Wave functions of $SU(3)$ pure gauge glueballs on the lattice

The Bethe-Salpeter wave functions of $SU(3)$ pure gauge glueballs are revisited in this study. The ground and the first excited states of the scalar and tensor glueballs are identified unambiguously through the variational method. We calculate the wave functions in the Coulomb gauge and use two lattice spacings to check the discretization artifacts. For the ground states, the radial wave functions are approximately Gaussian, and the size of the tensor glueball is roughly twice as large as that of the scalar glueball. For the first excited states, the radial nodes are clearly observed for both the scalar and the tensor glueballs, such that they can be interpreted as the first radial excitations. These observations may shed light on the theoretical understanding of the inner structure of glueballs.

Figure 1

Effective mass plateaus of the ground and the first excited states in $A_1^{++},E^{++}$, and $T_2^{++}$ channels. The left plot is from the $8^3\times 96$ lattice while the right one is from the ${12}^3\times 144$ lattice. The time and the mass axis are both in lattice units.

Figure 2

The comparison of the ground-state wave functions $\mathrm{\Phi }(r)$ (normalized to 1 at $r=0$) extracted through the different definitions of the two-gluon operators in the $A_1^{++}$, $E^{++}$, and $T_2^{++}$ channels. The finite lattice spacing effects are also checked by the comparison of the wave functions calculated on $8^3\times 96$ and ${12}^3\times 144$ lattices. In each channel, the wave functions lie on each other within errors and manifest the small lattice artifacts.

Figure 3

The ground-state wave functions derived on the ${12}^3\times 144$ lattice are plotted by data points. The curves are the fitting results using the parametrization in Eq. (14).

Figure 4

The wave functions of the first excited states. The data points are measured results, and the curves illustrate the parametrization in Eq. (16) with the best-fit parameters.