The glueball spectrum from an anisotropic lattice study

TL;DR

This work addresses the challenge of determining the glueball spectrum in SU(3) Yang–Mills theory below 4 GeV by employing anisotropic lattices and an improved action to suppress discretization errors. It combines a large operator basis with smearing and variational techniques to extract masses from correlation matrices, then uses finite-volume checks and two-glueball/torelon thresholds to identify genuine single-glueball states and assign continuum spins. The authors report thirteen confirmed glueballs (plus two tentative) with continuum-limit masses in units of $r_0$ and in physical units (assuming $r_0^{-1}=410$ MeV), finding good agreement with prior Wilson-action results for the lightest states and offering qualitative support for operator- and bag-model pictures. They also quantify finite-volume and anisotropy-related systematic uncertainties and outline plans to further improve the scalar sector and extend the study to include quarks. Overall, this work provides a markedly clearer map of the SU(3) glueball spectrum and establishes a robust methodology for future QCD investigations with dynamical quarks.

Abstract

The spectrum of glueballs below 4 GeV in the SU(3) pure-gauge theory is investigated using Monte Carlo simulations of gluons on several anisotropic lattices with spatial grid separations ranging from 0.1 to 0.4 fm. Systematic errors from discretization and finite volume are studied, and the continuum spin quantum numbers are identified. Care is taken to distinguish single glueball states from two-glueball and torelon-pair states. Our determination of the spectrum significantly improves upon previous Wilson action calculations.

Figures (8)

• Figure 1: The Wilson loop shapes used in making the basic glueball operators.
• Figure 2: Comparison of the pure-glue spectrum obtained from the $\beta=2.5$, $\xi=5$ simulation to the approximate locations of the two glueball states. The boxes are the simulation results; the standard deviations in these mass estimates are indicated by the vertical heights of the boxes. The dotted line segments shown in the upper shaded region indicate the approximate locations of states consisting of two free glueballs having zero total momentum. All energies are in terms of $a_t^{-1}$. The representations of the cubic point group which label the states are indicated along the horizontal axis. The most likely $J^{PC}$ interpretations of the states are also shown.
• Figure 3: Finite-volume effects on the results of the $\beta=2.4$, $\xi=5$ simulation. Each point shows the fractional change $\delta_G=1-m_G^S/m_G$ in the energy of a stationary state $G$, where $m_G$ is the energy of $G$ as measured on an $8^3\times 40$ lattice, and $m_G^S$ is the energy of $G$ as measured on a smaller $6^3\times 40$ lattice. The state $G$ corresponding to a given point is specified by combining the representation label below the point along the horizontal axis with the $PC$ value shown to its left along the vertical axis. The solid lines indicate $\delta_G=0$, the dotted lines above the solid lines indicate $\delta_G=0.02$, and the dotted lines below the solid lines indicate $\delta_G=-0.02$.
• Figure 4: Mass estimates of the $PC=++$ glueballs in terms of $r_0$ against the lattice spacing $(a_s/r_0)^2$. The solid symbols indicate results from the $\xi=5$ simulations, and the open symbols on the right-hand side of the vertical dashed line indicate results from the $\xi=3$ simulations. The solid curves are best fits to the simulation results for each state using $\varphi_1(a_s)$ from Eq. (\ref{['phiscalar']}) for the $A_1^{++}$ and $A_1^{\ast ++}$ levels and $\varphi_0(a_s)$ from Eq. (\ref{['phifour']}) for all other levels. The open symbols on the left-hand side of the vertical dashed line show the extrapolations to the continuum limit using the best-fit forms.
• Figure 5: Mass estimates (solid symbols) of the $PC=-+$ glueballs in terms of $r_0$ against the lattice spacing $(a_s/r_0)^2$. The solid curves are best fits of $\varphi_0(a_s)$ from Eq. (\ref{['phifour']}) to the results for each state; the open symbols are the continuum limit extrapolations.