Effective String Theory Revisited

TL;DR

The paper derives the Polchinski–Strominger interaction within static gauge for long relativistic strings, showing it arises as a one-loop effect from the Nambu–Goto action and is tied to the Polyakov determinant in conformal gauge. Dimensional regularization preserves the non-linearly realized Lorentz symmetry, while $\zeta$-function regularization requires non-covariant counterterms to maintain the algebra, with the evanescent Einstein term playing a key role in renormalization. The finite part of the one-loop amplitude reproduces the PS annihilation piece proportional to $(D-26)$, linking EFT results to the conformal-gauge picture, and the paper discusses implications for the string spectrum and exact S-matrix phase shifts, including a special, integrable structure at $D=26$. The work connects effective-string EFT to lattice QCD flux tubes, outlines how future studies could pin down UV completions and the role of string rigidity, and sets the stage for a companion analysis of the critical theory. Overall, it clarifies how Lorentz symmetry, regularization schemes, and non-local PS dynamics shape the quantum behavior of confining strings across dimensions.

Abstract

We revisit the effective field theory of long relativistic strings such as confining flux tubes in QCD. We derive the Polchinski-Strominger interaction by a calculation in static gauge. This interaction implies that a non-critical string which initially oscillates in one direction gets excited in orthogonal directions as well. In static gauge no additional term in the effective action is needed to obtain this effect. It results from a one-loop calculation using the Nambu-Goto action. Non-linearly realized Lorentz symmetry is manifest at all stages in dimensional regularization. We also explain that independent of the number of dimensions non-covariant counterterms have to be added to the action in the commonly used zeta-function regularization.

Figures (4)

• Figure 1: Quartic treel-level vertices following from the NG action. Solid lines follow the flow of flavor indices, and dashed lines show the contractions of momenta at the vertex.
• Figure 2: Sample one-loop diagrams contributing to the $2\to 2$ scattering in the NG theory. Solid and dashed lines have the same meaning as in Fig. \ref{['vertices']}.
• Figure 3: Connected diagrams contributing to the $R^{-3}$ order corrections to the energy levels.
• Figure 4: Sample connected diagrams contributing to the $R^{-5}$ order corrections to the energy levels.