Chiral Perturbation Theory

Abstract

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 $\epsilon$ 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 $\epsilon =0\mathrm{}$. We find (1) $S$-matrix elements go to finite limits as $\epsilon \to 0$. (2) They in general approach this limit in a nonanalytic $\epsilon \ln \epsilon$ 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 $f_{\pi }$) 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 $\epsilon \ln \epsilon$ and can be calculated exactly. (5) The matrix elements of two or more zero-energy operators can diverge as $\ln \epsilon$ or $\frac{1}{\epsilon }$ 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 $\pi \pi$ and $\pi N$ amplitudes in some detail.