New differential equations for on-shell loop integrals

TL;DR

This paper introduces second-order differential equations in momentum twistor space that reduce on-shell loop integrals by one loop level, enabling iterative solutions and revealing their simplified transcendental structure, with applications to N=4 super Yang-Mills amplitudes.

Contribution

The paper presents a new class of differential equations that systematically reduce loop order and can be solved iteratively, providing analytical tools for scattering amplitude calculations.

Findings

Two-loop integrals for planar MHV amplitudes are derived from a single master integral.

The integrals are shown to be less complex transcendental functions.

Full analytic solutions are provided for two specific two-loop integrals.

Abstract

We present a novel type of differential equations for on-shell loop integrals. The equations are second-order and importantly, they reduce the loop level by one, so that they can be solved iteratively in the loop order. We present several infinite series of integrals satisfying such iterative differential equations. The differential operators we use are best written using momentum twistor space. The use of the latter was advocated in recent papers discussing loop integrals in N=4 super Yang-Mills. One of our motivations is to provide a tool for deriving analytical results for scattering amplitudes in this theory. We show that the integrals needed for planar MHV amplitudes up to two loops can be thought of as deriving from a single master topology. The master integral satisfies our differential equations, and so do most of the reduced integrals. A consequence of the differential equations is that the integrals we discuss are not arbitrarily complicated transcendental functions. For two specific two-loop integrals we give the full analytic solution. The simplicity of the integrals appearing in the scattering amplitudes in planar N=4 super Yang-Mills is strongly suggestive of a relation to the conjectured underlying integrability of the theory. We expect these differential equations to be relevant for all planar MHV and non-MHV amplitudes. We also discuss possible extensions of our method to more general classes of integrals.