On Superplanckian Scattering on the Brane
The multidimensional space-time with $(D-4)$ compact extra space dimensions and SM fields confined on four-dimensional brane is considered. The elastic scattering amplitude of two particles interacting by gravitational forces is calculated at superplanckian energies. A particular attention is paid to a proper account of zero (massless) graviton mode. The renormalized Born pole is reproduced in the eikonal amplitude which makes a leading contribution at small momentum transfers. This singular part of the amplitude coincides with well-known $D$-dimensional amplitude taken at $D \to 4$. The expression for a contribution from massive graviton modes to the eikonal is derived, and it asymptotics in the impact parameter are calculated. Our formula gives correct four-dimensional result at $R_c \to 0$, where $R_c$ is the radius of the higher dimensions, contrary to formulae obtained in recent papers on collisions of particles living on the brane. The results are also compared with those obtained previously for the scattering of the bulk fields in flat higher dimensions.