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 $$\alpha $$ α and a logarithmic energy scale L —a reordering of terms as $$G(\alpha ,L) = 1 \pm \sum _{j \ge 0} \alpha ^j H_j(\alpha L)$$ G ( α , L ) = 1 ± ∑ j ≥ 0 α j H j ( α L ) is the corresponding log expansion. In a first part of this paper, we derive the leading log order $$H_0$$ H 0 and the next-to $$^{(j)}$$ ( j ) -leading log orders $$H_j$$ H j from the Callan–Symanzik equation. In particular, $$H_j$$ H j only depends on the $$(j+1)$$ ( j + 1 ) -loop $$\beta $$ β -function and anomalous dimensions. In two specific examples, our formulas reproduce the known expressions for the next-to-next-to-leading log approximation in the literature: 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. 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.

中文翻译：

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组合 Dyson-Schwinger 方程的对数展开式

我们给出了组合 Dyson-Schwinger 方程和量子场论中格林函数的对数展开之间的精确联系。后者是耦合常数 $$\alpha $$ α 和对数能量标度 L 中的三角幂级数——项的重新排序为 $$G(\alpha ,L) = 1 \pm \sum _{j \ge 0 } \alpha ^j H_j(\alpha L)$$ G ( α , L ) = 1 ± ∑ j ≥ 0 α j H j ( α L ) 是对应的对数展开。在本文的第一部分，我们推导出前导日志顺序 $$H_0$$ H 0 和下一个 $$^{(j)}$$ ( j ) - 前导日志顺序 $$H_j$$ H j来自 Callan-Symanzik 方程。特别地，$$H_j$$ H j 仅取决于$$(j+1)$$ ( j + 1 ) -loop $$\beta $$ β - 函数和异常维度。在两个具体示例中，我们的公式重现了文献中 next-to-next-to-leading 对数近似的已知表达式：对于量子电动力学和玩具模型中的光子传播器格林函数，其中所有具有顶点子散度的费曼图都被忽略了。在这项工作的第二部分，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。结合这些论点，我们的工作从相应的 Dyson-Schwinger 方程中推导出了格林函数的对数展开式。