Glueballs and fractional anomalous determinants at nonzero $\theta$, and the decays of the X(2370)

Abstract

Using fractional anomalous determinants, we construct a model of the trace and axial anomalies of a $SU(3)$ gauge theory at nonzero $\theta$ angle through the couplings to scalar and pseudoscalar glueballs. In the pure gauge theory, we reproduce the $\theta$ dependence of the vacuum energy and the scalar-glueball mass from the lattice, and estimate the jump in the topological charge density for the first-order transition at $\theta =\pi$. In QCD, the coupling of a pseudoscalar glueball to fractional anomalous determinants describes all of the four three-pseudoscalar decay channels of the $0^{-+}$ glueball candidate $X(2370)$ measured by the BESIII collaboration. Measuring the decay rates for $\eta \eta \eta$ and $\eta \eta {\eta }^{\prime }$ can provide a stringent test of the model.