Contribution of scalars to $\mathcal{N}=4$ supersymmetric Yang-Mills amplitudes

We analyze the contribution of $2n$ scalars to the operator product expansion series for Maximum Helicity Violating (MHV) gluon scattering amplitudes/polygonal Wilson loops in planar $\mathcal{N}=4$ supersymmetric Yang-Mills (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 also give explicit expressions for the first terms at large $\lambda$. For this analysis, we find an explicit computation of the $2n$-scalar term at any coupling by means of Young tableaux, similar to Nekrasov-Okounkov expressions in $\mathcal{N}=2$ SYM.