# Tests of the continuum limit for the $SO(4)$ Principal Chiral Model and the prediction for $\L_\MS$

## TL;DR

This study examines the continuum limit in SO(4) Principal Chiral Models, finding discrepancies with theoretical predictions for the mass gap, and suggests lattice artifacts may delay the onset of the continuum limit.

## Contribution

It provides the first detailed comparison of finite-size scaling results for SO(4) and SU(2)xSU(2) models, highlighting potential lattice artifacts affecting continuum extrapolation.

## Key findings

- Discrepancy in mass gap ratio for SO(4) model compared to theory.
- Excellent agreement between simulation and theory for SU(2)xSU(2).
- Evidence of lattice vortex artifacts delaying continuum limit in SO(4).

## Abstract

We investigate the continuum limit in $SO(N)$ Principal Chiral Models concentrating in detail on the $SO(4)$ model and its covering group SU(2)xSU(2). We compute the mass gap in terms of Lambda_MS and compare with the prediction of Hollowood of $m/\L_\MS = 3.8716$. We use the finite-size scaling method of L\"uscher et al. to deduce $m/\L_\MS$ and find that for the $SO(4)$ model the computed result of $m/\L_\MS \sim 14$ is in strong disagreement with theory but that a similar analysis of the SU(2)xSU(2) yields excellent agreement with theory. We conjecture that for $SO(4)$ violations of the finite-size scaling assumption are severe forthe values of the correlation length, $\xi$, investigated and that our attempts to extrapolate the results to zero lattice spacing, although plausible, are erroneous. Conversely, the finite-size scaling violations in the SU(2)xSU(2) simulation are consistent with perturbation theory and the computed $beta-$function agrees well with the 3-loop approximation for couplings evaluated at scales $L/a \le \xi$, where $\xi$ is measured in units of the lattice spacing, $a$. We conjecture that lattice vortex artifacts in the $SO(4)$ model are responsible for delaying the onset of the continuum limit until much larger correlation lengths are achieved notwithstanding the apparent onset of scaling. Results for the mass spectrum for SO(N) m, N=8,10 are given whose comparison with theory gives plausible support to our ideas.

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Source: https://tomesphere.com/paper/hep-lat/9511004