  We discuss the influence of a helicity imbalance on the phase diagram of
dense QCD at finite temperature. We argue that the helical chemical potential
is a thermodynamically relevant quantity in theories with the mass gap
generation. Using the linear sigma model coupled to quarks, we show that the
presence of the helical density substantially affects the phase diagram of
dense quark matter. A moderate helical density makes the chiral phase
transition softer while shifting the critical endpoint towards lower
temperatures and higher baryon chemical potentials. As the helical density
increases, the segment of the first-order transition disappears, and the chiral
transition becomes a soft crossover. At even higher helical chemical
potentials, the first-order transition reappears again at the zero-density
finite-temperature transition and extends into the interior of the phase
diagram. This evolution of the chiral transition reflects the existence of a
thermodynamic duality between helical and vector (baryonic) chemical
potentials. We also show that the presence of the helicity imbalance of quark
matter increases the curvature of the chiral pseudocritical line in QCD.
