The contribution of scalars to ${\cal N}=4$ SYM amplitudes

Abstract

We analyse the contribution of $2n$ scalars to the Operator Product Expansion series for MHV gluon scattering amplitudes/polygonal Wilson loops in planar ${\cal N}=4$ SYM. Hence, we sum up on $2n$ for large coupling $\lambda$: the logarithm of the amplitude is proportional to $\sqrt{\lambda}$, thus competing, unexpectedly, with the famous classical contribution. We give explicit expressions for the first terms at large $\lambda$ in case of two and four scalars. For finalising this analysis, we find profitable an explicit computation of the $2n$-scalar term at any coupling by means of Young tableaux, paralleling, under certain aspects, the Nekrasov-Okounkov expressions for $\mathcal N =2$ SYM prepotential.