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Nonvanishing Finite Scalar Mass in Flux Compactification

We study possibilities to realize a nonvanishing finite Wilson line (WL) scalar mass in flux compactification. Generalizing loop integrals in the quantum correction to WL mass at one-loop, we derive the conditions for the loop integrals and mode sums in one-loop corrections to WL scalar mass to be finite. We further guess and classify the four-point and three-point interaction terms satisfying these conditions. As an illustration, the nonvanishing finite WL scalar mass is explicitly shown in a six dimensional scalar QED by diagrammatic computation and effective potential analysis. This is the first example of finite WL scalar mass in flux compactification.


Introduction
It has been considered that the hierarchy problem is one of the guiding principles to search for the physics beyond the Standard Model (SM) of particle physics. In the SM, the quantum correction to the mass of Higgs field is sensitive to the square of the ultraviolet cutoff scale of the theory (for example, Planck scale or the scale of grand unified theory).
Since an experimental values of the Higgs mass is 125 GeV, the solution of the hierarchy problem requires an unnatural fine-tuning of parameters or exploring a new physics beyond the SM at order of TeV scale. Although the latter approaches have been mainly studied so far, no signature of new physics has been found, which is likely to increase the new physics scale, namely Higgs mass. Therefore, the solution seems to be desirable such that the Higgs mass is zero at the classical level and is generated by the quantum effects.
As one of the approaches to the hierarchy problem, higher dimensional theory with magnetic flux compactification has been studied. Magnetic flux compactification has been originally studied in string theory [1,2] Even in the field theories, flux compactification has many attractive properties: attempt to explain the number of the generations of the SM fermion [3], computation of Yukawa coupling [4][5][6]. Recently, it has been shown that the quantum corrections to the masses of zero-mode of the scalar field induced from extra component of higher dimensional gauge field (called Wilson-line (WL) scalar field) are canceled [7][8][9][10][11][12]. The physical reason of the cancellation is that the shift symmetry from translation in extra spaces forbids the mass term of scalar field. In that situation, the zero-mode of the scalar field can be identified with Nambu-Goldstone (NG) boson of spontaneously broken translational symmetry. It is not possible for these results to apply to the hierarchy problem as it stands since the scalar field is also massless at quantum level. If we identify Wilson-line scalar field with Higgs field, we need some mechanism to generate an explicit breaking term of the translational symmetry in compactified space and the scalar field must be a pseudo NG boson such as pion.
In this paper, we study the possibility to realize nonvanishing finite WL scalar mass in flux compactification. First, we generalize loop integrals in the quantum correction to WL scalar mass at one-loop. Then, the conditions for the loop integral and mode sum to be finite are derived. We further guess and classify the four-point and three-point interaction terms generating the finite one-loop quantum correction to WL scalar mass.
Of these interaction terms, we focus on a simplest interaction term and illustrate the finite quantum correction to the WL scalar mass in a six dimensional scalar QED in two ways: diagrammatic computation or effective potential analysis. This is the first example of finite WL scalar mass in flux compactification. This paper is organized as follows. We introduce a six-dimensional theory with flux compactification and derive Kaluza-Klein mass spectrum of scalar field, fermion field and SU(2) gauge field in section 2. In section 3, we generalize the loop integrals of quantum correction to the masses of Wilson-line scalar field. After deriving the conditions for the quantum correction to be finite, the interaction terms providing finite quantum corrections are classified. In section 4, we focus on an interaction term of all interaction terms classified in section 3 and calculate finite quantum correction to WL scalar mass in a six dimensional scalar QED. In the last section, we summarize our conclusion. The property of Hurwitz zeta function is summarized in appendix A.

Flux Compactification and Kaluza-Klein Mass
In this section, we introduce our setup and summarize the Kaluza-Klein mass spectrum of various fields, which are required for calculating the quantum correction to WL scalar mass.

Flux compactification
Let us first consider a six-dimensional U(1) gauge theory with a constant magnetic flux.
The six-dimensional spacetime is a product of four-dimensional Minkowski spacetime M 4 and two-dimensional torus T 2 . For later discussion, let us consider the following where the spacetime index is given by M, N = 0, 1, · · · , 6, µ, ν = 0, 1, 2, 3, m, n, = 5, 6 respectively and we follow the metric convention as η M N = (−1, +1, · · · + 1). The field strength and the covariant derivative of U(1) gauge field A M are defined by Ψ has ψ with a charge −g and χ with a charge +g and is expressed by We introduce the magnetic flux 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 . This background must satisfy their classical equation of motion ∂ m F mn = 0. In flux compactification, the background of A 5,6 is chosen as which introduces a constant magnetic flux density F 56 = f with a real number f . Note that this solution breaks an extra-dimensional translational invariance spontaneously.
Integrating over T 2 , the magnetic flux is quantized as follows where L 2 is an area of two-dimensional torus. In the following, we set L = 1 for simplicity.
It is useful to define ∂, z, and φ as In terms of these complex coordinates and variables, the VEV of φ is given by φ = fz/ √ 2. We expand φ around the flux background φ = φ + ϕ, where ϕ is a quantum fluctuation. To distinguish ϕ from an introduced bulk scalar Φ, we call ϕ Wilson line (WL) scalar field. Defining the covariant derivatives in the complex coordinates is also useful to obtain Kaluza-Klein (KK) masses later, which are defined as Note that D m means the covariant derivative with VEV.
Finally, we consider an SU(2) Yang-Mills theory with a constant magnetic flux. The Lagrangian of Yang-Mills theory is given by where a = 1, 2, 3 are gauge indices. The field strength and the covariant derivative of abc is a totally anti-symmetric tensor of SU (2). In the case of Yang-Mills theory, we introduce a flux background as For later convenience, we define the covariant derivatives in the complex coordinates as

Kaluza-Klein mass spectrum
To compute one-loop correction to WL scalar mass in flux compactification, we need to derive mass eigenvalues for fields propagating in a loop. In analogy to the quantum mechanics in magnetic field, we regard the covariant derivative D andD as creation and annihilation operators by which satisfy the commutation relation [a, a † ] = 1. Hereafter, we denote α = 2gf .
The ground state mode function is determined by aξ 0,j = 0, a †ξ 0,j = 0, where j = 0, · · · , |N | − 1 accounts for the degeneracy of the ground state. Creation operator and annihilation operator acts on mode functions as and we can construct the higher mode function ξ n,j in the same way as the harmonic oscillator (in detail, see [14]) where n = 0, 1, 2 · · · is Landau level. The higher mode function satisfies an orthonormality condition T 2 dx 2ξ n ,j ξ n,j = δ n,n δ j,j .

Scalar field
We decompose (2) into a four-dimensional part and an extra-dimensional part (see [7]) Now, we focus on the second term in (22) and extract mass term Then, the KK mass of scalar field is obtained by where the fact that a † a is a number operator is used. Note that the KK mass of WL scalar ϕ 2,3 induced from SU(2) Yang-Mills theory corresponds to (24) in Feynman gauge 2 .
Although SU(2) Yang-Mills theory involves a ghost field, the KK mass of the ghost field also agrees with (24) in Feynman gauge.

Fermion field
We decompose (3) as in the case of scalar field (see [9]) and we focus on the second and the third terms in (25). Decomposing these terms in terms of two-component Weyl spinors ψ and χ, the mass terms of fermion field are expressed by In this case, there are two pairs of annihilation and creation operators where a − , a † − act on ψ and a + , a † + act on χ. Using these annihilation and creation operators, we obtain the mass-squared operators for ψ and χ Note that (29) means an existence of chiral fermion in flux compactification since zeromode of ψ is massless but zero-mode of χ is massive. We can rewrite (25) in terms of Dirac fermion ψ Lj and Ψ n,j , and obtain We conclude that ψ Lj is massless and the KK mass of fermion Ψ n,j is given by

SU(2) gauge field
Decomposing (12) and focusing on F a µ5 F aµ5 + F a µ6 F aµ6 terms (see [10]), we obtain a mass term for an SU(2) gauge field, Diagonalizing the covariant derivatives, we find the KK mass of the SU(2) gauge field Note that Abelian gauge field is not expressed such as (33) since the part of commutator [A M , A N ] a is absent in the case of Abelian gauge theory. Therefore, the KK spectrum of Abelian gauge field is nothing but an ordinary KK mass spectrum m 2 U (1) = (n/R) 2 + (m/R) 2 (n, m is integer and R = L/(2π)).

Analysis on the divergence structure of loop integral and classification of interaction terms
In this section, we systematically analyze the divergence structure of the quantum corrections to WL scalar mass and classify possible interactions providing a finite mass.
3.1 The divergence structure of loop integral: part 1 In this subsection, we investigate the divergence structures for quantum correction to the WL scalar mass at one-loop. In general, there are two types of Feynman diagrams in figure 1. From these diagrams and the results of section 2.2, the general form of loop integral in the quantum correction can be written as where the dimensional regularization was employed for loop integral in the second line. Since the WL scalar cannot have a bare mass term, the loop integral and mode sum for one-loop correction to WL scalar mass must be finite to realize nonvanishing finite WL scalar mass. To clarify this point, we investigate in (35). For one-loop corrections, it is enough to consider the case b = 1 or b = 2. In the case of b = 1, the Gamma function part of (36) is expressed by Thus, J(x; a, 1) becomes In the case of b = 2, the same part of (36) is expressed by Thus, J(x; a, 2) becomes Here, Gamma function and Hurwitz zeta function can be expanded in where γ E = 0.5772 · · · is the Euler-Mascheroni constant, p is an arbitrary positive integer, being finite is that p is even, Applying this result to (38) and (40), J(x; a, 1) takes finite value at odd a, J(x; a, 2) does at even a.

Classification of interaction terms: part 1
We classify the interaction terms providing finite correction to WL scalar mass at oneloop. We consider interaction terms which has no derivatives acting on ϕ orφ because we consider one-loop corrections to WL scalar mass.

Four-point interaction
Four-point interaction term generates a correction to WL scalar mass of the left one in figure 1. The diagram corresponds to J(x; a, 1) (a: odd), from which we can guess the four-point interaction terms as follows, • scalar field loop • fermion field loop • SU(2) gauge field loop We did not consider a four-point interaction with such asφϕ∂ µ 1 · · · ∂ µaψ ∂ µ 1 · · · ∂ µa ψ since the fermion mass m f ermion = α(n + 1) is emerged from a numerator in the fermion propagator and then the form of Hurwitz zeta function is complicated. On the other hand, (/ k) 2a−1 is obtained by (45). Computing quantum correction, the trace of / k from a numerator in the propagator of fermion multiplied by (/ k) 2a−1 is given by k 2a . If a is odd, this term contributes to quantum correction to WL scalar mass.

The divergence structure of loop integral: part 2
In more general, we can consider the interaction term with coefficient depending on KK mode. The more general form of loop integral in the quantum correction is given by where f (n) is a coefficient generated by an interaction term depending on KK mode n.
Since the more complicated the form of f (n) is, the more difficult we express as Hurwitz zeta function, the discussion on the finiteness of loop integral becomes hard to proceed.
As a candidate, f (n) = ((α(n + q)) c (q and c are real numbers) is considered. In this section, we assume f (n) = α(n + q) for simplicity. Thus, I (x; a, b) is expressed by If q = x, the divergence will inevitably appears from either ζ To avoid the divergence and see whether the quantum correction is finite, we need to choose q = x (equivalent to the choice f (n) = KK mass), and then investigate in (52). Substituting b = 1 or b = 2 in (53) and using (37) or (39), we obtain Applying the result (43) to (54) and (55), K(x; a, 1) takes finite value at even a, K(x; a, 2) does at odd a.

Classification of interaction terms: part 2
We consider the case of four-point interaction terms (a: even) and guess their form providing finite quantum corrections to WL scalar mass, • scalar field loop • fermion field loop K(1; a, 1) →φϕψ(/ ∂) 2a−1 (a † a + 1)ψ (58) • SU(2) gauge field loop The case of three-point interaction term is hard to guess because the three-point interaction term cannot be expressed in terms of a mass-squared operator. Thus, we do not consider the three-point interaction terms in this section. If we compute loop integral by using f (n) = (α(n + x)) c , the meaning of c is the number of mass-squared operators.
One might think that it would be possible that finite quantum corrections from (47) and (

Comments on interactions between the field with different KK mode indices
Due to the presence of annihilation and creation operators, there are interactions between the field with different KK mode indices. For example, they correspond to ϕχ n,j ψ n+1,j (see [9]) and ϕA a µ,n,j A aµ n+1,j (see [10]). The generalization of loop integral with these interaction terms is very complicated because we must use two propagators in different KK modes and cannot systematically identify the coefficient depending on KK mode n.

Illustration of nonvanishing finite WL scalar mass
In section 3.2, we have classified the interaction terms generating finite quantum correction at one-loop. Now we focus on (47) since it has no derivatives and is the simplest interaction term of all interaction terms in section 3.2. Therefore, we explicitly calculate finite quantum corrections to WL scalar mass from (47) by diagrammatic calculation and effective potential analysis. This is the first example of WL scalar mass in flux compactification.

Set up
The Lagrangian we consider is given by (1), (2) and (47), where κ is a dimensionless coupling constant. Using the expansion of φ = φ + ϕ, the Lagrangian is deformed as where we note that the unnecessary terms are omitted. To derive a four-dimensional effective Lagrangian by KK reduction, we need to expand Φ in terms of mode functions Integrating over T 2 , the four-dimensional effective Lagrangian is obtained by − ig 2α(n + 1)ϕΦ n+1,j Φ n,j + ig 2α(n + 1)ϕΦ n,j Φ n+1,j − 2g 2φ ϕΦ n,j Φ n,j + κφΦ n,j Φ n,j + κϕΦ n,j Φ n,j + κ φ I Φ n,j Φ n,j + κ φ where φ I and φ I are expressed by When φ = fz/ √ 2, φ I and φ I lead to zero because of odd function with respect to integral variables z orz.

Diagrammatic computation
Before computing a finite quantum correction, we first review that the quantum correction to WL scalar mass is cancelled at one-loop in the case of κ = 0 (see [7]). Computation of Feynman diagram is expressed as (66) Note that I 3pt is the example of section 3.5. The sum of I 4pt and I 3pt is obtained by By the shift n → n + 1 in the second term, (67) becomes zero.
For κ = 0, we get a new quantum correction to WL scalar mass from the right diagram in figure 1. Computing the diagram results in Applying J(1/2; 0, 2) to (40), we obtain J(1/2; 0, 2) = Γ( )ζ[ , 1/2] and then where are used in the second line of (69). This correction is finite in → 0 limit. Thus, the quantum correction to WL scalar mass at one-loop is given by Note that we introduced a factor of torus area L 2 , which comes from the normalization factors for KK mode function. Obviously, δm 2 = 0 is reproduced for κ = 0 in sixdimensional scalar QED (see [7]). One of the interesting phenomenological applications is that the quantum correction δm 2 to WL scalar mass can be interpreted as Higgs mass.
This idea is based on gauge-Higgs unification, namely a zero-mode of ϕ is regarded as Higgs scale. This is analogous to the mass of pion as a pseudo NG boson for chiral symmetry.
The reason why the pion mass is not Planck scale is that chiral symmetry is dynamically broken at extremely lower energy scale comparing to the Planck scale, namely, QCD scale.
Note that the similar discussion cannot be applied to the ordinary gauge-Higgs unification since κ is replaced by an SU (2) L gauge coupling in this scenario.

Effective potential analysis
Next, we consider the quantum correction to WL scalar mass in terms of effective potential. In our setup (63), we read the KK mass spectrum of Φ to be α(n + 1/2) − κ φ I − κ φ I . Thus, the four-dimensional effective potential is given by where we take into account a degree of freedom of complex scalar field Φ. To obtain quantum correction to WL scalar mass from four-dimensional effective potential, we differentiate effective potential with respect to φ I and φ I . Thus, δm 2 is obtained as This result (74) agrees with (68) or (72).

Explicit breaking of translational invariance in extra space
where ≡ ( 5 + i 6 )/2 and 5 , 6 means constant parameters of translations in extra spaces. Focusing on the zero-mode of ϕ and noticing ∂ϕ =∂ϕ = 0, we obtain This shows that the zero-mode of ϕ is identified with a NG boson under the translation in extra spaces.
If κ = 0, ϕΦ n,j Φ n,j (orφΦ n,j Φ n,j ) in (63) is expected to break the translational invariance. To confirm it, we consider the following local six-dimensional transformation [9] where Λ = f ( z −¯ z). Infinitesimal transformation of ,¯ is expressed as Transformations of ϕ and Φ are the combination of translation δ T and infinitesimal trans- Using (19) and (62), we obtain For δΦ n,j , it is given by complex conjugate of (82), Let us confirm the explicit breaking of translational invariance of the interaction term ϕΦ n,j Φ n,j . First, a transformation of Φ n,j Φ n,j is Thus, the mass term of Φ n,j is invariant. For ϕΦ n,j Φ n,j , a transformation is δ n,j ϕΦ n,j Φ n,j = (δϕ) n,j Φ n,j Φ n,j + ϕδ can be written in terms of ϕ, ϕ and z,z as we find that the cubic terms introduced in this paper can be expressed by the non-local Wilson line operators where g 4 is a four dimensional gauge coupling constant. Note that the ΦΦ term cannot be included in (89). If this term is allowed, the WL scalar mass would be divergent.
We comment on how the finite WL scalar mass can be expressed in terms of the Wilson line operators. If the WL scalar mass is generated in the broken phase, where the VEV of the WL scalar field is non-zero, it is straightforward to express the WL scalar mass by the Wilson line operators as in the gauge-Higgs unification. As for the WL scalar field mass in the present paper, the mass is generated in the unbroken phase and is independent of the VEV of the WL scalar field. Therefore, we cannot express the WL scalar field mass by the Wilson line operators explicitly.
Under the constant shift of A 5 → A 5 − f 6 /2, A 6 → A 6 + f 5 /2, the operators are not obviously invariant, which means that the interaction terms (89) explicitly break the shift symmetry. Clarifying the origin of the interaction terms (89) is not easy and beyond the scope of this paper. We expect that the origin of the interaction terms would be connected to the quantum gravity effects, nontrivial backgrounds such as a vortex, or some non-perturbative dynamics. We leave this issue for our future work.

Conclusion and Discussion
We have studied the possibility to realize nonvanishing WL scalar mass in flux compactification. Using KK mass of various fields in the bulk, we have generalized loop integrals in the quantum correction to the WL scalar mass and have systematically analyzed their structure of divergence. The conditions for the loop integral and the mode sum to be finite were derived. Then, we have classified four-point and three-point interaction terms providing finite quantum corrections to WL scalar mass at one-loop.
Of these interaction terms, we focused onφΦΦ + ϕΦΦ type interaction since these interaction terms have no derivatives and are the simplest of all interaction terms. Using In particular, ζ[s, 1/2] is satisfied by ζ[0, 1/2] = 0.