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It is widely expected that the realization of scale invariance in thecritical regime implies conformal invariance for a large class of systems. Thisis known to be true if there exist no integrated operator which transforms likea vector under rotations and which has scaling dimension $-1$. In this articlewe give exact expressions for the critical exponents of some of these vectoroperators. In particular, we show that one operator has scaling dimensionexactly 3 in any space dimension. This operator turns out be the leadingoperator at least in $d=2$ and $d=4$. Moreover, we prove that the operatorpreviously considered in Monte-Carlo simulations has also scaling dimensionexactly $3$ in any dimension.

Exact critical exponents for vector operators in the 3d Ising model and conformal invariance