We investigate the impact of a uniform spin density on the critical temperature of the chiral phase transition in finite-temperature QCD in the scope of the linear $\sigma$ model. We demonstrate that at a finite spin potential ${\mu }_{\mathrm{\Sigma }}$, corresponding to a finite spin density, the predictive power of the model is challenged by an ambiguity associated with a contribution of the vacuum renormalization term to the free energy. Eliminating the regularization freedom through comparison with recent low-${\mu }_{\mathrm{\Sigma }}$ lattice data, we extend the phase diagram of QCD at finite spin density to regions inaccessible to lattice simulations. We show that, as the spin potential increases, the temperature of the chiral crossover transition diminishes and the chiral crossover turns into a first-order transition at a second-order critical end point $(T,{\mu }_{\mathrm{\Sigma }}{)}_{\mathrm{CEP}}\simeq (0.142,0.098)\mathrm{GeV}$. With increasing spin potential, the critical temperature touches the zero-temperature axis at ${\mu }_{\mathrm{\Sigma }}=0.310\mathrm{GeV}$, implying that the chiral symmetry is restored at higher potentials at any temperature.

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• Effective field theory
• Finite temperature field theory
• QCD phase transitions
• Sigma models
• Strong interaction