  The first two non-trivial moments of the distribution of the topological
charge (or gluonic winding number), i.e., the topological susceptibility and
the fourth cumulant, can be computed in lattice QCD simulations and exploited
to constrain the pattern of chiral symmetry breaking. We compute these two
topological observables at next-to-leading order in three-flavour Chiral
Perturbation Theory, and we discuss the role played by the eta propagation in
these expressions. For hierarchies of light-quark masses close to the physical
situation, we show that the fourth cumulant has a much better sensitivity than
the topological susceptibility to the three-flavour quark condensate, and thus
constitutes a relevant tool to determine the pattern of chiral symmetry
breaking in the limit of three massless flavours. We provide the complete
formulae for the two topological observables in the isospin limit, and predict
their values in the particular setting of the recent analysis of the RBC/UKQCD
collaboration. We show that a combination of the topological susceptibility and
the fourth cumulant is able to pin down the three-flavour condensate in a
particularly clean way in the case of three degenerate quarks.

High Energy Physics - Lattice;High Energy Physics - Phenomenology