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Explicit representations of chiral invariant Lagrangian theories of hadron dynamics

We study the general algebraic properties of and relations between the SUn tensors which characterize a Lagrangian theory invariant under chiral SUn × SUn. These tensors are the tensors Fij and vij which specify the chiral transformation laws of 0- meson and other multiplets of fields, together with SUn tensors which allow covariant derivatives of such fields to be built. Knowledge of these tensors is necessary and sufficient for the construction of SUn×SUn invariant Lagrangians and the related vector and axial vector currents. Our results include a vital, previously unnoticed, constraint on allowed Fij and completely new, and surprisingly simple, formulas for vij and the other relevant SUn tensors. We apply our general results for n = 3, giving explicit forms of possible Fij and proceeding thence to the creation of two simple models, the Cube Root and Rational Models of chiral SU3 × SU3, in which we can write down explicit closed forms for all SU3 tensors required for physical application. These models are quite new. Our work goes beyond that of Coleman, Wess & Zumino (1968) in that its general discussion is aimed towards and its SU3 part actually achieves the writing down of all results in closed form rather than giving only the early terms of power series expansions. It is possible to include explicitly and in closed form in each of our SU3 models terms which give rise to symmetry breaking according to the scheme of Gell-Mann, Oakes & Renner (1968).