Annihilation range and final-state interaction in the $\bar{p}-p$ annihilation into $\pi^-\pi^+$

11 pages, 11 figures (Revtex 4), revised version with one additional figure. Accepted for publication in PRCThe large set of accurate data on differential cross section and analyzing power from the CERN LEAR experiment on $\bar pp \to \pi^+\pi^-$ in the range from 360 to 1550 MeV/c is well reproduced within a distorted wave approximation approach. The initial $\bar pp$ scattering wave functions originate from a recent $\bar N N$ model. The transition operator is obtained from a combination of the $^3P_0$ and $^3S_1$ quark-antiquark annihilation mechanisms. A good fit to the data, in particular the reproduction of the double dip structure observed in the analyzing powers, requires quark wave functions for proton, antiproton, and pions with radii slightly larger than the respective measured charge radii. This corresponds to an increase in range of the annihilation mechanisms and consequently the amplitudes for total angular momentum J=2 and higher are much larger than in previous approaches. The final state $\pi\pi$ wave functions, parameterized in terms of $\pi\pi$ phase shifts and inelasticities, are also a very important ingredient for the fine tuning of the fit to the observables