Revisiting the local potential approximation of the exact renormalization group equation

The conventional absence of field renormalization in the local potential approximation (LPA) --implying a zero value of the critical exponent \eta -- is shown to be incompatible with the logic of the derivative expansion of the exact renormalization group (RG) equation. We present a LPA with \eta \neq 0 that strictly does not make reference to any momentum dependence. Emphasis is made on the perfect breaking of the reparametrization invariance in that pure LPA (absence of any vestige of invariance) which is compatible with the observation of a progressive smooth restoration of that invariance on implementing the two first orders of the derivative expansion whereas the conventional requirement (\eta =0 in the LPA) precluded that observation.