Closed flux tubes and their string description in D=3+1 SU(N) gauge theories

TL;DR

The paper investigates the spectrum of closed confining flux tubes wound on a spatial torus in SU(N) gauge theories in 3+1 dimensions, using lattice simulations across N=3,5,6 and a range of tube lengths. Most low-lying states are found to be well described by the Nambu-Goto free string spectrum, even at relatively short lengths where the tube is not string-like, and the leading Lüscher correction agrees with a bosonic string universality class. A notable exception is a set of anomalous 0^- states that deviate significantly from NG predictions, which may encode massive flux-tube modes and constrain the structure of the effective string action. The study also examines k=2 flux tubes and provides a comprehensive analysis of finite-volume and lattice-spacing effects, establishing NG as a robust starting point for the effective string description while highlighting potential corrections from massive modes and higher-order terms.

Abstract

We calculate the energy spectrum of a confining flux tube that is closed around a spatial torus, as a function of its length l. We do so for various SU(N) gauge theories in 3+1 dimensions, and for various values of spin, parity and longitudinal momentum. We are able to present usefully accurate results for about 20 of the lightest such states, for a range of l that begins close to the (finite volume) deconfining phase transition, and extends up to l.sqrt(K)~6 (where K is the string tension). We find that most of these low-lying states are well described by the spectrum of the Nambu-Goto free string theory in flat space-time. Remarkably, this is so not only at the larger values of l, where the gap between the ground state energy and the low-lying excitations becomes small compared to the mass gap, but also down to much shorter lengths where these excitation energies become large compared to sqrt(K), the flux-tube no longer `looks' anything like a thin string, and an expansion of the effective string action in powers of 1/l no longer converges. All this is for flux in the fundamental representation. We also calculate the k=2 (anti)symmetric ground states and these show larger corrections at small l. So far all this closely resembles our earlier findings in 2+1 dimensions. However, and in contrast to the situation in D=2+1, we also find that there are some states, with J,P=0,- quantum numbers, that show large deviations from the Nambu-Goto spectrum. We investigate the possibility that (some of) these states may encode the massive modes associated with the internal structure of the flux tube, and we discuss how the precocious free string behaviour of most states constrains the effective string action, on which much interesting theoretical progress has recently been made.

Figures (22)

• Figure 1: Effective central charge in SU(3): from Lüscher ($\bullet$) and Nambu-Goto ($\circ$) using eqns(\ref{['eqn_MLceff']},\ref{['eqn_NGceff']}).
• Figure 2: Effective central charge in SU(6): from Lüscher ($\bullet$) and Nambu-Goto ($\circ$) using eqns(\ref{['eqn_MLceff']},\ref{['eqn_NGceff']}).
• Figure 3: $\chi^2$ per degree of freedom for the best fit to $E_0(l)$ using eqn(\ref{['eqn_E0fitAH']}), for both SU(3), $\circ$, and SU(6), $\bullet$.
• Figure 4: Lightest flux tube energies for longitudinal momenta $q=0$, $\bullet$, $q=1$, $\bullet$, and $q=2$ in SU(3) at $\beta=6.0625$. The four $q=2$ states are $J^{P_t}= 0^+ (\star ), \ 1^\pm (\circ ), \ 2^+ (\Box ), \ 2^- (\bullet )$. Lines are Nambu-Goto predictions.
• Figure 5: As in Fig. \ref{['fig_Eqd4n3']} but for SU(5) at $\beta=17.63$.