We derive a multiscale generalisation of the Bakry--Émery criterion for a measure to satisfy a Log-Sobolev inequality. Our criterion relies on the control of an associated PDE well known in renormalisation theory: the Polchinski equation. It implies the usual Bakry--Émery criterion, but we show that it remains effective for measures which are far from log-concave. Indeed, using our criterion, we prove that the massive continuum Sine-Gordon model with $β< 6π$ satisfies asymptotically optimal Log-Sobolev inequalities for Glauber and Kawasaki dynamics. These dynamics can be seen as singular SPDEs recently constructed via regularity structures, but our results are independent of this theory.