a$, and $k_\mu=q_\mu/a$ is the momentum variable.

In the limit $a \to 0$, the correlation function (\[hyper\]) reduces to the standard continuum expression $$\left\langle \varphi^\alpha_x \varphi^\beta_{x'} \right\rangle = \delta_{\alpha,
\beta} \int_{-\infty}^{\infty} \frac{\mathrm{d}^d k}{(2 \pi)^d}
\frac{\mathrm{e}^{\mathrm{i} k \cdot (x-x')}}{k^2 + m^2}.$$ The momentum integral can be easily carried out by introducing the Schwinger proper time representation[@proptime] of the integrand, which is based on Euler’s form $$\label{Euler}
\frac{1}{A^z} = \frac{1}{\Gamma(z)} \int_0^\infty \frac{\mathrm{d} s}{s}
\, s^z \, \mathrm{e}^{- s A},$$ where $\Gamma(z)$ is the Euler gamma function. Specifically, $$\begin{aligned}
\label{continuum}
\left\langle \varphi^\alpha_x \varphi^\beta_{x'} \right\rangle &=&
\delta_{\alpha, \beta} \int_0^{\infty} \mathrm{d} s \, \int
\frac{\mathrm{d}^d k}{(2 \pi)^d} \, \mathrm{e}^{\mathrm{i} k \cdot
(x-x')} \, \mathrm{e}^{-s \left(k^2+m^2\right)} \nonumber \\ &=&
\delta_{\alpha, \beta} \int_0^{\infty} \mathrm{d} s \, \left( \frac{1}{4
\pi s} \right)^{d/2}  \mathrm{e}^{-(x-x')^2/4s} \, \mathrm{e}^{-s m^2}
\nonumber \\ &=& \delta_{\alpha, \beta} \frac{1}{(2 \pi)^{d/2}}
\left(\frac{m}{|x-x'|}\right)^{d/2-1} K_{d/2-1}(m|x-x'|),\end{aligned}$$ with $K_{d/2-1}$ a modified Bessel function. The integral in the second line is the continuum limit of the lattice expression (\[discrete\]), with the sum over the number of steps $b$ taken by the particle replaced by an integral over the Schwinger proper time parameter $s$, with the identification $$\label{identification}
b a^2/2d \equiv s.$$ The parameter $s$ is related to the proper time $\tau_\mathrm{p}$ 