Gravitational self-force correction to the innermost stable circular orbit of a Schwarzschild black hole

The innermost stable circular orbit (ISCO) of a test particle around a Schwarzschild black hole of mass $M$ is located at $r_{\rm isco}=6M G/c^2$ (Schwarzschild coordinate radius). If the particle is endowed with mass $\mu(\ll M)$, it experiences a gravitational self-force whose conservative piece alters the location of the ISCO. Here we calculate the resulting shifts $\Delta r_{\rm isco}$ and $\Delta\Omega_{\rm isco}$ in the ISCO's radius and frequency, at leading order in the mass ratio $\mu/M$. We obtain $\Delta r_{\rm isco}=-3.27 \mu G/c^2$ (in the Lorenz gauge) and $\Delta\Omega_{\rm isco}/\Omega_{\rm isco}=0.487 \mu/M$ (gauge invariant). We discuss the implications of our result within the context of extreme mass-ratio binary inspirals.