Quark masses and the KM matrix

In Section 4.2 we showed how the leptons could be coupled to the Higgs (scalar) field in such a way that the non-zero value of / in the vacuum generated mass terms for the leptons e, n, r while leaving their neutrinos massless. The interaction was of the form (4.2.48) 

A similar mechanism can be invoked to give mass to the quarks [though there may be other sources of mass for the quarks (see Section 20.3)], but this would leave the up-type quarks, the analogues of the neutrinos, massless. In order to construct an interaction invariant under weak SU(2) and weak U 1) which will give mass to the quarks with = 1/2, it turns out, on the basis of Table 9.2, that we need a Higgs doublet with Yw = —1-Recall that l = has Y y = 1, and is a weak isospin doublet. Recall also, that for ordinary isospin, the nucleon doublet N = ( ) and the antinucleon doublet N = (jp) (note the minus sign ) transform identically under isospin rotations. By analogy 

But now postulate that these bare quarks are not eigenstates of mass, i.e. that their mass-like couplings are non-diagonal. The mass-generating interaction can then be of the form 

We now wish to diagonalize D and U in order to find linear combinations of the bare quarks which have definite mass. Since D (and U) is an arbitrary matrix it cannot be diagonalized by one single unitary matrix as would be the case for a hermitian matrix. But it is possible to find pairs of unitary matrices U, U and C/l,such that 

In terms of these fields Cs-Qimrks in (9 J.6) becomes a sum of standard, diagonal, mass-like terms of the form one for each quark. 

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Quark masses and the KM matrix 9.7 Quark masses and the KM matrix... 

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