Traintrack Calabi-Yaus from twistor geometry

Abstract

We describe the geometry of the leading singularity locus of the traintrack integral family directly in momentum twistor space. For the two-loop case, known as the elliptic double box, the leading singularity locus is a genus one curve, which we obtain as an intersection of two quadrics in P 3. At three loops, we obtain a K3 surface which arises as a branched surface over two genus-one curves in P 1 × P 1. We present an analysis of its properties. We also discuss the geometry at higher loops and the supersymmetrization of the construction.

Cuts and isogenies

“…[52] from the point of view of direct integration in a Feynman parametric representation, and in ref. [56] from the point of view of the maximal cut in twistor space. In both papers the same elliptic curve was found using very different methods.…”

“…It turns out that the elliptic curve obtained in this way has the same j-invariant as those computed from twistor space in ref. [56] and from the parametric representation of ref. [52].…”

“…This curve has the same j-invariant as the one obtained by considering the curve embedded in momentum twistor space as described in ref. [56].…”

Building bases of loop integrands

“…Recently, the analysis of simple examples of such non-polylogarithmic pieces have attracted considerable interest in the high energy physics community, for both practical reasons and formal motivations alike (see, e.g. [105][106][107][108][109]).…”

“…Providing a precise definition of transcendental weight is complicated by the need for nonpolylogarithmic integrands in general. See e.g [105][106][107][108][109]…”