arXiv 1906.061312019-06-14hep-thmath-phmath.MP

组合Dyson-Schwinger方程的对数展开

Log-Expansions from Combinatorial Dyson-Schwinger Equations

Olaf Krüger

备注：: 26 pages
DOI：: 10.1007/s11005-020-01288-8

摘要：

给出了量子场论中组合Dyson-Schwinger方程与格林函数对数展开式之间的精确联系。后者是耦合常数$α$和对数能量标度$L$的三角幂级数--作为$G(α,L) = 1 \pm \sum_{j \geq 0} α^j H_j(αL)$的项的重新排序是相应的对数展开。在本文的第一部分中，我们从Callan-Symanzik方程导出了首对数阶$H_0$和次$^{(j)}$首对数阶$H_j$。特别地，$H_j$仅依赖于$(j+1)$-环$β$-函数和异常尺寸。对于量子电动力学中的光子传播子格林函数（以及在玩具模型中，忽略所有具有顶点子发散的费曼图），我们的公式再现了文献中的下一个下一个领先对数近似的已知表达式。在这项工作的第二部分中，我们回顾了Callan-Symanzik方程和Dyson-Schwinger方程之间的联系，即格林函数的不动点关系。结合参数，我们的工作提供了一个衍生的对数展开的格林函数从相应的戴森Schwinger方程。

英文摘要：

We give a precise connection between combinatorial Dyson-Schwinger equations and log-expansions for Green's functions in quantum field theory. The latter are triangular power series in the coupling constant $α$ and a logarithmic energy scale $L$ --- a reordering of terms as $G(α,L) = 1 \pm \sum_{j \geq 0} α^j H_j(αL)$ is the corresponding log-expansion. In a first part of this paper, we derive the leading-log order $H_0$ and the next-to$^{(j)}$-leading log orders $H_j$ from the Callan-Symanzik equation. In particular, $H_j$ only depends on the $(j+1)$-loop $β$-function and anomalous dimensions. For the photon propagator Green's function in quantum electrodynamics (and in a toy model, where all Feynman graphs with vertex sub-divergences are neglected), our formulas reproduce the known expressions for the next-to-next-to-leading log approximation in the literature. In a second part of this work, we review the connection between the Callan-Symanzik equation and Dyson-Schwinger equations, i.e. fixed-point relations for the Green's functions. Combining the arguments, our work provides a derivation of the log-expansions for Green's functions from the corresponding Dyson-Schwinger equations.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

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