DSF 94/59 INFN NA IV 94/59 A Geometric Picture for Fermion Masses

We describe a geometric picture for the pattern of fermion masses of the three generations which is invariant with respect to the renormalization group below the electroweak scale. Moreover, we predict the upper limit for the ratio between the Dirac masses of the and neutrinos, m =m (9:6 0:6) 10 . The Standard Model [1, 2] of strong and electroweak interactions describes accurately all the observed phenomena of elementary particles. In this framework, the fermion masses are essentially free parameters that must be xed by experiment. Many e orts have been devoted to nd relations among fermion masses, empirically and/or theoretically. For example, a composite model for \fundamental" fermions proposed in [3] predicts the following relation among the masses of the three generations of quarks and leptons mf (3) uutm3f (2) mf (1) (1) which is experimentally well veri ed only for charged leptons. Semi-empirical sum rules among masses can also be found, as the so-called \generationchanging mass-ratio sum rules" [4] s mc mu s ms md = s m me s mt mc s mb ms = s m m (2) which works fairly well for the experimental values given in [5]. Furthermore, the previous formulas may indicate the quarks and leptons in the second and third generations as the excited states of the corresponding ones of the rst generation. Other relations among fermion masses, more or less elegant and experimentally veri ed have been proposed; see for example [6]. Koide [7] has shown that the last experimental data about the tau lepton mass [5] are in excellent agreement with the following lepton mass formula me +m +m = 2 3 p me + p m + p m 2 (3) which he deduced in several di erent models [8]. An intriguing geometric interpretation of the previous formula has been recently proposed by Foot [9]. He considers a three-dimensional Euclidean vector space and the vectorM of components ( p m0; p m0; p m0) in a monometric, orthogonal, Cartesian reference frame and a vector L, associated to the charged leptons, with components ( p me; p m ; p m ). Here we generalize the suggestion of Foot which considers M= (1; 1; 1). Though the