  We define an effective action for spin foam models of quantum gravity by
adapting the background field method from quantum field theory. We show that
the Regge action is the leading term in the semi-classical expansion of the
spin foam effective action if the vertex amplitude has the large-spin
asymptotics which is proportional to an exponential function of the vertex
Regge action. In the case of the known three-dimensional and four-dimensional
spin foam models this amounts to modifying the vertex amplitude such that the
exponential asymptotics is obtained. In particular, we show that the ELPR/FK
model vertex amplitude can be modified such that the new model is finite and
has the Einstein-Hilbert action as its classical limit. We also calculate the
first-order and some of the second-order quantum corrections in the
semi-classical expansion of the effective action.
