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The generalized minimal subtraction scheme for ultraviolet renormalization(Kuznetsov and Tkachov, 1988) is fine-tuned with applications in mind. Theresulting $\text{MS}^4$ scheme obviates extraneous regularizations and rendersmomentum integrands integrable by subtracting troublesome asymptotic terms in aphysically correct fashion due to the use of special minimal subtractionoperators defined congruously with the physically natural Polchinski cutoffs. Adirect derivation of the Callan-Symanzik equations avoids divergentrenormalization factors or counterterms, and automatically yields explicitexact finite integral expressions for renormalization group functions.

Finite Z-less integral expressions for $\beta$-functions in the MS4 scheme