Spectral Parameters and Spectral Weights of Unstable Particles in Lattice QCD

We discuss the issues related to the spectral parameters (mass and width) of unstable particles in lattice calculations. Using a solvable model, the Lee Model, we show that the eigenvalues of the Hamiltonian, which is the spectrum measured from the Euclidean correlation functions in lattice simulations, need not be the same as the energy at which the scattering phase passes through π/2. It is in general also different from the real part of the actual pole position for the S-matrix element. These three quantities are close to one another only when the resonance is narrow. We also investigate the volume dependence of the so-called spectral weights for the energy eigenstates which can be obtained in lattice simulations. The volume dependence of this quantity helps to distinguish between one-particle and two-particle states in lattice calculations. Within the Lee model, we verify that this is indeed the case when the so-called V -particle is stable. For an unstable V -particle (resonance), this method is only applicable for narrow resonances. We argue that this conclusion should also be qualitatively correct in general theories like QCD.