On the Landau–Khalatnikov–Fradkin transformation in quenched $\mathrm{QED}_3$

We present the results of studies of the gauge covariance of the massless fermion propagator in three-dimensional quenched quantum electrodynamics in the framework of dimensional regularization in $d=3-2\varepsilon$. Assuming the finiteness of the perturbative expansion, i.e., the existence of the limit $\varepsilon\to 0$, we show that exactly for $d=3$ all odd perturbative coefficients starting from the third order must be equal to zero in any gauge. To test this, we calculate three- and four-loop corrections to the massless fermion propagator. Three-loop corrections are finite and gauge invariant, while four-loop corrections have singularities. The terms depending on the gauge parameter are completely determined by the lower orders in accordance with the Landau–Khalatnikov–Fradkin transformation.