Baryon Wilson loop area law in QCD

There is still confusion about the correct form of the area law for the baryonic Wilson loop (BWL) of QCD. Strong-coupling (i.e., finite lattice spacing in lattice gauge theory) approximations suggest the form $\exp [-K{A}_Y]$, where $K$ is the $q\overline{q}$ string tension and $A_Y$ is the global minimum area, generically a three-bladed area with the blades joined along a Steiner line ($Y$ configuration). However, the correct answer is $\exp [-(\frac{K}{2}\left)\right(A_{12}+A_{13}+A_{23}\left)\right]$, where, e.g., $A_{12}$ is the minimal area between quark lines 1 and 2 ($\Delta$ configuration). This second answer was given long ago, based on certain approximations, and is also strongly favored in lattice computations. In the present work, we derive the $\Delta$ law from the usual vortex-monopole picture of confinement, and show that, in any case, because of the ½ in the $\Delta$ law, this law leads to a larger value for the BWL (smaller exponent) than does the $Y$ law. We show that the three-bladed, strong-coupling surfaces, which are infinitesimally thick in the limit of zero lattice spacing, survive as surfaces to be used in the non-Abelian Stokes' theorem for the BWL, which we derive, and lead via this Stokes' theorem to the correct $\Delta$ law. Finally, we extend these considerations, including perturbative contributions, to gauge groups $\mathrm{SU}\mathrm{}\left(N\right)\mathrm{}$, with $N>\mathrm{}3\mathrm{}$.