Leptons participate in the weak interactions via the left-handed doublets:

Suppose and are the relative sizes
of the
and couplings. EM interactions
do not
distinguish **e** from . Similarly, in weak interactions,
[Got-84]. By redefining the weak
coupling constant, . This is the
symmetry of the weak interaction under the exchange
. Because the neutrinos
are massless,
. The inclusion of
the third doublet of left-handed leptons

leads to:

In the quark sector, one has the left-handed fermion states

It is not quite correct to extend the lepton universality to the
quarks. That will lead to transitions between
, and
only. However, decays such as
occur. Since is made up of
**u** and quarks, there must be a weak current which couples
**u** and , that is . In order to avoid the
introduction of a new coupling constant for quarks --- preserve
universality --- Cabibbo assumed that the weak interaction rotates quark
states. Subsequently, Kobayashi and Maskawa generalized Cabibbo's work to
three generations:

In other word, the quark states **u**, , **c**, , **t**, are
eigenstates of the weak interaction while the states **u**, **d**, **c**, **s**,
**t**, **b** are the mass eigenstates of the flavor-preserving strong
interaction. The weakly charged current of the quarks is:

where **U** is a matrix known as the
Cabibbo-Kobayashi-Maskawa (CKM) mixing matrix:

or

where , ;
are the Cabibbo angles and is a phase factor. At four-quark
level **U** reduces to a real matrix
and the theory can be shown to be **CP** invariant. However, with the
extension to **t** and **b** quarks, the mixing matrix contains a phase
factor and for that reason the theory violates [Hal-84].

The implication is that for nuclear beta decays involving
only vector interactions, one must replace the coupling
constant **G** of equation () by

In the case of a pure leptonic decay (-decay for instance), there is no mixing:

The top row the CKM matrix (equations and ) provides the unitarity test

which must be satisfied if the Standard Model with three generations is
correct.

Mon Jan 15 14:57:06 MET 1996