Production of $D_s\bar{D}_s$ and $D\bar{D}$ bound states in the $B$ decays within the Bethe-Salpeter framework

This work investigates the production of $D_s\bar{D}_s$- and $D\bar{D}$-bound states in $B$ decays using the Bethe-Salpeter framework and the one-boson-exchange model, finding that while $D\bar{D}$-bound states exist across all coupling sets, $D_s\bar{D}_s$-bound states are restricted to specific parameter ranges, with predicted branching ratios in the range of $10^{-6}$ to $10^{-4}$.

The Gist

Imagine the building blocks of the universe not as solid bricks, but as a bustling dance floor where particles constantly form pairs, break apart, and re-form. For decades, physicists believed these dancers (particles called hadrons) could only arise in two specific ways: either as a pair (one quark and one antiquark) or as a trio (three quarks). Yet in recent years, scientists have spotted some "exotic" dancers that seem to hold together in much looser, stranger formations.

This article is like a detective story investigating two specific types of these exotic dance pairs: one made of a "strange" charm pair ($D_s \bar{D}_s$) and another made of a "normal" charm pair ($D \bar{D}$). The authors want to know: Can these pairs stick together to form a stable "molecule," and if so, how often do we see them emerge during the decay of a heavier particle called a B-meson?

Here is the breakdown of their investigation, using simple analogies:

1. The Setup: The B-Meson Factory

Imagine a B-meson as a heavy, unstable parent particle. When it decays (dies), it doesn't simply vanish; it splits into smaller pieces. In this specific scenario, the B-meson splits into a K-meson and a pair of charm mesons.

• The Process: The B-meson breaks apart, and the two resulting charm mesons fly off. Normally, they would simply drift apart forever. But the authors ask: What if, for a tiny moment, they feel a strong magnetic pull that makes them stick together, forming a new, temporary "molecule" before separating again?

2. The Tool: The Bethe-Salpeter Framework

To determine whether these pairs can stick together, the authors use a mathematical tool called the Bethe-Salpeter (BS) framework.

• The Analogy: Imagine trying to predict whether two people holding hands will stay together or let go. You need to know how hard they pull (the force) and how fast they are spinning (their energy). The BS framework is like a super-advanced physics calculator that solves the "dance steps" of these particles. It calculates the wave function, which is essentially a map showing exactly how likely it is to find the two particles close together.

3. The Investigation: Two Different Pairs

The article examines two different pairs to see which is more willing to form a stable bond:

• Pair A: The $D \bar{D}$ pair (the "normal" charm)

  • The Result: This pair is very good at sticking together. The authors found that under almost all different "rules" (coupling sets) they tested, these two particles naturally formed a bound state.
  • The Metaphor: It is like two magnets perfectly aligned; they snap together easily. The math shows this bond is strong and stable within their model.

• Pair B: The $D_s \bar{D}_s$ pair (the "strange" charm)

  • The Result: This pair is much harder to hold together. They managed to form a bond only under very specific, restrictive conditions (using the strongest possible "glue" or coupling constant).
  • The Metaphor: These two are like magnets slightly misaligned. They can stick together, but only if you hold them very firmly in a very specific way. If the conditions aren't perfect, they drift apart.

4. The Prediction: How Often Does This Happen?

Once they knew the "dance steps" (the wave functions) for these pairs, the authors calculated the decay fraction.

• The Analogy: If you run a factory producing 100,000 B-mesons, how many of them would result in the birth of these exotic molecules?
• The Numbers:
  • For the $D \bar{D}$ molecule, they predict this happens about 1 to 400 times per million decays.
  • For the $D_s \bar{D}_s$ molecule, the prediction is somewhat higher, ranging from 10 to 2,000 times per million, depending on specific conditions.

5. Connection to the Real World: The X(3915) Puzzle

The article mentions a real mysterious particle called X(3915). Scientists are debating what this particle actually is.

• The Claim: If X(3915) is indeed a $D_s \bar{D}_s$ molecule, the authors calculate it should be produced in about 5.79 times per 10,000 B-decay cases.
• The Catch: This number is slightly higher than what current experiments have seen as an upper limit, but it falls within the same range as other theories. It suggests that while it is possible X(3915) is this molecule, it might be harder to produce than some other theories suggest.

Summary

In simple terms, this article says:
"We have simulated how heavy particles decay and attempt to form new, exotic 'molecules' using advanced mathematics. We found that the $D \bar{D}$ pair is a very natural candidate for forming a molecule, while the $D_s \bar{D}_s$ pair is a much more difficult connection requiring perfect conditions. We also calculated exactly how often we should expect these molecules to form in particle accelerators, helping experimental physicists know what to look for."

The authors conclude that B-meson decays are a great "factory" for hunting these exotic molecules, but the $D \bar{D}$ system looks like the more promising candidate for a stable, naturally occurring bound state.

Technical Summary

Technical Summary: Production of $D_s \bar{D}_s$- and $D \bar{D}$-bound states in $B$ decays within the framework of the Bethe-Salpeter equation

Problem Statement
The article addresses the nature of exotic hadrons and specifically investigates the possible existence of $D_s \bar{D}_s$ ($X_{s\bar{s}}$)- and $D \bar{D}$ ($X_{q\bar{q}}$)-bound states. Recent experimental observations by the LHCb collaboration of structures near the $D_s \bar{D}_s$ and $D \bar{D}$ thresholds (e.g., $X(3960)$ and $\chi_{c0}(3930)$) have intensified the debate regarding their interpretation as hadronic molecules. While theoretical studies employing various approaches (Lattice QCD, effective field theory, one-boson exchange) generally support the existence of a $D \bar{D}$-bound state, the status of the $D_s \bar{D}_s$-bound state remains unclear. Furthermore, although production rates of such molecules in $B$ and $B_s$ decays have been discussed, a rigorous calculation is required that incorporates the internal dynamics of these bound states via Bethe-Salpeter (BS) wave functions to provide precise predictions for experimental searches.

Methodology
The authors employ the Bethe-Salpeter (BS) framework to investigate the production of $X_{s\bar{s}}$ and $X_{q\bar{q}}$ in the decays $B^+ \to X_{s\bar{s}} K^+$ and $B^+ \to X_{q\bar{q}} K^+$. The theoretical formalism proceeds in three main steps:

1. Weak decay amplitudes: The process is modeled as a triangle diagram, where the $B^+$ meson first decays weakly into a pair of charmed mesons ($D_s^* \bar{D}^0$ or $D_s \bar{D}^*$). The effective weak Hamiltonian is constructed using the operator product expansion, and hadronic amplitudes are calculated within the factorization approximation using form factors from the covariant light-front quark model.
2. Bound state dynamics: The subsequent strong interaction scattering of the charmed meson pairs into the bound states is described using the One-Boson Exchange (OBE) model. The interaction kernels are derived from effective Lagrangian densities that respect heavy-quark and chiral symmetries and include the exchange of light vector mesons ($\rho, \omega, \phi$). The BS equations for S-wave systems are solved under the ladder and covariant instantaneous approximations.
3. Calculation of production rates: The normalized BS wave functions, which capture the internal dynamics of the bound states, are directly integrated into the decay amplitudes. The branching ratios are then calculated by integrating over phase space, varying the binding energy ($E_b$) and the coupling constants.

The study utilizes three sets of coupling constants (Sets A, B, and C), derived from QCD light-cone sum rules, to account for theoretical uncertainties. The cutoff parameter $\Lambda$ in the form factors is treated as a phenomenological input value, constrained by the requirement to find solutions for bound states.

Main Contributions and Results

• Existence of bound states:

  • $D \bar{D}$ system: Bound state solutions are found for all three coupling sets (A, B, and C). The required cutoff parameters ($\alpha$) are relatively small (in the range of $\sim 2$ to $6$), indicating that the effective attraction in this system is strong enough to naturally form shallow bound states within the OBE framework.
  • $D_s \bar{D}_s$ system: Bound state solutions are obtained only for Set C (the upper limit of the coupling constants). Even then, the solutions require large values of the cutoff parameter $\alpha$ ($6.64 \leq \alpha \leq 8.35$). The authors conclude that within this framework, the attraction generated solely by $\phi$ exchange is insufficient to naturally produce a shallow $D_s \bar{D}_s$ bound state without invoking additional mechanisms or large cutoff values.

• Branching ratios:

  • For the $D_s \bar{D}_s$-bound state ($B^+ \to X_{s\bar{s}} K^+$), the predicted branching ratios range from $1.09 \times 10^{-5}$ to $20.06 \times 10^{-4}$. Specifically, the predicted branching ratio, if $X(3915)$ is interpreted as a $D_s \bar{D}_s$ bound state with a mass of $3922.1$ MeV, is $5.79 \times 10^{-4}$. The authors note that this is slightly higher than the PDG upper limit ($2.8 \times 10^{-4}$) but comparable to other theoretical estimates based on the same molecular interpretation.
  • For the $D \bar{D}$-bound state ($B^+ \to X_{q\bar{q}} K^+$), branching ratios are predicted in the range of $1.56 \times 10^{-6}$ to $4.14 \times 10^{-4}$, depending on the binding energy and the coupling set. The results show that the branching ratio decreases with increasing coupling constants.

• Wave function analysis: The study demonstrates that as the binding energy increases, the peak value of the normalized BS wave functions in the low-momentum region decreases significantly. For the $D \bar{D}$ system, the wave functions at a fixed binding energy prove relatively insensitive to the specific choice of the coupling set.

Significance and Claims
The article claims that $B$-meson decays serve as a useful production environment for the search for heavy meson-molecule states. The primary significance of the work lies in its quantitative assessment of production rates using BS wave functions that include the essential internal dynamics of the bound states.

The authors modestly conclude that within their specific BS framework and the One-Boson Exchange model, the $D \bar{D}$ system is a more promising candidate for a bound state than the $D_s \bar{D}_s$ system, as the former supports bound states over a broader parameter range with more natural cutoff values. They suggest that further improvements in constraining heavy meson couplings and the inclusion of additional interaction mechanisms (such as exchanges beyond single vector mesons) would be valuable to enhance the quantitative reliability of these predictions. The work aims to provide guidelines for future experimental searches by establishing the dependence of production rates on binding energy and coupling strengths.