EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH CERN–TH.6801/93 Masses, Mixings, Yukawa Couplings and their Symmetries

We present a method to find the number of real and imaginary observable parameters coming from the Yukawa sector in an arbitrary gauge theory. The method leads naturally to a classification of Yukawa couplings according to their symmetries and suggests a new parametrization of masses and mixings that is useful to study the behaviour of Yukawa couplings under the renormalization group. We apply it to some examples based on the Standard Model with Yukawa couplings obeying various chiral symmetries. We also show how our method of parameter counting can be used in some models with an enlarged leptonic sector. CERN-TH.6801/93 February 1993 *) On leave of absence from Departament de F́ısica Teòrica, Universitat de València and IFIC, València, Spain. The most difficult part to understand of theories with spontaneously broken gauge theories is, together with the Higgs potential, the Yukawa sector. Its origin is unclear, and probably related to the origin of the spontaneous breaking of the symmetries itself. It contains a large number of free parameters that must be adjusted by hand to obtain a realistic spectrum of particles and mixings. From the 17 parameters needed to describe the electro-weak theory, 13 come from the Yukawa sector. In addition, the Yukawa couplings contain a lot of spurious parameters that cannot be observed because they can be removed from the Lagrangian with an appropriate choice of phases and mixings for the fermion fields. Thus, a simple question such as how many parameters are needed to describe a theory with Yukawa couplings cannot be answered at first sight. We cannot even say whether the couplings are real or not and so if the theory conserves CP or not. This represents a great complication when one tries to understand mass matrices in the Standard Model. In the last years this subject has received a lot of attention (for some popular mass-matrix models, see for example [1, 2, 3, 4]). Usually people make ansätze about a particular form of mass matrices and try to extract from them a reasonable spectrum of masses and mixings. However, completely different forms of mass matrices may be equivalent in their physical content. This has led to some confusion in the past. Moreover, some symmetries actually present in the Yukawa sector may be hidden when one does not work in the appropriate basis. For those reasons, an analysis based on symmetry should be superior and could clarify some of the issues present in particular forms of mass matrices. It would also allow a clear understanding of the number of parameters needed to describe the model. The standard method to calculate the number of parameters needed to describe the Yukawa sector of a spontaneously broken gauge theory is just by construction. After symmetry breaking one diagonalizes the mass matrices of all fields and after that one uses the freedom in the phase definition of the fields to reabsorb as many phases as possible. The remaining parameters in the Lagrangian are the physical parameters. As we know, the method works very well for the Standard Model. However, as the complication of the Yukawa sector increases, the method becomes intricate and can lead very easily to error; the question of CP-violation, for example, can then be difficult to answer. In addition, in many cases, phases and mixings can be moved from the gauge couplings to the Yukawa sector or even included in a redefinition of the fields [5, 6], which leads to further confusion. Here we propose a method, based on symmetry, to perform the counting before spontaneous symmetry breaking and without performing the explicit diagonalization of the mass matrices. The method clarifies the structure of the manifold of the parameters that are needed to describe the Yukawa sector of a gauge theory and suggests a parametrization that is useful when one wants to study the behaviour of the Yukawa couplings under the renormalization group. Basically the method consists in the study of the symmetries of both the full and the Yukawa Lagrangians. The number of parameters needed to describe the Yukawa sector comes from the balance between the number of parameters contained in the Yukawa matrices and the number of symmetries broken by the Yukawa couplings. As an example, we will start with the Standard Model with the most general set of Yukawa couplings. We will then impose to the Standard Model a set of chiral symmetries that could reduce the number of parameters and see how the method can be used in those cases. Finally we will apply the method to more involved models based on the same gauge group but different particle content. The number of parameters needed to describe the Yukawa sector of a theory is ob-