Shedding light on the nature of $φ(2170)$ with the parton and hadron cascade model PACIAE

Abstract: The nature of $φ(2170)$ remains open. We simulate its production in $e<sup>+e<sup>-$ collisions at $\sqrt{s}=4.95$ GeV using PACIAE 4.0, which sequentially generates the final partonic state (FPS) and the final hadronic state (FHS). While previous studies have interpreted $φ(2170)$ as an $ss\bar{s}\bar{s}$ or a $u\bar{u}s\bar{s}$ state, the $U(1)$ anomaly coupling allows non-strange quarks to couple to a vector $s\bar{s}$ component via soft-gluon interactions. This motivates us to also explore the $d\bar{d}s\bar{s}$ tetraquark configuration. In addition, we consider $φ(2170)$ as an excited strangeonium state, an $s\bar{s}g$ hybrid state, a $\barΛΛ$ bound state, and a $φK<sup>+K<sup>-$ resonance state. The strangeonium, hybrid, and tetraquark candidates are formed by coalescing their constituent partons in the FPS using the dynamically constrained phase-space coalescence model. The $\barΛΛ$ and $φK<sup>+K<sup>-$ states are produced via recombination of their constituent hadrons in the FHS. We calculate the orbital angular momentum quantum number of each candidate in its rest frame and perform spectral classification. Given $J<sup>{PC}=1<sup>{--}$, $φ(2170)$ can be interpreted as a $D$-wave $s\bar{s}$, a $P$-wave $s\bar{s}g$, a $P$-wave $u\bar{u}s\bar{s}/d\bar{d}s\bar{s}/ss\bar{s}\bar{s}$, an $S$-wave $\barΛΛ$, or an $S$-wave $φK<sup>+K<sup>-$ state. The yields of the $D$-wave $s\bar{s}$, $P$-wave $s\bar{s}g$, $u\bar{u}s\bar{s}$ and $d\bar{d}s\bar{s}$ states are of order $10<sup>{-4}$; those for the $S$-wave $\barΛΛ$ and $φK<sup>+K<sup>-$ states are of order $10<sup>{-5}$; while the $P$-wave $ss\bar{s}\bar{s}$ yield is of order $10<sup>{-6}$. Moreover, significant discrepancies are observed in the rapidity distributions and the $p_T$ spectra among the various candidates. These discrepancies could serve as valuable criteria for unraveling the nature of $φ(2170)$.

• The paper demonstrates that PACIAE+DCPC simulations yield comparable production rates for D-wave strangeonium, P-wave hybrid, and non-strange tetraquark states, with a notable suppression for fully strange tetraquarks.
• Simulated rapidity and transverse momentum spectra reveal configuration-dependent kinematic signatures, with hybrids showing a harder pT peak due to gluon dynamics.
• The study underscores the capability of dynamical coalescence methods in distinguishing exotic hadron structures through sensitivity to spatial and kinematic constraints.

Comprehensive Analysis of $\phi(2170)$ Structure via PACIAE-Based Coalescence Simulation

Introduction

The study addresses the unresolved problem of the internal structure of the $\phi(2170)$ resonance, a vector meson with $J^{PC}=1^{--}$ and mass around 2.17 GeV. Since its experimental discovery, $\phi(2170)$ has garnered significant attention as a candidate for nonstandard hadronic structures permitted by QCD, such as tetraquarks, strangeonium hybrids, hidden-strangeness molecular states, and baryon-antibaryon bound states. Empirically, the resonance exhibits comparable production rates in channels such as $\phi\eta'$ and $\phi\eta$ to the $X(2300)$, yet differs in quantum numbers and cannot be classified as a simple strangeonium-like hadro-charmonium. Theoretical assignments in the literature span $s\bar{s}$ excitations, $s\bar{s}g$ hybrids, tetraquark configurations ($ss\bar{s}\bar{s}$, $\phi(2170)$0, $\phi(2170)$1), $\phi(2170)$2 bound states, and $\phi(2170)$3 systems, but distinguishing amongst these is nontrivial.

To provide discriminating power, this work employs the PACIAE 4.0 parton and hadron cascade model in conjunction with the dynamically constrained phase-space coalescence (DCPC) approach. Simulations of $\phi(2170)$4 collisions at $\phi(2170)$5 GeV—motivated by high-luminosity BESIII data—probe the production yields, rapidity, and $\phi(2170)$6 spectra of $\phi(2170)$7 candidates in various internal configurations. This enables a systematic, dynamical QCD-based evaluation of the phenomenological implications of each scenario.

Methodology

PACIAE Simulation Framework

PACIAE 4.0 extends PYTHIA6 by disabling fragmentation and forced gluon splitting to isolate the parton-level state and retain gluons necessary for hybrid state construction. The event simulation propagates through several stages: generation of the initial partonic state, perturbative partonic rescattering, hadronization to form the final hadronic state (FHS), and subsequent hadronic re-scattering to kinetic freeze-out.

Dynamical Coalescence

Hadronic and exotic candidates are generated using the DCPC algorithm. The yields are estimated by phase-space integration under kinematic and spatial constraints. For a candidate system with $\phi(2170)$8 constituents, the formal yield,

$\phi(2170)$9

imposes invariant mass windowing and geometric constraints (e.g., maximum relative distance parameter $J^{PC}=1^{--}$0). For resonance constructions in the FHS (e.g., $J^{PC}=1^{--}$1, $J^{PC}=1^{--}$2) the algorithm exploits recombination of stable hadrons.

Quantum Number and Spectral Assignment

Through a covariant construction of rest-frame orbital angular momentum $J^{PC}=1^{--}$3 for each candidate, and explicit coupling of constituent spins, the spectral identity is assigned using standard $J^{PC}=1^{--}$4 notation. The methodology accommodates the internal symmetry differences among strangeonium, hybrid, tetraquark, molecular, and baryon-antibaryon candidates, and incorporates parity and $J^{PC}=1^{--}$5-parity projection in conformity with the quantum numbers of $J^{PC}=1^{--}$6.

Results

Spectral Classification and Production Yields

The calculation yields the following classifications for candidates compatible with $J^{PC}=1^{--}$7:

• $J^{PC}=1^{--}$8-wave $J^{PC}=1^{--}$9: $\phi(2170)$0 configuration,
• $\phi(2170)$1-wave $\phi(2170)$2: Hybrid structure with $\phi(2170)$3,
• $\phi(2170)$4-wave tetraquark ($\phi(2170)$5, $\phi(2170)$6, $\phi(2170)$7),
• $\phi(2170)$8-wave $\phi(2170)$9: $\phi\eta'$0 vector baryonium,
• $\phi\eta'$1-wave $\phi\eta'$2: Three-body molecular.

The event-averaged production yields for each configuration are summarized:

Configuration	Yield per Event ($\phi\eta'$3)
$\phi\eta'$4-wave $\phi\eta'$5	1.22
$\phi\eta'$6-wave $\phi\eta'$7	1.72
$\phi\eta'$8-wave $\phi\eta'$9	1.60
$\phi\eta$0-wave $\phi\eta$1	1.31
$\phi\eta$2-wave $\phi\eta$3	0.079
$\phi\eta$4-wave $\phi\eta$5	0.66
$\phi\eta$6-wave $\phi\eta$7	0.64

The results demonstrate that the $\phi\eta$8-wave $\phi\eta$9 hybrid and $X(2300)$0-wave $X(2300)$1, as well as non-strange tetraquark states, are produced at statistically similar rates ($X(2300)$2), whereas the fully strange $X(2300)$3 tetraquark is suppressed by about two orders of magnitude. Molecular ($X(2300)$4) and baryonium ($X(2300)$5) yields are intermediate ($X(2300)$6), but rise with increasing coalescence radius, reflecting their spatially extended structure.

Rapidity and $X(2300)$7 Spectra

The simulated rapidity distributions for all configurations show a prominent peak at $X(2300)$8, consistent with central production dominated by isotropic hadronization at BESIII-relevant energies. The peak heights distinguish the hierarchy of configurations: $X(2300)$9 and $s\bar{s}$0 leads over $s\bar{s}$1 and $s\bar{s}$2, with $s\bar{s}$3 being minimal.

The transverse momentum spectra further discriminate among candidate types. All coalesced quark states (strangeonium, tetraquarks) exhibit peaks near $s\bar{s}$4 GeV/$s\bar{s}$5, inherited from the primordial quark transverse momentum spectrum. The $s\bar{s}$6 hybrid exhibits a harder $s\bar{s}$7 spectrum with a peak at $s\bar{s}$8 GeV/$s\bar{s}$9, attributable to the harder gluon kinematics within the parton cascade. $s\bar{s}g$0 and $s\bar{s}g$1 candidates, formed in the hadron FHS, show even softer $s\bar{s}g$2 features due to mass-induced phase space suppression and the tight resonance mass windowing inherent to DCPC.

Parameter Dependence

The yields display strong dependence on the geometric constraint $s\bar{s}g$3 and the mass window $s\bar{s}g$4. Decrease in $s\bar{s}g$5 leads to reduced yields for compact multi-quark candidates, and increasing $s\bar{s}g$6 for hadronic molecules enhances their yields. Reduction of $s\bar{s}g$7 transitions the coalescence from extensive to exclusive, sharply lowering yields for all configurations, particularly baryonium. The methodology for extracting the orbital angular momentum quantum number (integer rounding or truncation) also impacts yield magnitudes but leaves the qualitative hierarchy invariant.

Implications and Prospects

The principal implication is that the production topology (rapidity and $s\bar{s}g$8 spectra) and the absolute yield—when compared among the various theoretically viable configurations—encode essential information about the microscopic structure of $s\bar{s}g$9. The similarity in yield between the $ss\bar{s}\bar{s}$0-wave strangeonium, $ss\bar{s}\bar{s}$1-wave hybrid, and $ss\bar{s}\bar{s}$2-wave non-strange tetraquarks suggests that, absent further input, production yield alone is insufficient for unambiguous discrimination. However, the $ss\bar{s}\bar{s}$3 suppression of $ss\bar{s}\bar{s}$4 production will be directly testable in future high-statistics experiments, allowing exclusion or confirmation of a fully strange tetraquark assignment.

The rapidity and $ss\bar{s}\bar{s}$5 shapes are direct consequences of parton-level kinematics and the hadronic recombination environment, hence are robust observables for experiment-theory comparison. Notably, the harder $ss\bar{s}\bar{s}$6 distribution for hybrids and the yield enhancement in non-strange tetraquark channels (due to initial state $ss\bar{s}\bar{s}$7-quark preference) are precise predictions of the PACIAE+DCPC scheme.

On the theoretical front, systematically contrasting these predictions with BESIII data offers discrimination power among QCD models proposing hybrid, tetraquark, or molecular nature for $ss\bar{s}\bar{s}$8. Additionally, the detailed sensitivity to spatial and kinematic constraints within the coalescence protocol can further be exploited to gain insight into the hadronization length scale for exotic states.

Conclusion

Through the combined use of parton-hadron transport and dynamically constrained coalescence, this study delivers quantitative predictions for the production and kinematic distributions of $ss\bar{s}\bar{s}$9 candidates with distinct internal structures in $\phi(2170)$00 collisions at charm threshold energies. The results highlight that both the overall yield and the differential spectra encode configuration-specific information, furnishing a roadmap for future experimental measurements to elucidate the QCD structure of $\phi(2170)$01. These findings underscore the utility of dynamical event generators combined with coalescence phenomenology as an incisive tool in the study of exotic hadrons.