Study of $B_{c} \to J/\psi (a_1(1260)$, $b_1(1235)$, $a_2(1320)$, $K_2^*(1430))$ decay with a perturbative QCD approach

Motivated by LHCb observations, this study employs the perturbative QCD factorization framework to analyze $B_c^+ \to J/\psi$ decays into axial-vector and tensor mesons, predicting branching ratios within experimental reach and establishing the dominance of longitudinal polarization amplitudes.

The Gist

Imagine the universe as a giant, bustling construction site. At the very top of the hierarchy are the "heavyweights" of the particle world: the Bottom quark and the Charm quark. Usually, these heavyweights hang out with lighter partners, but sometimes, they pair up to form a unique, double-heavy couple called the $B_c$ meson.

Think of the $B_c$ meson as a rare, high-performance sports car made of two different heavy engines. Because it's so heavy and unique, it doesn't just sit there; it eventually breaks down (decays) into lighter particles. Physicists are obsessed with watching how this car breaks apart because the way it falls apart tells us the secrets of the universe's fundamental forces.

The Mission: Watching the $B_c$ Break Up

This paper is like a detailed blueprint for a specific type of crash test. The researchers wanted to predict exactly what happens when the $B_c$ meson decays into two specific parts:

1. $J/\psi$: A heavy, stable "charmed" particle (like a heavy, solid core).
2. A "Light" Partner: Either an Axial-vector meson (like a spinning top, $a_1$ or $b_1$) or a Tensor meson (like a complex, wobbling gyroscope, $a_2$ or $K^*_2$).

The goal? To calculate how often this happens (Branching Ratio) and how the pieces spin as they fly apart (Polarization).

The Tool: The "pQCD" Microscope

To predict these crashes, the scientists used a powerful theoretical tool called perturbative QCD (pQCD).

• The Analogy: Imagine trying to predict the path of a leaf falling in a storm. It's chaotic. But if you zoom in with a super-microscope, you can see the individual air molecules hitting the leaf.
• In the Paper: The "storm" is the strong nuclear force (gluons) holding quarks together. The "microscope" is pQCD. It allows the scientists to calculate the interactions of these tiny particles by treating them like billiard balls bouncing off each other, but with complex math to account for the "wind" (quantum fluctuations).

They used a special technique called Sudakov suppression. Think of this as a "noise-canceling headphone" for the math. It filters out the impossible, chaotic scenarios where particles move too slowly or get stuck at the edges of the calculation, ensuring the prediction stays clean and accurate.

The Big Findings: What Did They Discover?

1. The "Spin" Matters (Polarization)

When the $B_c$ breaks apart, the pieces don't just fly away; they spin.

• The Finding: The researchers found that in almost every case, the pieces prefer to spin lengthwise (like a bullet) rather than sideways.
• The Analogy: Imagine throwing a football. It's much easier to throw it spinning end-over-end (longitudinal) than to make it wobble sideways. The math shows the universe prefers the "football throw" for these heavy particles.
• The Twist: The $b_1$ particle (a specific type of axial-vector) is a bit of a rebel. Because of its unique internal structure, it almost only spins lengthwise, whereas the $a_1$ particle is a bit more relaxed and allows for some sideways spinning.

2. The "Ghost" Particles (Tensor Mesons)

The paper looked at Tensor mesons ($a_2$ and $K^*_2$). These are particles with a spin of 2, which is very complex.

• The Finding: These decays are incredibly rare (about 100 times less likely than the others).
• The Reason: In the "standard" way particles usually form, these heavy spins can't be created directly. It's like trying to build a house using only a hammer when you actually need a crane.
• The Solution: These particles can only be formed through "non-factorizable" diagrams. Think of this as a secret handshake. The particles have to exchange a "gluon" (the force carrier) in a very specific, complex way to get the job done. Because this secret handshake is hard to do, these decays happen very rarely.

3. The "Twin" Problem ($a_1$ vs. $b_1$)

The researchers compared two very similar particles: $a_1$ and $b_1$. They look alike, but they have different internal "quantum numbers" (like different DNA).

• The Finding: The $a_1$ decay happens frequently. The $b_1$ decay is suppressed (happens less often).
• The Analogy: Imagine two twins trying to fit through a narrow door. Twin A ($a_1$) fits right through because their shape matches the door. Twin B ($b_1$) has a slightly different shape that makes the door feel much smaller, so they struggle to get through. However, the math shows that while Twin B struggles, the "secret handshake" (non-factorizable diagrams) helps them get through just enough to be seen by future experiments.

Why Should We Care?

You might ask, "Why calculate how often a rare particle breaks up?"

1. Testing the Rules: The Standard Model is the rulebook of physics. If the real-world experiments (like those at the LHCb at CERN) see these particles breaking up at a different rate than this paper predicts, it means the rulebook is wrong! It could signal New Physics—something we haven't discovered yet.
2. The Future is Bright: The paper predicts these events happen often enough (about 1 in 100 or 1 in 10,000 times) that the upgraded LHCb detector should be able to spot them soon.
3. Understanding the Invisible: By understanding how these heavy quarks dance and spin, we learn more about the "glue" (Strong Force) that holds the universe together.

Summary

This paper is a theoretical forecast for a high-stakes particle crash test. It uses advanced math to predict that the rare $B_c$ meson will break into specific spinning partners. It tells us that:

• The pieces mostly spin lengthwise.
• The "complex spin" particles are very rare and require a "secret handshake" to form.
• The difference between similar-looking particles ($a_1$ vs. $b_1$) is huge and depends on their internal quantum DNA.

These predictions are now a "shopping list" for experimentalists at the LHC. If they find exactly what this paper says, the Standard Model gets a gold star. If they find something different, we might just be on the verge of a massive discovery in physics!

Technical Summary

1. Problem Statement

The $B_c$ meson is a unique system composed of two heavy quarks with distinct flavors ($\bar{b}$ and $c$). Unlike other $B$ mesons, it decays exclusively via weak interactions because strong and electromagnetic annihilation are kinematically forbidden. While $B_c$ decays to charmonium states (like $J/\psi$) have been studied, there is a need for precise theoretical predictions for decays involving axial-vector mesons ($a_1, b_1$) and tensor mesons ($a_2, K^*_2$).

Key challenges addressed in this work include:

• Theoretical Consistency: Handling the double-heavy nature of the $B_c$ meson where the charm quark mass ($m_c$) is significant but not negligible compared to the bottom quark mass ($m_b$).
• Factorization Issues: Tensor mesons ($J^P=2^+$) cannot be created directly from the vacuum via standard $(V \pm A)$ or $(S \pm P)$ weak currents, causing factorizable emission amplitudes to vanish. This necessitates a rigorous treatment of non-factorizable diagrams.
• Endpoint Divergences: Controlling infrared divergences in the perturbative QCD (pQCD) calculation, particularly in the endpoint regions of light-cone distribution amplitudes (LCDAs).

2. Methodology

The authors employ the perturbative QCD (pQCD) factorization framework combined with $k_T$-factorization.

• Wave Functions:
  • $B_c$ Meson: A non-relativistic approximation is used for the $B_c$ light-cone wave function. Crucially, the longitudinal momentum fraction of the charm quark is fixed using a delta function: $\delta(x - m_c/M_{B_c})$. This approach naturally incorporates the double-heavy flavor nature and reduces sensitivity to endpoint uncertainties in light meson distribution amplitudes.
  • Light Mesons ($A, T$): Light-cone distribution amplitudes (LCDAs) are used for the axial-vector and tensor mesons, including twist-2 and twist-3 components.
  • $J/\psi$: Wave functions are described within the NRQCD framework with distinct longitudinal and transverse components.
• Factorization Theorem: The decay amplitude is factorized into a convolution of a hard kernel (calculable perturbatively) and meson wave functions.
  • Hard Scales: The hierarchy $\bar{\Lambda} \ll m_c \ll M_{B_c}$ allows for a systematic power expansion.
  • Sudakov Suppression: Double logarithmic divergences from overlapping collinear and soft regions are summed into a Sudakov factor, suppressing long-distance interactions and ensuring infrared safety.
• Decay Topologies:
  • Axial-Vector ($a_1, b_1$): Both factorizable and non-factorizable emission diagrams contribute.
  • Tensor ($a_2, K^*_2$): Factorizable emission diagrams vanish due to Lorentz covariance constraints ($\langle T | j_\mu | 0 \rangle = 0$). The decay proceeds entirely via non-factorizable emission diagrams.
• Helicity Analysis: The amplitudes are decomposed into longitudinal ($A_L$), normal ($A_N$), and transverse ($A_T$) components to calculate polarization fractions.

3. Key Contributions

1. Updated Theoretical Framework: The study updates previous calculations (e.g., Ref. [49]) by incorporating intrinsic $b$-dependence in the $B_c$ wave function to better account for transverse momentum effects and using updated CKM matrix elements and hadronic parameters.
2. Extension to New Channels: This is a comprehensive study of $B_c \to J/\psi$ decays into previously less explored channels: $b_1(1235)$, $a_2(1320)$, and $K^*_2(1430)$, in addition to $a_1(1260)$.
3. Distinction from Competing Models: The authors explicitly contrast their approach with recent work (Ref. [51]). While Ref. [51] modifies the Sudakov factor to include $m_c$, this paper optimizes the initial-state wave function using a non-relativistic delta-function approximation. Both approaches yield consistent orders of magnitude, validating the robustness of pQCD predictions for double-heavy systems.
4. Polarization Predictions: Detailed predictions for polarization fractions ($f_0, f_\parallel, f_\perp$) are provided, offering specific observables for experimental verification.

4. Key Results

Branching Ratios:

• Axial-Vector Decays:
  • $B(B_c^+ \to J/\psi a_1^+) \approx 1.1 \times 10^{-2}$
  • $B(B_c^+ \to J/\psi b_1^+) \approx 1.5 \times 10^{-3}$
  • Observation: The $a_1$ channel is dominated by factorizable contributions due to its large decay constant. The $b_1$ channel is suppressed in factorizable diagrams (due to its vanishing effective decay constant as a $1P_1$ state) but remains significant ($10^{-3}$) due to constructive interference in non-factorizable diagrams.
• Tensor Decays:
  • $B(B_c^+ \to J/\psi a_2^+) \approx 2.9 \times 10^{-4}$
  • $B(B_c^+ \to J/\psi K_2^{*+}) \approx 4.5 \times 10^{-5}$
  • Observation: These are significantly smaller ($10^{-4} \sim 10^{-5}$) because they rely entirely on non-factorizable mechanisms. The $K^*_2$ channel is further suppressed by the Cabibbo-suppressed $b \to s$ transition.

Polarization Fractions:

• Longitudinal Dominance: Longitudinal polarization ($f_0$) dominates all channels.
  • For tensor mesons ($a_2, K^*_2$), $f_0 \approx 95-96\%$, approaching complete longitudinal polarization. This is attributed to the orbital angular momentum ($L=1$) and the suppression of transverse modes by helicity selection rules.
  • For $b_1$, $f_0 \approx 90\%$, significantly higher than for $a_1$ ($f_0 \approx 69\%$). The suppression of factorizable diagrams in $b_1$ (which usually favor transverse modes) forces the decay to be dominated by non-factorizable longitudinal amplitudes.

Ratios:
The study defines ratios relative to benchmark decays ($B_c \to J/\psi \pi^+$ and $J/\psi \rho^+$) to test QCD dynamics:

• $R_{a_1/\pi} \approx 4.75$ (Enhanced factorizable contribution).
• $R_{b_1/\pi} \approx 0.65$ (Suppressed factorizable, enhanced non-factorizable).
• $R_{a_1/\rho} \approx 1.35$ and $R_{b_1/\rho} \approx 0.18$. These ratios clearly distinguish the $3P_1$ ($a_1$) and $1P_1$ ($b_1$) states based on their internal spin-parity configurations.

5. Significance

• Experimental Reach: The predicted branching ratios ($10^{-3} \sim 10^{-2}$ for axial-vectors and $10^{-5} \sim 10^{-4}$ for tensors) are within the reach of current and future LHCb data (High-Luminosity LHC).
• Testing QCD Factorization: The results provide a stringent test for the pQCD framework in the double-heavy quark sector, specifically validating the treatment of non-factorizable diagrams and the handling of the $m_c/m_b$ mass ratio.
• Probing Hadron Structure: The distinct polarization patterns and branching ratios offer a way to probe the internal structure of axial-vector mesons (distinguishing $1P_1$ vs $3P_1$) and tensor mesons, as well as flavor-SU(3) symmetry breaking effects (comparing $a_2$ and $K^*_2$).
• New Physics Sensitivity: Precise measurements of these channels can constrain hadronic parameters and potentially reveal deviations from the Standard Model, such as contributions from penguin operators (though absent here) or new physics in weak decay dynamics.

In conclusion, this paper provides a robust, updated theoretical baseline for $B_c$ decays to charmonium plus light mesons, highlighting the critical role of non-factorizable mechanisms and helicity conservation in determining decay rates and polarization.