Attempts to extract the order of the chiral transition of QCD at zero chemical potential, with two dynamical flavors of massless quarks, from simulations with progressively decreasing pion mass, have remained inconclusive because of their increasing numerical cost. In an alternative approach to this problem, we consider the path integral as a function of continuous number N f of degenerate quarks. If the transition in the chiral limit is first order for N f ≥ 3, a second-order transition for N f ¼ 2 then requires a tricritical point in between. This, in turn, implies tricritical scaling of the critical boundary line between the first-order and crossover regions as the chiral limit is approached. Noninteger numbers of fermion flavors are easily implemented within the staggered fermion discretization. Exploratory simulations at μ ¼ 0 and N f ¼ 2.8, 2.6, 2.4, 2.2, 2.1, on coarse N τ ¼ 4 lattices, indeed show a smooth variation of the critical mass mapping out a critical line in the (m, N f ) plane. For the smallest masses, the line appears consistent with tricritical scaling, allowing for an extrapolation to the chiral limit.

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 Nf ∈ [2, 8] mass-degenerate flavours on Nτ ∈ {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 Nf ≤ 6, and possibly up to the onset of the conformal window at 9 ≲ $$ {N}_{\mathrm{f}}^{\ast } $$ N f ∗ ≲ 12. A reanalysis of already published $$ \mathcal{O} $$ O (a)-improved Nf = 3 Wilson data on Nτ ∈ [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.

QCD with imaginary chemical potential is free of the sign problem and exhibits a rich phase structure constraining the phase diagram at real chemical potential. We simulate the critical endpoint of the Roberge-Weiss (RW) transition at imaginary chemical potential for N f " 2 QCD on Nτ " 6 lattices with standard Wilson fermions. As found on coarser lattices, the RW endpoint is a triple point connecting the deconfinement/chiral transitions in the heavy/light quark mass region and changes to a second-order endpoint for intermediate masses. These regimes are separated by two tricritical values of the quark mass, which we determine by extracting the critical exponent ν from a systematic finite size scaling analysis of the Binder cumulant of the imaginary part of the Polyakov loop. We are able to explain a previously observed finite size effect afflicting the scaling of the Binder cumulant in the regime of three-phase coexistence. Compared to Nτ " 4 lattices, the tricritical masses are significantly shifted. Exploratory results on Nτ " 8 as well as comparison with staggered simulations suggest that much finer lattices are needed before a continuum extrapolation becomes feasible.

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In the absence of a genuine solution to the sign problem, lattice studies at imaginary quark chemical potential are an important tool to constrain the QCD phase diagram. We calculate the values of the tricritical quark masses in the Roberge-Weiss plane, µ " ıπT {3, which separate mass regions with chiral and deconfinement phase transitions from the intermediate region, for QCD with N f " 2 unimproved staggered quarks on Nτ " 6 lattices. A quantitative measure for the quality of finite size scaling plots is developed, which significantly reduces the subjective judgement required for fitting. We observe that larger aspect ratios are necessary to unambiguously determine the order of the transition than at µ " 0. Comparing with previous results from Nτ " 4 we find a " 50% reduction in the light tricritical pion mass. The heavy tricritical pion mass stays roughly the same, but is too heavy to be resolved on Nτ " 6 lattices and thus equally afflicted with cut-off effects. Further comparison with other discretizations suggests that current cut-off effects on the light critical masses are likely to be larger than " 100%, implying a drastic shrinking of the chiral first-order region to possibly zero.