Systematic analysis of the form factors of $B_c\rightarrowη_c$, $J/ψ$ and corresponding weak decays

This paper employs three-point QCD sum rules, including higher-order vacuum condensate contributions, to calculate the form factors for $B_c \to \eta_c$ and $B_c \to J/\psi$ transitions, which are then used to predict the decay widths and branching ratios of various nonleptonic and semileptonic decay channels to aid in the study of heavy-quark dynamics and potential new physics beyond the Standard Model.

The Gist

Imagine the universe as a giant, bustling construction site. At the very bottom of this site, there are tiny, fundamental building blocks called quarks. Usually, these blocks stick together in pairs to form larger structures called mesons.

Most mesons are like "single-family homes," made of two blocks of the same type (like two heavy blocks or two light blocks). But there is a special, rare structure called the $B_c$ meson. It's like a unique "mixed-family home" built from two different heavy blocks: a bottom quark and a charm quark. Because it's made of heavy materials, it's a bit unstable, but it lasts longer than its neighbors because it can only fall apart through a specific, slow process called the "weak interaction."

Scientists want to understand exactly how this $B_c$ meson falls apart, specifically when it transforms into other heavy structures called charmonium (like the $\eta_c$ or $J/\psi$). To do this, they need to measure the "strength of the connection" during the transformation. In physics, this connection strength is called a form factor.

The Challenge: Seeing the Invisible

You can't just put a ruler on a quark to measure it. It's too small and moves too fast. So, the authors of this paper used a sophisticated mathematical tool called QCD Sum Rules.

Think of this tool like a sonar system or a CT scan for the subatomic world.

1. The Phenomenological Side (The "Echo"): They imagine the meson as a real object with specific properties (mass, decay rate) and calculate what the signal should look like.
2. The QCD Side (The "Source"): They calculate what the signal looks like based on the fundamental rules of quarks and gluons (the "glue" holding them together).
3. The Match: By matching the "echo" with the "source," they can deduce the hidden properties (the form factors) that link the two.

The Secret Ingredient: The "Coulomb-like" Correction

In their calculations, the authors discovered something crucial. When two heavy quarks orbit each other, they don't just float freely; they pull on each other strongly, similar to how planets orbit the sun. This is called a Coulomb-like interaction.

• Without the correction: The authors calculated the connection strength (form factors) and found values that were quite small. It was like measuring the strength of a bridge but forgetting to account for the heavy traffic it carries.
• With the correction: When they added this "traffic" factor (the Coulomb-like correction), the calculated strength jumped up significantly—about three times larger.

The Analogy: Imagine trying to guess how much weight a rope can hold. If you only look at the rope's thickness, you might guess it holds 10 pounds. But if you realize the rope is also being pulled tight by a heavy weight on the other end (the Coulomb effect), you realize it's actually holding 30 pounds. The authors found that ignoring this effect gives a misleadingly weak picture of the meson's behavior.

What They Found

Using this improved method, the authors calculated the "strength of the connection" for the $B_c$ meson turning into $\eta_c$ and $J/\psi$. They then used these numbers to predict how often the $B_c$ meson would decay into various other particles (like pions, kaons, or electrons and neutrinos).

• The Results: Their predictions for how often these decays happen (branching ratios) matched well with other theoretical methods only after they included the Coulomb-like correction.
• The Mystery: They also looked at a specific ratio involving a heavy particle called a tau ($\tau$). Theoretical models (including theirs) predicted this ratio to be around 0.25. However, real-world experiments (from the LHCb collaboration) measured it to be 0.71.

The Big Picture

This paper doesn't solve the mystery of why the experiment is so different from the theory, but it does two important things:

1. It proves that for heavy quark systems, you must include the "Coulomb-like" pull between quarks to get accurate numbers. Without it, your math is off by a factor of three.
2. It highlights a gap between our current understanding (the Standard Model) and reality. Since the theory (even with the new correction) still predicts a much lower number than what is seen in the lab, this gap might be a sign of "New Physics"—something in the universe we haven't discovered yet.

In short, the authors built a better "ruler" for measuring heavy quark interactions. They found that the old ruler was missing a crucial piece of the puzzle, and even with the new, more accurate ruler, the universe still seems to be doing something unexpected that our current theories can't fully explain.

Technical Summary

Problem Statement
The $B_c$ meson, being the only known quarkonium state composed of two different heavy flavors ($b$ and $c$), serves as a unique laboratory for studying heavy-quark dynamics and weak decay processes. A critical area of investigation involves the transitions of the $B_c$ meson to charmonium states ($\eta_c$ and $J/\psi$), which are governed by the $b \to c$ transition. These processes are essential for extracting the Cabibbo-Kobayashi-Maskawa (CKM) matrix element $V_{cb}$ and searching for New Physics (NP) beyond the Standard Model (SM), particularly in light of the long-standing discrepancies observed in the ratio $R(D^{(*)})$ and the recent LHCb measurement of $R(J/\psi) = \Gamma(B_c^- \to J/\psi \tau^- \bar{\nu}_\tau) / \Gamma(B_c^- \to J/\psi \mu^- \bar{\nu}_\mu)$. While various theoretical methods (Lattice QCD, Relativistic Quark Models, Light-Cone Sum Rules, etc.) have been applied to these transitions, existing results often show inconsistencies with experimental data or with each other. Furthermore, the $B_c \to \eta_c$ transition has not yet been observed experimentally, necessitating robust theoretical predictions.

Methodology
The authors perform a systematic analysis of the form factors for $B_c \to \eta_c$ and $B_c \to J/\psi$ transitions using the framework of three-point QCD Sum Rules (QCDSR).

• Correlation Functions: The analysis begins with the construction of three-point correlation functions involving interpolating currents for the $B_c$, $\eta_c$ (or $J/\psi$), and the weak transition current.
• Phenomenological and QCD Sides: The correlation functions are evaluated on two sides:
  • Phenomenological side: Inserted with a complete set of hadronic states to isolate the ground state contributions and define the form factors ($f_S, f_+, f_0, f_T$ for $B_c \to \eta_c$ and $V, A_0, A_1, A_2, T_0, T_1, T_2$ for $B_c \to J/\psi$).
  • QCD side: Calculated using the Operator Product Expansion (OPE) in the deep Euclidean region. The authors include contributions from perturbative terms and non-perturbative vacuum condensates, specifically the two-gluon condensate $\langle g_s^2 GG \rangle$ and the three-gluon condensate $\langle g_s^3 GGG f \rangle$.
• Coulomb-like Correction: A crucial aspect of this methodology is the inclusion of Coulomb-like $\alpha_s/v$ corrections. Given the non-relativistic nature of heavy quarkonium, ladder diagrams representing gluon exchanges between the heavy quarks are considered. This leads to a renormalization coefficient applied to the spectral density of the perturbative part.
• Sum Rule Extraction: By applying quark-hadron duality and performing double Borel transformations, the authors derive sum rules for the form factors. The stability of the results is ensured by satisfying the pole dominance condition (pole contribution $\geq 40\%$) and the convergence of the OPE.
• Extrapolation: The form factors are calculated in the space-like region ($Q^2 > 0$) and extrapolated to the physical time-like region ($Q^2 \leq 0$) using the $z$-series parameterization approach, which respects unitarity and analyticity.
• Decay Calculations: Using the obtained form factors and the factorization approach, the authors calculate the decay widths and branching ratios for various nonleptonic ($B_c^- \to \eta_c/J/\psi \pi^-, K^-, \rho^-, K^{*-}$) and semileptonic ($B_c^- \to \eta_c/J/\psi l^- \bar{\nu}_l$) decay channels.

Key Contributions and Results

• Form Factors: The paper provides a comprehensive set of form factors for $B_c \to \eta_c$ and $B_c \to J/\psi$ at $Q^2=0$ and across the space-like region.
  • Without the Coulomb-like correction, the calculated form factors are generally consistent with previous QCDSR results (e.g., Ref. [21]) but are smaller than those obtained by other methods (Lattice QCD, LFQM, etc.).
  • Significant Impact of Correction: Upon including the Coulomb-like $\alpha_s/v$ correction, the numerical values of the form factors at $Q^2=0$ increase by approximately a factor of three. With this correction, the results become compatible with predictions from other theoretical approaches.
• Decay Properties: The authors present detailed predictions for decay widths and branching ratios for 14 specific decay channels.
  • Results are provided both with and without the Coulomb-like correction. The corrected branching ratios align better with existing theoretical estimates from other collaborations.
  • Specific predictions include nonleptonic decays to $\pi, K, \rho, K^*$ and semileptonic decays to $e, \mu, \tau$.
• Ratio $R(J/\psi)$: The study calculates the ratio $R(J/\psi) = \Gamma(B_c^- \to J/\psi \tau^- \bar{\nu}_\tau) / \Gamma(B_c^- \to J/\psi \mu^- \bar{\nu}_\mu)$. The theoretical prediction is found to be $0.25$, which is consistent with other theoretical calculations (range $0.25 \sim 0.28$) but significantly lower than the experimental measurement of $0.71 \pm 0.17 \pm 0.18$.

Significance
The paper claims that its systematic analysis provides useful information for studying heavy-quark dynamics. The inclusion of the Coulomb-like correction is highlighted as a critical factor that significantly alters the magnitude of the form factors, bringing QCDSR results into agreement with other theoretical frameworks. The calculated decay widths and branching ratios serve as benchmarks for future experimental verification, particularly for the unobserved $B_c \to \eta_c$ modes.

Regarding the discrepancy in $R(J/\psi)$, the authors modestly note that while their SM-based prediction is lower than the experimental data, this deviation mirrors the situation in $B \to D^{(*)}$ decays. They suggest that this discrepancy may arise from uncertainties in experimental data and theoretical models, or potentially indicate the existence of New Physics beyond the Standard Model, though they do not claim to definitively identify the source of the deviation. The work aims to provide a solid theoretical foundation for further investigations into these heavy-quark transitions.