Closed flux tubes in D=2+1 SU(N) gauge theories: dynamics and effective string description

TL;DR

This work provides a comprehensive lattice study of closed confining flux tubes in SU($N$) gauge theories in $D=2+1$, testing the emergence of an effective string description. It demonstrates that the leading nonuniversal correction to the ground-state energy is consistent with $O(1/l^{\gamma})$ and requires $\gamma \geq 7$, reinforcing the universal NG-like structure predicted by effective-string theories. The study also identifies massive world-sheet modes on $k$-strings, with their masses tied to bulk physics via the bulk mass gap, and shows a remarkable near-orthogonality of flux tubes across different representations, implying strong suppression of screening even at small $N$. Together, these results bolster the view that confining flux tubes in $D=2+1$ SU($N$) theories are well captured by an effective string action, with clear implications for the bulk spectrum and for understanding representation dynamics on the world sheet.

Abstract

We extend our earlier calculations of the spectrum of closed flux tubes in SU(N) gauge theories in 2+1 dimensions, with a focus on questions raised by recent theoretical progress on the effective string action of long flux tubes and the world-sheet action for flux tubes of moderate lengths. Our new calculations in SU(4) and SU(8) provide evidence that the leading O(1/l^gamma) non-universal correction to the flux tube ground state energy does indeed have a power gamma greater than or equal to 7. We perform a study in SU(2), where we can traverse the length at which the Nambu-Goto ground state becomes tachyonic, to obtain an all-N view of the spectrum. Our comparison of the k=2 flux tube excitation energies in SU(4) and SU(6) suggests that the massive world sheet excitation associated with the k=2 binding has a scale that knows about the group and hence the theory in the bulk, and we comment on the potential implications of world sheet massive modes for the bulk spectrum. We provide a quantitative analysis of the surprising (near-)orthogonality of flux tubes carrying flux in different SU(N) representations, which implies that their screening by gluons is highly suppressed even at small N.

Figures (42)

• Figure 1: Best fits to SU(8) ground state energy with Nambu-Goto plus a $O(1/l^9)$ correction. Vertical line indicates the deconfining transition.
• Figure 2: Best fits to SU(8) ground state using Nambu-Goto with a $O(1/l^\gamma)$ correction for various $\gamma$: $p$-value for $l\in [14,17]$ for three mass fits as explained in text.
• Figure 3: Best fits to SU(8) ground state using Nambu-Goto with a $O(1/l^\gamma)$ correction for various $\gamma$: $p$-value for $l\in [14,18]$ for three mass fits as explained in text.
• Figure 4: Best fits to SU(6) ground state energy with Nambu-Goto plus a $O(1/l^7)$ correction. Vertical line indicates the deconfining transition.
• Figure 5: Best fits to SU(6) ground state energy using Nambu-Goto with a $O(1/l^\gamma)$ correction: $p$-value for $l\in [14,17],\, \bullet$, and for $l\in [14,18], \, \circ$, versus $\gamma$.