SU(2) X SU(2) BREAKING AND CABIBBO ANGLE

In a recent investigation of the transformation properties under chiral SU(3) × SU(3) of the hadron energy density H(X) Gell-Mann, Oakes and Renner have assumed the form H = H - H0 - CU8 where H is SU(3) × SU(3) invariant while U0 and U8 belong to the representation (3*, 3) + (3, 3*) and have found the constant C to be near the value - 2, which corresponds to the limit in which the subgroup SU(2) × SU(2) remains an exact symmetry with the conservation of the axial-vector currents being realized through massless pions. Here we pursue the question of the SU(3) × SU(3) transformation properties of H(X) further and investigate the deviations from the limiting case C = - 2 under specific assumptions regarding the origin of the breaking of SU(2) × SU(2). Our main assumption is that chiral SU(2) × SU(2) breaking in strong interactions (C ≠ - 2, mπ ≠ 0) and the non-conservation of strangeness in weak interactions (Cabibbo angle θ ≠ 0) derive from a common origin. We find that all the small parameters C + 2, mπ/mK and θ are simply related to each other and that the hadron energy density H(X) constains in addition a term U3, belonging to the same (3*, 3) + (3, 3*) representation of determined strength which also breaks SU(2) and thus contributes to certain mass shifts, e.g., K+ - Ko, along with the usual electromagnetic interaction. Numerically the comparison with experiment is favourable lending support to our assumptions regarding the breaking of SU(2) × SU(2). © 1969.