Measurement of the local and nonlocal amplitudes in $B^{+}\to K^{+}\mu^{+}\mu^{-}$ decays

Using 8.4 fb$^{-1}$ of LHCb data, this study performs an amplitude analysis of the $B^{+}\to K^{+}\mu^{+}\mu^{-}$ decay to simultaneously determine local and nonlocal amplitudes across the full dimuon mass spectrum, revealing that the compatibility of Wilson coefficient combinations with the Standard Model varies between 1.6$\sigma$ and 4$\sigma$ depending on the chosen form factors.

The Gist

Imagine you are a detective trying to solve a mystery at a massive, high-speed racetrack called the LHC (Large Hadron Collider). The racers are subatomic particles, and the specific case you are investigating is a very rare event: a heavy particle called a $B^+$ meson decaying (fall apart) into a lighter particle called a $K^+$ and a pair of muons (which are like heavy electrons).

This paper from the LHCb collaboration at CERN is the official report on how they solved this mystery. Here is the story in plain English, using some everyday analogies.

1. The Mystery: "Is the Rulebook Broken?"

For the last decade, physicists have noticed that these $B^+$ mesons don't always behave exactly how the "Rulebook" of the universe (called the Standard Model) predicts they should. It's like watching a soccer ball curve in a way that defies the laws of physics.

The team suspects that "New Physics" (a secret rule or a new character in the universe) might be involved. To find out, they need to measure exactly how the decay happens, down to the tiniest detail.

2. The Challenge: The "Ghost" in the Machine

The problem is that the decay isn't just a simple, direct path. It's messy.

• The Direct Path (Local): The particle decays directly into the final products. This is the "clean" signal we want to study.
• The Ghosts (Nonlocal): Sometimes, the particle briefly turns into other things (like a pair of charm quarks) before turning into the final muons. These are called "nonlocal" contributions. They are like ghosts that haunt the process, interfering with the direct path and making it hard to see the true signal.

In the past, physicists tried to guess how these ghosts behaved. In this paper, the LHCb team decided to stop guessing. They built a super-detailed map of these ghosts. They modeled them as if they were a choir of singers (resonances like the $J/\psi$ and $\psi(2S)$) and even a complex dance of two-particle interactions.

3. The Investigation: 8.4 "Years" of Data

The team looked at data collected from the LHCb detector, which corresponds to 8.4 inverse femtobarns of collisions.

• Analogy: Imagine trying to find a specific needle in a haystack. This dataset is like having a haystack the size of a mountain, but they have a magic magnet (the detector) that can pick out the needles. They collected enough data to be very confident in their findings.

They used a sophisticated computer model to separate the "Direct Path" from the "Ghostly Interference." They had to account for:

• The Camera Blur: The detector isn't perfect; it blurs the image slightly. They corrected for this.
• The Background Noise: Sometimes random particles crash together and look like the signal. They used a "Boosted Decision Tree" (a smart AI algorithm) to filter out the noise, like a bouncer at a club checking IDs to keep out the wrong people.

4. The Findings: A Tension with the Rulebook

After all the cleaning and modeling, they measured two key numbers (called Wilson Coefficients, $C_9$ and $C_{10}$). These numbers tell us how strong the forces are in the decay.

• The Result: When they compared their measurements to the Standard Model's prediction, they found a mismatch.
• The Significance: The mismatch is about 4 standard deviations (4 sigma) when using the most precise calculations for the "shape" of the particles (form factors).
  • Analogy: If you flip a coin 100 times and get 90 heads, you'd suspect the coin is rigged. A 4-sigma result is like getting 99 heads out of 100. It's very strong evidence that something is wrong with the coin (or the Rulebook), but it's not quite "100% proof" (which requires 5 sigma) yet.
• The Caveat: If they used a slightly different, less precise calculation for the particle shapes, the mismatch drops to 1.6 sigma (less suspicious). This tells us that the "New Physics" signal is real, but its strength depends heavily on how well we understand the shape of the particles involved.

5. The "Four Faces" of the Solution

One of the coolest parts of the paper is that the math allowed for four different solutions that fit the data almost equally well.

• Analogy: Imagine you are looking at a shadow on a wall. Depending on the angle of the light, the shadow could look like a rabbit, a dog, a bird, or a person. All four are valid interpretations of the same shadow.
  The team found that the "ghosts" (the nonlocal amplitudes) could be arranged in four different ways (phases) that all explained the data. They reported all four possibilities because the data isn't quite strong enough yet to say which one is the "true" shadow.

6. The Conclusion: A Step Closer to the Truth

The paper concludes that:

1. We are getting better: By modeling the "ghosts" (nonlocal effects) so carefully, we are getting a clearer picture of the "direct path."
2. The mystery remains: The data still hints that the Standard Model might be incomplete. The $B^+$ meson is behaving slightly differently than expected.
3. More data needed: To be 100% sure, we need more data from the LHC's future runs (Run 3). It's like needing more photos to be absolutely certain what the shadow really is.

In a nutshell: The LHCb team took a massive amount of data, built a incredibly detailed model to filter out the "noise" and "ghosts," and found strong hints that the universe's rulebook might have a few typos. They haven't found the new physics yet, but they've definitely narrowed down where to look next.

Technical Summary

1. Problem and Motivation

The decay $B^+ \to K^+\mu^+\mu^-$ is a flavor-changing neutral current (FCNC) process ($b \to s\ell^+\ell^-$) that is forbidden at tree level in the Standard Model (SM) and proceeds via loop diagrams (penguins and boxes). Over the last decade, measurements of $b \to s\mu^+\mu^-$ transitions have shown persistent tensions with SM predictions, particularly in the Wilson coefficient $C_9$.

However, a major source of uncertainty in interpreting these tensions is the hadronic nonlocal contributions. These arise from long-distance effects where four-quark operators interact electromagnetically with the muon pair (e.g., via charmonium resonances like $J/\psi$ or open-charm thresholds). Previous analyses often treated these nonlocal effects with simplified models or binned data, potentially masking new physics or misattributing hadronic effects to new physics.

This paper aims to perform a model-dependent amplitude analysis of the entire dimuon mass spectrum ($300 < m_{\mu\mu} < 4700$ MeV/$c^2$) to simultaneously determine:

1. Short-distance physics: The Wilson coefficients $C_9$ and $C_{10}$ (and their primed counterparts).
2. Long-distance physics: The magnitudes and phases of nonlocal amplitudes (one-particle resonances and two-particle rescattering).

2. Methodology

Dataset and Selection

• Data: The analysis uses $pp$ collision data collected by the LHCb experiment during Run 1 and Run 2, corresponding to an integrated luminosity of 8.4 fb$^{-1}$.
• Selection: Candidates are selected using a Boosted Decision Tree (BDT) classifier trained on kinematic and geometric variables. The selection requires a reconstructed $B^+$ mass window and tight particle identification (PID) to suppress backgrounds from misidentified pions and combinatorial sources.
• Regions: The analysis divides the dimuon mass spectrum into three regions based on dominant resonances:
  • Region 1: 300–1800 MeV/$c^2$ ($\phi(1020)$ dominant).
  • Region 2: 1800–3400 MeV/$c^2$ ($J/\psi$ dominant).
  • Region 3: 3400–4700 MeV/$c^2$ ($\psi(2S)$ dominant).

Theoretical Framework

The differential decay rate is modeled using an effective Hamiltonian. The key innovation is the treatment of the effective Wilson coefficient $C_9^{\text{eff}}$:
$$C_9^{\text{eff}} = C_9 + Y_{\text{light}}^{1p}(q^2) + Y_{cc}^{1p+2p}(q^2)$$

• Local Contributions: Described by form factors ($f_+, f_0, f_T$) calculated using Lattice QCD. Two sets of form factors are used:
  1. HPQCD (High Precision Lattice QCD) as the baseline.
  2. FNAL/MILC as a cross-check.
• Nonlocal Contributions: Modeled using dispersion relations:
  • One-particle (1p): Sum of relativistic Breit-Wigner functions for light resonances ($\rho, \omega, \phi$) and charmonium states ($J/\psi, \psi(2S), \psi(3770), \dots$). The $\psi(3770)$ uses a Flatté function to account for the open-charm threshold.
  • Two-particle (2p): Describes $B^+ \to K^+(D^{(*)}D^{(*)} \to \mu^+\mu^-)$ rescattering. Due to limited statistics, the $DD$, $DD^*$, and $D^*D^*$ contributions are approximated by a single effective amplitude with magnitude $\eta_{2p}$ and phase $\delta_{2p}$.

Fitting Procedure

• An unbinned maximum-likelihood fit is performed on the reconstructed dimuon mass spectrum ($m_{\mu\mu}^{\text{rec}}$).
• The fit accounts for detector resolution (modeled by Gaussian functions with power-law tails) and efficiency variations (parametrized as polynomials in $m_{\mu\mu}$).
• Systematic uncertainties are propagated via covariance matrices, including uncertainties from form factors, resonance parameters, and background modeling.

3. Key Contributions

1. Unified Amplitude Analysis: This is the first LHCb analysis to simultaneously fit local Wilson coefficients and a comprehensive model of nonlocal amplitudes (both 1p and 2p) across the entire accessible $q^2$ spectrum, rather than in bins.
2. Data-Driven Hadronic Modeling: The magnitudes and phases of the nonlocal amplitudes (except for the $J/\psi$ normalization) are measured directly from the data, reducing reliance on theoretical assumptions about hadronic effects.
3. Resolution of Ambiguities: The analysis identifies a four-fold ambiguity in the fit solutions, driven by the relative phases of the $J/\psi$ and $\psi(2S)$ resonances. The paper reports results for all four solutions, with the global minimum corresponding to a negative $J/\psi$ phase and positive $\psi(2S)$ phase.
4. Two-Particle Rescattering: It provides the first data-driven constraints on the effective two-particle charm rescattering amplitude in $B^+ \to K^+\mu^+\mu^-$, finding no significant evidence for a large contribution from $B^+ \to K^+(D^{(*)}D^{(*)} \to \mu^+\mu^-)$ within the current precision.

4. Results

Wilson Coefficients ($C_9, C_{10}$)

The fit extracts the combinations $C_V = C_9 + C_9'$ and $C_A = C_{10} + C_{10}'$.

• HPQCD Form Factors: The measured values show a tension with the Standard Model prediction at the 4.0$\sigma$ level. The data prefers $C_9 < C_9^{\text{SM}}$.
• FNAL/MILC Form Factors: Due to larger uncertainties in these form factors, the tension is reduced to 1.6$\sigma$.
• Consistency: The results are consistent with global analyses of $b \to s\mu^+\mu^-$ transitions that suggest new physics in $C_9$, though the shift is slightly smaller than previous measurements based solely on differential branching fractions.

Nonlocal Amplitudes

• Resonances: The magnitudes and phases of light ($\rho, \omega, \phi$) and charmonium resonances are measured. The $J/\psi$ phase is measured relative to the local amplitude.
• Two-Particle (2p) Amplitude: The fit finds the magnitude of the effective 2p amplitude ($\eta_{2p}$) to be small ($\sim 0.30$) with large statistical uncertainty. No significant evidence for a large 2p contribution was found.
• $C_9^\tau$: The analysis notes that the simplified treatment of the 2p amplitude prevents a precise determination of the lepton-flavor-violating coefficient $C_9^\tau$ (related to $\tau^+\tau^-$ intermediate states), requiring more data for a robust measurement.

$q^2$-Dependence Checks

Alternative fits introducing linear or step-like $q^2$ dependencies in the Wilson coefficients were performed:

• A mild preference (2$\sigma$) for a residual $q^2$ dependence in $C_A$.
• An indication (2.6$\sigma$) of a threshold effect in $C_V$ above the open-charm threshold.
  These features suggest potential missing components in the physics model or limitations in the current data precision.

5. Significance

• Clarifying the "New Physics" vs. "Hadronic" Debate: By measuring the nonlocal amplitudes directly from data, this analysis significantly reduces the theoretical uncertainty associated with hadronic effects. The persistence of the tension with the SM (up to 4$\sigma$) even after accounting for a data-driven nonlocal model strengthens the case for genuine new physics in the $C_9$ sector.
• Methodological Advancement: The approach of using dispersion relations to model the full spectrum sets a new standard for amplitude analyses in flavor physics, moving away from binned analyses which lose information on interference patterns.
• Future Prospects: The paper highlights that while current data constrains the effective 2p amplitude, future data from LHC Run 3 will be essential to disentangle the individual $DD$, $DD^*$, and $D^*D^*$ contributions and to probe lepton flavor universality violation ($C_9^\tau$) with higher precision.

In conclusion, this paper provides the most comprehensive amplitude analysis of $B^+ \to K^+\mu^+\mu^-$ to date, confirming tensions with the Standard Model while rigorously quantifying the long-distance hadronic contributions that could otherwise mimic new physics signals.