Working in the basis where the charged-lepton Yukawa matrix is diagonal and making the τ-dominance approximations, we analytically derive integral solutions to the one-loop renormalization-group equations (RGEs) for neutrino masses, flavor mixing angles, CP-violating phases and the Jarlskog invariant under the standard parametrization of the PMNS matrix in the standard model or its minimal supersymmetric extension for both Majorana and Dirac neutrinos. With these integral solutions, we carry out numerical calculations to investigate the RGE running of lepton flavor mixing parameters and the Jarlskog invariant, and also compare these integral solutions with the exact results obtained by numerically solving the one-loop RGEs. It is shown that these integral solutions coincide with the exact results and can well describe the evolution of lepton flavor mixing parameters and the Jarlskog invariant in most cases. Some important features of our integral solutions and the evolution behaviours of relevant flavor parameters are also discussed in detail both analytically and numerically.

在带电轻子汤河矩阵对角线的基础上进行工作，使τ-优势近似，我们在标准模型或标准模型中PMNS矩阵的标准参数化下，针对中微子质量，风味混合角，CP违反相和Jarlskog不变性，通过分析得出单环重归一化组方程（RGE）的积分解。它对Majorana和Dirac中微子的最小超对称扩展。利用这些积分解，我们进行了数值计算，以研究轻子风味混合参数和Jarlskog不变量的RGE运行，并将这些积分解与通过数值求解一回路RGE所获得的精确结果进行比较。结果表明，这些积分解与准确的结果相吻合，并且可以很好地描述大多数情况下轻子风味混合参数和Jarlskog不变量的演变。