  QCD with three degenerate quark flavours at zero baryon density exhibits a
first order thermal phase transition for small quark masses, which changes to a
smooth crossover for some critical quark mass m^c_0, i.e. the chiral critical
point. It is generally believed that as an (even) function of quark chemical
potential, m_c(mu), the critical point moves to larger quark masses,
constituting the critical endpoint of a first order phase transition in
theories with m\geq m^c_0. To test this, we consider a Taylor expansion of
m_c(mu) around mu=0 and determine the first two coefficients from lattice
simulations with staggered fermions on N_t=4 lattices. We employ two different
techniques: a) calculating the coefficients directly from a mu=0 ensemble using
a novel finite difference method, and b) fitting them to simulation data
obtained for imaginary chemical potentials. The mu^2 and mu^4 coefficients are
found to be negative by both methods, with consistent absolute values.
Combining both methods gives evidence that also the mu^6 coefficient is
negative. Hence, on coarse N_t=4 lattices a three-flavour theory with m > m^c_0
does not possess a chiral critical endpoint for quark chemical potentials
mu\lsim T. Simulations on finer lattices are required for reliable continuum
physics. Possible implications for the QCD phase diagram are discussed.
