Cancellation of One-loop Corrections to Scalar Masses in Yang-Mills Theory with Flux Compactification

We calculate one-loop corrections to the mass for the zero mode of scalar field in a six-dimensional Yang-Mills theory compactified on a torus with magnetic flux. It is shown that these corrections are exactly cancelled thanks to a shift symmetry under the translation in extra spaces. This result is expected from the fact that the zero mode of scalar field is a Nambu-Goldstone boson of the translational invariance in extra spaces.


Introduction
As one of the approaches to explore the physics beyond the Standard Model (SM), higher dimensional theories have been much attention to paid so far. Flux compactification has been extensively studied in string theory [1,2]. Even in the field theories, the flux compactification has been paid much attention. Many attractive aspects have been known in such a commpactification: spontaneously supersymmetry breaking [3], realization of four-dimensional chiral fermion zero-mode without any orbifold [4], explanation of the generation number of the SM fermions [5] and computation of Yukawa coupling [4,6,7].
Recently, corrections to the zero mode of the scalar field in effective field theory of flux compactification was studied [8][9][10]. According to their surprising results, the corrections at one-loop vanish thanks to the shift symmetry in extra spaces. These analyses have been done in an Abelian gauge theory with supersymmetry [8,9] or without supersymmetry [10].
In this paper, we extend the work of [10] to a Non-Abelian gauge theory, that is, a six-dimensional SU(2) Yang-Mills theory compactified on a torus with magnetic flux. The cancellation shown in [10] is nontrivial in this extension since we have to quantize the gauge theory and have to take into account the ghost field contributions, which are irrelevant in an Abelian gauge theory. Also, this extension is inevitable to apply to phenomenology.
We calculate one-loop corrections to the mass for the zero mode of the scalar field from the gauge boson, the scalar field and the ghost field loop contributions. These corrections are shown to be exactly cancelled. We discuss that a crucial point to show the cancellations is the shift symmetry under translation in extra spaces, as discussed in [10]. We also discuss that the scalar field is a Nambu-Goldstone (NG) boson of the translation in extra spaces.
This fact implies that the zero mode of the scalar field can only have derivative terms in the Lagrangian and the mass term is forbidden by the shift symmetry. This paper is organized as follows. In section 2, we introduce a six-dimensional Yang-Mills theory with flux compactification. In section 3, we consider mass eigenvalue and eigenstate for the gauge fields, scalar fields and ghost fields. An effective Lagrangian in four dimension is derived in section 4. In section 5, we show the cancellation of corrections to scalar mass at one-loop level. We discuss the physical reason why the corrections to scalar mass vanish. We provide our conclusions and discussion in the last section. The vertices needed for calculation are summarized in Appendix A.

Yang-Mills Theory with Flux Compactification
We consider a six-dimensional SU(2) Yang-Mills Theory with a constant magnetic flux.
Let us first discuss how the constant magnetic flux is introduced in our model. The magnetic flux is given by the nontrivial background (or vacuum expectation value (VEV)) of the fifth and the sixth component of the gauge field A 5,6 , which must satisfy their classical equation of motion In this paper, we choose a solution which introduces a magnetic field parametrized by a constant f , F 1 56 = f and breaks a six-dimensional translational invariance spontaneously. The magnetic flux is obtained by integrating on T 2 and is quantized.
where L 2 is an area of the square torus. For simplicity, we set L = 1 hereafter.
It is useful to define ∂ and φ as In this complex coordinate, the VEV of φ is given by φ = fz/ √ 2, and we expand it around flux background where ϕ a is a quantum fluctuation.
The Lagrangian (1) can be rewritten by using Eq. (7) as follows. where express the covariant derivatives with respect to the complex coordinates in compactified space. Φ a denotes an arbitrary field in the adjoint representation. We can get rid of the mixing terms between the gauge field and the scalars in the second line of Eq. (1) by introducing the gauge-fixing terms, The covariant derivatives D,D are defined by replacing φ a ,φ a in D,D with the VEV φ a , φ a , respectively.
Once we have gauge-fixed, we need to introduce the ghost fields by following Faddeev-Popov procedure to quantize gauge fields. The ghost Lagrangian reads Then, the total Lagrangian is 3

Kaluza-Klein Mass Spectrum
In this section, we discuss mass eigenstates and eigenvalues of the fields A a µ , ϕ a , c a , in which it reminds us of a discussion of Landau level in quantum mechanics.

Gauge Field
First, we find mass eigenvalue and eigenstate of the gauge field. The mass terms of gauge field correspond to the background part of the second line in Eq.(14).
We note that D andD can be identified with creation and annihilation operators (see [4], [8]): a ∝ iD, a † ∝ iD. Expressing them in a matrix form as we can calculate their commutator Thus, the creation and annihilation operators can be defined by and we obtain the commutation relation Diagonalizing the covariant derivatives are non-diagonal as the commutation relation is diagonalized.
Each component of creation and annihilation operators are summarized as follows.
We note that a 1 and a † 1 play no role of creation and annihilation operators. Although a 2 and a † 2 are ordinary annihilation and creation operators, the roles of creation and annihilation operators for a 3 and a † 3 are inverted because of [a 3 , a † 3 ] = −1. The ground state mode functions are determined by a 2 ψ 2 0,j = 0, a † 3 ψ 3 0,j = 0, where j = 0, · · · , |N | − 1 labels the degeneracy of the ground state. Higher mode functions ψ a na,j are constructed similar to the harmonic oscillator case, and satisfy a orthonormality condition The mass operator for gauge field (denoted by H from an analogy of harmonic oscillator with Landau level n 2,3 and mass eigenstate of gauge fields are defined bỹ with a unitary matrix

Scalar Field
Next, we find mass eigenvalues of the scalar fields. Extracting quadratic terms for ϕ a from Eq. (14), we obtain As the discussion in the previous subsection, we need to diagonalize them. In order to justify that the scalar masses can be simultaneously diagonalized by the same unitary as that of the gauge field, we give some arguments below. Because of Dφ aD ϕ a = −φ a DDϕ a , the second and the third terms in the first line of Eq. (30) can be diagonalized by the unitary matrix U .
Next, we focus on the first term in the first line of Eq. (30) Integrating out on square torus, the second term and third term in Eq. (32) vanish thanks to the orthogonality of the mode functions. The first term in Eq. (32) also vanishes since we will consider the zero mode ofφ 1 independent of z,z. This result is the same for Dϕ aD ϕ a . The last term in the first line of Eq. (30) can be also diagonalized by the Applying the same argument to the scalar mass terms from the gauge fixing terms in the second line of Eq. (30), we finally obtain mass eigenvalues of ϕ a :

Ghost Field
Finally, we find mass eigenvalues of ghost field. Extracting the quadratic terms for c a , we The differential operator D m D m can be rewritten as follows.
where we used [D 5 , D 6 ] = [D,D]/2i. Thus, the ghost mass matrix is diagonalized as Mass eigenstate of the ghost field is defined as

Effective Lagrangian
The purpose of this section is to derive effective Lagrangian in four dimensions by KK reduction. Although the gauge field A a µ and the ghost field c a are expanded for all components, only ϕ 2,3 are expanded as for the scalar field since we are interested in the corrections to the mass for the zero mode of ϕ 1 independent of z,z.
Notice that (A a µ,na,j ) † = A a µ,na,−j is satisfied because the gauge field is real A † µ = A µ and (ψ a na,j (z)) * = ψ a na,−j (z) is also satisfied. From Eq. (14) and the previous discussion, our Lagrangian is given by 1 1 Do not confuseF µν as the dual of F µν .
In the above expression, only the quadratic terms in the first and second lines are written in terms of mass eigenstate. In order to read vertices for Feynman diagram calculations, we must rewrite the remaining interaction terms in terms of the corresponding mass eigenstate, which will be done below. The relevant vertices required for our calculation in the next section are summarized in Appendix A.

Gauge Field
In this subsection, the interaction terms including the gauge field are considered. It is easy to expand the quartic term.
then the orthonormality condition for the mode functions leads to Next, we calculate the cubic term of ϕAA in a mass eigenstate. Expanding ∂A a µ [A µ ,φ] a in components, we have where a symbol ⊃ in the second line means that only the non-vanishing terms by the orthonormality condition are left. In the last line, the partial derivative is replaced by Eq. (23). Using the relation (28) and the orthonormality condition for mode functions, we find where α = 2gf . Similar procedure leads to

Scalar Field
Next, we calculate the cubic and quartic terms for the scalar field. It is also easy to compute the quartic term.
The reason why a facotr 2 appears is that there are two ways to choose a pair of KK expansions: ϕ bφc or ϕ b φ c since one of the two ϕ(φ) is taken to be ϕ 1 (φ 1 ). Hence, we obtain L ϕϕϕϕ = g 2 ε abc ε ab c δ bc The calculation of cubic terms including a single ϕ 1 can be done as in the case of gauge field, The first term and second term vanish because Dφ 1 = 0 for zero mode of ϕ 1 . Remaining non-vanishing terms are calculated as Lφφ ϕ = n 2 ,j g α(n 2 + 1) √ 2iφ

Ghost Field
Finally, we compute the cubic terms for the ghost and scalar fields, which include single ϕ 1 .
[ϕ, c] a∂ c a = iε abc ϕ acb∂ c c ⊃ −c 2∂c2 ϕ 1 +c 3∂c3 ϕ 1 where only non-vanishing terms are left in the first line and the partial derivative is replaced by Eq.(23). Using the relation (26) and the orthonormality condition for mode functions, we find

Cancellation of One-loop Corrections to Scalar Mass
In this section, we calculate one-loop corrections to scalar mass for the zero mode of ϕ 1 and show that they are exactly cancelled.
Thus, we conclude which implies that the corrections from the gauge boson loop are cancelled. We emphasize that this cancellation holds for an arbitrary ξ.

Ghost Loop
As for the ghost loop contributions, we have only to consider a diagram shown in Fig. 3.
of Feynman diagrams are expressed as where Wick rotation and a change of variable p 2 → ξp 2 are performed in momentum integral. Notice that we need to consider an overall sign (−1) for the ghost loop.

Cancellation between Scalar Loop and Ghost Loop Contributions
As you can see in subsection 5.1, one-loop corrections to the zero mode scalar mass are cancelled between two diagrams of gauge boson loop. In this subsection, we show the cancellation between the corrections from the scalar field and the ghost field loops.
First, let us consider the case ξ = 0. In this case, the contributions from the ghost field Eq.(70) and (71) trivially vanish since they are proportional to ξ 2 : I In the ξ = 0 case, the ghost field is massless, which implies no interaction with scalar fields, therefore it is natural to vanish the contributions from the ghost loops.
Next, let us see how the contributions from the scalar field loop become in ξ = 0 case.
Using the result Eq.(63) again, we conclude that these contributions are also zero.

Physical Reason of Cancellation
We have shown that one-loop corrections to the zero mode of ϕ 1 vanish. The physical reason of this remarkable result can be understood from the fact that the zero mode of ϕ 1 is a NG boson of translational invariance in extra spaces. The transformation of translation in extra spaces are given by where 5,6 means constant parameters of translations in extra spaces. These transformations can be rewritten in complex coordinate where ≡ 1 2 ( 5 + i 6 ). Focusing on the zero mode of ϕ 1 and noticing ∂ϕ 1 =∂ϕ 1 = 0, we find which is simply reduced to a constant shift symmetry. This shows that the zero mode of ϕ 1 is a NG boson under the translation in extra spaces. Therefore, only the derivative terms of the zero mode of ϕ 1 are allowed in the Langrangian and it is a natural result that one-loop corrections to the zero mode of ϕ 1 vanish. It is very interesting to note that the cancellations in the explicit calculations above have been shown by relying on the shift n → n + 1, which is a remnant of the shift symmetry discussed in this subsection.

Conclusion and Discussion
In this paper, we have studied that one-loop corrections to the scalar mass in a sixdimensional Yang-Mills theory compactified on a two-dimensional torus with a constant magnetic flux. Having performed KK expansion in terms of mode functions specified by the Landau level for the gauge field, the scaler field originated from the gauge field and the ghost field, the four-dimensional effective Lagrangian was derived.
Using this effective Lagrangian, we have calculated one-loop corrections to the mass for the zero mode of the scalar field from the gauge boson, the scalar field and the ghost field loop contributions. What a remarkable thing is that these corrections are shown to be cancelled. As for the gauge boson loop contributions, the cancellation was shown in an arbitrary gauge-fixing parameter ξ. As for the scalar and the ghost loop contributions, the cancellation is independently realized for each fields in ξ = 0 case, but the cancellation is not separable and nontrivially established in ξ = 1 case. Crucial point to show the cancellations was the shift of mode index n → n + 1 in the momentum integral and mode sum, which is a remnant of the shift symmetry described below.
The physical reason of cancellation is that the zero mode of the scalar field transforms as a constant shift under the translation in extra spaces. Therefore, the scalar field is a NG boson of the translation. This fact implies that the zero mode of the scalar field can only have derivative terms in the Lagrangian and the mass term is forbidden by the shift symmetry.
In this paper, we have taken a six-dimensional Yang-Mills theory compactified on a torus with the magnetic flux as an illustration. Our results are expected to be true at any order of loop calculations, for any other gauge theories and for any other models in extra even dimensions as far as the zero mode of the scalar field is the NG boson of translation in extra spaces. It would be interesting to extend our study along these lines.
As one of the more interesting phenomenological applications, we hit upon an appli-

Acknowledgments
This work is supported in part by JSPS KAKENHI Grant Number JP17K05420 (N.M.).

A Vertex
In appendix A, we summarize vertices which needed for calculations in section 4. A factor µ,n,jÃ 2 ⌫,n, jÃ 3 ⌫,n, j