To find the connections $A_i$, all one needs to do is to reduce ${\partial }_i{I}_a$ ($a=1\dots N$) again to a linear combination of master integrals.

A caveat concerns infrared divergences, i.e., when some of the $u_i^0$ are zero. In this case we replace the zero masses with a common regulator mass such that all IR divergences are contained within the master integrals themselves.

See Refs. [37, 38, 39] for related ideas.

As a rational function, $\mathcal{A}$ must have an expansion $\mathcal{A}={\sum }_{n=\ell }^{\infty }{\mathcal{A}}_n{t}^n$ for some integer $\ell$. In all the examples that we considered, $\ell \ge -1$, which we will assume in the following for simplicity. If $\ell <-1$ the formalism can be generalized, see Appendix pp1.

Notice that some naively different topologies [such as the diagram in Fig. 2] require the use of some algebra, i.e., $$\frac{1}{k_1^2-u_1}\frac{1}{k_1^2-u_1^{\prime }}=\frac{1}{u_1-u_1^{\prime }}(\frac{1}{k_1^2-u_1}-\frac{1}{k_1^2-u_1^{\prime }}),$$etc.

The incidence matrix $B$ of a graph with $V$ vertices and $E$ edges is defined to be the $V\times E$ matrix whose entries $B_{ij}$ are one if edge $j$ connects to vertex $i$, and zero otherwise. Internal lines starting and ending at the same vertex are by convention columns with all zeroes. In merlin we delete the rows of $B$ that correspond to external points of the graph. The code uses these incidence matrices to efficiently compute symmetries of the diagram.

To refresh all results, the corresponding data files must be deleted from the results folder.