Multiple Mellin-Barnes integrals with straight contours

We show how the conic hull method, recently developed for the analytic and noniterative evaluation of multifold Mellin-Barnes (MB) integrals, can be extended to the case where these integrals have straight contours of integration parallel to the imaginary axes in the complex planes of the integration variables. MB integrals of this class appear, for instance, when one computes the $\epsilon$-expansion of dimensionally regularized Feynman integrals, as a result of the application of one of the two main strategies (called A and B in the literature) used to resolve the singularities in $\epsilon$ of MB representations. We upgrade the Mathematica package MBConicHulls.wl which can now be used to obtain multivariable series representations of multifold MB integrals with arbitrary straight contours, providing an efficient tool for the automatic computation of such integrals. This new feature of the package is presented, along with an example of application by calculating the $\epsilon$-expansion of the dimensionally regularized massless one-loop pentagon integral in general kinematics and $D=4-2\epsilon$.

Figure 1

Singular structure, in the $(\Re (z_1),\Re (z_2))$-plane, of the integrand of Left: Eq. (2) and Right: Eq. (7). All the poles (represented as singular lines in the figures) of a given gamma function of the numerator of the corresponding MB integrands are plotted with the same color (for instance, the poles of $\mathrm{\Gamma }(\frac{3}{7}+z_1+z_2)$ are the oblique lines shown in light blue). In the right figure, $\mathrm{\Gamma }(-2+z_1)$ [resp. $\mathrm{\Gamma }(-\frac{3}{5}-z_2)$] has only 3 [resp. 1] singular lines, as the others are cancelled by the denominator of the MB integrand. The red point is $(c_1,c_2)=(-\frac{1}{7},-\frac{1}{9})$ and the blue one is $(c_1,c_2)=(\frac{7}{3},-\frac{3}{2})$.

Figure 2

Cone (shaded area in light blue) associated with the series representation of Eq. (2) computed in the text. We refer the reader to [24] for details about the derivation of the cone using the dashed black line.