  Using the hard-thermal-loop (HTL) resummation in real-time formalism, we
study the next-to-leading order (NLO) quark self-energy and corresponding NLO
dispersion laws. In NLO, we have replaced all the propagators and vertices with
the HTL-effective ones in the usual quark self-energy diagram. Additionally, a
four-point vertex diagram also contributes to the quark NLO self-energy. We
calculate the usual quark self-energy diagram and the four-point vertex diagram
separately. Using those, we express the NLO quark self-energy in terms of the
three- and four-point HTL-effective vertex functions. Using the Feynman
parametrization, we express the integrals containing the three- and four-point
HTL effective vertex functions in terms of the solid angles. After completing
the solid angle integrals, we numerically calculate the momentum integrals in
the NLO quark self-energy and plot them as a function of the ratio of momentum
and energy. Using the NLO quark self-energy, we plot the NLO correction to
dispersion laws.
