General Chiral Algebra andππScattering Lengths

Weinberg's and Khuri's current-algebra calculations for the low-energy $\pi \pi$ amplitude are generalized by allowing an isospin $I=2\mathrm{}$ component in the $\sigma$ commutator (as opposed to a pure $I=0\mathrm{}$ component given in the $\sigma$ model used by both authors). A consistency condition for the $\pi \pi$ amplitude is derived when one of the pions has zero four-momentum $q_1=0\mathrm{}$, and the others restricted so that $0<\mathrm{}{q_2}^2$, ${q_3}^2$, ${q_4}^2<{\mu }^2$. Using this consistency condition, the power-series expansion of the amplitude in the variables $s$, $t$, and $u$ is calculated up to fourth order in the momenta. It is found that the coefficients of the fourth-order terms are much smaller than the second-order ones in all models. The scattering lengths are calculated in several models without resorting to an expansion of the $\sigma$ commutator into a series of the pion field. It is found in each model that the current-algebra calculation of the scattering lengths agrees with the effective Lagrangian calculation in the tree-diagram approximation. The reason for the agreement can be traced to the negligible contributions of the fourth- and higher-order terms in the pion field when the $\sigma$ commutator is expanded in a series.