Effects of imaginary and real rotations on QCD matters

Inspired from perturbative calculations, this work introduces imaginary ($\Omega_{\rm I}$) and real ($\Omega$) rotation effects to the pure $SU(3)$ gauge potentials simply through variable transformations: The empirical Polyakov loop (PL) potentials can be rewritten as functions of the imaginary chemical potentials of gluons and ghosts $(q_{\rm ij})$, and the transformations are taken as $q_{\rm ij}\rightarrow q_{\rm ij}\pm\Omega_{\rm I}/T$ and $q_{\rm ij}\rightarrow q_{\rm ij}\pm i\,\Omega/T$, respectively. For the PL potential of Fukushima $(V_1)$, a smaller imaginary rotation $\Omega_{\rm I}$ tends to suppress PL at all temperature and the deconfinement transition keeps of first order. However, for the PL potential of Munich group $(V_2)$, $\Omega_{\rm I}$ tends to enhance PL at low temperature $T$, consistent with lattice simulations; but suppress PL at high $T$, consistent with perturbative calculations. Moreover, the deconfinement alters from first order to crossover with increasing $\Omega_{\rm I}$ as is expected from lattice simulations. On the other hand, the real rotation $\Omega$ tends to enhance PL at relatively low $T$ for both potentials, and the (pseudo-)critical temperature decreases with $\Omega$ as expected. Therefore, we find that analytic continuation of the phase diagram from imaginary to real rotation is not necessarily valid in the non-perturbative region. Finally, we apply the more successful PL potential $V_2$ to the Polyakov--Nambu-Jona-Lasinio (PNJL) model and discover that $\Omega_{\rm I}$ tends to break chiral symmetry while $\Omega$ tends to restore it. Especially, the modified model is even able to qualitatively explain the lattice result that a larger $T$ would catalyze chiral symmetry breaking for a large $\Omega_{\rm I}$.