Massive Scalar Propagator in Euclidean Space

In Euclidean space, the propagator for a massive scalar field with mass \(M > 0\) is\[G(x|y) \equiv \left( \frac{\hbar}{\mu} \right) \int\limits_{0}^{\infty} \mathrm{d}\xi \left( \frac{\mu}{\hbar \xi} \right)^{D/2} \exp{\left( -\frac{\mu (x - y)^{2}}{2 \hbar \xi} - \frac{M^{2} \xi}{2 \hbar \mu} \right)} \]where $D$ is the dimension of space and $\mu$ is a constant with units of mass. We have used a Schwinger parameter $\xi$ to write the propagator. This Schwinger parameter has units of length. The integral over $\xi$ can be performed explicitly to give\[G(x|y) = 2 \left( \frac{\mu^{2}}{\hbar^{2}} \right)^{\Delta} \left( \frac{\hbar}{\mu |x - y|} \right)^{\Delta} \left( \frac{M}{\mu} \right)^{\Delta} K_{\Delta} \left( \frac{M |x - y|}{\hbar} \right) \qquad \Delta \equiv \frac{D - 2}{2}\]where $K_{\Delta}$ is a modified Bessel function.

The semiclassical propagator is defined by taking $\hbar \rightarrow 0$. This corresponds to either using the asymptotic expansion for the Bessel function, or performing the integral over $\xi$ with saddle-point approximations. The result is\[G(x|y) \approx \left( \frac{\mu^{2}}{\hbar^{2}} \right)^{\Delta} \left( \frac{M}{\mu} \right)^{(D - 3) / 2} \left( \frac{\hbar^{2}}{\mu^{2} (x - y)^{2}} \right)^{(D - 1)/2} \exp{\left(- \frac{M |x - y|}{\hbar} \right)}\]We write this as an infinite series:\[G(x|y) \approx \left( \frac{\mu^{2}}{\hbar^{2}} \right)^{\Delta} \sum_{n = 0}^{\infty} \frac{(-1)^{n}}{\Gamma(n + 1)} \left( \frac{M}{\mu} \right)^{(2\Delta + 2n - 1)/2} \left( \frac{\hbar^{2}}{\mu^{2} (x - y)^{2}} \right)^{(2 \Delta - n + 1)/2}\]