Fermions and scalars in $N=4$ Wilson loops at strong coupling and beyond

We study the null polygonal Wilson loops/gluon scattering amplitudes in planar $N=4$ SYM by means of the pentagon Operator Product Expansion (OPE) series and its integrability features, with particular attention on the strong coupling expansion. Actually, at all couplings and for the hexagon we disentangle the $SU\left(4\right)$ matrix structure of the squared form factors for fermions, organising them in Young diagrams similar to those previously used for scalars. Then, we concentrate on the strong coupling regime and show the appearance of a new (effective) particle in the series: the fermion-antifermion bound state, the so-called 'meson'. Of course, we identify its interaction with itself in the OPE series, with formation of (effective) bound states, and with the gluons and bound states of them. All together these extract from the OPE series the $Ad{S}_5$ minimal area result for the Wilson loop, described by a set of Thermodynamic Bethe Ansatz (TBA)-like equations, and highlight also all the one-loop contributions. Moreover, the same strategy applies to $N=2$ Nekrasov partition function via the parallel meson/instanton. Eventually, to complete the strong coupling analysis, we consider the scalar sector for any polygon, confirming the emergence of a non-perturbative contribution from the string in $S^5$, which eludes and corrects the minimal area argument.