Closed k-strings in SU(N) gauge theories : 2+1 dimensions

TL;DR

This study computes ground-state energies for closed $k$-strings in $(2+1)$-D $SU(N)$ gauge theories (with $N=4,5,6,8$ and $k=2,3,4$) to test whether long flux tubes obey a bosonic string description, notably the Nambu-Goto model. Using lattice gauge theory with a variational, smearing-based operator basis, the authors extract $E_0(l)$ and determine continuum string tensions $\sigma_k$, comparing them to Casimir-scaling and alternative conjectures. They find that $k$-strings lie in the Nambu-Goto universality class with significant but controlled finite-$l$ corrections; the continuum $\sigma_k/\sigma_f$ ratios are within 1–2% of Casimir-scaling values, and the $N$-dependence favors a 1/N expansion over 1/N^2. Additionally, the low-lying spectrum falls into explicit SU($N$) representations, indicating that string binding respects the full gauge group rather than only its center, with excited states organizing into representation-specific towers and showing level crossings as a function of string length.

Abstract

We calculate the ground state energies of closed k-strings in (2+1)-dimensional SU(N) gauge theories, for N=4,5,6,8 and k=2,3,4. From the dependence of the ground state energy on the string length, we infer that such k-strings are described by an effective string theory that is in the same bosonic universality class (Nambu-Goto) as the fundamental string. When we compare the continuum k-string tensions to the corresponding fundamental string tensions, we find that the ratios are close to, but typically 1-2 percent above, the Casimir scaling values favoured by some theoretical approaches. Fitting the N-dependence in a model-independent way favours an expansion in 1/N (as in Casimir scaling) rather than the 1/N^2 that is suggested by naive colour counting. We also observe that the low-lying spectrum of k-string states falls into sectors that belong to particular irreducible representations of SU(N), demonstrating that the dynamics of string binding knows about the full gauge group and not just about its centre.

Figures (8)

• Figure 1: The effective coefficient of the $\pi/3\sigma l^2$ term in the Nambu-Goto expression for the ground state energy, eqn(\ref{['eqn_NGceff']}), for the range of lengths indicated. For $k=1$, $\bullet$, and $k=2$, $\circ$, strings in SU(4) at $\beta=32.0$.
• Figure 2: As in Fig.\ref{['fig_NGceffN4']} but for SU(5) at $\beta=80.0$.
• Figure 3: $k=2$ string tension in SU(5), extracted using single exponential, $S$, fits, as a function of the lattice spacing. Continuum extrapolations linear and quadratic in $a^2$ are shown.
• Figure 4: As in Fig.\ref{['fig_n5Scont']}, but using double exponential, $D$, fits.
• Figure 5: Calculated values of $r_2 = \sigma_{k=2}/\sigma_f$ and fits as discussed in Section \ref{['section_N']}.