Testing a Flavor-Singlet Radial-Hybrid Interpretation of the $X(2370)$: Glueball Ambiguity and Discriminating Amplitude Measurements

Abstract

The nature of the pseudoscalar $X(2370)$ remains unresolved: glueball, hybrid, conventional $q\bar q$, and mixed configurations are all viable at the level of present information. We test a flavor-singlet $J^{PC}=0^{-+}$ radial-hybrid interpretation using a channel-dependent one-node overlap surrogate at $m_X=2.359~\mathrm{GeV}$ and a separate flavor-singlet radiative production factor. The resulting decay template naturally suppresses $K^{*}\bar K$ and can be conditionally normalized to the measured width, although its dominant scalar and tensor channels remain model dependent. Reproducing the inferred central $J/\psi\to\gamma X$ rate requires an effective singlet-production amplitude about $4.4$ times the fixed-$V(0)$ continuation of the available $N_f=2$ lattice anchor. In a common-nuisance screening comparison, the hybrid-mixed-glueball integrated-score difference is $-1.29$, $+0.04$, and $+0.41$ for tight, baseline, and wide nuisance settings, respectively; these values constitute a conditional-score tie rather than a composition measurement. We identify coherent amplitudes in $f_0(980)\eta$, $a_0(980)\pi$, and $K_0^{*}(1430)\bar K$ as particularly informative. Under the stated proxy forecast covariance, including theory uncertainty for both templates, their joint separation is $\Delta\chi^2\simeq6.5$ from the shared glueball template and $\simeq7.6$ from the radial-$q\bar q$ surrogate. A flavor-singlet radial hybrid is therefore compatible with current data but remains indistinguishable from a mixed-glueball interpretation; a coherent amplitude analysis can test the competing decay patterns.