Understanding the enhanced $φη^{\prime}$ decay mode of the $φ(2170)$ through strange-meson loops

Abstract: The nature of the strangeonium-like $φ(2170)$ remains controversial. Recent measurements of $e<sup>+e<sup>-\toφη<sup>{(\prime)}$ reveal a striking puzzle: a broad $φ(2170)$-like structure appears in the $φη<sup>\prime$ channel but not in $φη$, despite the strong phase-space suppression of $φη<sup>\prime$. We investigate this unexpectedly large $φη<sup>\prime$ decay fraction within the excited-strangeonium assignment. A combined analysis of the $e<sup>+e<sup>-\toφη$ and $e<sup>+e<sup>-\toφη<sup>\prime$ cross sections is performed by including short-distance amplitudes from vacuum quark-pair creation and long-distance transitions mediated by strange-meson loops. The short-distance mechanism predicts too small a $φη<sup>\prime/φη$ strength ratio to explain the data. In contrast, strange-meson loops naturally enhance this ratio because the SU(3)-flavor factor at the $$K<sup>{(<em>)}K^{(</em>)}η<sup>{(\prime)}$$ vertex suppresses the transition to $φη$ relative to $φη<sup>\prime$. The resulting overall description of both cross sections supports an important role for long-distance dynamics in hidden-strangeness decays of excited vector strangeonia. More intriguingly, although the excited strangeonium contributions have been included, the current high-precision $φη$ data still favor the existence of an extra narrow vector state near $2.15~\mathrm{GeV}$ with a width of about $25~\mathrm{MeV}$. If confirmed, it would be a promising exotic-hadron candidate in the light-vector sector. Furthermore, we test $Γ(Y\toφη)/Γ(Y\toφη<sup>\prime)=0.25$, $1$, and $4$ as fit inputs and obtain similarly good descriptions of the cross sections in all three cases. The present data therefore do not allow this ratio to discriminate among different internal configurations of the narrow state.

• The paper explains the anomalous φη′ enhancement using coherent φ(3S), φ(2D), and continuum amplitudes with strange-meson rescattering, achieving χ²/d.o.f. = 1.023.
• SU(3)-flavor suppression makes the K^{(*)}K^{(*)}η coupling about 10 times smaller than the η′ coupling, producing loop branching fractions up to two orders of magnitude larger for φη′ than φη despite reduced phase space.
• The analysis also finds evidence, but not proof, for an additional narrow vector state near 2.15 GeV with a mass of roughly 2.14–2.15 GeV and width near 25 MeV, motivating finer energy scans.

The puzzle of the enhanced $\phi\eta'$ decay of the $\phi(2170)$

The $\phi(2170)$, first observed by BaBar in initial-state-radiation production of $\phi f_0(980)$ (2608.14440), remains one of the most contested assignments in light-vector spectroscopy, with proposed interpretations ranging from conventional excited $s\bar s$ states to strangeonium hybrids, fully strange tetraquarks, molecular configurations, and triangle-singularity effects. A particularly sharp puzzle has emerged from BESIII measurements: a broad structure near $2.18~\mathrm{GeV}$ with a width of about $150~\mathrm{MeV}$ appears clearly in $e^+e^-\to\phi\eta'$, while no corresponding broad structure is visible in $e^+e^-\to\phi\eta$, even though two-body phase space favors $\phi\eta$ by a factor of roughly $\phi(2170)$0. The paper under discussion addresses this anomaly within a conventional excited-strangeonium framework and shows that long-distance strange-meson loop dynamics can account for it without invoking exotic internal structure for the $\phi(2170)$1 itself.

Experimental context

The experimental record on the $\phi(2170)$2 exhibits substantial channel dependence. The 2026 PDG averages are $\phi(2170)$3 and $\phi(2170)$4, but the six measurements entering the width average have $\phi(2170)$5, corresponding to a confidence level below $\phi(2170)$6. Reported widths span approximately $\phi(2170)$7–$\phi(2170)$8: the $\phi(2170)$9 and neutral $\phi(2170)$0 channels favor narrow structures of about $\phi(2170)$1, whereas the $\phi(2170)$2, $\phi(2170)$3, and charged $\phi(2170)$4 analyses favor widths of $\phi(2170)$5–$\phi(2170)$6. The authors argue that this dispersion should not be read as a universal resonance width; rather, several nearby vector states with channel-dependent relative strengths and interference phases can each produce a single visible structure with a distinct line shape. This observation motivates analyzing cross sections directly instead of using resonance parameters extracted from separate experimental fits as theoretical inputs.

Theoretical framework

The analysis builds on the open-strange baseline of Wang et al., in which seven $\phi(2170)$7 annihilation cross sections into open-strange final states were described coherently through interference among the $\phi(2170)$8 ($\phi(2170)$9), $\phi f_0(980)$0 ($\phi f_0(980)$1), and nonresonant amplitudes. The $\phi f_0(980)$2 is thus treated as an interference manifestation of these two excited states, whose masses, widths, and dileptonic widths are fixed as inputs. The $\phi f_0(980)$3 and continuum complete the amplitude model.

Two decay mechanisms for excited vector strangeonia into $\phi f_0(980)$4 are considered: (i) short-distance tree-level decays via quark-pair creation (QPC) from the vacuum, and (ii) long-distance rescattering through intermediate $\phi f_0(980)$5 pairs, described with effective Lagrangians and regulated by a dipole form factor with cutoff $\phi f_0(980)$6, where $\phi f_0(980)$7 is the only free parameter of the decay dynamics.

The decisive point is the SU(3)-flavor structure at the $\phi f_0(980)$8 vertex. With the $\phi f_0(980)$9–$s\bar s$0 mixing angle $s\bar s$1, the coupling ratio evaluates to

$s\bar s$2

so the loop transition to $s\bar s$3 is suppressed by about one order of magnitude relative to $s\bar s$4 — roughly two orders of magnitude at the rate level. This hierarchy allows the loop mechanism to overcome the smaller $s\bar s$5 phase space, in marked contrast to the QPC amplitude, which favors $s\bar s$6.

Combined fit and origin of the enhancement

A simultaneous fit to the measured $s\bar s$7 and $s\bar s$8 cross sections yields $s\bar s$9, with a fitted cutoff parameter $2.18~\mathrm{GeV}$0, within the conventional range for hadronic-loop phenomenology. The broad enhancement in the $2.18~\mathrm{GeV}$1 channel arises from coherent interplay of the $2.18~\mathrm{GeV}$2, $2.18~\mathrm{GeV}$3, and continuum amplitudes, so the observed structure should not be interpreted as a single isolated resonance.

The branching-fraction results quantify the mechanism. Although the tree-level QPC branching fractions strongly favor $2.18~\mathrm{GeV}$4 (e.g., $2.18~\mathrm{GeV}$5 versus $2.18~\mathrm{GeV}$6 for the $2.18~\mathrm{GeV}$7), the loop-induced contribution to $2.18~\mathrm{GeV}$8 exceeds that to $2.18~\mathrm{GeV}$9 by one to two orders of magnitude: for the $150~\mathrm{MeV}$0, $150~\mathrm{MeV}$1 against $150~\mathrm{MeV}$2 for $150~\mathrm{MeV}$3; for the $150~\mathrm{MeV}$4, $150~\mathrm{MeV}$5 against $150~\mathrm{MeV}$6. After both mechanisms are combined, the total $150~\mathrm{MeV}$7 branching fraction substantially exceeds that for $150~\mathrm{MeV}$8. The implication is direct: the anomalously large $150~\mathrm{MeV}$9 strength associated with the $e^+e^-\to\phi\eta'$0 does not require a strangeness-rich exotic configuration, but follows from long-distance strange-meson loop dynamics within the same conventional $e^+e^-\to\phi\eta'$1 baseline that describes the open-strange line shapes.

Hints of an additional narrow vector state

The baseline fit leaves one high-precision BESIII data point near $e^+e^-\to\phi\eta'$2 in the $e^+e^-\to\phi\eta'$3 channel unexplained. Adding an extra narrow vector state $e^+e^-\to\phi\eta'$4 improves the description. Because the partial-width ratio $e^+e^-\to\phi\eta'$5 cannot be determined from the data, three schemes fixing $e^+e^-\to\phi\eta'$6, $e^+e^-\to\phi\eta'$7, and $e^+e^-\to\phi\eta'$8 were tested; all yield comparable fit qualities ($e^+e^-\to\phi\eta'$9, $e^+e^-\to\phi\eta$0, $e^+e^-\to\phi\eta$1) and consistent resonance parameters:

Quantity	Scheme I	Scheme II	Scheme III
$e^+e^-\to\phi\eta$2 (MeV)	$e^+e^-\to\phi\eta$3	$e^+e^-\to\phi\eta$4	$e^+e^-\to\phi\eta$5
$e^+e^-\to\phi\eta$6 (MeV)	$e^+e^-\to\phi\eta$7	$e^+e^-\to\phi\eta$8	$e^+e^-\to\phi\eta$9

The fitted width of about $\phi\eta$0 is consistent within uncertainties with the narrow structure reported by BESIII in $\phi\eta$1, with $\phi\eta$2 and $\phi\eta$3, although the fitted $\phi\eta$4 masses lie somewhat lower. If confirmed, such a narrow vector state could not be accommodated by the strangeonium baseline used here and would constitute a compelling exotic-hadron candidate in the light-vector sector.

An important caveat accompanies this result: because the major part of the anomalous $\phi\eta$5 strength is already explained by the loop dynamics of the broad excited strangeonium contributions, the ratio $\phi\eta$6 cannot presently discriminate among internal configurations of the narrow state — molecule, compact tetraquark, hybrid, or otherwise. The data constrain its mass and width far more tightly than its relative couplings.

Limitations and open questions

Several limitations qualify the conclusions. The loop amplitudes rely on an effective Lagrangian approach with a phenomenological dipole form factor, and the single free parameter $\phi\eta$7 absorbs off-shell and ultraviolet uncertainties; the extracted branching fractions inherit this model dependence. The coupling constants of the excited states are derived from QPC branching fractions, tying the hidden-strangeness predictions to the same quark-model inputs used in the open-strange baseline. Most significantly, the evidence for the additional narrow state rests on a localized deviation at essentially a single high-precision energy point, and the large bin size of the present measurements prevents the analysis from establishing a new resonance on its own. Whether the narrow structure near $\phi\eta$8 is genuine, and what its actual $\phi\eta$9 partial-width ratio is, remain open questions requiring finer energy scans in the $\phi(2170)$00–$\phi(2170)$01 region and combined analyses of the $\phi(2170)$02, $\phi(2170)$03, and open-strange channels, including improved measurements of $\phi(2170)$04

Conclusion

This work demonstrates that the unexpectedly prominent, phase-space-suppressed $\phi(2170)$05 decay mode associated with the $\phi(2170)$06 can be reproduced within a conventional excited-strangeonium framework once strange-meson loop transitions are included alongside short-distance QPC amplitudes. The SU(3)-flavor suppression of the $\phi(2170)$07 coupling relative to $\phi(2170)$08 is the dynamical origin of the enhancement, reversing the preference of the short-distance mechanism. The result carries a methodological lesson: unusual decay patterns should not be attributed to exotic internal structures before long-distance hadronic-loop effects are accounted for. At the same time, the residual deviation in the high-precision $\phi(2170)$09 data favors — without yet establishing — an additional narrow vector state near $\phi(2170)$10 with a width of about $\phi(2170)$11, whose confirmation would be a significant development for exotic-hadron spectroscopy in the light-vector sector.