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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?

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

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    $\begingroup$ Did you try the Feynman parameter $\frac{1}{A B C} = \int^1_0 dx dy dz \delta (x + y + z - 1) \frac{2}{(x A + y B + z C)^3}$? $\endgroup$ Oct 28, 2020 at 16:47
  • $\begingroup$ @MicheleGrosso After rewriting the expression using Feynman parameters, how do I complete the squares in order to go to spherical coordinates? $\endgroup$
    – ersbygre1
    Nov 2, 2020 at 1:47

1 Answer 1

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The answer is actually quite simple: one performs one loop integral after the other, so:

  1. Use Feynman parameters
  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
  6. Perform radial integration by using Beta and Gamma functions

...then do the same steps for the next loop momentum.

Finally, the integration over the Feynman parameters remains.

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