Abstract
The process of Compton decay of positronium by a twisted photon is considered within the \(A^{2}\) approximation. In the case of a Bessel cylindrical wave, the matrix element of such a process is closely related to a similar matrix element for a plane electromagnetic wave. Taking into account the momentum conservation law and within the framework of standard scattering theory, it is shown that in the expression for the probability of this process, virtually all key characteristics of the cylindrical wave of a twisted photon are lost. Moreover, it is demonstrated that the differential probability of positronium decay by a twisted photon equals the probability of decay by a plane wave photon with a certain momentum moving at a certain angle to the \(z\) -axis, averaged over the azimuthal angle of the incident photon.
