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High Energy Physics - Theory, hep-th
Abstract:
Dyson-Schwinger equations determine the Green functions $G^r(\alpha,L)$ in
quantum field theory. Their solutions are triangular series in a coupling
constant $\alpha$ and an external scale parameter $L$ for a chosen amplitude
$r$, with the order in $L$ bounded by the order in the coupling. Perturbation
theory calculates the first few orders in $\alpha$. On the other hand,
Dyson--Schwinger equations determine next-to$^{\{\mathrm{j}\}}$-leading log
expansions, $G^r(\alpha,L) = 1 + \sum_{j=0}^\infty \sum_{\mathcal{M}}
p_j^{\mathcal{M}}\alpha^j \mathcal{M}(u)$. $\sum_{\mathcal{M}}$ sums a finite
number of functions $\mathcal{M}$ in $u = \alpha L/2$. The leading logs come
from the trivial representation $\mathcal{M}(u) =
\begin{bsmallmatrix}\bullet\end{bsmallmatrix}(u)$ at $j=0$ with
$p_0^{\begin{bsmallmatrix}\bullet\end{bsmallmatrix}} = 1$. All non-leading logs
are organized by the suppression in powers $\alpha^j$. We describe an algebraic
method to derive all next-to$^{\{\mathrm{j}\}}$-leading log terms from the
knowledge of the first $(j+1)$ terms in perturbation theory and their
filtrations. This implies the calculation of the functions $\mathcal{M}(u)$ and
periods $p_j^\mathcal{M}$. In the first part of our paper, we investigate the
structure of Dyson-Schwinger equations and develop a method to filter their
solutions. Applying renormalized Feynman rules maps each filtered term to a
certain power of $\alpha$ and $L$ in the log-expansion. Based on this, the
second part derives the next-to$^{\{\mathrm{j}\}}$-leading log expansions. Our
method is general. Here, we exemplify it using the examples of the propagator
in Yukawa theory and the photon self-energy in quantum electrodynamics. The
reader may apply our method to any (set of) Dyson-Schwinger equation(s)
appearing in renormalizable quantum field theories.
