Two-loop octagons, algebraic letters and Q¯ equations

Song He,Chi Zhang,Zhenjie Li

doi:10.1103/physrevd.101.061701

Abstract

We compute the symbol of the first two-loop amplitudes in planar $\mathcal{N}=4$ SYM with algebraic letters, the eight-point Next to Maximally Helicity Violating (NMHV) amplitude (or the dual octagon Wilson loops). We show how to apply $\overline{Q}$ equations of [S. Caron-Huot and S. He, J. High Energy Phys. 07 (2012) 174] for computing the differential of two-loop $n$-point NMHV amplitudes and present the result for $n=8$ explicitly. The symbol alphabet for octagon consists of 180 independent rational letters and 18 algebraic ones involving Gram-determinant square roots. We comment on all-loop predictions for final entries and aspects of the result valid for all multiplicities.

Highlights

Scattering amplitudes are central objects in fundamental physics: do they connect to high energy experiments such as Large Hadron Collider, and provide new insights into quantum field theory itself
As we will review shortly, the differential of L-loop Maximally Helicity Violating (MHV) and Next to Maximally Helicity Violating (NMHV) amplitudes is given by a combination of generalized polylogarithmic functions of weight (2L − 1), dressed with certain universal objects that are independent of loop order; the latter are a collection of Yangian invariants times d log of final entries of the symbol, known for MHV to all n and NMHV for n 1⁄4 6, 7 [18,20]
The 180 rational letters are contained in the prediction of rational symbol alphabet from Landau equations [39], but we find that the last class of them is missing

Summary

INTRODUCTION

Scattering amplitudes are central objects in fundamental physics: do they connect to high energy experiments such as Large Hadron Collider, and provide new insights into quantum field theory itself. As we will review shortly, the differential of L-loop MHV and NMHV amplitudes is given by a combination of generalized polylogarithmic functions of weight (2L − 1), dressed with certain universal objects that are independent of loop order; the latter are a collection of Yangian invariants times d log of final entries of the symbol, known for MHV to all n and NMHV for n 1⁄4 6, 7 [18,20] We compute these universal NMHV final entries for all n [28], though here we only need the result for n 1⁄4 8. The generalized polylogarithm functions accompanying them have to be computed order by order, and we will present the weightthree functions for two-loop octagons which in turn give the complete symbol (there is no qualitative difference for higher n, which will be reported elsewhere [28])

REVIEW OF Q EQUATIONS

THE COMPUTATION

RESULTS

CONCLUSION AND DISCUSSIONS