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:
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 .