and U parameters associated to the vacuum polarization tensors of W , Z and photon.We should remember that these quantities are not directly-measured physical quantities.Directlymeasured physical quantities expressed in terms of four-fermi vertices, for instance, involve contributions coming from the KK modes of the gauge bosons in GHU.
In the previous section we have established three coupling sum rules to high accuracy.
For each fermion doublet β the sum rules are With these sum rules at hand we propose the following S, T and U for each fermion doublet β at the one loop level; where α * = α EM (m 2 Z ).In GHU m Z cos θ 0 W ̸ = m tree W .The terms proportional to h β − 1 in α * S β represent the improvement from the standard expression for S. It is straightforward to confirm that S, T, U defined by (6.2) are finite as a consequence of the sum rules in (6.1).In the numerical evaluation of finite S β , T β , U β by using the gauge coupling matrices of finite-dimensional rows and columns, one has to use the h factor defined by h β = A β 0 /B β , otherwise the result would be afflicted with the uncertainty associated with the divergence.Also notice that the weak mixing angle θ 0 W entering in (6.2) is the angle defined in (2.13).
To explicitly express S β , T β and U β in terms of the gauge couplings and mass spectra, in the SM framework are S RPP = −0.02±0.10,T RPP = 0.03±0.12and U RPP = 0.01±0.11 where the superscript RPP stands for Review of Particle Physics.[37] As emphasized in the beginning of this section, oblique corrections associated not only with W (0) , Z (0) and γ (0) , but also with the KK excited modes W (n) , Z (n) and γ (n) become important for physical observable quantities in GHU.In particular, the couplings of left-handed quarks and leptons to W (1) , Z (1) and γ (1) are large in the GUT inspired GHU.To compare with S RPP , T RPP and U RPP , one needs to include, in addition to S (0) , T (0) and U (0) , oblique corrections to the propagators of the KK gauge bosons.Contributions coming from internal fermions at the high KK levels equally affect the oblique corrections to the KK gauge boson propagators.To have definitive understanding of the contributions of KK fermions at the one loop level in GHU, it is necessary to directly evaluate observable quantities, which is left for future investigation.

Summary and discussions
In this paper we have examined the GUT inspired SO(5) × U (1) X × SU (3) C GHU model in the RS warped space.The W and Z couplings of quarks, leptons and their KK excited modes take the matrix form in the KK space.These coupling matrices have nontrivial offdiagonal elements, and have both vector and axial-vector components.Nevertheless these coupling matrices satisfy three sum rules (6.1).We have confirmed these coupling sum rules numerically from the evaluated W and Z coupling matrices.The rigorous derivation of the coupling sum rules would require the full treatment of the gauge bosons in the SO(5) × U (1) X theory.It is noteworthy that the sum rules hold even in the subspace of the W , Z and photon vacuum polarization tensors to very high accuracy.The appearance of the h β ̸ = 1 factor in the relation A β 0 = h β B β in (6.1) is anticipated from the vertex correction in the Z couplings at the tree level as exhibited in the approximate formula in (5.9).
With the coupling sum rules at hand, one can evaluate the finite oblique corrections unambiguously.The corrections are evaluated by using the mass spectrum and gauge coupling matrices determined numerically.We have found for θ H = 0.1 and m KK = 13 TeV that S ∼ 0.01, T ∼ 0.12 and U ∼ 0.00004 when the contributions of the fermion loops up to the n max = 12 level are taken into account.It was argued in the very early stage of the investigation [31] that there may arise a large correction to S in gauge theory in the RS space.We have found that the corrections in the GUT inspired GHU are small by direct evaluation of one loop diagrams.We note that Yoon and Peskin have evaluated the oblique corrections in a different SO(5) × U (1) X GHU model in a different method.[14] Their result also indicates small corrections for m KK = 13 TeV.However, to have definitive understanding of the contributions of KK fermions at the one loop level, it is necessary to evaluate observable quantities, by taking account of oblique corrections to the KK modes of the gauge fields.
The coupling sum rules presented in this paper are highly nontrivial.There must be some reason behind them, possibly originating from the 5D gauge invariance in the GHU scheme.Further investigation is necessary.

A Basis functions
We summarize the basis functions used for wave functions of gauge and fermion fields.
For gauge fields we introduce For fermion fields with a bulk mass parameter c, we define

B Wave functions
Wave functions of down-type quark and neutrino multiplets are given below.

B.1 Down-type quarks
The (d, d ′ , D + , D − ) fields are expanded as in (2.30).The wave functions are given by  for the d (n) mode where
3).The bulk mass parameter c of each doublet multiplet is determined such that the lowest value λ 0 reproduces m u , m c , m t , m e , m µ or m τ .For θ H = 0.1 and m KK = 13 TeV, (c u , c c , c t ) = (−0.859,−0.719, −0.275) and (c e , c µ , c τ ) = (−1.01,−0.793, −0.675).Although Eq. (2.24) is satisfied by either positive or negative c, the negative values for c are chosen in the B-model to have the spectra of the KK towers of down-type quarks and neutrinos consistent with observation as explained below.Wave functions of {u (n) } and {e (n) } in the first generation are contained in the SO(5) spinor multiplets Ψ (3,4) and Ψ (1,4) , which are denoted as (u, d, u ′ , d ′ ) and (ν e , e, ν ′ e , e ′ ), respectively.(u, d) and (ν e , e) are SU (2) L doublets, whereas (u ′ , d ′ ) and (ν ′ mD ), etc. Functions S L/Rj , C L/Rj are given in (A.5).c D is the bulk mass parameter of Ψ (3,1) ± field, and mD = m D /k where m D is a Dirac mass connecting D + and D − .The parameter µ represents the strength of a brane interaction among Ψ (3,4) , Ψ (3,1) ± and Φ, which is necessary to reproduce a mass of each down-type quark.As typical values we take ( md , ms , mb ) = (1, 1, 1) and (µ d , µ s , µ b ) = (0.1, 0.1, 1).We determine bulk mass parameters c D 's to reproduce a downtype quark mass in each generation, finding (c D d , c Ds , c D b ) = (0.6244, 0.6563, 0.8725).We have chosen the negative values for (c u , c c , c t ).With positive c u , c c > 1 2 there would arise an exotic extra light mode of charge Q EM = − 1 3 with a mass much less than m KK in the first and second generation from (2.29), which contradicts with the observation.One comment is in order.Eqs.(2.24) and (2.29) imply that the up-type quark mass is larger than the corresponding down-type quark mass, although in the first generation m u < m d .

21 )
The values in (3.21) are very close to those in the SM with sin 2 θ SM W = 0.2312.For (t, b)

10 − 13 .
(3.23)It is seen that the SO(5) structure is crucial to have consistent gauge couplings of quarks and leptons.Vanishingly small ĝZu,su2 R and ĝZd,su2 R are due to the cancellation among the a, b and c components, which is possible in the SO(5) × U (1) X gauge theory, but not in SU (2) L × SU (2) R × U (1) gauge theory.

V
and ĝZdd,su2 V defined in (3.10) and (3.14).Numerically all elements of ĝW 3 DD V , ĝZDD A , ĝZdD V /A and ĝZDd V /A are O(10 −6 ) or less.In the following we safely omit the contributions coming from the D modes in the expressions for the coupling sum rules.

Figure 1 :
Figure 1: The θ H -dependence of the S, T , U parameters is plotted for m KK = 13 TeV and n max = 12.S 0 , T 0 and U 0 are the values of S, T and U at θ H = 0.1, respectively.
One of the authors (Y.H.) would like to thank Masashi Aiko for many deep comments on the oblique corrections.This work is supported in part by Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, Grant No. JP19K03873 (Y.H.), European Regional Development Fund-Project Engineering Applications of Microworld Physics (No. CZ.02.1.01/0.0/0.0/16019/0000766) (Y.O.), and the Ministry of Science and Technology of Taiwan under Grant No. MOST-111-2811-M-002-047-MY2 (N.Y.).

F 1 =L
S L1 S R1 − S L2 S R2 .(B.2) For the D (n) mode the formulas are obtained by replacing d (n) by D (n) in (B.1) and (B.2).Except for the d (0) mode, namely d-quark, the d (n) (n ≥ 1) modes are mostly contained in (d, d ′ ) fields, whereas the D (n) (n ≥ 1) modes are mostly contained in (D + , D − ) fields.In Table7the norm of each component (N f = z L 1 dz |f | 2 etc.) is tabulated.For comparison we list the norms of u and u ′ components of the u (n) modes in Table8.One can see that (u ) is an SU (2) L doublet.On the other hand u nearly SU (2) L singlets.Further d (0) R ) has major components in the D ± fields.Its SU (2) R portion is small.Although the W boson acquires a small SU (2) R portion at θ H = 0.1

Table 1 :
The matter field content in the GUT inspired GHU model.The (SU (3) C , SO(5)) U (1) X content of each field is shown in the last column.

Table 2 :
The W -couplings of quarks and leptons in units of g w for θ H = 0.1, m KK = 13 TeV and M = 10 3 TeV.
Zν e21 R,nm , and |ĝ Zν eab R,nm − ĝZν eab L,nm | are O(10 −3 ) or less for n, m ≥ 1.The Z-couplings of the zero modes, namely those of leptons and quarks in three generations, are summarized in Table3.The deviations from the SM values are tiny.

Table 3 :
The Z-couplings of leptons and quarks in units of g w / cos θ 0 W for θ H = 0.1, m KK = 13 TeV and M = 10 3 TeV.For reference the SM values T 3 L − sin 2 θ SM W Q EM with sin 2 θ SM W = 0.2312 are listed as well.
All of them are important.For (u, d) quarks, for instance,

Table 6 :
Corrections to S β , T β , U β for θ H = 0.1 and m KK = 13 TeV.The numerical values are evaluated by including the contributions coming from the KK towers of fermions up to the n = n max = 12 level.The values in the neutrino sector are obtained by setting m νe = m νµ = m ντ = 10 −12 GeV and Majorana masses M e = M µ = M τ = 10 6 GeV.In the last row the average increments per level, namely (total)/n max , are listed.

Table 7 :
The norm of each component for the d(n)and D (n) modes.N f = z L 1 dz |f | 2 , etc.