CHIRAL SU(2) x SU(2) MIXING AND THE QUARK MODEL OF HADRONS

We present a simple form for the vector and axial-vector charges, transformed so that their matrix elements between (constituent) quark model states correspond to measurable transitions between physical states. A comparison with experiment of predictions for pionic transitions is made. (submitted for publication) *Work supported by the U. S. Atomic Energy Commission. TOn leave of absence from the Weizmann Institute of Science, Rehovot, Israel. The algebra of vector and axial-vector currents proposed by Gell-Mann’ has, for some time now, been one of the accepted “truths” of hadron physics. Given the correctness of the algebra, one is immediately led to consider how the observed particle and resonance states transform under this algebra. From the Adler-Weisberger relation2 itself, it is already clear that the observed hadron states at infinite momentum3 do not fall into irreducible representations of the chiral SU(2) X SU(2) algebra of charges. The axial-vector charge connects the nucleon to many higher mass N* states which contribute to the sum rule and must then share the same irreducible representation of SU(2) x SU(2) with the nucleon. Correspondingly, the nucleon must have components in several irreducible representations of the algebra. Although some progress and understanding have been achieved, 4 the problem of a complete classification of hadron states under the SU(2) X SU(2) charge algebra is unsolved. Furthermore, up to this point, much of the work on classifying the states has been on a case-by-case basis. For a systematic approach, one wants a transformation from the irreducible representations characteristic of the quark model to the reducible representations of the physical states. vll this paper, we assume such a transformation exists and choose to act with it on the charges rather than the states. Although the details of the transformation are unknown, we suggest that the transformed charges have a simple algebraic structure, allowing us to systematically relate many hadronic matrix elements. We start by defining the chiral SU(2) x SU(2) algebra of charges at equal times, 6 . . c I Q1, Q’ = i.e ijkQk, [Q;, Qj;l = i EijkQ”,, [Q;, Q;] = ic ‘jkQk, where i, j, k run from 1 to 3 and Q and Qs are the space integrals of the time component of the vector and axial-vector currents respectively. The operators (1)