Verifying the existence of bound states of gluons and distinguishing them from conventional quark--antiquark, hybrid or tetraquark states has remained a key problem in QCD. We show that QCD counting rules for the power-law fall-off of production cross sections at high momentum transfer can be used to distinguish gluonium states from conventional hadrons.

The valence two-gluon contribution to a $0^+$ gluonium bound state has $L=0$ and thus twist (dimension minus spin of their minimum interpolating operators) $\tau=2$. The competing twist assignments for scalar $f_0$ mesons have twist $\tau = 3$ for the valence $|q \bar q \rangle $ configuration or $|q\bar{q}g\rangle$ in an s-wave, and $\tau \ge 4$ for $|q q \bar q \bar q \rangle$ tetraquarks, etc. Thus, the production cross section for mesons with quark--containing valence wavefunctions relative to glueball production should be suppressed by at least a power of momentum transfer.

Distinguishing these processes is feasible in exclusive $e^-e^+ \to \phi f_0$ reactions at 9 and 11 GeV center of mass energy at Belle-II.

In the case of single--particle inclusive hadroproduction $ A B \to C X$, the cross section for scalar gluonium production at high transverse momentum $p_T$ and fixed $x_T = 2 {p_T\over \sqrt s} $ will dominate meson or tetraquark production by at least two powers of $p_T$.

SLAC-PUB-17359