Gauge-invariant propagators in QED_3

We discuss properties of various gauge-invariant propagators in QED_3. The frequently-encountered Schwinger propagator G_S(x-y)=langle exp(-iint_xy amu ds_mu) psi(x)barpsi(y)rangle, where the integral in the phase factor is calculated along a straight line, is shown to be an exponentially decreasing function of separation G_Ssim exp(-|x-y|/xi) in 3-epsilon dimensions rather than being described by a simple power-law. Near d=3, the correlation length xi vanishes as (d-3), rendering G_S an ill-defined object. We explicitly demonstrate that G_S(|x-y|) factorizes into a product of an average of Wilson phase factor involving the transverse component of a gauge field only, and Landau gauge-invariant propagator langle exp(-iint_xy amu_L ds_mu)psi(x)barpsi(y)rangle to the leading order in 1/N expansion. The root-cause of the problem is thus revealed as severe UV divergence in the gauge invariant Wilson factor. In contrast, we show that the appropriately regularized Brown propagator is a well-defined, gauge invariant object whose power-law decay is characterized by a positive anomalous dimension 16/(3pi2 N).