Growth estimates for Dyson-Schwinger equations

Dyson-Schwinger equations are integral equations in quantum field theory that describe the Green functions of a theory and mirror the recursive decomposition of Feynman diagrams into subdiagrams. Taken as recursive equations, the Dyson-Schwinger equations describe perturbative quantum field theory. However, they also contain non-perturbative information. Using the Hopf algebra of Feynman graphs we will follow a sequence of reductions to convert the Dyson-Schwinger equations to the following system of differential equations, $$\gamma_1^r(x) = P_r(x) - \sgn(s_r)\gamma_1^r(x)^2 + (\sum_{j \in \mathcal{R}}|s_j|\gamma_1^j(x)) x \partial_x \gamma_1^r(x)$$ where $r \in \mathcal{R}$, $\mathcal{R}$ is the set of amplitudes of the theory which need renormalization, $\gamma_1^r$ is the anomalous dimension associated to $r$, $P_r(x)$ is a modified version of the function for the primitive skeletons contributing to $r$, and $x$ is the coupling constant. Next, we approach the new system of differential equations as a system of recursive equations by expanding $\gamma_1^r(x) = \sum_{n \geq 1}\gamma^r_{1,n} x^n$. We obtain the radius of convergence of $\sum \gamma^r_{1,n}x^n/n!$ in terms of that of $\sum P_r(n)x^n/n!$. In particular we show that a Lipatov bound for the growth of the primitives leads to a Lipatov bound for the whole theory. Finally, we make a few observations on the new system considered as differential equations.