- #1

Milsomonk

- 96

- 17

So I'm trying to compute this Feynman integral:

$$ V=\dfrac {-i} {2} \int {\dfrac {d^4 k} {(2\pi)^4}} \dfrac {1} {k^2 - m^2} \dfrac {1} {(k+P_1)^2 -m^2} \dfrac {1} {(k+P_1 +P_2)^2 -m^2}$$

I have introduced the Feynman parameters and now have the integral:

$$ V=-i \int dx_1 \int dx_2 \int dx_3 \int {\dfrac {d^4 k} {(2\pi)^4}} \dfrac {\delta (1-x_1 -x_2 -x_3)} {-x_1 (k^2 - m^2) -x_2 ((k+P_1)^2 -m^2) -x_3 ((k+P_1 +P_2)^2 -m^2)}$$

Now focusing on the denominator I expand it out and need to complete the square and shift the integration variable according to Peskin Schroeder, but I'm not sure how to do this. Here is what I have so far:

$$ -x_1 k^2 -x_1 m^2 -x_2 k^2 -2x_2 k \cdot P_1 - x_2 P_1^2 -x_2 m^2 -x_3 k^2 -2 x_3 k \cdot P_1 -2 x_3 k \cdot P_2 - x_3 P_1 ^2 -x_3 P_2 ^2 -2x_3 P_1 \cdot P_2 -x_3 m^2 $$

I know that I need to get rid of the terms with dot products but I'm not sure how, any guidance would be awesome :)