Up Quark Masses from Down Quark Masses

The quark and charged lepton masses and the angles and phase of the CKM mixing matrix are nicely reproduced in a model which assumes O(3) ⊗ O(3) flavour symmetry broken by the v.e.v.’s of fields in its bi-fundamental representation. The relations among the quark mass eigenvalues, mu/mc ≈ mc/mt ≈ m2d/m2s ≈ m2s/m2b ≈ Λ2GUT/M2 Pl, follow from the broken flavour symmetry. Large tanβ is required which also provides the best fits to data for the obtained textures. Lepton-quark grandunification with a field that breaks both SU(5) and the flavour group correctly extends the predictions to the charged lepton masses. The seesaw extension of the model to the neutrino sector predicts a Majorana mass matrix quadratically hierarchical as compared to the neutrino Dirac mass matrix, naturally yielding large mixings and low mass hierarchy for neutrinos. In a bottom-up approach to the flavour problem, the first task is to decode the variety of experimental data on fermion mass ratios and mixings. By translating that information into simple mass matrix textures that reproduce the empirical relations one makes progress towards the interpretation of these textures in terms of flavour symmetries and their breakings. In spite of the many interesting textures and models present in the literature, the flavour issue still remains a totally open problem. We still ignore why mt ≫ mb while mu < md. In models based on abelian symmetries, the different scales present in the mass matrices are associated to powers of the small parameters defined in terms of the choice of the fermion abelian charges, all their O(1) complex pre-factors remaining arbitrary [1]. This intrinsically limits the predictivity of these models and their selection by the experimental progress. Instead, non-abelian flavour symmetries in principle establish exact relations and are more constrained but the overall hierarchies require more than one small parameter and more involved symmetry breaking schemes [2]. In this paper we develop a novel approach to these issues based on the following argumentation. Since the first suggestion of a relation between the Cabibbo angle and quark mass ratios [3], many models were built where the angles of the CKM mixing matrix are expressed in terms of the quark mass eigenvalues (in practice the angles are better known than some of the light quark masses). Many textures have been designed according to these lines, starting from those with a maximal number of vanishing matrix elements which are now excluded by the experiments [4, 5]. Some more recent ones [6] try to improve the fit to the data by enriching the textures, but are not quite consistent with the latest data [7]. Neutrino oscillation experiments have prompted a lot of work on the analogous approach to the lepton mass matrices including solutions consistent with lepton-quark (grand-)unification [8]. The considerable progress obtained in the last years in the measurements of the quark mixings, all quite consistent with the Standard Model (SM) description of flavour changing and CP violation effects [9], as well as in neutrino mixings, implies a much stronger selection on the allowed textures. One obvious difficulty in the definition of the mass textures is their dependence on the assumed basis for the fermions since only the family mixings inside the weak doublets of quarks or leptons is observable without physics beyond the SM. This allows for several different patterns of mass textures even after the number of free parameters are reduced by theoretical assumptions or educated guess. This fact notwithstanding, these efforts are an important step in the quest for the symmetries underlying the flavour theory which should naturally ensure these relations. Because of the hierarchical nature of the charged fermion masses and observed mixings, the flavour theory must also contain one or more small parameters whose existence would be natural only if protected by spontaneously broken flavour symmetry. In a recent publication [7], we have built fermion mass matrices by the identification of a few characteristic features of the mixing angles and phases and their implementation