Dynamical generation of fermion mass hierarchy in an extra dimension

We propose a new mechanism to produce a fermion mass hierarchy dynamically in a model with a singlet generation of fermions. A five dimensional gauge theory on an interval with point interactions (zero-width branes) takes responsibility for realizing three generations and each massless zero mode localizes at boundaries of the segments on the extra dimension. An extra-dimension coordinate-dependent vacuum expectation value of a scalar field makes large differences in overlap integrals of the localized zero modes and then an exponential fermion mass hierarchy can appear. The positions of the point interactions control the magnitude of the fermion mass hierarchy and are determined by the minimization condition of the Casimir energy. As a result of the minimization of the Casimir energy, an exponential mass hierarchy appears dynamically. We also discuss the stability of the extra dimension.


Introduction
The standard model (SM), which provides an articulate description of the nature around TeV energy scale, was completed by the discovery of the Higgs boson [1,2]. However, the SM still contains several mysteries and problems, which cannot be solved within the context of the SM. One is so-called the generation problem. The SM contains three sets of quarks and leptons, which have the exact same quantum numbers except for their Yukawa couplings. Three generations were introduced to the Kobayashi-Maskawa theory [3] by hand though the origin of the generations is not unveiled. Another problem is on the fermion mass hierarchy. Each generation of the quarks and the charged leptons has exactly the same quantum numbers though their masses have an exponential hierarchy around 10 5 . In the SM, the masses are generated by the Higgs mechanism and are determined by the dimensionless Yukawa couplings; however, there is no explanation to the question of why so large a hierarchy appears in the dimensionless parameters.
Because of the above circumstance, various theories beyond the SM have been explored.
One possibility in the context of four-dimensional (4d) gauge theory is a scenario with noncompact gauge symmetry, which can naturally produce the fermion mass hierarchy and three generations [4][5][6][7][8]. Another way is achieved by using extra dimensions. Extra dimension models with magnetic flux [9] can lead to both of the fermion mass hierarchy and three generations.
Magnetized orbifold models [10][11][12][13][14][15][16][17][18][19] are also fascinating to discuss within the fermion flavor structure and several achievements have been investigated 1 . However, some parameters of the models have to be chosen suitably by hand to make a fermion mass hierarchy. Moreover, in the case of extra dimension models, arguments for the stability of the extra dimension have mostly been postponed. Therefore, it is worth searching the dynamical generation mechanism of the fermion mass hierarchy and discussing the stability of the extra dimension, simultaneously.
In this paper, we propose a dynamical generation mechanism for the fermion mass hierarchy in a model with a single generation of fermions in five dimensions (5d). An interval extra dimension with point interactions [25][26][27][28] takes the responsibility to produce the generations. 2 Point interactions also play an important role to discuss the fermion mass hierarchy. In the previous model [25][26][27][28], the positions of the point interactions, which affect to the fermion mass hierarchy, have been controlled by hand. On the other hand, in this paper, the positions of the point interactions are determined dynamically through the minimization of the Casimir energy [29,30] (or say, Radion effective potential [31][32][33][34]). As a result, a large mass hierarchy appears dynamically in our model 3 . We also discuss the stability of the extra dimension from a Casimir energy point of view. This paper is organized as follows. In section 2, we review the 4d spectrum of a 5d U(1) gauge theory on an interval extra dimension. A general class of boundary conditions (BCs), which is important to determine the 4d spectrum of the fields and phase structure of the symmetries, is derived for gauge, fermion and scalar fields. Using the knowledge of the general boundary conditions, we display the 4d spectrum at low energies and the profiles of the mode functions with respect to the extra dimension. In section 3, we discuss the stability of the extra dimension. Evaluating the contribution of each field, we investigate the extra dimension length dependence of the total Casimir energy. In section 4, a theory with point interactions is reviewed and the 4d mass spectrum at low energies and the profiles of the mode functions are shown. In section 5, using all the results, we construct an SU(2) × U(1) model, which can lead to the fermion mass hierarchy dynamically with a single generation of fermions. The minimization of the Casimir energy determines the positions of the point interactions, which are important parameters to produce the fermion mass hierarchy, and leads to the stability of the extra dimension. After that, we find that a fermion mass hierarchy is realized dynamically. Section 6 is devoted to conclusion and discussion. In Appendix A, we provide a self-contained review on the formulation of wave functions of a 5d fermion under the presence of one point interaction in the bulk part of an interval.

4d spectum of a 5d U(1) gauge theory on an interval
In this section, we first summarize the results of allowed boundary conditions, which are consistent with the requirements from the action principle, the gauge invariance and 4d Lorentz invariance, for gauge, fermion and scalar fields on an interval. The boundary conditions are crucially important to determine the 4d mass spectrum at low energies and also the phase structure of symmetries [25][26][27][28][38][39][40][41][42][43][44][45][46][47][48]. We then derive the 4d mass spectrum of the gauge and fermion fields, which are necessary to evaluate Casimir energies. We further show that the scalar field can possess a coordinate-dependent vacuum expectation value (VEV) on the extra dimension [25][26][27][28][38][39][40], which is found to be a crucial ingredient of our dynamical generation mechanism for generating a fermion mass hierarchy.

Consistent BCs for the fields
In this subsection, we investigate the general class of BCs for an abelian gauge field, a fermion field and a scalar field on an interval, respectively.

BCs for Abelian gauge, ghost, and anti-ghost fields
First, we start from the gauge field: x µ (µ = 0, 1, 2, 3) denotes the four-dimensional Minkowski-spacetime coordinate and y is the coordinate of the extra dimension with 0 ≤ y ≤ L. Our choice of the 5d metric is η M N = diag(−1, 1, 1, 1, 1). We introduced the second term as a gauge fixing term and the third term as a kinetic term of ghost fields. The general class of boundary conditions for the gauge field is obtained from the action principle: We obtain the bulk field equation for A M , together with the following surface term from the first term of the action after taking the variation.
Since the boundary condition A µ = 0 at y = 0, L breaks 4d gauge symmetry explicitly, the general class of boundary conditions consistent with the 4d gauge invariance is given by the following: The BRST transformation leads us to BCs for the ghost field. The abelian gauge field A M and the ghost field c have a relation with each other through the Grassmann-odd BRST transformation This fact implies that ∂ y c (c) should obey the same boundary conditions as A y (A µ ). Thus we obtain the BCs for the ghost as The boundary condition for the anti-ghost fieldc can be derived from the action principle for the third term of the action. The variation for the third term produces the following surface term: c∂ y (δc) − (∂ yc )δc = 0 at y = 0, L.
Since c(x, y) obeys the boundary conditions (2.7), the following boundary condition should be imposed for the anti-ghost fieldc:

BCs for fermion
Next, we consider the BCs for the fermion with adding the following action to eq. (2.1): and Ψ is a 5d 4-component Dirac spinor. M F is a bulk mass of the fermion and we take the From the action principle δS F = 0, we obtain the following condition for the surface term: with the 5d Dirac equation, In terms of the chiral spinors Ψ R/L (Ψ = Ψ R + Ψ L ), which are defined as γ 5 Ψ R/L = ±Ψ R/L , we can rewrite the above equations as Since boundary conditions which consist of a linear combination of Ψ R and Ψ L break the 4d Lorentz invariance, the condition (2.16) should be reduced to the form which leads to the BCs: We should note that under the BC Ψ R = 0 (Ψ L = 0) at boundaries, the 5d Dirac equation automatically determines the BC for Ψ L (Ψ R ) as Thus we have the following four choices for the fermion BCs [25][26][27][28]: (2.23)

BCs for scalar field
Finally, we consider the general class of boundary conditions for a scalar field: where and Φ(x, y) denotes a 5d complex scalar field. As in the previous cases, we obtain the surface term from the action principle δS Φ = 0: Under the infinitesimal special variation δΦ = εΦ, we can rewrite the above surface term as where L 0 is an arbitral non-zero real constant, which possesses mass dimension −1. The above equation implies that Φ − iL 0 D y Φ and Φ + iL 0 D y Φ have a difference only up to a phase at the boundaries: With L + ≡ L 0 cot θ 0 2 and L − ≡ −L 0 cot θ L 2 , we obtain the general class of BCs for the scalar field [25][26][27][28], These boundary conditions are known as the Robin boundary condition. Note that the derived Robin boundary condition satisfies the condition (2.26) under the assumption that Φ and δΦ satisfy the same boundary condition.

4d spectrum
In the previous subsection, we investigated the general class of BCs for each field. Now, we derive the 4d spectrum of the gauge field and the fermion field under the derived boundary conditions, respectively. For the scalar field, we only investigate the vacuum expectation value for our purpose.

4d spectrum of Abelian gauge, ghost, and anti-ghost fields
First, we start from the abelian gauge, the ghost, and the anti-ghost fields. The action and the boundary conditions are given by eq. (2.1) and eqs. (2.5), (2.7), (2.9). The action S G can be rewritten as To obtain the 4d spectrum, we expand the fields as follows: Note that { f n }, {g n }, and {Ξ n } form complete sets, respectively, and can obey the orthonormal relations: Furthermore, { f n } and {g n } satisfy the quantum-mechanical supersymmetry (QM-SUSY) relations [46][47][48][49][50][51], D f n (y) = m n g n (y), D † g n (y) = m n f n (y).
A schematic figure of the 4d spectrum is depicted in Figure 1.

4d spectrum of fermion
Second, we investigate the 4d spectrum of the fermion on an interval. The action and BCs are given by eq. (2.10) and eq. (2.23). To evaluate the 4d spectrum of the fermion, we expand the fermion as and form complete sets. In the above, the operators D and D † are defined as Furthermore, {F (n) ψ R } and {G (n) ψ L } satisfy the QM-SUSY relations: (2.50) We can obtain the explicit forms of the wavefunctions after we solve the eigenvalue equations (2.47) while taking into account the BCs (2.23). However, we here concentrate on the existence of a chiral massless zero-mode and the form of its wavefunction. Zero-mode solutions are obtained from the QM-SUSY relations (2.50) with m ψ (0) = 0: The solutions of the above equations would be given as follows: Schematic figures of the zero-mode solutions are depicted in Figure 2. The zero-mode solution ψ L localizes to the boundary y = 0 (y = L) in the case of M F > 0 and localizes to y = L (y = 0) in the case of M F < 0.   It should be emphasized that the zero-mode solutions (2.53) (2.54) are consistent only with the type-(II) type-(I) BC given in (2.23) because of (2.21) and (2.22), respectively. Therefore we will concentrate on the type-(I) and type-(II) BCs in the following. The mass spectrum of both type-(I) and type-(II) is given by Inserting the mode expansions into the action and using the orthonormal relations of the mode functions, we have where for type-(I), for type-(II), (2.58) and A typical spectrum of the fermion is depicted in Figure 3. A chiral massless zero mode exists in the case of both type-(I) and type-(II).

Type-(I) case
Type-(II) case Figure 3: A typical mass spectrum of the fermion on an interval. Each black oval pair indicates a QM-SUSY pair to make a mass term.

Vacuum expectation value of the scalar
Finally, we comment on the vacuum expectation value of the scalar field. The action and the BCs are given by eq. (2.24) and eq. (2.30). It was found in Refs. [25,26] that under the Robin boundary condition (2.30), Φ(x, y) can possess a non-vanishing vacuum expectation value cn(y, a) is the Jacobi's elliptic function, and y 0 , Q are constants which are determined by the parameters L ± of the Robin BCs. Choosing suitable values of L ± , we can approximately take the form of the scalar VEV φ(y) as where A is a constant with mass dimension 3 2 .

Casimir energy and stability of the extra dimension
In the previous section, we succeeded in obtaining the 4d spectrum of the fields with the specified BCs. Taking the result into account, we evaluate the Casimir energy E[L] as a function of the length L of the extra dimension and show that the minimization of the Casimir energy provides a mechanism to stabilize the extra dimension.
For our purpose, we only concentrate on the gauge and fermion field contributions to the Casimir energy while ignoring the effect of the scalar field in this paper. We summarize the action and the BCs which we consider, We focus on the situation in which a chiral massless zero mode exists. As an example, we consider the type-(II) BC first. To evaluate the Casimir energy, we examine the partition The gauge field part of the partition function reads After moving to the Euclidian space, we obtain the Casimir energy of the gauge field: where and For further concrete discussions, we divide E U(1) [L] into two parts: Our interest is only in the L-dependence of the Casimir energy E U(1) [L] so that we simply ignore this part. We emphasize that this part actually does not affect any results of the L-dependence of the Casimir energy has an L-dependence and plays a crucial role when we discuss the L-dependence of the total Casimir energy. By using the formulas will help us to move on. Here, the index w is an integer which represents the winding number. By utilizing the Poisson summation formula, we obtain 20) and find that E U(1) part2 [L] contains a UV-divergence when t → 0. To remove this UV-divergence, we define the regularized total Casimir energy as We note that this regularization is equivalent simply to removing the w = 0 mode from the Casimir energy. The w 0 modes express winding modes and provide finite contributions to the L-dependence of the Casimir energy. On the other hand, w = 0 corresponds to an unwinding mode and it causes a UV-divergence. Since the regularized Casimir energy E U(1) part2 [L] reg. does not contain any unwinding mode, it has no UV-divergence and becomes finite. The explicit form where we performed the integration by substitution t ≡ w 2 L 2 t and used Γ( From the above analysis, we obtain the regularized Casimir energy E U(1) [L] reg. of the gauge field: (3.23) In the same way, we next evaluate the Casimir energy of the fermion with the type-(II) boundary condition. (It is found that the type-(I) boundary condition leads to the same conclusion as the type-(II) for the Casimir energy.) To move on, we introduce the chiral representation: The Gamma matrices are represented by and σ are Pauli matrices. The partition function of the fermion reads where the overall minus sign originates in the Grassmann property of fermions. After moving to the Euclidian space, we obtain the Casimir energy of the fermion: where into two parts as is the case of the gauge field: In the same way as the gauge field, E (F) part1 [M F , L] does not contain any L-dependence. Since we have an interest in the L-dependence of the Casimir energy, we just ignore this part.
can be also evaluated as the gauge field case. Using the formulas (3.15)-(3.17), (3.34) By using the Poisson summation formula (3.19), we obtain the following form for E (F) part2 [M F , L]: The regularized Casimir energy where the modified Bessel function is defined by Moreover, the modified Bessel function K D 2 (z) with D = odd integer can be expressed as Therefore the explicit form of E (F) part2 [M F , L] reg. is given by (3.40) From the analysis, we obtain the L-dependence of the regularized total Casimir energy of the fermion as (3.42)  We should give a comment for the above results. It was discussed in ref. [29] that, in the

Theory with point interactions
In the papers [26][27][28]52], a new way to produce generations and a mass hierarchy was proposed with introducing zero-width branes, so-called point interactions, to the extra dimension. In this section, we briefly review a theory with point interactions at first. In the theory, massless zero modes become degenerate and a nontrivial number of generations appears from a onegeneration 5d fermion (where a self-contained comprehensive review on the formulation is provided in Appendix A). In this section, we clarify the 4d mass spectrum of the theory with point interactions, which plays an important role in the calculation of the Casimir energy. where ε represents an infinitesimal positive constant. We should emphasize that the above BCs are consistent with the 5d gauge invariance since they are invariant under the 5d gauge transformation:

BCs and 4d mass spectrum
We expand a 5d fermion Ψ(x, y) with the BCs (4.1) or (4.2): It was found in Refs. [26][27][28] that we have three degenerate zero modes i,ψ R with i = 1, 2, 3 under the BC (4.1) the BC (4.2) and can obtain three degenerate massless chiral fermions where i,ψ L (y) and F (0) i,ψ R (y) are given by and θ(y) is the step function. Schematic figures of the localized zero modes G (0) i,ψ L (y) and F (0) i,ψ R (y) are depicted in Figure 8 and Figure 9. Each zero mode only lives in a segment and localizes to a boundary.
After substituting eq. (4.6) into the action (2.10) and using the orthonormal relations i,ψ L (y) only has a non-vanishing value within the segment L i−1 < y < L i and localizes to a boundary.
we obtain the 4d spectrum of the fermion, for the BC (4.2). (4.14) and the 4d mass spectrum m i,ψ (n) is given by where l i is defined by eq. (4.10).

Dynamical generation of fermion mass hierarchy
In this section, by using the previous results, we consider an SU(2) × U(1) model with a single generation of 5d fermions, which produces three generations of 4d chiral fermions by the point interactions, and discuss whether the model can dynamically generate a fermion mass hierarchy.
To this end, we first set an action and BCs of this model. The action consists of an SU(2) gauge i,ψ R (y) only has a non-vanishing value within the segment L i−1 < y < L i and localizes to a boundary. field, a U(1) gauge field, a single generation SU(2) doublet fermion, a single generation SU(2) singlet fermion, and an SU(2) doublet scalar field. The contents of our model mimic those of the SM without the color degree of freedom, where the U(1) (hyper)charges of Q and U take those of the quark doublet and the up-type singlet. Extra BCs via point interactions are a key ingredient to produce the three generations from one generation 5d fermion as we reviewed in Section 4. The positions of the point interactions crucially affect the fermion mass hierarchy through the overlap integrals, as we will see in Section 5.3. We will show that the positions of the point interactions can be determined dynamically through the minimization of the Casimir energy and then find that an exponential fermion mass hierarchy naturally appears. Following the results, we discuss the stability of the extra dimension.

Action and BCs
We start with the following action for the gauge fields and fermions: W a M , A M , c a , c andc a ,c denote an SU(2) gauge, a U(1) gauge, ghost and anti-ghost fields, respectively. g and g denote SU(2) and U(1) couplings of the SU(2) doublet fermion. Q and U indicate an SU(2) doublet fermion and an SU(2) single fermion, respectively. A bulk mass of the 5d fermion is denoted by M (Ψ) F (Ψ = Q, U). ε abc is a complete antisymmetric tensor and T a is a generator of SU(2) acts on a fundamental representation, which satisfies the following algebra and the orthogonal relation: According to the analysis given in Section 2, we choose boundary conditions for the fields as follows:

Determination of the positions of the point interactions
With the fixed length L, the minimization condition for the Casimir energy can determine the values of the parameters {L 1 , L 2 }. The above potential turns out to have the finite global To verify this statement, we consider the following function I(x, y, z): x, y, z > 0, (5.18) x + y + z = 1. I(x, y, z) imitates the function form of the fermion Casimir energy with the variables x = L 1 , We assume the function f (x) to be a monotonically decreasing function and also f (x) ≡ df (x) dx to be a monotonically increasing one with lim x→0 f (x) = +∞. We note that the fermion Casimir energy (3.41) turns out to satisfy those assumptions (see Figures 4 and 5). Substituting the condition eq. (5.19) into eq. (5.17), we obtain To investigate an extreme value of the above function, we examine ∂I ∂ x and ∂I ∂ y : From the conditions ∂I ∂ x = 0 and ∂I ∂ y = 0, we obtain the result Since we assumed that f (x) is a monotonically increasing function, the result (5.23) can be realized only when Thus we find that I(x, y, z) has an extreme value when x = y = z = 1 3 . Moreover, the function takes a local minimum at x = y = z = 1 3 . To show this, we consider the second-order differentials with the condition x = y = z = 1 3 : We now consider the Hessian matrix M: Since f (x) is a monotonically increasing function, f (x) > 0. Thus we find that The above results imply that the eigenvalues of the matrix M are positive and hence that the is a local minimum of the potential. Moreover, there is no other stationary point, we found that the position x = y = 1 3 is a global minimum of the function I(x, y, z).
From the above discussions, we conclude that the Casimir energy (5.16) has a global minimum

Fermion mass hierarchy
Under the above situation, we can produce the fermion mass hierarchy dynamically by introducing the Yukawa coupling to an SU(2) doublet scalar field Φ(x, y), which possesses the y-dependent VEV 4 A schematic figure is depicted in Figure 11. Since the minimization of the Casimir energy determines the positions of the point interactions as to make the distances between them equal, the exponential VEV of the scalar field makes an exponential mass hierarchy such as

Stability of the extra dimension
We have shown that for any fixed length L, the positions of the point interactions are determined dynamically to the value L 1 = L 3 , L 2 = 2L 3 from the minimization of the Casimir energy. Under this situation, we discuss the stability of the whole extra dimension. In our model the SU (2) doublet scalar field Φ(x, y) possesses the y-dependent VEV and breaks the gauge symmetry as SU(2) × U(1) → U(1) . Therefore, we will discuss the stability of the extra dimension in the broken phase.
As we investigated in Section 3, the extra dimension can be stabilized if the following two conditions are satisfied: (i) 5d massless gauge bosons exist and all 5d fermions have nonzero bulk masses. (ii) The degrees of freedom of fermions are sufficiently larger than those of bosons. The first condition (i) will ensure that the Casimir energy approaches to zero with negative values in L → ∞ limit, as in (3.42). The second condition (ii) will ensure that the Casimir energy goes to +∞ in L → 0 limit, as in (3.42).
In our model, the SU(2) × U(1) gauge symmetry is broken by the VEV of the scalar but a subgroup U(1) is still unbroken. Thus, the first condition (i) is satisfied in our model. The second condition (ii) seems to be satisfied in our model because the degrees of freedom of the fermions become three times the number of 5d fermions due to the point interactions. Moreover, there is still room for introducing extra fermions by using the type-(III) BCs, which do not produce any exotic chiral massless fermions. Therefore, in our setup, the extra dimension is expected to be stabilized by the Casimir energy. 6

Conclusion and Discussion
In this paper, we proposed a new mechanism to produce a fermion mass hierarchy dynamically We give a comment for the contribution of the scalar field to the Casimir energy at first. In this paper, we ignored the effect of the scalar field to the Casimir energy for simplicity because the contribution to the Casimir energy from the scalar field will have no exact analytic expression due to the Robin BC. However, the inclusion of the scalar field will not change the conclusions about the stability of the whole extra dimension and the positions of the point interactions if the degrees of freedom of the fermions are sufficiently larger than those of bosons.
Next, some comments are given to the flavor mixing of the fermions. In our model, we introduced the point interactions at y = L 1 , L 2 for both the SU(2) doublet and the singlet fermions. Here, mass matrices are diagonal and flavor mixing cannot appear. In general, however, there is no need to share the point interactions in fermions so that we can introduce the individual point interactions to each fermion, respectively, which means that (e.g.) the SU (2) doublet fermion feels the point interactions at y = L 1 , L 2 and the SU(2) singlet fermion feels the point interactions at y = L 1 , L 2 [26][27][28]. Then the mode functions of the SU(2)-doublet zero mode G (0) i,Q L (y) and the SU(2)-singlet zero mode F (0) j,U R (y) may have an overlap for i j. In other words, off diagonal components may appear in the mass matrix as and a flavor mixing can be realized.
If the minimization of the Casimir energy determines the positions of the point interactions as L 1 = L 1 , L 2 = L 2 , flavor mixing does not appear so that we need an idea to make L 1 L 1 , L 2 L 2 . One way to avoid the situation of L i = L i is to consider higher loop effects of the Casimir energy, which may make L 1 L 1 , L 2 L 2 through the interactions. To introduce exotic 5d fermions, where they contribute to the Casimir energy, is another way. Finally, we focus on the gauge universality. It was pointed out in Refs. [26][27][28] that the gauge symmetry breaking due to the y-dependent VEV of the scalar field would cause a gauge universality violation. That is because the y-dependent VEV of the scalar modifies the flat profile of the zero mode function of the gauge boson and thereby the values of the 4d gauge couplings change with respect to the generations through the overlap integrals. A way to avoid this crisis is to introduce two scalar fields; one is an SU(2) doublet scalar and another is a gauge-singlet scalar field. In the situation that the constant VEV of the SU(2) doublet scalar breaks the gauge symmetry and the y-dependent VEV of the gauge-singlet scalar provides a mass hierarchy, we can avoid the gauge universality violation. It would be of great interest to construct a more phenomenologically viable model along the lines discussed in this paper.

A Review of point interactions and derivation of fermion profiles under the existence of one point interaction in the bulk of an interval
In this Appendix, we first review a one-dimensional quantum mechanical system with a point interaction, and then apply the formulation for the five-dimensional Dirac action on an interval with a point interaction.
The well-known Dirac δ-function potential in quantum mechanics is an example of the point interaction, i.e., the interaction of zero range. The consistent manner to treat such a singularity has been given in Ref. [53]. According to the formulation [53], we regard a point interaction as an idealized long wavelength or infrared limit of localized interactions in one dimension, and hence it is a singular interaction with zero range at one point, say y = L 1 on a line R. A system with such an interaction can be described by the system on the line with the singular point removed, namely, on R\{L 1 }. In order to construct a quantum system on the domain D = R\{L 1 }, we require that the probability current j y (y) = −i (∂ y ϕ * )ϕ − ϕ * (∂ y ϕ) (y) is continuous around the singular point, i.e. [53] j y (L 1 − ε) = j y (L 1 + ε) , where ε represents an infinitesimal positive constant. We note that the above probability conservation guarantees the Hemiticity of the Hamiltonian.
The requirement (A.1) implies that any state in the domain D must obey a certain set of BCs at y = L 1 ± ε. For example, the Dirichlet BC satisfies the condition (A.1). The Dirichlet BC may be understood as a point interaction given by the Dirac δ-function potential V(y) = αδ(y) with the limit of the coupling α → ∞. Another type of boundary conditions, which also satisfies the condition (A.1), is known as the Robin BC, where ϕ (y) = ∂ϕ(y) ∂ y and M is a constant parameter with mass-dimension one. The above two types of BCs will become important in our analysis.
Interestingly, point interactions can appear not only in one-dimensional quantum mechanics but also in extra dimension scenarios [26], since finite-range-localized interactions will be regarded as the point interaction in an idealized long wavelength or infrared limit, like a domain wall potential or a brane in extra dimension models. Hence, as we will see below, we apply the above point interaction treatment to the five-dimensional Dirac action on an interval (y ∈ [0, L]) with a point interaction at y = L 1 , and explain how to decompose a five-dimensional Dirac field Ψ(x, y) into KK mass eigenmodes, in a self-contained way.
The five-dimensional free action for Ψ(x, y) that we focus on is Since the current form ΨΓ y Ψ is equivalent to Ψ L Γ y Ψ R + Ψ R Γ y Ψ L , the Dirichlet BC is found to satisfy the conditions (A.7) and (A.8). Thus, in the following analysis, we take the The consistency requirement will be obtained from the action principle δS = 0 [54]. In this case, the derived conditions are given by ΨΓ y δΨ y=0 = 0 = ΨΓ y δΨ y=L , (A.5) ΨΓ y δΨ y=L 1 −ε = ΨΓ y δΨ y=L 1 +ε , (A. 6) where δΨ means the variation of Ψ. Since Ψ and δΨ can be regarded as independent fields with the assumption that Ψ and δΨ take the same boundary conditions, it seems that the above conditions are more restrictive than those of (A.7) and (A.8). However, they turn out to reduce the same conclusions in our analysis given below, and in fact the Dirichlet BC (A.10) satisfies (A.5) and (A.6). The above conditions have been analyzed in Ref. [55] (see also [25,56,57]) and we will not discuss (A.5) and (A.6) here.
We should emphasize that once the BCs for the right-handed part of Ψ are fixed as above, the boundary conditions for the opposite chirality, i.e. the left-handed part of Ψ are automatically determined through the equation of motion as 8 It is worthwhile noticing that wavefunctions Ψ R/L (y) and/or their derivatives Ψ R/L (y) will become discontinuous at y = L 1 , as we will see later, because the continuity conditions of To perform the KK decomposition of the 5d fields Ψ L (x, y) and Ψ R (x, y), let us consider the following one-dimensional eigenvalue equations: with the BCs where the one-dimensional domain D is defined by The eigenfunctions of the equations (A.13) and (A.14) with the BCs (A.15) and (A.16) are found to be of the form 8 We cannot impose the Dirichlet BC for both Ψ R and Ψ L at a boundary because it is enough for Ψ R = 0 or Ψ L = 0 to satisfy the conditions (A.7) and (A.8), and in fact the requirement Ψ R = Ψ L = 0 at a boundary is overconstrained.
with n = 1, 2, · · · . We notice that even though the eigenfunctions G (n) 1,ψ L (y) and F (n ) 1,ψ R (y) G (n) 2,ψ L (y) and F (n ) 2,ψ R (y) entirely vanish on D 2 (on D 1 ), they are well defined on the whole domain D and satisfy the eigenvalue equations 26) and the BCs for n = 0, 1, 2, · · · , n = 1, 2, · · · , i = 1, 2 with the eigenvalues It should be emphasized that no eigenfunctions with non-zero eigenvalues take non-trivial values on both D 1 and D 2 . This is because there is no degeneracy for non-zero eigenvalues i.e. m i,ψ (n) m i ,ψ (n ) if n n or i i (except for n = n = 0), as long as L 1 is not equal to L/2. Hence, any linear combination of G (n) 1,ψ L (y) and G (n) 2,ψ L (y) for n 0 F (n ) 1,ψ R (y) and F (n ) 2,ψ R (y) cannot become a solution to the eigenvalue equation (A.13) ((A.14)).
The above observation shows that the five-dimensional fields Ψ L (x, y) and Ψ R (x, y) with the BCs can be decomposed, without any loss of generality, as where the coefficients of the decompositions ψ (n) i,L (x) and ψ (n) i,R (x) correspond to four-dimensional left-handed and right-handed chiral fermions, respectively.
• A simple generalization with multiple point interactions can be analyzed straightforwardly.
Especially, the case with two point interactions is attractive since threefold-degenerated chiral zero modes are realized (see Ref. [26]).
• An intrinsic profile of point interaction(s) can be arranged for each 5d fermion field individually. This property is one of the key ingredients of the flavor model proposed in Ref. [26]. At the two end points to the contrary, BCs should be arranged for all of the fields living in the bulk since the points are kinds of singularities on the background space.
• Since we removed the singular point from the interval, according to the formulation [53], no contribution via 'brane-localized terms' emerges in integrations along the y direction in the formulation. 1,ψ L ≡ a G (0) 1,ψ L + b G (0) 2,ψ L and G (0) 2,ψ L ≡ a G (0) 1,ψ L + b G (0) 2,ψ L (assuming a, a , b, b being real). Even after imposing the orthonormality condition in (A.32) in the set G (0) i,ψ L , G (n ) i,ψ L ; n = 1, 2, · · · , i = 1, 2 instead of G (n) i,ψ L ; n = 0, 1, 2, · · · , i = 1, 2 , one real degree of freedom remains to be unfixed, where we obtain a series of the zero-mode eigenfunctions parametrized by the remaining real degree of freedom. We focus on a concrete expression, Situations are changed when we switch on interactions, which includes mass perturbation.
When we introduce an additional mass term as mass perturbation, and if consequently the degeneracy is resolved, no redundancy remains in the form of the mass eigenstates. 9 Here, the rotational degree of freedom does not change the mass eigenvalues after the perturbation and does not affect physics (even though the diagonalizing matrix depends on θ). To take the simplest choice θ = 0 makes analyses transparent.
• It is noted that we can introduce point interactions in orbifolds. Here, we sketch how to obtain twofold degenerated localized chiral zero modes in the geometry of S 1 /Z 2 , where we consider that the fundamental region of y is [0 + ε, L], which is shrunken from that of where (y) represents the sign function, which is a compulsory factor to make the mass term Z 2 invariant (see e.g. [58]). Here, we select η Ψ = −1 for realizing left-handed chiral zero mode and introduce a point interaction at y = L 1 which put the additional BC for Ψ(x, y) as Ψ R = 0 at y = L 1 − ε, L 1 + ε . the whole system remains to be Z 2 symmetric.