  We calculate the longitudinal structure function of the deuteron up through
next-to-next-to-leading order in the framework of pionless effective field
theory. We use these results to compute the two-photon polarizability
contribution to Lamb shift in muonic deuterium, which can be utilized to
extract the nuclear charge radius of the deuteron. We present analytical
expressions order-by-order for the relevant transition matrix elements and the
longitudinal structure function, and we give numerical results for the
corresponding contributions to the Lamb shift. We also discuss the impact of
relativistic and other higher-order effects. We find agreement with previous
calculations and explain the accuracy of our calculation.
