The renormalization group relations for the higher-order hadronic vacuum polarization function perturbative expansion coefficients are studied. The folded recurrent and unfolded explicit forms of such relations are obtained. The explicit expression for the coefficients, which incorporate the contributions of the π 2 -terms in the perturbative expansion of the R -ratio of electron–positron annihilation into hadrons, is derived at an arbitrary loop level. The obtained results can be employed as an independent crosscheck of the higher-order perturbative calculations of the hadronic vacuum polarization function and in the studies of the renormalization scale setting in the relevant physical observables.

中文翻译：

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高阶强子真空极化函数微扰展开系数的归一化群关系的递归形式

研究了高阶强子真空极化函数摄动展开系数的归一化群关系。获得了这种关系的折叠的递归和展开的显式形式。系数的显式表达式是在任意循环水平上得出的，其中包含π2项对电子-正电子an灭的R-比值的扰动扩展到强子中的贡献。获得的结果可以用作强子真空极化函数的高阶微扰计算的独立交叉核对，以及在相关物理观测值中的重归一化尺度设置研究中。