Quasi-two-body decays $B^+\to D_s^+ (R\to) K^+K^-$ in the perturbative QCD approach

This paper presents the first perturbative QCD predictions for the branching fractions and direct CP asymmetries of the quasi-two-body decays $B^+\to D_s^+(R\to)K^+K^-$ involving various $S$-, $P$-, and $D$-wave resonances, finding results consistent with existing data and predicting vanishing CP asymmetries within the Standard Model.

The Gist

The Big Picture: A Particle Physics "Magic Trick"

Imagine a heavy, unstable particle called a $B^+$ meson (let's call it the "Grandparent"). In the world of particle physics, Grandparents don't just sit still; they love to break apart into smaller, lighter particles.

Usually, when a Grandparent breaks, it splits into two kids (a two-body decay). But sometimes, it splits into three kids (a three-body decay). This paper is about a specific three-kid family:

1. A $D_s^+$ meson (the "Bachelor" child).
2. A $K^+$ (a positive Kaon).
3. A $K^-$ (a negative Kaon).

The scientists at the LHCb experiment recently watched this happen, but they didn't look closely at how the two Kaons ($K^+$ and $K^-$) interacted with each other. They just saw the final result.

The Goal of This Paper:
The authors (Zou, Wang, Rui, and Li) want to look under the hood. They ask: "Did the two Kaons hang out together for a split second to form a temporary 'club' (a resonance) before flying apart?" They use a complex mathematical toolkit called Perturbative QCD (PQCD) to calculate exactly how often this happens and what those temporary clubs look like.

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The Analogy: The Dance Floor and the "Clubs"

Imagine the Grandparent ($B^+$) is a giant bouncer at a club who suddenly decides to retire. He needs to send three people out the door: the Bachelor ($D_s^+$) and a pair of dancers ($K^+$ and $K^-$).

In a simple two-body decay, the bouncer just hands two people a ticket, and they leave. Easy.

In this three-body decay, the bouncer hands out three tickets. But here's the twist: The two dancers ($K^+$ and $K^-$) might not just walk out separately. They might grab each other, spin around, and form a temporary dance couple (a Resonance) before letting go and walking away.

The paper investigates the different "dance styles" (quantum spins) these couples can form:

• S-wave (The Slow Waltz): The couple spins slowly. Examples: $f_0(980)$, $f_0(1370)$, $f_0(1500)$.
• P-wave (The Fast Foxtrot): The couple spins faster. Example: $\phi(1020)$.
• D-wave (The Complex Tango): The couple does a very intricate, high-energy spin. Examples: $f_2(1270)$, $f_2(1525)$.

How They Did the Math (The "Recipe")

To predict how often these dances happen, the authors used the PQCD approach. Think of this as a very precise recipe book for particle physics.

1. The Ingredients (Wave Functions): You can't just guess how the dancers move. You need to know their "personality" (wave functions). The authors used known data for the Grandparent and the Bachelor, but for the dancing pair ($K^+K^-$), they had to invent a new "personality profile" based on the type of dance (S, P, or D wave).
2. The Cooking Process (Hard Kernels): They calculated the "heat" of the interaction. This involves complex math to see how the particles exchange energy (gluons) while breaking apart.
3. The Flavor Profile (Resonance Models):
   • For most dancers, they used a standard "Breit-Wigner" model (like a standard bell curve for how long the dance lasts).
   • Special Case: The $f_0(980)$ is a tricky dancer. It's so close to the "door" (the energy threshold) that it gets stuck in the doorway. The standard model doesn't work. The authors had to use a special "Flatté model" (a custom-shaped door) to describe it correctly.

The Results: What Did They Find?

After doing all the heavy math, they got some numbers (Branching Fractions), which tell us the probability of this specific event happening.

• The Numbers: They found that these events are rare. Out of every billion $B^+$ mesons, only a few million (or even fewer) will do this specific dance.
• The "Off-Shell" Problem: For the $f_0(980)$, the math showed that the dance is barely possible. The energy of the Grandparent is just barely enough to create the couple, but not quite enough to let them spin freely. It's like trying to dance in a room that is slightly too small; the dance is awkward and happens very rarely.
• The "Annihilation" Surprise: For the fast-spinning $\phi(1020)$ dancer, the process is even rarer. It happens through a mechanism called "annihilation," which is like the Grandparent and the Bachelor canceling each other out to create the dancers. This is a "power-suppressed" process, meaning it's naturally much weaker than the standard dance moves.

The "Two-Body" Trick

Since the math for three-body decays is messy, the authors used a clever shortcut called the Narrow-Width Approximation (NWA).

• The Analogy: Imagine you want to know how many people leave a party by the front door. Instead of tracking every single person, you assume that anyone who forms a "couple" at the party leaves together as a single unit.
• By doing this, they could take their three-body results and "extract" the probability of the simpler two-body event ($B^+ \to D_s^+ + \text{Resonance}$).
• The Check: They compared their extracted numbers with previous theories and found they matched up perfectly. This proves their complex "dance floor" math is reliable.

The Big Conclusion: No "Magic" (CP Violation)

The paper ends with a very important "spoiler alert" for physicists.

In particle physics, scientists love to look for CP Violation. This is a phenomenon where matter and antimatter behave differently, which helps explain why the universe is made of matter and not just empty space. Usually, this happens when two different "paths" (like a tree path and a penguin path) interfere with each other.

The Finding:
The authors calculated that for this specific decay, there is only one path (the "tree" path). There is no "penguin" path to interfere with it.

• The Result: The direct CP asymmetry is zero.
• The Metaphor: If you flip a coin, it's fair. But if you have a trick coin that always lands on heads, that's CP violation. This paper says, "For this specific dance, the coin is fair."
• Why it matters: If future experiments at LHCb or Belle II see a non-zero CP asymmetry in this decay, it would be a massive shock. It would mean the Standard Model is wrong, and there is New Physics (something we don't understand yet) interfering with the dance.

Summary in One Sentence

This paper uses advanced math to map out the "dance moves" of a specific particle decay, confirming that while these dances are rare and complex, they follow the known rules of physics—and if they ever break those rules, it will be a sign of a brand-new discovery.

Technical Summary

1. Problem Statement

Three-body $B$-meson decays are significantly more complex than two-body decays due to their dependence on two independent kinematic variables and the presence of both resonant and non-resonant contributions. While experimental collaborations like LHCb have reported searches for the decay $B^+ \to D_s^+ K^+ K^-$, previous analyses often lacked a detailed amplitude analysis of the $K^+ K^-$ subsystem.

The specific challenges addressed in this work include:

• Theoretical Complexity: Accurately describing the dynamics of the $K^+ K^-$ pair, which involves intermediate resonances with different quantum numbers (S-wave, P-wave, D-wave).
• Factorization: Extending the perturbative QCD (PQCD) factorization framework, typically used for two-body decays, to quasi-two-body decays where a correlated two-meson system replaces a single particle.
• Resonance Modeling: Properly parameterizing the time-like form factors for various resonances, particularly the $f_0(980)$, which lies near the $K\bar{K}$ threshold and requires a model beyond the standard relativistic Breit-Wigner (RBW) approximation.
• CP Violation: Determining whether direct CP asymmetries exist in these specific tree-dominated channels within the Standard Model (SM).

2. Methodology

The authors employ the Perturbative QCD (PQCD) factorization approach to calculate the decay amplitudes. The methodology involves the following key steps:

• Kinematic Framework: The calculation is performed in the $B$-meson rest frame using light-cone coordinates. The process is treated as a quasi-two-body decay where the $K^+ K^-$ pair forms a clustered system with invariant mass $\omega^2$.
• Effective Hamiltonian: The weak interaction is governed by the effective Hamiltonian containing tree-level operators $O_1$ and $O_2$. Notably, the Hamiltonian lacks penguin operators, which is crucial for the CP asymmetry analysis.
• Wave Functions and Distribution Amplitudes (LCDAs):
  • The initial $B^+$ and final $D_s^+$ meson wave functions are taken from established PQCD literature.
  • Two-Meson Wave Functions: The core innovation is the introduction of specific two-meson distribution amplitudes ($\Phi_{KK}$) for the $K^+ K^-$ system. These are derived for:
    • S-wave: Scalar resonances $f_0(980)$, $f_0(1370)$, $f_0(1500)$.
    • P-wave: Vector resonance $\phi(1020)$.
    • D-wave: Tensor resonances $f_2(1270)$ and $f_2'(1525)$.
  • These wave functions include twist-2 and twist-3 LCDAs expanded in Gegenbauer polynomials, with moments ($a_n$) determined by fitting to previous experimental data (e.g., $B \to KKK$).
• Resonance Modeling:
  • RBW Model: Used for narrow-width resonances ($f_0(1370)$, $f_0(1500)$, $\phi(1020)$, $f_2(1270)$, $f_2'(1525)$) to describe their time-like form factors.
  • Flatté Model: A modified Flatté model is specifically adopted for the $f_0(980)$ to account for its proximity to the $K\bar{K}$ threshold and strong coupled-channel effects, which the RBW model fails to describe accurately.
• Feynman Diagrams: The amplitude is calculated by summing contributions from four topologies:
  1. Factorizable emission.
  2. Non-factorizable emission.
  3. Factorizable annihilation.
  4. Non-factorizable annihilation.
     The authors note that while annihilation diagrams are power-suppressed, they provide the dominant strong phase necessary for potential CP violation (though absent here due to the lack of penguin interference).

3. Key Contributions

• First PQCD Predictions: This work provides the first complete PQCD predictions for the branching fractions of the quasi-two-body decays $B^+ \to D_s^+ (R \to) K^+ K^-$, covering S, P, and D-wave resonances.
• Systematic Treatment of $K^+ K^-$ Dynamics: The paper systematically incorporates the two-meson distribution amplitudes for scalar, vector, and tensor resonances, extending previous studies that focused primarily on charmless decays or specific resonance types.
• Extraction of Two-Body Branching Fractions: By applying the Narrow-Width Approximation (NWA), the authors extract the branching fractions for the underlying two-body decays $B^+ \to D_s^+ R$. These results serve as a consistency check against previous theoretical calculations and provide targets for future experiments.
• CP Asymmetry Analysis: The authors rigorously demonstrate that within the Standard Model, direct CP asymmetries vanish for these specific decay modes because the process is mediated exclusively by tree-level operators without interference from penguin diagrams.

4. Results

The numerical results are summarized in Table II (Quasi-two-body) and Table III (Extracted two-body) of the paper:

• Branching Fractions:
  • The predicted branching fractions for the quasi-two-body modes lie in the range of $10^{-8}$ to $10^{-6}$.
  • $B^+ \to D_s^+ (\phi(1020) \to) K^+ K^-$: $\approx 7.52 \times 10^{-8}$. This is small because the process proceeds purely through annihilation topologies, which are power-suppressed.
  • $B^+ \to D_s^+ (f_0(980) \to) K^+ K^-$: $\approx 9.90 \times 10^{-8}$. Despite having color-allowed emission contributions, the rate is suppressed because the $K^+ K^-$ invariant mass slightly exceeds the $f_0(980)$ pole mass, restricting the phase space to the off-shell tail.
  • $B^+ \to D_s^+ (f_0(1370) \to) K^+ K^-$: $\approx 1.04 \times 10^{-6}$ (The largest predicted rate).
  • $B^+ \to D_s^+ (f_2(1270) \to) K^+ K^-$: $\approx 6.33 \times 10^{-7}$.
• Extracted Two-Body Decays:
  • Using the NWA, the branching fraction for $B^+ \to D_s^+ f_2(1270)$ is extracted as $\approx 27.5 \times 10^{-6}$, which is consistent with previous PQCD predictions ($\approx 29.9 \times 10^{-6}$).
  • The NWA was not applied to the $f_0(980)$ channel because the decay $f_0(980) \to K^+ K^-$ is kinematically forbidden at the resonance pole; the observed process relies on the finite-width tail, violating NWA assumptions.
• Uncertainties: The dominant theoretical uncertainty arises from hadronic inputs (wave functions and Gegenbauer moments), followed by scale variations and CKM matrix element uncertainties.

5. Significance

• Experimental Guidance: The predicted branching fractions ($10^{-8} - 10^{-6}$) are within the sensitivity reach of current and future experiments like LHCb and Belle II. This provides a concrete target for experimentalists to perform amplitude analyses of the $K^+ K^-$ subsystem.
• New Physics Probe: The finding that direct CP asymmetries vanish in the SM for these modes is of critical importance. Any experimentally observed non-zero CP asymmetry in $B^+ \to D_s^+ K^+ K^-$ would constitute a clear and unambiguous signal of physics beyond the Standard Model, as the SM prediction is strictly zero due to the absence of penguin contributions.
• Theoretical Validation: The consistency between the NWA-extracted two-body branching fractions and previous independent PQCD calculations validates the two-meson wave function formalism used in this study, reinforcing its applicability to complex three-body decay dynamics.