Dyson–Schwinger equations in minimal subtraction

• Published in: Annales de l'Institut Henri Poincaré. D. Combinatorics, physics and their interactions Vol. 12; no. 1; pp. 1 - 50
• Main Author: Balduf, Paul-Hermann
• Format: Journal Article
• Language: English
• Published: 01.01.2025
• ISSN: 2308-5827; 2308-5835
• DOI: 10.4171/aihpd/169

We compare the solutions of one-scale Dyson–Schwinger equations (DSEs) in the minimal subtraction (MS) scheme to the solutions in kinematic momentum subtraction (MOM) renormalization schemes. We establish that the MS-solution can be interpreted as a MOM-solution, but with a shifted renormalization point, where the shift itself is a function of the coupling. We derive relations between this shift and various renormalization group functions and counterterms in perturbation theory. As concrete examples, we examine three different one-scale Dyson–Schwinger equations: one based on the 1-loop multiedge graph in D=4 dimensions, one for D=6 dimensions, and one for mathematical toy model. For each of the integral kernels, we examine both the linear and nine different non-linear Dyson–Schwinger equations. For the linear cases, we empirically find exact functional forms of the shift between MOM and MS renormalization points. For the non-linear DSEs, the results for the shift suggest a factorially divergent power series. We determine the leading asymptotic growth parameters and find them in agreement with the ones of the anomalous dimension. Finally, we present a tentative exact solution to one of the non-linear DSEs of the toy model.