Algebraic Realizations of Chiral Symmetry

The sum of tree graphs for forward pion scattering, generated by any chiral-invariant Lagrangian, is required to grow no faster at high energies than the actual scattering amplitude. In consequence, algebraic restrictions must be imposed on the axial-vector coupling matrix X and the mass matrix m²: For each helicity, X must, together with the isospin T, form a representation of SU(2)SU(2), and m² must behave with respect to commutation with T and X as the sum of a chiral scalar and the fourth component of a chiral four-vector. If it is further assumed that the contribution of tree graphs to inelastic forward pion scattering vanishes at high energy, the two parts of the mass matrix must commute; this fixes various mixing angles, and leads to predictions like m_σ=m_ρ, m_A₁=√2m_ρ, Γ_ρ=135 MeV, etc. If all pion transitions involved only p-wave pions, then X would form part of the algebra of SU(4), and the mass matrix would behave as the sum of a 1- and a 20-dimensional representation of SU(4); if s-wave transitions are allowed, then the algebra must be enlarged to at least SO(7).