The Quantized $O(1,2)/O(2)\times Z_2$ Sigma Model Has No Continuum Limit in Four Dimensions. II. Lattice Simulation

TL;DR

This paper investigates a lattice formulation of the $O(1,2)/O(2)\times Z_2$ sigma model, demonstrating through simulations that it lacks a continuum limit, implying it reduces to free fields without the original model's symmetries.

Contribution

The study develops a lattice action suitable for noncompact curved spaces and shows that the model does not possess a continuum limit, contrasting with prior theoretical expectations.

Findings

No value of $eta$ yields a vanishing $eta_R$.

The continuum limit corresponds to free massless fields.

The model's continuum limit does not preserve original symmetries.

Abstract

A lattice formulation of the $O(1,2)/O(2)\times Z_2$ sigma model is developed, based on the continuum theory presented in the preceding paper. Special attention is given to choosing a lattice action (the ``geodesic'' action) that is appropriate for fields having noncompact curved configuration spaces. A consistent continuum limit of the model exists only if the renormalized scale constant $\beta_R$ vanishes for some value of the bare scale constant~$\beta$. The geodesic action has a special form that allows direct access to the small-$\beta$ limit. In this limit half of the degrees of freedom can be integrated out exactly. The remaining degrees of freedom are those of a compact model having a $\beta$-independent action which is noteworthy in being unbounded from below yet yielding integrable averages. Both the exact action and the $\beta$-independent action are used to obtain $\beta_R$ from Monte Carlo computations of field-field averages (2-point functions) and current-current averages. Many consistency cross-checks are performed. It is found that there is no value of $\beta$ for which $\beta_R$ vanishes. This means that as the lattice cutoff is removed the theory becomes that of a pair of massless free fields. Because these fields have neither the geometry nor the symmetries of the original model we conclude that the $O(1,2)/O(2)\times Z_2$ model has no continuum limit.