Volume dependence of the pion mass

Volume effects on meson masses are important for understanding the results of QCD lattice simulations, which are still limited to large meson masses and small volumes. Such finite volume effects have been studied in different approaches [2, 3, 4], mainly using chiral perturbation theory (chPT) . Since quark effects involving higher momentum scales also seem to play a role [5], we study the pion mass volume dependence in a model involving quark effects with renormalization group (RG) methods, which provides a way to account for these effects in the low-energy theory. We consider the quark-meson-model, as an effective low-energy model of QCD, in a finite three-dimensional volume, using the Schwinger proper-time renormalization group. We use a local potential approximation of the effective action, which incorporates fermionic as well as bosonic fluctuations to the potential density U [1]. In order to break chiral symmetry in the finite volume, we introduce a small current quark mass. The corresponding effective meson potential breaks chiral O(4) symmetry explicitly and depends on σ and ~π fields separately. Therefore the following ansatz for the mesonic potential is justified: