The effective potential in the presence of several mass scales

We consider the problem of improving the effective potential in mass independent schemes, as e.g. the $\MSbar$ or $\DRbar$ renormalization scheme, in the presence of an arbitrary number of fields with $\phi$-dependent masses $M_i(\phi_c)$. We use the decoupling theorem at the scales $\mu_i=M_i(\phi_c)$ such that the matching between the effective (low energy) and complete (high energy) one-loop theories contains no thresholds. We find that for any value of $\phi_c$, there is a convenient scale $\mu^*\equiv\min_i\{M_i(\phi_c)\}$, at which the loop expansion has the best behaviour and the effective potential has the least $\mu$-dependence. Furthermore, at this scale the effective potential coincides with the (improved) tree-level one in the effective field theory. The decoupling method is explicitly illustrated with a simple Higgs-Yukawa model, along with its relationship with other decoupling prescriptions and with proposed multi-scale renormalization approaches. The procedure leads to a nice suppression of potentially large logarithms and can be easily adapted to include higher-loop effects, which is explicitly shown at the two-loop level.