Improved perturbative QCD study of the decay $B_c^+ \to \eta_c L^+$

This paper presents an improved perturbative QCD study at leading order that calculates the branching ratios and relative ratios for $B_c^+ \to \eta_c L^+$ decays (where $L$ represents various light mesons), finding results consistent with existing predictions and highlighting distinctive patterns for scalar meson modes to facilitate future experimental tests at the Large Hadron Collider.

The Gist

Imagine the subatomic world as a bustling, chaotic construction site. In this site, there are massive, heavy workers called quarks. Usually, these workers pair up with their own kind (like a heavy "bottom" quark with another bottom quark) or with light workers (like a bottom quark with a light "up" quark).

But there is a special, rare worker called the $B_c$ meson. It's unique because it's the only one built from two different heavy bosses: a bottom quark and a charm quark. Because it has two heavy bosses, it's like a double-decker bus that can fall apart in two different ways, making it a fascinating object for physicists to study.

This paper is a detailed blueprint for predicting how this special "double-heavy" bus ($B_c$) breaks apart into a specific set of parts: a $\eta_c$ (a heavy charm-anticharm pair) and a light meson (a lighter particle like a pion, rho, or various other shapes).

Here is the breakdown of their study, translated into everyday language:

1. The Goal: Predicting the Crash

Physicists want to know: "If we smash a $B_c$ meson, how often does it turn into an $\eta_c$ plus a light particle?"
Think of it like predicting the odds of a specific car model breaking down into a specific set of spare parts. The authors used a sophisticated mathematical tool called Improved Perturbative QCD (iPQCD).

• The Analogy: Imagine trying to predict the weather. You can't just guess; you need a super-complex model that accounts for wind, pressure, humidity, and history. This paper uses a "super-computer" model of the strong nuclear force (the glue holding quarks together) to calculate the odds of these particle crashes.

2. The Main Findings: The "Big" and the "Small"

The team calculated the probabilities (called Branching Ratios) for many different outcomes.

• The "Common" Crashes (Pions and Rho mesons):
  They found that the $B_c$ meson frequently breaks into an $\eta_c$ and a pion (a very light particle) or a rho (a slightly heavier, spinning particle).

  • The Result: Their prediction for the pion crash is about 0.2%. This is a "big" number in the world of rare particle physics.
  • The Comparison: They compared this to a similar, well-known crash ($B_c \to J/\psi + \pi$). They found the new crash is actually more likely than the old one, which is surprising because the particles involved are slightly different. It's like finding out your car is more likely to lose a tire than a headlight, even though the headlight is bigger. This tells us the "glue" (QCD dynamics) works differently than we thought.

• The "Exotic" Crashes (Scalars and Tensors):
  They also looked at crashes involving "scalar" and "tensor" particles. These are like particles with weird shapes or internal structures that physicists are still arguing about.

  • The Surprise: For some of these shapes (specifically the "scalar" ones), the odds of the crash happening are tiny (one in a billion).
  • The Mystery: They found a huge difference between crashes involving "strange" particles and those that don't. It's like finding that your car is 100 times more likely to lose a strange, exotic part than a normal one. This suggests that the internal structure of these "scalar" particles is very complex and perhaps not what we thought they were.

3. The "Secondary" Crashes (The Domino Effect)

The $\eta_c$ particle doesn't just sit there; it immediately falls apart again into other things, like protons and antiprotons, or pions.

• The Analogy: Imagine the $B_c$ meson is a Russian nesting doll. When you open it, you find an $\eta_c$. But the $\eta_c$ is also a doll that opens up to reveal even smaller toys (like protons).
• The Prediction: The authors calculated the odds of seeing the entire chain reaction (e.g., $B_c \to \eta_c + \pi$, and then $\eta_c \to \text{proton} + \text{antiproton}$). They predict these "multi-step" events happen often enough (about 1 in a million) that the LHCb experiment (a giant particle detector at CERN) should be able to spot them soon.

4. Why This Matters

• Testing the Theory: By comparing their predictions to real data coming from the LHC (the Large Hadron Collider), scientists can check if their "weather model" (the iPQCD theory) is accurate. If the real data matches their numbers, it means we finally understand how the strong force works in these heavy particles.
• Solving the Mystery of Light Scalars: There is a long-standing debate about what "scalar" particles really are. Are they simple pairs of quarks, or are they complex "tetraquarks" (four quarks stuck together)? The fact that the authors found such tiny probabilities for certain scalar crashes might help solve this puzzle. It's like finding a fingerprint that proves who the culprit is.

Summary

In short, this paper is a high-precision prediction manual for how a rare, heavy particle ($B_c$) breaks apart.

1. It predicts where to look for these particles in future experiments.
2. It suggests that some crashes are much more common than previously thought.
3. It highlights strange differences in how particles with "strange" quarks behave compared to those without, offering clues to the deep structure of matter.

The authors are essentially saying: "We've done the math. Here are the numbers. Go to the LHC, look for these specific patterns, and let's see if nature agrees with our blueprint."

Technical Summary

1. Problem Statement

The $B_c^+$ meson is unique among heavy mesons as it contains two different heavy quarks ($b$ and $c$), allowing both constituents to decay. While decays involving the $J/\psi$ meson (e.g., $B_c^+ \to J/\psi \pi^+$) have been extensively studied and observed, the decays into the $\eta_c$ meson (the ground-state $c\bar{c}$ pseudoscalar) remain largely unobserved experimentally, particularly the channel $B_c^+ \to \eta_c \pi^+$.

Existing theoretical predictions for the branching ratios (BRs) of $B_c^+ \to \eta_c L^+$ (where $L$ is a light meson) vary widely across different frameworks (ranging from $0.25 \times 10^{-3}$ to $4.22 \times 10^{-3}$ for the pion mode). This discrepancy indicates an incomplete understanding of the underlying QCD dynamics, specifically the transition form factors and the interplay between perturbative and non-perturbative effects. Furthermore, the nature of light scalar mesons (whether they are $q\bar{q}$ states or tetraquarks) remains a long-standing puzzle. The paper aims to resolve these discrepancies and provide precise predictions using a refined theoretical framework to guide future experimental searches at the LHC.

2. Methodology

The authors employ the Improved Perturbative QCD (iPQCD) formalism at the leading order (LO) in the strong coupling constant $\alpha_s$. Key methodological features include:

• Framework: The calculation utilizes the $k_T$ factorization theorem, which incorporates transverse momentum dependence to handle soft gluon emissions and suppress long-distance contributions via Sudakov resummation.
• Wave Functions:
  • $B_c$ Meson: Uses a newly proposed transverse-momentum-dependent wave function that includes finite charm quark mass effects in both the hard kernel and the Sudakov resummation.
  • $\eta_c$ Meson: Described using non-relativistic QCD (NRQCD) derived wave functions, including both twist-2 and twist-3 distribution amplitudes.
  • Light Mesons ($L$): Covers pseudoscalars ($P$), vectors ($V$), axial-vectors ($A$), scalars ($S$), and tensors ($T$). For light scalars, the study assumes a $q\bar{q}$ assignment within two scenarios (S1 and S2) regarding the mass ordering of ground and excited states.
• Factorization: The decay amplitude is decomposed into factorizable emission ($F_e$) and non-factorizable emission ($M_e$) contributions. The calculation explicitly evaluates the hard kernels for these topologies.
• Secondary Decays: The authors apply the narrow-width approximation to predict multibody decay rates (e.g., $B_c^+ \to \pi^+ \eta_c(\to p\bar{p})$) by inputting measured $\eta_c$ decay branching ratios.

3. Key Contributions

• Systematic Calculation: This is the first comprehensive iPQCD study of $B_c^+ \to \eta_c L^+$ decays covering a full spectrum of light mesons ($P, V, A, S, T$).
• Resolution of Discrepancies: The study provides a consistent set of predictions that align with several existing models (specifically the covariant confined quark model) while highlighting significant deviations in others, helping to pinpoint where theoretical understanding of $c\bar{c}$ dynamics needs refinement.
• Scalar Meson Insights: The paper offers a unique perspective on light scalar mesons by predicting their production rates in $B_c$ decays. It highlights a peculiar phenomenon where $\Delta S = 0$ scalar decays are heavily suppressed compared to $\Delta S = 1$ modes, contrary to naive expectations.
• Experimental Guidance: By calculating multibody decay chains via $\eta_c$ resonances, the authors provide specific targets (branching ratios $\sim 10^{-6}$ to $10^{-5}$) that are accessible to the upgraded LHCb detector.

4. Key Results

A. Pseudoscalar and Vector Modes ($P, V$)

• Branching Ratios:
  • $BR(B_c^+ \to \eta_c \pi^+) = (2.03^{+0.53}_{-0.41}) \times 10^{-3}$
  • $BR(B_c^+ \to \eta_c \rho^+) = (5.37^{+1.42}_{-1.08}) \times 10^{-3}$
• Ratios:
  • The ratio $R_{\eta_c/J/\psi}^{\pi} = BR(B_c^+ \to \eta_c \pi^+) / BR(B_c^+ \to J/\psi \pi^+) = 1.74^{+0.66}_{-0.50}$.
  • The ratio $R_{\rho/\pi}^{\eta_c} = BR(B_c^+ \to \eta_c \rho^+) / BR(B_c^+ \to \eta_c \pi^+) = 2.65^{+0.02}_{-0.03}$.
• Dynamics: Unlike $B_c \to J/\psi$ decays (dominated by twist-2 amplitudes), $B_c \to \eta_c$ decays are governed by twist-3 distribution amplitudes of the $\eta_c$, leading to different QCD dynamics.
• Multibody Predictions: Predicted rates for channels like $B_c^+ \to \pi^+ \eta_c(\to p\bar{p})$ are $\approx 2.7 \times 10^{-6}$, which are within the reach of LHCb.

B. Axial-Vector Modes ($A$)

• $K_1$ Mixing: The study analyzes $B_c^+ \to \eta_c K_1(1270, 1400)^+$ for two mixing angles ($\theta_K \approx 33^\circ$ and $58^\circ$).
  • At $\theta_K \approx 58^\circ$, a destructive interference significantly suppresses the $K_1(1400)$ mode relative to $K_1(1270)$.
  • Precise measurement of these ratios could determine the mixing angle $\theta_K$.
• $b_1(1235)$: The BR for $B_c^+ \to \eta_c b_1(1235)^+$ is predicted to be $\approx 7.88 \times 10^{-4}$, comparable to the $J/\psi b_1$ mode, despite the latter having three polarization states.

C. Scalar Modes ($S$) - Surprising Findings

• Suppression: Due to the near-zero vector decay constants ($f_S \approx 0$) for light scalars, factorizable contributions are suppressed.
• $\Delta S$ Ratios: The paper finds a surprisingly large ratio between $\Delta S = 1$ and $\Delta S = 0$ modes:
  • $BR(B_c^+ \to \eta_c \kappa^+) / BR(B_c^+ \to \eta_c a_0(980)^+) \sim \mathcal{O}(10^2)$.
  • This is drastically different from the $B_c \to J/\psi S^+$ sector and the naive expectation based on CKM elements alone.
• Mechanism: This is attributed to destructive interference between factorizable and non-factorizable amplitudes in the $\eta_c$ channel, which is not present in the $J/\psi$ channel. The interference is stronger in Scenario 1 (S1) than Scenario 2 (S2).

D. Tensor Modes ($T$)

• Factorizable contributions vanish for tensor mesons at LO. The decays are driven entirely by non-factorizable emissions.
• Predicted $BR(B_c^+ \to \eta_c a_2(1320)^+) \approx 1.33 \times 10^{-4}$.

5. Significance and Outlook

• Experimental Validation: The predictions provide concrete targets for the LHCb experiment (post-2022 upgrade). The multibody channels (e.g., via $\eta_c \to p\bar{p}$) offer a viable path to observe $B_c^+ \to \eta_c \pi^+$ despite the challenging background.
• QCD Dynamics: The results highlight the critical role of twist-3 contributions in $\eta_c$ production and the complex interference patterns in scalar meson production.
• Scalar Meson Nature: The distinct behavior of scalar meson production in $B_c \to \eta_c S$ versus $B_c \to J/\psi S$ offers a new probe to test the $q\bar{q}$ vs. tetraquark nature of light scalars.
• Theoretical Consistency: The agreement of iPQCD predictions with covariant quark models for $P$ and $V$ modes, and the specific deviations found for $S$ modes, helps validate the iPQCD formalism as a reliable tool for heavy meson decays involving charmonia.

In conclusion, this work bridges the gap between theoretical uncertainty and experimental capability, offering a roadmap to explore the QCD dynamics of the $B_c$ meson and the internal structure of the $\eta_c$ and light hadrons.