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 epsilon-expansion is made, highlighting the key differences that make the new expansion much more appropriate to correctly describe strongly disordered systems, especially those controlled by a T = 0 RG fixed point. This new loop expansion produces an effective theory with cubic vertices. We compute the one-loop corrections due to cubic vertices, finding new terms that are absent in the epsilon-expansion. However, these new terms are subdominant with respect to the standard, supersymmetric ones, therefore dimensional reduction is still valid at this order of the loop expansion