Contributions of the subprocesses $ρ(770,1450,1700)\to K \bar{K}$ and $ω(782,1420,1650)\to K \bar{K}$ for the three-body decays $B\to η^{(\prime)} K\bar{K}$

Using the perturbative QCD approach, this paper analyzes resonance contributions from $\rho$ and $\omega$ mesons and their excited states to the three-body decays $B\to \eta^{(\prime)} K\bar{K}$, revealing that virtual contributions from the tails of $\rho(770)$ and $\omega(782)$ are comparable to those from higher resonances and thus constitute a significant component of these decays.

The Gist

Imagine the subatomic world as a bustling, high-energy dance floor. In this paper, physicists are trying to understand a very specific, complex dance move: a heavy "B-meson" (a particle made of a bottom quark) breaking apart into three lighter particles: an eta meson and a pair of kaons (a "kaon-antikaon" pair).

The authors, Ming-Yue Jia and colleagues, are essentially acting as forensic accountants for this particle dance. They want to know exactly how much credit to give to different "intermediate dancers" (resonances) that briefly appear during the breakup.

Here is the breakdown of their work using simple analogies:

1. The Problem: The "Ghost" Dancers

When a B-meson decays, it doesn't always just split into three pieces instantly. Often, it splits into two pieces first, and one of those pieces is a short-lived "resonance" (like a $\rho$ or $\omega$ particle) that immediately falls apart into the final two kaons.

Think of it like a relay race:

• Runner A (The B-meson) passes the baton to Runner B (The Resonance).
• Runner B sprints a short distance and then passes the baton to Runners C and D (The Kaons).

Usually, scientists focus on the "official" runners: the heavy, well-known resonances like $\rho(1450)$ or $\omega(1420)$. These are the big, strong athletes who are clearly visible on the track.

However, this paper points out that there are "ghost" runners too. Specifically, the lighter, famous resonances $\rho(770)$ and $\omega(782)$ are too light to naturally turn into a pair of kaons (it's like a small child trying to carry a heavy adult). In normal physics, we would say, "They can't do that; it's impossible."

But in the quantum world, things are fuzzy. These light particles can still "virtually" try to become kaon pairs, even if they don't have enough energy to do it perfectly. This is called the Breit-Wigner tail. It's like a ghostly echo of the runner that lingers on the track even after they've technically finished their race.

2. The Discovery: The Ghosts Are Loud

The authors used a sophisticated mathematical toolkit called Perturbative QCD (think of it as a high-precision calculator for particle interactions) to measure these "ghost" contributions.

The Big Surprise:
They found that these "ghost" contributions from the light $\rho(770)$ and $\omega(782)$ are not negligible. In fact, they are just as important as the contributions from the heavier, "official" runners ($\rho(1450)$ and $\omega(1420)$).

The Analogy:
Imagine you are trying to calculate the total noise in a concert hall. You measure the loud drums (the heavy resonances) and realize they are very loud. You might ignore the faint hum of the air conditioning (the light resonances).
This paper says: "Wait! The air conditioning hum is actually just as loud as the drums!" If you ignore it, your calculation of the total noise is wrong.

3. The Shape of the Data

The authors also looked at how these particles behave.

• Normal Resonances: Usually, when a particle decays, the data looks like a perfect bell curve (a hill). The peak is right where the particle's mass is.
• The "Ghost" Resonances: Because the light $\rho(770)$ is too light to make kaons, it can't form a nice hill. Instead, the data looks like a broad, flat bump that starts low and rises slowly, peaking at a much higher energy than the particle's actual mass.

It's like trying to push a heavy boulder up a hill. A normal particle rolls up the hill easily. The "ghost" particle is like a boulder that gets stuck at the bottom but somehow manages to push the ground up a little bit further away, creating a weird, wide shape rather than a sharp peak.

4. Why Does This Matter?

The authors calculated the "branching fractions" (essentially, the probability of this specific dance happening) and the "CP asymmetries" (a measure of whether matter and antimatter behave differently).

• The Prediction: They predict these probabilities will be in the range of $10^{-10}$ to $10^{-8}$. This is tiny, but measurable.
• The Future: They are telling the big particle detectors at LHCb (in Europe) and Belle-II (in Japan): "Look for these specific patterns! Don't ignore the light resonances, because they are doing half the work."

Summary

This paper is a correction to the physics community's "to-do list."

• Old View: "Only count the heavy, obvious resonances when calculating B-meson decays."
• New View: "You must also count the 'virtual' contributions from the light resonances. They are invisible to the naked eye but contribute significantly to the final result. If you ignore them, your math is off."

It's a reminder that in the quantum world, even the things that shouldn't happen (like a light particle becoming a heavy pair) can happen just enough to change the outcome of the universe's most complex dances.

Technical Summary

1. Problem Statement

The paper addresses a gap in the theoretical understanding of three-body hadronic $B$ meson decays, specifically the processes $B \to \eta^{(\prime)}K\bar{K}$. While previous studies within the Perturbative QCD (PQCD) framework had analyzed similar decays (e.g., $B \to \pi K\bar{K}$), the specific contributions of intermediate vector resonances decaying into kaon pairs were missing for the $\eta^{(\prime)}$ modes.

Key issues addressed include:

• Virtual Contributions: The natural decay modes of $\rho(770)$ and $\omega(782)$ into $K\bar{K}$ are kinematically forbidden because their pole masses are below the $K\bar{K}$ threshold. However, their "virtual" contributions via the Breit-Wigner (BW) tails in the off-shell region are non-negligible. Previous works suggested these tails might be comparable to contributions from excited resonances like $\rho(1450)$ and $\omega(1420)$, but a systematic calculation for $B \to \eta^{(\prime)}K\bar{K}$ was lacking.
• Resonance Spectrum: The study aims to include not only the ground states $\rho(770)$ and $\omega(782)$ but also their excited states ($\rho(1450, 1700)$ and $\omega(1420, 1650)$) to provide a complete picture of the $K\bar{K}$ invariant mass spectrum in these decays.

2. Methodology

The authors employ the Perturbative QCD (PQCD) approach combined with a quasi-two-body framework.

• Theoretical Framework:

  • The decay is treated as a cascade process: $B \to \eta^{(\prime)} \rho/\omega \to \eta^{(\prime)} K\bar{K}$.
  • The weak interaction is described by the effective Hamiltonian $H_{eff}$ involving current-current, QCD penguin, and electroweak penguin operators.
  • The factorization formula for the decay amplitude is expressed as a convolution of the $B$-meson wave function, the hard scattering kernel ($H$), and the distribution amplitudes (DAs) of the final states.

• Handling the $K\bar{K}$ System:

  • Since the subprocesses $\rho/\omega \to K\bar{K}$ cannot be calculated perturbatively (due to non-perturbative strong dynamics at the resonance scale), the authors introduce kaon vector time-like form factors ($F_K(s)$) into the distribution amplitudes of the $K\bar{K}$ system.
  • These form factors are constructed using a sum of Breit-Wigner functions for the relevant resonances ($\rho, \rho', \rho''$ and $\omega, \omega', \omega''$) and the $\phi$ family (though $\phi$ contributions are noted as natural decays and treated separately).
  • The form factors incorporate isospin decomposition ($I=0$ and $I=1$) and coupling constants ($c_R^K$) derived from experimental data and previous theoretical works.

• Kinematics and Inputs:

  • Calculations are performed in the $B$-meson rest frame using light-cone coordinates.
  • The differential branching fraction is calculated by integrating over the invariant mass squared $s$ of the $K\bar{K}$ pair.
  • Inputs include CKM matrix elements, Wilson coefficients, meson decay constants, and Gegenbauer moments for the distribution amplitudes.

3. Key Contributions

• First Systematic Analysis: This is the first work to systematically calculate the contributions of $\rho(770, 1450, 1700) \to K\bar{K}$ and $\omega(782, 1420, 1650) \to K\bar{K}$ specifically for the $B \to \eta^{(\prime)}K\bar{K}$ decay channels within the PQCD framework.
• Quantification of Virtual Effects: The paper rigorously quantifies the "virtual" contributions from the BW tails of the lightest resonances ($\rho(770)$ and $\omega(782)$). It demonstrates that these off-shell contributions are not negligible artifacts but significant physical components.
• Differential Distribution Analysis: The authors provide differential branching fractions as a function of the $K\bar{K}$ invariant mass, revealing the shape of the distribution for virtual contributions versus resonant peaks.

4. Key Results

• Branching Fractions:

  • The predicted CP-averaged branching fractions for the quasi-two-body decays range from $10^{-10}$ to $10^{-8}$.
  • Crucial Finding: The virtual contributions from the BW tails of $\rho(770)$ and $\omega(782)$ are found to be comparable to or even larger than the contributions from the excited resonances $\rho(1450, 1700)$ and $\omega(1420, 1650)$.
  • For example, in $B^+ \to \eta [\rho(770)^+ \to] K^+ \bar{K}^0$, the branching fraction is $\approx 8.03 \times 10^{-8}$, which is significantly smaller than the two-body $B^+ \to \eta \rho(770)^+$ decay (due to phase space suppression) but comparable to the excited state contributions.

• Differential Branching Fractions:

  • The differential distribution for the virtual $\rho(770) \to K\bar{K}$ process does not follow a typical Breit-Wigner peak near the pole mass. Instead, it exhibits a broad bump peaking around $\sqrt{s} \approx 1.35$ GeV.
  • This peak location is driven by the interplay between the BW tail and the strong phase space suppression factor $|\vec{q}_h|^3$ in the decay rate formula.
  • The shape of this distribution is found to be largely insensitive to the decay width of the intermediate state because, at high invariant masses ($s \gg m_R^2$), the imaginary part of the BW denominator becomes negligible.

• CP Asymmetries:

  • Direct CP asymmetries ($A_{CP}$) are predicted for various channels. Most are small (near zero), but some modes (e.g., $B^0 \to \eta' [\rho(770)^0 \to] K^+ K^-$) show significant asymmetries (up to $\sim 60\%$), though with large theoretical uncertainties.

• Reliability Check:

  • The authors verify that the majority of the branching ratio comes from the low invariant mass region (around 1.35 GeV for $\rho(770)$ tails), ensuring that the PQCD framework (valid for large energy release) remains applicable and reliable for these calculations.

5. Significance

• Completeness of Theoretical Models: The study emphasizes that ignoring the virtual tails of light resonances ($\rho(770), \omega(782)$) in three-body $B$ decays leads to an incomplete and potentially inaccurate description of the $K\bar{K}$ spectrum. These contributions must be included in any comprehensive analysis.
• Experimental Guidance: The predictions provide specific targets for high-statistics experiments like LHCb and Belle II. The distinct shape of the differential branching fraction (the broad bump at 1.35 GeV) offers a unique signature to distinguish virtual contributions from genuine resonant states.
• Future Interference Studies: While this paper calculates individual contributions, the authors note that the relative phases between different resonances are currently unknown. This work lays the necessary groundwork for future studies that will include interference effects to determine total branching fractions and CP asymmetries more precisely.

In summary, this paper establishes that the "off-shell" tails of the lightest vector mesons are a dominant or comparable source of $K\bar{K}$ pairs in $B \to \eta^{(\prime)}K\bar{K}$ decays, challenging the assumption that only excited resonances contribute significantly in the off-shell region.