On the order of the QCD chiral phase transition for different numbers of quark flavours

The nature of the QCD chiral phase transition in the limit of vanishing quark masses has remained elusive for a long time, since it cannot be simulated directly on the lattice and is strongly cutoff-dependent. We report on a comprehensive ongoing study using unimproved staggered fermions with $N_\text{f}\in[2,8]$ mass-degenerate flavours on $N_\tau\in\{4,6,8\}$ lattices, in which we locate the chiral critical surface separating regions with first-order transitions from crossover regions in the bare parameter space of the lattice theory. Employing the fact that it terminates in a tricritical line, this surface can be extrapolated to the chiral limit using tricritical scaling with known exponents. Knowing the order of the transitions in the lattice parameter space, conclusions for approaching the continuum chiral limit in the proper order can be drawn. While a narrow first-order region cannot be ruled out, we find initial evidence consistent with a second-order chiral transition in all massless theories with $N_\text{f}\leq 6$, and possibly up to the onset of the conformal window at $9\lesssim N_\text{f}^*\lesssim 12$. A reanalysis of already published $\mathcal{O}(a)$-improved $N_\text{f}=3$ Wilson data on $N_\tau\in[4,12]$ is also consistent with tricritical scaling, and the associated change from first to second-order on the way to the continuum chiral limit. We discuss a modified Columbia plot and a phase diagram for many-flavour QCD that reflect these possible features.