⚛️ phenomenology

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

Using the PACIAE 4.0 model to simulate $e^+e^-$ collisions at $\sqrt{s}=4.95$ GeV, this study evaluates various structural interpretations of the $\phi(2170)$ resonance—including strangeonium, hybrid, tetraquark, and molecular configurations—by calculating their production yields and identifying distinct rapidity and transverse momentum spectral signatures that can distinguish their true nature.

Original authors: Jian Cao, Wen-Chao Zhang, Bo Feng, Ya-Hui Hou, An-Ke Lei, Zhi-Lei She, Hua Zheng, Li-Lin Zhu, Dai-Mei Zhou, Yu-Liang Yan, Ben-Hao Sa

Published 2026-07-02

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Jian Cao, Wen-Chao Zhang, Bo Feng, Ya-Hui Hou, An-Ke Lei, Zhi-Lei She, Hua Zheng, Li-Lin Zhu, Dai-Mei Zhou, Yu-Liang Yan, Ben-Hao Sa

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the universe is a giant, chaotic kitchen where tiny ingredients called quarks and gluons are constantly being tossed around, trying to stick together to form bigger dishes called hadrons (like protons and neutrons). Most of the time, they stick together in simple pairs (like a sandwich) or triplets (like a triple-decker). But sometimes, they form weird, exotic shapes that physicists have never seen before, like a four-ingredient salad or a sandwich with a ghost inside.

One of these mysterious "dishes" is a particle called φ(2170). For years, scientists have been arguing over what exactly this particle is made of. Is it a simple pair of strange quarks? Is it a four-quark salad? Is it a hybrid with a gluon inside? Or is it two different particles stuck together like magnets?

This paper is like a team of chefs using a super-computer simulation (called PACIAE) to cook up this particle in a virtual lab and see which recipe works best. Here is how they did it, explained simply:

1. The Virtual Kitchen

The researchers set up a digital simulation of a high-energy collision (like smashing two particles together at BESIII, a real-world particle collider). They didn't just watch the final result; they watched the whole cooking process in two stages:

Stage 1 (The Raw Ingredients): First, they simulated the chaotic soup of raw quarks and gluons flying around.
Stage 2 (The Plating): Then, they watched how these raw ingredients cooled down and stuck together to form the final dishes (hadrons).

2. Testing Different Recipes

The team tried to "cook" the φ(2170) particle using five different recipes (configurations) to see which one produced the most realistic results:

The Simple Pair: A standard pair of strange quarks ( $s\bar{s}$ ) spinning in a specific way (like a dancer doing a D-spin).
The Hybrid: A strange quark pair with a "glue" particle (gluon) stuck inside ( $s\bar{s}g$ ).
The Four-Quark Salads:
- Two strange quarks and two anti-strange quarks ( $ss\bar{s}\bar{s}$ ).
- Two strange quarks mixed with two up quarks ( $u\bar{u}s\bar{s}$ ).
- Two strange quarks mixed with two down quarks ( $d\bar{d}s\bar{s}$ ).
The Bound Couples: Two heavy particles (Lambda baryons) stuck together ( $\bar{\Lambda}\Lambda$ ).
The Resonance Trio: A phi particle hugging a pair of Kaons ( $\phi K^+ K^-$ ).

3. The "Coalescence" Rule

To decide if a group of ingredients actually became a φ(2170), the scientists used a rule called DCPC. Think of this like a "dance floor" rule:

If the ingredients are close enough to each other (within a certain radius, like 1 to 2 femtometers) and have the right energy, they are considered to have "danced" together and formed the particle.
If they are too far apart or moving too wildly, they don't count.

4. The Results: Who Won the Cooking Contest?

After running billions of virtual collisions, the team counted how many φ(2170) particles they made with each recipe. Here is what they found:

The Most Common Recipes: The "Simple Pair" (D-wave $s\bar{s}$ ), the "Hybrid" ( $s\bar{s}g$ ), and the "Four-Quark Salads" with up or down quarks ( $u\bar{u}s\bar{s}$ and $d\bar{d}s\bar{s}$ ) were the most frequent. They appeared about 100 times more often than the rarest recipes.
The Rare Recipes: The "Bound Couples" ( $\bar{\Lambda}\Lambda$ ) and the "Resonance Trio" ( $\phi K^+ K^-$ ) were harder to make, appearing about 10 times less often than the winners.
The Rarest Recipe: The "Four-Quark Salad" made entirely of strange quarks ( $ss\bar{s}\bar{s}$ ) was the hardest to cook, appearing about 100 times less often than the up/down versions.

5. The "Fingerprint" Differences

The most exciting part of the paper isn't just how many they made, but how they moved. The different recipes left different "fingerprints" on the particle's movement:

The Speed (Momentum):
- The Hybrid recipe ( $s\bar{s}g$ ) was the "fastest" and "heaviest" moving. Because it contains a gluon (which is very energetic), this particle tended to move faster than the others.
- The Bound Couples ( $\bar{\Lambda}\Lambda$ ) were the "slowest." Because they are made of heavy ingredients, they moved more sluggishly.
- The Simple Pair and Four-Quark Salads moved at a medium speed.
The Direction (Rapidity):
- All recipes tended to cluster in the middle of the collision zone, but the Hybrid recipe spread out more widely, while the others were more tightly packed.

The Conclusion

The paper concludes that we cannot yet say for sure which recipe is the "real" φ(2170). However, the different recipes leave very distinct fingerprints in their speed and direction.

The Takeaway: If real-world scientists at the BESIII collider measure the speed and direction of the φ(2170) particles they find, they can compare those measurements to the "fingerprints" calculated in this paper. If the real particles move fast and spread out like the Hybrid recipe, then that's what φ(2170) is. If they move slowly and are heavy like the Bound Couple, then that's the answer.

Essentially, this paper provides a menu of possibilities with clear descriptions of what each dish looks like, so real scientists can go to the lab, taste the particle, and finally identify the recipe.

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

Problem Statement
The intrinsic nature of the $\phi(2170)$ resonance (also known as $Y(2175)$ ) remains an unresolved issue in hadron spectroscopy. While its quantum numbers are established as $J^{PC} = 1^{--}$ , its internal structure is debated among various theoretical interpretations. Proposed candidates include an excited strangeonium ( $s\bar{s}$ ) state, an $s\bar{s}g$ hybrid state, a tetraquark state (specifically $ss\bar{s}\bar{s}$ , $u\bar{u}s\bar{s}$ , or the newly explored $d\bar{d}s\bar{s}$ ), a $\bar{\Lambda}\Lambda$ bound state, or a $\phi K^+ K^-$ resonance. A distinguishing feature of this work is the motivation to explore the $d\bar{d}s\bar{s}$ tetraquark configuration, driven by the U(1) anomaly coupling which allows non-strange quarks to couple to a vector $s\bar{s}$ component via soft-gluon interactions. The paper aims to distinguish between these competing hypotheses by simulating their production in $e^+e^-$ collisions at $\sqrt{s} = 4.95$ GeV and comparing their observable characteristics.

Methodology
The study utilizes the Parton and Hadron Cascade Model (PACIAE) version 4.0 to simulate $e^+e^-$ annihilation events. The simulation proceeds in two sequential stages:

Final Partonic State (FPS): The process begins with PYTHIA6 (with hadronization disabled) to generate an initial partonic state. Gluon splitting into quark-antiquark pairs is disabled to preserve gluons for hybrid state formation. This is followed by an energy de-excitation process and partonic $2 \to 2$ rescattering using leading-order perturbative QCD cross-sections.
Final Hadronic State (FHS): The FPS is hadronized via the Lund string fragmentation scheme to create an intermediate hadronic state, which then undergoes hadronic $2 \to 2$ rescattering to reach kinetic freeze-out.

To form specific $\phi(2170)$ candidates, the Dynamically Constrained Phase-Space Coalescence (DCPC) model is applied:

Partonic Coalescence (FPS): Excited strangeonium ( $s\bar{s}$ ), hybrid ( $s\bar{s}g$ ), and tetraquark ( $u\bar{u}s\bar{s}$ , $d\bar{d}s\bar{s}$ , $ss\bar{s}\bar{s}$ ) states are formed by coalescing constituent partons in the FPS.
Hadronic Recombination (FHS): The $\bar{\Lambda}\Lambda$ bound state and the $\phi K^+ K^-$ resonance are formed by recombining constituent hadrons in the FHS.

The formation of a candidate is governed by dynamical constraints on the invariant mass ( $m_{inv}$ ), cluster radius ( $R_0$ ), and the identities of the constituents. The orbital angular momentum quantum number ( $L$ ) for each candidate is calculated in its rest frame using the vector sum of constituent momenta and positions. Spectral classification is performed using standard $n^{2S+1}L_J$ notation. Given the experimental $J^{PC} = 1^{--}$ , the study restricts analysis to configurations compatible with this assignment: D-wave $s\bar{s}$ , P-wave $s\bar{s}g$ , P-wave tetraquarks, S-wave $\bar{\Lambda}\Lambda$ , and S-wave $\phi K^+ K^-$ .

Key Contributions and Results
The paper provides the first estimates of production yields for all these candidate configurations at $\sqrt{s} = 4.95$ GeV, based on a sample of 2.2 billion simulated events. The key findings include:

Production Yields:
- Order $10^{-4}$ : The D-wave $s\bar{s}$ , P-wave $s\bar{s}g$ , P-wave $u\bar{u}s\bar{s}$ , and P-wave $d\bar{d}s\bar{s}$ states exhibit the highest yields.
- Order $10^{-5}$ : The S-wave $\bar{\Lambda}\Lambda$ and S-wave $\phi K^+ K^-$ states show intermediate yields.
- Order $10^{-6}$ : The P-wave $ss\bar{s}\bar{s}$ tetraquark state has the lowest yield, approximately two orders of magnitude smaller than the $u\bar{u}s\bar{s}$ state. This suppression is attributed to the lower initial production probability of strange quarks and the difficulty of recombining multiple rare quark pairs.
- The yield of the $u\bar{u}s\bar{s}$ state is slightly higher than that of $d\bar{d}s\bar{s}$ (ratio $\approx 1.23:1$ ) due to the higher charge production probability and lower threshold of the $u$ quark in initial hard scattering.
Spectral and Kinematic Distributions:
- Rapidity ( $y$ ): All configurations show a peak at mid-rapidity. The peak heights follow the hierarchy $u\bar{u}s\bar{s} > d\bar{d}s\bar{s} > s\bar{s} > s\bar{s}g > \bar{\Lambda}\Lambda > \phi K^+ K^- > ss\bar{s}\bar{s}$ . Notably, the $s\bar{s}g$ hybrid state, despite having a high total yield, exhibits a broader rapidity distribution, resulting in a lower peak height compared to other configurations.
- Transverse Momentum ( $p_T$ ): Most configurations (D-wave $s\bar{s}$ , tetraquarks, $\bar{\Lambda}\Lambda$ , $\phi K^+ K^-$ ) peak around $p_T \approx 0.3$ GeV/c. In contrast, the P-wave $s\bar{s}g$ hybrid state displays a harder spectrum with a peak shifted to $p_T \approx 0.5$ GeV/c. This shift is attributed to the intrinsically harder $p_T$ spectra of gluons in parton cascades. The $\bar{\Lambda}\Lambda$ and $\phi K^+ K^-$ states exhibit softer spectra due to the heavy mass of constituent baryons and kinematic suppression of high- $p_T$ components in the three-body recombination, respectively.
Parameter Sensitivity: The study investigates the dependence of yields on the coalescence radius ( $R_0$ ) and mass uncertainty ( $\Delta m$ ). Yields for partonic coalescence states (strangeonium, tetraquarks) are strongly dependent on the upper bound of $R_0$ , while hadronic recombination states ( $\bar{\Lambda}\Lambda$ , $\phi K^+ K^-$ ) show opposite trends regarding the lower bound of $R_0$ . Reducing $\Delta m$ significantly suppresses yields, with the $\bar{\Lambda}\Lambda$ state disappearing entirely if $\Delta m$ is reduced below 0.64 times the total width of $\phi(2170)$ .

Significance
The paper claims that the distinct differences in production yields, rapidity distributions, and transverse momentum spectra among the various theoretical configurations serve as valuable criteria for identifying the true nature of the $\phi(2170)$ . By providing specific predictions for these observables, the work offers a framework for experimental verification. The authors suggest that measuring these quantities in $e^+e^-$ collisions at BESIII energies and comparing them with the presented predictions could help resolve the ambiguity surrounding the $\phi(2170)$ 's structure. The study explicitly rules out the $\phi(2170)$ being a $\phi\eta'/\phi\eta$ hadro-strangeonium state based on parity and charge-conjugation constraints, focusing the investigation on the remaining viable candidates.