I=2 Two-Pion Wave Functions with Non-zero Total Momentum

We calculate the two-pion wave function for the I=2 $S$-wave two-pion system with a finite scattering momentum and estimate the interaction range between two pions. It allows us to examine the validity of the necessary condition for the finite-volume method for the scattering phase shift. A calculation is carried out with a plaquette gauge action for gluons and a clover-improved Wilson action for quarks at $1\mathrm{/}a=1.63\mathrm{G}\mathrm{e}\mathrm{V}$ on ${32}^3\times 120$ lattice in the quenched approximation. We conclude that the necessary condition is satisfied within statistical errors for the lattice size $L\ge 32$, when the quark mass is in the range $m_{\pi }^2=0.176-0.345{\mathrm{G}\mathrm{e}\mathrm{V}}^2$ and the scattering momentum in $k^2<0.026{\mathrm{G}\mathrm{e}\mathrm{V}}^2$. We also find that the energy dependence of the interaction range is small and it takes $1.2-1.7\mathrm{f}\mathrm{m}$ for our simulation parameters. We obtain the phase shift from the two-pion wave function with a smaller statistical error than that from the conventional analysis with the two-pion time correlator.