  In this work we have studied the QCD phase structure and critical dynamics
related to the 3-$d$ $O(4)$ and $Z(2)$ symmetry universality classes in the
two-flavor quark-meson low energy effective theory within the functional
renormalization group approach. We have employed the expansion of Chebyshev
polynomials to solve the flow equation for the order-parameter potential. The
chiral phase transition line of $O(4)$ symmetry in the chiral limit, and the
$Z(2)$ line of critical end points related to the explicit chiral symmetry
breaking are depicted in the phase diagram. Various critical exponents related
to the order parameter, chiral susceptibilities and correlation lengths have
been calculated for the 3-$d$ $O(4)$ and $Z(2)$ universality classes in the
phase diagram, respectively. We find that the critical exponents obtained in
the computation, where a field-dependent mesonic nontrivial dispersion relation
is taken into account, are in quantitative agreement with results from other
approaches, e.g., the conformal bootstrap, Monte Carlo simulations and $d=3$
perturbation expansion, etc. Moreover, the size of the critical regime in the
QCD phase diagram is found to be very small.
