Pion-Pion Scattering Based on Current Algebra, Analyticity, and Unitarity

Using SU(2) × SU(2) current algebra and pion-pole dominance, we derive from the Ward identities an exact crossing-symmetric expression for the $\pi \pi$ scattering amplitude. We make approximations which are suitable at low energy for those three- and four-point functions of the problem which cannot be determined from the constraints of current algebra. We parametrize these functions in terms of propagators and polynomials exhibiting the correct analyticity properties. Form factors, analytic in the cut plane, are expressed in effective-range form, and the $s$- and $p$-wave amplitudes are constructed in terms of them. The existence of resonances in the $\pi \pi$ system is not assumed, and soft-pion estimates are not used. Instead all the parameters are free to be varied. We determine all the free parameters of the problem self-consistently by imposing the constraints that follow from elastic unitarity. The scheme predicts all the features of low-energy $\pi \pi$ scattering, the only input parameters being $m_{\pi }$ and $F_{\pi }$, the pion mass and decay constant. Among our principal results are the $s$- and $p$-wave scattering lengths, the corresponding phase shifts, and the determination of an important parameter which measures the isospin $T=2\mathrm{}$ component of the $\sigma$ commutator, ${\sigma }^{\mathrm{ab}}$. The details of the method predispose scattering lengths to be small. We find that unitarity prefers the $T=2\mathrm{}$ component of ${\sigma }^{\mathrm{ab}}$ to be small relative to the $T=0\mathrm{}$ component. As a consequence, our scattering lengths are in excellent agreement with those obtained by Weinberg. The $T=J=1\mathrm{}$ phase shift exhibits a $\rho$ resonance around 915 MeV with a width of 210 MeV. The $T=2\mathrm{}$, $J=0\mathrm{}$ phase shift is small and in agreement with experimental results. The $T=J=0\mathrm{}$ phase shift displays acceptable behavior at low energy; we offer physical arguments to say that its higher-energy behavior is less reliable than that of the $p$ wave at the same energies. We discuss our results and analyze the predictive power of the method presented. Finally, we suggest some improvements on our calculations, including possible applications to related problems.