Inflation from charged scalar and primordial magnetic fields?

A model of inflation is presented where the inflaton field is a complex scalar field coupled to a $U(1)$ gauge field. Because of the axial symmetry of the potential, the inflation is driven by the radial direction while the angular field is gauged by $U(1)$. Because of the coupling of the inflaton to the gauge field, a time-dependent mass term for the gauge field is generated dynamically and conformal invariance is broken. We study whether a significant amount of primordial magnetic fields can be generated during inflation by allowing a time-dependent $U(1)$ gauge kinetic coupling.