Rotor return-kernel fluctuation prefactor ($Z_0=\sqrt{I/(2\pi {\hslash }^2\beta )}$)

Writing $\varphi ={\varphi }_0+2\pi m\tau /\beta +\eta$ with endpoint-fixed $\eta$ separates winding action and Gaussian fluctuations. The latter give the free return kernel on the lifted line, $Z_0=\sqrt{I/(2\pi {\hslash }^2\beta )}$. The starting-angle integral supplies $2\pi$. Zero winding is not the same as the zero-momentum ground-state sector.