Cabibbo Theory, SU(3) Symmetry, and Weak N–Y Transition FormFactors

For the ∆S = 0 processes,

and for the |∆S| = 1 processes,

the matrix elements of the vector (Vμ) andthe axial vector (Aμ) currents between anucleon 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 arethe vector, weak magnetic and induced scalar N − Ytransition form factors and and are the axial vector, induced tensor (orweak electric), and induced pseudoscalar form factors, respectively.

In the Cabibbo theory, the weak vector(Vμ) and the axial vector(Aμ) currents corresponding tothe ∆S = 0 and ∆S = 1hadronic currents whose matrix elements are defined between the states areassumed 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-knownGell–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 obtainedusing the λi given in Eq. (B.9) and have beentabulated in Table B.1.

From the property of the SU(3) group, it follows that there are threecorresponding SU(2) subgroups of SU(3) which must be invariant under theinterchange of quark pairs ud, ds, andus respectively, if the group is invariant under theinterchange of u, d, ands quarks. Each of these SU(2) subgroups has raising andlowering operators. One of them is SU(2)I , generated bythe generators (λ1,λ2,λ3) to be identified with the isospinoperators (I1, I2,I3) in the isospin space. For example,I± of isospin space is givenby

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 ofSU(2)U and SU(2)V .