Two-nucleon systems in a finite volume: Quantization conditions
The quantization condition for interacting energy eigenvalues of the two-nucleon system in a finite cubic volume is derived in connection to the nucleon-nucleon scattering amplitudes. This condition is derived using an auxiliary (dimer) field formalism that is generalized to arbitrary partial waves in the context of nonrelativistic effective field theory. The quantization condition presented gives access to the scattering parameters of the two-nucleon systems with arbitrary parity, spin, isospin, angular momentum and center-of-mass motion, from a lattice QCD calculation of the energy eigenvalues. In particular, as it includes all noncentral interactions, such as the two-nucleon tensor force, it makes explicit the dependence of the mixing parameters of nucleon-nucleon systems calculated from lattice QCD when there is a physical mixing among different partial waves, e.g. $S\mathrm{\text{\ensuremath{-}}}D$ partial wave mixing in the deuteron channel. We provide explicit relations among scattering parameters and their corresponding point group symmetry class eigenenergies with orbital angular momentum $l\ensuremath{\le}3$, and for center-of-mass boost vectors of the form $\frac{2\ensuremath{\pi}}{L}(2{n}_{1},2{n}_{2},2{n}_{3})$, $\frac{2\ensuremath{\pi}}{L}(2{n}_{1},2{n}_{2},2{n}_{3}+1)$ and $\frac{2\ensuremath{\pi}}{L}(2{n}_{1}+1,2{n}_{2}+1,2{n}_{3})$. $L$ denotes the special extent of the cubic volume and ${n}_{1}$, ${n}_{2}$, ${n}_{3}$ are integers. Our results are valid below inelastic thresholds up to exponential volume corrections that are governed by the pion mass.