Lessons from LHCb and Belle II measurements of $B\to J/ψπ$ and $B\to J/ψK$ decays

This paper utilizes recent LHCb and Belle II measurements of CP asymmetries in $B\to J/\psi\pi$ decays, combined with flavor-SU(3) relations and first-order breaking corrections, to generate new Standard Model predictions for CP violation and decay rates in related $B\to J/\psi K$ and $B_s$ processes.

The Gist

Imagine the universe as a giant, complex machine built from tiny, invisible building blocks called particles. Physicists are like mechanics trying to understand the blueprints of this machine. One of the most important blueprints is the "Standard Model," which predicts how these particles should behave.

This paper is about two teams of mechanics (the LHCb and Belle II collaborations) who recently took very precise measurements of a specific type of particle breakdown. They looked at how a heavy particle called a B-meson decays (breaks apart) into a J/ψ particle and a lighter particle (either a pion or a kaon).

Here is the story of what they found and what it means, explained simply:

1. The Mystery of the "Mirror World" (CP Violation)

In the universe, there is a subtle rule called CP symmetry. Think of it like looking in a mirror. If you watch a movie of a particle decay in a mirror, it should look exactly the same as the real movie.

However, nature has a tiny glitch. Sometimes, the "mirror movie" plays slightly differently than the real one. This is called CP violation. It's like a clock that ticks slightly faster in the mirror than in reality. This glitch is crucial because it helps explain why our universe is made of matter instead of being empty space where matter and antimatter canceled each other out.

2. The Six "Twins" and the Rulebook

The paper focuses on six specific decay modes (ways the particles can break apart). Imagine these six decays as six identical twins. Because of a fundamental symmetry in physics called SU(3) flavor symmetry, these twins are supposed to behave in very similar, predictable ways.

• The Twins: Some twins decay into pions, others into kaons. Some are charged, some are neutral.
• The Rulebook (SU(3) Relations): The authors use a mathematical "rulebook" that says, "If Twin A behaves this way, Twin B must behave that way, unless there is a small, known exception."

3. The New Measurements

Recently, the LHCb and Belle II teams measured a few of these twins with high precision:

• They measured how often a specific charged B-meson breaks into a J/ψ and a pion.
• They measured how often a neutral B-meson breaks into a J/ψ and a neutral pion.
• They found a tiny difference in how these twins behave compared to their "antimatter" versions (the CP violation).

4. Predicting the Unknown

The main goal of the paper is to use these new measurements of the "known twins" to predict the behavior of the "unknown twins" that haven't been measured yet.

Using their rulebook, the authors made several predictions:

• The Missing Link: They predicted the CP violation for a decay involving a Kaon ($B^+ \to J/\psi K^+$). They found it should be very small, almost zero, but slightly negative.
• The "Golden" Difference: There is a famous measurement in physics called $\sin 2\beta$ (a value that describes the universe's matter-antimatter imbalance). The authors calculated the difference between this famous value and the new measurements. Their result suggests the difference is tiny—almost zero. This is a good sign for the Standard Model, as it means the "blueprint" is holding up.
• The Ghost Particle: They predicted the behavior of a very rare decay ($B_s \to J/\psi \pi^0$) that is currently too hard to measure. They set a "lower bound," saying, "If you look hard enough, you will find this happening at least this often."

5. The "Small Cracks" in the Rulebook

The authors are honest about the limitations. The "rulebook" (SU(3) symmetry) isn't perfect; it's like a map that is 95% accurate but has some small errors because the particles aren't perfectly identical (one is slightly heavier than the other).

• The Analogy: Imagine the twins are wearing shoes. The rulebook assumes they all wear the same size. In reality, one twin wears a size 9 and another a size 9.5. The authors calculated how much this "shoe size difference" (called symmetry breaking) messes up the predictions. They found that while it adds some noise, the main predictions still hold up.
• They also discussed "higher-order corrections," which are like accounting for the wind or temperature affecting the twins' walk. They concluded that while these factors exist, they don't ruin the main conclusions, though future, more precise measurements will be needed to be 100% sure.

Summary

In short, this paper is a cross-check. The LHCb and Belle II teams measured a few pieces of a puzzle. The authors used a mathematical framework (the SU(3) rulebook) to fill in the missing pieces.

Their findings suggest that:

1. The Standard Model's predictions for these specific particle decays are working well.
2. The "glitch" (CP violation) in these decays is consistent with what we expect.
3. We can now predict the behavior of particles we haven't even seen clearly yet, guiding future experiments on what to look for.

It's a story of using a few known facts to solve a larger mystery, confirming that our current understanding of the universe's building blocks is still on solid ground.

Technical Summary

Technical Summary: Lessons from LHCb and Belle II measurements of $B \to J/\psi\pi$ and $B \to J/\psi K$ decays

Problem Statement
The paper addresses the need to interpret recent experimental measurements of CP violation and decay rates in the $B_q \to J/\psi P$ system (where $q=u,d,s$ and $P=\pi, K$) within the Standard Model (SM). Specifically, the LHCb collaboration recently reported the first evidence for direct CP violation in $B^\pm \to J/\psi \pi^\pm$, and Belle II has measured CP asymmetries in $B^0 \to J/\psi \pi^0$. The authors aim to utilize these new data points, combined with flavor-SU(3) symmetry relations, to predict unmeasured observables and refine the extraction of the CKM parameter $\sin 2\beta$ from the "golden mode" $B^0 \to J/\psi K_S$. A central goal is to quantify the difference between the measured CP asymmetry $S_{\psi K_S}$ and the theoretical parameter $\sin 2\beta$, which is sensitive to potential new physics or higher-order SM corrections.

Methodology
The authors employ a formalism based on flavor-SU(3) symmetry to relate the amplitudes of six decay modes: $B^+ \to J/\psi \pi^+$, $B^+ \to J/\psi K^+$, $B^0 \to J/\psi \pi^0$, $B^0 \to J/\psi K^0$, $B_s \to J/\psi K^0$, and $B_s \to J/\psi \pi^0$.

1. Amplitude Decomposition: The decay amplitudes are expanded in terms of CKM factors ($\lambda^q_i = V^*_{ib}V_{iq}$) and hadronic matrix elements. The authors expand the amplitudes to first order in the SU(3) breaking spurion $\epsilon$ (where $\epsilon \sim f_K/f_\pi - 1 \approx 0.2$) and zeroth order in isospin breaking $\delta$.
2. Parameter Counting: The analysis identifies 16 observables (branching ratios and CP asymmetries) across the six modes. Under the stated approximations (neglecting terms of order $\mathcal{O}(\epsilon R_u)$, $\mathcal{O}(R_u^2)$, and $\mathcal{O}(\delta)$), the system depends on 12 real parameters, yielding four independent relations.
3. Input Data: The analysis incorporates recent measurements from LHCb and Belle II, including branching ratios for $B^0 \to J/\psi \pi^0$ and CP asymmetries ($C$ and $S$ parameters) for neutral decays.
4. Higher-Order Corrections: Section 4 explicitly investigates second-order corrections in $\eta, \rho, \epsilon$ to the relation between $S_{\psi K_S}$ and $\sin 2\beta$. The authors demonstrate how certain theoretical amplitude ratios can be replaced by experimentally measurable observables (such as $R^{sd}_{K^0 K^0}$ and $C_{\psi K_S}$) to assess the magnitude of these corrections.

Key Contributions and Results

• Direct CP Violation Predictions:
  Using the measured difference $\Delta A_{CP} = A_{\psi \pi^+} - A_{\psi K^+}$ and the SU(3) relation $A_{\psi K^+}/A_{\psi \pi^+} = -|V_{cd}/V_{cs}|^2$, the authors extract individual CP asymmetries:

  • $A_{\psi \pi^+} \simeq (+1.23 \pm 0.47) \times 10^{-2}$
  • $A_{\psi K^+} \simeq (-6.5 \pm 2.5) \times 10^{-4}$
    They also predict the direct CP asymmetry for $B_s \to J/\psi K_S$ ($C_{\psi K_S} = -0.17 \pm 0.19$) and $B_s \to J/\psi \pi^0$ ($C_{\psi \pi^0} = 0$), noting that the latter prediction relies on neglecting isospin-breaking terms of order $\lambda^s_c \delta A^{(2)}_c$.

• Indirect CP Violation and $\sin 2\beta$:
  The paper derives a relation connecting the difference $S_{\psi K_S} - \sin 2\beta$ to the measured asymmetry in $B_s \to J/\psi K_S$ ($S^s_{\psi K_S}$):
  $$S^d_{\psi K_S} - \sin 2\beta = -|V_{cd}/V_{cs}|^2 \frac{\cos 2\beta}{\cos 2\beta_s} (S^s_{\psi K_S} + \sin 2\beta_s)$$
  Using current data, this yields a constraint of $S^d_{\psi K_S} - \sin 2\beta = +0.001 \pm 0.015$. The authors emphasize that reducing the uncertainty on $S^s_{\psi K_S}$ (currently $\sim 0.4$) to below $0.3$ would determine this difference to better than one percent.

  Alternatively, using a relation involving $S^d_{\psi \pi^0}$ and isospin asymmetries ($\Delta_K, \Delta_\pi$), they estimate $\sin 2\beta - S^d_{\psi K_S} \approx +0.05 \pm 0.03$. They note this central value is surprisingly large but consistent with zero within $2\sigma$.

• $B_s \to J/\psi \pi^0$ Decay:
  The authors establish a lower bound on the branching ratio $\mathcal{B}(B_s \to J/\psi \pi^0) \gtrsim 9 \times 10^{-7}$, which is a factor of 13 below the current experimental upper bound. They also predict the indirect CP asymmetry $S^s_{\psi \pi^0} = -\sin(2\gamma - 2\beta_s) = -0.78 \pm 0.07$.

• Higher-Order Analysis:
  The paper provides a framework to express higher-order corrections to the $S_{\psi K_S} - \sin 2\beta$ relation in terms of measurable quantities like $R^{sd}_{K^0 K^0}$ and $C_{\psi K_S}$. This allows future experiments to assess the significance of unknown $\mathcal{O}(\epsilon)$ and $\mathcal{O}(\rho)$ terms without relying solely on theoretical estimates.

Significance and Claims
The paper claims that the system of six $B_q \to J/\psi P$ decays offers a robust test of the SM and a precision tool for determining CKM parameters. By leveraging the new LHCb and Belle II data, the authors provide specific predictions for unmeasured observables that can be tested in the near future.

The authors modestly state that their predictions for $B_s \to J/\psi \pi^0$ and the extraction of $\sin 2\beta$ rely on specific approximations (neglecting certain isospin-breaking and higher-order SU(3) breaking terms). They explicitly note that the prediction $C_{\psi \pi^0} = 0$ is subject to uncertainty if isospin-breaking terms are not negligible. Furthermore, they highlight that the precision of the $\sin 2\beta$ extraction is currently limited by experimental uncertainties in $S^s_{\psi K_S}$ and isospin violation measurements in $\Upsilon(4S)$ decays. The paper concludes that further experimental improvements in the eight currently measured observables, and the measurement of the eight unmeasured ones, will significantly enhance the ability to test the SM and constrain new physics.