  When free vortices are present in the pseudogap region of underdoped
cuprates, it is shown that conventional Anderson-Higgs mechanism does not work
because the Goldstone field is not an analytic function. However, after
decomposing the Goldstone field into longitudinal and transverse components, we
find that the former can be eliminated by a special gauge transformation and
the gauge field becomes massive, but the latter persists in the theory in any
case. Thus we obtain an extended Anderson-Higgs mechanism which exhibits
completely new physics comparing with the conventional one.
