We investigate the radiative QED corrections to the lepton (L = e, μ and τ) anomalous magnetic moment due to the contributions of diagrams with insertions of the photon vacuum polarisation operator consisting solely of four closed lepton (l = e, μ and τ) loops. Moreover, we focus on specific operators with two loops formed by leptons L of the same type as the external one, the other two formed by leptons ℓ different from L. The approach is essentially based on the employment of the Mellin–Barnes representation of the x-parametrization of the corresponding Feynman diagrams. This allows one to obtain, for the first time, exact analytical expressions for the radiative corrections of the tenth order w.r.t. the electromagnetic coupling constant e. Analytically, the radiative corrections are expressed in terms of the ratio r = m _ℓ /m _L of the internal to external lepton masses. The dependence on r is investigated numerically in the whole interval of r, 0 < r < ∞ . To make comparisons with earlier published results possible, our exact analytical expressions are expanded as r → 0 and r → ∞ and compared with the corresponding asymptotic expansions known in the literature.

中文翻译：

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梅林-巴恩斯表示中轻子异常的十阶 QED 辐射校正的分析计算

我们研究了轻子的辐射 QED 校正（L=e,μ和τ）由于插入仅由四个闭合轻子组成的光子真空极化算符的图的贡献而产生反常磁矩（我=e,μ和τ) 循环。此外，我们关注具有由轻子形成的两个环的特定算子L与外部的类型相同，另外两个由轻子形成ℓ不同于L。该方法本质上是基于使用 Mellin-Barnes 表示法X-相应费曼图的参数化。这使得人们第一次获得电磁耦合常数十阶辐射校正的精确解析表达式e。从分析上讲，辐射校正以比率表示r=米 _ℓ /米 _L 内部到外部轻子质量。依赖于r在整个区间内进行数值研究r, 0 <r< ∞ 。为了与早期发布的结果进行比较，我们的精确解析表达式扩展为r→ 0 和r→ ∞ 并与文献中已知的相应渐近展开式进行比较。