The discrete flavor symmetry A 4 explains very well neutrino data at low energy, but it seems difficult to extend it to grand unified models since, in general, left-handed and right-handed fields belong to different A 4 representations. Recently a model has been proposed where all the fermions equally transform under A 4 . We study here a concrete SO10 realization of such a model providing small neutrino masses through the see-saw mechanism. We fit the charged fermion masses run up to the unification scale. Some fermion masses properties come from the SO10 symmetry while lepton mixing angles are a consequence of the A 4 properties. Moreover, our model predicts the absolute value of the neutrino masses; these are in the range m ' 0:005-0:052 eV.

This chapter of the report of the "Flavor in the era of the LHC" Workshop discusses the theoretical, phenomenological and experimental issues related to flavor phenomena in the charged lepton sector and in flavor conserving CPviolating processes. We review the current experimental limits and the main theoretical models for the flavor structure of fundamental particles. We analyze the phenomenological consequences of the available data, setting constraints on explicit models beyond the standard model, presenting benchmarks for the discovery potential of forthcoming measurements both at the LHC and at low energy, and exploring options for possible future experiments.

The existence of a SL(2, R) symmetry is discussed in SU (N ) Yang-Mills in the maximal Abelian Gauge. This symmetry, also present in the Landau and Curci-Ferrari gauge, ensures the absence of tachyons in the maximal Abelian gauge. In all these gauges, SL(2, R) turns out to be dynamically broken by ghost condensates.It is widely believed that the dual superconductivity mechanism [1, 2] can be at the origin of color confinement. The key ingredients of this mechanism are the Abelian dominance and the monopoles condensation. According to the dual superconductivity picture, the low energy behavior of QCD should be described by an effective Abelian theory in the presence of monopoles. The condensation of the monopoles gives rise to the formation of Abrikosov-Nielsen-Olesen flux tubes which confine all chromoelectric charges. This mechanism has received many confirmations from lattice simulations in Abelian gauges, which are very useful in order to characterize the effective relevant degrees of freedom at low energies.Among the Abelian gauges, the so called maximal Abelian gauge (MAG) plays an important role. This gauge, introduced in [2, 3], has given evidences for the Abelian dominance and for the monopoles condensation, while providing a renormalizable gauge in the continuum. Here, the Abelian degrees of freedom are identified with the components of the gauge field belonging to the Cartan subgroup of the gauge group SU(N). The other components correspond to the (N 2 − N) off-diagonal generators of SU(N) and, being no longer protected by gauge invariance, are expected to acquire a mass, thus decoupling at low energies. The understanding of the mechanism for the dynamical mass generation of the off-diagonal components is fundamental for the Abelian dominance.A feature to be underlined is that the MAG is a nonlinear gauge. As a consequence, a quartic self-interaction term in the Faddeev-Popov ghosts is necessarily required for renormalizability [4,5]. Furthermore, as discussed in [6,7] and later on in [8], the four ghost interaction gives rise to an effective potential whose vacuum configuration favors the formation of off-diagonal ghost condensates cc , cc and cc 1 . However, these ghost condensates were proven [9] to originate an unwanted effective tachyon mass for the off-diagonal gluons, due to the presence in the MAG of an off-diagonal interaction term 1 The notation for the ghost condensates cc , cc and cc stands for f iab c a c b , f iab c a c b and f iab c a c b , where f iab are the structure constants of the gauge group. The index i runs over the Cartan generators, while the indices a, b correspond to the off-diagonal generators.

The effective potential for an on-shell BRST invariant gluon-ghost condensate of mass dimension 2 in the Curci-Ferrari gauge in SU (N ) Yang-Mills is analysed by combining the local composite operator technique with the algebraic renormalization. We pay attention to the gauge parameter independence of the vacuum energy obtained in the considered framework and discuss the Landau gauge as an interesting special case.