  As was discovered some time ago by Giddings and Strominger (GS), an axion can
support a wormhole geometry in the presence of a massless dilaton, as long as
the dilaton coupling remains below a critical value. We find that when the
dilaton becomes massive, the set of solutions is vastly increased: not only do
solutions exist above the critical value of the coupling, but new branches of
solutions with several minima in the geometry also appear. All of these
generalised GS-like solutions possess the property that, when analytically
continued, they lead to a contracting baby universe. We show that in addition
there exist families of solutions which, upon analytic continuation, lead to
expanding baby universes. A curious property of axion-dilaton wormhole families
is that their Euclidean action often decreases when the solutions acquire
additional oscillations in the fields. When we replace the dilaton by an
ordinary scalar field with a double well potential, we find analogous wormhole
families leading to expanding baby universes. This time the Euclidean action
has the expected behaviour of increasing with the number of oscillations in the
fields, although it also contains a puzzling aspect in that some solutions
possess a negative action.
