I would like to evaluate integrals of the following type (in position space):

$$\int \frac{d^{2\omega}z}{\left[(x_1-z)^2 (x_2-z)^2 (x_3-z)^2 \right]^A} \tag{1}$$

I can introduce three Feynman parameters using the last equation of the section "Formulas" in this wikipedia article, integrate over $z$, then integrate over one Feynman parameter with the delta function in order to obtain:

$$\pi^\omega \frac{\Gamma(3A-\omega)}{\Gamma^3(A)} \int_0^1 d\gamma \int_0^{1-\gamma} d\beta \frac{\left[ \gamma\beta (1-\gamma-\beta) \right]^{A-1}}{\left[\gamma\beta x_{12}^2+\gamma(1-\gamma-\beta)x_{13}^2 +\beta(1-\gamma-\beta)x_{23}^2\right]^{3A-\omega}} \tag{2}$$

where $x_{ij}:=x_i-x_j$. Is it possible to go further with this integral? If yes, in what way? If not, is there at least a way to extract the divergent part for $\omega \rightarrow 2$ and $A=\omega-1$?


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.