The Leutwyler-Smilga relation on the lattice

According to the Leutwyler-Smilga relation, in Quantum Chromodynamics (QCD) the topological susceptibility vanishes linearly with the quark masses. Calculations of the topological susceptibility in the context of lattice QCD, extrapolated to zero quark masses, show a remnant non-zero value as a lattice artefact. Employing the Atiyah-Singer theorem in the framework of Symanzik’s effective action and chiral perturbation theory, we show the validity of the Leutwyler-Smilga relation in lattice QCD with lattice artefacts of order $a^2$ in the lattice spacing $a$.