QCD at Fixed Topology

TL;DR

The paper tackles finite-size effects in lattice QCD when simulations are performed at fixed topology. It derives a saddle-point relation between the fixed-Q partition function and the θ-dependent vacuum, and shows how the fixed-Q hadron mass $M_Q$ deviates from the infinite-volume mass $M(0)$ by a term proportional to $M''(0)$ that scales as $1/(βV)$. Through chiral perturbation theory, it analyzes θ-dependence of the vacuum energy and Goldstone masses, and discusses large-$N_c$ behavior of the η′, along with a qualitative instanton-gas model that aligns with lattice observations. The framework provides practical formulas to correct fixed-topology data and extract physical parameters, with insights into θ-dependence across regimes and potential limitations near the chiral limit.

Abstract

Since present Monte Carlo algorithms for lattice QCD may become trapped in a fixed topological charge sector, it is important to understand the effect of calculating at fixed topology. In this work, we show that although the restriction to a fixed topological sector becomes irrelevant in the infinite volume limit, it gives rise to characteristic finite size effects due to contributions from all $θ$-vacua. We calculate these effects and show how to extract physical results from numerical data obtained at fixed topology.