# How to deal with two-loop integrals in dimensional regularization?

I've done several calculations on one-loop diagrams in dimensional regularization, involving things like Feynman parameters, or using hyperspherical coordinates after a Wick rotation, s.t. you can drop terms like $$p^\mu a_\mu$$ in numerators, or replacing $$p^\mu p^\nu$$ with $$\eta^{\mu\nu}/d$$.

However, when it comes to two-loop calculations, I'm stumped. I would like to understand how to do a simple integral like this one: $$\int\mathrm{d}^dp\int\mathrm{d}^dk \frac{1}{p^2-m^2}\frac{1}{k^2-m^2}\frac{c_1 p^2 + c_2k^2+c_3p\cdot k}{(k-p)^2-m^2}$$ so that I can apply similar steps to my own calculations. $$d=4-\epsilon$$.

In particular:

1. How do I use Feynman parameters in this case? edit: I know about the $$1/(ABC)=...$$ formula. The question should read: how do I complete the squares after employing the Feynman parameters $$x,y,z$$?
2. How do I deal with terms like $$p\cdot k$$?

If I use the Feynman parameters, is this the correct ansatz?

$$\frac{1}{p^2-m^2}\frac{1}{k^2-m^2}\frac{1}{(k-p)^2-m²} = \int_0^1 \delta(x+y+z-1)\frac{2}{[ a_1(p+a_2)² + a_3(k+a_4)² -a_5^2 ]^3}$$

• Did you try the Feynman parameter $\frac{1}{A B C} = \int^1_0 dx dy dz \delta (x + y + z - 1) \frac{2}{(x A + y B + z C)^3}$? Oct 28, 2020 at 16:47
• @MicheleGrosso After rewriting the expression using Feynman parameters, how do I complete the squares in order to go to spherical coordinates? Nov 2, 2020 at 1:47

2. Complete the square for one (!) loop momentum, e.g. $$a_1(p-a_2)^2-a_3^2$$
3. Wick rotation, $$p^0=\text{i}k^0_E$$, $$k^2=-k_E^2$$
4. Dimensional regularization: $$d^4p_E \to \lim\limits_{\epsilon\to 0^+}d^dp_E$$ with $$d=4-\epsilon$$ (or another regularization)
5. Use generalized spherical coordinates, s.t. $$\int d\Omega$$ is trivial