Analysis of the form factors of $B_c\rightarrow D^{(*)}$, $D_{s}^{(*)}$ and their nonleptonic decays

This paper calculates the form factors for $B_c \to D^{(*)}$ and $B_c \to D_s^{(*)}$ transitions using three-point QCD sum rules with various condensate contributions, and subsequently predicts the decay widths and branching ratios for several two-body nonleptonic decay processes to provide insights into heavy-quark dynamics.

The Gist

Imagine the universe as a giant, bustling construction site. In this site, there are tiny, heavy bricks called quarks. Usually, these bricks pair up to build stable structures called mesons.

This paper is about a very special, rare building called the $B_c$ meson. Think of it as a unique house built from two very heavy, different types of bricks: a "bottom" brick and a "charm" brick. Because both bricks are heavy, this house is heavy, and because they are different, they can't just sit still; they eventually have to break apart or change.

The scientists in this paper wanted to understand exactly how this house falls apart and turns into other, lighter houses. Specifically, they looked at the process where the $B_c$ house transforms into a "charmonium" house (like a $J/\psi$ or $\eta_c$) plus a "D" or "D-s" house.

Here is a breakdown of their work using simple analogies:

1. The Challenge: The Invisible Blueprint

In the world of tiny particles, you can't just take a ruler and measure how fast a house breaks apart. The rules are governed by Quantum Chromodynamics (QCD), which is like the physics of how these bricks stick together. It's incredibly complex and "non-perturbative," meaning you can't just use simple math to guess the outcome; you have to account for the messy, sticky glue holding everything together.

To predict how fast these decays happen, scientists need to know the "Form Factors."

• The Analogy: Imagine trying to predict how much water flows through a pipe. The "Form Factor" is like the width and shape of the pipe. If you don't know the pipe's shape, you can't calculate the flow. In this paper, the "pipe" is the transition from the heavy $B_c$ meson to the lighter particles. The scientists needed to calculate the exact shape of this "pipe" at every possible speed.

2. The Method: The Three-Point Sum Rule

The authors used a powerful tool called Three-Point QCD Sum Rules.

• The Analogy: Imagine you are trying to figure out the weight of a hidden object inside a sealed box. You can't open it, but you can shake the box and listen to the sound it makes (the "phenomenological" side) and also calculate what the sound should be based on the physics of the materials inside (the "QCD" side).
• By matching the sound you hear with the sound you calculate, you can deduce the hidden object's properties.
• In this paper, they matched the "sound" of the particle decay with the complex math of quarks and gluons. They didn't just look at the basic math; they included "condensates," which are like accounting for the background noise or the "vacuum energy" of empty space that affects how the bricks interact.

3. The Results: Mapping the Pipe

The team calculated these "Form Factors" (the pipe shapes) for several different transitions:

• $B_c$ turning into $D$ or $D^*$ (and their strange cousins, $D_s$ and $D_s^*$).
• They calculated these values at different energy levels (momentum transfers).
• The Fitting: Since they calculated the values for a specific range of energies, they used a mathematical "stretching" technique (called the z-series parameterization) to smoothly connect the dots. This allowed them to predict the values even for energies they didn't calculate directly, creating a complete map of how the transition works.

Key Finding: They found that their calculated "pipe widths" (form factors) were generally smaller than some other scientists' predictions. This is likely because they accounted for a specific type of "Coulomb-like" correction (a specific way heavy quarks attract each other) that others might have missed or treated differently.

4. The Application: Predicting the Decay Rates

Once they had the "pipe shapes" (form factors), they could finally answer the big question: How often does this happen?

They used these numbers to predict the Decay Widths (how fast the house falls apart) and Branching Ratios (what percentage of the time it turns into one specific type of house versus another).

• They predicted the rates for 8 specific decay channels (e.g., $B_c \to J/\psi D_s$).
• The Comparison: They compared their predictions to real-world data from the LHCb experiment (a giant particle detector at CERN).
  • The LHCb has already seen the $B_c$ turning into $J/\psi$ plus a pion.
  • The authors calculated the ratio of how often $B_c$ turns into $J/\psi$ + $D_s$ compared to $J/\psi$ + pion.
  • The Result: Their prediction ($3.3$) was very close to the experimental measurement ($2.90$). This suggests their "blueprint" is accurate.

Summary

In short, this paper is a detailed engineering report on a rare, heavy particle.

1. They built a mathematical model to understand the invisible "glue" holding the particle together.
2. They calculated the "shape" of the transition (form factors) using a method that accounts for the messy quantum vacuum.
3. They used these shapes to predict how often this particle breaks apart into specific lighter particles.
4. Their predictions match existing experimental data, giving physicists more confidence in understanding how heavy quarks behave and providing a roadmap for future experiments to look for these specific decay patterns.

The paper concludes that these results are useful for future experiments to verify and study the dynamics of heavy quarks, essentially helping us understand the fundamental rules of how matter is built and breaks down.

Technical Summary

Technical Summary: Analysis of the Form Factors $B_c \to D^{(*)}, D^{(*)}_s$ and Relevant Nonleptonic Decays

Problem Statement
The $B_c$ meson, a unique bound state of two heavy quarks ($b$ and $c$) with different flavors, serves as an excellent laboratory for studying heavy-quark dynamics. Unlike other $B$ mesons, both heavy quarks in the $B_c$ can decay individually, leading to a richer spectrum of decay channels. While experimental progress, particularly by the LHCb collaboration, has confirmed several nonleptonic decay modes (e.g., $B_c \to J/\psi \pi$, $B_c \to J/\psi D_s$), a comprehensive theoretical understanding of the transition form factors and decay rates for processes involving charmonium and open-charm mesons ($D^{(*)}$ and $D^{(*)}_s$) remains necessary. Specifically, the nonleptonic decays $B_c \to \eta_c D^{(*)}$, $B_c \to J/\psi D^{(*)}$, and their strange counterparts involve $b \to c\bar{c}d$ and $b \to c\bar{c}s$ transitions. These processes are governed by weak form factors that encapsulate complex non-perturbative QCD effects, which must be calculated to predict decay widths and branching ratios accurately.

Methodology
The authors employ the framework of three-point QCD Sum Rules (QCDSR) to systematically calculate the form factors for the transitions $B_c \to D$, $B_c \to D^*$, $B_c \to D_s$, and $B_c \to D^*_s$.

1. Correlation Function Construction: A three-point correlation function is constructed involving interpolating currents for the initial $B_c$ meson, the final state meson ($D^{(*)}$ or $D^{(*)}_s$), and the transition current derived from the low-energy effective Hamiltonian.
2. Phenomenological Side: The correlation function is expressed in terms of hadronic degrees of freedom, isolating the ground state contributions and parameterizing the transition matrix elements using various form factors ($f_+, f_0, f_T, V, A_0, A_1, A_2$).
3. QCD Side (OPE): The correlation function is calculated in terms of quark and gluon fields using the Operator Product Expansion (OPE). The calculation includes contributions from:
   • Perturbative terms (calculated via Cutkosky's rules).
   • Non-perturbative vacuum condensates: quark condensate $\langle \bar{q}q \rangle$, quark-gluon mixed condensate $\langle \bar{q}g_s\sigma G q \rangle$, gluon condensate $\langle g_s^2 G^2 \rangle$, triple-gluon condensate $\langle f^3 G^3 \rangle$, and the mixed term $\langle \bar{q}q \rangle \langle g_s^2 G^2 \rangle$.
   • Note: The authors find that contributions from the quark condensate and quark-gluon mixed condensate vanish after the double Borel transformation for these specific transitions.
4. Sum Rule Extraction: By equating the phenomenological and QCD sides and applying a double Borel transformation to suppress excited and continuum states, sum rules for the form factors are derived.
5. Numerical Analysis:
   • Input parameters (meson masses, quark masses, decay constants, condensate values) are taken from the Particle Data Group (PDG) and previous QCDSR studies.
   • Borel windows are determined based on pole dominance ($\ge 40\%$) and OPE convergence.
   • Form factors calculated in the space-like region ($1 \text{ GeV}^2 \le Q^2 \le 9 \text{ GeV}^2$) are fitted using the $z$-series parameterization approach to extrapolate results to the time-like region ($Q^2 < 0$), specifically obtaining values at $Q^2=0$.
6. Decay Width Calculation: Using the factorization approach and the obtained form factors, the decay widths and branching ratios for eight two-body nonleptonic channels are calculated.

Key Contributions and Results

• Form Factor Calculations: The paper provides numerical predictions for all relevant form factors ($f_S, f_+, f_0, f_T, V, A_0, A_1, A_2$) at $Q^2=0$ for the four transition channels.
  • The results for $f_+$ and $f_0$ in $B_c \to D$ are found to be smaller than many other theoretical predictions (e.g., pQCD, LCSR), though they show rough compatibility with the BSW relativistic quark model. The authors attribute this discrepancy partly to the lack of Coulomb-like $\alpha_s/v$ corrections for the heavy quarkonium $B_c$ in the current QCDSR framework.
  • For $B_c \to D^*$ and $B_c \to D^*_s$, the results are consistent with several other models (BSW, CLFQM), though differences exist with Ref. [15].
  • A characteristic trend is observed where the scalar form factor $f_S$ is significantly larger than other form factors for pseudoscalar transitions, and the vector form factor $V$ is larger than the axial-vector form factors $A_{0,1,2}$ for vector transitions.
• Nonleptonic Decay Predictions: The authors predict the decay widths and branching ratios for:
  • $B_c \to \eta_c D, \eta_c D^*$
  • $B_c \to J/\psi D, J/\psi D^*$
  • $B_c \to \eta_c D_s, \eta_c D^*_s$
  • $B_c \to J/\psi D_s, J/\psi D^*_s$
• Comparison with Experiment:
  • The predicted branching ratios are generally smaller than other theoretical predictions, primarily due to the smaller values of the calculated form factors.
  • The ratios $\mathcal{B}(B_c \to J/\psi D_s) / \mathcal{B}(B_c \to J/\psi \pi)$ and $\mathcal{B}(B_c \to J/\psi D^*_s) / \mathcal{B}(B_c \to J/\psi D_s)$ are calculated as $3.3^{+2.5}_{-1.4}$ and $5.4^{+3.9}_{-2.3}$, respectively. These values are noted to be roughly consistent with experimental measurements of $2.90 \pm 0.57 \pm 0.24$ and $2.37 \pm 0.56 \pm 0.10$.
  • The branching ratios for most channels fall within the range $10^{-5} \sim 10^{-3}$, suggesting they are within the detection capabilities of the LHCb experiment.

Significance
The authors state that the results regarding the form factors and decay properties of the $B_c$ meson provide useful information for the study of heavy-quark dynamics. By systematically analyzing these transitions using three-point QCDSR and including higher-order condensate contributions, the work offers theoretical inputs that can be verified by future experimental data. The predictions for branching ratios, which lie within the sensitivity of current experiments, serve as a benchmark for testing the factorization approach and understanding the non-perturbative aspects of $B_c$ decays.