Imprints of asymptotic freedom on confining strings

TL;DR

This work develops a framework to imprint asymptotic freedom onto confining strings by exploiting the Polyakov loop correlator, recast as the cylinder partition function $Z_{ ext{cyl.}}( au,R)$ in large $N$ YM theories. By exploiting dual open/closed string descriptions, it links perturbative short-distance behavior to the high-energy spectrum of winding closed strings, deriving explicit asymptotic densities $ ho_v(m,R)$ that grow slower than the Hagedorn rate in $D=3,4$. In a toy integrable model for $D=3$, causality and analyticity imply a bound on boundary reflection $K(p)$, $|K(p)|^2 o rac{ ilde\lambda}{2p}$ at large $p$, and exclude an asymptotically linear phase shift $S(s) ightarrow e^{i c s}$. The results open new avenues for flux-tube sum rules, boundary S-matrix bootstrap ideas, and potential non-perturbative constraints on Wilson-line dynamics, bridging UV gauge theory data with IR string behavior through rigorous causality/analyticity arguments.

Abstract

We consider the Polyakov loop correlator in the confining phase of large $N$ Yang-Mills theory in three and four dimensions. It can be computed by summing over the exchange of closed flux tubes winding around the thermal cycle. At short separations, the leading divergence is controlled by perturbation theory. Combining these two facts allows us to determine the asymptotic spectral density of string states contributing to the correlator. This sharply relates the weakly-coupled UV of the gauge theory to the dynamics of highly energetic flux tubes. Then, in a toy integrable setting, we explore how this can bound the scattering data of the Goldstone modes on top of a long string. We derive a bound on the asymptotic behavior of the reflection amplitude of Goldstones against the flux tube boundary sourced by the Polyakov line, and rule out an asymptotically linear phase shift for the S-matrix. Along the way, we discuss how causality can impose bounds on thermodynamic quantities, and show how the positivity of time delays follows from unitarity and analyticity of $2d$ massless elastic S-matrices. We include a review on reflection amplitudes, and their computation in the theory of long effective strings.

Figures (6)

• Figure 1: The correlator of two Polyakov loops is equivalent to the worldsheet cylinder partition function.
• Figure 2: Summary of the different limits of $Z_{\text{cyl.}}(\tau,R)$.
• Figure 3: The analytic structure of $R(p)$.
• Figure 4: Order $p^2$ contributions to the R-matrix.
• Figure 5: Order $p^3$ contributions to the R-matrix.