Effective Lagrangians in $2 + \epsilon$ Dimensions

Abstract

The failure of the the loop expansion and effective lagrangians in two dimensions, which traditionally hinges on a power counting argument is considered. We establish that the book keeping device for the loop expansion, a role played by (the reciprocal of) the pion-decay constant itself vanishes for $d=2$, thereby going beyond the power counting argument. We point the connection of our results to the distinct phases of the candidate for the effective lagrangians, the non-linear sigma model, in $d=2+\epsilon$, and eventually for $d=2$. In light of our results, we recall some of the relavant features of the multi-flavor Schwinger and large $N_f$ $QCD_2$ as candidates for the underlying theory in $d=2$.