Rooting out letters: octagonal symbol alphabets and algebraic number theory

Jacob L. Bourjaily, Andrew J. McLeod, Cristian Vergu, Matthias Volk, Matt von Hippel, Matthias Wilhelm

Research output: Contribution to journalArticlepeer-review

16 Scopus citations


It is widely expected that NMHV amplitudes in planar, maximally supersymmetric Yang-Mills theory require symbol letters that are not rationally expressible in terms of momentum-twistor (or cluster) variables starting at two loops for eight particles. Re- cent advances in loop integration technology have made this an ‘experimentally testable’ hypothesis: compute the amplitude at some kinematic point, and see if algebraic symbol letters arise. We demonstrate the feasibility of such a test by directly integrating the most difficult of the two-loop topologies required. This integral, together with its rotated image, suffices to determine the simplest NMHV component amplitude: the unique component finite at this order. Although each of these integrals involve algebraic symbol alphabets, the combination contributing to this amplitude is — surprisingly — rational. We describe the steps involved in this analysis, which requires several novel tricks of loop integration and also a considerable degree of algebraic number theory. We find dramatic and unusual simplifications, in which the two symbols initially expressed as almost ten million terms in over two thousand letters combine in a form that can be written in five thousand terms and twenty-five letters.

Original languageEnglish (US)
Article number25
JournalJournal of High Energy Physics
Issue number2
StatePublished - Feb 1 2020

All Science Journal Classification (ASJC) codes

  • Nuclear and High Energy Physics


Dive into the research topics of 'Rooting out letters: octagonal symbol alphabets and algebraic number theory'. Together they form a unique fingerprint.

Cite this