Integrals

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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:
 * 1) The hard integral**

math \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}}}\ , math

where

math \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 \ , math

and

math \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 \ . \end{equation} math

The numerator is given by

math \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] \ . \end{equation} math

The momenta

math \begin{equation*} p_1, p_2, p_3 \end{equation} math

are lightlike, whereas in the most general configuration

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

Finally,

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