Volume 98B, number 6 PHYSICS LETTERS 29 January 1981 AMBIGUITIES IN ZEROTH ORDER FERMION MASS MATRICES M. GRONAU and R. YAHALOM Department of Physics, Technion - Israel Institute of Technology, Haifa, Israel Received 15 September 1980 The six quark Weinberg-Salam model with the symmetry of the three generation permutation group S3 has an extra 0(2) pseudosymmetry when the Higgs fields lie in an $3 doublet. We use this model as a prototype to question the deriva- tion of zeroth order relations between fermion mixing angles and fermion masses. It has been suggested for some time that relations between fermion mixing angles 0.e. generalized Cabibbo angles) and fermion masses may be obtained by im- posing discrete symmetries on the least understood part of the weak and electromagnetic interactions - i.e. the couplings of the Higgs fields to the fermions and to themselves [1]. When applying this idea one calculates the fermion mass matrix in the lowest order of perturbation theory and (with appropriately chosen Higgs vacuum expectation values) expects the results to be stable under higher orders. Some of these appli- cations have led to rather interesting results. Here we would like to address a crucial point which has been overlooked more than once in these applications. Since the Higgs potential V(0) is at most a quartic polynomial, il may acquire some continuous symmetry by imposing the discrete symmetry condition. This new symmetry (pseudosymmetry [2]) is not necessarily shared by the rest of the lagrangian. In this case one expects two phe- nomena [2]: (1) The existence of so-called pseudo-Goldstone bosons, i.e. bos0ns which are massless in zeroth order and which pick up a finite mass from higher order ef- fects. (2) There is an infinity of solutions of the absolute minimum condition of V(~b), all obtained from each other by acting with the pseudosymmetry transforma- tions. The mixing angle and fermion mass (MAFM) re- lations may in principle depend on the specific choice of one of these solutions as the vacuum expectation values of the Higgs fields. The physical solution is in fact determined by higher order corrections [3]. In this short note we would like to illustrate the second point by an example which has already been previously studied in the literature [4] however over- looking the effect of a pseudosymmetry. Different MAFM relations will be shown to follow from two apparently (by lowest order) possible choices of the Higgs vacuum expectation values. (No attempt will be made to determine the physical solution from higher order corrections.) We will also illustrate by our exam- ple the possibility that all the infinity of solutions of the minimum condition of V(~b) are physically equivalent. This is the case when these solutions may as well be obtained from each other by acting with gauge group transformations. Now, the MAFM relations are unique- ly determined by zeroth order calculations and no pseudo-Goldstone boson is expected. We will consider the phenomenologically successful and by now the almost standard six quark SU 2 X U 1 gauge model [5,6], in which the weak eigenstates are assigned to three left-handed doublets and six right- handed singlets: (u0) (co) (to) L 1 = , L2 = , L3 = , do L So L b0 L U0R , d0R ; C0R , S0R ; t0R , b0R . (1) The Higgs fields belong to weak isodoublets. Since the theory, prior to the generation of quark masses, is in- variant under permutation of the three generations of quarks, it is tempting to assume that also the Higgs sec- 0 031-9163/81/0000-0000/$ 02.50 © North-Holland Publishing Company 441