Topological susceptibility with a single light quark flavour

One of the historical suggestions to tackle the strong CP problem is to take the up quark mass to zero while keeping $m_d$ finite. The $\theta$ angle is then supposed to become irrelevant, i.e. the topological susceptibility vanishes. However, the definition of the quark mass is scheme-dependent and identifying the $m_u=0$ point is not trivial, in particular with Wilson-like fermions. More specifically, up to our knowledge there is no theoretical argument guaranteeing that the topological susceptibility exactly vanishes when the PCAC mass does. We will present our recent progresses on the empirical check of this property using $N_f=1+2$ flavours of clover fermions, where the lightest fermion is tuned very close to $m^{PCAC}_u$=0 and the mass of the other two is kept of the order of magnitude of the physical $m_s$. This choice is indeed expected to amplify any unknown non-perturbative effect caused by $m_u\not=m_d$. The simulation is repeated for several $\beta$s and those results, although preliminary, give a hint about what happens in the continuum limit.