Remarks on Stable and Quasi-stable k-Strings at Large N

TL;DR

This work develops a systematic large-$N$ framework for stable and quasi-stable $k$-strings in Yang–Mills theories and their ${\cal N}=1$ SUSY extensions. It shows that bona fide string tensions depend only on $N$-ality, while representation-dependent tensions arise for quasi-stable strings with lifetimes suppressed by powers of $1/N$ or by exponential factors in $N$, depending on the decay channel. The authors advocate the sine formula for stable $k$-string tensions in the saturation limit, provide physical arguments and supporting evidence, and relate these results to lattice measurements by analyzing Euclidean observables and the contour areas required to detect decays. They also discuss Casimir scaling and orientifold versus SUSY theories, connecting large-$N$ predictions to practical lattice observations at $N=3$. Overall, the paper clarifies how to interpret representation-dependent tensions and decay rates in both Minkowski and Euclidean frameworks, with implications for lattice studies of confinement.

Abstract

We discuss k-strings in the large-N Yang-Mills theory and its supersymmetric extension. Whereas the tension of the bona fide (stable) QCD string is expected to depend only on the N-ality of the representation, tensions that depend on specific representation R are often reported in the lattice literature. In particular, adjoint strings are discussed and found in certain simulations. We clarify this issue by systematically exploiting the notion of the quasi-stable strings which becomes well-defined at large N. The quasi-stable strings with representation-dependent tensions decay, but the decay rate (per unit length per unit time) is suppressed as Lambda^2 F(N) where F(N) falls off as a function of N. It can be determined on the case-by-case basis. The quasi-stable strings eventually decay into stable strings whose tension indeed depends only on the N-ality. We also briefly review large-N arguments showing why the Casimir formula for the string tension cannot be correct, and present additional arguments in favor of the sine formula. Finally, we comment on the relevance of our estimates to Euclidean lattice measurements.

Figures (10)

• Figure 1: A flux tube for the 2-string.
• Figure 2: (a). A screening of an adjoint Wilson loop by a dynamical adjoint loop. (b). The same process described in 't Hooft double index notation. The process in non-planar.
• Figure 3: The interaction of two "fundamental" strings: (a) Field-theory picture --- (two)gluon exchange; (b) String-theory picture --- exchange of a closed string between two world sheets.
• Figure 4: The spatial slice of two overlapping fundamental strings.
• Figure 5: a. A dynamical pair of soft quarks near the external charges. b. Creation of a dynamical pair from the vacuum in the process of the string breaking. The dynamical quarks are hard.