Localized Gravitons, Gauge Bosons and Chiral Fermions in Smooth Spaces Generated by a Bounce

The paper asks whether the Randall-Sundrum mechanism — trapping four-dimensional physics on a thin slice of a five-dimensional curved space — really needs the sharp, infinitely thin brane, or whether a smooth lump of scalar field can do the same job. The authors solve Einstein's equations coupled to a scalar field with a double-well potential and find a bounce solution $φ_B(y)=v tanh(ay)$ that generates a smooth warp factor $A(y)=-β ln cosh^2(ay)-β^2 tanh^2(ay)$, which falls off exponentially at large distance exactly like the Randall-Sundrum warp factor but has no delta-function anywhere. On this background they show that the graviton zero mode $ψ_0 = N e^{2A(y)}$ is normalizable, that a bulk fermion coupled to the bounce deposits exactly one chiral massless mode on the wall, and that a constant gauge zero mode, normally unable to see the warp factor at all, becomes normalizable once a dilaton couples to the gauge kinetic term. They then repeat the gauge argument starting from an ordinary brane, showing the dilaton trick does not depend on the smooth bounce. The point of the exercise is that brane-like localization of all the Standard Model degrees of freedom can be recovered from a fully regular field configuration, so the brane need only be an approximation rather than a fundamental object.