The curvature of the critical surface (m(u,d),m(s))**crit(mu): A Progress report

At zero chemical potential mu, the order of the temperature-driven quark-hadron transition depends on the quark masses m_{u,d} and m_s. Along a critical line bounding the region of first-order chiral transitions in the (m_{u,d},m_s) plane, this transition is second order. When the chemical potential is turned on, this critical line spans a surface, whose curvature at mu=0 can be determined without any sign or overlap problem. Our past measurements on N_t=4 lattices suggest that the region of quark masses for which the transition is first order shrinks when mu is turned on, which makes a QCD chiral critical point at small mu/T unlikely. We present results from two complementary methods, which can be combined to yield information on higher-order terms. It turns out that the O(mu^4) term reinforces the effect of the leading O(mu^2) term, and there is strong evidence that the O(mu^6) and O(mu^8) terms do as well. We also report on simulations underway, where the strange quark is given its physical mass, and where the lattice spacing is reduced.