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Higher order quantization conditions for two-body scattering with spin

Lucas Chandler, Frank X. Lee, Andrei Alexandru

Abstract

We examine the Lüscher quantization condition to high order for the scattering of a spinless particle and a spin-1/2 particle in a periodic box. First, we derive the quantization conditions in a non-relativistic framework up to total angular momentum $J=11/2$ in both cubic and elongated geometries, and for both rest and moving frames. Then, we introduce a method to transparently cross-check their convergence, using both quantized energy levels in the box and infinite-volume phase shifts for the same potential. We clarify how to incorporate spin-orbit coupling into the formalism and show in detail how the quantization conditions converge order by order in the various irreducible representations. In all, we validated 19 quantization conditions (12 in cubic box, 7 in elongated box). This is a necessary step in applying the method in precision studies of systems in finite volume with half-integer spin, such as meson-baryon scattering.

Higher order quantization conditions for two-body scattering with spin

Abstract

We examine the Lüscher quantization condition to high order for the scattering of a spinless particle and a spin-1/2 particle in a periodic box. First, we derive the quantization conditions in a non-relativistic framework up to total angular momentum in both cubic and elongated geometries, and for both rest and moving frames. Then, we introduce a method to transparently cross-check their convergence, using both quantized energy levels in the box and infinite-volume phase shifts for the same potential. We clarify how to incorporate spin-orbit coupling into the formalism and show in detail how the quantization conditions converge order by order in the various irreducible representations. In all, we validated 19 quantization conditions (12 in cubic box, 7 in elongated box). This is a necessary step in applying the method in precision studies of systems in finite volume with half-integer spin, such as meson-baryon scattering.
Paper Structure (21 sections, 88 equations, 3 figures, 9 tables)

This paper contains 21 sections, 88 equations, 3 figures, 9 tables.

Table of Contents

  1. Introduction
  2. Quantization Condition
  3. Symmetry-adapted quantization conditions
  4. Kinematics
  5. Group symmetry of rest and moving frames
  6. Simplification of QCs
  7. From non-relativistic to relativistic QCs
  8. Two-particle energies in the box
  9. Reduction of the lattice Hamiltonian
  10. Symmetry-adapted energy levels in the box
  11. Convergence checks
  12. Infinite volume phase shifts
  13. Results and discussion
  14. Summary and Outlook
  15. Matrix elements for quantization conditions
  16. ...and 6 more sections

Figures (3)

  • Figure 1: Phase shifts as a function of CM $k$ from the test potential for partial waves up to $l = 5$ for spin-up branch in Eq.\ref{['eq:pot2']} (top) and spin-down branch in Eq.\ref{['eq:pot3']} (bottom) states. The s-wave ($l = 0$) resides in the spin-up branch.
  • Figure 2: Continuum extrapolation of the 4th lowest level in the $G_{1g}$ irrep rest frame in a cubic box of $L=36$ fm. Three lattices are used: $40^3$ with $a = 0.9 \mathop{\hbox{fm}}$, $48^3$ with $a = 0.75 \mathop{\hbox{fm}}$, and $60^3$ with $a = 0.6 \mathop{\hbox{fm}}$. The red points are from the three-point stencil in Eq.\ref{['eq:latH3']} and the blue points are from the seven-point stencil In Eq.\ref{['eq:latH7']}. The fitted curves and forms are also displayed.
  • Figure 3: (top) Phase shift reconstruction from Lus̈cher formula (lowest partial wave) in $G_{1g}$ and $H_u$ irreps in the rest frame, and $G_{2}$ irrep of $^2C_{4v}$ for moving frame $\bm d = (0,0,1)$, all in the cubic box. Black points are predicted phase shifts from box energy levels. The red curve is infinite-volume phase shift. Vertical gray lines indicate non-interacting levels. (bottom) Elongated box: $G_{1g}$, $G_{2u}$ irreps in rest frame, and $G_{2}$ of $^2C_{4v}$ for moving frame $\bm d = (0,0,1)$.