In typical statistical mechanical systems the grand canonical partition function at finite volume is proportional to a polynomial of the fugacity e µ T . The zero of this Lee-Yang polynomial closest to the origin determines the radius of convergence of the Taylor expansion of the pressure around µ = 0. The computationally cheapest formulation of lattice QCD, rooted staggered fermions, with the usual definition of the rooted determinant, does not admit such a Lee-Yang polynomial. We show that the radius of convergence is then bounded by the spectral gap of the reduced matrix of the unrooted staggered operator. We suggest a new definition of the rooted staggered determinant at finite chemical potential that allows for a definition of a Lee-Yang polynomial, and therefore of the numerical study of Lee-Yang zeros. We also describe an algorithm to determine the Lee-Yang zeros and apply it to configurations generated with the 2-stout improved staggered action at Nt = 4. We perform a finite volume scaling study of the leading Lee-Yang zeros and estimate the radius of convergence of the Taylor expansion extrapolated to an infinite volume. We show that the limiting singularity is not on the real line, thus giving a lower bound on the location of any possible phase transitions at this lattice spacing. In the vicinity of the crossover temperature at zero chemical potential, the radius of convergence turns out to be at µ B T ≈ 2 and roughly temperature independent.

All approaches currently used to study finite baryon density lattice QCD suffer from uncontrolled systematic uncertainties in addition to the well-known sign problem. We formulate and test an algorithm, sign reweighting, that works directly at finite µ = µ B /3 and is yet free from any such uncontrolled systematics. With this algorithm the only problem is the sign problem itself. This approach involves the generation of configurations with the positive fermionic weight |Re det D(µ)| where D(µ) is the Dirac matrix and the signs sign(Re det D(µ)) = ±1 are handled by a discrete reweighting. Hence there are only two sectors, +1 and −1 and as long as the average ±1 = 0 (with respect to the positive weight) this discrete reweighting by the signs carries no overlap problem and the results are reliable. The approach is tested on N t = 4 lattices with 2 + 1 flavors and physical quark masses using the unimproved staggered discretization. By measuring the Fisher (sometimes also called Lee-Yang) zeros in the bare coupling on spatial lattices L/a = 8, 10, 12 we conclude that the cross-over present at µ = 0 becomes stronger at µ > 0 and is consistent with a true phase transition at around µ B /T ∼ 2.4.

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The phase diagram and the location of the critical endpoint (CEP) of lattice QCD with unimproved staggered fermions on a Nt = 4 lattice was determined fifteen years ago with the multiparameter reweighting method by studying Fisher zeros. We first reproduce the old result with an exact algorithm (not known at the time) and with statistics larger by an order of magnitude. As an extension of the old analysis we introduce stout smearing in the fermion action in order to reduce the finite lattice spacing effects. First we show that increasing the smearing parameter ρ the crossover at µ = 0 gets weaker, i.e., the leading Fisher zero gets farther away from the real axis. Furthermore as the chemical potential is increased the overlap problem gets severe sooner than in the unimproved case, therefore shrinking the range of applicability of the method. Nevertheless certain qualitative features remain, even after introducing the smearing. Namely, at small chemical potentials the Fisher zeros first get farther away from the real axis and later at around aµq = 0.1 − 0.15 they start to get closer, i.e., the crossover first gets weaker and later stronger as a function of µ. However, because of the more severe overlap problem the CEP is out of reach with the smeared action.

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