Localization of Vector Field on Dynamical Domain Wall

In the previous works (arXiv:1202.5375 and 1402.1346), the dynamical domain wall, where the four dimensional FRW universe is embedded in the five imensional space-time, has been realized by using two scalar fields. In this paper, we consider the localization of vector field in three formulations. The first formulation was investigated in the previous paper (arXiv:1510.01099) for the $U(1)$ gauge field. In the second formulation, we investigate the Dvali-Shifman mechanism (hep-th/9612128), where the non-abelian gauge field is confined in the bulk but the gauge symmetry is spontaneously broken on the domain wall. In the third formulation, we investigate the Kaluza-Klein modes coming from the five dimensional graviton. In the Randall-Sundrum model, the graviton was localized on the brane. We show that the $(5,\mu)$ components $\left(\mu=0,1,2,3\right)$ of the graviton are also localized on the domain wall and can be regarded as the vector field on the domain wall. There are, however, some corrections coming from the bulk extra dimension if the domain wall universe is expanding.


I. INTRODUCTION
There is a long history in the scenarios that our universe could be a brane or domain wall embedded in a higher dimensional space-time [1,2]. After the discovery of the so-called D-brane solution in string theories [3,4], the brane world scenarios [5][6][7][8][9] or the domain wall senario [10][11][12][13][14][15][16][17][18][19][20] have been well studied. In the studies, the models of the inflationary brane using the trace anomaly have been proposed [21][22][23]. The brane can be regarded with a limit that the thickness of the domain wall vanishes. Recently a model where the general FRW universe is embedded in the five dimensional space-time with an arbitrary warp factor by using two scalar fields [24,25] 1 In this paper, we investigate the localization of the vector field in the model [24,25] by using three formulations. The localization of the graviton has been shown in [25] and the localizations of the spinor field and vector field have been also investigated and shown in [27]. The first formulation of the localization of the vector field in this paper is just the review of the work in [27] about the U (1) gauge field by using the action of the five dimensional vector field. The second formulation is an extension of the Dvali-Shifman mechanism in [28], where the non-abelian gauge field is confined in the four dimensional bulk but the gauge symmetry is spontaneously broken on the three dimensional domain wall. An extension of the work in [28] on the static four dimensional domain wall has been investigated in [29] and in this paper, we further extend the mechanism to the dynamical domain wall model. As the third formulation, we investigate the Kaluza-Klein modes coming from the five dimensional graviton. In the third formulation, we consider the vector field coming from the Kaluza-Klein reduction. In the second Randall-Sundrum model [6], the graviton was localized on the brane. The localized graviton can be regarded as a zero mode of the five dimensional graviton. We show that the (5, µ) components (µ = 0, 1, 2, 3) of the graviton are also localized and can be regarded as the vector field on the four dimensional domain wall. We show that, however, there appear some corrections coming from the bulk extra dimension if the domain wall is dynamical.
In the next section, we briefly review on the formulation of the dynamical domain wall based on [24,25]. In section III, we also review on the localization of the vector field in [27]. In section IV, we extend the formulation in [28] and [29] to the four dimensional dynamical domain wall model. In section V, we consider the Kaluza-Klein vector field coming from the five dimensional graviton. The last section VI is devoted to the summary of the obtained results.

II. DOMAIN WALL MODEL WITH TWO SCALAR FIELDS
In [24,25], the formulation of the dynamical domain wall model have been proposed by using two scalar fields. The formulation could be regarded as an extension of the formulation in [30] 2 .
The metric of the five dimensional space-tme embedded a general spacially flat FRW universe with an arbitrary warp factor is given by Here ds 2 FRW is the metric of the FRW universe, In [24,25], the following action with two scalar fields φ and χ were considered, We can construct a model to realize the arbitrary metric (1) by using the model (3). The energy-momentum tensor for the scalar fields φ and χ in the model (3) are given by The variations of φ and χ give the following field equations, Here A φ = ∂A(φ, χ)/∂φ, etc. By choosing φ = t and χ = w, we obtain By using the Einstein equation and the equations in (7), we find A, B, C, and V can be expressed as follows, Here G µν is the Einstein tensor. The explicit forms of A(φ, χ), B(φ, χ), C(φ, χ), and V (φ, χ) can be obtained by replacing t and w in the r.h.s. of Eqs. (8) by φ and χ. The obtained expressions in the action (3) gives a model which realize the metric (1). Eqs. (5) and (6) are satisfied automatically, which can be seen by using the Bianchi identity ∇ ν R µν − 1 2 Rg µν = 0.

III. LOCALIZATION OF VECTOR FIELD
In this section, we review on the localization of the vector field by using the formulation in [27]. We consider the following action of the five dimensional vector field, In the background (1) with (2), we assume that e u(t,w) is given by the product of the t-dependent part and w-dependent part, e u(t,w) = T (t) W (w), Under the assumption (10), the action (9) has the following form, The variations of A 5 , A 0 , and A i give the following equations, If we assume and choose we rewrite Eqs. (12), (13), and (14) as follows, Eqs. (18) and (19) are nothing but the field equations of the vector field in four dimensions. On the other hand, Eq. (17) can be regarded as a gauge condition, which is a generalization of the Landau gauge, ∂ µ A µ = 0. By choosing X(w) decreases rapidly enough for large |w|, A µ becomes normalizable. Then if we choose m(χ) as in (16), the vector field localizes on the domain wall.

IV. DVALI-SHIFMAN MECHANISM
In [28], the non-abelian vector field on the three dimensional domain wall embedded in the four dimensional spacetime was considered. In the bulk space-time, the vector field is confined but on the domain wall, the scalar field which generates the domain wall also change the potential of the Higgs field and there occurs the spontaneous breakdown of the gauge symmetry and massless U (1) gauge field appears on the domain wall. An extension of the scenario was proposed in [29], where the four dimensional domain wall in the five dimensional space-time was considered and by the mechanism similar to that in [28], the standard model could be realized on the domain wall. In this section, we consider a similar mechanism on the dynamical domain wall.
We consider the following action for the SU (2) gauge field, Here G a MN is the field strength of the SU (2) field and we also include the scalar field η a which is the adjoint representation of SU (2). The parameters κ and v have the dimension of mass and λ is a dimensionless positive parameter. We assume κ 2 − v 2 < 0 and also the metric in (10).
In the limit of |w| = |χ| → ∞, because the potential for the scalar field η a is given by 1 2 λ(η 2 + κ 2 ) 2 , there does not occur the breakdown of the SU (2) gauge symmetry if the gauge coupling is strong enough. On the other hand, on the brane, w = χ ∼ 0, the potential becomes 1 2 and therefore SU (2) guge symmetry is spontaneously broken. By substituting the expression of η a into (21) into the equation of the motion we obtain which tells Because the η a has a vacuum expectation value, the gauge field obtains a mass, Then by using (25), we find By substituting (10) into the action (20), we obtain Then the equations for the gauge fields are given by For the massless vector field A 3 M , by choosing and in the order of O g 0 , Eqs. (29), (30), and (31) reduce to the equations for the vector field and the gauge fixing condition, Therefore the massless gauge field appears on the domain wall.
In [29], the confinement in the bulk space-time was assumed but in the dimensions higher than four, there could be a phase transition and the confinement could occur only in the strong coupling region. Then we may consider the scalar field, which also plays a role of the gauge coupling. The scalar field depends on the coordinate w in the extra dimension and the gauge coupling can become strong and the confinement always occurs in the bulk space-time.

V. KALUZA-KLEIN REDUCTION
In the Randall-Sundrum model, the massless graviton in four dimensions appears as a zero mode, or normalized and localized mode, of the five dimensional graviton. On the other hand, in the standard Kaluza-Klein model, the vector field appears as the fluctuation h 5µ of the (5, µ) components of the metric (µ = 0, 1, 2, 3). Therefore if the (5, µ) components are also localized on the brane or the domain wall, the modes can be regarded as the vector field in four dimensions. In this section, we investigate the possiblity that the vector field appears due to the Kaluza-Klein reduction.
We consider the fluctuation arround the background space-time, g AB = g AB + h AB . Then we obtain the following expressions, We now impose the gauge condition ∇ A h AB = 0. Then the action has the following form, Then by the variation of the action with respect to h AB , we obtain the following equation, Because we are interested in the (5, µ) component, we put h µν = h 55 = 0. Then (5, µ) component of Eq. (40) has the following form, A. Localization on Flat Domain Wall Before considering the FRW universe, we first consider the case that the four dimensional domain wall is flat as in the Randall-Sundrum model. Then the metric has the following form, Then Eq. (41) has the following form, The deviation of (43) is given in the Appendix A. By assuming h 5µ (x ν , w) = N (w)A µ (x ν ), we consider the following Lagrangian density of the scalar field instead of (3) (see Ref. [24]), The Lagrangian density is given by putting A(φ, χ) = B(φ, χ) = 0, C(χ) = C(φ, χ)| φ=0 , and V(χ) = V (φ, χ)| φ=0 . We also used (8) in the second equality in (44). Then we obtain, If we choose Eq. (45) coincides with the expression of the standard equation for the vector field in four dimensions, For example, we consider the case, u(w) = −2 w 2 + w 2 0 , we find N ∝ e −2 √

B. Localization on the Dynamical Domain Wall
We now consider the case that the domain wall is dynamical, that is, the FRW universe is embedded in five dimensional bulk space-time as in (1). Then the equation for the graviton is given by The derivation of (48) is given in Appendix B. We assume the Lagrangian density L m is given by (3). By substituting the expression of L m into (48) and using the gauge fixing condition ∇ A h A5 = 0, again, we find By assuming h 5µ (x ν , w) = N (w)A µ (x ν ), Eq. (49) can be rewritten as In case that u can be separated into a sum of w-dependent part u w (w) and t-dependent part u t (t), that is, u(w, t) = u w (w)+u t (t) as in (10) (46), the first term vanishes in (50) and we obtain On the other hand, the vector field in the four dimensional FRW space-time obeys the following equation, There are some differences between (51) and (52) even if we choose the gauge condition∇ ν A ν = 0. Therefore the vector field can localized on the domain wall even if dynamical but there appear some corrections from the extra dimensions.

VI. SUMMARY
In summary, the localization of vector field in the model [24,25] has been investigated by using three formulations.
1. The first formulation was just the review of the work in [27], where we have used the action of the five dimensional vector field.
2. The second formulation was an extension of those in [28] and [28]. In this formulation, the non-abelian gauge field is confined in the bulk space-time but massless U (1) gauge field appears due to the spontaneous breakdown of the gauge symmetry. In [29], the confinement in the bulk space-time was assumed in the five dimensional bulk space-time. It is known, however that there could be a phase transition in the dimensions higher than four and the confinement could occur only in the strong coupling region. Then we may consider the model where the scalar field plays a role of the gauge coupling. The strong coupling phase can be always realized if the gauge coupling is given by the scalar field depending on the corrdinate in the extra dimension and the coupling becomes strong enough and the confinement always occurs in the bulk space-time.
3. The third formulation was given by the Kaluza-Klein modes coming from the five dimensional graviton. In the second Randall-Sundrum model [6], the graviton was localized on the brane. The localized graviton can be regarded as a zero mode of the five dimensional graviton. We have shown that the (5, µ) components (µ = 0, 1, 2, 3) of the graviton are also localized on the domain wall and can be regarded as the vector field on the four dimensional domain wall. We found that, however, some corrections appear from the bulk extra dimension if we consider the dynamical domain wall. An interesting point is that if we have several extra dimensions and the extra dimensions have a symmetry under the non-ablian group transformation, there could appear the non-abelian gauge theory localized on the domain wall.
Then it might be interesting to realize the GUT on the domain wall.
This work is supported (in part) by MEXT KAKENHI Grant-in-Aid for Scientific Research on Innovative Areas "Cosmic Acceleration" (No. 15H05890) and the JSPS Grant-in-Aid for Scientific Research (C) # 23540296 (S.N.).
Appendix A: Derivation of (43) Here we consider the derivation of of (43). We have the following expressions of the connection the Riemann tensor, Ricci tensor, and the scalar curvature Then we find Then by substituting the above expressions into (41), we obtain Eq. (43).