Boundary localized symmetry breaking and topological defects

We discuss the structure of topological defects in the context of extra dimensions where the symmetry breaking terms are localized. These defects develop structure in the extra dimension which differs from the case where symmetry breaking is not localized. This new structure can lead to corrections to the mass scale of the defects which is not captured by a simple effective theory obtained by integrating out the extra dimension. We also consider the Higgsless model of symmetry breaking and show that no finite energy defects appear in some situations where they might have been expected.

Figure 1

The value of the string profile on the symmetry breaking brane, $\rho (r,L)$, for the two solutions: $n=0$ (flat line) and $n=1$. All quantities are in units of the bulk mass parameter. We have chosen $n=1$, $L=5/2$, and $\tilde{\lambda }=3$.

Figure 2

The value of $\rho (r,y)$ throughout the bulk with the same parameters as before. The value for $r$ goes from 0 to 8, while $y$ goes from 0 to 2.5.

Figure 3

A comparison of the energy density as a function of $r$ in the cases where symmetry breaking happens on the brane (lower curve) and the pure 4D model (upper curve). The string tension comes from multiplying these curves by $2\pi v^{3}r$ and integrating over $r$. The difference goes to zero faster than $1/r^2$ at large $r$, so they have identical large distance log divergences as they must if the effective theory is to capture the IR physics.

Figure 4

The profile for the Higgs field as represented by the function, $f(r)$. The bottom curve is from the 4-dimensional model. The parameters for this solution are $\beta =1$ and for the 5-dimensional model, $e^2{L}^2=5$.

Figure 5

The profile for the gauge field. The bottom curve is $\alpha (r)$ from the 4-dimensional model while the top curve is the gauge field on the symmetry breaking brane: $\alpha (r,L)$. Both curves are going to zero as $r^2$. The model parameters are the same as before.

Figure 6

The value of $\alpha (r,y)$ throughout the bulk. The model parameters are the same as in Fig. 4. The value of $r$ goes from 0 to 8, while $y$ goes from 0 to 1.

Figure 7

The energy density of the string solution as a function of $r$. This is the quantity in square brackets in Eq. (63) and this figure is analogous to Fig. 3. The 5D model peaks at a higher energy density in the core of the string, but the string tension comes from multiplying these functions by $2\pi r{v}^2$ and then integrating. As a result, the string tension for the 5D model is smaller. The parameters are the same as Fig. 4.

Figure 8

The string tension, $g(\beta )$. The top line is the 4D model where $g(1)=1$. The next three lines down are for the 5D model with $e^2{L}^2=0.2$, 1.0, 5.0, respectively.