Massless bosons on domain walls: Jackiw-Rebbi-like mechanism for bosonic fields

It is important to obtain (nearly) massless localized modes for the low-energy four-dimensional effective field theory in the brane-world scenario. We propose a mechanism for bosonic zero modes using the field-dependent kinetic function in the classical field theory setup. As a particularly simple case, we consider a domain wall in five dimensions and show that massless states for scalar (0-form), vector (1-form), and tensor (2-form) fields appear on a domain wall, which may be called topological because of the robustness of their existence (insensitive to continuous deformations of parameters). The spin of localized massless bosons is selected by the shape of the nonlinear kinetic function, analogously to the chirality selection of the fermion by the well-known Jackiw-Rebbi mechanism. Several explicitly solvable examples are given. We consider not only (anti) Bogomol’nyi-Prasad-Sommerfield (BPS) domain walls in a noncompact extra dimension but also non-BPS domain walls in a compact extra dimension.

Figure 1

The field-dependent “mass” $\mathcal{M}(T(y))$ and the associated topological edge states (fermion zero modes). The solid curves are $\mathcal{M}=\pm (\tanh y+\frac{1}{10}{e}^{-y^2/100}\sin y)$, and the broken red curves correspond to the mode functions of the fermion zero modes.

Figure 2

The normalized mode functions of bound states. The black solid curves show the Schrödinger potentials appearing in $Q^{†}Q$ and $Q{Q}^{†}$. (a1) and (b1) correspond to $Q^{†}Q$ in the sine-Gordon model and the $T^4$ model, respectively. Similarly, (a2) and (b2) correspond to $Q{Q}^{†}$ in the sine-Gordon model and the $T^4$ model, respectively. The red dashed curves are the mode functions of the corresponding zero modes while the blue dotted curves are the lightest massive bound states.

Figure 3

Non-BPS domain wall solutions with the BPS and anti-BPS walls at $y=0$ and $\pi R$ in the sine-Gordon model. The left-most column shows the background configuration $gT/\mathrm{\Lambda }$, and the other three columns show mode functions of the small fluctuations for the case of ${\epsilon }_{\mathrm{F},\mathrm{B}}=+1$. In the figures, only the half period is shown. The case of ${\epsilon }_{\mathrm{F},\mathrm{B}}=-1$ can be obtained by exchanging $(f_0,h_0)$ by $({\tilde{f}}_0,{\tilde{h}}_0)$.

Figure 4

The mode functions of the topological bosons for the noncanonical fields $\mathrm{\Phi },{\mathcal{A}}_M,{\theta }_{MN}$ around the non-BPS domain wall solution. The case of ${\epsilon }_B=1(-1)$ is shown in the left (right) panel.