Topological charge and topological susceptibility in connection with translation and gauge invariance

TL;DR

This paper investigates the expectation value of topological charge density in quantum chromodynamics, revealing it to be zero in certain definitions and leading to a modified understanding of the Witten-Veneziano formula for the eta prime meson mass.

Contribution

It establishes an equation for the Chern-Simons term EV and shows how different definitions of topological susceptibility affect physical predictions.

Findings

EV of topological charge density at arbitrary θ is zero.

Topological susceptibility differs depending on the vacuum definition.

The Witten-Veneziano formula for η' mass is modified.

Abstract

It is shown that the evaluation of the expectation value (EV) of topological charge density over $\theta$-vacuum is reduced to investigation of the Chern-Simons term EV. An equation for this quantity is established and solved. EV of the topological charge density at an arbitrary $\theta$ occurs equal to zero and, as a consequence, topological susceptibility of both QCD and pure Yang-Mills vacua defined in a Wick sense is equal to zero, whereas when defined in a Dyson sense it differs from zero by the quantity proportional to the respective condensate of the chromomagnetic field. Thus, the usual Witten-Veneziano formula for the $\eta^{'}$ meson mass is modified.