Abstract
We derive analytic results for the symbol of certain two-loop Feynman integrals relevant for seven- and eight-point two-loop scattering amplitudes in planar mathcal{N}=4 super-Yang-Mills theory. We use a bootstrap inspired strategy, combined with a set of second-order partial differential equations that provide powerful constraints on the symbol ansatz. When the complete symbol alphabet is not available, we adopt a hybrid approach. Instead of the full function, we bootstrap a certain discontinuity for which the alphabet is known. Then we write a one-fold dispersion integral to recover the complete result. At six and seven points, we find that the individual Feynman integrals live in the same space of functions as the amplitude, which is described by the 9- and 42-letter cluster alphabets respectively. Starting at eight points however, the symbol alphabet of the MHV amplitude is insufficient for individual integrals. In particular, some of the integrals require algebraic letters involving four-mass box square-root singularities. We point out that these algebraic letters are relevant at the amplitude level directly starting with N2MHV amplitudes even at one loop.
Highlights
Task, including Mellin-Barnes integration [4,5,6,7] as well as solving differential equations for the Feynman integrals under consideration [8,9,10,11,12,13]
Planar N = 4 sYM theory shows a number of simplifying features; one of the most remarkable is the existence of a hidden dual conformal symmetry (DCI) [19,20,21] which effectively reduces the number of kinematic variables a given amplitude can depend on
The bootstrap approach to scattering amplitudes crucially relies on the knowledge of the appropriate function space
Summary
There exist two seven-point generalizations of Ω(62) which appear in MHV amplitudes. The second-order differential equations satisfied by these integrals are reviewed in appendix B. The larger dimensionality (6 = 3 × 7 − 15) of the kinematic DCI seven-point space makes a direct integration of the second order differential equations challenging. Instead of directly integrating up the second order partial differential equations, we turn to an alternative strategy that is inspired by the bootstrap approach for planar scattering amplitudes in N = 4 sYM, see e.g. For the two-loop seven-point cases discussed below, it turns out that we have sufficiently many constraints to find unique solutions within our initial weight four function space. We note that upgrading the symbol-level expressions to full function level results is possible, but requires more work and would involve fixing certain multiple-zeta valued ambiguities (e.g. ζ3 × log s, etc.) that are in the kernel of the symbol map. This is beyond the scope of our current work where are content with obtaining the symbol
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