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The first two non-trivial moments of the distribution of the topologicalcharge (or gluonic winding number), i.e., the topological susceptibility andthe fourth cumulant, can be computed in lattice QCD simulations and exploitedto constrain the pattern of chiral symmetry breaking. We compute these twotopological observables at next-to-leading order in three-flavour ChiralPerturbation Theory, and we discuss the role played by the eta propagation inthese expressions. For hierarchies of light-quark masses close to the physicalsituation, we show that the fourth cumulant has a much better sensitivity thanthe topological susceptibility to the three-flavour quark condensate, and thusconstitutes a relevant tool to determine the pattern of chiral symmetrybreaking in the limit of three massless flavours. We provide the completeformulae for the two topological observables in the isospin limit, and predicttheir values in the particular setting of the recent analysis of the RBC/UKQCDcollaboration. We show that a combination of the topological susceptibility andthe fourth cumulant is able to pin down the three-flavour condensate in aparticularly clean way in the case of three degenerate quarks.

Determining the chiral condensate from the distribution of the winding number beyond topological susceptibility