For an infrared-safe observable, loop corrections and integrated unresolved real emissions have compensating infrared divergences and collinear divergences. Keep a common regulator until they are added. At the first QCD correction to photon-mediated quark production, virtual gluon exchange corrects the two-parton state and real emission supplies the three-parton state. Their inclusive sum is finite after ultraviolet renormalization; resolved three-jet rates additionally depend on an infrared-safe resolution prescription.

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