In search for signals of the $D\bar{D}$ bound state $X(3700)$ from study of the $B^+ \to D^+ D^- K^+$, $B^0 \to D^+ D^- K^0$ and $\Lambda_b \to D^+ D^- \Lambda$ reactions

This theoretical study proposes that the $B^+ \to D^+ D^- K^+$ reaction offers a significantly more promising signal for detecting the predicted $D\bar{D}$ bound state $X(3700)$ compared to $\Lambda_b$ decays, urging experimental verification via upcoming LHCb upgrades to confirm the state's existence.

The Gist

Imagine the subatomic world as a bustling, chaotic dance floor where particles constantly bump into each other, pair up, and sometimes stick together to form new, temporary couples.

This paper is a theoretical investigation into a very specific, elusive couple: a $D$ meson and an anti-$D$ meson (let's call them "D-couples"). Scientists have long suspected that under the right conditions, these two particles can stick together so tightly they form a bound state—like two dancers who refuse to let go, creating a new, stable entity. The authors call this hypothetical new partner $X(3700)$.

Here is a simple breakdown of what the researchers did and what they found:

1. The Setup: Three Different Dance Halls

To see if this $X(3700)$ couple exists, the scientists looked at three different "dance halls" (particle reactions) where these D-couples are created:

• Hall A: A $B^+$ particle decaying into a $D^+ D^- K^+$ trio.
• Hall B: A $B^0$ particle decaying into a $D^+ D^- K^0$ trio.
• Hall C: A heavy $\Lambda_b$ particle decaying into a $D^+ D^- \Lambda$ trio.

In all these halls, the $D$ and anti-$D$ particles are born close together. The researchers wanted to see if, as they danced apart, they would show signs of having been a tight-knit couple (the $X(3700)$) before separating.

2. The Problem: A Loud Musician Drowns Out the Signal

There is a major obstacle. In all three halls, there is a very loud, famous musician playing right next to the dance floor: a particle called $\psi(3770)$.

• Think of the $\psi(3770)$ as a massive, booming bass drum. It creates a huge spike in the data right near where the $D$-couples are born.
• The signal for the quiet, shy $X(3700)$ couple is right next to this bass drum. Because the drum is so loud, it's very hard to hear the whisper of the $X(3700)$ in the current data.

3. The Insight: Comparing the Halls

The researchers realized that while the "loud music" (the $\psi(3770)$) is present in all three halls, the background noise (the way the particles interact before forming the final state) is different in each hall.

• In Hall A ($B^+$ decay), the background conditions are such that the "whisper" of the $X(3700)$ gets amplified. It's like being in a room with perfect acoustics where a quiet voice carries far.
• In Hall C ($\Lambda_b$ decay), the background conditions are different. The whisper is much quieter, almost drowned out by the bass drum.

4. The Prediction: A 13-to-1 Ratio

The authors performed a clever calculation. They asked: "If we turn down the volume of the loud bass drum ($\psi(3770)$) so that it sounds the same in both Hall A and Hall C, what happens to the quiet whisper?"

Their answer is striking:

• In Hall A, the whisper (the signal for the $X(3700)$ bound state) becomes 13 times louder than in Hall C.
• Specifically, in the tiny energy range just above where the $D$-couples are born (between 3739 and 3750 MeV), the $B^+$ reaction should show a massive "bump" or enhancement that the $\Lambda_b$ reaction simply does not have.

5. The Call to Action

The current data from the LHCb experiment (a giant particle detector) isn't precise enough to see this difference yet. There is only one data point in that specific quiet zone, and the error bars are too big to tell the difference between a whisper and silence.

The Conclusion:
The paper doesn't claim to have found the $X(3700)$ yet. Instead, it acts as a blueprint for a future experiment. The authors are calling on the LHCb team to upgrade their equipment and take much more precise measurements in that specific energy range.

If they measure the $B^+$ and $\Lambda_b$ reactions again with better precision and find that the $B^+$ reaction is indeed 13 times stronger near the threshold, it would be the "smoking gun" proving that the $D\bar{D}$ bound state ($X(3700)$) really exists. It's like finally hearing the quiet dancer clearly because we finally turned down the bass drum and listened in the right room.

Technical Summary

Technical Summary: Search for $D\bar{D}$ Bound State Signals in $B$ and $\Lambda_b$ Decays

Problem Statement
The existence of a $D\bar{D}$ bound state in the isospin $I=0$ channel, predicted by chiral dynamics and various phenomenological models (including lattice QCD), remains experimentally elusive. While a corresponding $D_s\bar{D}_s$ bound state (identified as $X_{c0}(3930)$) has been observed, the $D\bar{D}$ analog (referred to here as $X(3700)$) has not been definitively confirmed. Previous proposals to observe this state include radiative decays of $\psi(3770)$, $\gamma\gamma$ fusion, and various $B$ and $\Lambda_b$ decays. However, experimental data remains inconclusive, often obscured by the strong $\psi(3770)$ resonance peak located just above the $D\bar{D}$ threshold. This paper addresses the challenge of isolating the signal of the $X(3700)$ bound state by performing a comparative theoretical study of three specific reactions: $B^+ \to D^+ D^- K^+$, $B^0 \to D^+ D^- K^0$, and $\Lambda_b \to D^+ D^- \Lambda$.

Methodology
The authors employ a theoretical framework based on the local hidden gauge approach extended to the charm sector. The study proceeds through the following steps:

1. Quark-Level Mechanisms: The production mechanisms for the three reactions are analyzed at the quark level. The authors distinguish between external emission (involving hadronization of $c\bar{u}$, $c\bar{d}$, or $s\bar{c}$ pairs) and internal emission (involving hadronization of the $c\bar{c}$ pair).

   • For $\Lambda_b \to D^+ D^- \Lambda$, the process involves the hadronization of a $c\bar{c}$ pair (internal emission, color-suppressed) and a $c\bar{u}/c\bar{d}$ pair with an $\bar{s}s$ pair (external emission).
   • For $B^+ \to D^+ D^- K^+$ and $B^0 \to D^+ D^- K^0$, similar topologies are considered, with specific weights assigned to external and internal emission mechanisms based on color suppression factors ($\beta \approx 1/3$).

2. Final State Interaction (FSI): The core of the analysis focuses on the final state interaction of the charmed meson pairs ($D\bar{D}$, $D_s\bar{D}_s$, and $\eta\eta$). The authors utilize the Bethe-Salpeter equation ($T = [1 - VG]^{-1}V$) to generate resonant states dynamically. This interaction generates two key states:

   • $X_{c0}(3930)$: Coupling strongly to $D_s\bar{D}_s$.
   • $X(3700)$: Coupling strongly to $D\bar{D}$, identified as the $D\bar{D}$ bound state.

3. Amplitude Construction: The total transition amplitude ($t_{tot}$) for each reaction is constructed by summing the tree-level contributions (direct production) and the rescattering terms (FSI). The $\psi(3770)$ contribution is explicitly included as a $P$-wave excitation. The differential decay widths ($d\Gamma/dM_{12}$) are calculated by integrating over the relevant phase space.

4. Data Fitting: The theoretical model is fitted to existing experimental data from LHCb for the $D^+ D^-$ invariant mass distributions in the three reactions. The fit is restricted to the region from threshold up to $\approx 3800$ MeV to avoid complications from the $\chi_{c2}(3930)$ and to focus on the threshold behavior where the $X(3700)$ signal is expected. Parameters such as normalization constants ($A, A', A''$) and the $\psi(3770)$ coupling ($\gamma$) are determined via a combined fit.

Key Contributions and Results

• Comparative Analysis of Production Mechanisms: The study reveals that the final state interaction responsible for producing the resonances is significantly more important in the $B^+ \to D^+ D^- K^+$ reaction compared to the $\Lambda_b \to D^+ D^- \Lambda$ reaction. This difference arises from the relative weights of the tree-level (non-resonant) terms versus the rescattering terms in the respective amplitudes.
• Threshold Enhancement Prediction: By normalizing the $D^+ D^-$ mass distributions of the $B^+$ and $\Lambda_b$ reactions to the same value at the peak of the $\psi(3770)$, the authors demonstrate a distinct difference in the region just above the $D^+ D^-$ threshold.
  • The $B^+ \to D^+ D^- K^+$ distribution shows a clear enhancement close to the threshold.
  • The $\Lambda_b \to D^+ D^- \Lambda$ distribution does not show this enhancement.
• Quantitative Ratio: In the mass range $M_{D^+D^-} \in [3739, 3750]$ MeV (approximately 10 MeV above threshold), the normalized mass distribution for the $B^+$ reaction is found to be approximately 13 times larger than that of the $\Lambda_b$ reaction. This factor is attributed to the presence of the $X(3700)$ bound state below the threshold, which enhances the production cross-section more effectively in the $B^+$ channel due to the specific interference of tree and rescattering amplitudes.

Significance and Claims
The paper claims that while current experimental data lacks the precision to definitively resolve the $X(3700)$ signal due to the overwhelming $\psi(3770)$ peak and limited statistics, the theoretical framework provides a clear strategy for its discovery.

• Experimental Proposal: The authors call for precise measurements of the differential decay widths in the invariant mass range $[3739, 3750]$ MeV for both $B^+ \to D^+ D^- K^+$ and $\Lambda_b \to D^+ D^- \Lambda$ reactions.
• Feasibility: They assert that the upcoming upgrades of the LHCb facility will provide the necessary statistics and resolution to observe this predicted relative enhancement.
• Implication: Observing a ratio of approximately 13 between the normalized distributions in the specified mass range would provide "great support" for the existence of the $D\bar{D}$ bound state. The work serves as a motivation for these specific measurements to resolve the long-standing question of the $D\bar{D}$ bound state's existence.