  The use of a running coupling constant in renormalizable theories is well
known, but the implementation of this idea for effective field theories with a
dimensional coupling constant is in general less useful. Nevertheless there are
multiple attempts to define running couplings including the effects of gravity,
with varying conclusions. We sort through many of the issues involved, most
particularly the idea of operator mixing and also the kinematics of crossing,
using calculations in Yukawa and lambda phi^4 theory as illustrative examples.
We remain in the perturbative regime. In some theories with a high permutation
symmetry, such as lambda phi^4, a reasonable running coupling can be defined.
However in most cases, such as Yukawa and gauge theories, a running coupling
fails to correctly account for the energy dependence of the interaction
strength. As a byproduct we also contrast on-shell and off-shell renormalizaton
schemes and show that operators which are normally discarded, such as those
that vanish by the equations of motion, are required for off-shell
renormalization of effective field theories. Our results suggest that the
inclusion of gravity in the running of couplings is not useful or universal in
the description of physical processes.
