When the center matters: color screening and gluelumps in dihedral lattice gauge theories

Abstract

Confinement is one of the hallmarks of quantum chromodynamics (QCD). Yet, its first-principle characterization, even in simpler models, remains elusive. Through a combination of group-theoretical arguments and numerical analysis, we show that the physical consequences of confinement in a class of discrete non-Abelian lattice gauge theories (LGTs), the dihedral groups $D_N$, are intimately connected with the presence of a $\mathbb{Z}_2$ central subgroup. When the center is trivial (for odd $N$), static charges are screened by a gluon cloud, forming composite objects known in SU$(N)$ gauge theories as gluelumps. This finding implies that string breaking can occur through fluctuations of the electric field only, without the need to nucleate particle--antiparticle pairs from the vacuum. Furthermore, numerical analysis hints at finite-range interactions between the gluelumps in the continuum limit. Our results showcase how the rich and intricate physics typically associated with QCD can emerge in much simpler discrete non-Abelian LGTs, making them ideal settings to test this phenomenology both in numerical calculations and in near-term quantum devices.

Figures (6)

• Figure 1: (a) Ground-state configurations at strong coupling of $D_{N_{\rm odd}}$ LGTs on a ladder with static color charges on the corners. The string in the fundamental representation $\tau$ breaks when the charge separation $R$, in lattice units, is greater than 8, making the anti-fundamental $\tilde{\tau}-$glueballs energetically favourable. The charge-glueball configuration is gauge-invariant due to the screening fusion rule of Eq. \ref{['eq:fusion']}. (b) In $D_{N_{\rm even}}$, the GS at strong coupling always comprises the shortest $\tau$-string connecting the charged corners, since there is no equivalent to the screening fusion rule. (c) Ordering of links in a plaquette operator and highlighting of incoming and outgoing links associated with a vertex of the directed lattice.
• Figure 2: Magnetic energy density for ${\rm D}_3$ and ${\rm D}_4$ LGTs on a ladder. Continuous lines with filled markers indicate the presence of two static charges on the upper corners of the ladder, while dashed lines with empty markers stand for the neutral sector. The results are almost independent of the number of rungs ($N_r \in [4,60]$, increasing for darker colors). The plaquette expectation value $\langle B_p \rangle \sim 2$ at $g^2\to 0$, corresponds to the absence of magnetic fluxes.
• Figure 3: String tension as a function of the coupling $g^2$. We compare different charge separations $R$, indicated by line shading, for $D_3$ (warm colours and circles) and $D_4$ (cold colours and squares). The weak confinement $\sigma \propto g^6$ at $g^2\lesssim 0.2$ is qualitatively similar for both models. For $D_4$, the string tension $\sigma$ is independent of $R$, indicating the presence of a field string connecting them. Instead, for $D_3$ at strong coupling, the string breaks, indicated by $\sigma$ decreasing with growing $R$. The inset shows the energy difference $E_{q=1}(g,R) - E_{q=0}(g,R)=\sigma(g,R) R$ [see Eq. \ref{['eq: string tension']}]. Its independence of $R$ at strong coupling highlights the mutual screening of the two gluelumps, and $\sigma R=2M_0(g)$.
• Figure 4: Rescaled interaction potential $V(g,Rg^\kappa)$, with $\kappa=5/2$ as a function of the separation between the two charges. The excellent collapse suggests that the screening distance scales as $R_s \sim g^{-5/2}$. Inset: deviation of the renormalized mass $M(g)$ from the behaviour $M_0(g)=4g^2$ expected at strong-coupling. The leftmost point corresponds to a coupling strength where $R$ is too small to see the screening effect.
• Figure 5: Graphic comparison of the fusion rules of $D_3$ (left) and $D_4$ (right). Arrows indicate nonvanishing Clebsch--Gordan coefficients. For $D_3$, the fundamental representation $\tau$ has a non-vanishing Clebsch--Gordan coefficient with itself, while for $D_4$, only terms connecting $\tau$ with the one-dimensional representations are finite.