Resonance masses from Monte Carlo simulations (with emphasis on theρmeson)

All lattice QCD simulations performed to date have been done on finite-volume lattices with parameter values such that $m_{\rho }<2\mathrm{}{E}_{\pi }$, where $E_{\pi }$ is the energy of the lightest p≠0 pion state on the lattice. As the lattice volume increases or the quark mass falls the $\rho$ mass will show irregular behavior as a result of avoided level crossings with the lightest $\pi \pi$ state. By modeling the $\rho \pi \pi$ interaction on a lattice of finite volume, determining a small set of parameters from lattice data, and taking the infinite-volume limit of the model, one can determine the value of the $\rho$ mass in the limit. It is argued that extrapolations from lattice QCD simulations with parameter values where $m_{\rho }<2\mathrm{}{m}_{\pi }$ to $m_{\pi }=0\mathrm{}$ may be misleading.