On the constituent counting rules for hard exclusive processes involving multiquark states

Abstract

At high energies, the cross section of a hard exclusive process at finite scattering angle falls off as a negative power of the center-of-mass energy $\sqrt{s}$. If all involved quark-gluon compositions undergo hard momentum transfers, the scaling of the fall-off is determined by the underlying valence structures of the initial and final states, known as the constituent counting rules. It was argued in the literature that the counting rules are a powerful tool to determine the valence degrees of freedom inside multiquark states when applied to exclusive production processes. However, we demonstrate that for hadrons with hidden flavors the naive application of the constituent counting rules is problematic, since it is not mandatory for all components to participate the hard scattering at the scale $\sqrt{s}$. The correct scaling rules can be obtained easily by using effective field theory. A few examples involving the $Z_c(3900)^\pm$ and $X(3872)$ are discussed.