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Upcoming experiments on the interaction of electrons with intense laserfields are envisaged to become more and more accurate, which calls fortheoretical computations of rates and probabilities with correspondingly higherprecision. In strong-field QED this requires the knowledge of radiativecorrections to be added to leading-order results. Here, we first derive themass operator in momentum space of an off-shell electron in the presence of anarbitrary plane wave. By taking the average of the mass operator in momentumspace over an on-shell electron state, we obtain a new representation,equivalent to but more compact than the known one computed in [Sov. Phys. JETP\textbf{42}, 400 (1975)]. Moreover, we use the obtained mass operator todetermine the electron mass shift in an arbitrary plane wave, which generalizesthe already known expression in a constant crossed field. The spin-dependentpart of the electron mass shift can be related to the anomalous magnetic momentof the electron in the plane wave. We show that within the locally constantfield approximation it is possible to conveniently define a local expression ofthe electron anomalous magnetic moment, which reduces to the known expressionin a constant crossed field. Beyond the locally constant field approximation,however, the interaction between the electron and the plane wave is non-localsuch that it is not possible to conveniently introduce an electron anomalousmagnetic moment.

The electron mass shift in an intense plane wave