We describe the on-shell method to derive the Renormalization Group (RG) evolution of Wilson coefficients of high dimensional operators at one loop, which is a necessary part in the on-shell construction of the Standard Model Effective Field Theory (SMEFT), and exceptionally efficient based on the amplitude basis in hand. The UV divergence is obtained by firstly calculating the coefficients of scalar bubble integrals by unitary cuts, then subtracting the IR divergence in the massless bubbles, which can be easily read from the collinear factors we obtained for the Standard Model fields. Examples of deriving the anomalous dimensions at dimension six are presented in a pedagogical manner. We also give the results of contributions from the dimension-8 H⁴D⁴ operators to the running of V⁺V⁻H² operators, as well as the running of B⁺B⁻H²D²ⁿ from H⁴D²ⁿ⁺⁴ for general n.

我们描述了一种带壳方法，可在一环上推导高维算子的威尔逊系数的重归一化组（RG）演化，这是标准模型有效场理论（SMEFT）的带壳构造的必要部分，并且基于振幅的基础，效率极高。UV散度是通过首先通过单一切口计算标量气泡积分系数，然后减去无质量气泡中的IR散度而获得的，这可以从我们为标准模型字段获得的共线因子中轻松读取。以教学方式介绍了在尺寸6处得出异常尺寸的示例。我们还给出了维度8 H ⁴ D ⁴的贡献结果。运营商的运行V ⁺ V ^- ħ ²的运营商，以及的运行乙⁺乙^- ħ ² d ^{2 Ñ}从ħ ⁴ d ^{2 Ñ 4}一般Ñ。