[@Mitter_1975; @Ritus_1985; @Ehlotzky_2009; @Reiss_2009; @Di_Piazza_2012]). However, even under the plane-wave approximation, the radiative corrections of these processes have never been computed. The reason is that calculations including the effects of the external laser field exactly are significantly more complex than the corresponding calculations in vacuum. The standard technique, in fact, is to work within the so-called Furry picture [@Furry_1951], where the electron-positron field is quantized in the presence of the background field [@Fradkin_b_1991; @Landau_b_4_1982]. This requires that the Dirac equation can be solved analytically in the presence of the background field, which has been achieved in Ref. [@Volkov_1935] in the case of a plane wave (see also Ref. [@Landau_b_4_1982]), the corresponding states being known as Volkov states. An alternative, equivalent technique is the so-called operator technique, first proposed by Schwinger [@Schwinger_1951] and then developed for the case of a background plane wave [@Baier_1976_a; @Baier_1976_b; @Di_Piazza_2007; @Di_Piazza_2008_b; @Di_Piazza_2013; @Di_Piazza_2018_d], which does not require the explicit solution of the Dirac equation in the plane-wave field.

Going back to the radiative corrections, a systematic study has been only carried out in the special case of a zero-frequency plane wave or a constant crossed field, i.e., a constant and uniform electromagnetic field with electric 