Landau Singularities and Symbology: One- and Two-loop MHV Amplitudes in SYM Theory

TL;DR

This work investigates the link between Landau singularities and symbol alphabets for one- and two-loop MHV amplitudes in planar $\mathcal{N}=4$ SYM by applying Landau equations to the relevant integrals. It shows a striking one-loop pattern where the sub-sub-leading ($\mathrm{S^2LLS}$) and sub-leading ($\mathrm{SLLS}$) singularities align with the first and second entries of the pentagon's symbol, while the leading singularity is canceled by the chiral pentagon numerator. Remarkably, all known two-loop symbol letters are already present among the Landau singularities of the one-loop pentagon, even though the two-loop double pentagon introduces many additional Landau loci that do not appear in the symbol. The paper discusses caveats, including the role of numerators and second-type singularities, and highlights a potential connection between Landau geometry and cluster algebras, suggesting directions for extending these ideas to NMHV amplitudes.

Abstract

We apply the Landau equations, whose solutions parameterize the locus of possible branch points, to the one- and two-loop Feynman integrals relevant to MHV amplitudes in planar $\mathcal{N}=4$ super-Yang-Mills theory. We then identify which of the Landau singularities appear in the symbols of the amplitudes, and which do not. We observe that all of the symbol entries in the two-loop MHV amplitudes are already present as Landau singularities of one-loop pentagon integrals.

Figures (3)

• Figure 1: The scalar bubble and triangle integrals. All propagators are taken to be massless.
• Figure 2: The scalar box integral and its degenerations. All propagators are taken to be massless.
• Figure 3: The four-mass box integral.