I=2 pion scattering length from two-pion wave functions

We calculate the two-pion wave function in the ground state of the I=2 S-wave system and find the interaction range between two pions, which allows us to examine the validity of the necessary condition for the finite-volume method for the scattering length proposed by Lüscher. We work in the quenched approximation employing a renormalization group improved gauge action for gluons and an improved Wilson action for quarks at 1/a=1.207(12) GeV on 16³×80, 20³×80, and 24³×80 lattices. We conclude that the necessary condition is satisfied within the statistical errors for the lattice sizes L≥24 (3.92 fm) when the quark mass is in the range that corresponds to m²_π=0.273 0.736 GeV². We obtain the scattering length with a smaller statistical error from the wave function than from the two-pion time correlator.