⚛️ phenomenology

Quasi-two-body decays $B_c^+ \to \chi_{c0,c1} [\rho(K^*) \to] \pi\pi(K\pi)$ in the PQCD approach

This paper employs the perturbative QCD framework to analyze quasi-two-body decays of $B_c^+$ mesons into charmonium states ( $\chi_{c0}, \chi_{c1}$ ) and pion or kaon-pion pairs via intermediate P-wave resonances, revealing that resonant $\rho$ production significantly enhances branching fractions and alters the relative yield ratios between charmonium modes compared to direct two-body decays.

Original authors: Jun Deng, Xian-Qiao Yu

Published 2026-07-28

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Jun Deng, Xian-Qiao Yu

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the universe as a giant, chaotic construction site where the smallest building blocks—quarks—are constantly being glued together to form larger structures called particles. Among these, there's a special, rare duo known as the $B_c$ meson. Think of it as a unique marriage between two very different heavy partners: a bottom quark and a charm quark. Unlike other particles that might fall apart instantly or get smashed by strong forces, this couple is surprisingly stable, living long enough to dance through the weak force before finally breaking up. Scientists are obsessed with watching how this $B_c$ meson decays because it's like a secret laboratory. By studying exactly how it splits apart, we can test the rules of the Standard Model—the rulebook of particle physics—and see if there are any hidden cracks that might point to new, undiscovered physics.

The key to this investigation lies in understanding how these heavy particles transform. Sometimes, they don't just break into two pieces; they create a fleeting, intermediate "dance partner" that quickly spins into two other particles. In the language of physics, this is called a "quasi-two-body" decay. It's like a parent handing a gift to a child, but the gift is actually a box containing two smaller toys that were stuck together for a split second. To predict how often this happens, physicists use a powerful tool called Perturbative Quantum Chromodynamics (pQCD). Think of pQCD as a high-precision calculator that tracks the messy, invisible interactions of quarks and gluons, allowing scientists to simulate these decays on a computer before they even happen in a real collider.

In this study, the authors dive deep into a specific set of these rare dances: the decay of the $B_c$ meson into a charmonium particle (a heavy charm-anticharm pair) and a pair of lighter pions or kaons. Specifically, they looked at cases where the light particles are produced via a "resonance," a short-lived state like the $\rho$ or $K^*$ meson, which acts as the temporary bridge between the heavy $B_c$ and the final products. Using the pQCD framework, the team calculated the likelihood of these events happening. They found that the most common path involves the $\rho(770)$ resonance, leading to a branching ratio (the probability of this specific decay occurring) of about $3.24 \times 10^{-3}$ for the $\chi_{c0}$ mode and $4.19 \times 10^{-3}$ for the $\chi_{c1}$ mode. Interestingly, the interference between different $\rho$ resonances adds about 25% to the total yield, acting like a constructive echo that boosts the signal.

The study also revealed that the $K\pi$ channels are much rarer, suppressed by the "Cabibbo" effect to a probability of roughly $10^{-6}$ , making them significantly harder to spot. A major discovery here concerns the "spin" or orientation of the particles. For the $\chi_{c1}$ mode, the decay is overwhelmingly dominated by a longitudinal polarization (about 94%), meaning the particles are aligned in a specific direction, and this alignment tends to decrease as the resonance gets heavier. Perhaps most surprisingly, the ratio of $\chi_{c1}$ to $\chi_{c0}$ production in these resonant decays is about 1.30. This is a stark contrast to the simple two-body decays, where that ratio is around 4.7. This suggests that the presence of the resonant $\rho$ particle dramatically reshapes how these heavy charmonium states are produced, challenging simpler theoretical expectations. The authors also confirmed that the ratio of $K\pi$ to $\pi\pi$ production remains consistent at around $2 \times 10^{-3}$ across different modes. While these results are based on theoretical calculations and simulations rather than direct new measurements, they provide a crucial, detailed map for experimentalists at facilities like the LHCb to look for these specific patterns in the data.

Technical Summary: Quasi-two-body decays $B^+_c \to \chi_{c0,c1}[\rho(K^*) \to]\pi\pi(K\pi)$ in the PQCD approach

Problem Statement
The $B_c$ meson, as the lightest bound state of two different heavy quarks ( $b$ and $\bar{c}$ ), offers a unique laboratory for testing the Standard Model (SM) and probing non-perturbative Quantum Chromodynamics (QCD). While theoretical investigations into two-body decays $B_c \to \chi_{cJ}\pi$ have revealed significant challenges within conventional collinear factorization frameworks—particularly regarding the vanishing leading-twist decay constant of the axial-vector charmonium $\chi_{c1}$ and the suppression of $\chi_{c0}$ production via color-singlet currents—recent improved perturbative QCD (iPQCD) calculations for two-body modes have shown tension with experimental data (e.g., the ratio $R_{\chi_{c0}/J/\psi}$ ). Specifically, the interference patterns between twist-2 and twist-3 contributions observed in $\chi_{c1}$ modes are absent in $\chi_{c0}$ and $\chi_{c2}$ channels, motivating a deeper investigation into production mechanisms. This work addresses the need to extend these analyses to multi-body decay channels where resonant contributions may alter the relative yields of P-wave charmonia.

Methodology
The authors employ the perturbative QCD (PQCD) factorization approach at leading order to study the quasi-two-body decays $B^+_c \to \chi_{c0,c1}[\rho(K^*) \to]\pi^+\pi^0(K^0\pi^+)$ . The methodology relies on the following key components:

Quasi-two-body Approximation: The three-body decay is treated within a framework where the final-state meson pair ( $\pi\pi$ or $K\pi$ ) is collimated, allowing the non-perturbative dynamics of the pair production to be absorbed into two-meson distribution amplitudes (DAs). The hard kernel involves a single virtual gluon exchange at leading order in $\alpha_s$ .
Resonance Parametrization: The $\pi\pi$ and $K\pi$ pairs are produced via P-wave resonances. The broad $\rho$ resonances ( $\rho(770)$ , $\rho(1450)$ , $\rho(1700)$ ) are described using the Gounaris-Sakurai (GS) model, while the narrow $K^*(892)$ is modeled using the relativistic Breit-Wigner (RBW) function. Normalization factors ( $N_\rho = 1.05$ , $N_{K^*} = 1.48$ ) are introduced to reconcile theoretical form factors with resonance parameters, justified by the Watson theorem.
Distribution Amplitudes (DAs):
- $B_c$ Meson: Modeled using a standard PQCD wave function with a shape parameter $\beta_{Bc}$ .
- $\chi_{cJ}$ Mesons: Both twist-2 and twist-3 light-cone distribution amplitudes (LCDAs) are included. For $\chi_{c1}$ , separate amplitudes for longitudinal and transverse polarizations are defined. The models utilize asymptotic forms derived from P-wave Schrödinger states in a Coulomb potential, incorporating relativistic corrections.
- Two-meson DAs: The $\pi\pi$ and $K\pi$ systems are expanded in terms of Gegenbauer polynomials up to twist-3, with coefficients taken from previous literature.
Amplitude Calculation: The decay amplitudes are computed by evaluating factorizable and non-factorizable emission diagrams. The formalism retains the transverse momentum ( $k_T$ ) of valence quarks to regulate endpoint singularities and ensure non-vanishing contributions for $\chi_{c1}$ production.

Key Contributions and Results
The paper presents the first PQCD analysis of these specific quasi-two-body channels, yielding the following results:

Branching Ratios: The dominant channels involving the $\rho(770)$ resonance yield branching fractions of $\mathcal{B}(B^+_c \to \chi_{c0}\pi^+\pi^0) = 3.24 \times 10^{-3}$ and $\mathcal{B}(B^+_c \to \chi_{c1}\pi^+\pi^0) = 4.19 \times 10^{-3}$ . Constructive interference among $\rho$ resonances contributes approximately 25% to the total rate.
Cabibbo Suppression: The $K\pi$ channels are found to be Cabibbo-suppressed, with branching ratios on the order of $10^{-6}$ .
Polarization: For the $\chi_{c1}$ modes, the longitudinal polarization fraction ( $f_L$ ) is dominant at approximately 94% and decreases as the resonance mass increases.
Relative Ratios:
- The ratio $R^{\pi\pi}_{\chi_{c1}/\chi_{c0}} \approx 1.30$ is calculated. This contrasts significantly with the two-body result of $\sim 4.7$ , indicating that resonant $\rho$ production dramatically reshapes the relative yields of P-wave charmonia.
- The ratio $R_{K/\pi} \approx 2 \times 10^{-3}$ is found to be consistent across both $\chi_{c0}$ and $\chi_{c1}$ modes.
Two-body Extraction: Under the narrow-width approximation, the authors extract two-body branching fractions for $B^+_c \to \chi_{cJ}\rho^+(K^{*+})$ , enabling a direct comparison with existing two-body iPQCD calculations.

Significance
The paper claims that its primary significance lies in demonstrating how resonant dynamics in multi-body decays alter the phenomenological landscape compared to simple two-body decays. Specifically, the study highlights that the inclusion of resonant $\rho$ production significantly modifies the ratio of $\chi_{c1}$ to $\chi_{c0}$ yields, a finding that challenges naive extrapolations from two-body results. By successfully applying the PQCD framework to these quasi-two-body channels and resolving the issues associated with $\chi_{c1}$ production (via higher-twist LCDAs and $k_T$ factorization), the work provides a consistent theoretical baseline for future experimental comparisons with LHCb data. The results serve to refine the understanding of heavy quarkonium production mechanisms in $B_c$ decays and offer a more complete picture of the interplay between perturbative and non-perturbative QCD effects in these processes.

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