Does Compton/Schwarzschild duality in higher dimensions exclude TeV quantum gravity?

Abstract

In three spatial dimensions, the Compton wavelength $(R_C \propto M^{-1}$) and Schwarzschild radius $(R_S \propto M$) are dual under the transformation $M \rightarrow M_{P}^2/M$, where $M_{P}$ is the Planck mass. This suggests that there could be a fundamental link -- termed the Black Hole Uncertainty Principle or Compton-Schwarzschild correspondence -- between elementary particles with $M < M_{P}$ and black holes in the $M > M_{P}$ regime. In the presence of $n$ extra dimensions, compactified on some scale $R_E$ exceeding the Planck length $R_P$, one expects $R_S \propto M^{1/(1+n)}$ for $R_P < R < R_E$, which breaks this duality. However, it may be restored in some circumstances because the {\it effective} Compton wavelength of a particle depends on the form of the $(3+n)$-dimensional wavefunction. If this is spherically symmetric, then one still has $R_C \propto M^{-1}$, as in the $3$-dimensional case. The effective Planck length is then increased and the Planck mass reduced, allowing the possibility of TeV quantum gravity and black hole production at the LHC. However, if the wave function of a particle is asymmetric and has a scale $R_E$ in the extra dimensions, then $R_C \propto M^{-1/(1+n)}$, so that the duality between $R_C$ and $R_S$ is preserved. In this case, the effective Planck length is increased even more but the Planck mass is unchanged, so that TeV quantum gravity is precluded and black holes cannot be generated in collider experiments. Nevertheless, the extra dimensions could still have consequences for the detectability of black hole evaporations and the enhancement of pair-production at accelerators on scales below $R_E$. Though phenomenologically general for higher-dimensional theories, our results are shown to be consistent with string theory via the minimum positional uncertainty derived from $D$-particle scattering amplitudes.