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We isolate the two-step mechanism involving a real intermediate photon from the one-step mechanism involving a virtual photon for the trident process in a constant crossed field. The two-step process is shown to agree with an integration over polarised sub-processes. At low to moderate quantum non-linearity parameter, the one-step process is found to be suppressed. When the parameter is large, the two decay channels are comparable if the field dimensions are not much greater than the formation length.
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We reexamine the process $gammato e^++ e^-$ in a background magnetic field comparable to $B_cequiv m_e^2/e$. This process is known to be non-perturbative in the magnetic-field strength. However, it can be shown that the {it moments} of the above pair production width is proportional to the derivatives of photon polarization function at the zero energy, which is perturbative in $B$. Hence, the pair-production width can be easily obtained from the latter by the inverse Mellin transform. The implications of our approach are discussed.
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