Interaction and correlation functions for $\pi f_1(1285)$, $\eta f_1(1285)$

This paper investigates the interaction and correlation functions of $\pi^0 (\eta) f_1(1285)$ systems by modeling the $f_1(1285)$ as a $K^* \bar K$ molecular state within a fixed-center approximation framework, successfully reproducing experimental proton-$f_1(1285)$ data while predicting a structure near 1500–1600 MeV and a threshold cusp, but failing to support the existence of the $\pi_1(1400)$, $\pi_1(1600)$, or $\eta_1(1855)$ resonances.

The Gist

The Big Picture: Building with Lego Bricks

Imagine the subatomic world as a giant construction site. Physicists are trying to figure out how tiny particles stick together to form larger, more complex structures.

In this paper, the authors are studying a specific construction project involving three main characters:

1. The "Guest": A particle called a pion ($\pi^0$) or an eta ($\eta$). Think of these as small, energetic visitors.
2. The "Host": A particle called $f_1(1285)$. The authors treat this not as a single solid brick, but as a molecule made of two smaller bricks stuck together (specifically, a $K^*$ and a $\bar{K}$).
3. The Mystery: There are some "ghost" particles (called resonances like $\pi_1$ and $\eta_1$) that scientists have seen in experiments but don't fully understand. Some theories suggest these ghosts are actually formed when the "Guest" and the "Host" interact.

The authors wanted to see what happens when the Guest visits the Host. Do they get along? Do they form a new, stable structure (a resonance)? Or do they just bounce off each other?

The Method: The "Fixed Center" Game

To figure this out, the authors used a mathematical tool called the Fixed Center Approximation (FCA).

The Analogy:
Imagine the "Host" ($f_1(1285)$) is a double-decker bus made of two people holding hands. The "Guest" ($\pi$ or $\eta$) is a person trying to bump into the bus.

• The Old Way: Some theories treated the bus as a solid, unbreakable wall.
• The Authors' Way: They realized the bus is actually two people. So, they calculated what happens when the Guest bumps into the first person, and then what happens if that person bumps into the second person, all while the two people in the bus stay holding hands (the "cluster" remains intact).

They used a sophisticated set of equations (Faddeev equations) to map out every possible way the Guest could interact with the two parts of the Host without breaking the Host apart. They then solved a "traffic flow" equation (Lippmann-Schwinger) to see how the particles move and scatter.

What They Found: The Results

1. The "Handshake" (Scattering Length)
The authors calculated how "friendly" or "sticky" the interaction is.

• For the Pion ($\pi^0$): The interaction is very weak. It's like two people passing each other on a sidewalk and barely nodding. The "scattering length" (a measure of how much they interact) is tiny.
• For the Eta ($\eta$): The interaction is slightly stronger, but still relatively gentle.

2. The "Ghost" Hunt (Resonances)
This is the most critical part. Scientists have been looking for specific "ghost" particles (like $\pi_1(1400)$, $\pi_1(1600)$, and $\eta_1(1855)$) that some theories claim are formed by this exact interaction.

• The Result: The authors did not find clear evidence of these ghosts in their calculations.
• The Nuance:
  • Around 1500–1600 MeV (an energy level), the Pion interaction showed a "curious bump." It looked a little bit like a resonance, but it wasn't a strong, clear signal. It's like hearing a faint hum in a room—you aren't sure if it's a machine or just the wind.
  • Around 1855 MeV (where the $\eta_1$ ghost is supposed to be), they found nothing.
  • However, right at the moment the Eta particle hits the energy threshold to interact with the Host (around 1833 MeV), they saw a sharp "cusp" (a sudden spike). Imagine a car hitting a speed bump; the graph jumps up sharply. This is a real effect, but it's not a new particle; it's just a reaction to the threshold.

3. The "Correlation" (How they move together)
The authors also calculated a "correlation function."

• The Analogy: Imagine taking a photo of two people walking out of a party. If they are friends, they walk close together. If they are strangers, they walk apart.
• The Finding: For the Pion and the Host, the "photo" shows they are almost strangers. They don't stick together much. The correlation is very close to 1 (which means "no interaction"). This is much weaker than what was seen in previous experiments with protons and the same Host.

The Conclusion

The authors conclude that while their method is reliable (it worked well when tested against previous proton experiments), this specific interaction does not seem to be the "factory" that creates the mysterious $\pi_1$ or $\eta_1$ particles.

They found some interesting wiggles and bumps in the data, but they are not the strong, clear signals of new particles that some other theories predicted. It's as if they went looking for a specific type of bird in a forest, heard some rustling, but ultimately concluded that the bird they were looking for isn't nesting there.

In short: They built a detailed map of how these particles interact, found that the interaction is generally weak, and determined that this specific dance does not create the exotic particles some scientists were hoping to find.

Technical Summary

Technical Summary: Interaction and Correlation Functions for $\pi f_1(1285)$ and $\eta f_1(1285)$

Problem Statement
The paper investigates the interaction between pseudoscalar mesons ($\pi^0$ and $\eta$) and the axial-vector meson $f_1(1285)$. The study is motivated by the recent experimental observation of the exotic state $\eta_1(1855)$ by the BESIII Collaboration and the ongoing theoretical debate regarding the nature of exotic $1^{-+}$ states, specifically $\pi_1(1400)$, $\pi_1(1600)$, and $\eta_1(1855)$. Previous theoretical approaches have proposed these states as dynamically generated resonances arising from the interaction of pseudoscalar mesons with axial-vector mesons (e.g., $\pi f_1(1285)$ or $\eta f_1(1285)$). However, these approaches often treat the axial-vector mesons as fundamental matter fields using the Weinberg-Tomozawa interaction at the $O(1/f^2)$ level. In contrast, this work operates under the hypothesis that the $f_1(1285)$ is a molecular state composed of $K^*\bar{K} - \bar{K}^*K$. The authors aim to determine if the interaction of a pseudoscalar meson with this molecular cluster reproduces the claimed exotic resonances or yields different structural features.

Methodology
The authors employ a framework analogous to the interaction of a particle with a nucleus, treating the $f_1(1285)$ as a two-particle cluster ($K^*\bar{K}$) and the incident particle ($\pi^0$ or $\eta$) as an external projectile. The methodology proceeds as follows:

1. Molecular Structure: The $f_1(1285)$ is defined as a superposition of $K^*\bar{K}$ and $\bar{K}^*K$ components with isospin $I=0$.
2. Fixed Center Approximation (FCA): The interaction is modeled using the FCA to the Faddeev equations. In this approximation, the cluster ($f_1(1285)$) remains intact during the interaction; the external particle scatters off the constituents ($K^*$ and $\bar{K}$) without breaking the cluster.
3. Optical Potential and Unitarization: Unlike previous studies that stopped at the FCA level, this work constructs an optical potential from the FCA diagrams and solves the Lippmann-Schwinger equation. This step is crucial for restoring elastic unitarity at the particle-cluster threshold, a known deficiency in standard FCA applications.
4. Input Amplitudes: The scattering amplitudes for the elementary interactions ($\pi K^*$, $\pi \bar{K}$, $\eta K^*$, $\eta \bar{K}$) are derived from coupled-channel calculations involving the Bethe-Salpeter equation, ensuring unitarity in the coupled channels.
5. Observables: The authors calculate the total scattering amplitude ($T_{tot}$), the scattering length ($a$), the effective range ($r_0$), and the correlation functions $C(p)$ for various source radii ($R$).

Key Contributions

• Formalism Advancement: The paper applies a refined formalism that combines the Fixed Center Approximation with the solution of the Lippmann-Schwinger equation to ensure elastic unitarity. This addresses limitations in earlier FCA applications that failed to satisfy unitarity near thresholds.
• Molecular vs. Fundamental Approach: The study explicitly contrasts with approaches treating axial-vector mesons as fundamental fields. By treating $f_1(1285)$ as a $K^*\bar{K}$ molecule, the authors incorporate higher-order terms beyond the leading-order Weinberg-Tomozawa interaction.
• Systematic Calculation: The work provides the first calculation of scattering lengths, effective ranges, and correlation functions for the $\pi^0 f_1(1285)$ and $\eta f_1(1285)$ systems within this specific molecular framework.

Results

• Scattering Parameters:
  • For the $\pi^0 f_1(1285)$ system, the scattering length is found to be very small ($a \approx -0.06$ fm) with a large, complex effective range ($r_0 \approx -46.00 - i0.43$ fm).
  • For the $\eta f_1(1285)$ system, the scattering length is $a \approx (0.29 - i0.07)$ fm and the effective range is $r_0 \approx (0.42 + i0.45)$ fm.
• Scattering Amplitudes:
  • $\pi^0 f_1(1285)$: The amplitude exhibits a smooth behavior at threshold but displays a "curious structure" around 1500–1600 MeV. The real and imaginary parts interchange roles, resembling a resonance with a phase of $\pi/2$. However, the authors note it is unclear if this corresponds to the $\pi_1(1400)$ or $\pi_1(1600)$ states.
  • $\eta f_1(1285)$: No structure is observed around 1855 MeV, which would be associated with the $\eta_1(1855)$. Instead, a strong cusp is observed at the $\eta f_1(1285)$ threshold (1833 MeV). A structure is noted around 1450 MeV, but the authors express reluctance to trust results in this region due to the large energy gap from the threshold.
• Correlation Functions:
  • The $\pi^0 f_1(1285)$ correlation function remains close to unity, indicating a weak interaction. A "kink" is observed around $p_{cm} \approx 110$ MeV, corresponding to the opening of the $\pi K$ threshold in the sub-amplitude.
  • The $\eta f_1(1285)$ correlation function also stays close to unity, suggesting an interaction weaker than that of the $K f_1(1285)$ system. The shape is similar to the $p f_1(1285)$ correlation function measured by ALICE but with significantly weaker strength.

Significance and Claims
The authors claim that their approach, which treats the $f_1(1285)$ as a molecular state and rigorously enforces elastic unitarity, does not provide clear signals for the $\pi_1(1400)$, $\pi_1(1600)$, or $\eta_1(1855)$ resonances as dynamically generated states from the $\pi f_1$ or $\eta f_1$ interactions alone. While a structure appears in the $\pi^0 f_1$ amplitude near 1500–1600 MeV, the authors remain cautious about identifying it with the PDG-listed states.

The paper emphasizes that the theoretical explanation for the $\pi_1(1600)$ and $\eta_1(1855)$ remains elusive within this specific three-body molecular overlap framework. The authors suggest that extending the calculation to include coupled channels with other pseudoscalar-axial vector mesons (as done in other works) would require handling a three-body overlap of molecular components, a task deemed more difficult and likely to yield different results than two-body overlaps.

The study validates the reliability of the unitarized FCA method by noting its previous success in reproducing the $p f_1(1285)$ correlation function measured experimentally. The authors conclude that their predictions for $\pi^0 f_1(1285)$ and $\eta f_1(1285)$ correlation functions and scattering parameters await future experimental verification, which would further test the molecular nature of the $f_1(1285)$ and the dynamics of exotic meson interactions.