$$ I_{n,m}(\boldsymbol{x},\boldsymbol{\tau}) = \dfrac{1}{(2\pi)^{n+m}} \int_{-\infty}^{\infty} \cdots \int_{-\infty}^{\infty} \dfrac{e^{i(\boldsymbol{p}\cdot \boldsymbol{x}-\boldsymbol{q}\cdot \boldsymbol{\tau})}}{|\boldsymbol{p}|^2-|\boldsymbol{q}|^2} d^n\boldsymbol{p} \ d^m\boldsymbol{q} $$ All bold variables are real euclidean vectors. Vectors $\boldsymbol{p}$ and $\boldsymbol{x}$ have dimension $n$. Vectors $\boldsymbol{\tau}$ and $\boldsymbol{q}$ have dimension $m$
I am having trouble finding documentation on this integral, mostly because I do not know what its commonly named as in the literature. I know its related to propagators in QFT, but I can only find special cases of $n$ and $m$.
Does anyone know if it has a formal name?
My first attempt
$$ I_{n,m} \propto\frac{1}{x^{n}\tau^{m}} \int_{0}^{\infty} \int_{0}^{\infty} (p)^{n/2}(q)^{m/2}\dfrac{J_{n/2-1}(p)J_{m/2-1}(q)}{(p/x)^2-(q/\tau)^2} \ dq \ dp $$ Contour integration fails if both dimensions are even.
Second Attempt
$$ I_{n,m} = \frac{1}{x^{n/2-1}\tau^{m/2-1}}\int_{-\infty}^{\infty}\phi(a)J_{n/2-1}(ax)J_{m/2-1}(a\tau)da $$ where $\phi(a)$ is a distribution satisfying $$ \int(\Box I_{n,m}) d^n p d^m q = 1 $$ and $$ \Box = \frac{1}{x^{n-1}} \frac{\partial}{\partial x} \left( x^{n-1} \frac{\partial}{\partial x} \right) - \frac{1}{\tau^{m-1}} \frac{\partial}{\partial \tau} \left( \tau^{m-1} \frac{\partial}{\partial \tau} \right) $$ I do not know how to solve for $\phi(a)$.