Consider the Euclidean propagator
\begin{equation} \Delta_E (x_1-x_2)\equiv \int \frac{d^4p}{(2\pi)^4} \frac{e^{ip\cdot(x_1-x_2)}}{p^2+m^2}. \end{equation}
I am a bit confused as to whether the 4-momentum, $p$, in the propagator contains implicit mass dependence in this scenario.
e.g., suppose one took a derivative of the propagator with respect to mass. Would this simply yield the following?
\begin{align} \frac{\partial}{\partial m} \Delta_E (x_1-x_2)&=-2m\int \frac{d^4p}{(2\pi)^4}\frac{e^{i p\cdot(x_1-x_2)}}{(p^2+m^2)^2} \\ &=-\frac{m}{4\pi}K_0 \left(m|x_1-x_2|\right), \end{align} where $K_0$ is the modified Bessel function of the second kind.
Or does the 4-momentum carry some mass dependence that would complicate this?
