Three-body final state interactions in $B^+\to D\bar{D}K^+$ decays

This paper utilizes the dispersive Khuri-Treitan formalism combined with Heavy Quark Spin Symmetry to analyze $B^+\to D\bar{D}K^+$ decays, demonstrating that incorporating three-body final state interactions is essential for accurately describing experimental data and extracting the pole structures of the $\chi_{c0}(3930)$ and $\psi(3770)$ resonances, which are found to originate from bare input states.

The Gist

Imagine you are watching a high-speed billiard game, but instead of balls, you have subatomic particles. In this specific game, a heavy particle called a $B^+$ meson crashes and breaks apart into three smaller particles: a $D$ meson, an anti-$D$ meson, and a $K^+$ meson.

This paper is like a detective story trying to figure out what happened immediately after the crash, before the particles flew away.

The Problem: The "Crowded Room" Effect

Usually, when physicists study these crashes, they look at just two particles at a time (like watching the $D$ and anti-$D$ bounce off each other). They assume the third particle (the $K^+$) just stands in the background, watching.

But in this specific crash, the three particles are packed so tightly together in the "room" (the available energy space) that they can't ignore each other. The $K^+$ is constantly bumping into the $D$ and anti-$D$, changing how they bounce. This is called Three-Body Final State Interaction.

If you ignore the third particle, your math is like trying to predict the path of a billiard ball while pretending the other two balls on the table don't exist. You get the wrong answer.

The Detective's Tool: The "Khuri-Treiman" Map

To solve this, the authors used a sophisticated mathematical tool called the Khuri-Treiman formalism.

Think of this like a 3D topographic map of a mountain range.

• The Peaks: These represent "resonances" or temporary particles that form for a split second before disappearing. In this paper, the detectives are looking for two specific peaks: one called $\chi_{c0}(3930)$ and another called $\psi(3770)$.
• The Terrain: The map isn't flat; it's warped by the interactions between the three particles.

The authors used this map to "smooth out" the data from real-world experiments (from labs like LHCb, BABAR, and Belle) to see the true shape of the peaks, rather than the distorted shape caused by the crowded room.

The Ingredients: Heavy Quark "Legos"

To build their model, the scientists used a concept called Heavy Quark Spin Symmetry.

Imagine the heavy particles (the $D$ mesons) are like LEGO bricks with a specific magnetic orientation. Because they are so heavy, they behave in a very predictable, symmetrical way, almost like a spinning top that refuses to wobble. The authors built a "potential" (a set of rules for how these LEGOs stick together) based on this symmetry. They combined:

1. Contact Terms: Like Velcro, where the particles stick together just because they are close.
2. Bare States: Like pre-made LEGO structures (the "bare" particles) that the scientists put into the model to see if they match the data.

The Investigation: Fitting the Puzzle

The team took their mathematical model and tried to fit it to the actual data collected from millions of particle collisions. They tested two scenarios:

1. Scenario A: They included the influence of a mysterious particle called $X_0(2900)$ (a "ghost" in the $K$-meson channel) that might be messing with the data.
2. Scenario B: They ignored that ghost and only looked at the $D$ and anti-$D$ interaction.

The Result: Scenario A (including the three-body interactions and the ghost particle) fit the data perfectly. It successfully described the "invariant mass distributions" (which is just a fancy way of saying "how often we see these particles at specific speeds") for all three possible combinations of particles.

The Big Discovery: Are they Real or Fake?

The most exciting part of the paper is answering a deep question: Are the peaks we see ($\chi_{c0}$ and $\psi$) real, brand-new particles, or are they just illusions created by the way the particles bounce off each other?

To find out, the authors performed a "Pole Trajectory Analysis."

Imagine the peaks are balloons floating in a room. The authors slowly let the air out of the "bare state" balloons (the ones they put in the model at the start).

• If the peak disappears when the air is let out, it was just a fake illusion created by the interaction.
• If the peak stays and just moves slightly, it means the peak is a real, fundamental particle that was there all along.

The Verdict: The peaks for $\chi_{c0}(3930)$ and $\psi(3770)$ stayed put. They are real particles that originated from the "bare states" the scientists put in. They aren't just accidental bumps in the data; they are genuine, compact particles (like a tight bundle of quarks), though they do have a small "molecular" component (about 35%) where they act a bit like two particles loosely stuck together.

Summary in One Sentence

This paper used advanced math to account for the chaotic "crowded room" effect of three particles interacting, proving that two specific subatomic particles ($\chi_{c0}$ and $\psi$) are real, fundamental objects, not just mathematical mirages.

Technical Summary

Here is a detailed technical summary of the paper "Three-body final state interactions in $B^+ \to \bar{D}DK^+$ decays":

1. Problem Statement

The paper addresses the challenge of precisely extracting resonance parameters (mass, width, and quantum numbers) in the three-body decay processes $B^+ \to \bar{D}DK^+$ (specifically $\bar{D} = D^0, D^-, D_s^-$).

• Context: Recent experiments (LHCb, BABAR, Belle) have observed structures in the invariant mass spectra of these decays, notably $X(3960)$ (in $D_s^+D_s^-$) and $\chi_{c0/2}(3930)$ (in $D^+D^-$ and $D^0\bar{D}^0$).
• The Issue: The phase space for these decays is small, making Three-Body Final State Interactions (3B-FSI) significant. Traditional analyses often focus on two-body subsystems or ignore the coupled-channel effects of the third particle ($K^+$).
• Specific Challenge: Direct experimental extraction of $D\bar{D}$, $\bar{D}K$, and $DK$ phase shifts is currently infeasible. Furthermore, there is ambiguity regarding whether the observed structures ($X(3960)$ vs. $\chi_{c0}(3930)$) are the same state and whether they are dynamically generated or originate from bare charmonium states.

2. Methodology

The authors employ a rigorous theoretical framework combining Khuri-Treiman (KT) equations with Heavy Quark Spin Symmetry (HQSS).

• Khuri-Treiman Formalism:

  • The decay amplitude is decomposed into isobar amplitudes ($s, t, u$ channels) representing rescattering in the $D\bar{D}$, $\bar{D}K$, and $DK$ subsystems.
  • A dispersive representation is used to solve the integral equations, projecting singularities from the $t$ and $u$ channels (involving the $K^+$) onto the complex $s$-plane of the $D\bar{D}$ system.
  • The $t$-channel contribution includes potential resonances like $X_0(2900)$ or $X_1(2900)$ in the $D^-K^+$ channel, parametrized using Breit-Wigner or Flatté formulas.

• $D\bar{D}$ Interaction Model:

  • The $D\bar{D}$ scattering amplitude is constructed using HQSS.
  • The potential includes two components:
    1. Contact Terms: Derived from a leading-order Lagrangian involving low-energy constants ($F_A, F_A^\lambda$).
    2. Bare States: Explicit contributions from bare charmonium states (e.g., $\psi(3770)$, $\chi_{c0}$) coupled to the open-charm meson pairs.
  • The full scattering amplitude is obtained by solving the Lippmann-Schwinger Equation (LSE) with these potentials, incorporating form factors to ensure convergence.

• Fitting Strategy:

  • The model performs a simultaneous fit to experimental data from LHCb, BABAR, and Belle for three channels: $B^+ \to D^0\bar{D}^0K^+$, $B^+ \to D^+D^-K^+$, and $B^+ \to D_s^+D_s^-K^+$.
  • Two schemes are compared:
    • Scheme I: Includes 3B-FSI and the $X_0(2900)$ resonance in the $t$-channel.
    • Scheme II: Considers only 2B-FSI ($D\bar{D}$) without the $t$-channel resonance.

• Pole Analysis:

  • Poles are searched for on all 8 Riemann sheets (RS) of the complex $\sqrt{s}$ plane.
  • To distinguish between dynamically generated states and renormalized bare states, the authors perform a pole trajectory analysis on a uniformized complex $z$-plane (using Jacobi elliptic functions). They track pole movement as the coupling constants of the input bare states are varied from fitted values to zero.

3. Key Contributions

• Unified Description: The paper provides a unified description of three distinct $B^+$ decay channels ($D^0\bar{D}^0$, $D^+D^-$, $D_s^+D_s^-$) within a single framework, successfully describing the invariant mass distributions of all three two-body subsystems.
• Resolution of State Identity: The analysis confirms that the structures observed as $X(3960)$ in $D_s^+D_s^-$ and $\chi_{c0}(3930)$ in $D\bar{D}$ are manifestations of the same pole ($\chi_{c0}(3930)$), with mass and width variations arising from channel-dependent threshold effects.
• Nature of Resonances: Through pole trajectory analysis, the authors demonstrate that the observed poles ($\chi_{c0}(3930)$ and $\psi(3770)$) stem from the input bare charmonium states rather than being purely dynamically generated molecular states.
• Compositeness Calculation: Using the Weinberg criterion and effective range expansion (corrected for coupled-channel effects), the authors calculate the compositeness of $\chi_{c0}(3930)$ to be approximately 34%, indicating it is predominantly a compact charmonium state with a significant molecular component.

4. Results

• Resonance Parameters (Scheme I):
  • $\chi_{c0}(3930)$: Pole position at $3.913 - 0.016i$ GeV.
  • $\psi(3770)$: Pole position at $3.761 - 0.006i$ GeV.
  • $X_0(2900)$: Included as a $t$-channel resonance with mass $\approx 2.893$ GeV, though its impact on the $D\bar{D}$ spectrum is found to be small.
• Fit Quality:
  • Scheme I (with 3B-FSI) yields a $\chi^2/\text{d.o.f} = 2.488$.
  • Scheme II (without 3B-FSI) yields $\chi^2/\text{d.o.f} = 1.758$ but fails to describe the full dataset (specifically the $D\bar{K}$ and $DK$ spectra) consistently.
• Stability: The results are shown to be stable against variations in the cutoff parameter $\Lambda$ (tested from 1.432 to 2.068 GeV).
• Pole Trajectories: As the bare state couplings are reduced to zero, the poles corresponding to $\chi_{c0}(3930)$ and $\psi(3770)$ move back to the positions of the input bare states, confirming their origin. A third pole at $4.106 - 0.092i$ is identified as a dynamically generated state but is far from the physical region.

5. Significance

• Methodological Advancement: This work demonstrates the necessity of including three-body final state interactions in the analysis of heavy-flavor decays with small phase spaces. It validates the use of the dispersive Khuri-Treiman formalism for systems involving heavy mesons where direct phase shift data is unavailable.
• Clarification of Exotic Candidates: By rigorously analyzing the pole structure, the paper clarifies the nature of the $X(3960)$ and $\chi_{c0}(3930)$, resolving debates about whether they are distinct states or the same resonance seen in different channels.
• Insight into QCD: The finding that these states are primarily renormalized bare charmonia (with ~34% molecular component) provides crucial constraints for QCD models of charmonium and tetraquark states, suggesting that not all near-threshold structures are purely molecular.
• Future Applications: The framework established here can be applied to other three-body heavy-flavor decays to extract precise resonance parameters and study the interplay between bare states and continuum dynamics.