Transplanckian bremsstrahlung and black hole production

Classical gravitational bremsstrahlung in particle collisions at transplanckian energies is studied in ${\mathcal M}_4\times {\mathcal T}^d$. The radiation efficiency $\epsilon\equiv E_{\rm rad}/E_{\rm initial}$ is computed in terms of the Schwarzschild radius $r_S(\sqrt{s})$, the impact parameter $b$ and the Lorentz factor $\gamma_{\rm cm}$ and found to be $\epsilon=C_d (r_S/b)^{3d+3} \gamma_{\rm cm}^{2d+1}$, larger than previous estimates by many powers of $\gamma_{\rm cm}\gg 1$. The result is reliable for impact parameters in the overlap of $r_S<b<b_c {\rm and} b>\lambda_C$, with $b_c$ marking (for $d\neq 0$) the loss of the notion of classical trajectories and $\lambda_C\equiv \hbar/mc$ the Compton length of the scattered particles. The condition on $s$ and $m$ for extreme radiation damping and (presumably) no black hole production is also derived.