We systematically analyze quantum corrections in see-saw scenarios, including effects from above as well as below the see-saw scales. We derive approximate renormalization group equations for neutrino masses, lepton mixings and CP phases, yielding an analytic understanding and a simple estimate of the size of the effects. Even for hierarchical masses, they often exceed the precision of future experiments. Furthermore, we provide a software package allowing for a convenient numerical renormalization group analysis, with heavy singlets being integrated out successively at their mass thresholds. We also discuss applications to model building and related topics.This relation emerges from integrating out heavy, singlet neutrinos with mass matrix M. The Dirac neutrino mass m Dirac ν is proportional to the neutrino Yukawa coupling Y ν . Clearly, the see-saw operates at high energy scales while its implications are measured by experiments at low scales. Therefore, the neutrino masses given by Eq. (1) are subject to quantum corrections, i.e. they are modified by renormalization group (RG) running.The running of neutrino masses and lepton mixing angles has been investigated intensively in the literature. For non-hierarchical neutrino mass spectra, RG effects can be very large and they can have interesting implications for model building. For example, lepton mixing angles can be magnified [6,7,8,9,10], bimaximal mixing at high energy can be compatible with low-energy experiments [11,12,13] or the small mass splittings can be generated from exactly degenerate light neutrinos [14,15,16,17,18,19]. On the other hand, facing the high precision of future neutrino experiments, rather small RG corrections are important as well. For instance, deviations from θ 13 = 0 or maximal mixing θ 23 = π/4 are induced by RG effects [20,21,22] also for a hierarchical spectrum. However, in most studies only the running of the dimension 5 operator has been considered, which is only appropriate for the energy range below the mass scale of the heavy singlets.The importance of including the effects from energy ranges above and between these mass thresholds when analyzing RG effects in GUT models has been pointed out in [23,24,25,26,27,11,8,12,13,21]. They are typically at least as important as the effects from below the thresholds since the relevant couplings, i.e. the entries of Y ν , can be of order one, regardless of tan β. 1 Previous studies have investigated the RG effects above the see-saw scales mainly numerically.In this paper we derive formulae which allow to understand the running of the neutrino parameters above the see-saw scales analytically. We further provide a software package for analyzing the RG evolution (with correct treatment of non-degenerate see-saw scales) numerically. We apply our results to investigate consequences of the running above the see-saw scales for model building and leptogenesis and compare the size of RG corrections to the precision of future experiments.The paper is organized as follows: In Sec. 2, we review ho...
We give a consistent definition of generalised CP transformations in the context of discrete flavour symmetries. Non-trivial consistency conditions imply that every generalised CP transformation can be interpreted as a representation of an automorphism of the discrete group. This allows us to give consistent generalised CP transformations of popular flavour groups. We are able to clear up issues concerning recent claims about geometrical CP violation in models based on T , clarify the origin of "calculable phases" in ∆(27) and explain why apparently CP violating scalar potentials of A 4 result in a CP conserving ground state.
A plausible explanation for the lightness of neutrino masses is that neutrinos are massless at tree level, with their mass (typically Majorana) being generated radiatively at one or more loops. The new couplings, together with the suppression coming from the loop factors, imply that the new degrees of freedom cannot be too heavy (they are typically at the TeV scale). Therefore, in these models there are no large mass hierarchies and they can be tested using different searches, making their detailed phenomenological study very appealing. In particular, the new particles can be searched for at colliders and generically induce signals in lepton-flavor and lepton-number violating processes (in the case of Majorana neutrinos), which are not independent from reproducing correctly the neutrino masses and mixings. The main focus of the review is on Majorana neutrinos. We order the allowed theory space from three different perspectives: (i) using an effective operator approach to lepton number violation, (ii) by the number of loops at which the Weinberg operator is generated, (iii) within a given loop order, by the possible irreducible topologies. We also discuss in more detail some popular radiative models which involve qualitatively different features, revisiting their most important phenomenological implications. Finally, we list some promising avenues to pursue.
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