Matter Fields and Non-Abelian Gauge Fields Localized on Walls

Massless matter fields and non-Abelian gauge fields are localized on domain walls in a (4+1)-dimensional $U(N)_c$ gauge theory with $SU(N)_{L}\times SU(N)_{R}\times U(1)_{A}$ flavor symmetry. We also introduce $SU(N)_{L+R}$ flavor gauge fields and a scalar-field-dependent gauge coupling, which provides massless non-Abelian gauge fields localized on the wall. We find a chiral Lagrangian interacting minimally with the non-Abelian gauge field together with nonlinear interactions of moduli fields as the (3+1)-dimensional effective field theory up to the second order of derivatives. Our result provides a step towards a realistic model building of brane-world scenario using topological solitons.


§1. Introduction
Gauge hierarchy problem is a good guiding principle to construct theories beyond the Standard Model (SM). Brane world scenario 1), 2), 3) is one of the most attractive proposals to solve this problem, besides models with supersymmetry (SUSY). 4) In the brane world scenario, it is assumed that all fields except the graviton field are localized on (3+1)-dimensional world volume of a defect called 3-brane, immersed in a many-dimensional space-time called bulk. In order to realize such a scenario dynamically, we may use a topological soliton. For instance, let us consider a domain wall solution as the simplest soliton. To obtain (3+1)-dimensional world volume on the domain wall, we need to consider a theory in a (4+1)-dimensional space-time. Bulk fields in (4+1)-dimensions can provide massless modes localized on the domain wall, besides many massive modes in general. After integrating over massive modes, one obtains low-energy effective field theory describing the effective interactions of massless modes. Massless matter fields have been successfully localized on domain walls, 5) but localization of the gauge field on domain walls in field theories has been difficult. 6) It has been noted that the broken gauge symmetry in the bulk outside of the soliton inevitably makes the localized gauge field massive with the mass of the order of inverse width of the wall. 7), 8) To localize a massless gauge field, one needs to have the confining phase rather than the Higgs phase in the bulk outside of the soliton. Earlier attempts used a tensor multiplet in order to implement Higgs phase in the dual picture, but this approach successfully localize only U(1) gauge field. 9) More recently, a classical realization of the confinement 10), 11) through the position-dependent gauge coupling has been successfully applied to localize the non-Abelian gauge field on domain walls. 12) The nontrivial profile of this position-dependent gauge coupling was naturally introduced on the domain wall background through a scalar-field-dependent gauge coupling function resulting from a cubic prepotential of supersymmetric gauge theories. The appropriate profile of the position-dependent gauge coupling was obtained from domain wall solutions using two copies of the simplest model or from a model with less fields and a particular mass assignment. However, it was still a challenge to introduce matter fields in nontrivial representations of the gauge group of the localized gauge field.
Parameters of soliton solutions are called moduli and can be promoted to fields on the world volume of the soliton. Massless fields in the low-energy effective field theory on the soliton background are generally given by these moduli fields. Moduli with non-Abelian global symmetry is often called the non-Abelian cloud, and has been explicitly realized in the case of domain walls using Higgs scalar fields with degenerate masses in U(N) c gauge theories. 13) This model also has a non-Abelian global symmetry SU(N) L ×SU(N) R ×U(1) A , 2 which is somewhat similar to the chiral symmetry of QCD. If we turn this global symmetry into a local gauge symmetry, we should be able to obtain the usual minimal gauge coupling between these moduli fields and the gauge field. Since we wish to localize the gauge field on the domain wall, it is essential to choose the global symmetry of moduli fields to be unbroken in the vacua (of both left and right bulk outside of the wall). This choice will guarantee that the bulk outside of the domain wall is not in the Higgs phase. Therefore we are led to an idea where we introduce gauge fields corresponding to a flavor symmetry group of scalar fields which will be unbroken in the vacuum. If we introduce the additional scalar-field-dependent gauge coupling function similarly to the supersymmetric model, we should be able to localize both massless matter fields and the massless gauge field at the same time on the domain wall.
The purpose of this paper is to present a (4+1)-dimensional field theory model of localized massless matter fields minimally coupled to the non-Abelian gauge field which is also localized on the domain wall with the (3+1)-dimensional world volume. We also derive the low-energy effective field theory of these localized matter and gauge fields. To introduce non-Abelian flavor symmetry (to be gauged eventually) in the domain wall sector, we replace one of the two copies of the U(1) c gauge theory with the flavor symmetry U(1) L × U(1) R in Ref. 13 for the (subgroup of) the flavor SU(N) L+R symmetry. In order to obtain the field-dependent gauge coupling function, for the gauge field localization mechanism, 12) we also introduce a coupling between a scalar field and gauge field strengths inspired by supersymmetric gauge theories, although we do not make the model fully supersymmetric at present. This scalarfield-dependent gauge coupling function gives appropriate profile of position-dependent gauge coupling through the background domain wall solution. With this localization mechanism for gauge field, we find massless non-Abelian gauge fields localized on the domain wall.
We also obtain the low-energy effective field theory describing the massless matter fields in the non-trivial representation of non-Abelian gauge symmetry. Since our flavor symmetry resembles the chiral symmetry of QCD before introducing the gauge fields that are localized, we naturally obtain a kind of chiral Lagrangian as the effective field theory on the domain wall. We find an explicit form of full nonlinear interactions of moduli fields up to the second order of derivatives. Moreover, these moduli fields are found to interact with SU(N) L+R flavor gauge fields as adjoint representations. In analyzing the model, we use mostly the strong coupling limit for the domain wall sector. The strong coupling is merely to describe our result explicitly at every stage. Even if we do not use the strong coupling, the physical features are unchanged. It is easy to expect that (the part of) the gauge symmetry is broken when the walls separate in each copy of the domain wall sector. Our results of the lowenergy effective field theories shows that flavor gauge symmetry SU(N) L+R is broken on the non-coincident wall and the associated gauge bosons acquire masses as walls separate. This geometrical Higgs mechanism is quite similar to D-brane systems in superstring theory. So our domain wall system provides a genuine prototype of field theoretical D3-branes. This is an interesting problem, which we plan to analyze more in future. We also find indications that additional moduli will appear in the supersymmetric version of our model, which is also an interesting future problem to study.
The organization of the paper is as follows. In section 2, we explain the localization mechanism by taking Abelian gauge theory as an illustrative example. In section 3, we introduce the chiral model with the non-Abelian flavor symmetry for the domain wall sector and then also introduce gauge fields for the unbroken part of the flavor symmetry. By introducing the scalar-field-dependent gauge coupling function, we arrive at the localized massless gauge field interacting with the massless matter field in a nontrivial representation of flavor gauge group. The low-energy effective field theory is also worked out. In section 4, an attempt is made to make the model supersymmetric. New additional features of the supersymmetric models are also described. In section 5, we summarize our results and discuss remaining issues and future directions. In Appendix A we discuss domain wall solution for gauged massive CP 1 sigma model. Appendix B describes derivation of effective Lagrangian which includes full nonlinear interactions between moduli fields. Appendix C contains derivation of positivity condition for the potential appearing in section 4. §2. Abelian-Higgs model of gauge field localization

The domain wall sector
Let us illustrate the localization mechanism for the gauge fields and the matter fields on the domain walls by using a simplest model in (4+1)-dimensional spacetime : two copies (i = 1, 2) of U(1) models, each of which has two flavors (L, R) of charged Higgs scalar fields H i = (H iL , H iR ) : We use the metric η M N = diag(+, −, · · · , −), M, N = 0, 1, · · · , 4. The Higgs field H i is charged with respect to the U(1) i gauge symmetry and the covariant derivative is given by where w i M is the U(1) i gauge field with the field strength Since we want domain walls, we will choose resulting in the U(1) iA flavor symmetry * ) . We have included the neutral scalar fields σ i in this Abelian-Higgs model. The gauge coupling g i appears not only in front of the kinetic terms of the gauge fields and σ i , but also as the the quartic coupling constant of H i . Both these features are motivated by the supersymmetry. Indeed, we can embed this bosonic Lagrangian into a supersymmetric model with eight supercharges by adding appropriate fermions and bosons, which will not play a role to obtain domain wall solutions. We have taken this special relation among the coupling constants only to simplify concrete computations below. One may repeat the following procedure in models with more generic coupling constants without changing essential results. The first term of the potential is the wine-bottle type and the Higgs fields develop nonzero vacuum expectation values. There are two discrete vacua for each copy i Thanks to the special choice of the coupling constants in L i motivated by the supersymmetry, there are Bogomol'nyi-Prasad-Sommerfield (BPS) domain wall solutions in these models. Let y be the coordinate of the direction orthogonal to the domain wall and we assume all the field depend on only y. Then, as usual, the Hamiltonian can be written as follows Thus the Hamiltonian is bounded from below. This bound is called Bogomol'nyi bound, and is saturated when the following BPS equations are satisfied In order to obtain the domain wall solution interpolating the two vacua in Eq. (2 . 6), we impose the boundary conditions : Tension T i of the domain wall is given by a topological charge as The second equation of the BPS equations (2 . 8) can be solved by the moduli matrix For a given H i0 , the scalar function ψ i is determined by the master equation The asymptotic behavior of the field ψ i is determined by the condition that the configuration reaches the vacuum at left and right infinities: There exists redundancy in the decomposition in Eq. (2 . 11), which is called the V -transformation: (2 . 14) For example, a single domain wall solution centered at y = 0 can be generated by a moduli matrix Then the master equation is No analytic solutions for the master equation have been found for finite gauge couplings g i , so we must solve it numerically. The corresponding solution is shown in Fig. 1. The generic solutions of the domain wall are generated by the generic moduli matrices (after fixing the V -transformation) The complex constants C iL , C iR are free parameters containing the moduli parameters of the BPS solutions. The moduli parameter can be defined by The other degree of freedom in C iL , C iR can be eliminated by the V -transformation in Eq. (2 . 14) and has no physical meaning. Then the master equation is found to be It is obvious that the real parameter y i is the translational moduli of the domain wall.
The other parameter α i is an internal moduli which is the Nambu-Goldstone (NG) mode associated with the U(1) iA flavor symmetry spontaneously broken by the domain walls. One can take, if one wishes, the strong gauge coupling limit of the Lagrangian L i . As is well-known, the U(1) gauge theory with two flavors of Higgs scalars in the strong gauge coupling limit becomes a non-linear sigma model whose target space is CP 1 : The gauge fields and the neutral scalar field become infinitely massive and lose their kinetic terms. They are mere Lagrange multipliers in the limit, and are solved as Plugging these into L i , we get with a projection operator Let us introduce an inhomogeneous coordinate φ i of CP 1 by Then the Lagrangian of the CP 1 model in terms of φ i is Let us reconsider the domain wall solutions in this limit. The Hamiltonian can be written as (2 . 26) The BPS equation and the boundary conditions are given by The tension of the domain wall is This is the same as the one in the finite gauge coupling model.
In this way, the strong gauge coupling limit has a great advantage compared to the finite gauge coupling case. One can exactly solve the BPS equation and see the moduli parameter in the analytic solutions. Furthermore, there is no important differences between domain wall 8 solutions in the finite coupling (Abelian-Higgs model) and the strong coupling (non-linear sigma model). Both solutions have the same tension of domain wall and the same number of the moduli parameters. To see the difference explicitly, let us compare the configuration of the neutral scalar field σ i . In the strong gauge coupling limit, it can be written as where we have used In Fig. 1, we show the configurations of σ i in two cases, the one in the small finite gauge coupling and the one in the strong gauge coupling limit. As can be seen from the figure, there are no significant differences.
Let us next derive the low energy effective theory on the domain wall. We integrate all the massive modes while keeping the massless modes. We use the so-called moduli approximation where the dependence on (3+1)-dimensional spacetime coordinates comes into the effective Lagrangian only through the moduli fields: (2 . 32) The effective Lagrangian for the moduli field C i (x µ ) can be obtained by plugging this into the Lagrangian L i and integrate it over y. This can be done explicitly as follows.
With Eq. (2 . 31), the effective Lagrangian is given by where energy of soliton solution is neglected since it does not contribute to dynamics of moduli. Note that 2m i v 2 i is precisely the domain wall tension. This is the free field Lagrangian. Although we have derived this effective Lagrangian in the strong gauge coupling limit, we can obtain the same Lagrangian in the finite gauge coupling constant. In other words, the effective Lagrangian cannot distinguish the infinite versus finite coupling cases at least in the quadratic order of the derivative expansion.

Localization of the Abelian gauge fields
In the previous subsection, we have seen that the NG modes of the translation and U(1) global symmetry are the only massless modes in the Abelian-Higgs model. They are localized on the domain wall. There are no massless gauge field on the domain wall and all the modes contained in the gauge field are massive. The mass of the lightest mode of the gauge field is of the order of the inverse of the width of the domain wall, since the bulk outside of the domain wall is in the Higgs phase. The low energy effective Lagrangian for the massless fields is obtained after integrating out the massive modes including gauge fields.
In order to obtain the massless gauge field to be localized on the domain wall, we need a new gauge symmetry which is unbroken in the bulk. Recently, a new mechanism was proposed to localize gauge fields on domain walls. 12) A key ingredient is the so-called dielectric coupling constant 10), 11) for the new gauge symmetry. To illustrate the new localization mechanism, let us introduce a new U(1) gauge field a M which we wish to localize on the domain wall. Since this gauge symmetry should be unbroken in the bulk, we consider the case where all the Higgs fields are neutral under this newly introduced U(1) gauge symmetry. The gauge field a M is assumed to couple to the neutral scalar fields σ i only in the following particular combination where a real constant λ with the unit mass dimension, in accordance with the (4+1)dimensional spacetime and the field strength is defined by The field-dependent gauge coupling function is given by except for the additional kinetic term (the last term) of the (3+1)-dimensional gauge field w µ , which is the zero mode (y-independent mode) of the (4+1)-dimensional field w µ . The (3+1)-dimensional gauge coupling constant is given by where we have used the asymptotic behavior ψ i → log 2 cosh 2m i (y − y i ) as |y| → ∞. Note that this result is again independent of the gauge couplings g i in the domain wall sector. In summary, the low energy effective Lagrangian is Now we separate the quantum fields (fluctuations) from the classical background moduli parameters by Then the effective Lagrangian up to the second order of the small quantum fluctuations is given by We note that the massless gauge field a µ has a positive finite gauge coupling squared * ) 1/(4λ(y 0 2 − y 0 1 )) provided y 0 2 − y 0 1 > 0. Although we succeeded in localizing the massless U(1) gauge field a µ on the domain walls, the Lagrangian Eq. (2 . 42) has no charged matter fields minimally coupled with the localized gauge field a µ . To obtain matter fields interacting with the localized gauge field, one may be tempted to identify the Higgs fields H i = (H iL , H iR ) as matter fields * * ) with * ) Here we are content with the fact that the positivity of the gauge kinetic term is assured at least in finite region of moduli space, instead of just at a point. However, it is possible to make a more economical model where one has less moduli, and the positivity of the gauge kinetic term is assured. 12) * * ) We consider the diagonal subgroup U (1) A of U (1) 1A and U (1) 2A . Actually the U (1) iA global symmetries are broken by the domain wall solution, we consider this gauging to leading order of gauge coupling only to illustrate the Higgs mechanism for the broken symmetry. charges (1, −1). The minimal gauge interaction of Higgs fields with the a M is introduced through the modified covariant derivatives as Since the moduli field C i is charged, the derivatives in the low energy effective theory Eq. (2 . 33) should be replaced by the covariant derivative It is straightforward task to derive the effective Lagrangian with the covariant derivative above along the same line of reasoning for the previous case This clearly shows that the new gauge field a µ is not massless due to the Higgs mechanism, and should be integrated out together with the other massive fields. Namely the low energy effective Lagrangian does not include the massless gauge fields, since the U(1) symmetry which we gauged is broken by the domain wall. A more explicit example at the strong gauge coupling limit is described in Appendix A. Thus the Abelian-Higgs model in this section gives an important lesson that we should not gauge a symmetry which is broken by the domain wall solution, since the corresponding gauge fields may be localized on the domain walls but they become massive and should be integrated out from the low energy effective theory. In the next section, we will give a model with a non-Abelian global symmetry whose unbroken subgroup can be gauged to yield massless localized gauge fields on the domain wall. §3. The chiral model In this section we study domain walls in the chiral model which is a natural extension of the Abelian-Higgs model in the previous section. This chiral model leads to two important consequences 1) massless non-Abelian gauge fields are localized on the domain wall and moreover 2) the scalar fields which are non-trivially interacting are also localized on the domain walls.  13) To localize the gauge field in a simple manner, we again introduce two sectors L 1 and L 2 , but only the former is extended to Yang-Mills-Higgs system and the latter is the same form as (2 . 1). The second sector couples to the first sector through the coupling as described in (2 . 35) after gauging the flavor symmetry it plays a role as localization of gauge fields, combined with the first sector. The matter contents are summarized in Table I. Since the presence of two factors of SU(N) global symmetry resembles the chiral symmetry of QCD, we call this Yang-Mils-Higgs system as the chiral model. The Lagrangian is then given by , and adjoint scalar as Σ 1 . The covariant derivative and the field strength are denoted as The mass matrix is given by Let us note that the chiral model reduces to the Abelian-Higgs model in the limit of N → 1, by deleting all the SU(N) groups.
The second sector is just necessary to realize the field-dependent gauge coupling function similar to (2 . 35) as we will discuss in the subsequent subsection. In the rest of this subsection, we focus only on the first sector (i = 1) and suppress the index i = 1. The symmetry transformations act on the fields as There exist N + 1 vacua in which the fields develop the following VEV with r = 0, 1, 2, · · · , N. We refer these vacua with the label r. In the r-th vacuum, both the local gauge symmetry U(N) c and the global symmetry are broken, but a diagonal global symmetries are unbroken (color-flavor-locking) As in the Abelian-Higgs model, the BPS equations for the domain walls can be obtained through the Bogomol'nyi completion of the energy density with the assumption that all the fields depend on only the fifth coordinate y and W µ = 0: This bound is saturated when the following BPS equations are satisfied The tension of the domain wall is given by Let us concentrate on the domain wall which connects the 0-th vacuum at y → ∞ and the N-th vacuum at y → −∞. Its tension can be read as 14) where ψ is the solution of the master equation (2 . 12) in the Abelian-Higgs model. Eq. (3 . 8) shows that the unbroken global symmetry for N-th vacuum ( The domain wall solution further breaks these unbroken symmetries because it interpolates the two vacua. The breaking pattern by the domain wall is * ) This spontaneous breaking of the global symmetry gives NG modes on the domain wall as massless degrees of freedom valued on the coset similarly to the chiral symmetry breaking in QCD : Since our model can be embedded into a supersymmetric field theory, these NG modes (U(N) chiral fields) appear as complex scalar fields accompanied with additional N 2 pseudo-NG modes * * ) . * ) The unbroken generators of U (1) A+c for r-th vacuum contains different combination of U (N ) c generators depending on r. Therefore the right and left vacua preserve actually different U (1) A+c , and the wall solution does not preserve any of these U (1) A+c . * * ) One of them is actually a genuine NG mode corresponding to the broken translation.

Localization of the matter fields
In the remainder of this subsection, we will give the low-energy effective Lagrangian on the domain walls where the massless moduli fields (the matter fields) are localized. The best way to parametrize these massless moduli fields is to use the moduli matrix formalism 13), 14), 15) where S ∈ GL(N, C) and Ω = SS † is the solution of the following master equation where We have used the V -transformation to identify the moduli e φ , which is a complex N by N matrix. It can be parametrized by an N × N hermitian matrixx and a unitary matrix U where U is nothing but the U(N) chiral fields associated with the spontaneous symmetry breaking Eq. (3 . 18) andx is the pseudo-NG modes whose existence we promised above.
In the strong gauge coupling limit g → ∞, solution of master equation is simply Ω = Ω 0 . After fixing the U(N) c gauge, we obtain Let us denote, for brevityŷ the Higgs fields are then given as From this solution, one can easily recognize that eigenvalues ofx correspond to the positions of the N domain walls in the y direction. Now we promote moduli parametersx and U to fields on the domain wall world volume, namely functions of world volume coordinates x µ . We plug the domain wall solutions H L,R (y;x(x µ ), U(x µ )) into the original Lagrangian L in Eq.(3 . 2) at g → ∞ and pick up the terms quadratic in the derivatives. Thus the low energy effective Lagrangian is given by Here we have eliminated the massive gauge field W µ by using the equation of motion. Using the solutions for H L and H R we have found a closed formula for the effective Lagrangian up to the second order of derivatives but with full nonlinear interactions involving moduli fieldŝ x and U. Detailed derivation is given in Appendix B.
Here we exhibit the result only in the leading orders of U − 1 andx: The quantum numbers are summarized in Table II. We now introduce a field-dependent gauge coupling function g 2 (Σ) for A M , which is inspired by the supersymmetric model in Ref.12). The Lagrangian is given by The next step is to derive the low energy effective theory on the domain wall worldvolume in the moduli approximation as in the previous subsections. Again, we promote the moduli parameters as the fields on the domain wall world-volume and pick up the terms up to the quadratic order of the derivative ∂ µ . Similarly to section 3.2, we utilize the strong gauge coupling limit g i → ∞, to simplify the computation without changing the final result. Let us emphasize that we keep the field-dependent gauge coupling function e(Σ) finite.
The spectrum of massless NG modes is unchanged by switching on the SU(N) L+R gauge interactions * ) .
We just repeat the similar computation to those in section 3.2. Again we shall focus on the first sector L 1 and suppress the index i = 1 of fields. Since color gauge fields W µ becomes auxiliary fields and eliminated through their equations of motion, it is convenient to define the covariant derivative only for the flavor (SU(N) L+R ) gauge interactions aŝ (3 . 36) Then we obtain the effective Lagrangian of the first sector as Eliminating W µ , we obtain the following expression for the integrand of the effective Lagrangian after some simplification where we defined fields H ab with the label ab of adjoint representation of the flavor gauge group SU(N) L+R+c and the covariant derivative as In Appendix B, we will describe fully the procedure to derive the effective Lagrangian by substituting (3 . 26) and (3 . 27) and rewriting in terms of moduli fieldsx and U. Here we merely state the result: is a Lie derivative with respect to A. The covariant derivative D µ is defined by Eqs. (3 . 19) and (3 . 20) show that The complex moduli e φ is decomposed into hermitian part ex and unitary part U in Eq.(3 . 23).
Since we can express e 2x = e φ e φ † , and U = e −φ ex, we find that they transform as adjoint where y i is the wall position for the i-th domain wall sector. Summarizing, we obtain the following effective Lagrangian

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where L 2,eff is given in (2 . 34). This is the main result of this paper. We have succeeded in constructing the low energy effective theory in which the matter fields (the chiral fields) and the non-Abelian gauge fields are localized with the non-trivial interaction. We show the profile of "wave functions" of localized massless gauge field and massless matter fields as functions of the coordinate y of the extra dimension in Fig. 2. As is seen from Eq.(3 . 47), the flavor gauge symmetry SU(N) L+R+c is further (partly) broken and the corresponding gauge field A µ becomes massive, when the fluctuation φ = exU develops non-zero vacuum expectation values. Especially,x is interesting because its nonvanishing (diagonal) values of the fluctuation has the physical meaning as the separation between walls away from the coincident case. For instance, if all the walls are separated, SU(N) L+R+c is spontaneously broken to the maximal U(1) subgroup U(1) N −1 . However, if r walls are still coincident and all other walls are separated, we have an unbroken gauge symmetry SU(r) × U(1) N −r+1 . Then, a part of the pseudo-NG modesx turn to NG modes associated with the further symmetry breaking SU(N) L+R+c → SU(r) × U(1) N −r+1 , so that the total number of zero modes is preserved 13) * ) . These new NG modes, called the non-Abelian cloud, spread between the separated domain walls. 13) The flavor gauge fields eat the non-Abelian cloud and get masses which are proportional to the separation of the domain walls. This is the Higgs mechanism in our model. This geometrical understanding of the Higgs mechanism is quite similar to D-brane systems in superstring theory. So our domain wall system provides a genuine prototype of field theoretical D3-branes. §4. Embedding into supersymmetric theory A crucial point to localize gauge field around domain wall is the coupling between scalar and gauge kinetic term. Such a coupling is naturally realized in (4+1)-dimensional supersymmetric gauge theory. 12) This theory generally consists of hypermultiplet part and vector multiplet part. The latter is specified by the so-called prepotential. In (4 + 1)-dimensional theory the prepotential generally allows up to cubic terms in vector multiplets, 17) which serves interactions among vector mutiplets such as (3 . 33).

Supersymmetric model
In embedding the model into supersymmetric gauge theories in (4+1) dimensions, we will give non-Abelian global flavor symmetry SU(N i ) V for each copy (i = 1, 2) of the domain wall sector, instead of only one copy as in (3 . 34) of the previous section. This contains the model (3 . 34) as a limiting case of N 2 → 1, and may offer more general situation phenomenologically. To formulate supersymmetric gauge theories, we need to introduce Y i as auxiliary fields of U(N i ) c vector multiplet, and Φ i and Y i as adjoint scalar fields and auxiliary fields of SU(N i ) V vector multiplet. As bosonic fields of theories with eight supercharges, we also need to double the scalar fields H i , by introducing another setH They are in the same representations as H i under U(N i ) c and U(1) iA .
Explicit charge assignments for hypermultiplets matter fields and adjoint scalar fields are summarized in Table III. The resultant supersymmetric Lagrangian is written as where where α, β · · · denote all gauge groups and their generators collectively. We label them with the ordering where 0 i denotes U(1) i parts of U(N i ) c gauge group, while I i = 1, · · · , N 2 i − 1 are color indices of SU(N i ) c and A i = 1, · · · , N 2 i − 1 denotes flavor indices of SU(N i ) V gauge group. The scalar fields Σ α and auxiliary fields Y α are explicitly written by and similarly the field strength F α M N and gauge field W α M are written by We adopt the convention of U(N i ) c and SU(N i ) V matrices such as iR . Covariant derivatives of Σ I i , Φ A i are defined as the adjoint representation. We will not display the Chern-Simons term L iCS and the fermionic term L ifermion , since we do not need them for our analysis.
Functions a αβ (Σ) are gauge coupling functions, which are given as second derivative of the prepotential a αβ (Σ) = ∂ 2 a(Σ) ∂Σ α ∂Σ β . (4 . 12) From the above prepotential, we see the coupling constants of U(1) i and SU(N i ) c are given byĝ i and g i , respectively * ) . We denote the coupling function of SU(N i ) V corresponding to but will suppress the argument Σ to write e i in the following.
The constants c α are coefficients of the Fayet-Iliopoulos (FI) terms, allowed to be non-zero only for the U(1) part of the gauge groups * * ) (4 . 14) We have assumed both the FI parameters c 0 1 and c 0 2 to be positive in the same direction in SU(2) R , which is chosen to be along the third component. In this setup, theH fields will vanish in the classical solution. Moreover, they do not contribute to the desired order of effective Lagrangian. Similarly we have neglected the auxiliary fields Y other than the third component in SU(2) R , and we have denoted as Y α . Hence we can call the potential after eliminating the auxiliary fields Y 's to be D-term potential.
The F-term potential V iF can be worked out from the following superpotential where we restored the tilde fieldsH's to facilitate writing the superpotential. After eliminating the auxiliary fields F 's, and with the use of we have (4 . 3).
Finally, let us work out explicit forms of the D-term potential V D . Collecting terms containing the auxiliary fields Y 's, we obtain where (4 . 19) are Hermitian matrices, with the decomposition We observe, that in the potential (4 . 17), Y I i do not couple to the rest of auxiliary fields and can be easily eliminated. Having this done, we collect the U(1) i and SU(N i ) V terms into a matrix form labeled by α, β = 0 1 , Eliminating remaining auxiliary fields we obtain: Matrix G = (G αβ ) is explicitly given by with the inverse where we abbreviated:g 2 =ĝ 2 1 m 2 2 +ĝ 2 2 m 2 1 , (4 . 26)

Positivity of Potential
The F-term potential (4 . 3) is manifestly positive. The D-term potential (4 . 21) is positive definite under certain conditions. To find the condition we shall decompose (4 . 21) to: It is clear that the V 1D is positive definite by itself. Therefore we can only focus on V 2D , which is positive if and only if G is positive definite. It is easy to recognize that positivity of G is manifest once the adjoint scalars vanish Φ i = 0. Nevertheless, it is instructive and assuring if we consider the potential as well as the BPS equations keeping the adjoint scalars Φ i nonzero.
To ascertain positivity of G we need to compute its eigenvalues. This is most easily done by looking at its determinant (We leave the derivation of this result to the Appendix C): (4 . 33) Requiring det G > 0, we have In Appendix C we show that this condition is both necessary and sufficient to ensure positivity of matrix G in Eq.(4 . 24).

BPS equations
Let us denote the codimension of the domain wall as y. Since we assume Lorentz invariance for other dimensions, we obtain gauge field to vanish for component other than y.

The energy density H for domain walls is given by
where color-flavor indices α, β span all values as in Eq.(4 . 4) and we have incorporated color sector α = I 1 , I 2 into the definition of matrix G for brevity. Accordingly, we have incorporated the definition, (r − c) I i = r I i . Since there is no mixing of color sector with the rest, the inverse is calculated trivially and non-color part remains the same as in (4.1).
Now we observe that the mixing due to the cubic prepotential occurs only in the kinetic term and potential of the vector multiplets. Moreover, they appear as G and G −1 respectively. Therefore the cross term coming out of the Bogomol'nyi completion has no dependence on the metric G. This fact implies that the cancellation of cross terms to give topological charge goes through unaffected by the mixing of the vector multiplets.
More explicitly, we obtain the Bogomol'nyi completion as The last term gives the usual Bogomol'nyi bound and becomes the topological charge. The line before that is the total derivative which give vanishing contribution for an infinite line −∞ < y < ∞. More explicitly,

BPS equations for H's andH's of hypermultiplets arẽ
We can easily solve the BPS equation for hypermultiplets, by using the moduli matrix approach. We define S ic , S iF and ψ i as where S ic , S iF ∈ SL(N i , C). The hypermultiplet fieldsH iL andH iR do not contribute to domain wall solution and they are therefore vanishing. We write down (4 . 42)-(4 . 44) in terms of the gauge invariant fields The adjoint scalar fields of the vector multiplets are given by Also, we have (4 . 53)

28
BPS equations for vector multiplets (4 . 42)-(4 . 44) can be now rewritten as the following master equations: (4 . 56) Here we have used a notation Irrespective of the possible additional moduli, we can demonstrate that the BPS equations admit the coincident wall solution. Since the hypermultiplet parts are already solved as in (4 . 47)-(4 . 48), our main task is to solve the master equations (4 . 54)-(4 . 56) associated to the vector multiplet. In order to solve them explicitly, we take strong gauge coupling limit g i , g i → ∞, where the master equations give just the algebraic constraints for Ω ic , Ω iF and η i . In principle, they can be solved algebraically. Furthermore, Eq.(4 . 34) with the limit g i → ∞ tells us that positivity is maintained only if Φ i vanishes. In the following we will, therefore, consider a special point in the solution space where Φ i = 0, i = 1, 2 (4 . 58) which implies from Eq.(4 . 51) that Ω iF are constant matrices. Then the differential equations (4 . 55)-(4 . 56) reduce to the set of algebraic equations: Notice, that for both sectors i = 1, 2 these equations are the same and do not couple to each other. We can, therefore, focus our discussion only on one sector, since all results are equivalent in both of them. So in the remaining discussion we will drop the index i from all fields. Now we consider moduli matrix for the coincident walls corresponding to the most symmetric point of the moduli space.  Note that in this solution we restore a moduli parameter y 0 corresponding to the position of the coincident wall. A similar construction of domain wall solution works for the second sector (i = 2), besides the first sector (i = 1) given above. Let us note that the field-dependent gauge coupling function similarly to (3 . 33) is automatically obtained as a bosonic part of the Lagrangian specified by the cubic prepotential in Eq.(4 . 11), Restoring the index i = 1, 2 for both of the domain wall sectors, and by using (4 . 65) with (4 . 50), we finally conclude that the appropriate profile of the field-dependent gauge coupling function Σ 0 1 /m 1 − Σ 0 2 /m 2 , similarly to (3 . 33) is achieved. When we make (a part of) the global flavor symmetry as a local gauge symmetry, we can have several options. Since the first flavor group SU(N 1 ) is generally different from the second flavor group SU(N 2 ), we can naturally introduce two different gauge fields for the i = 1 and 2. This option leads to two decoupled sectors in the low-energy effective Lagrangian, which can only be coupled by higher derivative terms induced by massive modes. Another interesting option is to introduce a gauge field only for the diagonal subgroup of isomorphic subgroups of two different flavor groups, such as SU(Ñ ) ∈ SU(N 1 ), SU(Ñ ) ∈ SU(N 2 ) withÑ ≤ N 1 , N 2 . This option is interesting in the sense that the massless gauge field exchange will communicate between two domain wall sectors. We hope to come back to these issues in near future. Let us make a few comments. First we have shown that the chiral model analyzed in section 3 can be extended to a supersymmetric gauge theory with eight supercharges and that the field-dependent gauge coupling function which is a clue for localization is naturally explained by taking the cubic prepotential. Second, there may be more moduli not contained in to (H 0 L , H 0 R ), which require further studies. Third, here we have presented a solution at a special point Φ = 0. It would be interesting to consider the case or Φ = 0, but in this case, we need to take a finite gauge coupling limit, on which we will investigate in future work. §5. Conclusions and discussion In this paper we have successfully localized both massless non-Abelian gauge fields and massless matter fields in non-trivial representation of the gauge group. We first considered a (4+1)-dimensional U(N) gauge theory with additional SU(N) L × SU(N) R × U(1) A flavor symmetry. We introduced the flavor gauge field for the diagonal flavor group SU(N) L+R , which is unbroken in the coincident wall background. The flavor gauge fields are localized on the wall by introducing the scalar-field-dependent gauge coupling function. Then we studied the low-energy effective Lagrangian and showed that massless localized matter fields interact minimally with localized SU(N) L+R gauge field as adjoint representations. Moreover, full nonlinear interaction between the moduli containing up to the second derivatives, was worked 31 out. The field-dependent gauge coupling function is naturally realized in supersymmetric gauge theories using the so-called prepotential. For this reason, we also explored bosonic part of N = 1 supersymmetric extension of our model. Main result of this paper is the effective Lagrangian (3 . 49). The moduli field U appearing in the effective theory, is a chiral N × N matrix field like a pion, since it is a NG boson of spontaneously broken chiral symmetry. Other moduli in (3 . 49), denoted by N ×N Hermitian matrixx, has the physical meaning of positions of N domain walls as its diagonal elements. We argued that the fluctuations of moduli fieldx, can develop VEV corresponding to splitting of walls, and the Higgs mechanism will occur as a result. Namely, the flavor gauge fields get masses by eating the non-Abelian cloud. Therefore, in this model, Higgs mechanism has a geometrical origin like low energy effective theories on D-branes in superstring theory.
Amongst the possible future investigations, we would like to study non-coincident solution to further clarify this geometrical Higgs mechanism.
We have noticed that our effective moduli fields resemble the pion in QCD. Similar attempts have been quite successful using D-branes. 25) We believe that our methods can provide more insight in various aspects of low-energy hadron physics. We plan to explore this direction more fully in subsequent studies.
In the discussion of supersymmetric extension of our model in section 4, we employed a general setup where both sectors possessed their own domain wall solution, preserving the same half of the supercharges. But another alternative approach is also possible. We can consider a model, where different halves of supercharges are preserved at each sector (BPS and anti-BPS walls), and the SUSY is completely broken in the system as a whole. It has been proposed that the coexistence of BPS and anti-BPS walls gives the supersymmetry breaking in a controlled manner. 26) In our present case, BPS and anti-BPS sectors interact only weakly. If we choose flavor gauge field for each sector separately, we have only higher derivative interactions induced by massive modes. If we choose the diagonal subgroup of (subgroups of) each sector as flavor gauge group, we have a more interesting possibility of the massless gauge field as a messenger between two sectors. We plan to address this issue elsewhere.
In order to construct a realistic brane-world scenario with the SM fields on the domain wall, we need the localization of fields in the fundamental representation of the gauge group. This is still an open problem and one of the priorities of our future investigations. In particular, the SM contains chiral fermions. Localization of chiral fermions is a particularly challenging problem. Anomaly associated with the chiral fermion is also an interesting issue to be addressed. We would also like to clarify these problems in subsequent studies.
Two more issues remain to be addressed. First is the question of sign of gauge kinetic term. In our present model, the positivity of the gauge coupling function is assured only when positions of walls are properly ordered (see Eq.(3 . 48)), namely only in a region of the moduli space, More economical models such as given in Ref. 12) may not have such moduli and, therefore, the effective gauge coupling may be always positive. And lastly, as discussed in section 4, we have not succeeded in exhausting all moduli in the supersymmetric extension of our model. We would also like to investigate these aspects in the future.

Acknowledgements
This work is supported in part by Japan Society for the Promotion of Science ( Here we consider the domain wall solutions in the gauged massive CP 1 sigma model. The model is obtained as the strong gauge coupling limit of a model similar to that we have studied in section 2.2. Namely, we start with the Lagrangian which has U(1) × U(1) gauge symmetry with two flavors where H = (H L , H R ). The covariant derivative is given bỹ The mass matrix is chosen M = diag(m, −m) as before. We next take the strong gauge coupling limit g → ∞ of only one of the gauge coupling which results in the non-linear sigma model coupled to the other gauge field with the finite gauge coupling e. In the limit the gauge field w M and the neutral scalar field σ become Lagrange multipliers. After solving their equations of motion, we have where we have introduced the covariant derivativê Plugging these into the original Lagrangian at g → ∞, we get the gauged massive CP 1 sigma model with the projection operator As before, let us rewrite this Lagrangian with respect to the inhomogeneous coordinate Then the charge matrix should be chosen as which leads to a natural expression that the complex scalar field φ has the U(1) charge 1 for the gauge field a M : Plugging these into Eq.(A . 6), we finally get the Lagrangian Let us next consider a domain wall solution in this model. We assume all the fields depend on only the extra-dimensional coordinate y. Then the four dimensional components of the Maxwell equation can be immediately solved by a µ = 0, µ = 0, 1, 2, 3.
(A . 14) The fifth component is Now the Hamiltonian reduces to the following form (A . 16) Thus the reduced Hamiltonian is minimized when the following first order equation is satisfied Lagrangian is given by where we have promoted the moduli parameter y 0 , α to the fields y 0 (x µ ), α(x µ ) on the wall, and we have introduced the covariant derivativê where α is the function of the (3+1)-dimensional coordinate x µ . Assuming a µ to be yindependent (zero mode), we finally obtain Thus we find that the gauge field a µ (x) absorbs the scalar field α(x) to become massive via the Higgs mechanism. Since that the U(1) gauge field a µ is massive in the effective Lagrangian, we have to integrate it out according to the spirit of the low energy effective theory.

Appendix B Effective Lagrangian on the domain wall
In this appendix we derive our main result (3 . 41) of the effective Lagrangian for the gauged Chiral model introduced in §3.

B.1. Compact form of gauged nonlinear model
Starting from the Lagrangian using the Einstein summation convention for a = {L, R} we first eliminate the gauge fields W µ to obtain a simple expression for gauged nonlinear sigma model. Gauge fields W µ are given by equations of motion as The effective Lagrangian (B . 1) should also contain kinetic term for gauge field A µ , but we will not explicitly write it here, for brevity. Eq. (B . 1) can be further simplified by using the following identities After some algebra we find: Plugging above expression back into the (B . 1) we arrive at:
(B . 13) In the following we would like to carry out the integration over the extra-dimensional coordinate y. This can be done in two steps. First, we must factorize all quantities depending on y (or onŷ) to one term inside the trace, effectively reducing our problem to fit the following form: where M is some matrix, independent of y and f some function. In the second step we diagonalizex:x = P −1 diag(λ 1 , . . . , λ N )P , and use the fact that f (P −1ŷ P ) = P −1 f (ŷ)P . This transformation leads to For every term in the sum we can perform substitutionỹ = my − λ i . The key observation is that in each term the integration will be the same and independent on a particular value of λ i . Thus we arrive at an identity It appears as if we just made a substitutionŷ =ỹ1 N . This is possible, of course, only thanks to the diagonalization trick and properties of the trace. In the subsequent subsections, however, we will refer to this procedure as if it is just a 'substitution', for brevity. Let us decompose the effective Lagrangian (B . 13) into three pieces L eff = Tx + T U + T mixed (B . 16) and see the outlined procedure for each term.

B.2.1. Kinetic term for U
First, let us concentrate only on terms containing double derivatives of U, which we denote T U : where we have used the fact that inside commutator it is possible to freely interchange e −ŷ cosh(ŷ) → − eŷ cosh(ŷ) , since the difference is just a constant matrix. In this way we made T U manifestly invariant under exchangeŷ → −ŷ.
Since in the first factor of T U allŷ-dependent quantities are on the right side, we can, according to our previous discussion, make use of the identity (B . 15) and carry out the integration: For the second term, however, we first use the identity:  . Now allŷ-dependent factors are standing on the right and we can formally exchangeŷ →ỹ. The summation can be carried out to get: The formula for T U now reads: Since we started with T U invariant under the transformationŷ → −ŷ, we should take only even part of the above formula (under exchange Lx → −Lx) as the final result: Now we can carry out the integration using primitive function dy cosh(y − α) cosh(y) = 1 sinh(α) ln 1 1 − tanh(α) tanh(y) .
Therefore we obtain the result to all orders inx as: we can easily read off coefficients of terms beyond the leading one. For example, the first three terms reads:
With use of the identity (B . 17) and one can prove the following: We can use this result to factorize allŷ-dependent quantities to the right and make the substitutionŷ =ỹ1 N : .
Now we are free to perform summation and integration to obtain: we can easily read off coefficients of terms beyond the leading order in the series expansion: Kinetic term forx is given by We are going to need the identity With the aid of this we arrive at where we again employed diagonalization trick and identity (B . 15). Let us carry out the summation and the integration to obtain: leading to the power series: Putting all pieces together as L eff = Tx + T U + T mixed , we obtain our final result (3 . 41).