Brane-Anti-Brane Solution and SUSY Effective Potential in Five Dimensional Mirabelli-Peskin Model

A localized configuration is found in the 5D bulk-boundary theory on an $S_1/Z_2$ orbifold model of Mirabelli-Peskin. A bulk scalar and the extra (fifth) component of the bulk vector constitute the configuration. $\Ncal=1$ SUSY is preserved. The effective potential of the SUSY theory is obtained using the background field method. The vacuum is treated in a general way by allowing its dependence on the extra coordinate. Taking into account the {\it supersymmetric boundary condition}, the 1-loop full potential is obtained. The scalar-loop contribution to the Casimir energy is also obtained. Especially we find a {\it new} type which depends on the brane configuration parameters besides the $S_1$ periodicity parameter.

1 Introduction Through the development of the recent several years, it looks that the higher-dimensional approach begins to obtain the citizenship as an important building tool in constructing a unified theory. Among many ideas in this approach, the system of bulk and boundary theories becomes a fascinating model of the unification. The boundary is regarded as our 4D world. It is inspired by the M, string and D-brane theories [1]. One pioneering paper, giving a concrete field-theory realization, is that by Mirabelli and Peskin [2]. They consider 5D supersymmetric Yang-Mills theory with a boundary matter. The boundary couplings with the bulk world are uniquely fixed by the SUSY requirement. They demonstrated some consistency of the bulk and boundary quantum effects by calculating self-energy of the scalar matter field. Here we examine the vacuum configuration and the effective potential.
Contrary to the motivation of ref. [2], we do not seek the SUSY breaking mechanism, rather we make use of the SUSY-invariance properties in order to make the problem as simple as possible. The SUSY symmetry is so restrictive that we only need to calculate some small portion of all possible diagrams.
In the calculation of the effective potential of the 5D model, we recall that of the Kaluza-Klein model. The dynamics quantumly produces the effective potential which describes the Casimir effect [3,4]. The situation, however, is different from the present case in the following points: 1) the 4D reduction mechanism; 2) Z 2 -symmetry; 3) treatment of the vacuum with respect to the extra-coordinate dependence; 4) supersymmetry; 5) characteristic length scales. We will compare the present result with the KK case.
2 Mirabelli-Peskin Model Let us consider the 5 dimensional flat space-time with the signature (-1,1,1,1,1). 3 The space of the fifth component is taken to be (S 1 ), with the periodicity 2l, and has the Z 2 -orbifold condition.
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 boundary Lagragians are, in the bulk action, described by the delta-functions along the extra axis x 5 .
We consider both bulk and boundary quantum effects. 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 (its 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. As the 5D gauge-fixing term, we take the Feynman gauge: The corresponding ghost Lagrangian is given by where c andc are the complex ghost fields. We take the following bulk action.
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: P = +1 for A m , λ L , X 3 ; P = −1 for A 5 , Φ, λ R , X 1 , X 2 (m = 0, 1, 2, 3). Then (A m , λ L , X 3 − ∇ 5 Φ) constitute an N = 1 vector multiplet. Especially D ≡ X 3 − ∇ 5 Φ plays the role of D-field on the wall. We introduce one 4 dim 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 [2]. 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 φ, φ † . 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: (7)). We note the interaction between the bulk fields and the boundary ones is definitely fixed from SUSY.
3 SUSY Boundary Condition, Background Expansion and Generalized vacuum First we point out an important fact about the SUSY effective potential. The 1-loop SUSY effective potential can be calculated only by the scalar loop 4 up to the F -and D-independent terms in the off-shell treatment. If we trace the origin of this phenomenon, it is simply that the auxiliary fields have the higher physical dimension of M 2 . They cannot have the Yukawa coupling with fermions and vectors. F and D-dependence in the SUSY effective potential is very important to determine the vacuum behaviour. The above fact means that dV ef f 1−loop /dD ( or dV ef f 1−loop /dF ) is definitely determined only by the scalar loop. Miller [6,7], using the above fact, obtained F-tadpole or D-tadpole [8] (F and Dtadpole correspond to dV ef f 1−loop /dF and dV ef f 1−loop /dD, respectively.) in general 4D SUSY theories. He noticed, if the theory preserve SUSY at the quantum level, the F and D-independent parts in V ef f 1−loop can be obtained, instead of calculating diagrams, by a boundary condition on the effective potential. This is because, in the SUSY-preserving case, the effective potential should satisfy: V ef f (F = 0, D = 0) = 0 -supersymmetric boundary condition-. He confirmed the correctness by comparing his results with the results in the ordinary method. (See ref. [9] for an application to unified models.) We follow Miller's idea.
Hence we may put, for the purpose of obtaining the 1-loop SUSY effective potential, the following conditions: Here the extra (fifth) component of the bulk vector A 5 does not taken to be zero because it is regarded as a 4D scalar on the wall. The extra coordinate x 5 is regarded as a parameter. Then L blk reduces to where we have dropped terms of 2tr where we have dropped F † F -terms as the irrelevant terms. α ′ , β ′ are the suffixes of the fundamental representation. In the same way, we obtain L red bnd on the x 5 = l wall. Now we take the background-field method [10,11,12] to obtain the effective potential. We expand all scalar fields (Φ, X 3 , A 5 ; φ, φ ′ ), except ghosts, into the quantum fields (which are denoted again by the same symbols) and the background fields (ϕ, χ 3 , a 5 ; η, η ′ ).
We treat the ghosts c andc as quantum fields.
We state a new point in the present use of the background-field method. Usually we take the following procedure in order to obtain the vacuum [13].
[Ordinary procedure of the vacuum search] 1) First we obtain the effective potential assuming the scalar property of the vacuum (as described in (8)) and the constancy of the scalar vacuum expectation values.
2) Then we take the minimum of the effective potential.
In the present case, however, we have the extra coordinate x 5 . We have "freedom" in the treatment of the vacuum expectation values because x 5 is regarded as a simple parameter. We require that the background fields may be constant only in 4D world, not necessarily in 5D world. We may allow the background fields to depend on the extra coordinate x 5 . This standpoint gives us an interesting possibility to the higher dimensional model and generalizes the vacuum of the system.
When the background fields (ϕ, χ 3 , a 5 ; η, η ′ ) satisfy the field equations derived from (9) and (10), we say they satisfy the on-shell condition. The equations are , in the order of the variations (δΦ α , δA 5α , δχ 3 α , δφ † α ′ , δφ ′ † α ′ ), respectively given as, where we assume, based on the standpoint of the previous paragraph, ϕ = ϕ(x 5 ), In the above derivation, we use the fact that total divergences, in the action, vanish from the periodicity condition. Because we seek the effective potential (an off-shell quantity), we generally do not need to assume the above on-shell condition. 5 The quadratic part w.r.t. the quantum fields (Φ, X 3 , A 5 ; φ, φ ′ ) give us the 1-loop quantum effect. This part is given as where d α = χ 3 α − ∇ 5 ϕ α is the background (4 dimensional) D-field and We obtain the final "1-loop Lagrangian", necessary for the present purpose, as where δ(x 5 − l) part is dropped because we need not to consider the quantum propagation in the x 5 = l brane. 6 4 Mass-Matrix and the Localized Background Configuration We are now ready for the full ( with respect to the coupling order) calculation of the 1loop (we call this "1-loop full") effective potential. The "1-loop action" can be expressed as where S ghost is decoupled from others, and the components M ′ s are read from (14). Now we restrict the form of the background fields in the present 5D approach. The relevant scalars are a 5 and ϕ in the bulk. We should take into account the x 5 -dependence and the Z 2 -property of the background fields.
(i) Brane-anti-brane solution We take the following forms of a 5 (x 5 ) and ϕ(x 5 ), which describe the localized (around x 5 = 0) configurations and a natural generalization of the ordinary treatment stated before.
where ǫ(x 5 ) is the periodic sign function with the periodicity 2l. 7ā γ andφ γ are some positive constants. See Fig.1. It is the thin-wall limit of a (periodic) kink solution and shows the localization of the fields.
The background fields, (16), satisfy the required boundary condition. We show they also satisfy the on-shell condition (12) for an appropriate choice of a,φ, η, η ′ and χ 3 . The assumed background forms are summarized as where "const"'s mean some constants which generally may be different. 8 We 7 We define the values at x 5 = nl to be 0 in (16) in order to make the function ǫ(x 5 ) piece-wise continuous and also to make it Fourier expandable. 8 Although Dα is made of the bulk fields, it behaves as a boundary field (D-field of N = 1 SUSY multiplet), hence we consider the case that its background value dα is independent of x 5 .  note the relation where δ(x 5 ) is the periodic delta function with the periodicity 2l. The above equation expresses the localization of the bulk scalar at x 5 = 0 and x 5 = l. It is considered to be the field theoretical version of the brane-anti-brane configuration. See Fig.2. Using this relation, the first two equations of (12) are replaced by We note here the following things.
Then we can conclude that (17) is a solution of the field equation (12) for the following choice.
where c is a free parameter. 10 In this choice d α = 0 is concluded. Hence the final two equations of (12) are satisfied. We can regard these as the new onshell condition due to the restriction of the background fields (16). The present vacuum (minimum point of the effective potential) should be consistent with (20).
(ii) Sawtooth-wave solution We consider another solution.
Taking the localized solution (i), we evaluate S (2) , (15), furthermore. 11 From the periodicity (x 5 → x 5 + 2l) and the Z 2 property, the bulk quantum fields Φ(X), A 5 (X) and c(X) can be KK-expanded as (The Z 2 -parity of the ghost field is even because it should be the same as that of the gauge parameter Λ : . ) Now we use the Fourier expansion of the periodic sign function, and the relation: (A)η = 0, η † = 0(Bulk-Boundary decoupled case) We look at the potential from the vanishing scalar-matter point. In this case the singular terms, δ(0)-terms, disappear and the matrix M decouples to the boundary part (φ, φ † ) and the bulk part (Φ, A). The former part gives the following eigen values.
where we take G=SU(2) and the doublet representation for the boundary matter fields. k m is the 4D momentum. This gives, taking the supersymmetric boundary condition, the following potential before the renormalization: The last perturbative (w.r.t. g ) form is logarithmically divergent. It can be checked by the perturbative calculation. It is renormalized by the bulk wave function of X 3 and Φ. Here the 4D world's connection to the Bulk world appears. The quantum fluctuation within the boundary influence the bulk world through the renormalization. The form of (33) is similar to the 4D super QED [7]. We see the present model produces a desired effective potential on the brane.
The bulk part of M and the ghost part do not depend on the field d. They and their eigenvalues depend only on the brane parameters,ā andφ, and the size of the extra space, l. In the SUSY boundary condition, their contribution to the vacuum energy is zero. The scalar loop contribution is expected to be cancelled by the quantum effect of the non-scalar fields. Let us, however, examine the scalar-loop contribution to the Casimir energy (potential). General case is technically difficult. We consider the large circle limit:ĝ 2 ≡ g 2 l = fixed ≪ 1 ,â = √ lā = fixed ,φ = √ lφ = fixed , l → ∞. This is the situation where the circle is large compared with the inverse of the domain wall height. (â and ϕ have the dimension of M . ) We notice, in this limit, Q mn -terms disappear. In the "propagator" terms of the bulk quantum fields, KK-mass terms m 2 π 2 /l 2 disappear. All KK-modes equally contribute to the vaccum energy. The eigen values of the bulk part of M can be easily obtained. In particular, for the special caseâ = 0, the nontrivial factor is only k 2 +ĝ 2φ2 . Hence each KK-mode equally contributes to the vacuum energy as This quantity is quadratically divergent. After an appropriate normalization, the final form should become, based on the dimensional analysis, the following one.
where c 1 , c 2 and c 3 are some finite constants which are calculable after we know the bulk quantum dynamics sufficiently. This is a new type Casimir energy. This is the reason why we have examined the scalar-loop contribution. Comparing the ordinary one (37) explained soon, it is new in the following points: 1) it depends on the brane parametersφ andâ besides the extra-space size l; 2) it depends on the gauge couplingĝ; 3) it is proportional to 1/l 2 . We expect the above result of Casimir energy are cancelled by the spinor and vector-loop contribution in the present SUSY theory. The unstable Casimir potential do not appear in SUSY theory.
(B)ā = 0,φ = 0 In this case, Q mn -terms disappear and we do have no localized (brane) configuration. The bulk background configuration is trivial: a 5 (x 5 ) = 0, ϕ(x 5 ) = 0. 5D bulk quantum fields fluctuate with the periodic boundary condition in the extra space. This is similar to the 5D Kaluza-Klein case mentioned in the introduction. The eigen values for the bulk part, c(X),c(X), Φ(X) and A 5 (X) are commonly given by, The eigen values are basically the same as the KK case [3]. They depend only on the radius (or the periodicity) parameter l. It gives the scalar-loop contribution to the Casimir potential. From the dimensional analysis, after the renormalization, it has the following form.
We expect again this contribution is cancelled by the spinor and vector fields. The eigenvalues for the boundary part is obtained as a complicated expression involving the following terms: We have the full expression in the computer file. In the manipulation of eigenvalues search (determinant calculation), we face the following combination of terms.
The first term comes from the singular terms in M, the second from the KKmode sum. Using the relation m∈Z 1 = 2lδ(0), the above sum leads to a regular quantity.