Family of Singular Solutions in A SUSY Bulk-Boundary System

A set of classical solutions of a singular type is found in a 5D SUSY bulk-boundary system. The"parallel"configuration, where the whole components of fields or branes are parallel in the iso-space, naturally appears. It has three {\it free} parameters related to the {\it scale freedom} in the choice of the brane-matter sources and the {\it"free"wave} property of the {\it extra component} of the bulk-vector field. The solutions describe brane, anti-brane and brane-anti-brane configurations depending on the parameter choice. Some solutions describe the localization behaviour even after the non-compact limit of the extra space. Stableness is assured. Their meaning in the brane world physics is examined in relation to the stableness, localization, non-singular (kink) solution and the bulk Higgs mechanism.

This is a typical model of the spontaneous symmetry breaking. The symmetry, in this simple example, is Z 2 -symmetry (a discrete symmetry): y ↔ −y.
The stableness is guaranteed by that the kink solution connects two degenerate vacua: φ = φ 0 at y = ∞ and φ = −φ 0 at y = −∞. For the static configuration on this background, the leading value of the action, S, in the "thin-wall" limit (kL ≫ 1, L: infrared regularization parameter of y-axis) : φ kink ∼ φ 0ǫ (ky), ∂ y φ kink ∼ 2φ 0δ (y) ,is estimated as, where the infrared regularization parameters, L and T , are introduced: −T /2 < t < T /2, −L/2 < y < L/2.ǫ(y) andδ(y) are the ordinary (non-periodic) sign and delta functions respectively. 3 In the recent development of the brane world, it has become clear that the kink-type configuration plays a very important role in the extra-space behaviour of the higher dimensional models. This is because it describes the stable localization configuration. In the Randall-Sundrum model I(wall-anti-wall model) [2], they considered the following bulk-boundary theory in the AdS 5 space-time on S 1 /Z 2 orbifold.
where Λ, M, V hid(vis) are 5D cosmological constant, 5D Planck mass and the brane tension at x 5 = 0(π). δ(s) is the periodic delta function. The Einstein 3ǫ (y) andδ(y) are defined bỹ These should be compared with periodic ones, ǫ(y) and δ(y), used later. equation and Z 2 -symmetry (even) of σ(x 5 ) requires where k is a scale with mass dimension. They applied this result to the mass hierarchy problem and give rich possibilities in the unified models. In the Randall-Sundrum model II [3] (one wall model), partly from the stability assurance, they considered the l → ∞ limit of the model I.
In this model, the stability is guaranteed by the same reason as the first example of the kink solution. 4 In fact the above solution can be obtained by the "thinwall" limit of the generalized kink solution in the bulk Higgs model [5,6].
This model makes it possible to treat the brane system in the non-singular way. Both models explained above are non-supersymmetric. The first one is a flat theory, whereas the second one is the model of a 5D gravitational space-time (AdS 5 ). The "curvedness" simply comes from the "warped" factor, e −2σ . For a fixed x 5 -slice, the 4D space-time is the flat one. The dilaton (σ) field part controls the bulk-scale of the each 4D Minkowski (flat) slice at the point x 5 . In the present paper, we examine a 5D SUSY flat theory where two scalar fields , which come from the 5D SUSY multiplet, play the similar role to the scalar and dilaton fields in the above two examples.
2 Mirabelli-Peskin Model Inspired by the Horava-Witten model [7] (11D supergravity on S 1 /Z 2 -orbifold, the strong coupling limit of the 10D heterotic string theory), Mirabelli and Peskin [8] proposed a field theory model describing a bulk-boundary system which mimics the brane(-anti-brane) configuration in the string theory. Let us consider the 5 dimensional flat space-time with the signature (-1,1,1,1,1). 5 The space of the fifth component is taken to be S 1 , with the periodicity 2l, and has the Z 2 -orbifold condition.
4 Another way out was suggeted in [2] and was analysed by Goldberger and Wise [4]. They try to stabilize the system, keeping the compact extra-space, by regarding the length parameter l as an expectation value of some scalar field (radion). 5 Notation is basically the same as ref. [9].
We take a 5D bulk theory L bulk which is coupled with a 4D matter theory L bnd on a "wall" at x 5 = 0 and with L ′ bnd on the other "wall" at x 5 = l.
The bulk dynamics is given by the 5D super YM theory which is made of a vector field A M (M = 0, 1, 2, 3, 5), a scalar field Φ, a doublet of symplectic Majorana fields λ i (i = 1, 2), and a triplet of auxiliary scalar fields X a (a = 1, 2, 3): where all bulk fields are the adjoint representation (suffixes: α, β, · · ·) of the gauge group G. The bulk Lagrangian L SY M is invariant under the 5D SUSY transformation. This system has the symmetry of 8 real super charges.
It is known that we can consistently project out N = 1 SUSY multiplet, which has 4 real super charges, by assigning Z 2 -parity to all fields in accordance with the 5D SUSY. A consistent choice is given as: plays the role of D-field on the wall. We introduce one 4D chiral multiplet (φ, ψ, F ) on the x 5 = 0 wall and the other one (φ ′ , ψ ′ , F ′ ) on the x 5 = l wall: complex scalar fields φ, φ ′ , Weyl spinors ψ, ψ ′ , and auxiliary fields of complex scalar F, F ′ . These are the simplest matter candidates and were taken in the original theory [8]. Using the N = 1 SUSY property of the fields (A m , λ L , X 3 − ∇ 5 Φ), we can find the following bulk-boundary coupling on the x 5 = 0 wall. where We take the fundamental representation for Θ = (φ, ψ, F ). The quadratic (kinetic) terms of the vector A m , the gaugino spinor λ L and the 'auxiliary' field D = X 3 − ∇ 5 Φ are in the bulk world. In the same way we introduce the coupling between the matter fields (φ ′ , ψ ′ , F ′ ) on the x 5 = l wall and the bulk fields: (12)). We note the interaction between the bulk fields and the boundary ones is definitely fixed from SUSY.
3 Vacuum of Mirabelli-Peskin Model We now examine the vacuum structure. Generally the vacuum is determined by the potential part of scalar fields. We first reduce the previous system to the part which involves only scalar fields or the extra component of the bulk vector.
where we have dropped terms of 2tr The field c is the ghost field which is introduced in the usual procedure of fixing the gauge freedom of L SY M . While L bnd , on the x 5 = 0 wall, reduces to α ′ , β ′ · · · are the suffixes of the fundamental representation. In the same way, on the x 5 = l wall by replacing, in (14), φ and φ † by φ ′ and φ ′ † , respectively. The vacuum is usually obtained by the constant solution of the scalar-part field equation. In higher dimensional models, however, extra-coordinate(s) can be regarded as parameter(s) which should be separately treated from the 4D space-time coordinates. In this standpoint, it is the more general treatment of the vacuum that we allow the x 5 -dependence on the bulk-part of the solution. We generally call the classical solutions (ϕ, χ 3 , a 5 ; η, η ′ , f, f ′ ) the background fields. 6 They satisfy the field equations derived from (13) and (14) are given by, for the bulk-fields variation, where The field equations for the boundary-fields part are given by where d α = (χ 3 − ∂ 5 ϕ + ga 5 × ϕ) α is the background D-field. From the equation (17), we obtain Then we know Before systematically solving the equations above, we note a simple structure involved in them. Under the "parallel" circumstance, a 5α ∝ ϕ α ∝ η † T α η ∝ η ′ † T α η ′ , the equations for δΦ α (15) and δA 5α (16) are The first one is a static wave equation with "source" fields located at x 5 = 0 and l. It is easily integrated once.
This result was used in the original paper [8]. The second equation of (22) is a (static) "free" wave equation (no source fields). a 5α do not receive, in the "parallel" environment, any effect from the boundary sources η, η ′ . This characteristically shows the difference between the bulk scalar Φ α and the extra component of the bulk vector A 5α in the vacuum configuration. We first solve (15), (16) and (17) with respect to a 5α and ϕ α . They also give the solutions for χ 3 α and d α = Z α . Using these results we solve (18) and (19) with respect to η, η ′ , f and f ′ for given values of m α ′ β ′ and λ α ′ β ′ γ ′ . Here we seek a natural solution by requiring that d α is independent of x 5 .
Then, from the equation of (15), we have Z × a 5 = 0. It says that we may consider the three cases : 1) a 5α = 0, 2) Z α = 0, 3) a 5α ∝ Z α ( = 0). It turns out that the case 3) includes the case 1) and 2). Hence we explain case 3). Before proceeding the analysis furthermore, we note here a mathematical fact about the solution of the "free" field equation in S 1 /Z 2 space. A) Z 2 -odd The two independent solutions are given by the periodic sign function (see Fig.1) and the sawtooth-wave function (see Fig.2), Both functions are piece-wise continuous. A useful relation is [y − l] p = [y] p − lǫ(y). Their derivatives are given by BB − type : dǫ(y) dy = 2(δ(y) − δ(y − l)) , where δ(y) is the periodic (periodicity 2l) delta function. We have named the above three distributions Brane-Anti-Brane(BB), Anti-Brane(B) and Brane(B) respectively for a later purpose. See Fig.3.

B) Z 2 -even
The two independent solutions are given by the identity function (see Fig.4), and the periodic "absolute-linear" function (see    Both functions are continuous. The first one is smooth and the second one is piece-wise smooth. Their derivatives are given by di(y) dy = 0 , dv(y) dy = ǫ(y) .
The even solution v(y) appears as the dilaton in the Randall-Sundrum of Sec.1.
In the first example of Sec.1 and in the present model, the odd ones appear.
The mathematical fact explained above shows the important connection among the brane configuration, the boundary condition and Z 2 -symmetry. Let us examine the case 3) a 5α ∝ Z α ( = 0). Noting (24), we may put the following forms for Z α and a 5α .
whereZ α andā α are constants and j(x 5 ) is a function of x 5 which is to be specified below. The first equation of (32) says The equation (16) saysā First we solve (33) with the requirement: whereφ α is a constant and h(x 5 ) is a function of x 5 to be determined. The second relation says the two scalars, a 5 and ϕ, are (anti)parallel in the iso-space. Then (33) reduces to From the equation (15), h satisfies the "free" field equation except the fixed points. Hence we have where c 1 is a free parameter. Next we solve (34). Because ϕ α =φ α h(x 5 ) ∝Z α , the equation reduces to the "free" one: The solution is given by where c 2 is another free parameter. Because ∂ 5 2 j = −2lc 2 δ ′ (x 5 − l) − 2lδ ′ (x 5 ), the solution j of (39), by itself, does not satisfy (38) on the points x 5 = 0, l. In order to correct it, we must require the variation δA 5α , on the points x 5 = 0, l, to satisfy the Neumann boundary condition(the second relation of (39)). This condition "absorbs" the singularities appearing in the variation equation (used to derive the field equation) at the points x 5 = 0, l and makes the "free" wave property (38) consistent everywhere in the extra space.
Summarizing the case 3) solution, we have with the boundary condition: where c 1 , c 2 and c 3 are three free parameters. The meaning of c 1 is the scale freedom in the "parallel" condition of brane sources η ′ † T α η ′ ∝ η † T α η , and that of c 2 and c 3 is the "free" wave property of a 5α . The bulk scalar configuration influences the boundary source fields through the parameter c 1 , whereas the bulk vector (5th component) does not have such effect. Instead the latter one satisfies the field equation only within the restricted variation (Neumann boundary condition).
This solution includes the cases 1) and 2) as described below. Some special cases are listed as follows.
(3A) c 3 = 0 This is the case 1). There are some special cases.
3A-a) Another special cases are given by fixing two parameters, c 1 and c 2 (keeping the c 3 -freedom), as shown in Table 1. Table 1 Various vacuum configurations of the Mirabelli-Peskin model. BB,B and B correspond to brane-anti-brane, anti-brane and brane respectively. See Fig.3.
We see the bulk scalar Φ is localized on the wall(s) where the source(s) exists, whereas the extra component of the bulk vector A 5 on the wall(s) where the Neumann boundary condition is imposed. The two cases, (c 1 = −1, c 2 = −1) and (c 1 = 1/0, c 2 = 1/0), are treated in [11].
4 Fermion Localization, Stability, and Bulk Higgs Mechanism The vacuum is basically determined by the scalar fields as explained so far. Let us examine the small fluctuation of bulk fermions (gauginos) around the background solution obtained previously. We take (c 1 = −1, c 2 = −1) solution as a representative one. We assume η = 0, η ′ = 0. The relevant part of the Lagrangian We consider a simple case of G=U(1). The field equation for λ L is given by (The same thing can be said for λ R .) We focus on the fermion zero-mode with chirality ±1: λ L = σ(x m )ω(x 5 ), γ m ∂ m σ = 0, γ 5 σ = ±σ. Then the extra-space behaviour ω(x 5 ) is obtained as As far as 1 ± c 3 > 0, the fermion zero mode is localized around the brane. (If we require fermions with both chiralities to be localized, we must choose the parameter c 3 as −1 < c 3 < 1.) In the present approach, (N = 1)SUSY is basically respected. If SUSY is preserved, the solutions obtained previously are expected to be stable, because the force between branes (Casimir force) vanish from the symmetry. In some cases, we can more strongly confirm the stableness from the topology (or index) as follows. We can regard the extra-space size (S 1 radius) l as an infrared regularization parameter for the non-compact extra-space R(−∞ < y < ∞). By letting l → +∞ in the previous result, we can obtain the vacuum solutions in this case. First we note ǫ(y) →ǫ(y) , An interesting case is the l → ∞ limit of (c 1 = −1, c 2 = −1) in Table 1.