Mass and width of $T_{c\bar c}(4020)$ in the developed Bethe-Salpeter theory

This paper employs a developed Bethe-Salpeter theory within relativistic quantum field theory to model the exotic resonance $T_{c\bar c}(4020)$ as an unstable $D^{*}\bar{D}^{*}$ molecular state, successfully calculating its mass and width above the threshold to match experimental observations.

The Gist

Imagine the universe of subatomic particles as a bustling dance floor. Usually, dancers (particles) pair up to form stable couples (bound states) that stay together tightly. However, sometimes a new, exotic dancer appears on the floor called $T_{c\bar{c}}(4020)$. Scientists have been trying to figure out exactly what kind of dancer this is.

Here is a simple breakdown of what this paper does to solve the mystery:

1. The Mystery: A Dancer Standing on the Wrong Side of the Line

In the world of particle physics, there is a "threshold line" (a specific energy level).

• The Rule: If two dancers are holding hands in a stable, permanent bond (a "bound state"), they must be standing below this line.
• The Problem: The exotic dancer $T_{c\bar{c}}(4020)$ was observed standing above this line.
• The Conflict: Previous theories tried to explain this dancer as a stable couple made of two heavy mesons (let's call them $D^*$ and $\bar{D}^*$). But you can't have a stable couple standing above the line where they are supposed to fall apart. It's like trying to explain a rock sitting on top of a hill as a rock that is "stuck" to the ground. Physics says that's impossible; if it's above the line, it should be unstable and rolling down.

2. The New Approach: The "Unstable Molecular State"

Instead of forcing the dancer to be a stable couple, the authors of this paper say: "Let's treat this dancer as an unstable, fleeting molecular state."

Think of it like a soap bubble.

• A stable bound state is like a solid rock. It sits still and doesn't change.
• An unstable molecular state is like a soap bubble. It exists for a moment, has a shape, but is constantly trying to pop (decay).

The authors used a sophisticated mathematical tool called the Bethe-Salpeter theory (which is like a complex rulebook for how particles interact). However, the standard rulebook only works for stable rocks. So, they used a "Developed" version of this rulebook (DBST) that can handle wobbly, unstable bubbles.

3. The Experiment: Calculating the "Pop"

The researchers didn't just guess; they ran a detailed simulation with two main steps:

• Step 1: The "Prepared" State. They first calculated what the bubble would look like if it were perfectly stable (ignoring the fact that it wants to pop). This gave them a baseline mass (weight) of 4016 MeV.
• Step 2: The "Time Evolution." Then, they let the bubble breathe. They asked, "What happens when this bubble tries to decay into other particles?"
  • They looked at two ways the bubble could pop:
    1. Turning into a heavy particle called $h_c$ and a pion ($\pi$).
    2. Splitting back into the two original heavy mesons ($D^*\bar{D}^*$).

4. The Result: The Bubble Pops Up

When they added the energy effects of these "popping" (decay) channels into their math, something magical happened. The energy level of the bubble shifted upward.

• Before correction: The mass was 4016 MeV (below the line).
• After correction: The mass became 4019 MeV.

This new number is above the threshold line, which matches exactly what experimentalists see in the real world.

5. The Conclusion

The paper concludes that the exotic particle $T_{c\bar{c}}(4020)$ is indeed a molecular state made of two heavy mesons ($D^*$ and $\bar{D}^*$), but it is unstable.

• Why this matters: It solves the paradox. You can't explain it as a stable rock because it's above the line. But if you explain it as a wobbly, unstable bubble that is constantly trying to decay, the math works perfectly, and the numbers match the experiments.
• The Width: The paper also calculated how fast this "bubble" pops (its "width"). They found it pops mostly into the $h_c + \pi$ channel, while the channel where it splits back into the two original mesons is incredibly rare (almost non-existent).

In short: The authors took a particle that seemed to break the rules of stability, applied a new mathematical lens that accounts for instability, and showed that it fits the rules perfectly once you realize it's a fleeting, unstable molecular state rather than a permanent one.

Technical Summary

Technical Summary: Mass and Width of $T_{c\bar{c}}(4020)$ in the Developed Bethe-Salpeter Theory

Problem Statement
The exotic meson resonance $T_{c\bar{c}}(4020)$ (also known as $Z_c(4020)$) has been observed experimentally lying above the $D^*\bar{D}^*$ threshold. Standard theoretical approaches utilizing the homogeneous Bethe-Salpeter equation (BSE) treat resonances as bound states. However, a fundamental constraint of the standard homogeneous BSE is that bound-state masses must be real numbers lying below the corresponding threshold. Consequently, previous works that modeled $T_{c\bar{c}}(4020)$ as a $D^*\bar{D}^*$ bound state faced a theoretical inconsistency with experimental data, as they could not naturally account for a state existing above the threshold.

Methodology
To resolve this discrepancy, the authors apply the Developed Bethe-Salpeter Theory (DBST), a framework designed to describe unstable two-body systems within relativistic quantum field theory. The methodology proceeds in two distinct stages:

1. Preparation of the Bound State: The system is initially treated as a stable $D^*\bar{D}^*$ bound state with spin-parity $J^P = 1^+$. The authors solve the standard homogeneous BSE in the ladder approximation to obtain the "prepared state" mass ($M_0$) and the Bethe-Salpeter (BS) wave function.

   • Interaction Kernel: Unlike previous molecular models treating heavy mesons as point-like objects, this work treats the heavy mesons ($D^*$ and $\bar{D}^*$) as quark-antiquark bound states, effectively modeling the system as an unstable four-body state. The interaction kernel is derived from one-light-meson exchange ($\pi^0, \eta, \sigma, \rho^0, \omega$) between the light quarks and one-heavy-meson exchange ($J/\psi$) between the heavy quarks, based on an effective low-energy QCD Lagrangian.
   • Wave Function: The BS wave function for the $D^*\bar{D}^0$ system is constructed using five independent scalar functions derived from the general tensor structures for vector-vector bound states.

2. Time Evolution and Resonance Calculation: The prepared state is then subjected to time evolution governed by the total Hamiltonian, which includes decay interactions.

   • Decay Channels: Two decay channels are considered: $h_c(1P)\pi^+$ and $D^*\bar{D}^0$.
   • Scattering Matrix: Using a generalized Mandelstam approach, the authors calculate the T-matrix elements $T_{(c';b)a}(\epsilon')$ for arbitrary final-state energies $\epsilon'$. This involves evaluating matrix elements between the initial four-quark bound state and the final states, incorporating form factors that depend on the final-state energy.
   • Dispersion Relation: The real part of the self-energy correction, $D(\epsilon)$, is calculated via a dispersion relation involving the imaginary part $I(\epsilon)$, which is derived from the sum of squared T-matrix elements over all open and closed channels.
   • Physical Mass and Width: The physical mass $M$ and width $\Gamma$ are determined by locating the pole on the second Riemann sheet of the scattering amplitude, given approximately by $\epsilon_{pole} \approx M_0 + (2\pi)^3[D(M_0) - iI(M_0)]$.

Key Contributions

• Theoretical Framework: The paper applies DBST to the $T_{c\bar{c}}(4020)$ resonance, explicitly demonstrating how an unstable molecular state can naturally lie above the threshold of its constituent mesons, a feature impossible in the standard homogeneous BSE.
• Four-Body Structure: The authors treat the $D^*\bar{D}^*$ molecule not as a system of point-like mesons but as a four-quark system ($c\bar{c}u\bar{d}$), deriving the interaction kernel from quark-level exchanges (light and heavy meson exchanges).
• Energy Level Correction: The work quantifies the energy level correction ($\Delta M$) arising from coupling to decay channels, showing that this correction shifts the mass of the prepared bound state ($M_0$) to a physical mass ($M$) that exceeds the $D^*\bar{D}^*$ threshold.

Results

• Mass Calculation: The numerical solution yields a prepared state mass $M_0 \approx 4016.15$ MeV, which is below the $D^*\bar{D}^*$ threshold ($\approx 4017$ MeV). After calculating the energy level correction due to decay channels, the physical mass is found to be $M \approx 4019.01$ MeV.
• Comparison with Experiment: The calculated physical mass ($4019.01$ MeV) is in good agreement with the experimental value ($4024.1 \pm 1.9$ MeV) and, crucially, lies above the $D^*\bar{D}^*$ threshold, consistent with experimental observations.
• Decay Widths: The total decay width is calculated to be $\Gamma \approx 14.32$ MeV. The analysis indicates that the decay width for the $D^*\bar{D}^0$ channel ($\Gamma_2 \approx 0.2 \times 10^{-3}$ MeV) is negligible compared to the $h_c(1P)\pi^+$ channel ($\Gamma_1 \approx 14.32$ MeV).
• Sensitivity: Numerical tests varying the constituent quark masses ($m_{u,d}$) show that the calculated mass and width depend on these parameters but not sensitively, suggesting the robustness of the results within the model's uncertainties.

Significance
The paper concludes that it is more reasonable to regard the exotic resonance $T_{c\bar{c}}(4020)$ as an unstable meson-meson molecular state ($D^*\bar{D}^*$) rather than a stable bound state. The study provides a theoretical verification of the molecular hypothesis for $T_{c\bar{c}}(4020)$ with $J^P=1^+$ by demonstrating that the developed Bethe-Salpeter theory can successfully reconcile the existence of the resonance above the threshold with its molecular structure. The results align with experimental data, supporting the view that the resonance is an unstable system whose physical properties are significantly modified by its coupling to decay channels.