Numerical Approach to the Low-Energyπ−πBootstrap

Abstract

The $\ensuremath{\pi}\ensuremath{-}\ensuremath{\pi}$ amplitude in the low-energy region is parametrized in a crossing-symmetric way as the sum of the $\ensuremath{\rho}$ and ${f}^{0}$ resonance poles plus a polynomial background. The parametrization is flexible and capable of producing amplitudes having quite different features in the energy region below 1 GeV. The parameters are then varied so as to minimize the deviation from elastic unitarity on a set of closely spaced points. In addition, negative-moment finite-energy sum rules are used to connect the low-energy region with assumed Regge asymptotic behavior in the $I=1, 2$ amplitudes. With the mass and width of the ${f}^{0}$, the mass of the $\ensuremath{\rho}$, and the slope of the $\ensuremath{\rho}$ trajectory fixed, an approximate solution satisfying the constraints is found, yielding a $\ensuremath{\rho}$ width of 80\ifmmode\pm\else\textpm\fi{} 30 MeV. The solution displays the usual characteristics of the low-energy $\ensuremath{\pi}\ensuremath{-}\ensuremath{\pi}$ amplitudes suggested by other analyses, namely, small scattering lengths and a large $I=0$ $S$-wave phase shift near the mass of the $\ensuremath{\rho}$. This resonantlike behavior is found without introducing an $S$-wave pole in the parametrization, while the small scattering lengths are obtained as results of the numerical bootstrap, although no current-algebra constraints are included. Our proposed solution is also found to satisfy various inequalities proposed by Martin for the ${\ensuremath{\pi}}^{0}\ensuremath{-}{\ensuremath{\pi}}^{0}$ scattering amplitude.