We investigate the predictions of Nambu-Goto (NG) and Polyakov-Kleinert (PK) effective string actions for the Casimir energy and the width of the quantum delocalization of the string at two loop order in 4-dim pure SU(3) Yang-Mills lattice gauge theory. Intermediate and large color source separation distance, before the string breaks in full QCD, at two temperature scales are considered near the deconfinement point.

At a temperature closer to the critical point T/Tc=0.9, we found that the next to leading-order (NLO) contributions from the expansion of the NG string to improve the match to lattice data in the intermediate distance scales for both the quark-antiquark potential and broadening of the color tube compared to the free string approximation. Nevertheless, the Nambu-Goto string action in the next-to-leading order approximation does not provide a precise match with the numerical data for both the quark-antiquark potential and broadening profile.

We conjecture possible stiffness of the QCD string through studying the effects of extrinsic curvature term in Polyakov-Kleinert action. The consequences of adding a smoothing term proportional to the extrinsic curvature to the Nambu-Goto string action, as suggested by Polyakov, are investigated. The mean square width of the flux tube of the smooth open-string is derived considering Dirichelet boundary condition. We find that the theoretical predictions derived based on this smooth string formalism return a good fitting behavior for the lattice Mont-Carlo data at both long and intermediate quark separations regions.

Vacuum Structure and Confinement