Abstract
We propose an alternative approach to Lüscher’s formula for extracting two-body scattering phase shifts from finite volume spectra with no reliance on the partial wave expansion. We use an effective-field-theory-based Hamiltonian method in the plane wave basis and decompose the corresponding matrix elements of operators into irreducible representations of the relevant point groups. The proposed approach allows one to benefit from the knowledge of the long-range interaction and avoids complications from partial wave mixing in a finite volume. We consider spin-singlet channels in the two-nucleon system and pion-pion scattering in the ρ-meson channel in the rest and moving frames to illustrate the method for non-relativistic and relativistic systems, respectively. For the two-nucleon system, the long-range interaction due to the one-pion exchange is found to make the single-channel Lüscher formula unreliable at the physical pion mass. For S-wave dominated states, the single-channel Lüscher method suffers from significant finite-volume artifacts for a L = 3 fm box, but it works well for boxes with L > 5 fm. However, for P-wave dominated states, significant partial wave mixing effects prevent the application of the single-channel Lüscher formula regardless of the box size (except for the near-threshold region). Using a toy model to generate synthetic data for finite-volume energies, we show that our effective-field-theory-based approach in the plane wave basis is capable of a reliable extraction of the phase shifts. For pion-pion scattering, we employ a phenomenological model to fit lattice QCD results at the physical pion mass. The extracted P-wave phase shifts are found to be in a good agreement with the experimental results.
Highlights
We propose an alternative approach to Lüscher’s formula for extracting twobody scattering phase shifts from finite volume spectra with no reliance on the partial wave expansion
In the upper row of figures 5 and 6, we show the corresponding positive- and negativeparity phase shifts extracted from the FV energies using the single-channel Lüscher formula, along with the results in the infinite volume
The FV energy levels are computed by solving the Lippmann-Schwinger equation (LSE) or Bethe-Salpeter equation (BSE) for nonrelativistic or relativistic systems, respectively, for a given irrep and finding the poles of the resulting amplitude
Summary
Discretization of three-momenta of two particles in a cubic box with periodic boundary conditions is considered e.g. in refs. [4, 9]. For non-interacting systems, the momentum and energy in the CMF are related to those in the BF by the Lorentz transformation p∗1 =. In figure 1, we show a set of discrete momenta with the corresponding meshes in different frames. In this work we are only interested in the d2 = 0, 1, 2, 3, 4 cases and, consider only the Oh, D4h, D2h and D3d groups and the corresponding subgroups O, D4, D2 and D3 containing only proper rotations. The C4v group contains two improper rotations and is the symmetry group for systems with particles of unequal masses. For non-relativistic systems, we will end up with several degenerate states in different irreps reflecting the additional degeneracy of the more symmetric group Oh
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