Palatini formulation for gauge theory: implications for slow-roll inflation

We consider a formulation of gauge field theory where the gauge field $A_\alpha$ and the field strength $F_{\alpha\beta}$ are independent variables, as in the Palatini formulation of gravity. For the simplest gauge field action, this is known to be equivalent to the usual formulation. We add non-minimal couplings between $F_{\alpha\beta}$ and a scalar field, solve for $F_{\alpha\beta}$ and insert it back into the action. This leads to modified gauge field and scalar field terms. We consider slow-roll inflation and show that because of the modifications to the scalar sector, adding higher order terms to the inflaton potential does not spoil its flatness, unlike in the usual case. Instead they make the effective potential closer to quadratic. The modifications also solve the problem that Higgs inflation in the Palatini formulation is sensitive to higher order terms.