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.
