Summation of logarithms in an arbitrary local quantum field theory

The Bogolyubov–Parasyuk theorem yields a set of recurrence relations, independent of the specific model and renormalizability condition, which allows summing counterterms to the considered theory, as well as summing logarithmic quantum corrections in arbitrary order according to the perturbation theory. The renormalizability condition is irrelevant in this method because only the locality condition for the counterterms is used, which always has a local characteristic after subtracting divergent subgraphs in an arbitrary divergent graph.

The resulting equations for the scattering amplitudes are integro-differential equations, while for the effective potential, they are partial differential equations. Although a large set of previously obtained integro-differential equations exists, this report considers an example with an effective potential due to its simplicity and clarity. The simplicity of these equations leads to a simple analysis of these equations, which may be important in specific applications of this method. The talk demonstrates equations summing both leading and subleading counterterms and logarithms to an arbitrary scalar effective potential. Interestingly, the resulting equations are linear, and their limit to the renormalizable model yields different solutions relative to the renormalizable theory in the corresponding order. This discrepancy leads to a special method for summing logarithms in renormalizable theory, which allows for the simple extraction of a complete series of logarithmic quantum corrections.