The use of unitarity bounds for a stable extrapolation of low energy data ( pi pi scattering)

The properties of the scattering amplitude allow one to define a function f(z) satisfying the following conditions: 1) f(z) is holomorphic in a simply connected domain D, which can be mapped conformally onto the unit disk; 2) mod Im f(z) mod is bounded by some constant M in D; 3) mod Re f(z) mod is known not to exceed some constant m on a certain part Gamma /sub 1/ of the boundary Gamma of D; f(z) is continuously extensible onto Gamma . Using these properties, constraints are derived on the real part of f(z) valid at any point z in D union Gamma . The result is used for performing a stable extrapolation of low energy pion-pion scattering data to any finite energy. A bound on energy averaged values of the real part of the scattering amplitude is derived. The bound depends on m, M, on the energy variable s and on the energy average interval s/sub 2/-s/sub 1 /. Generalizations of the method are discussed. (22 refs).