String-inspired higher-curvature terms and the Randall-Sundrum scenario

We consider the $O\left({\alpha }^{\prime }\right)$ string effective action, with Gauss-Bonnet curvature-squared and fourth-order dilaton-derivative terms, which is derived by a matching procedure with string amplitudes in five space-time dimensions. We show that a non-factorizable metric of the Randall-Sundrum (RS) type, with a four-dimensional conformal factor $e^{-2k|z|},$ can be a solution of the pertinent equations of motion. The parameter k is found to be proportional to the string coupling $g_s$ and thus the solution appears to be non-perturbative. It is crucial that the Gauss-Bonnet combination have the right (positive in our conventions) sign, relative to the Einstein term, which is the case necessitated by compatibility with string (tree) amplitude computations. We study the general solution for the dilaton and metric functions, and thus construct the appropriate phase-space diagram in the solution space. In the case of an anti–de Sitter bulk, we demonstrate that there exists a continuous interpolation between (part of) the RS solution at $z=+\infty$ and an (integrable) naked singularity at $z=0.\mathrm{}$ This implies the dynamical formation of domain walls (separated by an infinite distance), thus restricting the physical bulk space-time to the positive z axis. Some brief comments on the possibility of fine-tuning the four-dimensional cosmological constant to zero are also presented.