  We investigate the Euclidean vacuum mode functions of a massive vector field
in a spatially open chart of de Sitter spacetime. In the one-bubble open
inflationary scenario that naturally predicts a negative spatial curvature
after a quantum tunneling, it is known that a light scalar field has the
so-called supercurvature mode, i.e. an additional discrete mode which describes
fluctuations over scales larger than the spatial curvature scale. If such
supercurvature modes exist for a vector field with a sufficiently light mass,
then they would decay slower and easily survive the inflationary era. However,
the existence of supercurvature mode strongly depends on details of the system.
To clarify whether a massive vector field has supercurvature modes, we consider
a U(1) gauge field with gauge and conformal invariances spontaneously broken
through the Higgs mechanism, and present explicit expressions for the Euclidean
vacuum mode functions. We find that, for any values of the vector field mass,
there is no supercurvature mode. In the massless limit, the absence of
supercurvature modes in the scalar sector stems from the gauge symmetry.
