The chiral and deconfinement crossover transitions: PNJL model beyond mean field

The Polyakov loop extended Nambu and Jona-Lasinio model (PNJL model) in a mean field framework shows astonishingly good agreement with lattice QCD calculations which needs to be better understood. The present work reports on further developments concerning both Polyakov loop and mesonic fluctuations beyond mean field approximation. Corrections beyond mean field are of special interest for the thermal expectation values of the Polyakov loop $\Phi$ and its conjugate $\Phi^*$, which differ once the quark chemical potential is non-zero. Mesonic fluctuations are also considered with emphasis on the role of pionic modes.