Corrections to Nambu-Goto energy levels from the effective string action

TL;DR

This work derives the leading Lorentz-invariant corrections to Nambu-Goto energy levels for long open and closed strings using a Hamiltonian perturbation framework with Weyl ordering; it identifies a boundary term responsible for $O(1/R^4)$ open-string corrections and a bulk $c_4$ term yielding $O(1/R^5)$ and $O(1/R^7)$ corrections in various sectors, and cross-validates with annulus/torus partition functions. For open strings, degeneracies are largely preserved by the c2,c3 sector, while boundary terms lift them and set the dominant subleading correction; for closed strings, c2,c3 produce NG-consistent shifts and c4 generates state-dependent splittings among left-right mixed states, with $D=3$ sometimes simplifying the spectrum. The results provide a concrete, testable link between effective-string theory and lattice data, outlining how measured energy splittings can distinguish universal from model-dependent coefficients and highlighting the need for higher-precision lattice studies and possible resummations beyond the $1/r$ expansion. Overall, the paper clarifies how Lorentz invariance constrains the universal corrections to NG and offers a framework to confront these predictions with numerical simulations of confining strings.

Abstract

The effective action on long strings, such as confining strings in pure Yang-Mills theories, is well-approximated by the Nambu-Goto action, but this action cannot be exact. The leading possible corrections to this action (in a long string expansion in the static gauge), allowed by Lorentz invariance, were recently identified, both for closed strings and for open strings. In this paper we compute explicitly in a Hamiltonian formalism the leading corrections to the lowest-lying Nambu-Goto energy levels in both cases, and verify that they are consistent with the previously computed effective string partition functions. For open strings of length R the leading correction is of order 1/R^4, for excited closed strings of length R in D>3 space-time dimensions it is of order 1/R^5, while for the ground state of the closed string in any dimension it is of order 1/R^7. We attempt to match our closed string corrections to lattice results, but the latter are still mostly outside the range of convergence of the 1/R expansion that we use.

Figures (1)

• Figure 1: The expected energy levels, and the ones computed in lattice simulations of $SU(3)$ gauge theory Teper, for level 2 with $q=0$ ($N_L=N_R=1$), and level 3 with $q=1$ ($N_L=2$, $N_R=1$). The discrete points are the lattice results from Teper, annotated as $(|J|,P_\perp,P_\parallel)$ for $q=0$ and as $(|J|,P_\perp)$ for $q=1$; the solid lines are the corresponding Nambu-Goto energy levels, and the other lines include the shifts we calculated from $\mathcal{H}_4$ (for the lightest $n=2,3$ state with the given quantum numbers), using the specific value $c_4=(D-26)/192\pi^2 T^2$. The vertical line is the expected radius of convergence for each level, we expect a matching only for points that are well to the right of this line.