  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 Coleman-Weinberg effective potential, which
describes the vacuum of the theory at the quantum level, has to be supplemented
with a prescription to handle the hierarchy in mass scales. In any quantum
field theory involving scalar fields and multiple, highly differing mass
scales, it is in general not possible to choose a single renormalization scale
that will remove all the large logarithms in the effective potential. In this
paper, we focus on the so-called decoupling method, which freezes the effects
of heavy particles on the renormalization group running of the light degrees of
freedom at low energies. We study this for a simple two-scalar theory and 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 an
alternative implementation of the decoupling approach, which gives different
results for the shape of the potential, but still leads to similar conclusions
on the amount of fine-tuning in the model. We suggest a way to avoid running
into this fine-tuning problem by adopting a prescription on how to fix
parameters in such decoupling approaches.
