This is more a question of the validity or feasibility of using the Feynman trick to compute integrals of the above form. I'm sorry I haven't worked out a lot of what I've done since it's quite involved, not easy to type out, and probably completely useless for the questions I'm asking.
As part of an attempt at approximating an integral for work, I have managed to reduce it to the following form:
$$\int \frac{d^3 p}{(2\pi)^3} \frac{|p|^4}{(|p|^2 + m_1^2)\,\,\,(|p|^4 + m_2^4)} \frac{\sin^4\theta}{(|p-k|^2 + m_1^2)\,\,\,(|p-k|^4 + m_2^4)}$$
I'm a lot more hopeful of finding a "closed form" solution of this in terms of $m_1$ and $m_2$, two positive constants that I'd like to vary to describe different physical situations.
First attempt at a solution:
Understandably, I moved into spherical coordinates and the angular part of the integral is doable, I can plug it into Mathematica and get the results. The radial part is also technically doable, but far more complicated and involved. I will of course resort to this if all else fails. However, I wanted to try something cleverer, if possible.
I was wondering if the Feynman trick or something like it would work here, since it is something we use very often in Quantum Field Theory integrals of the same form. Of course, we now have a $|p|^4$ in the denominator. As far as I've understood, the parametrisation works because we're able to complete a square at some point and then shift the variable from $\vec{p} \to (\vec{p}-x\vec{k})$ which greatly simplifies the integral. However, if I used the same technique to the four terms here, I would need at some point to "complete a fourth-power" and I'm not quite sure how to do that. Also, terms such as $|\vec{p}-x\vec{k}|^4$ would have pesky factors of $x^3$ and $x^4$ which I would have to add by hand and the integral becomes overall just as complicated (unless I'm doing something wrong).
Questions:
1) Would anyone know if it is feasible to use something like the Feynman or Schwinger parametrisation to evaluate integrals of this kind? Or would it be too difficult because of the quartic term?
2) Could I simply break up the quartic term into
$$|p|^4 + m_2^4 = (|p|^2 + i m_2^2)(|p|^2 - i m_2^2)$$
or am I being to naive and would this not be allowed (the integral is completely Euclidean, over $\mathbb{R}_3$). And even if I could, this would be an integral with 6 terms within it. Has anyone attempted to calculate something similar? Would it be doable?
3) Also, does the existence of the $\sin^4 \theta$ rule out the possibility of using such a trick? I've never encountered it so far, and I suspect that the change of variables will no longer hold in this case.
4) Any other tips for elegantly evaluating integrals such as these would be greatly appreciated. As I said, I would like the results in terms of the two parameters $m_1$ and $m_2$, where $m_1$ is of the order of $10^{-2}$ and $m_2$ of the order of $3$.
Thanks!
