A model-independent measurement of the CKM angle $\gamma$ in the decays $B^\pm\to[K^+K^-\pi^+\pi^-]_D h^\pm$ and $B^\pm\to[\pi^+\pi^-\pi^+\pi^-]_D h^\pm$ ($h = K, \pi$)

Using 9 fb$^{-1}$ of LHCb proton-proton collision data, this paper presents the first phase-space-binned, model-independent measurement of the CKM angle $\gamma$ in $B^\pm\to[K^+K^-\pi^+\pi^-]_D h^\pm$ and $B^\pm\to[\pi^+\pi^-\pi^+\pi^-]_D h^\pm$ decays, yielding a result of $53.9^\circ$that, when combined with existing integrated measurements, achieves one of the most precise determinations of$\gamma$to date at$52.6^\circ$.

The Gist

Imagine the universe as a grand, cosmic dance floor. For decades, physicists have been trying to understand why the dance floor is filled mostly with "matter" (us, stars, planets) and almost no "antimatter" (the mysterious mirror image of matter that should have been created in equal amounts at the Big Bang). If the dance had been perfectly symmetrical, matter and antimatter would have annihilated each other instantly, leaving nothing behind.

The secret to this imbalance lies in a subtle "twist" in the rules of the dance, known as CP Violation. To measure this twist, physicists look for a specific angle in the mathematical map of particle physics called the CKM angle $\gamma$ (gamma). Think of $\gamma$ as the "secret handshake" that tells us how much the universe prefers one dancer over the other.

This paper from the LHCb collaboration at CERN is a major step forward in measuring that secret handshake with incredible precision. Here is how they did it, explained simply:

1. The Detective Work: Catching Particles in the Act

The scientists used the LHCb detector, a massive, high-tech camera at the Large Hadron Collider, to watch protons smash into each other. When these protons collide, they sometimes create heavy particles called B-mesons. These B-mesons are unstable and quickly decay (fall apart) into other particles.

The team focused on a specific, rare decay chain:

• A B-meson splits into a D-meson and a K-meson (or a pion).
• The D-meson then splits into four particles: either two pions and two kaons, or four pions.

2. The "Quantum Coin Flip" Analogy

Here is where it gets tricky. The D-meson is a "quantum chameleon." It can exist as two different versions at the same time (like a coin spinning in the air, being both heads and tails).

• When the B-meson decays, it can create a "heads" D-meson or a "tails" D-meson.
• These two versions can interfere with each other, like two waves in a pond crashing together.
• The way they crash together depends on the angle $\gamma$.

If we just looked at the average result of millions of these crashes, the waves would cancel each other out, and we'd see nothing. It's like trying to hear a whisper in a noisy room.

3. The "Smart Sorting" Strategy (Binning)

To hear the whisper, the scientists didn't just count the particles; they sorted them into buckets.
Imagine you have a giant jar of mixed jellybeans. If you just count them, you get a total number. But if you sort them by color and flavor, you might notice a pattern: "Hey, all the red ones are on the left side of the jar!"

The scientists divided the "phase space" (a complex map of how the particles fly apart) into bins (buckets).

• The Problem: In the past, to sort these buckets, they had to guess the rules of the dance using a theoretical model. If their guess was wrong, the measurement would be biased. It was like sorting jellybeans based on a guess of what flavor they should be.
• The Breakthrough: This paper is model-independent. Instead of guessing, they used real-world data from a different experiment (BESIII in China) to tell them exactly how the "strong force" (the glue holding the particles together) behaves in each bucket. It's like having a friend who actually ate the jellybeans and told you exactly how they tasted, rather than guessing.

4. The Result: Pinpointing the Angle

By using this "smart sorting" with real-world data, they measured the angle $\gamma$.

• The Measurement: They found $\gamma$ to be approximately 53.9 degrees.
• The Combination: When they combined this new, precise method with older, simpler measurements, the result sharpened to 52.6 degrees.

This is one of the most precise measurements of this angle ever made. It's like going from guessing the time of day to looking at a high-precision atomic clock.

5. Why Does This Matter?

• Testing the Standard Model: The "Standard Model" is our current best theory of how the universe works. This measurement acts as a "stress test." If the angle $\gamma$ measured here doesn't match the angle calculated from other methods, it means our theory is missing something. It could be a hint of New Physics—perhaps a new particle or force we haven't discovered yet.
• Solving the Mystery: While this specific measurement doesn't fully explain why the universe is made of matter, it tightens the constraints on where that explanation might be hiding.

Summary

Think of this paper as the scientists finally putting on 3D glasses to watch a 2D movie. By using a new, model-free way to sort particle data and combining it with external "cheat codes" (data from BESIII), they have measured a fundamental angle of the universe with unprecedented clarity. They haven't solved the whole mystery of the matter-antimatter imbalance yet, but they have cleared away a lot of the fog, bringing us one step closer to understanding why we exist.

Technical Summary

Here is a detailed technical summary of the LHCb paper CERN-EP-2025-199, titled "A model-independent measurement of the CKM angle $\gamma$ in the decays $B^\pm \to [K^+K^-\pi^+\pi^-]Dh^\pm$ and $B^\pm \to [\pi^+\pi^-\pi^+\pi^-]Dh^\pm$ ($h = K, \pi$)".

1. Problem Statement and Physics Motivation

The primary goal of this analysis is to determine the Cabibbo-Kobayashi-Maskawa (CKM) angle $\gamma$ (defined as $\arg(-V_{ud}V^*_{ub}/V_{cd}V^*_{cb})$) with high precision.

• Significance: $\gamma$ is the only CKM angle measurable directly in tree-level $b$-quark decays, making it theoretically clean (negligible hadronic uncertainties). It serves as a "standard candle" for the Standard Model (SM). Comparing direct tree-level measurements with indirect loop-level determinations (which are sensitive to New Physics) allows for stringent tests of the SM and searches for CP violation beyond the SM.
• Challenge: Previous measurements using four-body charm decays ($D \to K^+K^-\pi^+\pi^-$ and $D \to \pi^+\pi^-\pi^+\pi^-$) relied on amplitude models to estimate strong-phase differences across the phase space. This introduced model-dependent systematic uncertainties that were difficult to quantify accurately.
• Objective: To perform the first model-independent, phase-space binned measurement of $\gamma$ using these four-body decay modes, utilizing external strong-phase inputs from the BESIII experiment to eliminate model dependence.

2. Methodology

A. Decay Channels and Strategy

The analysis utilizes the interference between $b \to c\bar{u}s$ and $b \to u\bar{c}s$ transitions in $B^\pm \to Dh^\pm$ decays, where the neutral $D$ meson decays into self-conjugate four-body final states:

1. $D \to K^+K^-\pi^+\pi^-$
2. $D \to \pi^+\pi^-\pi^+\pi^-$

The analysis employs a phase-space binned approach. The $D$-meson phase space is divided into bins ($i$) based on the predicted strong-phase difference between $D^0$ and $\bar{D}^0$ decays. The yields of $B^+$ and $B^-$ decays in each bin are sensitive to $\gamma$ and the strong-phase parameters.

B. External Inputs (BESIII)

Crucially, the analysis uses model-independent measurements of the strong-phase parameters ($c_i, s_i$) provided by the BESIII collaboration:

• For $D \to K^+K^-\pi^+\pi^-$: Inputs from 20 fb$^{-1}$ of $e^+e^-$ data at the $\psi(3770)$ threshold [Ref 18].
• For $D \to \pi^+\pi^-\pi^+\pi^-$: Inputs from 3 fb$^{-1}$ of data [Ref 21].
• These inputs constrain the amplitude-averaged cosine ($c_i$) and sine ($s_i$) of the strong-phase difference in each bin, removing the need for an internal amplitude model.

C. Data Set and Selection

• Data: Proton-proton collision data collected by the LHCb experiment at $\sqrt{s} = 7, 8,$ and $13$TeV, corresponding to an integrated luminosity of **9 fb$^{-1}$**.
• Selection: Candidates are reconstructed with five charged tracks. A kinematic fit constrains the $D$ mass and the $B$ vertex.
• Background Suppression:
  • Multivariate analysis (Boosted Decision Trees) reduces combinatorial background.
  • Particle Identification (PID) distinguishes between $K$ and $\pi$.
  • Vertexing and mass windows suppress charmless backgrounds and misidentified $K_S^0$ decays.
• Normalization: The $B^\pm \to D\pi^\pm$ channel is used as a normalization mode due to its higher branching fraction and negligible CP violation ($r_B^{D\pi} \approx 0.005$).

D. Statistical Framework

• Yield Extraction: An unbinned extended maximum-likelihood fit is performed on the invariant mass spectra ($m(Dh^\pm)$) to extract signal yields in each phase-space bin.
• Parameterization: The yields are parameterized in terms of CP-violating observables ($x_{\pm}^{DK}, y_{\pm}^{DK}, x_{\xi}^{D\pi}, y_{\xi}^{D\pi}$), which are then transformed into physics parameters ($\gamma, \delta_B, r_B$).
• Handling Systematics: Unlike previous analyses where $c_i$ and $s_i$ were fixed, this analysis treats them as constrained nuisance parameters with Gaussian priors based on BESIII measurements. This accounts for their uncertainties directly in the likelihood function.
• Confidence Intervals: Due to the large uncertainties on $s_i$ causing non-Gaussian behavior in the $\gamma$ distribution, the authors use the Feldman-Cousins (Plugin) method via pseudoexperiments to assign robust confidence intervals, rather than relying solely on Wilks' theorem.

3. Key Contributions

1. First Model-Independent Binned Measurement: This is the first time these specific four-body $D$ decay modes have been used for a binned $\gamma$ measurement without relying on an internal amplitude model for strong-phase inputs.
2. Integration of BESIII Data: The analysis successfully incorporates the latest, high-precision strong-phase measurements from BESIII, significantly reducing model-dependent systematic uncertainties.
3. Cross-Checks: The analysis validates the $D \to \pi^+\pi^-\pi^+\pi^-$ results using an alternative binning scheme based on older CLEO-c data, finding excellent agreement.
4. Combined Result: It provides a combined result of the binned analysis with existing phase-space integrated measurements of the same modes.

4. Results

A. Binned Analysis Only

Using the phase-space binned approach for both decay modes:

• $\gamma = (53.9^{+9.5}_{-8.9})^\circ$
• This result includes both statistical and systematic uncertainties. The systematic uncertainty is dominated by the external strong-phase inputs.

B. Combined Analysis (Binned + Integrated)

When combining the new binned results with existing phase-space integrated measurements (from Ref. [16]):

• $\gamma = (52.6^{+8.5}_{-6.4})^\circ$
• This is one of the most precise determinations of $\gamma$ to date.
• The central value is consistent with the global LHCb combination of other decay modes ($\gamma \approx 64.4^\circ$) and indirect loop-level determinations, though the uncertainty remains larger than the indirect methods.

C. Other Physics Parameters

The analysis also extracted:

• $r_B^{DK} = 0.101^{+0.021}_{-0.018}$
• $\delta_B^{DK} = (111.0^{+8.0}_{-7.9})^\circ$
• The parameter $r_B^{D\pi}$ was found to be consistent with zero, with a confidence interval covering the physical boundary.

5. Significance and Future Prospects

• Precision: The result represents a significant step forward in precision for tree-level $\gamma$ measurements using four-body charm decays.
• Model Independence: By removing model dependence, the systematic uncertainty is better controlled, making the result more robust for global CKM fits.
• Non-Gaussianity: The study highlights that large uncertainties on external strong-phase inputs can induce non-Gaussian distributions in $\gamma$, necessitating the use of advanced statistical methods (Plugin method) for accurate error estimation.
• Future Improvements:
  • The measurement is currently statistically limited. The LHCb Upgrade I will provide larger datasets, improving statistical precision.
  • The dominant systematic uncertainty comes from the BESIII strong-phase inputs. Future BESIII data (full 20 fb$^{-1}$) is expected to reduce these uncertainties by a factor of ~2.5, further enhancing the precision of $\gamma$.
  • Complementary measurements of charm mixing by LHCb will help constrain the system further.

In conclusion, this paper establishes a new benchmark for model-independent $\gamma$ measurements, demonstrating the power of combining LHCb's high-statistics $B$-physics data with precision charm-physics inputs from BESIII.