Abstract
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.
Highlights
In the last two decades, compelling evidences obtained from a number of successful neutrino oscillation experiments have proved that neutrinos are massive and lepton flavor mixing exists [1], and this demonstrates that the standard model (SM) of particle physics is incomplete
It is very important and useful to investigate the evolution of relevant flavor parameters or the stability of some specific textures against the energy scale by means of the renormalization-group equations (RGEs), especially in the cases where nearly degenerate neutrino masses or large tan β in the minimal supersymmetric standard model (MSSM) is taken into account, so as to establish some correlations between physical phenomena at high and low energy scales and reveal some underlying structures of lepton mass matrices or flavor mixing pattern which are instructive for model building
As for the running strengths of these phases, Eqs. (40)—(44) indicate that in the Inverted neutrino mass ordering (IMO) case, the absolute values of ∆δ, ∆ρ and ∆σ with δ (Λ) = ρ (Λ) = 0 are nearly equal to those with δ (Λ) = σ (Λ) = 0; but in the neutrino mass ordering (NMO) case, the former ones are slightly larger than the latter ones since there is an additional term proportional to ζ32 − ζ3−21 in the formulas given by Eq (40) compared with those in Eq (43), which can enlarge the absolute values of ∆δ, ∆ρ and ∆σ
Summary
In the last two decades, compelling evidences obtained from a number of successful neutrino oscillation experiments have proved that neutrinos are massive and lepton flavor mixing exists [1], and this demonstrates that the standard model (SM) of particle physics is incomplete. With some specific parametrizations of the PMNS matrix U , the individual RGEs for neutrino masses, flavor mixing angles and CP-violating phases have been derived in Refs. Our main purpose is to analytically derive integral solutions to the one-loop RGEs for neutrino masses, flavor mixing angles, CP-violating phases and the Jarlskog invariant of U both in the Majorana case and in the Dirac case, with the τ -dominance approximations. We consider the most general case with the popular parametrization of U given in Eq (5) below the cutoff scale, and the integral results for neutrino masses, flavor mixing angles, CP-violating phases and the Jarlskog invariant are exhaustively derived.
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