$C\!P$ violation analysis of local and nonlocal amplitudes in the $\overline{B}^0 \to \overline{K}^{*0}\mu^+\mu^-$ decay

Using 8.4 fb$^{-1}$ of LHCb Run 1 and Run 2 data, this study performs a comprehensive $C\!P$ violation analysis of the $\overline{B}^0 \to \overline{K}^{*0}\mu^+\mu^-$ decay by fitting angular observables with nonlocal hadronic amplitudes, achieving an order-of-magnitude improvement in precision for $C\!P$-violating Wilson coefficients while finding no significant deviation from the Standard Model.

The Gist

Imagine the universe as a grand, cosmic dance floor. For a long time, physicists have been trying to figure out why there is so much more "matter" (the stuff we are made of) than "antimatter" (its mysterious, opposite twin). If the rules of the dance were perfectly symmetrical, matter and antimatter should have been created in equal amounts and annihilated each other instantly, leaving an empty universe. But we are here, so the dance must have had a slight, uneven step.

This paper is a report from the LHCb experiment at CERN, a massive particle collider in Switzerland. They are looking for that uneven step, known as CP violation, by watching a very specific, rare dance move performed by a subatomic particle called the $B^0$ meson.

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

1. The Rare Dance Move

The scientists watched a specific particle decay (break apart) into a set of other particles: a $K^{*0}$ meson and two muons (heavy electrons).

• The Analogy: Imagine a rare, complex dance routine where a dancer spins and splits into three specific partners. This happens very rarely in nature.
• Why it matters: In the "Standard Model" (the current rulebook of physics), this dance should look almost exactly the same whether the dancer is made of matter or antimatter. If the dance looks different, it means the rulebook is incomplete, and there might be new, hidden forces at play.

2. The "Full Spectrum" Approach

Previous experiments tried to find this difference by looking at specific slices of the dance, avoiding the "loud" parts where other particles (like charmonium resonances) interfere. It was like trying to hear a whisper in a quiet room by only listening when the music stops.

• What this paper did differently: This team looked at the entire dance floor, including the loud, chaotic parts where the "charmonium" particles are dancing.
• The Analogy: Instead of waiting for the music to stop, they turned up the volume and analyzed the entire song, including the heavy bass and the complex harmonies. By using a sophisticated mathematical filter (called "nonlocal amplitudes"), they were able to separate the specific "whisper" of the CP violation from the "noise" of the other particles.

3. The "Weak Phase" and the Compass

To find the difference between matter and antimatter, the scientists looked at the angles at which the particles flew apart.

• The Analogy: Imagine the particles are arrows shot from a bow. The direction they fly depends on a hidden "compass" inside the particle, called a weak phase.
• The Goal: They wanted to see if the compass for the "matter" dancer pointed in a slightly different direction than the compass for the "antimatter" dancer. If the compasses pointed differently, that would be the "uneven step" causing the matter-antimatter imbalance.

4. The Results: A Perfectly Symmetric Dance

After analyzing a massive amount of data (equivalent to 8.4 "inverse femtobarns"—a unit representing billions of collisions), the team made a precise measurement.

• The Finding: The compasses for matter and antimatter pointed in the exact same direction, within the limits of their measurement tools.
• The Analogy: They watched the dance from every angle, in every lighting condition, and found that the matter dancer and the antimatter dancer performed the routine with perfect symmetry. There was no detectable "uneven step."
• The Precision: Their measurement was incredibly sharp—about 10 times more precise than previous attempts. They could now measure the "imaginary" parts of the physics (the hidden phases) even better than the "real" parts.

5. What This Means

• No New Physics Found (Yet): The results match the current "Standard Model" predictions perfectly. The universe is still behaving according to the known rules for this specific dance move.
• A Stronger Baseline: Even though they didn't find new physics, they set a much tighter "fence" around where new physics could be hiding. If there is a new force causing the matter-antimatter imbalance, it must be hiding in a place even more subtle than they could detect with this experiment.
• The "Nonlocal" Success: The paper proves that their new method of analyzing the "whole song" (including the charmonium resonances) works. It's a successful test of their mathematical tools, even if the result was "nothing new."

Summary

The LHCb team performed the most precise check yet on how a specific particle behaves compared to its antimatter twin. They looked at the angles of the debris from billions of collisions, using advanced math to filter out background noise. They found no difference. The dance is perfectly symmetrical, consistent with our current understanding of the universe, but the tools they used to check are now sharper than ever before.

Technical Summary

Technical Summary: CP Violation Analysis of Local and Nonlocal Amplitudes in the $B^0 \to K^{*0}\mu^+\mu^-$ Decay

Problem Statement
The observed dominance of matter over antimatter in the universe remains unexplained by the Standard Model (SM), as the SM's built-in mechanism for CP violation is insufficient to account for the observed asymmetry. Flavor-changing neutral-current (FCNC) decays, such as $B^0 \to K^{*0}\mu^+\mu^-$, are highly sensitive probes for physics beyond the Standard Model (BSM) due to the suppression of SM contributions. While previous analyses of this decay have produced results difficult to explain within the SM, firm conclusions regarding BSM physics have been hindered by theoretical uncertainties in calculating strong force effects (nonlocal hadronic amplitudes). Furthermore, prior measurements of CP violation in this channel were limited to regions of the dimuon mass squared ($q^2$) far from charmonium resonances, restricting sensitivity primarily to T-odd asymmetries and leaving T-even asymmetries (proportional to the sine of the strong phase difference) largely unexplored.

Methodology
This analysis utilizes proton-proton collision data collected by the LHCb experiment during Run 1 (2011–2012) and Run 2 (2016–2018), corresponding to an integrated luminosity of $8.4 \text{ fb}^{-1}$. The study performs a search for CP violation in the $B^0 \to K^{*0}\mu^+\mu^-$ decay (with $K^{*0} \to K^-\pi^+$) by exploiting the full five-dimensional angular distribution of the decay products ($\cos \theta_\ell, \cos \theta_K, \phi, q^2, m^2_{K\pi}$) across the entire kinematic range $0.1 \le q^2 \le 18.0 \text{ GeV}^2/c^4$.

Key methodological features include:

• Unbinned Maximum-Likelihood Fit: The analysis employs an unbinned fit to the mass and angular distributions, simultaneously fitting $B^0$ and $\bar{B}^0$ samples.
• Inclusion of Nonlocal Amplitudes: Unlike previous studies that vetoed charmonium regions, this analysis includes the full $q^2$ spectrum, incorporating the interference between FCNC amplitudes and nonlocal amplitudes arising from charmonium resonances ($J/\psi, \psi(2S)$, etc.) and open-charm intermediate states. This allows sensitivity to T-even asymmetries.
• Weak Effective Theory Framework: The decay amplitudes are modeled using the Weak Effective Theory Hamiltonian. The analysis determines complex Wilson coefficients ($C_7, C_9, C_{10}$ and their primed counterparts) by parameterizing the effective coefficient $C_9^{\text{eff}}(q^2)$ to include nonlocal contributions via dispersion relations involving spectral densities for $c\bar{c}$ and $q\bar{q}$ loops.
• Systematic Control: Detection asymmetries between matter and antimatter are mitigated by reversing the magnetic field polarity. Systematic uncertainties, including those from acceptance differences and resolution smearing, are evaluated using pseudoexperiments and data-driven calibration. The signal fraction is determined via a one-dimensional mass fit, and the final fit constrains 153 free parameters, including Wilson coefficients, nonlocal parameters, form factors, and background shapes.

Key Contributions

• First Full-Spectrum CP Analysis: This work presents the first direct measurement of the phases of the $b \to s \ell^+\ell^-$ Wilson coefficients $C_9$ and $C_{10}$ using an unbinned $q^2$ analysis that includes charmonium resonance regions.
• Enhanced Sensitivity: By including the interference between charmonium resonances and the FCNC process, the analysis gains sensitivity to T-even asymmetries, which were previously inaccessible in binned analyses excluding resonance regions.
• Precision Improvement: The precision of the CP-violation observables is improved by an order of magnitude relative to previous measurements. Notably, the imaginary parts of the Wilson coefficients are now determined with greater precision than the real parts.
• Model-Independent Constraints: The fit does not assume SM values for the other Wilson coefficients during the profiling procedure, allowing for a more robust extraction of CP-violating phases.

Results
The analysis finds no significant evidence for direct CP violation in the $B^0 \to K^{*0}\mu^+\mu^-$ decay. The results are consistent with the Standard Model.

• Wilson Coefficients: The measured values for the magnitudes and phases of $C_9$ and $C_{10}$ are compatible with SM expectations within $1\sigma$.
  • $\delta\phi_{C9} = -0.067 \pm 0.032 \text{ (stat)} \pm 0.011 \text{ (syst)}$
  • $\delta\phi_{C10} = 0.043 \pm 0.035 \text{ (stat)} \pm 0.010 \text{ (syst)}$
• Weak Phase: Under the assumption that the SM is the sole source of CP violation, the averaged weak phase for $C_9$ and $C_{10}$ is determined to be $-0.012 \pm 0.025 \text{ (stat)} \pm 0.008 \text{ (syst)}$. This is consistent with the value derived from $B_s^0$ oscillations in the decay $B_s^0 \to J/\psi\phi$.
• Real Part of $C_9$: The real part of $C_9$ is measured to be approximately $0.8$ units lower than the SM expectation, a trend consistent with other analyses but with a significance of only $2.1\sigma$ in this specific context, largely due to the treatment of nonlocal contributions as fit parameters rather than theoretical estimates.

Significance
The paper claims that this analysis represents a significant step forward in the search for CP violation in flavor-changing neutral-current decays. It achieves a level of sensitivity to CP violation comparable to SM expectations, a capability previously unattained in this specific decay channel. By exploring a set of CP-violating couplings that $B_s^0$ oscillations are not sensitive to, and by improving the precision of the imaginary components of the Wilson coefficients by an order of magnitude, this work provides stringent constraints on BSM physics models that would induce new CP-violating phases in $b \to s$ transitions. The results reinforce the consistency of the SM in this sector while establishing a new benchmark for precision in the determination of weak phases in rare decays.