Scattering in the quenched approximation

We study, in the quenched approximation, Luescher’s relation between pion scattering lengths and the finite-volume energy of two pions at rest. The quenched relation is drastically different from the full theory one; in particular, ``enhanced finite-volume corrections" of order $L^0=1$ and $L^{-2}$ occur at one loop ($L$ is the linear size of the box), due to the special properties of the ${\eta }^{\mathrm{\prime }}$ in the quenched approximation. Numerical examples show that the size of these effects can be substantial.