We consider perturbation theory for SU(2) × SU(2) × and SU(3) × SU(3) chiral symmetries realized by Nambu-Goldstone bosons. Exact expressions are derived for the derivatives with respect to the symmetry-breaking parameter ε of Green's functions, scattering amplitudes, and the matrix elements of operators, including the effects of renormalization and the external mass-shell constraints. These expressions are used to systematically classify all leading nonanalytic behavior in the expansion of these quantities around ε=0. We find (1) S-matrix elements go to finite limits as ε→0. (2) They in general approach this limit in a nonanalytic εlnε manner. (3) At exceptional momentum points, corresponding to the low-energy theorems of current algebra, the leading nonanalytic corrections can be absorbed into the renormalization of the parameters (such as fπ) of the theory by the symmetry-breaking interaction. Hence leading-order corrections to low-energy theorems are expected to be analytic. (4) The errors in off-shell partial-conservation-of-axial-vector-current extrapolations are often of order εlnε and can be calculated exactly. (5) The matrix elements of two or more zero-energy operators can diverge as lnε or 1ε or worse in the chiral limit. (6) The leading corrections in SU(2) × SU(2) expansions are very small (a few percent). (7) Expansions around SU(3) × SU(3) are marginal. The corrections are often 30% and in one case are larger than the leading term. We calculate the leading renormalization of the meson decay constants and consider the ππ and πN amplitudes in some detail.
All Science Journal Classification (ASJC) codes
- Physics and Astronomy (miscellaneous)