We analyze mass renormalization in massive Dirac-like systems in ($2+1$) dimensions arising from electron-phonon interactions at finite temperatures, employing the large-$N$ expansion. Our model combines the low-energy description of charge carriers in a buckled honeycomb lattice with the low-energy approximation for phonons and electron-phonon interactions in two-dimensional materials. Hence the system is modeled as a massive Dirac-like field coupled to a two-component vector field ${\mathcal{A}}_i$, representing the phonon modes. This framework allows us to compute the one-loop electron self-energy at finite temperature, from which we derive the renormalized electronic band gap, $2m^R$. The effective model is subsequently applied to describe the renormalized optical band gap $\left(E_{\mathrm{opt}}\right)$ in monolayers of transition metal dichalcogenides (TMDs), including ${\mathrm{MoS}}_2$, ${\mathrm{MoSe}}_2$, ${\mathrm{WS}}_2$, and ${\mathrm{WSe}}_2$, using the relation $E_{\mathrm{opt}}=2m^R-\left|E_b\right|$, where $|E_b|$ is the exciton binding energy that remains constant at the examined temperature. A good agreement is observed with experimental data for reasonable values of the ultraviolet cutoff, $\mathrm{\Lambda }\approx 1$ eV. Our main findings indicate that $E_{\mathrm{opt}}$ remains nearly constant at low temperatures, whereas at higher temperatures it decreases linearly with the temperature $T$. Specifically, we find that $E_{\mathrm{opt}}$ reduces by approximately $\approx [0.1,0.2]$ eV as the temperature increases from $\approx 4$ K to 500 K, consistent with recent experimental observations. Furthermore, we estimate the temperature range at which the transition to the linear regime occurs, obtaining typical values within $\approx [110,150]$ K for the four materials under consideration.

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• Band gap
• Dirac fermions
• Electron-phonon coupling
• Finite temperature field theory
• Large-N expansion in field theory