  In many realizations of beyond the Standard Model theories, new massive
particles are introduced, leading to a multi-scale system with widely separated
energy scales. In this setting the effective potential, which takes into
account quantum corrections for the scalar sector, has to be supplemented with
a prescription to handle the hierarchy in mass scales. In this paper, we focus
on the so-called decoupling method, which freezes the effects of heavy
particles on the RG running of the light degrees of freedom at low energies.
For a two-scalar theory, we disentangle the effects of the high-energy degrees
of freedom on the shape of the potential and on the fine-tuning of the model
parameters. We find that, while the decoupling method leads to an acceptable
and convergent effective potential, the method does not solve the fine-tuning
problem that is inherent to the hierarchy problem of multi-scale theories. We
also consider two alternative implementations of the decoupling, which give
different results for the shape of the potential, but still lead to similar
conclusions on the amount of fine-tuning in the model.
