  We investigate the gauge invariance of the dynamic (ac) Stark shift under
"hybrid" gauge transformations from the "length" (E.r) to the "velocity" (A.p)
gauge. By a "hybrid" gauge transformation, we understand a transformation in
which the scalar and vector potentials are modified, but the wave function
remains unaltered. The gauge invariance of the leading term is well known,
while we here show that gauge invariance under perturbations holds only if one
takes into account an additional correction to the transition current, which
persists only in the velocity gauge. We find a general expression for this
current, and apply the formalism to radiative and relativistic corrections to
the dynamic Stark effect, which is described by the sum of two polarizability
matrix elements.
