Research Article | Open Access
A. Tofighi, "Domain Wall Splitting in Extra Dimension", International Scholarly Research Notices, vol. 2012, Article ID 505708, 11 pages, 2012. https://doi.org/10.5402/2012/505708
Domain Wall Splitting in Extra Dimension
We study the splitting of domain walls in 5-dimensional flat Minkowski spacetime. To construct a domain wall structure we utilize a potential of the form and for simplicity we assume the scalar field to be real. We also assume the coefficient a to be temperature-dependent. We find exact analytical expressions for the energy density of the domain wall and from this expression we find close analytical form for the separation of the divided domain walls. We find that near critical temperature the domain walls split. At the critical temperature, the domain walls rejoin. In the region above this critical temperature, the kink solution is nontopological. We find that the phenomena of splitting of domain walls occurs in this region as well. This effect is especially manifest near the inflection point of the potential.
The subject of extra dimension has become an active area of research in the past decade and many models have been constructed to address fundamental problems of particle physics and cosmology, for instances the issue of hierarchy and the cosmological constant problem [1–3].
An essential idea in the Ekpyrotic model of universe  is that the hot big bang is as a result of collisions of two branes. An important feature of this model is nucleation and splitting of the branes. To realize the phenomena of splitting of branes, a complex scalar field is coupled to gravity in a five-dimensional world with warped geometry . In this work, a numerical solution is found and the region close to the critical temperature is studied. In a recent attempt , a real scalar field is utilized to describe brane splitting and localization of fermions in five dimensions. They consider the case in the Newtonian limit as well as the case with the inclusion of gravity. Numerical techniques are also employed to obtain their results.
In the present work, we attempt an analytical treatment for the discussion of the brane splitting, and for simplicity we mainly consider the Newtonian limit. In this limit, it is possible to obtain analytical expressions [7, 8] for the equations of motions in the static case. Having these exact solutions, we compute the energy density of the domain wall, and by mapping out the structure of the critical points of this quantity we find analytical expressions for the distance between the separated domain walls.
The plan of this paper is as follows. In Section 2, we describe the model. And we unravel the critical structure of the potential . In Section 3, we discuss the equation of motion in the static case. We find the domain wall solutions in different regions and we compute the energy density for each case. We find that the general form of this energy density has two symmetric maxima with respect to a minima at the origin. We notice that away from a critical value of the mass parameter the solution is a single kink, but the energy density of this kink is not sharp and has a slight distortion manifested as two maxima of this quantity. But very close to this critical value of the mass parameter these two maxima of energy density comprise two separate kinks. Finally we conclude in Section 4. Technical details are explained in the appendix.
2. The Model
In this work, we assume a flat background metric, , where the capital Latin letters run over five dimensions and the Greek letters over the four dimensions. The space coordinates are decomposed into the subset and the extra dimension direction . The action contains only one real scalar field and it is defined by where the potential is given by , and are real and nonnegative quantities. Now if we assume that all of these parameters be strictly positive, then the potential will have three local minima, one is corresponding to a disordered bulk phase and the other two are at with They correspond to ordered bulk phases. The potential also contains maxima at , where The parameters and may not be constant. In this work, we only investigate the variation of the mass parameter on the structure of the domain wall . We also assume that is only a function of the extra dimension , namely, .
3. Effects of the Variation of the Mass Parameter on the Thick Brane Solutions
The equation of motion from the action is While resorting to numerical solutions [5, 6], the strategy is to prescribe some boundary conditions. However, in the exact treatment, the boundary conditions are a by-product of the solutions. We distinguish three different regions.
3.1. The Region below Critical Temperature
In this region, . We present the exact solution of (3.1) in the following form: where the coefficients , are and finally the coefficient is given by The energy density of the scalar field is given by In the appendix, we show that this function has two maxima which are symmetric with respect to . We also compute the distance between these two maxima, which is given by where . If these maxima are well separated then they designate the case of two separate domain walls. In this work, all of the figures are drawn for the case , , and we only consider the effect of variation of the mass parameter on physical quantities. Figure 1 shows the separation between two maxima versus for . We see that close to the critical point , the separation between domain walls is increased. To investigate this region, we assume with small positive . Figure 2 shows this phenomena of domain wall splitting when .
Figure 3 shows the scalar field for (solid curve), (dashed curve), and (dotted curve). Our analytical results confirm the previous result that near the phase transition the domain wall splits to two domain walls, and each of these domain walls connects an ordered phase to a common disordered phase .
3.2. At Critical Temperature
The case of critical temperature corresponds to . From (3.4), the value of becomes unity, and hence from (3.3) the coefficient is identically zero. Therefore, the solution as described by (3.2) does not exist at critical temperature. The treatment of  is numerical and its discussion in the case of is incorrect. Namely its discussion is valid only for . The scalar field in this case is given by Figure 4 shows the shape of the scalar field (dashed curve), and the associated energy density solid curve) is shown. The splitting of the domain walls disappears at this temperature and we have a scalar field which connects an ordered phase to a disordered phase.
3.3. The Region above Critical Temperature
Here we will discuss the region . The scalar field in this case is given by where and again .
The distance between the two maxima of the energy density in this region is The structure of critical points of the potential as stated in Section 2 holds if . Therefore, the mass parameter is bounded from above. Hence in this situation, . Figure 7 shows the the variation of versus . We see that away from the boundaries we have two maxima but the distance between these maxima is such that we do not have two separate kinks. However, again we see that close to the boundaries we have an enhancement in the distance between the two kinks. To study this effect, we assume . Figure 8 represents this effect.
With and we will have . By choosing , we can have an estimate of splitting of domain wall near the inflection point (). Figure 9 shows this result.
In this paper, we have provided an analytical treatment of the phenomena of domain wall splitting. For simplicity, we have neglected gravity in this paper. In the case of gravity, the expression for the matter energy density is where is the warp factor and it is found from the solution of the Einstein equations. In general, it is not easy to obtain an analytical expression for the warp factor for the potential. For , we suppose that an analytical treatment presented here, with inclusion of gravity, is not possible due to the complicated nature of . However the numerical work on this subject [5, 6] demonstrates that this phenomena occurs with inclusion of gravity as well. But these studies do not address the case of critical temperature or beyond the critical temperature.
We only considered the static case in this work. It will be of interest to consider the time-dependent solutions and its implications. We hope to report on these issues in the future.
Profile of the Energy Density of the Scalar Field
From (3.5), we can obtain At points , , we have . Therefore, they are the extremal points of the function . To see the nature of these extremal points we take derivative of (A.1) with respect to , and we have But for and , the first term in the right-hand side of (A.2) vanishes. So for these points , and they are the two minima for the function . A well-known result that a topological kink connects two distinct minima of the potential . We can repeat the argument for and show that this point is also a minima of the energy density profile. The energy density of the scalar field versus has three minima: , , and . These are associated to the three minima of the . So from (3.2), we have The energy density has two maxima, and , for the case of , their values are obtained from We can see that at these points as at . And as is negative for . Therefore and are indeed the two maxima for the energy density. So we have managed to map out the structure of the critical point of the function . Now utilizing (A.3) and (A.4), we find the separation of the two maxima as and the result is
But when , the values of and are determined from And by repeating the same procedure, we find the distance between the two maxima of the energy density for the nontopological solution is
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Copyright © 2012 A. Tofighi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.