Precision measurement of CP violation and branching fractions in $B^{\pm} \to K^0_{\mathrm{S}} h^{\pm}$ $(h = \pi, K)$ decays and search for the rare decay $B_c^{\pm} \to K^0_{\mathrm{S}} K^{\pm}$

Using 13 TeV proton-proton collision data from the LHCb experiment, this study presents the most precise measurements to date of CP asymmetries and the branching fraction ratio for $B^{\pm} \to K^0_{\mathrm{S}} \pi^{\pm}$ and $B^{\pm} \to K^0_{\mathrm{S}} K^{\pm}$ decays, while also setting upper limits on the rare $B_c^{\pm} \to K^0_{\mathrm{S}} K^{\pm}$ decay.

The Gist

Imagine the universe as a giant, cosmic kitchen where particles are the ingredients. Most of the time, when these ingredients mix and react, they follow a strict recipe book called the Standard Model. This book predicts exactly how things should behave. But sometimes, physicists suspect there might be a "secret ingredient" or a "ghost chef" (New Physics) tweaking the recipe in ways we can't see yet.

This paper from the LHCb collaboration at CERN is like a team of master chefs who have just finished a very precise taste test of a specific, rare dish: the decay of a B meson (a heavy particle containing a bottom quark) into lighter particles.

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

1. The Main Dish: $B^{\pm} \to K^0_S \pi^{\pm}$ and $K^0_S K^{\pm}$

Think of a B meson as a heavy, unstable balloon. When it pops (decays), it releases smaller particles.

• The Goal: They wanted to see if the balloon pops in a perfectly symmetrical way or if it favors one side over the other.
• The "CP Violation" Concept: In physics, there's a rule called CP symmetry. It's like a mirror. If you look at a particle decay in a mirror, it should look exactly the same as the real thing, just flipped. If the mirror image behaves differently, that's called CP Violation.
  • The Analogy: Imagine a coin toss. If you flip a coin 1,000 times, you expect roughly 500 heads and 500 tails. If you get 900 heads and 100 tails, something is "violating" the symmetry. In the Standard Model, this specific "coin toss" (the decay of the B meson) is predicted to be perfectly balanced (50/50).
  • The Twist: If the universe is rigged by "New Physics," the coin might be slightly weighted. The LHCb team wanted to weigh the coin with extreme precision to see if it's truly fair.

2. The Experiment: The Great Particle Hunt

The team used the Large Hadron Collider (LHC), a massive particle accelerator that smashes protons together like two high-speed trains colliding.

• The Data: They collected data from 2016–2018, which is like gathering 5.4 "buckets" of collision data (measured in inverse femtobarns).
• The Filter (The BDT): The collisions create a chaotic mess of debris. To find the specific "balloon pops" they were looking for, they used a Boosted Decision Tree (BDT).
  • The Analogy: Imagine trying to find a specific, rare red marble in a giant pile of mixed gravel, sand, and other marbles. The BDT is a super-smart robot sieve that learns what the red marble looks like and filters out everything else, leaving only the best candidates.

3. The Results: The "Fair Coin" Check

After filtering the data, they counted the results:

• The First Decay ($B^{\pm} \to K^0_S \pi^{\pm}$): They found that the "coin" is indeed very fair. The imbalance was tiny (about -2.8%). This matches the Standard Model's prediction that this specific decay should be almost perfectly symmetrical.
  • Significance: This is the most precise measurement ever made for this specific decay. It sets a new "gold standard" for how clean this test is.
• The Second Decay ($B^{\pm} \to K^0_S K^{\pm}$): This one is rarer and more complex. They found a larger imbalance (about 11.8%), but the error bars (the "fuzziness" of the measurement) are still a bit wide.
  • Significance: This result is interesting because it's starting to show a difference between what the Standard Model predicts and what they see. It's like the coin is slightly weighted, but they need to flip it a few more times to be sure it's not just a fluke.

4. The Side Quest: The "Ghost" Particle ($B_c^+$)

They also looked for a very rare, "ghostly" decay called $B_c^+ \to K^0_S K^{\pm}$.

• The Analogy: This is like looking for a specific, almost impossible flavor of ice cream in a shop that doesn't usually sell it. Theoretically, this decay happens through a mechanism called "weak annihilation," which is very poorly understood.
• The Outcome: They didn't find any "ghosts." No signal was detected.
• The Takeaway: While they didn't find the particle, they set a strict "speed limit" (an upper limit) on how often this could possibly happen. This helps rule out some wild theories about how these particles interact.

5. Why Does This Matter?

Think of the Standard Model as a map of the world. It's a great map, but we know there are unexplored territories (Dark Matter, gravity, etc.).

• Precision is Key: You can't find a tiny island on a map if your compass is wobbly. By making the most precise measurements ever of these decays, the LHCb team is sharpening their compass.
• The Verdict: So far, the map looks correct. The "coin" is mostly fair. However, the slight wobble in the second decay suggests there might be a tiny island nearby. If future experiments (with even more data) confirm that wobble, it means the Standard Model is incomplete, and we've found a crack in the foundation of our understanding of the universe.

In summary: The LHCb team used a giant particle microscope to weigh a cosmic coin with incredible precision. The coin looks mostly fair, confirming our current laws of physics, but the measurement is so sharp that it might just be the first hint of a new, hidden law of nature waiting to be discovered.

Technical Summary

1. Problem and Motivation

The paper addresses the need for high-precision tests of the Standard Model (SM) in the flavor sector, specifically focusing on charmless hadronic $B$ meson decays.

• $B^\pm \to K^0_S \pi^\pm$: This decay is theoretically "clean" because it is dominated by a single gluonic penguin amplitude with no tree-level contribution and highly CKM-suppressed subleading terms. The SM predicts its direct CP asymmetry ($A_{CP}$) to be very close to zero. Any significant deviation would be a clear signal of New Physics (NP).
• $B^\pm \to K^0_S K^\pm$: This decay proceeds via a doubly CKM-suppressed $b \to ssd$ transition. Its theoretical predictions suffer from large uncertainties due to the interplay of perturbative and non-perturbative QCD effects (specifically annihilation topologies). Precision measurements here are crucial to discriminate between theoretical frameworks like QCD Factorization (QCDF) and Perturbative QCD (pQCD).
• $B^\pm_c \to K^0_S K^\pm$: This rare decay proceeds purely via weak annihilation. It offers a unique probe to isolate and constrain these poorly understood annihilation contributions, which are difficult to calculate theoretically.

Previous measurements by LHCb (Run 1), BaBar, and Belle had limited precision. This analysis aims to significantly improve these constraints using Run 2 data.

2. Methodology

Data Sample:
The analysis utilizes proton-proton collision data collected by the LHCb experiment at a center-of-mass energy of $\sqrt{s} = 13$ TeV, corresponding to an integrated luminosity of 5.4 fb$^{-1}$ (collected between 2016 and 2018).

Event Selection and Reconstruction:

• Candidates: $B^\pm$ candidates are reconstructed by combining a $K^0_S \to \pi^+\pi^-$ candidate with a charged track identified as a pion ($\pi^\pm$) or kaon ($K^\pm$).
• Kinematic Cuts: Strict requirements are applied to particle momenta ($p > 2$ GeV/c for pions from $K^0_S$, $p > 8$ GeV/c for $K^0_S$, $p > 25$ GeV/c for $B^\pm$), transverse momentum ($p_T > 1$ GeV/c), and vertex quality (flight distance significance, impact parameters).
• Background Suppression:
  • A Boosted Decision Tree (BDT) and Gradient Boosted Decision Tree (BDTG) are employed to separate signal from background. These classifiers use kinematic properties, vertex quality, lifetime information, and isolation variables.
  • Mass Vetoes: Dedicated vetoes are applied to suppress peaking backgrounds, including misidentified $\Lambda \to p\pi^-$, cross-feed from $B^\pm \to K^0_S \pi^\pm$ (where $\pi$ is mis-ID as $K$), and $B^\pm \to D^0 \pi^+$ ($D^0 \to K^-\pi^+$).
• Simulation: Monte Carlo samples generated with Pythia, EvtGen, and Geant4 are used to model detector acceptance and efficiency. Data-driven corrections (using $B^\pm \to K^0_S \pi^\pm$ and sPlot techniques) are applied to align simulation with data.

Analysis Strategy:

• Yield Extraction: A simultaneous unbinned extended maximum-likelihood fit is performed on the $B^\pm$ mass distributions ($5000 < m_{B^\pm} < 5700$ MeV/$c^2$) for four samples: $B^+ \to K^0_S \pi^+$, $B^- \to K^0_S \pi^-$, $B^+ \to K^0_S K^+$, and $B^- \to K^0_S K^-$.
• Fit Components: The mass model includes:
  • Signal: Modeled by a Double-Sided Crystal Ball (DSCB) function plus a Gaussian.
  • Cross-feed: Modeled by DSCB functions (accounting for particle misidentification).
  • Partially Reconstructed Background: Modeled by an ARGUS function convolved with a Gaussian.
  • Combinatorial Background: Modeled by a first-order polynomial.
• CP Asymmetry Calculation: Raw charge asymmetries are corrected for detection and production asymmetries ($A_{det+prod}$) using the control channel $B^\pm \to J/\psi K^\pm$. Corrections for CP violation in the neutral kaon system ($A_{K^0}$) are also applied.
• Branching Fraction Ratio: Calculated as the ratio of signal yields corrected by the ratio of total efficiencies (derived from simulation with data-driven corrections).
• $B^\pm_c$ Search: A similar BDTG-based selection is applied to search for $B^\pm_c \to K^0_S K^\pm$. An upper limit is set using the profile-likelihood method.

3. Key Contributions and Results

A. CP Asymmetries ($A_{CP}$)
The analysis yields the most precise measurements to date:

• $B^\pm \to K^0_S \pi^\pm$:
  $$A_{CP} = -0.028 \pm 0.009 \text{ (stat)} \pm 0.009 \text{ (syst)}$$
  This represents a factor-of-two improvement in precision over previous LHCb Run 1 results. The result is consistent with the SM prediction of zero but shows a $2.3\sigma$ tension with the recent Belle II measurement.
• $B^\pm \to K^0_S K^\pm$:
  $$A_{CP} = 0.118 \pm 0.062 \text{ (stat)} \pm 0.031 \text{ (syst)}$$
  This measurement begins to discriminate between theoretical models (QCDF and pQCD), showing deviations of $2.5\sigma$ and $1.4\sigma$ respectively.

B. Branching Fraction Ratio
The ratio of branching fractions is measured with high precision:
$$\frac{\mathcal{B}(B^\pm \to K^0_S K^\pm)}{\mathcal{B}(B^\pm \to K^0_S \pi^\pm)} = 0.055 \pm 0.004 \text{ (stat)} \pm 0.002 \text{ (syst)}$$
This result agrees well with previous LHCb measurements and theoretical predictions.

C. Search for $B^\pm_c \to K^0_S K^\pm$

• No significant signal was observed (significance of 1.0$\sigma$).
• An upper limit was set on the product of the branching fraction ratio and the fragmentation fraction ratio ($f_c/f_u$):
  $$\frac{f_c}{f_u} \cdot \frac{\mathcal{B}(B^\pm_c \to K^0_S K^\pm)}{\mathcal{B}(B^\pm \to K^0_S \pi^\pm)} < 0.015 \quad (90\% \text{ CL})$$
  $$< 0.016 \quad (95\% \text{ CL})$$

4. Significance

• Theoretical Benchmark: The measurement of $A_{CP}(B^\pm \to K^0_S \pi^\pm)$ with a total uncertainty of 0.013 establishes a new benchmark for a "null-test" channel in the $b \to s$ sector. The precision is now sufficient to probe potential New Physics contributions that were previously hidden by experimental uncertainty.
• Hadronic Dynamics: The improved precision on $A_{CP}(B^\pm \to K^0_S K^\pm)$ provides critical data to refine theoretical models of hadronic annihilation and penguin amplitudes, helping to resolve tensions between different QCD factorization approaches.
• Annihilation Constraints: The upper limit on the $B^\pm_c$ decay provides the first experimental constraint on the weak annihilation contribution in this specific channel, which is vital for understanding non-leptonic heavy-meson decays.
• Future Outlook: These results set a new standard for precision in flavor physics. They provide a foundational dataset for future global analyses and will serve as a crucial reference for the upcoming high-luminosity runs of the upgraded LHCb and Belle II experiments.