The RG flow between neighboring minimal CFT models $A_2^{(p)}$ and $A_2^{(p-1)}$ with $W_3$ symmetry is explored.
Although in perturbed theory dilatation current is no longer conserved it is still possible to get an exact operator expression for its divergence. Exploring this anomalous conservation law one can express the leading order anomalous dimensions of local fields in terms of structure constants of OPE in the original CFT. We generalize these line of argument for the case when a higher spin $W$ current is present.
We introduce the notion of anomalous $W$ zero mode matrix which again can be expressed in terms of OPE coefficients of the original CFT. Diagonalization of this matrix provides an additional independent confirmation that indeed
$A_2^{(p)}$ flows to $A_2^{(p-1)}$.