Quantization conditions in the finite volume within the plane wave basis expansion

Abstract

In this work, we propose an alternative approach of Lüscher's formula to extract the two-body interaction from the finite volume energy levels. We adopt the plane wave expansion rather than the partial wave expansion in calculation. In this framework, the exponential suppressed effect and partial wave mixing effect are embedded naturally. We use the spin singlet NN system and $\pi\pi$ system in $\rho$ channel as examples of non-relativistic and relativistic examples. For spin singlet NN system, the one-pion-exchange interaction at physical pion mass will make the single channel Lüscher's formula unreliable. For S-wave dominant states, Lüscher's formula gives rise to significant deviation for the $L=3$ fm box. For the $^1P_1$ dominant states, the mixing effect from the higher partial wave components is significant even in the box with $L=8$ fm. We adopt a toy model to illustrate the combination of effective field theory and the plane wave expansion approach can solve the problems. For the $\rho$ channel $\pi\pi$ interaction, we use a phenomenological model to fit the lattice QCD results. The fitting results can depict the $\pi\pi$ scattering phase shift well.