Two-Nucleon Systems in a Finite Volume: (I) Quantization Conditions

TL;DR

This paper derives a quantization condition linking finite-volume energy levels to nucleon-nucleon scattering parameters, including non-central interactions, enabling lattice QCD calculations of two-nucleon systems with arbitrary quantum numbers.

Contribution

It introduces a generalized quantization condition using an auxiliary field formalism for two-nucleon systems in finite volumes, accounting for all partial waves and non-central forces, including tensor interactions.

Findings

Provides explicit relations between scattering parameters and energy eigenvalues.

Includes all non-central interactions like tensor forces in the formalism.

Applicable to various partial waves and boost vectors in finite volume calculations.

Abstract

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 non-relativistic 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 non-central 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-D 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 smaller than or equal to 3, and for center of mass boost vectors of the form 2\pi (2n_1, 2n_2, 2n_3)/L, 2\pi (2n_1, 2n_2, 2n_3+1)/L and 2\pi (2n_1+1, 2n_2+1, 2n_3)/L. 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.