Chiral Perturbation Theory

We consider perturbation theory for SU(2) \ifmmode\times\else\texttimes\fi{} SU(2) \ifmmode\times\else\texttimes\fi{} and SU(3) \ifmmode\times\else\texttimes\fi{} SU(3) chiral symmetries realized by Nambu-Goldstone bosons. Exact expressions are derived for the derivatives with respect to the symmetry-breaking parameter $\ensuremath{\epsilon}$ 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 $\ensuremath{\epsilon}=0$. We find (1) $S$-matrix elements go to finite limits as $\ensuremath{\epsilon}\ensuremath{\rightarrow}0$. (2) They in general approach this limit in a nonanalytic $\ensuremath{\epsilon}\mathrm{ln}\ensuremath{\epsilon}$ 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 ${f}_{\ensuremath{\pi}}$) 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 $\ensuremath{\epsilon}\mathrm{ln}\ensuremath{\epsilon}$ and can be calculated exactly. (5) The matrix elements of two or more zero-energy operators can diverge as $\mathrm{ln}\ensuremath{\epsilon}$ or $\frac{1}{\ensuremath{\epsilon}}$ or worse in the chiral limit. (6) The leading corrections in SU(2) \ifmmode\times\else\texttimes\fi{} SU(2) expansions are very small (a few percent). (7) Expansions around SU(3) \ifmmode\times\else\texttimes\fi{} 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 $\ensuremath{\pi}\ensuremath{\pi}$ and $\ensuremath{\pi}N$ amplitudes in some detail.

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