Cancellation of unequal meson couplings in the mixed-state sum rule

Determine how the unequal couplings of the molecule and four-quark components disappear from the left-hand side of the ratio of moments ${\cal R}_{10}$ used to determine the masses of the mixed $J/\psi J/\psi$ and $cc\bar c\bar c$ states.

Background

The paper discusses a previous QCD spectral-sum-rule analysis that predicts an averaged molecule and four-quark mass through mixing. That analysis uses the ratio of moments R10{\cal R}_{10}, but the present authors state that they cannot understand how the couplings of the two meson components— which are not equal—are eliminated from the expression used to extract the masses. This leaves the treatment of the couplings in the mixed-state sum rule unresolved.

References

However, we fail to understand how the meson couplings which are not equal disappear in the LHS of ${\cal R}_{10}$ used to get the masses.

— New Dynamics (?) of $J/ψJ/ψ$ and $ΥΥ$ families from QCD Laplace sum rules at NLO  (2609.18906 - Narison et al., 16 Sep 2026) in Section “Comments and comparison with some QSSR results,” subsection “Charm quark channel”