Couplings of low-lying glueballs to light quarks, gluons, and hadrons

We derive a set of QCD sum rules for the operator $\int \mathrm{dx}{G}_{\mu }^{\nu }{G}_{\nu }^{\mu }$, which is in a sense a generalization of the trace anomaly. Combining these with plausible phenomenology, we arrive at a semi-quantitative picture of the couplings between the two lowest-lying glueballs (called $S$ for $J^{\mathrm{PC}}=0^{++}$ and $P$ for $0^{-+}$) and light quarks, gluons, and hadrons. These couplings suggest the existence of an approximate chiral/dilatation symmetry carried by the glueball interactions, with large and calculable symmetry-breaking terms arising primarily from the glueball masses. The couplings are of typical hadronic size, even though suppressed in the large-$N$ limit. The QCD sum rules mentioned above are closely related to the one-loop effective action, although derived quite differently; we believe the sum rules partially justify use of the effective action for small field strengths. A somewhat speculative generalization of sum rules to axial-anomaly densities gives an approximate evaluation of the Witten sum rule for the $\theta$ dependence of the vacuum energy. The $S-P$ chiral symmetry is related to the quark ${\mathrm{U}}_A\mathrm{}\left(1\right)\mathrm{}$ symmetry through the Peccei-Quinn mechanism; this symmetry is badly broken by the $P$ mass.