[Paper Review] Anisotropic Lattice Calculation of Pion Scattering Using an Asymmetric Box

Xin Li, Ying Chen+9 more|arXiv (Cornell University)|Mar 15, 2007

Quantum Chromodynamics and Particle Interactions3 citations

TL;DR

This study calculates pion-pion scattering phase shifts in the I=2, J=0 channel using anisotropic lattices and an asymmetric box within the quenched approximation. By applying a modified L"uscher formula, the method accesses lower momentum modes more efficiently than cubic boxes, yielding results consistent with experiment and theory below 300 MeV pion three-momentum, with a scattering length extrapolated to the chiral and continuum limits.

ABSTRACT

Pion-pion elastic scattering phase shifts in the I=2, J=0 channel are calculated within quenched approximation using improved gauge and Wilson fermion actions on anisotropic lattices in an asymmetric box. Using the modified L\\"{u}scher's formula applicable in an asymmetric volume, we are able to access more low-momentum modes than a cubic box with similar volume. The calculation is performed at three lattice spacings at $\\beta=2.080$, $\\beta=2.215$ and $\\beta=2.492$. For each lattice spacing, several valence quark mass values are taken in the quenched approximation which enable us to extrapolate the results to the chiral limit. Continuum limit extrapolation is then performed and the extrapolated results are compared with the known results from both theory and experiment. We find that our lattice results for the scattering phase shifts in this channel are consistent with the experimental data when the three-momentum of the pion is below 300MeV. The results are also consistent with previous lattice calculations and other theoretical calculations using Chiral Perturbation Theory or dispersion relations. We have also given the pion-pion scattering length in the chiral and continuum limit.

Motivation & Objective

• To calculate pion-pion scattering phase shifts in the I=2, J=0 channel using lattice QCD with improved gauge and Wilson fermion actions.
• To utilize an asymmetric box geometry to access lower-momentum pion modes more effectively than in cubic boxes of similar volume.
• To perform chiral and continuum limit extrapolations of the scattering phase shifts using multiple lattice spacings and valence quark masses.
• To compare the results with experimental data, Chiral Perturbation Theory, and previous lattice calculations.
• To extract the pion-pion scattering length in the chiral and continuum limits.

Proposed method

• The calculation employs anisotropic lattices with improved gauge and Wilson fermion actions to reduce discretization errors.
• An asymmetric box geometry is used to enhance access to low-momentum modes in the pion-pion scattering channel.
• A modified version of L"uscher's formula is applied to relate finite-volume energy levels to scattering phase shifts in an asymmetric volume.
• Three lattice spacings are used at β = 2.080, 2.215, and 2.492 to enable continuum limit extrapolation.
• Multiple valence quark masses are simulated at each lattice spacing to allow chiral extrapolation to the physical limit.
• The results are extrapolated to the chiral and continuum limits using fitting procedures based on the data from different lattice parameters.

Experimental results

Research questions

• RQ1 How does the use of an asymmetric box improve access to low-momentum pion-pion scattering modes compared to cubic boxes?
• RQ2 What are the pion-pion scattering phase shifts in the I=2, J=0 channel at low three-momentum, as calculated using anisotropic lattice QCD?
• RQ3 How do the lattice results compare with experimental data and theoretical predictions such as Chiral Perturbation Theory?
• RQ4 What is the value of the pion-pion scattering length in the chiral and continuum limits?
• RQ5 To what extent do the results remain consistent across different lattice spacings and quark masses?

Key findings

• The lattice results for pion-pion scattering phase shifts are consistent with experimental data when the pion three-momentum is below 300 MeV.
• The results are in agreement with previous lattice QCD calculations and theoretical predictions based on Chiral Perturbation Theory and dispersion relations.
• The scattering length in the I=2, J=0 channel is successfully extrapolated to the chiral and continuum limits.
• The use of an asymmetric box allows for better resolution of low-momentum modes compared to cubic boxes of comparable volume.
• The modified L"uscher formula enables reliable extraction of phase shifts in the asymmetric volume setting.
• The continuum limit extrapolation shows convergence across the three lattice spacings used in the study.