# Integrals

If you have an integral you can't solve, but think that someone else in the workshop might, try adding it on this page. Likewise, if you see an integral on this page which you know how to solve, write in the answer (or the procedure for solving it.) If you need help in adding math to a wikipage take a look at Math Mode on the practice page.

1) The hard integral
The integral below (the "hard" integral topology) appears in the two-loop expansion of the lightlike polygonal Wilson loop at weak coupling (see for example Equation (B.1) of this paper). It would be very interesting to have its analytic expression for generic values of the kinematic invariants. Here it is:

$\hspace{-0.5cm}f_{\rm Hard} (p_1,p_2,p_3;Q_1,Q_2,Q_3) \ := \ \nonumber \\ \\ {1 \over 8}\,{\Gamma (2-2\epsilon_\mathrm{UV}) \over \Gamma(1-\epsilon_\mathrm{UV})^2} \int_{0}^{1}\! \Big( \prod_{i=1}^{3} d\tau_i \Big) \int_{0}^{1}\!\Big(\prod_{i=1}^{3} \, d\alpha_i \Big) \delta ( 1 - \sum_{i=1}^3 \alpha_i ) \nonumber \\ \\ \ (\alpha_1 \alpha_2 \alpha_3)^{-\epsilon_\mathrm{UV}} { \mathcal{N} \over \mathcal{D}^{2-2\epsilon_\mathrm{UV}}}\ ,$

where

$\mathcal{D} := -\alpha_1 \alpha_2 (z_1 - z_2)^2 -\alpha_2 \alpha_3 (z_2 - z_3)^2 -\alpha_1 \alpha_3 (z_1 - z_3)^2 \ ,$

and

$\begin{equation*} \hspace{-0.54cm}(z_1 - z_2)^2 &=& Q_3^2 + 2 (p_1 p_2) (1 - \tau_1) \tau_2 + 2 (Q_3 p_1 ) ( 1 - \tau_1) + 2 (Q_3 p_2 ) \tau_2 \ , \\ \\ (z_2 - z_3)^2 &=& Q_1^2 + 2 (p_2 p_3) (1 - \tau_2) \tau_3 + 2 (Q_1 p_2 ) ( 1 - \tau_2) + 2 (Q_1 p_3 ) \tau_3 \ , \\ \\ (z_3 - z_1)^2 &=& Q_2^2 + 2 (p_3 p_1) (1 - \tau_3) \tau_1 + 2 (Q_2 p_3 ) ( 1 - \tau_3) + 2 (Q_2 p_1 ) \tau_1 \ .$

The numerator is given by

$\begin{equation*} \hspace{-0.54cm}\mathcal{N} &=& 2 (p_1 p_2) (p_1 p_3) \Big[ \alpha_1 \alpha_2 ( 1 - \tau_1) + \alpha_3 \alpha_1 \tau_1 \Big] \\ \\ &+& 2 (p_1 p_2) (p_2 p_3) \Big[ \alpha_2 \alpha_3 ( 1 - \tau_2) + \alpha_1 \alpha_2 \tau_2 \Big] \\ \\ &+& 2 (p_1 p_3) (p_2 p_3) \Big[ \alpha_3 \alpha_1 ( 1 - \tau_3) + \alpha_2 \alpha_3 \tau_3 \Big] \\ \\ &+& 2 \alpha_1 \alpha_2 \Big[ 2 (p_1 p_2) (p_3 Q_3) - (p_2 p_3) (p_1 Q_3) - (p_3 p_1 ) (p_2 Q_3) \Big] \\ \\ &+& 2 \alpha_2 \alpha_3 \Big[ 2 (p_2 p_3) (p_1 Q_1) - (p_3 p_1) (p_2 Q_1) - (p_1 p_2 ) (p_3 Q_1) \Big] \\ \\ &+& 2 \alpha_3 \alpha_1 \Big[ 2 (p_3 p_1) (p_2 Q_2) - (p_1 p_2) (p_3 Q_2) - (p_2 p_3 ) (p_1 Q_2) \Big] \ .$

The momenta

$\begin{equation*} p_1, p_2, p_3$

are lightlike, whereas in the most general configuration

$\begin{equation*} Q_i^2 \neq 0, \qquad i=1, 2, 3 \, .$

Finally,

$\epsilon_\mathrm{UV} >0 \, .$