Finite-volume two-pion energies and scattering in the quenched approximation

We investigate how Luescher close-quote s relation between the finite-volume energy of two pions at rest and pion scattering lengths has to be modified in quenched QCD. We find that this relation changes drastically, and in particular, that open-quote open-quote enhanced finite-volume corrections close-quote close-quote of order L0=1 and L-2 occur at one loop (L is the linear size of the box), due to the special properties of the η' in the quenched approximation. We define quenched pion scattering lengths, and show that they are linearly divergent in the chiral limit. We estimate the size of these various effects in some numerical examples, and find that they can be substantial. copyright 1995 The American Physical Society