Spectrum in the presence of brane-localized mass on torus extra dimensions

The lightest mass eigenvalue of a six-dimensional theory compactified on a torus is numerically evaluated in the presence of the brane-localized mass term. The dependence on the cutoff scale $\Lambda$ is non-negligible even when $\Lambda$ is two orders of magnitude above the compactification scale, which indicates that the mass eigenvalue is sensitive to the size of the brane, in contrast to five-dimensional theories. We obtain an approximate expression of the lightest mass in the thin brane limit, which well fits the numerical calculations, and clarifies its dependence on the torus moduli parameter $\tau$. We find that the lightest mass is typically much lighter than the compactification scale by an order of magnitude even in the limit of a large brane mass.


Introduction
Many extra-dimensional models have four-dimensional (4D) brane-like defects on the compact space, such as orbifold fixed points or solitonic objects [1]- [4]. We can freely introduce 4D terms localized at the branes 1 [5,6,7]. Such brane-localized terms are induced by quantum effect even if they are absent at tree level [8,9]. They change the Kaluza-Klein (KK) spectrum and deform the profiles of the mode functions [10,11,12]. In particular, the brane-localized mass terms are often introduced in order to remove unwanted modes from the 4D effective theory [13,14,15]. In five-dimensional (5D) theories, the effects of such brane masses can be translated into the change of the boundary conditions for the bulk fields. This is because the branes in 5D can be regarded as the boundaries of the extra dimension. In this case, large brane masses can make zero-modes of the bulk fields heavy enough up to half of the compactification scale.
In contrast, the branes are no longer the boundaries of the extra compact space in higher-dimensional theories. Since effects of the brane terms spread over higherdimensional space and are diluted, they are expected to be smaller than those in the 5D case. Therefore, it is important to check whether the brane mass can make unwanted modes heavy enough or not. In this paper, we evaluate the lightest mass eigenvalue of a six-dimensional (6D) theory in the presence of the brane-localized mass term. The authors of Ref. [11] discussed a closely related issue in the case of the T 2 /Z 2 compactification whose torus moduli parameter is τ = i, and obtained the result that the inverse of the lightest mass eigenvalue has a logarithmic dependence on the cutoff scale. Here we generalize their setup and consider a generic torus whose moduli parameter is arbitrary. Then we can explicitly see the relation to the well-known results in the 5D theories by squashing or stretching the torus. Besides, we are interested in a different parameter region from that discussed in Ref. [11]. We mainly focus on the limit of a large brane mass, in which the dependence of the mass eigenvalues on the brane mass is negligible, and evaluate the ratio of the lightest mass to the compactification scale by numerical calculations.
The paper is organized as follows. After explaining the setup in the next section, we will see the dependences of the lightest mass eigenvalue on the cutoff scale of the theory and on the brane mass in Sec. 3. In Sec. 4, we find an approximate expression of the lightest mass as a function of the torus moduli parameter τ , and estimate its ratio to the 1 Here we do not consider branes spread over other dimensions. The word "brane" is understood as the "3-brane" in this paper. compactification scale. Sec. 5 is devoted to the summary. We provide a brief review of the case of a 5D theory in Appendix A, and discuss theories with fermion or vector field in Appendix B.

Setup
We consider a 6D theory of a complex scalar field φ as a simple example. 2 The Lagrangian is given by where M = 0, 1, 2, · · · , 5, and the ellipsis denotes interaction terms, which are irrelevant to the following discussion. The brane mass parameter c is a real dimensionless constant.
The extra dimensions are compactified on a torus T 2 . 3 The background metric is assumed to be flat, for simplicity. For the coordinates of the extra dimensions, it is convenient to use a complex (dimensionless) coordinate z ≡ 1 2πR (x 4 + ix 5 ), where R > 0 is one of the radii of T 2 . The torus is defined by identifying points in the extra dimensions as z ∼ z + n 1 + n 2 τ, (n 1 , n 2 ∈ Z) where τ is a complex constant that satisfies Im τ > 0.
The Lagrangian (2.1) is then rewritten as where µ = 0, 1, 2, 3, and we have used that We can expand φ as are normalized as and satisfy ∂ z ∂zf n,l = −λ 2 n,l f n,l ,λ n,l = π |n + lτ | Im τ . (2.8) This corresponds to the KK expansion in the absence of the brane-localized mass term.
Since the 6D theory is non-renormalizable, it should be regarded as an effective theory valid only below the cutoff scale Λ. Here we relabel the KK modes by using the KK label a = 0, 1, 2, · · · defined in such a way that The correspondence of the labels (n, l) and a depends on the value of τ , as shown in Tables I   and II. The number of the KK excited modes below Λ, i.e., N Λ , grows as except for regions: arg τ ≃ 0, π, |τ | ≪ 1 and |τ | ≫ 1, in which the spacetime approaches 5D and thus N Λ ∝ Λ.
Then, the mass squared eigenvalues, which are denoted as m 2 0 , m 2 1 , · · · , m 2 N Λ , are obtained as eigenvalues of the finite matrix M 2 ab (a, b = 0, 1, · · · , N Λ ). Sincem 2 n,l =m 2 −n,−l , all the nonzero modes have degenerate modes when the brane mass is absent. Especially,m 2 1 =m 2 2 . This means that M 2 ab has the eigenvaluem 2 1 with the eigenvector (0, 1, −1, 0, 0, · · · , 0). In fact, this is the second smallest eigenvalue ofM 2 ab . Namely, the mass of the first KK excited mode m 1 is independent of c and Λ: Thus we take m 1 as the compactification scale throughout the paper.
Plots in Fig. 1 show the Λ-dependence of the lightest eigenvalue m 0 in the cases of τ = exp( πi 120 ), exp( 2πi 3 ), and 50 exp( 2πi 3 ) and c = 10.0, in the unit of m 1 . The right end of the horizontal axis in each plot corresponds to the value of Λ such that N Λ ≃ 4000. For a given value of c, the ratio m 0 /m 1 can be approximated by The horizontal axes in Fig. 1 denote the asymptotic lines that the curves approach. Typically, m 0 approaches to the limit value much more slowly compared with the 5D case (see Fig. 6 in Appendix A.2). Thus the cutoff dependence of the spectrum cannot be neglected even when Λ/m 1 = O(100). This cutoff dependence becomes smaller when arg τ ≃ 0, π or |τ | ≪ 1 or |τ | ≫ 1. This is because the torus is squashed or stretched in such cases, and the spacetime approaches to 5D. In fact, as we can see from Fig. 1, Note that the curve for Λ < 40m 1 in the top-left plot or in the bottom plot are almost the same as that of the 5D case (shown in Fig. 6). The cusp at Λ = 40m 1 indicates that the field begins to feel the width of the squashed torus or the smaller cycle of the long thin torus.
In the following, we focus on the limit value (3.3). Fig. 2 shows its dependence on the brane mass c. The unit here is taken as 1/(πR). For small values of c, the lightest mass eigenvalue m 0 is approximated as which is plotted as the solid line in Fig. 2. This is because the brane mass can be treated as a perturbation in this region, and the mixing among the KK modes induced by it is negligible. As the brane mass grows, such mixing effect becomes significant, and m 0 saturates and is almost independent of c when c > ∼ 5. This situation is the same as the 5D case (see Fig. 5 in Appendix A.1). In the following discussion, we take c = 10.0 as a representative of c ≫ 1. The solid lines represent the approximate expression (4.12).

Approximate expression
since the theory is defined on the torus. Besides, from (2.8) and (2.13), we also find that As mentioned in the previous section, there are two limits in which the spacetime approaches to 5D, i.e., arg τ → 0, π (squashed torus) and |τ | → 0, ∞ (stretched torus). In these cases, the low-lying KK masses in the absence of the brane mass are approximately expressed as follows.
where a < ∼ 2 |τ |, and n(a) ≡ (−1) a floor a+1 Especially when |τ | is a rational number, i.e., |τ | = p/q (p and q are relatively prime integers and q > 0), the light masses are approximated as As for the cases of |τ | ≪ 1 and of π − arg τ ≪ 1, approximate expressions of m a are obtained from (4.3) and (4.5) by using (4.1) and (4.2), respectively. Then, we identify the effective radius of S 1 as . (4.6) Using this, the low-lying KK masses m a can be expressed as (see Appendix A.1) where the "effective 5D brane mass"ĉ eff is defined aŝ which is identified from the condition that (3.5) is reproduced. When c is sufficiently large, the solutions of (4.8) are Especially, the lightest mass eigenvalue is This is plotted as solid lines in Fig. 3.
Finally, we evaluate the ratio of m 0 to the compactification scale m 1 . As Fig. 4 shows, this ratio is much smaller than the value of the 5D case, 1/2, except for the extreme cases in which the spacetime is 5D-like. Typically, m 0 is lighter than m 1 by one order of magnitude. Namely, the brane-localized mass cannot make the zero-modes as heavy as the compactification scale. This is an important fact in model building.

Summary
We have evaluated the mass eigenvalues of a 6D theory compactified on a torus in the presence of the brane-localized mass term. Especially we focus on the lightest mode that becomes massless in the zero brane-mass limit.
From the numerical calculations, we confirmed that the lightest mass eigenvalue m 0 has non-negligible dependence on the cutoff scale Λ even when Λ is larger than the compactification scale by two orders of magnitude. This indicates that m 0 is sensitive to the internal structure of the brane when the brane has a finite size. This is consistent with the results in Ref. [11].
We find an approximate expression of m 0 which is valid for a large brane mass. It clarifies the dependence on the size and the shape of the torus, and reduces to the known result in the 5D case when the torus is squashed or stretched.
In contrast to the 5D case, m 0 is much smaller than the compactification scale unless the torus is squashed or stretched. Their ratio is typically O(0.1). This is because the effects of the brane term are spread out over the codimension two compact space and diluted. Hence we should be careful in model building especially when we introduce the brane mass terms in order to decouple unwanted modes.
Although we have not discussed in this paper, the brane mass also deforms the profiles of the mode functions. They can be obtained by calculating the eigenvectors of M 2 ab in (2.13). The main effect of the brane mass on the mode functions is to push them out from the position of the brane. Namely, it reduces their absolute values at the brane to zero.

On more general setups
We have discussed in a theory of a scalar field because it is the simplest case. However, the properties of the spectrum clarified in the text are also found in cases of fermion and vector fields, as shown in Appendix B. So our result is valid in a wider class of 6D theory.
Besides, we have assumed that the bulk mass is zero and the brane squared mass is positive. In the presence of the bulk mass M bk , the mass matrix (2.13) becomes wherem n,l = |n + lτ | /(RIm τ ). Thus the bulk mass just raises the whole spectrum.
However, if we allow a tachyonic brane mass, i.e., c 2 < 0, a light mode may appear below the compactification scale. If |c| 2 is large enough, m 0 becomes tachyonic and thus φ = 0 is no longer the vacuum. In such a case, φ has a nontrivial background that depend on the extra-dimensional coordinates z andz, and we have to expand φ around it in order to obtain the mass matrix M 2 ab . It is not an easy work to find such a nontrivial background. Here we do not discuss this issue further, but give a comment on it. Note that the smallest diagonal element M 2 00 = M 2 bk + c 2 /(4π 2 R 2 Im τ ) provides the upper bound on m 0 . Thus, 2πR √ Im τ M bk > |c| must be satisfied in order to avoid the vacuum instability for φ = 0.
In other words, there is a value of c that leads to a tachyonic mass eigenvalue no matter how large M bk is. This indicates that the effect of the brane mass on the spectrum does not saturate, which is in contrast to the non-tachyonic brane mass. The mode function is attracted toward the brane by the tachyonic brane mass.
We considered the scalar field with the periodic boundary condition. Twisted boundary conditions are also allowed, but they just raise the mass spectrum. This can be understood from the fact that imposing the twisted boundary conditions is equivalent to introducing a non-vanishing background gauge field coupled to the scalar field with the periodic boundary conditions. Such a background gauge field play the same role as the bulk scalar mass M bk mentioned above.
We have also assumed that the spacetime is flat, no background magnetic fluxes exist, 4 and there is only one brane, for simplicity. It is an interesting and useful extension to relax these assumptions. This will be discussed in separate papers.

Acknowledgements
The author would like to thank Yukihiro Fujimoto for valuable information. This work was supported in part by Grant-in-Aid for Scientific Research (C) No. 25400283 from Japan Society for the Promotion of Science (Y.S.).

A 5D case
Here we summarize the effects of the brane-localized mass in a 5D complex scalar theory.
The Lagrangian is whereM = 0, 1, 2, 3, 4, and the ellipsis denotes interaction terms. The brane mass parameterĉ is a real dimension 1/2 constant. The extra dimension is compactified on S 1 whose radius is R.

A.1 Analytic expressions
The KK expansion of φ is The mode function f n (x 4 ) satisfies the mode equation, which is read off from (A.1) as By integrating this over an infinitesimal interval [−ǫ, ǫ], we obtain Thus the brane mass changes the boundary condition of the bulk field.
In the bulk region [ǫ, 2πR − ǫ], (A.3) is solved as where C ±n are complex constants. From the periodic condition f n (x 4 + 2πR) = f n (x 4 ) and (A.4), we obtain The solutions of these equations are or C −n = e 2πimnR C +n , m n =ĉ  As the brane mass grows, such mixing effect becomes significant, and m 0 saturates and is almost independent ofĉ when c ≡ πR/2ĉ > ∼ 5. In the limit ofĉ → ∞, the spectrum determined by (A.8) is Since the second smallest solution of the second equation in (A.8) are greater than 1/R, the first KK excited mass is m 1 = 1/R, which is independent ofĉ and taken as the compactification scale.

A.2 Numerical evaluation
In order to see the cutoff dependence of the spectrum and compare it with that in the 6D case, we follow the same procedure as in Sec. 3.
We relabel the KK modes by using the KK label a = 0, 1, 2, · · · , which is defined as , (A. 10) and expand φ as where n(a) ≡ (−1) a floor a+1 2 . This is the KK expansion in the absence of the brane mass.
Then, we can rewrite the 5D Lagrangian (A.1) in terms of φ a as We consider only the modes whose masses are below the cutoff scale Λ. (A.14) Then the KK mass eigenvalues m 2 0 , m 2 1 , · · · , m 2 N Λ are calculated as eigenvalues of the finite matrixM 2 ab (a, b = 0, 1, · · · , N Λ ).

B Cases of spinor and vector fields B.1 Brane mass for spinor fields
We consider a theory which has a 6D Weyl spinor field Ψ + whose 6D chirality is +. We can introduce the following brane mass term with the 4D spinor field localized on the brane.
where χ is a 4D left-handed Weyl spinor, and the 2-component spinor ψ is the 4D righthanded component of the 4-component spinorΨ, which is defined as The 6D gamma matrices Γ M are defined as The brane mass parameter c is dimensionless, and assumed to be real.
In the 2-component notation, (B.1) is rewritten as where λ is the 4D left-handed component ofΨ. Thus, the equations of motion are From these, we obtain This has the same form as the equation of motion for φ derived from (2.3). Hence the spectrum in this system is the same as that of the scalar field discussed in the text [11].
In the case that Ψ + does not have any charges, the following Majorana mass term is allowed on the brane.
where the brane mass parameter h has the mass dimension −1. In contrast to the above Dirac mass term, the equation of motion in this case does not have the form of (B.6). Thus the spectrum in this case has to be discussed separately.

B.2 Brane mass for vector field
Here we consider a theory of a vector field A M . We can introduce the following brane mass term.
where the real scalar S is the Stueckelberg's scalar field, which is localized on the brane.
The brane mass parameter c is real and dimensionless. The above Lagrangian is invariant under the gauge transformation: where Λ(x, z) is the gauge transformation parameter.
In order to fix the gauge, we add the following gauge-fixing term.
where ξ is the gauge parameter.
Performing the partial integral, the total Lagrangian becomes We expand the 6D gauge field into the 4D KK modes as µ (x) = 0. The mode functions u a (z) and w a (z) are real, but v a (z) is complex. They are normalized as 5 (B.13) Here we choose the mode functions as where a = (n, l) (see Sec. 2),â = 1, 2, · · · , and λ n,l ≡ π(n + lτ ) Im τ .