  We analyse the breaking of conformal invariance for null polygonal Wilson
loops in ${\cal N}=4$ SYM beyond that induced by the UV divergences due to the
cusps. It only shows up in exceptional configurations, where the polygon
intersects the critical light cone of an inversion or a special conformal
transformation. In comparison with the related study for the Euclidean version
by Drukker and Gross, we find different leading terms both for weak as well as
for strong coupling. Hence the conformal anomaly due to intersections of a null
polygon with a critical light cone defines a new universal function of the
coupling constant.
