New differential equations for on-shell loop integrals

TL;DR

This work introduces a novel framework of second-order differential equations that lower the loop order by one and apply directly to on-shell loop integrals in planar N=4 SYM, enabling iterative, analytic progress. Formulated in momentum twistor space, the method reveals that many MHV integrals up to two loops derive from a single master topology and satisfy recursive relations, with explicit solutions for select two-loop cases expressed in harmonic polylogarithms. The authors demonstrate the approach across ladders, pentaladders, pentagon/penta-box, hexagon, and double-pentagon topologies, including both finite and regulator-regularized divergent integrals, and validate results against Mellin–Barnes representations. These results hint at a deeper integrable structure behind planar amplitudes and suggest broader applicability to higher-loop and non-MHV configurations, as well as extensions to other regulators and integral classes.

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.

Figures (8)

• Figure 1: 'Two-mass-hard' integral in momentum space, dual space, and momentum twistor space variables, see equations (\ref{['twistorbox']}) and (\ref{['eq-2mh1loop']}).
• Figure 2: Integrals contributing to the one-loop MHV amplitude. The box integrand is obtained from the pentagon integrand by taking the soft limit $p_{n} \to 0$. The dashed line represents here the numerator which is denoted by wiggly line in ArkaniHamed:2010kv.
• Figure 3: Integrals contributing to the two-loop MHV amplitude. The integrands of the double box and penta-box integrals are obtained from the double pentagon integrals by taking the soft limits. The dashed line have again the same meaning as in 1-loop case.
• Figure 4: Ladder integrals defined in equations (\ref{['1box']}) and (\ref{['def-boxes']}).
• Figure 5: Figure (a) represents the seven-point integrals defined in (\ref{['1pent']}) and (\ref{['loopspent']}), and (b) shows the eight-point integrals defined in (\ref{['1pentmassive']}) and (\ref{['looppentmassive']}).