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We consider double-winding, triple-winding and multiple-winding Wilson loopsin the $SU(N)$ Yang-Mills gauge theory. We examine how the area law falloff ofthe vacuum expectation value of a multiple-winding Wilson loop depends on thenumber of color $N$. In sharp contrast to the difference-of-areas law recentlyfound for a double-winding $SU(2)$ Wilson loop average, we show irrespective ofthe spacetime dimensionality that a double-winding $SU(3)$ Wilson loop followsa novel area law which is neither difference-of-areas nor sum-of-areas law forthe area law falloff and that the difference-of-areas law is excluded and thesum-of-areas law is allowed for $SU(N)$ ($N \ge 4$), provided that the stringtension obeys the Casimir scaling for the higher representations. Moreover, weextend these results to arbitrary multi-winding Wilson loops. Finally, we arguethat the area law follows a novel law, which is neither sum-of-areas nordifference-of-areas law when $N\ge 3$. In fact, such a behavior is exactlyderived in the $SU(N)$ Yang-Mills theory in the two-dimensional spacetime.

Double-winding Wilson loops in the $SU(N)$ Yang-Mills theory