Modified Extrapolation to Determine theππCross Section

The dependence of the $\pi +N\to \left(2\mathrm{}\pi \right)+N$ cross section for low invariant $\pi \pi$ mass and small $\left|t\right|$ is discussed with a view toward extrapolation to the pion pole and extraction of $\pi \pi$ phase shifts. There is a complicated $t$-dependent term in the cross section which makes extrapolation inaccurate. This term can be isolated by consideration of certain moments (in the distribution with respect to the pion angles $\theta$, $\varphi$ in the dipion rest frame), if nucleon helicity flip is assumed to dominate. Indirect empirical evidence and the absorption model with $\pi$ exchange both support this assumption in the $5-10-\frac{\mathrm{BeV}}{c}$ region. The assumption is weaker than the popular assumption that the Chew-Low pion-exchange formula holds, and can also be directly tested by examining the $t$ dependence of certain moments at 0° (forward production of the dipion). For $s$- and $p$-wave $\pi \pi$ scattering, one considers the coefficients of 1, $cos\theta$, and ${sin}^2\theta$ in addition to the usual $\varphi$-dependent terms. The coefficient of ${sin}^2\theta$ is the complicated term. It is reasonable to hope that the remaining moments are smoothly varying for $0\ge -t\gtrsim 4{\mu }^2$, and are suitable for extrapolation. They are constrained to a definite $t$ dependence at 0° by angular momentum conservation. We feel that reliable determination of the $\pi \pi$ cross section in the $\rho$ region and below is attainable by this method in high-statistics experiments currently under way.