  Recent calculations of nuclear structure corrections to the Lamb shift in
light muonic atoms are based on an expansion in a parameter eta, where only
terms up to second order are retained. The parameter eta can be shown to be
proportional to the square root of the muon/proton mass ratio, so that it is
small and the expansion is expected to converge. However, practical
implementations show that the eta convergence may be slower than expected. In
this work we probe the uncertainties due to this expansion using a different
formalism, which is based on a multipole expansion of the longitudinal and
transverse response functions and was first introduced by Leidemann and
Rosenfelder. We refer to this alternative expansion as the eta-less formalism.
We generalize this formalism to account for the cancellation of elastic terms
such as the third Zemach moment (or Friar moment) and embed it in a
computationally efficient framework. We implement and test this approach in the
case of muonic deuterium. The comparison of results in the point nucleon limit
for both methods achieve sub-percent agreement. When nucleon form factors are
introduced we find a 4% and 2% difference in the third Zemach moment and
nuclear polarizability, respectively, compared to the eta-less expansion,
indicating that the nucleon form factor approximations should be improved.
However, we find that the sum of these terms removes this dependence and the
uncertainty due to the eta-expansion and the related second-order approximation
in the nucleon form factors amounts only to 0.2% and thus is fully justified in
muonic deuterium. This computationally efficient framework paves the way to
further studies in light muonic systems with more than two nucleons, where
controlling and reducing uncertainties in nuclear structure corrections is key
to the experimental efforts of the CREMA collaboration.
