First measurement of time-dependent $CP$ violation in the decay flavor-changing neutral-current decay $B^{0}\rightarrow K_{S}^{0}\mu^{+}\mu^{-}$

Using 9 fb⁻¹ of proton-proton collision data collected by the LHCb experiment, this study presents the first time-dependent CP violation measurement in the flavor-changing neutral-current decay $B^{0}\rightarrow K_{S}^{0}\mu^{+}\mu^{-}$, determining the parameters $C$ and $S$ to be consistent with Standard Model predictions.

The Gist

The Big Picture: Catching a Ghost in the Machine

Imagine the universe as a giant, complex clockwork machine. For decades, physicists have had a very detailed instruction manual for how this machine works, called the Standard Model. It predicts how tiny particles behave with incredible accuracy.

However, sometimes the clockwork makes a sound that doesn't quite match the manual. These "glitches" are called anomalies. They suggest there might be a hidden gear or a secret mechanic (New Physics) that we haven't discovered yet.

This paper is about LHCb scientists trying to catch one of these glitches in action. They are looking at a very rare event: a heavy particle called a $B^0$ meson decaying into a specific set of lighter particles (a neutral kaon and two muons).

The Main Character: The "Flavor-Changing" Particle

In the world of particle physics, particles have "flavors" (like up, down, strange, charm, bottom). Usually, a particle keeps its flavor. But sometimes, a "bottom" quark can magically turn into a "strange" quark. This is called a Flavor-Changing Neutral Current (FCNC).

Think of it like a magician pulling a rabbit out of a hat, but the rabbit is actually a completely different animal, and the magician didn't touch the hat. This is so rare in our current understanding of physics that if it happens more often than expected, or in a weird way, it's a huge red flag that something new is going on.

The Mystery: Time-Dependent CP Violation

The core of this paper is a measurement of CP Violation.

• C (Charge): Swapping matter for antimatter.
• P (Parity): Flipping left and right (like looking in a mirror).

The Standard Model says that the laws of physics should mostly be the same whether you are looking at matter or antimatter, or looking in a mirror. But they aren't exactly the same. This difference is CP Violation.

The scientists are looking at Time-Dependent CP violation. Imagine you have two identical twins, one made of matter ($B^0$) and one made of antimatter ($\bar{B}^0$). They are born at the same time. As they travel, they can spontaneously turn into each other (this is called mixing). Eventually, they both decay.

The question is: Does the matter twin decay at a slightly different rate or in a slightly different way than the antimatter twin, depending on how long they lived?

If the answer is "yes," and the difference is bigger than the Standard Model predicts, it means there is a new force or particle influencing them.

The Experiment: The Great Particle Race

The LHCb team acted like high-speed race officials. They used the Large Hadron Collider to smash protons together, creating millions of these $B^0$ mesons.

1. The Setup: They collected data from 2011 to 2018 (a massive amount of data, equivalent to 9 "inverse femtobarns"—a unit of particle collision volume).
2. The Filter: They had to find the specific race where the $B^0$ turned into a neutral kaon ($K_S^0$) and two muons ($\mu^+\mu^-$). This is like finding one specific grain of sand on a beach.
3. The "Tagging": To know if they were watching the matter twin or the antimatter twin, they used "flavor tagging." It's like looking at the crowd around the twins at the starting line. If the crowd has a lot of positive pions, the twin was likely an antimatter twin. This helps them sort the race results.
4. The Stopwatch: They measured exactly how long each particle lived before decaying.

The Results: The Scoreboard

After crunching the numbers, the scientists found:

• The "S" parameter (Mixing-induced CP violation): $+0.82$
• The "C" parameter (Direct CP violation): $-0.13$

The Verdict: These numbers are consistent with the Standard Model.

Think of it like this: The Standard Model predicted the twins would finish the race with a specific time difference. The LHCb team measured the time difference, and it matched the prediction perfectly (within the margin of error).

Why is this a Big Deal if they found "nothing"?

You might think, "If they found nothing new, why publish?"

1. It's the First Time: This is the first time ever that scientists have measured this specific type of time-dependent CP violation in this specific type of decay ($b \to s \ell^+\ell^-$). It's like opening a new door in a house you've been exploring for years. Even if the room is empty, you needed to check it.
2. Ruling Out Theories: Many "New Physics" theories predicted that this specific decay would show a huge deviation from the Standard Model. By showing that the result matches the Standard Model, the scientists have effectively closed the door on those specific theories. They have narrowed down the search for the "hidden mechanic."
3. The Method Works: They proved that their complex method of measuring time-dependent asymmetry in this rare decay works. This sets the stage for even more precise measurements in the future as they collect more data.

The Analogy: The Broken Watch

Imagine the Standard Model is a watch that tells time perfectly.

• Previous studies suggested the watch might be running 5 minutes fast.
• This study is like taking a stopwatch to a specific second hand on that watch to see if it's ticking at the right speed.
• The result: The second hand is ticking exactly as the manual says it should.

Conclusion: The watch is still ticking right, at least for this second hand. But now we know exactly how it ticks, and we can be sure that if the watch does break later, we'll know exactly where to look. The search for the "New Physics" continues, but this particular clue didn't lead to a treasure chest this time.

Technical Summary

1. Problem and Motivation

• Physics Context: Flavor-changing neutral-current (FCNC) decays, specifically the $b \to s \ell^+ \ell^-$ transition, are highly suppressed in the Standard Model (SM) and occur only via loop diagrams. This makes them sensitive probes for physics beyond the Standard Model (BSM).
• The Anomaly: Previous LHCb measurements of branching fractions and angular observables in semileptonic $B$-meson decays have revealed persistent tensions with SM predictions, suggesting potential new physics contributions to effective four-fermion couplings.
• The Gap: While time-dependent CP violation analyses are well-established in hadronic $b \to s \bar{q}q$ channels (e.g., $B^0 \to \phi K^0_S$), they have not yet been performed in semileptonic $b \to s \ell^+ \ell^-$ processes.
• Objective: This paper presents the first experimental study of time-dependent CP violation in the $B^0 \to K^0_S \mu^+ \mu^-$ decay. The goal is to measure the mixing-induced ($S$) and direct ($C$) CP violation parameters to constrain new CP-violating phases that are complementary to those derived from time-integrated rates.

2. Methodology

• Dataset: The analysis utilizes proton-proton collision data collected by the LHCb experiment during Run 1 (2011–2012) and Run 2 (2015–2018) at center-of-mass energies of 7, 8, and 13 TeV. The total integrated luminosity is 9 fb$^{-1}$.
• Signal Selection:
  • Reconstruction: $K^0_S$ candidates are reconstructed from $\pi^+\pi^-$ pairs (categorized as "long" or "downstream" tracks based on decay vertex position). These are combined with oppositely charged muons to form $B^0$ candidates.
  • Kinematic Cuts: The dimuon mass squared ($q^2$) is required to be outside the charmonium resonance regions ($J/\psi$ and $\psi(2S)$) to isolate the short-distance FCNC contribution. The analysis covers the full kinematically allowed $q^2$ range excluding these regions.
  • Background Suppression: A Gradient Boosted Decision Tree (BDTG) classifier is used to distinguish signal from combinatorial backgrounds and misidentified decays (e.g., $\Lambda_b^0 \to \Lambda \mu^+ \mu^-$ where the proton is misidentified as a pion, or $B^0 \to D^- \pi^+$ where pions are misidentified as muons).
• Flavor Tagging:
  • The initial flavor ($B^0$ or $\bar{B}^0$) at production is determined using both Same-Side (SS) and Opposite-Side (OS) tagging algorithms.
  • The effective tagging efficiency ($\epsilon_{tag} \langle D^2 \rangle$) is approximately 2.7% for Run 1 and 4.1% for Run 2.
• Fit Strategy:
  • An unbinned extended maximum-likelihood fit is performed on the $B^0$ mass distribution to determine signal yields.
  • A simultaneous weighted fit is applied to the decay-time distribution of tagged $B^0$ and $\bar{B}^0$ candidates.
  • The time-dependent CP asymmetry is modeled as:
    $$A_{CP}(t) \approx S \sin(\Delta m_d t) - C \cos(\Delta m_d t)$$
  • Systematic effects (decay-time resolution, acceptance, flavor-tagging calibration, and background correlations) are carefully modeled and constrained using control channels (e.g., $B^0 \to J/\psi K^0_S$) and simulation.
  • The $B^0$ lifetime ($\tau_{B^0}$) and oscillation frequency ($\Delta m_d$) are constrained to world-average values.

3. Key Contributions

• First Measurement: This is the world's first measurement of time-dependent CP violation parameters in a $b \to s \ell^+ \ell^-$ process.
• Theoretical Cleanliness: The analysis provides a theoretically clean observable sensitive to the imaginary parts of Wilson coefficients governing $b \to s$ transitions, offering constraints on new CP-violating phases that are more direct than those from angular analyses.
• Comprehensive Systematics: The paper details a robust treatment of systematic uncertainties, including fit biases due to limited statistics, flavor-tagging calibration differences between control and signal channels, and mass-time correlations.

4. Results

The CP violation parameters $C$ (direct) and $S$ (mixing-induced) were determined for the total $q^2$ range (excluding resonances):

• $C = -0.13 \pm 0.32 \text{ (stat)} \pm 0.04 \text{ (syst)}$
• $S = +0.82 \pm 0.29 \text{ (stat)} \pm 0.05 \text{ (syst)}$

Interpretation:

• The results are consistent with the Standard Model prediction, where $C \approx 0$ and $S \approx \sin(2\beta) \approx 0.72 \pm 0.02$.
• The statistical correlation coefficient between $C$ and $S$ is 0.50.
• The analysis was also performed in low-$q^2$ and high-$q^2$ bins separately, with results consistent with the total sample and the SM, though with larger uncertainties due to lower statistics.

5. Significance

• Validation of SM: The measurement confirms the SM prediction for CP violation in this rare decay mode, placing tight constraints on BSM models that predict large new CP-violating phases in $b \to s$ transitions.
• New Probe: It establishes time-dependent CP violation as a viable and powerful complementary probe to branching fractions and angular observables for investigating the $b \to s \ell^+ \ell^-$ anomalies.
• Future Potential: While current uncertainties are dominated by statistics, this analysis demonstrates the methodology required for future high-luminosity runs (Run 3 and beyond), where precision will significantly improve, potentially revealing subtle deviations from the SM.

In conclusion, this paper marks a milestone in flavor physics by successfully applying time-dependent analysis techniques to semileptonic rare decays, providing a new window into the search for new physics in the flavor sector.