Continuum limit of a qubit-regularized SU(3) lattice gauge theory with glueballs

Abstract

We show that a simple qubit-regularized $\mathrm{SU}(3)$ lattice gauge theory (LGT) on a plaquette chain admits a continuum limit with massive glueball excitations, providing a minimal toy model of strong interactions without quarks. By mapping the plaquette-chain Hamiltonian to the three-state quantum clock model in a magnetic field, we demonstrate that the theory can be tuned to a continuum limit governed at short distances by the $\mathbb{Z}_3$ parafermion conformal field theory (CFT), which serves as the ultraviolet (UV) fixed point. A small relevant magnetic perturbation then drives the system to a massive continuum quantum field theory in the infrared (IR). The resulting relativistic massive particles can be interpreted as quasi one-dimensional analogues of glueballs. In the continuum theory we compute the ratio of the lowest glueball masses with opposite charge conjugation to be $m^{-}/m^{+} = \,1.459(2)$ and find $\sqrtσ/m^{+}\,= 0.2648(2)$, where $σ$ is the string tension between a static quark and antiquark.

Figures (14)

• Figure 1: Physical basis state of the qubit-regularized $\mathrm{SU}(3)$ on a plaquette chain in the basis. Links are represented by dimer tensor states, which may equivalently be viewed as color flux states $\left| \lambda\right\rangle$, where $\lambda={\mathbf{1}}$ (blue), ${\mathbf{3}}$ (red), and ${\mathbf{\bar{3}}}$ (yellow) are irreducible representations of $\mathrm{SU}(3)$. The two indices of each tensor are associated with the lattice sites connected by the link and transform in the irreps $\lambda$ and its conjugate $\bar{\lambda}$. Gauge invariance enforces that the irreps meeting at each site combine to a singlet, implying for a plaquette chain the only possibilities are (i) a singlet, a triplet, and an anti-triplet index, (ii) three triplet indices, or (iii) three anti-triplet indices meeting at a site (top). An equivalent representation in terms of color fluxes is shown in the bottom panel, where a dashed line denotes the state $\left| {\mathbf{1}}\right\rangle$, arrows pointing to the right or upward denote $\left| {\mathbf{3}}\right\rangle$, and oppositely oriented arrows denote $\left| {\mathbf{\bar{3}}}\right\rangle$.
• Figure 2: Action of the plaquette operators $\hat{\mathcal{U}}_P$ and $\hat{\mathcal{U}}_P^\dagger$ on a representative state. Traversing the plaquette clockwise or counterclockwise updates the links according to the cyclic rule ${\mathbf{1}} \rightarrow {\mathbf{3}} \rightarrow {\mathbf{\bar{3}}} \rightarrow {\mathbf{1}}$, applied to the irrep at the first site of each link; the irrep at the opposite end is then fixed.
• Figure 3: Plot of the mass ratio $R(\mu)$ in \ref{['eq:mass_scaling_ansatz']} as a function of the scaling variable $\mu = h^{15/28}L$. The data points are obtained by extrapolating the data at each fixed value of $\mu$ to the joint limit $L \to \infty$ and $h \to 0$. The and regimes correspond to $\mu \to 0$ and $\mu \to \infty$, respectively. In the limit, the degeneracy of the lightest spin primaries related by charge conjugation at the $\mathbb{Z}_3$ parafermion leads to the result that $R(0) = 1$CARDY1986186Lepori:2009ip. In the limit, no closed-form expression for $R(\infty)$ is known since the theory is non-integrable.
• Figure 4: Relativistic dispersion relation defined in \ref{['eq:dispersion']}. The straight line is a linear fit to all data points (87 in total) with $48 \leq L \leq 96$ and $10.0 \leq \mu \leq 14.0$. The slope yields $\zeta^2$, giving $\zeta = 2.5907(2)$ with $\chi^2/\mathrm{d.o.f.} = 1.73$.
• Figure 5: Static quark potential $V({\mathsf{w}})$ versus separation ${\mathsf{w}}$ for $L=96$ and $\mu=14.0$. The solid line is a linear fit whose slope gives $\bar{\sigma} = 0.009500(3)$.