Matching heavy-quark fields in QCD and HQET at three loops

The relation between the heavy-quark field in QCD and the corresponding field in HQET is derived up to three loops, and to all orders in the large-${\beta }_0$ limit. The corresponding relation between the QED electron field and the Bloch–Nordsieck one is gauge invariant to all orders. We also prove that the $\overline{\text{MS}}$ anomalous dimension of the QED electron field depends on the gauge parameter only at one loop.