Preprint 2025-01
B. Eden
The off-stell one- and two loop recovered from intersection theory.
Abstract:
We advertise intersection theory for generalised hypergeometric functions as a means of evaluating Mellin-
Barnes representations. As an example, we study two-parameter representations of the off-shell one- and
two-loop box graphs in exactly four-dimensional configuration space. Closing the integration contours for the
MB parameters we transform these into double sums. Polygamma functions in the MB representation of the
double box and the occurrence of higher poles are taken into account by parametric differentiation. Summing
over any one of the counters results into a p+1Fp that we replace by its Euler integral representation. The
process can be repeated a second time and results in a two- or four-parameter Euler integral, respectively. We use intersection theory to derive Pfaffian systems of equations on related sets of master integrals and solve for the box and double box integrals reproducing the known expressions. Finally, we use a trick to re-derive the double box from a two-parameter Euler integral. This second computation requires only very little computing resources.
20 pp.
Preprint series: Institut für Mathematik, Humboldt-Universität zu Berlin (ISSN 0863-0976), 2025-01