Loop expansion around the Bethe solution for the random magnetic field Ising ferromagnets at zero temperature

We apply to the random-field Ising model at zero temperature (T = 0) the perturbative loop expansion around the Bethe solution. A comparison with the standard ε expansion is made, highlighting the key differences that make the expansion around the Bethe solution much more appropriate to correctly describe strongly disordered systems, especially those controlled by a T = 0 renormalization group (RG) fixed point. The latter loop expansion produces an effective theory with cubic vertices. We compute the one-loop corrections due to cubic vertices, finding additional terms that are absent in the ε expansion. However, these additional terms are subdominant with respect to the standard, supersymmetric ones; therefore, dimensional reduction is still valid at this order of the loop expansion.