Abstract: Motivated by discrete flavour symmetry models, we analyse Spontaneous CP Violation (SCPV) for potentials involving three or six Higgs fi elds (both electroweak doublets and singlets) which fall into irreducible triplet representations of discrete symmetries belonging to the Delta(3n(2)) and Delta(6n(2)) series, including A(4), S-4, Delta(27) and Delta(54). For each case, we give the potential and fi nd various global minima for di ff erent regions of the parameter space of the potential. Using CP-odd basis Invariants that indicate the presence of Spontaneous CP Violation we separate the VEVs into those that do or do not violate CP. In cases where CP is preserved we reveal a CP symmetry of the potential that is preserved by those VEVs, otherwise we display a non-zero CP-odd Invariant. Finally we identify interesting cases where there is Spontaneous Geometrical CP Violation in which the VEVs have geometrical phases.

Abstract: In this paper we present a novel CPT symmetry test in the neutral kaon system based, for the first time, on the direct comparison of the probabilities of a transition and its CPT reverse. The required interchange of in <-> out states for a given process is obtained exploiting the Einstein-Podolsky-Rosen correlations of neutral kaon pairs produced at a phi-factory. The observable quantities have been constructed by selecting the two semileptonic decays for flavour tag, the pi and 3 pi(0) decays for CP tag and the time orderings of the decay pairs. The interpretation in terms of the standard Weisskopf-Wigner approach to this system, directly connects CPT violation in these observables to the violating R delta parameter in the mass matrix of K-0 – (K) over bar (0), a genuine CPT violating effect independent of Delta Gamma and not requiring the decay as an essential ingredient. Possible spurious effects induced by CP violation in the decay and/or a violation of the Delta S = Delta Q rule have been shown to be well under control. The proposed test is thus fully robust, and might shed light on possible new CPT violating mechanisms, or further improve the precision of the present experimental limits. It could be implemented at the DA Phi NE facility in Frascati, where the KLOE-2 experiment might reach a statistical sensitivity of O (10(-3)) on the newly proposed observable quantities.

Abstract: The so-called Golden Mass Relation provides a testable correlation between charged-lepton and down-type quark masses, that arises in certain flavor models that do not rely on Grand Unification. Such models typically involve broken family symmetries. In this work, we demonstrate that realistic fermion mass relations can emerge naturally in modular invariant models, without relying on ad hoc flavon alignments. We provide a model-independent derivation of a class of mass relations that are experimentally testable. These relations are determined by both the Clebsch-Gordan coefficients of the specific finite modular group and the expansion coefficients of its modular forms, thus offering potential probes of modular invariant models. As a detailed example, we present a set of viable mass relations based on the Gamma 4 approximately equal to S4 symmetry, which have calculable deviations from the usual Golden Mass Relation.