Stabilizing matter and gauge fields localized on walls

Both non-Abelian gauge fields and minimally interacting massless matter fields are localized on a domain wall in the five-dimensional spacetime. Field-dependent gauge coupling naturally gives a position-dependent coupling to localize non-Abelian gauge fields on the domain wall. An economical field content allows us to eliminate a moduli for a instability, and to demonstrate the positivity of the position-dependent coupling in the entire moduli space. Effective Lagrangian similar to the chiral Lagrangian is found with a new feature of different coupling strengths for adjoint and singlet matter that depend on the width of the domain wall.


Introduction
Localization of massless gauge fields on a domain wall with 3 + 1 dimensional world volume has been a long-standing problem to achieve the dynamical compactification in the braneworld scenario [1]. If the gauge symmetry is unbroken only inside the domain wall and broken outside, the gauge field inevitably acquires a mass proportional to the inverse of the width of the wall [2][3][4]. The localization of gauge fields can be achieved if the gauge theory is in the confining phase outside of the domain wall [2,3]. This requirement can be translated into a position-dependent gauge coupling [5][6][7].
The supersymmetric gauge theories in 4 + 1 spacetime dimensions allows a cubic coupling [8] between adjoint scalar fields Σ α and gauge field strengths F β with coupling constants C αβγ . In Ref. [9], a domain wall solution is chosen so that the scalar field Σ is positive inside and vanishes asymptotically outside of the domain wall. In this way, non-Abelian gauge fields have been localized on the domain wall. More recently a method to localize non-Abelian gauge fields together with minimally interacting matter fields on domain walls has been introduced and a particular model has been presented in five-dimensional spacetime [10,11], by gauging the unbroken global symmetry associated to the degenerate scalar fields to form the domain wall [12][13][14]. This mechanism provides a step towards a realistic model of branes as soliton solutions of higher dimensional field theories, and realizes one of the most important characteristics of D-branes: massless U (N ) gauge fields emerge when N walls are coincident, and become massive as walls separate. In Ref. [9], two real scalar fields σ 1 and σ 2 were introduced to form the usual kink profile in the extra-dimensional coordinate y as shown in the left panel of Fig. 1, and the cubic coupling in Eq.(1.1) is chosen as (1.2) whereG M N is the non-Abelian gauge field strength to be localized on the domain wall.
In this way, one achieved the desired profile of the position-dependent coupling which is positive and vanishes asymptotically outside of the domain wall as shown in the right panel of Fig.1. More recently, matter fields have also been localized on the domain wall interacting minimally with the localized gauge fields [10]. Although the model realized the localization mechanism in a simple setup, the width of the position-dependent coupling is a modulus and can become negative implying an instability of the gauge kinetic term.
Moduli of domain walls arise as remaining degrees of freedom of scalar fields constrained by gauge symmetry to form the domain wall. This observation prompts us to construct a model with smaller number of scalar fields with a different charge assignments in order to eliminate the unwanted modulus in the position-dependent coupling [9].
The purpose of this paper is to present models in 4 + 1 dimensional spacetime with domain wall solutions which allow stable localized gauge fields interacting minimally with localized matter fields. For any values of the moduli parameters, we show that the position-dependent gauge coupling is positive everywhere in the extra-dimensional coordinate y. In particular, its width is no longer a modulus and is fixed by the parameters of the theory. We thus find that the gauge kinetic term is stable for all values of moduli, namely for all possible domain wall configurations. As localized matter fields, we obtain scalar fields in singlet and adjoint representations of the gauge group. They are associated with the broken part of the global symmetry, and interact minimally with the localized gauge fields. The effective Lagrangian on the domain wall is also worked out, using the method in Ref. [15]. It resembles the chiral Lagrangian associated with the chiral symmetry breaking of QCD. A new feature of the effective Lagrangian is that the coupling strength of the adjoint matter fields is larger than that of the singlet matter fields. We examine generality of models with a stable kinetic term for gauge fields to identify the class of models with desirable properties. We also discuss ways to build more realistic models for a phenomenology of the brane-world scenario. The organization of the paper is as follows. Section 2 is devoted to construct a model in 4 + 1 spacetime dimensions with domain wall solutions which localizes matter and gauge fields. The position-dependent gauge coupling is shown to be positive. In Section 3, the lowenergy effective Lagrangian is obtained. In Section 4, we examine the generality of models with the stable gauge kinetic term. Conclusion and discussion are in Section 5. Some details to derive the low-energy effective Lagrangian is given in Appendix A. In Appendix B, we summarize geometrical features of the three-flavor model.

A model with localized matter and gauge fields
In this section we first present a model allowing the domain wall solution with unbroken non-Abelian global symmetry [14]. As the second step, we introduce non-Abelian gauge fields for the unbroken symmetry. As the third step, we consider the cubic coupling [9] in Eq.(1.1) to localize non-Abelian gauge fields : we use expectation values of a singlet scalar field to give the position-dependent gauge coupling, whose positivity is demonstrated for any values of moduli parameters. A broken part of the global symmetry provides matter fields in singlet and adjoint representations of the localized gauge fields [10].

Lagrangian with global symmetry and domain wall solutions
Let us consider a five-dimensional SU (N ) c × U (1) 1 × U (1) 2 gauge theory and N scalar fields H 1 (H 2 ) in the fundamental representation with the degenerate mass m (−m), together with a singlet scalar field H 3 , whose charge assignments are summarized in Tab. 1. Therefore we obtain global symmetry SU (N ) L × SU (N ) R × U (1) A . In addition, we introduce adjoint and singlet scalars Σ and σ associated with the gauge group SU (N ) c × U (1) 1 and U (1) 2 , respectively. We assume the following Lagrangian with the signature (+, −, −, −, −) The U (N ) c = SU (N ) c × U (1) 1 gauge coupling and gauge fields are denoted by g and an N × N matrix W M with M = 0, 1, 2, 3, 4. The U (1) 2 gauge coupling and gauge field are denoted by e and A M . Covariant derivatives and field strengths are defined by and The global symmetry U L ∈ SU (N ) L , U R ∈ SU (N ) R , e iα ∈ U (1) A , and the local gauge symmetry U c ∈ U (N ) c , e iβ ∈ U (1) 2 act on the fields as Let us note that the Lagrangian (2.2) can be embedded into the five-dimensional N = 1 supersymmetric gauge theory (with 8 supercharges). This fact allows us to obtain the so-called Bogomol'nyi-Prasad-Sommerfield (BPS) domain wall solution as we will show in the next subsection. We stress, however, that this particular choice is merely to simplify calculations and is not required for our results to hold. Without loss of generality, we can assume the mass parameter m to be positive. We also assume c 1 > 0 and c 2 > 0. Then there exist N + 1 discrete vacua with r = 0, 1, . . . , N , where scalar fields develop vacuum expectation value (VEV) Local gauge symmetry is completely broken and only a subgroup of global symmetry remains in these vacua. The breaking patterns in the r = 0 and r = N vacua are Let us consider domain wall solutions connecting N -th (0-th) vacuum at left (right) infinity y = −∞ (y = ∞). Then the (coincident) domain wall solutions preserve the diagonal subgroup as the largest global * symmetry SU (N ) L+R , providing the Nambu-Goldstone modes associated to the breaking of the global symmetry We assume that fields depend only on extra-dimensional coordinate y and that all gauge fields except W y and A y vanish. * Local gauge symmetry SU (N ) c contains global symmetry as a constant gauge transformation, which is displayed in the above symmetry breaking pattern of r-th and N -th vacuua. However, Nambu-Goldstone modes only come from the genuine global symmetry which is not locally gauged. Hence we do not count the SU (N ) c transformations to preserve the color-flavor-locked vacua (2.6). The discrete symmetry in the denominator corresponds to a simultaneous rotation of the U (1) A by an integer multiple of the angle 2π/N and the center Z N of SU (N ) L−R . Physically it represents indistinguishability of N domain walls.
Thanks to the special choice of the potential, we can rewrite the energy density as Thus, we obtain the Bogomol'nyi bound for the total energy (per unit world volume) E where T is the tension of the domain wall. This bound is saturated when the following BPS equations are satisfied To use the moduli-matrix formalism [16][17][18], we introduce S(y) ∈ GL(N, C) and ψ(y) ∈ C Then the matter part (2.11)-(2.13) can be solved by with complex constant N × N moduli matrices H 0 1 , H 0 2 and a moduli constant H 0 3 , which describe moduli of the solution. The rest of the BPS equations (2.14) and (2.15) turn into the master equations for gauge-invariant Hermitian fields Ω ≡ SS † and η ≡ Re(ψ).
Moduli matrices related by the following V -transformations give identical physical fields where V ∈ GL(N, C) and v ∈ C. The equivalence class quotiented by this V -transformation defines the moduli space of domain walls. We can use this freedom to choose the form of the moduli matrices Let us also decompose H 0 2 as where φ is a Hermitian N × N matrix and U is a unitary N × N matrix. With this choice, the master equations (2.20) and (2.21) become where Ω 0 = e 2my 1 N + e 2φ e η .
No analytic solution of this system of the differential equations is known in general. However, one can study essential features of solutions, if one takes the strong gauge coupling limit g 2 , e 2 → ∞. Eqs. (2.25) and (2.26) reduce to a system of algebraic equations in this limit: It turns out that the effective theory describing massless excitations localized on the background solution of the equations of this system precisely coincides (at least in the lowest order of approximation) with the one obtained from (2.25) and (2.26) (see the detailed discussion in Ref. [10]). It is therefore sufficient just to study solutions of (2.27) and (2.28).
(2.29) Then (2.28) reduces to a polynomial equation of order N + 1 for x := e −η If this equation is solved, one can supply its solution into Eq.(2.27) to obtain Ω.
In the simplest case, where all walls are coincident φ = my 0 1 N , we can solve equation (2.30) explicitly (e 0 := e 2m(y−y0) ) to find Physical fields can be expressed in terms of Ω and σ as where we fixed the gauge such that S = Ω 1/2 and Im(ψ) = 0. This set of solutions is not invariant under the symmetry transformations (2.4) in general. For a choice of U = 1 N , however, the solutions (2.33)-(2.38) are invariant under the action of the diagonal global symmetry SU (N ) L+R+c . We show the y-dependence of e −η and of σ = ∂ y η/2 for the coincident case in Fig.2.
For more general case such as non-coincident walls, the dependence of e −η and σ on y is more complicated. Furthermore, the equation (2.30) cannot be solved in a closed form in general, except for first few values of N . Thus, one has to use numerical techniques. In Fig.  3, we present an example of five non-coincident walls.

Localization of non-Abelian gauge fields
In order to obtain massless gauge fields localized on the domain wall we need to introduce a new gauge symmetry which is not broken in the bulk.
(2.39)  where we assume that λ is a positive constant If we denote the field strength for the new gauge fields asG M N the Lagrangian for the gauged model is given by As discussed in Ref. [9], a term that is linear in adjoint scalars, appearing in front of the kinetic term for gauge field, naturally arises in five-dimensional N = 1 supersymmetric gauge theories [8]. In fact, our model can be embedded into N = 1 supersymmetry. It is not hard to see that the solution (

Positivity of position-dependent gauge coupling
We wish to show that the field-dependent gauge coupling (2.40) assures the positive definiteness of the position-dependent gauge coupling for any configurations of the domain-wall.
Since we do not need the effective theory in full, we reserve to derive the rest of the effective Lagrangian for the next section.
For the moment it is sufficient to know, that the field-dependent gauge coupling 1/g 2 (σ) is given by its value in the background solution This leads to the formula which is indeed positive in the whole range of y-coordinate. Integrating 1/g 2 (y) over the extra-dimensional coordinate y we obtain the effective gauge coupling in 3 + 1-dimensional world volume (2.47) The asymptotic values of η are found from (2.31) as The easiest way to see these asymptotic values is to note that in (2.32) we can take the limits of (2.30) to obtain  Thus, the effective gauge coupling is given as It is interesting to observe that the effective gauge coupling is proportional to the width of the domain wall ln(1 + N c 1 /c 2 ), which is not a modulus, but is fixed by parameters of the theory. This feature is in sharp contrast to that in Ref. [10] where the effective gauge coupling constant is proportional to the domain wall width, which is a modulus undetermined by the theory. We have now confirmed the stability of the gauge kinetic term (by choosing the parameters of the theory as λ, m, c 1 , c 2 > 0). Eq. (2.47) shows that the effective four-dimensional gauge coupling constantg is determined only by the boundary conditions at infinity. It can be interpreted as a kind of topological charge of the η-kink, whose profile is shown in Fig. 4.
Note that we have considered only the BPS solutions so far, and confirmed the stability of the model. The anti-BPS solutions are also stable, since they can are obtained by the parity transformation x → −x.

Effective Lagrangian
In this section we calculate the low-energy effective Lagrangian on the background domain wall solution in the moduli approximation [19], where the moduli are promoted to fields on the world-volume with coordinates x µ and are assumed to depend only weakly on x µ . As the background we consider the coincident walls φ bg = my 0 1 N . Our model have two moduli: a Hermitian N × N matrix φ and a unitary N × N matrix U . We study in two steps: taking only U as moduli fields, and then both φ as well as U .

Effective Lagrangian for chiral fields
Here we ignore the moduli fields φ, and take only U moduli fields, that describe the Nambu-Goldstone modes associated with the symmetry breaking in Eq.

2)
where Ω and η are defined in Eqs.(2.31) and (2.32) with the replacement y 0 → y 0 (x µ ) and U → U (x µ ). We plug these into the Lagrangian (2.42), where the gauge fields W µ and A µ are no longer dynamical and should be eliminated as auxiliary fields. After integrating over the extra-dimensional coordinate and taking up to quadratic terms in the derivatives, we obtain the low energy effective Lagrangian (the detailed calculation is given in Appendix A) and Let us define the decay constants f π for the adjoint field and f η for the singlet field Then the canonically normalized adjoint and singlet fields can be defined asπ and η respectively 1 The effective Lagrangian Eq. (3.4) can be rewritten as A similar effective Lagrangian has been obtained in studying Skyrmions realized inside domain walls [13]. A new feature of (3.10) compared with our previous work [10] is that the coupling strength f π of the adjoint field is larger ‡ than f η of the singlet field by a factor √ α + 1. ‡ Difference of factors of 2 and N is due to a convention of SU (N ) matrix normalization Tr(T I T J ) = δ IJ /2.

Full effective Lagrangian
As the final step, we consider the general case of both U and φ as moduli fields. Then scalar fields are given in terms of moduli fields U (x µ ) and φ(x µ ) as § 14) (3.15) To obtain the effective Lagrangian, we need to repeat the same procedure as in the previous subsection, where the covariant derivatives acting on functions of matrices require more care (see e.g. Appendix B in Ref. [10]) and causes difficulty when deriving the closed form of the effective Lagrangian. However, we have a convenient parameter to expand the effective Lagrangian, the ratio c 1 /c 2 whose logarithm has a physical meaning as the width of the domain wall (see Eq.(2.50)). As given in Appendix A, the effective Lagrangian up to the order of c 1 /c 2 is given as where with a Lie derivative with respect to A, L A (B) = [A, B]. Interestingly, this is the same effective Lagrangian we obtained previously [10]. The rest of terms are of order of c 1 /c 2 , given by , (3.20) and . (3.26) All the above integrals can be obtained in closed forms.

More general models of stable position-dependent coupling
In this section, we wish to show that there are more models with the stable positiondependent coupling. We will illustrate the point by extending the model to include more fields and more gauge symmetry. The position-dependent gauge coupling comes from the cubic coupling between a singlet scalar field and field strengths of non-Abelian gauge fields in 4 + 1 dimensions where the function C(σ i ) of singlet scalar fields σ i should be linear, if it is to be embeddable into a supersymmetric gauge theory in 4 + 1 dimensions where γ i are constant coefficients. Usually each domain wall has one complex moduli: a position and a phase. For example, both the massive CP 2 model and the massive CP 1 × CP 1 model have two free domain walls, corresponding to the two complex moduli. Although two free domain walls can produce the desired profile of position-dependent gauge coupling by an appropriate choice of parameters in Eq.(4.2), they provide the undesired modulus for the width of the profile. To avoid this problem, we are led to consider models with a single complex moduli. The simplest one of such models is the three flavor model in Ref. [9], where three scalar fields are constrained by the two Abelian gauge symmetry U (1) × U (1). Geometry of this three-flavor model is examined in Appendix B. Our model in this paper is an extension of this model to non-Abelian gauge group: U (1) × U (1) → U (N ) × U (1). The next-simplest possibility is to consider four scalars constrained by three Abelian gauge symmetry U (1) × U (1) × U (1), which we call four-flavor models. We give quantum numbers of fields of a typical four-flavor model in Tab.3. In the limit of strong gauge couplings, the gauge theory becomes a nonlinear sigma model whose target space is given by an intersection of three conditions as The vacuum condition is given by The moduli matrix formalism [18,20] is powerful enough to give generic solutions of the BPS equations of this nonlinear sigma model. Especially, we are interested in the kink profiles of σ 1 , σ 2 and σ 3 . They can be expressed as derivatives of real functions η i where the real functions η i are determined by the following algebraic conditions e −η2−η3+2m3y + e −η3+2m4y = c 3 , (4.10) with a real constant a. The parameter a is (the real part of) the unique modulus of the solution, corresponding to the position of the domain wall. As we expected, we find only a single modulus. The width of the domain wall is not a modulus, but fixed by the theory.
where γ 2,3 can be any non-negative real numbers. This class of models can be easily made to localize non-Abelian gauge fields and minimally interacting matter fields by extending two of the U (1) factor groups to (possibly different) U (N ) gauge groups with N scalar fields in the fundamental representations, similarly to our model in previous sections. A new interesting feature of the four-flavor model (and its non-Abelian extensions) is that two possible profile of singlet fields σ 1 , σ 2 can provide a different profile for different non-Abelian gauge groups such as SU (3), SU (2) and U (1) with their associated matter fields.

Conclusion and discussion
We have successfully stabilized gauge fields and matter fields that are localized on domain walls. The low-energy effective Lagrangian on the domain wall has been worked out, where the adjoint matter fields is found to couple more strongly than singlet matter fields. We also explored possible generalization of our stabilization mechanism by including more fields and more gauge symmetries and found a class of more generic models with an added flexibility for model building with different localization profile for different gauge groups.
To build realistic models of brane-world with our scenario of localized gauge fields and matter fields, we should address several questions. Perhaps the most important question is to obtain (massless) matter fields in representations like the fundamental rather than the adjoint of localized gauge fields. One immediate possibility is to use the localization mechanism of fermions in a kink background. It has been found that zero modes of such fermions are localized in such a way to give automatically chiral fermions [21]. We will pursue this direction and associated anomaly questions further.
Secondly, we should devise a way to give small masses to our matter fields in order to do phenomenology. Since some of our matter fields are the Nambu-Goldstone modes of a broken global symmetry, we need to consider an explicit breaking of such global symmetry.
Thirdly, another question is to study possibility of supersymmetric model of gauge field localization. We need to settle the issues of possible new moduli in that case. Moreover, we should examine the mechanism of supersymmetry breaking.
Another interesting possibility for model building is to localize gauge fields of different gauge groups with different profiles. This situation is often proposed in recent brane-world phenomenology, for instance in Ref. [22].
Finally, let us examine similarities and differences of our domain walls compared to Dbranes. The most interesting similarity of our domain wall with D-branes is the realization of geometrical Higgs mechanism, where massless gauge fields in the coincident wall become massive as walls separate. Detailed analysis of this phenomenon is worth doing. On the other hand, there are differences as well. D-branes in string theory are defined by the Dirichlet condition for fundamental string attached to it, but no such condition is visible in our domain walls. Domain walls similar to ours have been constructed in Ref. [23], where a probe magnetic charge is placed in the bulk Higgs phase. The magnetic flux from the probe magnetic monopole is carried by a vortex which can end on the domain wall. They observed that this wall-string junction configuration resembles an open string ending on D-branes in string theory. However, this phenomenon is in theories with one dimension less, namely the world volume of domain walls has only 1 + 2 dimensions (fundamental theory is in 1 + 3 dimensions). In our model based on a theory in 1 + 4 dimensions, monopole (codimension three) is a string-like soliton, and might possibly be a candidate of something similar to the fundamental string. It is worth pursueing a possibility of composite solitons consisting of domain walls and other solitons such as monopoles, in order to clarify more similarities of solitons with D-branes. Another interesting issue of the effective action on D-branes is the non-Abelian generalization of the Dirac-Born-Infeld (DBI) action. There have been studies to obtain first few corrections to Yang-Mills action in string theories. Since we are at present interested in up to quadratic terms in derivatives, we have found the ordinary quadratic action of Yang-Mills fields together with the action of moduli fields interacting minimally with the Yang-Mills fields in addition to the nonlinear interactions among themselves. Our result is trivially consistent with the quadratic approximation of DBI action, but does not give us informations on non-Abelian generalization of DBI action. In order to shed lights on that issue, we need to compute higher derivative corrections to our effective Lagrangian, which is an interesting future problem.

A. Derivation of the effective Lagrangian
In this appendix, we derive the effective Lagrangians (3.4) and (3.16) in the strong coupling limit. Since we are interested in the low energy effective Lagrangian, we focus on up the quadratic terms in derivatives, which we call L (2) .

A.1. Eliminating gauge fields
Here we eliminate gauge fields W µ , A µ to obtain a nonlinear sigma model. Starting withL in (2.42) in the strong gauge coupling limit g → ∞ and e → ∞, L (2) is given as where the covariant derivatives are given as: Here we singled-out covariant derivativesD µ containing V µ fields associated with the gauged part of the flavor symmetry SU (N ) L+R . Notice that W µ and A µ are no longer dynamical, but they are merely auxiliary fields to be eliminated. Moreover, the constraints are satisfied by the scalar fields (2.33)-(2.35) which depend on the moduli fields φ(x µ ) and U (x µ ) The equation of motion for W µ gives where the sum over the index a = 1, 2 is implied (Einstein summation convention). Plugging this back into (A1) we obtain Next step is to eliminate the auxiliary fields A µ . By using the following identities derived from the constraint (A3) and by solving equations of motion for A µ , we obtain By using the following identities we obtain a simpler expression for A µ Using (A12), we can rewrite the effective Lagrangian as an integral of a nonlinear sigma model over y After some algebra, covariant derivatives of these can be rewritten as where Putting this into Eq. (A12) we obtain: Substituting this result into (A13), we obtain Finally integration over y is most easily done by substituting x = e −η and using the identity: leading to Eq.(3.4) (including the kinetic term for the gauge fields).
A.3. Effective Lagrangian for all terms: φ, U and y 0 Next we derive the effective Lagrangian including all fluctuations. Plugging the solution (3.11)-(3.13) into the Lagrangian (A13), we obtain where The only work which remains to be done is to perform the integration. This, however, turns out to be difficult. The main source of difficulty lies in the fact the we cannot solve (3.15) explicitly. We therefore use some approximation technique, such as the Taylor expansion. We find it convenient to use as an expansion parameter c 1 /c 2 which determines the width of the coincident wall. In the lowest order of approximation we see that Eq.(3.15) reduces to e −η = 1 and we obtain the effective Lagrangian in Eq.(3.16).

A.4. Complexity of the Effective Lagrangian
The formula (3.16) illustrates the complexity of the interactions between moduli fields φ and U in the general case. Let us offer some explanation for this complexity. The structure, which is new in (3.16) compared to (3.17) is of the general form where M (y) and N (y) are some matrix-valued functions, containing either derivatives of φ or derivatives of U . If we assume that φ = mP −1 diag(y 1 , . . . , y N )P is diagonalizable we can rewrite the above as which is in a sense trivial, since we already encountered this kind of terms in (3.4). Thus, the new kind of complexity in our result (3.16) can be understood as a manifestation of the fact, that the interaction does not depend only on various moments of the fluctuation as in (3.17) but also on their relative size.

B. Geometry of the three-flavor model
Quantum numbers of three-flavor model are shown in Tab. B1. Since we are interested in the domain wall solutions, it is enough to consider the strong gauge coupling limit where