Resummation methods for Master Integrals

Abstract

We present in detail two resummation methods emerging from the application of the Simplified Differential Equations approach to a canonical basis of master integrals. The first one is a method which allows for an easy determination of the boundary conditions, since it finds relations between the boundaries of the basis elements and the second one indicates how using the x → 1 limit to the solutions of a canonical basis, one can obtain the solutions to a canonical basis for the same problem with one mass less. Both methods utilise the residue matrices for the letters {0, 1} of the canonical differential equation. As proof of concept, we apply these methods to a canonical basis for the three-loop ladder-box with one external mass off-shell, obtaining subsequently a canonical basis for the massless three-loop ladder-box as well as its solution.

Pentagon integrals to arbitrary order in the dimensional regulator

“…In appendix A we present the letters of the alphabet for this particular family. Notice that here we follow [16][17][18] for the definition of the letters of the alphabet, which is different from the standard notation [28][29][30]. Usually the so-called d log form of a system of canonical differential equations is given as dg( x,…”

“…Using the methods described in [17], we can readily obtain a pure basis of 11 Master Integrals for the massless pentagon family from the x → 1 limit of (3.7). The results are by construction in terms of Goncharov Polylogarithms up to weight four, however following the arguments of the last section, we can obtain results in terms of Goncharov Polylogarithms of arbitrary weight.…”

“…To be more specific, as described in [17] the x → 1 limit of the solution for the pentagon with one off-shell leg can be given by the following formula…”

“…The second input in (4.1), g trunc , is just the regular part of the solution (3.7) at x → 1 where we have set x = 1 explicitly [17].…”

“…Having the solution of the pentagon with one off-shell leg at hand, we can obtain in an algorithmic way a fully analytical solution for the massless pentagon by taking the x → 1 limit of the former result, with x being a dimensionless parameter that is introduced in the SDE approach. This is a special feature of the SDE approach first described in [4,16] and in more details in [17].…”