Finite volume QCD at fixed topological charge

TL;DR

This work develops a rigorous 1/$V$ saddle-point expansion to relate correlation functions computed at fixed topological charge $Q$ to their θ-dependent counterparts in QCD, enabling systematic extraction of the topological susceptibility $\chi_t$ and higher cumulants from fixed-Q lattice data. By deriving both bosonic and fermionic (Ward–Takahashi) formulations, the authors show that fixed-Q observables converge to their θ=0 values in the infinite-volume limit, while finite-volume corrections scale as $1/V$ and involve $\chi_t$ and $c_4$. They provide explicit formulas for two-, three-, and four-point functions, including CP-even and CP-odd channels, and demonstrate practical routes to evaluate CP-odd quantities like the neutron electric dipole moment from fixed-topology ensembles. The approach offers a practical, short-distance-free method to determine $\chi_t$ from lattice data in the fixed-topology regime, with clear conditions for validity and paths for extension to other observables.

Abstract

In finite volume the partition function of QCD with a given $θ$ is a sum of different topological sectors with a weight primarily determined by the topological susceptibility. If a physical observable is evaluated only in a fixed topological sector, the result deviates from the true expectation value by an amount proportional to the inverse space-time volume 1/V. Using the saddle point expansion, we derive formulas to express the correction due to the fixed topological charge in terms of a 1/V expansion. Applying this formula, we propose a class of methods to determine the topological susceptibility in QCD from various correlation functions calculated in a fixed topological sector.