Fermions and scalars in $\mathcal{N} = 4$ Wilson loops at strong coupling and beyond

Abstract: We study the strong coupling behaviour of null polygonal Wilson loops/gluon amplitudes in $\mathcal{N} = 4$ SYM, by using the OPE series and its integrability features. For the hexagon we disentangle the $SU(4)$ matrix structure of the form factors for fermions, organising them in a pattern similar to the Young diagrams used previously for the scalar sector \cite{BFPR2,BFPR3}. Then, we complete and extend the discussion of \cite{BFPR1} by showing, at strong coupling, the appearance of a new effective particle in the series: the fermion-antifermion bound state, the so-called meson. We discuss its interactions in the OPE series with itself by forming (effective) bound states and with the gluons and bound states of them. These lead the OPE series to the known $AdS_5$ minimal area result for the Wls, described in terms of a set of TBA-like equations. This approach allows us to detect all the one-loop contributions and, once the meson has formed, applies to $\mathcal{N}=2$ Nekrasov partition function via the parallel meson/instanton (in particular, they share the mechanism by which their bound states emerge and form the TBA node). Finally, to complete the strong coupling analysis, we consider the scalar sector for any polygon, confirming the emergence of a leading contribution from the non-perturbative theory on the sphere $S^5$.