Landau-Khalatnikov-Fradkin transformation in three-dimensional quenched QED

Abstract

We study the gauge-covariance of the massless fermion propagator in three-dimensional quenched Quantum Electrodynamics in the framework of dimensional regularization in d = 3 − 2ε. Assuming the finiteness of the quenched perturbative expansion, that is the existence of the limit ε → 0, we state that, exactly in d = 3, all odd perturbative coefficients, starting with the third order one, should be zero in any gauge.1 At the next-to-leading order of the 1/N -expansion, the polarization operator reads: Π(q 2 ) = N Ce 2 /(8q) where C = 1 + (184/9π 2 − 2)/N . This result was obtained independently and by different methods in Refs. [12][13][14]. At the interacting IR fixed point, the beta function therefore retains the form (3) with N → N C. This in turn implies that the IR fixed point remains stable as it is only weakly affected by next-to-leading order corrections: α → 8/(N C) ≈ (8/N )(1 − 0.071/N ), in agreement with Eq. ( 30) of [13]. To the best of our knowledge, calculations at the order 1/N 3 have not been performed yet.