The topological susceptibility of QCD: from Minkowskian to Euclidean theory

TL;DR

This paper explores the relationship between Minkowskian and Euclidean topological susceptibilities in QCD, highlighting conditions for their equality and the impact of finite temperature and infrared regularization.

Contribution

It clarifies the connection between Minkowskian and Euclidean topological susceptibilities in QCD, especially at finite temperature, and introduces a method to restore their equality via infrared regularization.

Findings

At zero temperature, the susceptibilities are equal.

Finite temperature introduces complexities due to Kogut-Susskind poles.

Infrared regularization can restore the equality of susceptibilities.

Abstract

We show how the topological susceptibility in the Minkowskian theory of QCD is related to the corresponding quantity in the Euclidean theory, which is measured on the lattice. We discuss both the zero-temperature case (T = 0) and the finite-temperature case (T > 0). It is shown that the two quantities are equal when T = 0, while the relation between them is much less trivial when T > 0. The possible existence of ``Kogut-Susskind poles'' in the matrix elements of the topological charge density between states with equal four-momenta turns out to invalidate the equality of these two quantities in a strict sense. However, an equality relation is recovered after one re-defines the Minkowskian topological susceptibility by using a proper infrared regularization.