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Working in the basis where the charged-lepton Yukawa matrix is diagonal andmaking the $\tau$-dominance approximations, we analytically derive integralsolutions to the one-loop renormalization-group equations (RGEs) for neutrinomasses, flavor mixing angles, CP-violating phases and the Jarlskog invariantunder the standard parametrization of the PMNS matrix in the standard model orits minimal supersymmetric extension for both Majorana and Dirac neutrinos.With these integral solutions, we carry out numerical calculations toinvestigate the RGE running of lepton flavor mixing parameters and the Jarlskoginvariant, and also compare these integral solutions with the exact resultsobtained by numerically solving the one-loop RGEs. It is shown that theseintegral solutions coincide with the exact results and can well describe theevolution of lepton flavor mixing parameters and the Jarlskog invariant in mostcases. Some important features of our integral solutions and the evolutionbehaviors of relevant flavor parameters are also discussed in detail bothanalytically and numerically.

Integral solutions to the one-loop renormalization-group equations for lepton flavor mixing parameters and the Jarlskog invariant