The fixed point action for the Schwinger model: a perturbative approach

We compute the fixed point action of a properly defined renormalization group transformation for the Schwinger model through an expansion in the gauge field. It is local, with couplings exponentially suppressed with the distance. We check its perfection by computing the 1-loop mass gap at finite spatial volume, finding only exponentially vanishing cut off effects, in contrast with the standard action, which is affected by large power-like cut off effects. We point out that the 1-loop mass gap calculation provides a check of the classical perfection of the fixed point action, and not of the 1-loop perfection, as could be naively expected.