Connecting dilaton thermal fluctuation with the Polyakov loop at finite temperature

Understanding the character of the deconfinement phase transition is one of the fundamental challenges in particle physics. In this work, we derive a formula for the expectation value of the Polyakov loop -- the order parameter of the deconfinement phase transition -- in pure $\mathrm{SU(N_{\mathrm{c}})}$ gauge systems at finite temperatures starting from the Coleman\textendash Weinberg-type effective potential encoding the trace anomaly of QCD. Our results are in good agreement with the Lattice QCD data and can effectively describe the large-$N_{\mathrm{c}}$ behaviors of the expectation value of the Polyakov loop. Notably, our findings predict the strongest first-order deconfinement phase transition as $N_{\mathrm{c}} \to +\infty$. Furthermore, to establish a relation between the dilaton field and the Polyakov loop, we also derive the scale transformation rule for temperature based on quantum statistical mechanics. The results of this work may shed a light on the connection between deconfinement phase transition and evolution of scale symmetry in the thermal system.