Stability of a fixed point in the replica action for the random field Ising model

We reconsider stability of the non-trivial fixed point in $6-\epsilon$ dimensional effective action for the random field Ising model derived by Br\'{e}zin and De Dominicis. After expansion parameters of physical observables are clarified, we find that the non-trivial fixed point in $6-\epsilon$ dimensions is stable, contrary to the argument by Br\'{e}zin and De Dominicis. We also computed the exponents $\nu$ and $\eta$ by the $\epsilon$ expansion. The results are consistent with the argument of the dimensional reduction at least in the leading order.