Cabibbo Theory, SU(3) Symmetry, and Weak N–Y Transition Form
Factors
For the ∆S = 0 processes,
and for the |∆S| = 1 processes,
the matrix elements of the vector (Vμ) and
the axial vector (Aμ) currents between a
nucleon or a hyperon and a nucleon N = n,
p are written as:
and
where and are the masses of the nucleon and hyperon, respectively. and are
the vector, weak magnetic and induced scalar N − Y
transition form factors and and are the axial vector, induced tensor (or
weak electric), and induced pseudoscalar form factors, respectively.
In the Cabibbo theory, the weak vector
(Vμ) and the axial vector
(Aμ) currents corresponding to
the ∆S = 0 and ∆S = 1
hadronic currents whose matrix elements are defined between the states are
assumed to belong to the octet representation of SU(3).
Accordingly, they are defined as:
Where are the generators of flavor SU(3) and is are the well-known
Gell–Mann matrices written as
The generators obey the following algebra of SU(3) generators
are the structure constants, and are antisymmetric and symmetric,
respectively, under the interchange of any two indices. These are obtained
using the λi given in Eq. (B.9) and have been
tabulated in Table B.1.
From the property of the SU(3) group, it follows that there are three
corresponding SU(2) subgroups of SU(3) which must be invariant under the
interchange of quark pairs ud, ds, and
us respectively, if the group is invariant under the
interchange of u, d, and
s quarks. Each of these SU(2) subgroups has raising and
lowering operators. One of them is SU(2)I , generated by
the generators (λ1,
λ2,
λ3) to be identified with the isospin
operators (I1, I2,
I3) in the isospin space. For example,
I± of isospin space is given
by
The other two are defined as SU(2)U and SU(2)V generated by the generators ,
respectively, in the U-spin and V-spin space with (d s) and
(u s) forming the basic doublet representation of
SU(2)U and SU(2)V .