We revisit the time evolution of the lepton family number for a SU(2) doublet consisting of a neutrino and a charged lepton. The lepton family number is defined through the weak basis of the SU(2) doublet, where the charged lepton mass matrix is real and diagonal. The lepton family number carried by the neutrino is defined by the left-handed current of the neutrino family. For this work we assume the neutrinos have Majorana mass. This Majorana mass term is switched on at time t = 0 and the lepton family number is evolved. Since the operator in the avor eigenstate is continuously connected to that of the mass eigenstate, the creation and annihilation operators for the two eigenstates are related to each other. We compute the time evolution of all lepton family numbers by choosing a specific initial avor eigenstate for a neutrino. The evolution is studied for relativistic and nonrelativistic neutrinos. The nonrelativistic region is of particular interest for the Cosmic Neutrino Background predicted from big bang models. In that region we find the lepton family numbers are sensitive to Majorana and Dirac phases, the absolute mass, and mass hierarchy of neutrinos.
We study the time evolution of lepton family number for neutrino which forms SU(2) doublet with charged lepton. The lepton family number is defined through a weak basis of SU(2) doublet in which the charged lepton mass matrix is a real and diagonal one. The lepton family number carried by the neutrino is defined with a left-handed current of the neutrino family. We study the time evolution of the lepton family number operator for Majorana neutrino. To be definite, we introduce the mass term at t = 0 and study the time evolution of the lepton family number for the later time. Since the operator in flavor eigenstate is continuously connected to that of the mass eigenstate, the creation and annihilation operators for flavor eigenstates are related to those of mass eigenstates. The total lepton number of the Majorana neutrino is conserved. By choosing a specific flavor eigenstate of neutrino as an initial state, we compute the time evolution of all lepton family numbers. They are sensitive to Majorana and Dirac phases and also are sensitive to the absolute mass and mass hierarchy of neutrinos.
In this talk, we have reviewed the recent development on the time evolution of lepton family number carried by Majorana neutrinos [1]. This article focuses on the subtle points of the derivation of the lepton family numbers and their time evolution. We also show how the time evolution is sensitive to m ee and m eµ components of the effective Majorana mass matrix by applying the formula to the two family case. The dependence on the Majorana phase is clarified and the implication on CNB (cosmic neutrino background) is also discussed.
The low energy effective potential for the model with a light scalar and a heavy scalar is derived. We perform the path integration for both heavy and light scalars and derive the low energy effective potential in terms of only the light scalar. The effective potential is independent of the renormalization scale approximately. By setting the renormalization scale equal to the mass of the heavy scalar, one finds the corrections with the logarithm of the ratio of the two scalar masses. The large logarithm is summed with the renormalization group (RG) and the RG improved effective potential is derived. The improved effective potential includes the one-loop correction of the heavy scalar and the leading logarithmic corrections due to the light scalar. We study the correction to the vacuum expectation value of the light scalar and the dependence on the mass of the heavy scalar.
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