One loop radiative correction to Kaluza-Klein masses in $S^2/Z_2$ universal extra dimensional model

We investigate a radiative correction to the masses of Kaluza-Klein(KK) modes in a universal extra dimensional model which are defined on a six-dimensional spacetime with extra space as a two-sphere orbifold $S^2/Z_2$. We first define the Feynman rules which are necessary for the calculation. We then calculate the one-loop diagrams which contribute to the radiative corrections to the KK masses, and obtain one-loop corrections to masses for fermions, gauge bosons and scalar bosons. We estimate the one-loop corrections to KK masses for the first KK modes of standard model particles as a function of momentum cut-off scale, and we determine the lightest KK particle which would be a promising candidate of a dark matter.


I. INTRODUCTION
The Standard Model(SM) has been well established. It has indeed passed test of the accelerator experiments. It is, however, not a satisfactory theory for all the physicists. There seems to be several flaws, e.g., the hierarchy problems, no candidate of dark matter, and so on. With the fact that relic abundance of the dark matter is well explained by weakly interacting massive particle, these problems strongly indicate a new physics beyond the SM at TeV scale.
There are many candidates of such models, say, models with supersymmetry, little higgs, extra dimensions, and so on. Since the Large Hadron Collider experiment is now operating, which will explore the physics at TeV scale, it is urgent to investigate possible models at that scale.
Among these, the idea of Universal Extra Dimensional(UED) model is very interesting [1,2]. The minimal version of UED has recently been studied very extensively. It is a model with one extra dimension defined on an orbifold S 1 /Z 2 . This orbifold is given by identifying the extra spatial coordinate y with −y and hence there are fixed points y = 0, π. By this identification chiral fermions are obtained. It is shown that this model is free from the current experimental constraints if the scale of extra dimension 1/R, which is the inverse of the compactification radius R, is larger than 400 GeV [1,3]. The dark matter can be explained by the first or second Kaluza-Klein (KK) mode [4], which is often the first KK photon, and this model can be discriminated from other models [5]. This model can also give plausible explanations for SM neutrino masses which are embedded in extended models [6].
In contrast with the UED models in five dimensions, UED models in six dimensions have interesting properties which would explain some problems in the SM. For example, in six dimensions, the number of generations of quarks and leptons is derived by anomaly cancellations [7] and the proton stability is guaranteed by a discrete symmetry of a subgroup of six dimensional Lorentz symmetry [8]. Candidate of UED models in six dimensional model are the one with extra dimensions of T 2 , a torus [1], or the one with S 2 , a sphere [9,10]. In these two classes of models, the latter is quite new and its phenomenology has been studied recently in Refs. [11][12][13][14].
To study it in detail, first of all we have to calculate the quantum correction to mass spectrum. It is well known that at tree level all the particles in the same Kaluza-Klein (KK) mode are degenerate in mass and therefore it is impossible to predict even a decay mode. It is hence inevitable to calculate the quantum correction to their mass to find its phenominological consequences, say, collider signatures. This small correction relative to their tree mass is crucial since mass differences by it is "infinitely" large campared with it at tree level, and hence it determines physics. The aurthors of [15] have calculated it in S 1 and T 2 but there is no calculation for the S 2 model [9].
In models with two sphere, fermions cannot be massless because of the positive curvature and hence they have a mass of O(1/R) [16,17]. We cannot overcome the theorem simply by the orbifolding of the extra spaces. In usual cases, we have no massless fermion on the curved space with positive curvature, but we know a mechanism to obtain a massless fermion on that space by introducing a nontrivial background gauge field [18,19]. The nontrivial background gauge field can cancel the spin connection term in the covariant derivative. As a result, a massless fermion naturally appears. Furthermore, we note that the background gauge field configuration is energetically favorable since the background gauge kinetic energy lowers a total energy. In order to realize chiral fermions, the orbifolding is required, for instance. Unfortunately it makes the calculation of the quantum correction very difficult.
In this paper, we study a new type of UED with S 2 /Z 2 extra dimensions. We treat this theory as cut-off theory. We show feynmann rules for it and calculate the quantum correction as a function of the cut-off. By this we show the mass spectrum for the theory and offer a basics for studying phenominology such as LHC physics.
The paper is organized as follows. In Section II, we recapitulate the model [9]. We then specify the Feynman rules for propagators and vertices on the six dimensional spacetime with S 2 /Z 2 extra space. In Section III, we discuss the one loop calculation for KK mass correction and derive a formulas to estimate corrected KK masses. In Section IV, we estimate the corrected first KK masses for each SM particles and determine the lightest KK particle of the model. Section V is devoted to the summary and discussions.
II. S 2 /Z 2 UED MODEL In this section, we first review the universal extra dimensions defined on the six dimensional spacetime which has extra space as two-sphere orbifold S 2 /Z 2 [9]. We then define Feynman rules relevant to our calculation.

A. Structure of the model
The model is defined on the six-dimensional spacetime M 6 which has extra dimensional space compactified as two-sphere orbifold S 2 /Z 2 . The coordinate of M 6 is denoted by X M = (x µ , y θ = θ, y φ = φ), where x µ and {θ, φ} are the M 4 coordinates and the S 2 spherical coordinates, respectively. The orbifold is defined by identifying the point (θ, φ) with (π − θ, −φ).
We introduce a gauge field A M (x, y) = (A µ (x, y), A α (x, y)), SO(1,5) chiral fermions Ψ ± (x, y), and complex scalar field H(x, y) as the SM Higgs field. The chiral fermion is defined by the action of SO(1,5) chiral operator Γ 7 = γ 5 ⊗ σ 3 , where σ i (i = 1, 2, 3) are Pauli matrices and γ 5 is SO(1,3) chiral operator, such that Γ 7 Ψ ± (x, y) = ±Ψ ± (x, y), so that the chiral projection operator is given by Γ ± = (1 ± Γ 7 )/2. The boundary conditions for each field can be defined as where Υ 5 = γ 5 ⊗I 2 with I 2 being 2×2 identity, requiring the invariance of an action in six-dimensions under the Z 2 transformation. The gauge symmetry of the model is G =SU(3)×SU(2)×U(1) Y ×U(1) X defined on the sixdimensional spacetime with gauge coupling constants g 6a in six dimensions where index a = {X, 1, 2, 3} distinguishes the gauge symmetries U(1) X , U(1) Y , SU (2) and SU (3). The extra U(1) X is introduced, which is associated with a background gauge field A B φ given by [19][20][21] A B φ =Q X cos θ, whereQ X is U(1) X charge operator, in order to obtain massless chiral fermions in four dimensions. We then assign the U(1) X chargeQ X = 1 2 to fermions as the simplest case in which massless SM fermions appear in four dimensions. Fermions in six dimensions are thus introduced as in Table I whose zero modes are corresponding to SM fermions. Then the action of our model in six dimensions is written as where the Γ M in the first line in the RHS are gamma matrices in six-dimensions defined as Γ M = {γ µ ⊗ I 2 , γ 5 ⊗ iσ 1 , γ 5 ⊗ iσ 2 } with the four-dimensional Gamma matrices γ µ , the third to the fourth lines are gauge fixing terms with gauge fixing parameter ξ, the fifth line corresponds to a ghost term for non-Abelian gauge group, the sixth line denotes a Lagrangian for Higgs field, and the last line denotes a Yukawa interactions with Yukawa couplings Y u,d,e in six dimensions. In this model, we assume that U(1) X gauge symmetry is broken at some high scale and corresponding gauge bosons including KK modes are heavy so that they are decoupled from the low energy sector. The KK masses for each particles are obtained from kinetic terms in Eq. eigenfunctions of the Dirac operator on S 2 which are given as where P (m,n) ℓ (z) is Jacobi polynomial with z = cos θ, C ℓm α(β) are the normalization constants determined by dΩα * α (β * β ) = 1, and the indices {ℓ, m} corresponds to angular momentum quantum numbers on two-sphere specifying KK modes [9,17]. These mode functions satisfy where M ℓ = √ ℓ(ℓ+1) R corresponding to KK mass and the iD is Dirac operator on S 2 /Z 2 with background gauge field in Eq. (7). They also satisfy the completeness relation A fermion Ψ , are expanded such as where a ψ ℓm shows an SO(1,3) Dirac femion, P L(R) are Chiral projection operators in fourdimensions, and {P L , P R } are interchanged in RHS for Ψ (±γ 5 ) − (x, θ, φ). Then the kinetic and KK mass terms for KK modes are given in terms of ψ ℓm such that where ǫ ℓm is 0 for {ℓ = odd, m = 0} and unity for other modes. We also need to take into account Yukawa couplings of Higgs zero mode and fermion non-zero KK modes to obtain mass spectrum of the KK particles after the electroweak symmetry breaking. The Yukawa coupling with Higgs zero mode H 00 is written as where ψ F = Q, L correspond to SU (2) doublets and ψ f = U, D, E correspond to SU(2) singlets.
After the zero mode Higgs getting vacuum expectation value (VEV), we have the mass term of the KK modes of the form where m SM s express the masses in the SM, and the mass spectrum is obtained by diagonalizing the mass matrix. The four-dimensional components of gauge fields A µ (x, θ, φ) are expanded in terms of linear combination of spherical harmonics, satisfying boundary condition Eq. (4), such that where for m = 0 with ℓ =even(odd), respectively. Then the kinetic and KK mass terms for KK modes are obtained as is the square of angular momentum operator and ǫ ℓm is the same as in Eq. (14). After electroweak symmetry breaking, KK modes of W ± and Z boson also obtain the contribution of SM mass m 2 W W +ℓm µ W µ−ℓm and m 2 Z Z ℓm µ Z µℓm respectively. The Lagrangian quadratic in {A θ , A φ } is given by whereÃ φ = A φ / sin θ and the second line in the RHS corresponds to a gauge fixing term. This Lagrangian is not diagonal for {A θ , A φ } so that we carry out the substitution to diagonalize the quadratic terms. Then the quadratic terms become Thus the mode functions for extra dimensional components gauge filed φ i is given as where factor of 1/ ℓ(ℓ + 1) is required for normalization due to extraL 2 factor in Eq. (22). Then φ i (θ, φ, x) is expanded as taking into account the boundary condition Eq. (5). Therefore KK mass terms for φ i are obtained as where ǫ ℓm is 0 for {ℓ = odd, m = 0} and unity for other modes, the gauge fixing parameter ξ is taken as 1 in our analysis applying Feynman-t'Hooft gauge, and the KK modes of φ 2 are interpreted as Nambu-Goldstone (NG) bosons. These NG bosons will be eaten by KK modes of four-dimensional components of gauge field giving their longitudinal component. and m 2 Z Z ℓm 1,2 Z ℓm 1,2 respectively. The mode function for the scalar field is also given by the Spherical Harmonics Y ℓm (θ, φ), and taking into account the boundary condition Eq. (6). Thus the kinetic and KK mass terms are obtained as and we also have SM Higgs mass contribution m 2 H (H ℓm ) † H ℓm . Therefore the KK masses are given in general, at tree level, as which is characterized by angular momentum number ℓ but independent of m. Also the KK parity is defined as (−1) m for each KK particles due to discrete Z ′ 2 symmetry of (θ, φ) → (θ, φ + π) which is understood as the symmetry under the exchange of two fixed points on S 2 /Z 2 orbifold ( π 2 , 0) and ( π 2 , π). Thus we can confirm the stability of the lightest KK particle with odd parity.
B. Propagators on the six-dimensions with S 2 /Z 2 extra space Here we discuss the Feynman rules for propagators and vertices on the M 4 × S 2 /Z 2 for a fermion and a gauge boson. We first consider propagators on M 4 × S 2 and then derive those of after orbifolding taking into account the boundary conditions of fields on S 2 /Z 2 . The propagator of here Ψ being Dirac fermion, is defined as an inverse matrix of the Dirac operator on six dimensions where the iΓ α ∂ α = γ 5 ⊗ iD is the extra-dimensional components of the Dirac operator in six dimensions with iD given in Eq. (11). Then the propagator should satisfy where R 2 factor on the RHS is appeared to compensate mass dimensions. Then the can be written, using mode function in Eq. (9) and (10), as where we can confirm that it satisfies Eq. (30) applying Eq. (11) and completeness condition Eq. (12) for mode functions. The propagator on M 4 × S 2 /Z 2 is obtained in terms of the linear combination where each ΨΨ are given by Eq. (31). Then it can be simplified as where the sign factor (−1) ℓ+m is derived from the relation Ψ ℓm (−z, −φ) = (−1) ℓ+m Ψ ℓ−m (z, φ). This propagator has similar structure as the minimal UED case except for the sign factor [15,22]. The propagator of gauge boson, D µν in Feynman-t'Hooft gauge, where the operator in LHS comes from the kinetic term shown in Eq. (18). Then the propagator is expressed by Spherical Harmonics as which satisfies Eq. (34) with completeness relation of Spherical Harmonics. The propagator on M 4 × S 2 /Z 2 is obtained by taking into account the boundary condition Eq. (4) as the case of fermion propagator. Substituting Y ℓm (θ, φ) to the linear combination (Y ℓm (θ, φ)+Y ℓm (π −θ, −φ))/2, satisfying the boundary condition , we obtaiñ where the factor of (−1) ℓ is derived from the relation The propagators for φ 1,2 , the linear combination of extra dimensional component of gauge field A θ,φ , are similarly given as four dimensional component which satisfy where the minus sign in the RHS comes from the minus sign in θθ and φφ components of the metric g M N , and extraL 2 factor comes from LHS of Eq. (25) due to the substitutions of Eqs. (20) and (21). Thus the propagators before and after orbifolding are obtained as whereỸ ℓm = Y ℓm / ℓ(ℓ + 1), for both φ 1 and φ 2 in the Feynman-t'Hooft gauge, and the propagator on M 4 × S 2 /Z 2 is obtained in the same way as four dimensional components using (Ỹ ℓm (θ, φ) + Y ℓm (π − θ, −φ))/2 for mode functions. The propagator for scalar boson is obtained similar to the gauge boson case as This propagator satisfies the condition Then after orbifolding, we obtaiñ C. Vertices on the six-dimensions with S 2 /Z 2 extra space The Feynman rules for the vertices in the model, along with the propagators, are obtained in terms of mode functions. As an example, we consider a gauge interaction of a chiral fermion Ψ (±γ 5 ) ± written by where T a s are generators of a gauge symmetry. When the fieldsΨ (±γ 5 ) , Ψ (±γ 5 ) and A µ on the interaction make contractions, applying the propagators of Ψ and A µ given in Eq. (33) and Eq. (36), we get the propagators on momentum space, and the mode functions Ψ ℓm (z, φ)(Ψ † ℓm (z, φ)) and Y ℓm (z, φ) are left as a vertex factor. Thus the vertex factor of ψ ℓ 1 m 1 A i ℓ 2 m 2 ψ ℓ 3 m 3 coupling for each KK modes combinaton is derived as where the integration in RHS is over S 2 /Z 2 , the mode functions for fermion are given in Eq. (9) and (10), and indices of T a are understood to be contracted with KK fermions ψ ℓ 1 m 1 and ψ ℓ 3 m 3 .
Other vertex factors are derived in the same way and the full list of the vertex factors in the model is given in Appendix. A.

III. ONE LOOP QUANTUM CORRECTION OF KK MASSES
In this section, we derive one-loop corrections to KK mass M ℓ which does not include SM mass m SM , being purely obtained from extra-dimensional kinetic term, by calculating relevant one-loop diagrams. Readers who are interested in our results might skip this section discussing technical details.

A. Correction to KK masses of fermion
The one loop diagrams relevant to correction to KK masses of fermion are given as Fig. 1 and 2 where corresponding quantum numbers {ℓ, m} are shown for each external and internal lines. The Fig. 1 shows the virtual gauge boson contributions including both four-dimensional components and extra-dimensional components, and the Fig. 2 shows the virtual Higgs boson contribution. Here we FIG. 1: One loop diagram for correction to KK masses of fermion with virtual gauge bosons including extra dimensional components.
where P L and P R on the RHS will be interchanged for Ψ (±γ 5 ) − and the underline for the KK numbers in the vertex factor I α(β) ℓm;ℓ 2 m 1 −m;ℓ 1 m 1 shows the corresponding spherical harmonics is conjugate one Y * ℓ 2 m 1 −m . The quantum numbers ℓ 1 , ℓ 2 , m 1 and m ′ 1 are associated with the KK modes in the loop which are summed over. In this paper we employ, for all one-loop diagram calculations, the cut-off Λ for four momentum integration and ℓ max for angular momentum sum, in order to regularize the loop diagram. Summing over m ′ 1 , the products of Kronecker delta and vertex factors I x (x = α or β) are arranged as where we used the relation I x ℓ 1 −m 1 ;ℓ 2 −m 2 ;ℓ 3 −m 3 = (−1) ℓ 1 +ℓ 2 +ℓ 3 +m 1 +m 3 I x ℓ 1 m 1 ;ℓ 2 m 2 ;ℓ 3 m 3 obtained from the definition of the vertex factor given in Eq. (46). The first term in RHS of Eq. (48) corresponds to bulk contribution conserving KK-number m, and the second term corresponds to the boundary contribution violating the conservation of the KK-number m. Thus the one-loop diagram contribution can be separated to the bulk and boundary contributions such that −iΣ Fig.1 where bulk contribution is proportional to m conserving Kronecker deltas {δ m,m ′ , δ −m,m ′ }, and the boundary contribution is proportional to m violating Kronecker deltas {δ 2m 1 ,m+m ′ , δ 2m 1 ,m−m ′ }. As usual we combine denominator in the momentum integral using Feynman parameter α, and carry out Wick rotation. The momentum integral becomes where we performed the substitution k−αp → k, k E is a Euclidean 4-momentum, and the integration over k 2 E is regularized by cut-off Λ. We estimate the momentum integration numerically taking cut-off Λ as parameter for bulk contribution, while we only left leading log-divergent part with renormalization scale µ for boundary contribution such that assuming vanishing contribution at the cut-off scale Λ. Then after arranging the terms, Eq. (47) is organized as where the coefficients Σ L(R) andΣ L(R) correspond to the terms proportional to / p and iγ 5 respectively, associated with P L(R) . These coefficients are explicitly given by iΣ iΣ We also note that there appear sign factors of (−1) {ℓ,m} from propagators in S 2 /Z 2 orbifold and vertices, which lead different sign contributions from each KK modes in the loop. Then the contributions depend on ℓ max differently for even or odd ℓ max as we will see below. Also we can numerically check that external ℓ and m are conserved so that ℓ = ℓ ′ and m = m ′ cases give non-zero contribution, which is satisfied for other diagrams. The other diagrams are also calculated in the same manner, and we list the contributions from each diagrams in Table IX. Therefore we obtain one-loop level correction to the Lagrangian such that where each coefficients Σ bound are understood as the sum of corresponding contributions from all diagrams.
To obtain one-loop corrections to KK mass, we take into account a renormalization condition for each KK modes. The kinetic and KK mass terms of each KK modes of fermion are given in Eq. (14). Defining as a renormalization of fields, where we have taken up to the first order in δ R(L) , which corresponds to the counter terms. Thus the contribution from these counter terms is shown by Then we employ the renormalization conditions at a cut off scale Λ as which corresponds to the requirement that the kinetic term has canonical form at the cut-off scale, and the counter terms δ R(L) are determined from these conditions. Combining contributions from the one-loop diagrams and the counter terms, we obtain the corrected kinetic and KK mass terms such that Therefore, normalizing the kinetic terms by we obtain the corrected KK mass of for each KK modes of fermions, where we have taken the first order of Σ and δ in RHS.

B. Correction to KK masses of gauge boson
The one loop diagrams corresponding to correction to KK masses of gauge bosons are given in  case of KK mass of fermions, and we summarize the results in Appendix B. Separating bulk and boundary contributions, we can organize the one loop diagram contributions generally as iΠ bulk(bound) µν (p 2 (p 2 ; µ); ℓm; ℓm) = (p 2 g µν − p µ p ν )iΠ bulk(bound) (p 2 (p 2 ; µ); ℓm; ℓm) where we also separated the terms proportional to (p 2 g µν − p µ p ν ) contributing the correction of kinetic term of A µ and the terms proportional to g µν in RHS of second line. The contributions to the coefficients Π(Π)s from each diagrams are summarized in Table X and XI. Thus the correction to the Lagrangian is where Π(Π)s in the RHS are understood as the sum of corresponding contributions from all diagrams.
As in the fermion case, we consider the renormalization condition to obtain one-loop corrections to KK masses for KK gauge bosons. The kinetic and KK mass terms of each KK modes of gauge boson are given in Eq. (18). Defining as a renormalization of gauge field, these terms become where we have taken up to the first order in δ 3 corresponding to counter terms. Thus the contribution from these counter terms is Then we employ the renormalization condition at a cut-off scale Λ as which requires the canonical form of kinetic term at the cut-off scale as in the fermion case, and the counter term δ 3 is determined by the condition. Combining contributions from one-loop diagrams and counter terms, we obtain the one-loop corrections to kinetic and KK mass terms such that Therefore, normalizing the kinetic terms by we obtain one-loop corrections to KK mass for each KK modes of gauge bosons, where we have taken the first order in Π and δ 3 in RHS.
The extra dimensional components of gauge fields A θ,φ are considered as scalar fields in four dimensions, and the KK mass eigenstates are given by φ i which are defined as Eq. (20) and (21). Then the kinetic and KK mass term of φ i are the same form as that of scalar field. For φ i , we write the contributions from one-loop diagrams to corrections of diagonal mass terms φ i φ i such that where the bulk and the boundary contributions are separated, and each numerical factors can be calculated in the same manner as the above cases. The RHS of Eq. (76) corresponds to the expansion in terms of p 2 , and we omit p 2 = 0 in Θ(Θ) hereafter. We note that there are contributions to offdiagonal φ 1 φ 2 terms. However this contributions are negligibly small compared to diagonal terms, and we just ignore these contributions in this paper. The contributions to these coefficients Θ(Θ)s from each diagrams are summarized in Table XII, XIII, XIV and XV. Thus the correction to the Lagrangian is where Θs in the RHS are understood as the sum of corresponding contributions from all diagrams. Then we consider the renormalization condition to obtain the one-loop corrections to KK masses for φ 1,2 . Defining as a renormalization of fields φ i → Z (i) where we have taken up to the first order of δ (i) 3 corresponding to counter terms. Thus the contribution from the counter terms is Then we employ the renormalization condition at a cut-off scale Λ as in the same way as four dimensional component case, and the counter term δ (i) 3 is determined by the condition. Combining contributions from the one-loop diagrams and counter terms, we obtain the one-loop corrections to kinetic and KK mass terms as Therefore, normalizing the kinetic terms by we obtain the one-loop corrections to KK mass for each KK modes of φ i , where we took first order of Θ and δ in RHS.
FIG. 9: One loop diagram for correction to KK masses of Higgs bosons with virtual gauge bosons including extra components.
FIG. 10: One loop diagram for correction to KK masses of Higgs bosons with virtual fermions.
where we list the contribution from each diagrams in Table XVI and XVII. The RHS of Eq. (85) corresponds to the expansion in terms of p 2 as in the previous case, and we omit p 2 = 0 in Ξ(Ξ) hereafter. Thus the correction to the Lagrangian is where Ξ(Ξ)s in the RHS are understood as the sum of corresponding contributions from all diagrams.
The renormalization condition and one-loop corrections to KK mass are also discussed as in the case of φ i . Repeating the same procedures as the case of φ i , we obtain counter terms by the renormalization of field H → √ Z H H = √ 1 + δ H . We then apply the renormalization condition at a cut-off scale Λ, requiring the canonical form of kinetic term at the cut-off scale. Then we carry out the normalization of fields same as above cases. Therefore we obtain the one-loop corrections to KK mass for each KK mode of scalar boson, where we have taken the first order in Ξ and δ in RHS.

IV. KK MASS SPECTRUM AND LKP
In this section, we estimate the one-loop corrections to KK masses for the first KK modes of SM fermions shown in Table I, SM gauge bosons, and Higgs boson. In estimating these corrections, we take the renormalization scale µ and the external momentum p 2 to be the first KK mass µ 2 = 2/R 2 and p 2 = −2/R 2 respectively, and four-momentum cut-off scale Λ as a parameter. Then angular momentum cut-off ℓ max is taken to be the maximum integer satisfying ℓ max (ℓ max + 1)/R ≤ Λ. We specify the lightest KK particle which is expected to be the dark matter candidate.
To estimate the one-loop corrections to KK masses for SM fermions, we need to calculate coefficients Σ(Σ)s in Eq. (58) from one-loop diagrams. For each fermions in Table I, the coefficients Σ(Σ)s obtain contributions from each diagrams such that whereΣs are given in the same way, and each terms in the RHS are the contributions from different loop diagrams with particles running in a loop shown as superscripts; the G µ , W µ and B µ denote the SU (3), SU(2) and U(1) Y gauge fields respectively, the G i , W i and B i denote corresponding extra dimensional components φ i , and the H denotes the Higgs boson. Here we take into account contribution from Higgs-loop only for fermions in the 3rd generations since the Yukawa couplings for the 1st and the 2nd generations are small. The contributions to each coefficients are obtained from Table IX applying corresponding gauge couplings g 6a and (T a ) 2 factor regarding the particles inside a loop; g 6a=1,2,3 for U(1) Y , SU(2) and SU (3) gauge coupling, and ( We also note that each gauge couplings, Yukawa couplings and the Higgs self coupling in six-dimensions {g 6a , Y u,d,e , λ 6 } are related to that in four-dimensions as g a (y u,e,d ) = g 6a (Y u,d,e )/ √ 4πR 2 and λ = λ 6 /(4πR 2 ) since the factor 4πR 2 comes from the surface area of S 2 and 1/ √ 4π factor comes from mode functions for zero mode. The mass matrix for fermion Eq. (16) is corrected by quantum correction as ℓm is given by Eq. (65) and indices F (f ) show corresponding fermion comes from SU(2) doublet(singlet). Here we write one-loop correction to KK mass (δM) ℓm as a product of 1/R and dimensionless numerical factor ∆ f (F ) ℓm;Λ which depends on {ℓ, m} and Λ, such as and we list the numerical factor ∆ Table II for the first KK mode. The correction to offdiagonal components are induced by normalization of field Eq. (64) such that where we have taken the first order of {Σ, δ}, and the values of δN ℓm;Λ is shown in Table IV for the first KK mode. We thus obtain the KK mass eigenvalues   Π(Π)s obtain contributions from each diagrams such that    whereΠs are given in the same way, each terms in the RHS are the contributions from different loop diagrams with particles running in a loop shown as superscripts, and c G,W are ghost fields corresponding to G µ and W µ respectively. The contributions to each coefficients are obtained from Table X and XI. For a U(1) Y gauge field B µ and a SU(2) gauge field W 3 µ , the mass matrix at one-loop level is given as Aµ ℓm as a product of 1/R 2 and numerical factor as in the fermion case, where A µ = {B µ , W µ , G µ } , such as and we list the numerical factors ∆ Aµ ℓm;Λ in Table II for the first KK mode. The correction to non-diagonal elements is induced by renormalization of field Eq. (73) such that where we have taken up to the first order in Πs, and the values of δN W,B are shown in Table IV for the first KK mode. Diagonalizing the mass matrix, we obtain the KK masses of KK-photon and KK-Z boson at one-loop level such that where we ignored terms proportional to δN B,W G in the square root in the second line of RHS for both equation due to the smallness of the factor. For KK modes of W ± bosons and Gluon, the one-loop corrections to KK masses are where (δM 2 ) G ℓm is given by Eq. (74) with coefficients Eq. (101). The corrected masses of the first KK modes for each gauge bosons are shown in the Table V and the right-panel of Fig. 12.
To estimate the one-loop corrections to KK masses for physical extra-dimensional components of gauge fields φ 1 , we need to calculate coefficients Θ(Θ)s in Eq. (77) from one-loop diagrams. For each gauge bosons, the coefficients Θ(Θ)s obtain contributions from each diagrams such that

+ Θ
(1)c G c G Fig.4 , whereΘs are given in the same way, and each terms in the RHS are the contributions from different loop diagrams with particles running in a loop shown as superscripts. The contributions to each coefficients are obtained from and we list the numerical factor ∆ φ 1 ℓm in Table II for the first KK mode. The correction to off-diagonal elements is induced by the renormalization of field Eq. (82) such that where we have taken up to the first order of Θ(Θ), and the values of δN W 1 ,B 1 ℓm;Λ are shown in Table IV for the first KK mode. Diagonalizing the mass matrix, we obtain the one-loop corrections to KK

masses of extra dimensional components KK-photon and KK-Z boson such that
where we ignored the terms proportional to δN B,W φ in square root in the second line of RHS for both equation due to the smallness of the factor. For KK modes of W ± bosons and Gluon, the one-loop corrections to KK masses are where (δM 2 ) G 1 ℓm is given by Eq. (83) with coefficients in Eq. (111). The one-loop corrections to masses of the first KK modes for each φ 1 are shown in Table V and the right-panel of Fig. 12.
To estimate the one-loop corrections to KK masses for SM Higgs boson, we need to calculate coefficients Ξ(Ξ) in Eq. (86) from one-loop diagrams. The coefficients Ξ(Ξ) obtain contributions from each diagrams such that  Table IV. The (δM 2 ) H ℓm is also written as Eq. (105) such that and we list the numerical factor ∆ H ℓm;Λ in Table II. Also the one-loop corrections to masses of the first KK modes for Higgs bosons are shown in Table V and the right-panel of Fig. 12.
The Table V shows the one-loop corrections to KK mass of the first KK modes ℓ = 1, |m| = 1 for each SM particles where we take cut off ℓ max = 2, 3, 4, 5 as reference values. From the Table we can see that the first KK photon has the lightest KK mass for any cut-off scale. The first KK photon thus is a promising candidate of the dark matter in the model. The smallest correction to the KK photon mass is due to the negative contributions from fermion loops while the non-Abelian gauge bosons obtain large contribution from the loops associated with self-interactions. We also find that negative contribution from fermion loops to KK masses of φ 1 tends to be less effective compared with the case of four-dimensional components, and the one-loop corrections to the KK masses become larger than that of four-dimensional components. We show the cut-off scale dependence of the first KK masses in the Fig. 12, where we ignored the SM masses of e, µ, u, d, s, c and write as E R , L, u R , d R , Q L for SU(2) singlet charged lepton, doublet lepton, singlet quarks and doublet quarks respectively. Here the discrete shape of the plot corresponds to increase of ℓ max at ΛR = ℓ max (ℓ max + 1). We also note that the corrected masses tend to be small for even ℓ max compared to odd one, which is due to (−1) {ℓ,m} factors appearing in loop diagram as we mentioned below Eq. (57). The quantum corrections tend to be large for the large cut-off scale since the six dimensional loop expansion parameter ǫ = g 2 a 32π 2 (RΛ) 2 becomes large. This indicates that the perturbation is not reliable and we cannot take a large cut-off scale, which is a generic feature of non-renormalizable theories.

V. SUMMARY AND DISCUSSIONS
We have investigated the quantum correction of Kaluza-Klein masses in the S 2 /Z 2 universal extra dimensional model in order to determine the lightest KK particle which would be the dark matter candidate.
We first derived the Feynman rules for propagators and vertices on the six dimensional spacetime M 4 × S 2 /Z 2 . We then calculated the one-loop diagrams relevant to one-loop corrections to the KK masses with the Feynman rules. The contributions from the one-loop diagrams can be separated to the bulk contribution conserving KK number m and the boundary contribution violating the KK number conservation, which is the similar structure as in the minimal UED case. In calculating these diagrams, we introduced the cut-off for both 4-momentum integration and angular momentum sum inside a loop by defining the 4-momentum cut-off scale Λ and the angular momentum number cut-off ℓ max . Thus the bulk contribution is estimated with cut-off scale Λ as an upper bound of momentum integration while the boundary contribution is estimated by taking leading log-divergent part assuming vanishing contribution at the cut-off scale. The contributions from the one loop diagrams are then organized as the form of Eq. (58), (68), (77) and (86) for fermions, gauge bosons, extra dimensional components gauge bosons and scalar bosons respectively, and the explicit form of the contributions are summarized in the Table IX, X, XI, XII,  At last we estimated the corrected KK masses of first KK mode (ℓ = 1, |m| = 1) for each SM particles applying the formulas in Section III. The resulting corrected masses, taking 1/R = 500GeV as a reference value, are summarized in Table V for each cut-off. We then find that the lightest KK particle is the 1st KK photon γ 11 µ as in the minimal UED case, and it can be identified as the promising DM candidate. We have also shown the numerical plot of the one-loop correction to the first KK modes as a function of cut-off scale Λ in the Fig. 12. We have found that the corrected masses tend to be small for even ℓ max compared to odd one due to the sign factor (−1) {ℓ,m} which appears in loop diagram calculation. The quantum correction tend to be large for higher cut-off since the number of modes inside a loop becomes very large, which indicates that we can not take cut-off scale too large.
It would be very interesting to study the relic density of the dark matter candidate, γ 11 µ , with the corrected mass spectrum, which will give constraints for the S 2 /Z 2 radius R and the cut-off scale Λ. Furthermore the one-loop corrections to the mass spectrum would lead a prediction of the experimental signature of our model and the analysis is left for future work.
Here we summarize the Feynman rules for vertices in our model. The vertices including fermions are derived from gauge interactions and Yukawa interactions where A θ,φ in Eqs. (20) and (21) are substituted. The Feynman rules for these vertices are listed in Table VI where the non-trivial factors in these vertices are given by integrations of mode function Interaction vertex factor  as Eq. (46) for I α(β) , and The vertices for self-interactions of gauge boson are derived from the terms in the Lagrangian where i is the index of gauge group. The first term of Eq. (A5) in the bracket is expanded as where the second line of RHS gives three-and four-point self-interaction of gauge boson summarized in Table VIII. The second and the third term of Eq. (A5) give interactions between A µ and A θ,φ .
By substituting A θ,φ of Eqs. (20) and (21), we obtain interaction terms providing vertex factors from the third line to the seventh line in Table VIII. The last term of Eq. (A5) gives interactions between φ i s after substituting A θ,φ of Eqs. (20) and (21) such that providing vertex factors from the eighth line to the thirteenth line in Table VIII. The vertex factor associated with ghost fields c i is also obtained from the guage-ghost interaction in Eq. (8), and it is summarized in both the fourteenth and the fifteenth vertex factors in  these coefficients are given by where we adopt the convention that we put an underline for indices ℓm of Y * ℓm . The coefficients for four point vertices are also given by  , ℓmax ℓ 2 =0 , ℓ 1 m 1 =−ℓ 1 } and the overall factor (T a ) 2 g 2 6a /64π 2 R 2 ((Y u,d,e ) 2 /64π 2 R 2 ) for fermion(Higgs) loop are omitted for all contributions in the third column. Also ln(Λ 2 /µ 2 ) factor is omitted for boundary contributions in the third column. The subscripts f A µ etc. show the propagating particles inside a loop.
The vertices including Higgs field are derived from the interactions which are the same structure as in the SM. The Feynman rules of these interactions are listed in Table VII, where the non-trivial coefficients in the vertex factors have been already given above.  the Table IX. The overall factor T r[T a i T a j ]g 2 6a /64π 2 R 2 is also omitted for each expression. A coefficient providing zero contribution is omitted.

One loop corrections to KK masses of fermion
For fermions, the corresponding one-loop diagrams are shown in Figs. 1 and 2, and the contributions from each diagram are expressed as Eqs. (52) and (53). We have calculated a diagram in Fig. 1 at Sec. III for four dimensional components of gauge boson inside a loop and got Eqs. (54)-(57), which are summarized in the first part of Table IX. The contribution from φ i to the diagram in Fig. 1 is obtained by usingΨφ i Ψ vertices in Table VI and φ i propagator Eq. (39). We then obtain the 2nd and the 3rd part of Table IX. The Fig. 2 can be calculated in the same way by using Yukawa coupling constants in Table VI and scalar boson propagator Eq. (42), where Y f is the corresponding Yukawa coupling constant in six-dimensions. The RHS is expressed as the case of Fig. 1 and the results are summarized in the 4th part of Table IX 2

. One loop corrections to KK masses of four dimensional components gauge boson
For gauge bosons, the corresponding one-loop diagrams are shown in Figs. 3-7, and they are calculated as in the fermion case. The Fig. 3 for four dimensional components of gauge bosons can be obtained by using relevant propagators of fermion Eq. (33) andΨA µ Ψ vertices in Table VI, where the sign ± corresponds to Ψ (±γ 5 ) + in the loop. Taking the sum over m ′ 1 , the products of Kronecker delta become m-conserving {δ m,m ′ , δ −m,m ′ } and m-violating {δ 2m 1 ,m+m ′ , δ 2m 1 ,m−m ′ }, and we separate the bulk and the boundary contribution as in the femion case, iΠ Fig.3 µν (p; ℓm, ℓ ′ m ′ ) = iΠ  The overall factor C 2 (G)g 2 6a /64π 2 R 2 is also omitted where C 2 (G) is obtained as f lmi f lmj = C 2 (G)δ ij for corresponding gauge group G.
in Table VII such that iΠ Fig.4 µν (p; ℓm; ℓ ′ m ′ ) = which is also expressed as the form of Eqs. (66) and (67), and the explicit form of each coefficients are summarized in the 2nd part of which is expressed as the form of Eqs. (66) and (67), and the explicit form of each coefficients are summarized in the 3rd part of Table X. The Fig. 6 for two virtual A µ is calculated by making use of propagator of A µ Eq. (36) and (A µ ) 3 vertex in Table VIII such that which is expressed as the form of Eqs. (66) and (67) which is expressed as the form of Eqs. (66) and (67) and the explicit form of each coefficients are summarized in the 3rd part of Table XI.

One loop corrections to KK masses of extra dimensional components gauge boson
Here we calculate one loop corrections to φ 1 φ 1 term. The Fig. 3 for extra dimensional components of gauge bosons φ 1 can be obtained by making use of regarding propagator of fermion in Eq. (33) andΨφ i Ψ vertex in Table VI, such that where the sign ± corresponds to Ψ (±γ 5 ) + in the loop. After taking the sum over m ′ 1 , it is separated into bulk and boundary contribution as in the previous cases. We then arrange the terms in the form of Eqs. (75) and (76) and each coefficients are given in the first line of Table XII. In arranging the terms, we expanded the denominator of Eq. (B8) to extract the terms proportional to p 2 such as where ∆ M = (1 − α)M 2 ℓ 1 + αM 2 ℓ 2 , and k E is a Euclidean momentum.

Diagram Coefficients
(1)Aµ bulk (1)φ 1 bulk   Table VII such that which is expressed as the form of Eqs. (75) and (76), and each coefficients are summarized in the 3rd part of Table XII. The Fig. 6 for two virtual A µ is calculated by making use of propagator of A µ in Eq. (36) and (A µ ) 2 φ 1 vertex in Table VIII such that which is expressed as the form of Eqs. (75) and (76) and each coefficients are summarized in the 3rd part of
which is expressed as the form of Eqs. (75) and (76) and each coefficients are summarized in the 2nd part of Table XIII. Fig. 7 for virtual φ i is calculated in the same way by making use of propagator of φ i in Eq. (39) and (φ i ) 4 vertex in Table VIII, and the results are summarized in the 7th and the 8th parts of Table XIII. The one loop corrections to φ 2 φ 2 term and off-diagonal φ 1 φ 2 term are carried out in the same way, and the results are summarized in Table XIV and XV respectively.     the Table IX. The overall factor C 2 (G)g 2 6a /64π 2 R 2 is also omitted.

One loop corrections to KK mass of Higgs boson
For Higgs boson, the corresponding one-loop diagrams are shown in Fig. 8-11. These diagrams are calculated as in the previous cases. Fig. 11 for virtual A µ can be obtained by making use of propagators in Eq. (42) and H † HA µ vertex in After summing over m ′ 1 , it is separated into the bulk and the boundary contribution as in the previous cases. We then arrange the terms in the form of Eqs. (84) and (85) using Eq. (B9) to extract terms proportional to p 2 , and each coefficients are given in the first part of the Table XVI. Fig. 11 for one virtual φ i is calculated in the same way by applying propagator of φ i in Eq. (39), propagator of scalar boson in Eq. (42) and H † Hφ i vertex in Table VII, and the results are summarized in the 2nd part of Table XVI. The Fig. 10 for virtual A µ is calculated by making use of propagator of A µ in Eq. (36) and Aµ bulk   the Table IX. The overall factor (T a ) 2 g 2 6a /64π 2 R 2 and λ 6 /64π 2 R 2 are also omitted for loops with gauge interaction and Higgs-self interaction respectively.
H † H(A µ ) 2 vertex in Table VII such that iΞ Fig.10(Aµ) (p; ℓm; ℓ ′ m ′ ) =i g 2 6a R 2 T 2 a g µν g µν ℓmax ℓ 1 =0 which is expressed as the form of Eqs. (84) and (85) and each coefficients are summarized in the 3rd part of Table XVI. The Fig. 10 for virtual φ i is calculated in the same way by applying propagator of φ i in Eq. (39) and H † H(φ i ) 2 vertex in Table VII, and the results are summarized in the 4th part of Table XVI. The Fig. 8 is calculated in the same manner by making use of propagator of scalar boson in