Cutting the traintracks: Cauchy, Schubert and Calabi-Yau

Author:

Cao Qu,He Song,Tang YichaoORCID

Abstract

Abstract In this note we revisit the maximal-codimension residues, or leading singularities, of four-dimensional L-loop traintrack integrals with massive legs, both in Feynman parameter space and in momentum (twistor) space. We identify a class of “half traintracks” as the most general degenerations of traintracks with conventional (0-form) leading singularities, although the integrals themselves still have rigidity $$ \left\lfloor \frac{L-1}{2}\right\rfloor $$ L 1 2 due to lower-loop “full traintrack” subtopologies. As a warm-up exercise, we derive closed-form expressions for their leading singularities both via (Cauchy’s) residues in Feynman parameters, and more geometrically using the so-called Schubert problems in momentum twistor space. For L-loop full traintracks, we compute their leading singularities as integrals of (L−1)-forms, which proves that the rigidity is L−1 as expected; the form is given by an inverse square root of an irreducible polynomial quartic with respect to each variable, which characterizes an (L−1)-dim Calabi-Yau manifold (elliptic curve, K3 surface, etc.) for any L. We also briefly comment on the implications for the “symbology” of these traintrack integrals.

Publisher

Springer Science and Business Media LLC

Subject

Nuclear and High Energy Physics

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