Measurement of $\gamma$ using $B^{\pm}\rightarrow DK^{\pm}$ and $B^{\pm}\rightarrow D\pi^{\pm}$ decays with $D\rightarrow K_{\rm S}^{0}\pi^{+}\pi^{-}$ and $D\rightarrow K_{\rm S}^{0}K^{+}K^{-}$

Using the upgraded LHCb detector with 5.8 fb$^{-1}$ of 2024 data, this paper presents the first measurement of the CKM angle $\gamma$ via $B^{\pm}\rightarrow DK^{\pm}$ and $B^{\pm}\rightarrow D\pi^{\pm}$ decays with $D$ mesons decaying to $K_{\rm S}^{0}\pi^{+}\pi^{-}$ or $K_{\rm S}^{0}K^{+}K^{-}$, yielding a value of $\gamma=(68.1\pm 6.7)^{\circ}$ through the observation of $C\!P$ violation in Dalitz plot distributions.

The Gist

Imagine the universe as a giant, complex clockwork machine. For decades, physicists have been trying to understand why this machine sometimes runs slightly differently when you look at it in a mirror. This phenomenon is called CP violation (Charge-Parity violation). It's the reason why the universe is made of matter rather than being an empty void where matter and antimatter canceled each other out after the Big Bang.

This paper from the LHCb collaboration at CERN is like a team of master watchmakers who have just finished a high-precision inspection of one specific gear in that cosmic clock. They measured a specific angle, named gamma ($\gamma$), which is a crucial piece of the puzzle explaining how matter and antimatter behave differently.

Here is a breakdown of what they did, using simple analogies:

1. The Goal: Measuring the "Twist"

In the Standard Model of physics (our best rulebook for how particles work), there is a shape called the "Unitary Triangle." Think of this triangle as a map of the rules governing how particles mix and change. One of the corners of this triangle is the angle $\gamma$.

If we measure this angle perfectly and it matches our predictions, our rulebook is correct. If it doesn't match, it means there are hidden forces or "new physics" we haven't discovered yet. This paper reports a new, very precise measurement of that angle.

2. The Experiment: A Cosmic Dance Floor

To measure this angle, the scientists used the LHCb detector, a massive, high-tech camera and sensor array at CERN. They looked at a specific "dance" performed by particles:

• The Dancers: They watched B-mesons (heavy particles) decay into D-mesons and either a Kaon or a Pion.
• The Spin: The D-meson then decays further into other particles.
• The Mirror Trick: The scientists compared the dance of the "matter" version of the particle ($B^+$) with the "antimatter" version ($B^-$).

If the laws of physics were perfectly symmetrical, these two dances would look identical. But because of CP violation, the "matter" dancer and the "antimatter" dancer take slightly different steps. The difference in their steps reveals the value of the angle $\gamma$.

3. The Method: The "Dalitz Plot" Map

To see these subtle differences, the scientists didn't just count the particles; they mapped out where the particles landed. They used a tool called a Dalitz plot, which is like a scatter plot or a map of a dance floor.

• The Binning Strategy: Imagine the dance floor is a giant pizza. The scientists cut this pizza into many slices (bins). They counted how many "matter" dancers landed in each slice versus how many "antimatter" dancers landed there.
• The Interference: The particles behave like waves. When the waves from the matter and antimatter paths overlap, they interfere with each other (like ripples in a pond). This interference creates a pattern on the pizza slices that changes depending on the angle $\gamma$.

4. The Data: A Fresh Look

This paper is special because it uses data collected in 2024 by the upgraded LHCb detector.

• The Upgrade: Think of the old detector as a standard camera, and the new one as a high-speed, 4K camera with better lenses. It can see faster and catch more details.
• The Sample: They analyzed data equivalent to 5.8 inverse femtobarns of collisions. To put that in perspective, it's like watching billions of particle collisions to find a few thousand specific "golden tickets" (the signal events) amidst a sea of noise.

5. The Results: The Final Number

After crunching the numbers and accounting for all the possible errors (like background noise or slight imperfections in the camera), they arrived at their result:

$\gamma = 68.1^\circ \pm 6.7^\circ$

• What this means: The angle is approximately 68 degrees. The "$\pm 6.7$" is the margin of error, like saying a measurement is "about 68 degrees, give or take a few."
• The Comparison: This result is consistent with previous measurements and with indirect predictions from other parts of physics. It's like checking a new thermometer against an old one; they agree, which gives us confidence that our "rulebook" (the Standard Model) is still holding up.

6. Why This Matters (According to the Paper)

The paper emphasizes that this is the first measurement of $\gamma$ using the upgraded LHCb detector. It proves that the new, faster detector works exactly as promised.

• No "New Physics" Found (Yet): The result fits perfectly with the current Standard Model. This is good news for the theory, but it also means the scientists haven't found the "smoking gun" for new, unknown physics in this specific measurement.
• Precision: The measurement is limited by statistics (they need even more data to shrink the error bar), not by a lack of understanding of the theory.

Summary

In short, the LHCb team used a super-powered microscope to watch heavy particles dance. By comparing the steps of matter and antimatter dancers on a mapped-out dance floor, they measured a fundamental angle of the universe to be 68.1 degrees. This confirms that our current understanding of the universe's rules is solid, even as they continue to hunt for the tiny cracks where new physics might hide.

Technical Summary

Technical Summary: Measurement of $\gamma$ using $B^\pm \to DK^\pm$ and $B^\pm \to D\pi^\pm$ Decays

Problem and Motivation
The origin of CP violation in the quark sector is described within the Standard Model (SM) by the Cabibbo-Kobayashi-Maskawa (CKM) matrix. A primary goal of particle physics is to overconstrain the Unitary Triangle to test the SM's consistency; a non-closure would indicate physics beyond the SM. The angle $\gamma \equiv \arg(-V_{ud}V^*_{ub}/V_{cd}V^*_{cb})$ is the only CKM angle measurable in tree-level decays without significant loop contributions, offering a benchmark with negligible theoretical uncertainties. While previous LHCb measurements have established a combined direct determination of $\gamma = (62.8 \pm 2.6)^\circ$, this paper presents the first measurement of $\gamma$ using the upgraded LHCb detector (Run 3), utilizing data collected in 2024 at a center-of-mass energy of $\sqrt{s} = 13.6$ TeV.

Methodology
The analysis utilizes a model-independent approach based on the interference between $b \to c\bar{u}s$ and $b \to u\bar{c}s$ decay amplitudes in $B^\pm \to DK^\pm$ and $B^\pm \to D\pi^\pm$ decays, where the $D$ meson decays to $K^0_S\pi^+\pi^-$ or $K^0_S K^+K^-$.

• Observables: The analysis measures six CP observables: $x^{DK}_\pm$ and $y^{DK}_\pm$ for the $B^\pm \to DK^\pm$ modes, and $x^{D\pi}_\xi$ and $y^{D\pi}_\xi$ for the $B^\pm \to D\pi^\pm$ modes. These are related to the weak phase $\gamma$, the amplitude ratio $r_B$, and the strong phase difference $\delta_B$.
• Dalitz Plot Binning: The $D$-decay phase space is partitioned into bins symmetric around $m^2_- = m^2_+$ (where $m^2_\pm$ are the squared masses of the $K^0_S h^\mp$ combinations). For $D \to K^0_S\pi^+\pi^-$, an "optimal" 8-bin scheme is used; for $D \to K^0_S K^+K^-$, a 2-bin scheme is employed.
• Strong-Phase Inputs: To avoid model-dependent assumptions about the strong-phase variation across the Dalitz plot, the analysis incorporates external measurements of the average cosine ($c_i$) and sine ($s_i$) of the strong-phase difference in each bin. These values are taken from quantum-correlated charm measurements by the BESIII and CLEO collaborations.
• Data Sample: The measurement uses an integrated luminosity of $5.8 \text{ fb}^{-1}$ collected in 2024. The analysis distinguishes between "long" and "downstream" $K^0_S$ reconstruction categories based on decay vertex position.
• Fit Strategy: A two-step fit procedure is employed:
  1. Global Mass Fit: An unbinned extended maximum-likelihood fit to the invariant-mass spectra of selected $B^\pm$ candidates determines signal and background yields, as well as mass shape parameters. This step accounts for misidentified backgrounds (e.g., $B^\pm \to D\pi^\pm$ misidentified as $B^\pm \to DK^\pm$) and partially reconstructed backgrounds.
  2. CP Fit: A simultaneous fit to the invariant-mass distributions across 160 subsets (defined by $B$ charge, decay mode, $K^0_S$ type, and Dalitz bin) determines the six CP observables. The signal yields in each bin are constrained by the theoretical relations involving $x, y, c_i, s_i$, and the fractional yields $F_i$.

Key Contributions

• First Run 3 Measurement: This is the inaugural measurement of $\gamma$ using the upgraded LHCb detector, demonstrating the capability of the new all-software trigger system and improved tracking to select low transverse momentum tracks, resulting in a signal selection efficiency improvement of approximately a factor of 2.7 for the "long" $K^0_S$ category compared to Run 1/2.
• Model Independence: By utilizing external strong-phase measurements ($c_i, s_i$) rather than amplitude models, the analysis minimizes theoretical uncertainties associated with the $D$-decay dynamics.
• Control Mode Utilization: The $B^\pm \to D\pi^\pm$ channel, while having low sensitivity to $\gamma$ due to a small amplitude ratio ($r^{D\pi}_B \approx 0.005$), is used as a control mode to constrain normalization and efficiency effects, reducing dependence on simulation.

Results
The CP observables are measured as follows (units of $10^{-2}$):

• $x^{DK}_- = 4.81 \pm 0.88 \text{ (stat)} \pm 0.20 \text{ (syst)} \pm 0.23 \text{ (strong-phase)}$
• $y^{DK}_- = 6.70 \pm 1.26 \pm 0.44 \pm 0.56$
• $x^{DK}_+ = -7.63 \pm 0.88 \pm 0.28 \pm 0.15$
• $y^{DK}_+ = -1.20 \pm 1.34 \pm 0.35 \pm 0.44$
• $x^{D\pi}_\xi = -9.44 \pm 2.51 \pm 0.57 \pm 0.69$
• $y^{D\pi}_\xi = 2.76 \pm 2.99 \pm 0.19 \pm 1.21$

Interpreting these observables yields the CKM angle:
$$\gamma = (68.1 \pm 6.7)^\circ$$
The uncertainty is dominated by statistical limitations. The measurement also determines the hadronic parameters:

• $r^{DK}_B = 0.0781^{+0.0078}_{-0.0079}$
• $\delta^{DK}_B = (121.5^{+6.9}_{-7.4})^\circ$
• $r^{D\pi}_B = 0.0073^{+0.0016}_{-0.0015}$
• $\delta^{D\pi}_B = (286^{+20}_{-23})^\circ$

The result is consistent with previous LHCb measurements and indirect determinations from global CKM fits. A comparison with the previous LHCb combined result yields a $p$-value of 12%.

Significance and Claims
The paper claims that this measurement provides a robust SM benchmark for $\gamma$ with minimal theoretical uncertainty. The authors note that the precision of this specific measurement is approximately 20% lower than the previous LHCb combined result, despite higher signal yields. This is attributed to two factors:

1. Updated central values for the strong-phase parameters ($c_i, s_i$) from BESIII, which reduced the per-event sensitivity in pseudoexperiments.
2. A lower determined value of $r^{DK}_B$, which scales inversely with the uncertainty on $\gamma$.

The analysis successfully validates the upgraded LHCb detector's performance for precision flavor physics, particularly its ability to handle high-luminosity data with complex trigger strategies and to utilize external strong-phase inputs to achieve a model-independent determination of $\gamma$. The results are consistent with the Standard Model, and no evidence of non-closure of the Unitary Triangle is found within the current precision.