We consider a scalar quantum field ϕ with arbitrary polynomial self-interaction in perturbation theory. If the field variable ϕ is repaced by a global diffeomorphism ϕ(x) = ρ(x) + a1ρ2(x) + …, this field ρ obtains infinitely many additional interaction vertices. We propose a systematic way to compute connected amplitudes for theories involving vertices which are able to cancel adjacent edges. Assuming tadpole graphs vanish, we show that the S-matrix of ρ coincides with the one of ϕ without using path-integral arguments. This result holds even if the underlying field has a propagator of higher than quadratic order in the momentum. The diffeomorphism can be tuned to cancel all contributions of an underlying ϕt-type self interaction at one fixed external offshell momentum, rendering ρ a free theory at this momentum. Finally, we mention one way to extend the diffeomorphism to a non-diffeomorphism transformation involving derivatives without spoiling the combinatoric structure of the global diffeomorphism.

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We examine selected one-scale Dyson-Schwinger-equations in the minimal subtraction renormalization scheme and dimensional regularization. They are constructed from three different integral kernels; two one-loop Feynman integrals and one toy model. For each of the kernels, we examine both linear and non-linear DSEs. We confirm that the solutions in minimal subtraction are equal to the ones in kinematic renormalization, but with a shifted renormalization point. We compute various series expansions symbolically to at least ten loop order and numerically to higher order. In the linear cases, we identify the so-obtained sequences and find an analytic generating function for the shift of the renormalization point. In 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 compute coefficients of the MS-counter terms Z and confirm, for the linear DSEs, a closed form expression consistent with the literature.

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.