Dyson-Schwinger方程的增长估计
Growth estimates for Dyson-Schwinger equations
摘要：
戴森-施温格方程（英语：Dyson-Schwinger equations）是量子场论中的积分方程，它描述了理论的格林函数，并反映了费曼图到子图的递归分解。Dyson-Schwinger方程作为递归方程描述微扰量子场论。然而，它们也包含非微扰信息。使用费曼图的霍普夫代数，我们将遵循一系列约简以将戴森-施温格方程转换为以下微分方程组，\[ γ_1^r(x) = P_r(x) - \sgn(s_r)γ_1^r(x)^2 + (\sum_{j \in \mathcal{R}}|s_j|γ_1^j(x)) x \partial_x γ_1^r(x) \]其中$r \in \mathcal{R}$，$\mathcal{R}$是需要重整化的理论振幅的集合，$γ_1^r$是与$r$相关联的反常维数，$P_r(x)$是对$r$有贡献的原始骨架的函数的修改版本，并且$x$是耦合常数。其次，我们通过扩展$γ_1^r(x) = \sum_{n \geq 1}γ^r_{1,n} x^n$将新的微分方程组作为递归方程组来处理。我们根据$\sum P_r(n)x^n/n!$的收敛半径得到了$\sum γ^r_{1,n}x^n/n!$的收敛半径。特别是，我们表明，Lipatov界的原始增长导致整个理论的Lipatov界。最后，我们对新的微分方程系统作了一些观察。
英文摘要：
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, \[ γ_1^r(x) = P_r(x) - \sgn(s_r)γ_1^r(x)^2 + (\sum_{j \in \mathcal{R}}|s_j|γ_1^j(x)) x \partial_x γ_1^r(x) \] where $r \in \mathcal{R}$, $\mathcal{R}$ is the set of amplitudes of the theory which need renormalization, $γ_1^r$ 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 $γ_1^r(x) = \sum_{n \geq 1}γ^r_{1,n} x^n$. We obtain the radius of convergence of $\sum γ^r_{1,n}x^n/n!$ 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.
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