The k=2 string tension in four dimensional SU(N) gauge theories

TL;DR

The paper measures the $k=2$ string tensions in SU(4) and SU(5) gauge theories in 3+1 dimensions using lattice simulations to test the MQCD conjecture that $\sigma_k/\sigma_f = \sin(k\pi/N)/\sin(\pi/N)$. By forming closed $k$-strings and analyzing their correlation functions with the Luscher correction, the authors obtain continuum estimates: $\sigma_{k=2}/\sigma_f \approx 1.35$–$1.41$ (SU(4)) and $1.56(10)$ (SU(5)). The results are broadly consistent with the MQCD prediction and indicate that $k=2$ strings are bound states with $\sigma_{k=2} < 2\sigma_f$, informing confinement dynamics and potential implications for the SU($N$) glueball spectrum and center-vortex densities. These lattice findings support a universal pattern for string tensions in QCD-like theories and motivate further exploration of higher-$k$ flux tubes.

Abstract

We calculate the k=2 string tensions in SU(4) and SU(5) gauge theories in 3+1 dimensions, and compare them to the k=1 fundamental string tensions. We find, from the continuum extrapolation of our lattice calculations, that K(k=2)/K(k=1) = 1.40(8) in the SU(4) gauge theory, and that K(k=2)/K(k=1) = 1.56(10) in SU(5). We remark upon the way this might constrain the dynamics of confinement and the intriguing implications it might have for the mass spectrum of SU(N) gauge theories. We also note that these results agree closely with the MQCD-inspired conjecture that the SU(N) string tensions satisfy K(k)/K(1) = sin(k.pi/N)/sin(pi/N).

Figures (3)

• Figure 1: The masses of the fundamental, $\bullet$, and $k=2$, $\circ$, flux loops versus their length, at $\beta=10.7$ for SU(4). Shown also is the variation one expects from eqn(\ref{['eqn_poly']}).
• Figure 2: The mass of the fundamental flux loop as a function of its length, at $\beta=10.9$ for SU(4). Shown also is the variation one expects from eqn(\ref{['eqn_poly']}).
• Figure 3: The ratio of $k=2$ and $k=1$ string tensions in our SU(4)($\bullet$) and SU(5) ($\circ$) lattice calculations plotted as a function of $a^2\sigma_f$. Extrapolations to the continuum limit, using a leading $O(a^2)$ correction, are displayed.