Is there supercurvature mode of massive vector field in open inflation?

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.