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Approaches to finite baryon density lattice QCD usually suffer fromuncontrolled systematic uncertainties in addition to the well-known signproblem. We test a method - sign reweighting - that works directly at finitechemical potential and is yet free from any such uncontrolled systematics: withthis approach the only problem is the sign problem itself. In practice theapproach involves the generation of configurations with the positive fermionicweights given by the absolute value of the real part of the quark determinant,and a reweighting by a sign. There are only two sectors, +1 and -1 and as longas the average $\left\langle \pm \right\rangle \neq 0$ (with respect to thepositive weight) this discrete reweighting has no overlap problem - unlikereweighting from $\mu=0$ - and the results are reliable. We also presentresults based on this algorithm on the phase diagram of lattice QCD with twodifferent actions: as a first test, we apply the method to calculate theposition of the critical endpoint with unimproved staggered fermions at$N_\tau=4$; as a second application, we study the phase diagram with 2stoutimproved staggered fermions at $N_\tau=6$. This second one is already areasonably fine lattice - relevant for phenomenology. We demonstrate that themethod penetrates the region of the phase diagram where the Taylor andimaginary chemical potential methods lose predictive power.

New approach to lattice QCD at finite density: reweighting without an overlap problem