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

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 2π(2n_1, 2n_2, 2n_3)/L, 2π(2n_1, 2n_2, 2n_3+1)/L and 2π(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.

49 pages, 2 figures. Fixed typos. Added footnotes. Two missing equations in Appendix C are included. Added a reference to the PRD supplemental material to this article. This supplemental material is in a mathematica notebook format which contains all the NN energy quantization conditions presented in this work to facilitate their use by lattice QCD practitioners. Version is published in PRD

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