The two factors cancel each other, leaving a finite correction which depends on the sensitivity with which photons can be detected in the experiment.
2.
The physical constant, the electron's charge, can then be defined in terms of some specific experiment; we set the renormalization scale equal to the energy characteristic of this experiment, and the first term gives the interaction we see in the laboratory ( up to small, finite corrections from loop diagrams, providing such exotica as the high-order corrections to the magnetic moment ).
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