Systematic analysis of the form factors of $B_{c}$ to $P$-wave charmonia and corresponding weak decays

This paper employs three-point QCD sum rules to calculate the form factors for $B_c$ transitions to $P$-wave charmonia, which are then used within the naive factorization approach to predict the decay widths and branching ratios of corresponding semileptonic and nonleptonic weak decays.

The Gist

Imagine the subatomic world as a bustling, high-energy construction site. In this paper, a team of physicists acts as structural engineers trying to understand how a very specific, rare building block called the $B_c$ meson transforms into other structures.

Here is a simple breakdown of what they did, using everyday analogies:

1. The Star of the Show: The $B_c$ Meson

Most particles are like couples made of two identical twins (two up-quarks) or two different people who are both "heavy" (like two heavyweights). The $B_c$ meson is unique because it's a "mixed marriage" of two different heavy flavors: a bottom quark and a charm quark.

• Why it matters: Because it's so heavy and unique, it doesn't just fall apart easily. It has to decay (break down) via the "weak force," which is like a slow, deliberate demolition process. This makes it a perfect laboratory to test the rules of the universe (the Standard Model).

2. The Mission: Predicting the "Shape" of the Transformation

The researchers wanted to predict exactly how the $B_c$ meson changes into a specific family of particles called P-wave charmonia (think of these as the "children" or "offspring" made of a charm quark and an anti-charm quark).

To do this, they needed to calculate something called Form Factors.

• The Analogy: Imagine you are trying to predict how a rubber ball deforms when you squeeze it. The "Form Factor" is the mathematical description of that shape-shifting. It tells you how "stiff" or "flexible" the particle is during the transition. Without knowing this shape, you can't predict how fast the ball will roll or how far it will go.

3. The Tool: The "Three-Point QCD Sum Rules"

Calculating these shapes is incredibly hard because the forces holding these particles together (Quantum Chromodynamics, or QCD) are messy and chaotic at low energies. You can't just use simple math; you need a special tool.

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

• The Analogy: Think of this like a blindfolded detective. The detective can't see the crime scene directly (the particle interaction). Instead, they look at the "echoes" or "shadows" left behind by the event.
  • They build a mathematical "shadow" based on what we know about quarks (the ingredients).
  • They compare it to a "shadow" based on what we know about the final particles (the result).
  • Where the shadows match, they find the truth about the form factors.

4. The "Coulomb-like" Correction: Adding the Fine Details

In their calculations, they included a special tweak called the Coulomb-like correction.

• The Analogy: Imagine you are baking a cake. You have the main recipe (flour, sugar, eggs). But if you ignore the fact that the oven is slightly hotter than usual, your cake might burn.
  • The "main recipe" is the standard calculation.
  • The "Coulomb-like correction" is realizing that the heavy quarks inside the particle are moving so fast and are so close together that they attract each other strongly, like magnets.
  • The Surprise: When they added this "oven temperature" correction, the predicted "shape" (form factor) became three times larger than before! This is a huge difference, like realizing your cake is actually a giant wedding cake, not a cupcake.

5. The Results: What Happens When the $B_c$ Breaks Down?

Using these new, more accurate "shape" calculations, the team predicted how often the $B_c$ meson would decay into different particles.

• Semileptonic Decays: The $B_c$ turns into a charmonium particle plus a lepton (like an electron or a tau) and a neutrino.
  • Prediction: They found that decays into the $h_c$ and $\chi_{c2}$ particles are much more common (have larger "branching ratios") than previously thought, especially when you include that "Coulomb" correction.
• Nonleptonic Decays: The $B_c$ turns into a charmonium particle plus a light meson (like a pion or a kaon).
  • Prediction: They calculated the odds of these specific combinations happening.

6. Why This Matters (The "So What?")

The authors compared their results with other scientists' predictions.

• The Conflict: Some other models predicted different numbers. The authors found that their results (especially with the Coulomb correction) were significantly higher.
• The Takeaway: This isn't a bad thing; it's exciting! It means there is a lot of work to be done.
  • The "Recipe" Debate: Different teams are using slightly different "recipes" (different assumptions about mass, energy scales, or how to handle the math).
  • The Future: The Large Hadron Collider (LHC) is currently churning out billions of these $B_c$ mesons. Soon, experimentalists will be able to measure exactly how often these decays happen.
  • The Goal: When the real-world data arrives, it will tell us which "recipe" is correct. If the data matches the authors' "Coulomb-corrected" predictions, it proves that the strong attraction between the heavy quarks is a critical piece of the puzzle we've been missing.

Summary

In short, this paper is a theoretical blueprint. The authors built a more detailed map of how a rare particle ($B_c$) transforms into its "children" (P-wave charmonia). They discovered that if you account for the intense "magnetic" pull between the heavy ingredients, the transformation happens much more vigorously than we thought. Now, it's up to the experimentalists at the LHC to check if their map is accurate.

Technical Summary

1. Problem Statement

The $B_c$ meson is unique as the only known quarkonium composed of two different heavy flavors ($b$ and $\bar{c}$). While its properties and decays to $S$-wave charmonia (like $J/\psi$ and $\eta_c$) have been studied extensively, a systematic theoretical analysis of its transitions to P-wave charmonia ($\chi_{cJ}$ with $J=0,1,2$ and $h_c$) remains incomplete.

The primary challenges in this domain are:

• Non-perturbative QCD: Calculating hadronic transition form factors requires handling non-perturbative strong interactions, which standard perturbation theory cannot address at low energy scales.
• Lack of Experimental Data: Experimental measurements for $B_c \to$ P-wave charmonia decays are scarce, making theoretical predictions crucial for guiding future experiments at the LHC.
• Theoretical Discrepancies: Existing theoretical models (e.g., Light-Front Quark Model, NRQCD, previous Sum Rules) often yield inconsistent results for these specific transitions, particularly regarding the magnitude of form factors and the impact of higher-order corrections.

2. Methodology

The authors employ the Three-Point QCD Sum Rules (QCDSR) framework, a non-perturbative method that bridges the gap between quark-gluon degrees of freedom and hadronic observables.

• Correlation Functions: They construct three-point correlation functions involving interpolating currents for the $B_c$ meson, the final P-wave charmonia ($\chi_{c0}, \chi_{c1}, \chi_{c2}, h_c$), and the transition currents (vector, axial-vector, and tensor).
• Phenomenological vs. QCD Sides:
  • Phenomenological Side: The correlation function is saturated by a complete set of hadronic states, isolating the ground state contributions to extract form factors.
  • QCD Side: The Operator Product Expansion (OPE) is performed. The calculation includes:
    • Perturbative contributions: Calculated using Cutkosky's cutting rules.
    • Non-perturbative contributions: Specifically the gluon condensate $\langle g_s^2 GG \rangle$. Higher-dimensional condensates are neglected as they are suppressed.
    • Coulomb-like Correction: A crucial addition in this work is the inclusion of the Coulomb-like $\alpha_s/v$ correction (where $v$ is the relative velocity of heavy quarks). This accounts for the non-relativistic nature of the heavy quark system and resums leading-order Coulomb interactions.
• Borel Transformation: Double Borel transformations are applied to suppress contributions from higher resonances and continuum states, ensuring the stability of the sum rules.
• Extrapolation: Since QCDSR calculations are valid in the space-like region ($Q^2 > 0$), the authors use the $z$-series parameterization (Bourrely-Caprini-Lellouch approach) to extrapolate the form factors into the time-like region ($Q^2 < 0$) required for physical decay processes.
• Decay Analysis:
  • Semileptonic Decays: Calculated using the derived form factors and the effective weak Hamiltonian.
  • Nonleptonic Decays: Analyzed using the Naive Factorization Approach (NFA) to estimate branching ratios for processes like $B_c^- \to \chi_{cJ} \pi^-, K^-, \rho^-, K^{*-}$ and $h_c \pi^-, K^-, \rho^-, K^{*-}$.

3. Key Contributions

• Systematic Form Factor Calculation: This is the first comprehensive QCDSR study covering vector, axial-vector, and tensor form factors for $B_c$ transitions to all P-wave charmonia states ($\chi_{c0}, \chi_{c1}, \chi_{c2}, h_c$).
• Inclusion of Coulomb-like Corrections: The authors explicitly quantify the impact of the $\alpha_s/v$ correction. They demonstrate that this correction significantly enhances the form factors (by a factor of $\sim 3$) compared to calculations without it, highlighting the importance of non-relativistic effects in heavy-heavy systems.
• Comprehensive Decay Predictions: The paper provides a complete set of predictions for both semileptonic ($B_c \to X l \bar{\nu}_l$) and nonleptonic ($B_c \to X P/V$) decay widths and branching ratios, including channels involving the $h_c$ meson which are often overlooked.
• Critical Comparison: The work offers a detailed comparison with other theoretical models (LFQM, NRQCD, previous QCDSR), identifying sources of discrepancies such as different decay constants, energy scales, and interpolating currents.

4. Key Results

• Form Factors:
  • The form factors are extracted at $Q^2=0$ and fitted using the $z$-series.
  • Impact of Coulomb Correction: Including the Coulomb-like correction increases the form factors by approximately 3 times. Consequently, decay widths (which scale with the square of form factors) increase by roughly 9 times.
  • The authors argue that while the correction is large, the results without the correction might be more phenomenologically reasonable given the current lack of higher-order perturbative calculations, though they present both sets of data.
• Semileptonic Decays:
  • Predicted hierarchy of decay widths: $\Gamma(B_c \to h_c l \bar{\nu}_l) > \Gamma(B_c \to \chi_{c2} l \bar{\nu}_l) > \Gamma(B_c \to \chi_{c0} l \bar{\nu}_l) > \Gamma(B_c \to \chi_{c1} l \bar{\nu}_l)$.
  • Branching ratios for $B_c \to h_c l \bar{\nu}_l$ and $B_c \to \chi_{c2} l \bar{\nu}_l$ are found to be significantly larger than in other models, especially when Coulomb corrections are included.
• Nonleptonic Decays:
  • Using NFA, the authors calculate branching ratios for various final states.
  • The predicted hierarchy for nonleptonic widths is $\chi_{c2} > h_c > \chi_{c0} > \chi_{1}$.
  • Ratios of branching fractions (e.g., $Br(B_c \to \chi_{c0}\pi)/Br(B_c \to \chi_{c2}\pi)$) are compared with recent factorization studies, showing some agreement but also notable differences, particularly for vector meson final states.
• Comparison with Experiment: The paper notes that the experimental upper limit for $B_c \to \chi_{c0}\pi$ from LHCb is consistent with their predictions (within uncertainties), providing a useful constraint for the total $B_c$ production cross-section.

5. Significance

• Theoretical Benchmark: This work provides a rigorous benchmark for the $B_c \to$ P-wave charmonia sector, filling a gap in the literature regarding the $h_c$ and $\chi_{c2}$ transitions.
• Guidance for Experiments: With the high luminosity of the LHC (Run 3 and HL-LHC), abundant $B_c$ events are expected. These predictions are essential for identifying rare decay channels and distinguishing between different theoretical models of heavy quark dynamics.
• Insight into Heavy Quark Dynamics: The significant impact of the Coulomb-like correction underscores the necessity of treating heavy quarkonia as non-relativistic systems in QCD sum rule calculations, refining our understanding of the interplay between perturbative and non-perturbative QCD.
• Future Directions: The authors highlight that the large uncertainty introduced by the NFA in nonleptonic decays and the sensitivity to the Coulomb correction suggest a need for more advanced techniques (like QCD Factorization or pQCD) and higher-order perturbative calculations in future studies.