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Filtrations in Dyson-Schwinger equations: next-to^{j} -leading log expansions systematically

Krueger, O., & Kreimer, D. (2015). Filtrations in Dyson-Schwinger equations: next-to^{j} -leading log expansions systematically. Annals of Physics, 360, 293-340. doi:10.1016/j.aop.2015.05.013.

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Creators:
Krueger, Olaf1, Author
Kreimer, Dirk, Author
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1Quantum Gravity & Unified Theories, AEI-Golm, MPI for Gravitational Physics, Max Planck Society, ou_24014

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Free keywords: 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.

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Dates: 2014-12-042015
Publication Status: Published in print
Pages: \$2 pages, 1 Figure
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Identifiers: arXiv: 1412.1657
DOI: 10.1016/j.aop.2015.05.013
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Title: Annals of Physics
Source Genre: Journal
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Pages: - Volume / Issue: 360 Sequence Number: - Start / End Page: 293 - 340 Identifier: -