Comparison of the O(3) Bootstrap $\sigma$-Model with the Lattice Regularization at Low Energies

Abstract

The renormalized coupling $\text{gr}$ defined through the connected 4-point function at zero external momentum in the non-linear O(3) sigma-model in two dimensions, is computed in the continuum form factor bootstrap approach with estimated error $\sim 0.3$. New high precision data are presented for $\text{gr}$ in the lattice regularized theory with standard action for nearly thermodynamic lattices $L/\xi \sim 7$ and correlation lengths $\xi$ up to $\sim 122$ and with the fixed point action for correlation lengths up to $\sim 12$. The agreement between the form factor and lattice results is within $\sim 1$. We also recompute the phase shifts at low energy by measuring the two-particle energies at finite volume, a task which was previously performed by Lüscher and Wolff using the standard action, but this time using the fixed point action. Excellent agreement with the Zamolodchikov S-matrix is found.

Comments: 23 pages, 4 figures