Test of lepton flavour universality with $B^0\to K^{*0}\ell^+\ell^-$ decays at large dilepton invariant mass

Using 9 fb⁻¹ of proton-proton collision data collected by the LHCb detector, this study presents the first hadron collider measurement of the lepton flavour universality ratio $R_{K^{*0}}$ in $B^0 \to K^{*0} \ell^+ \ell^-$ decays at high dilepton invariant mass, finding a value of $1.08\,^{+0.14}_{-0.12}\text{(stat)} \ \pm 0.07\text{(syst)}$ that is consistent with Standard Model predictions.

The Gist

The Big Picture: Checking the Rules of the Universe

Imagine the universe is a giant, complex video game with strict rules. One of the most important rules in the "Standard Model" (the game's rulebook) is called Lepton Flavor Universality.

Think of Leptons (like electrons and muons) as two different types of players in this game. They have different weights (masses), but the rulebook says: "The game's physics engine should treat both players exactly the same, regardless of their weight." If you throw a heavy ball and a light ball with the same force, they should react to gravity in the same way.

However, recently, players at CERN noticed something weird. When they watched these particles decay (break apart), the "heavy" muons seemed to be behaving slightly differently than the "light" electrons. It was like the game engine was secretly giving the heavy ball a slight boost. This hinted that there might be New Physics (a hidden cheat code or a new character) messing with the rules.

The Experiment: The "High-Speed" Race

This specific paper is about a new race the LHCb team ran to check if the rules are still fair.

• The Players: They looked at a specific type of particle decay called $B^0 \to K^{*0} \ell^+ \ell^-$. Imagine a heavy, unstable car ($B^0$) crashing and breaking apart into a smaller car ($K^{*0}$) and two passengers ($\ell^+ \ell^-$).
• The Passengers: Sometimes the passengers are an electron pair (lightweight), and sometimes they are a muon pair (heavyweight).
• The Track: The team focused on a very specific part of the race track: the high-energy zone. In the past, they checked the low-energy zone and found the rules held up. But the "high-energy" zone (where the passengers are moving very fast) had been a mystery for a long time.

The Challenge: The "Ghost" Interference

Why is this high-energy zone tricky? Imagine you are trying to time a race, but there is a massive, noisy crowd (called resonances) standing right in the middle of the track. This crowd is so loud and chaotic that it drowns out the sound of the racers.

• The Problem: In the middle of the track, there are "ghosts" (particles called charmonium) that make it impossible to tell if the racers are following the rules or if the crowd is just making noise.
• The Solution: The LHCb team decided to run the race past the crowd. They looked at the section of the track after the noisy crowd, where the signal is clean. This is the "large dilepton invariant mass" region mentioned in the title.

The Method: The "Double-Check" Scale

To measure if the rules are fair, they didn't just count the electrons and muons. They used a clever trick called a Double Ratio.

Imagine you have a scale to weigh two different fruits (Apples and Oranges), but your scale is slightly wobbly and might be off by a few grams.

1. Step 1: You weigh the Apples.
2. Step 2: You weigh the Oranges.
3. Step 3 (The Trick): You also weigh a Reference Fruit (a standard 1kg weight) alongside both.

By comparing the Apples to the Reference, and the Oranges to the Reference, you cancel out the wobble in the scale. You get a perfect ratio of Apples to Oranges, even if your scale is broken.

In this experiment:

• Apples/Oranges: The rare decays (the ones they are studying).
• Reference Fruit: A very common, well-known decay ($B^0 \to K^{*0} J/\psi$) that happens all the time.
• By comparing the rare decays to the common one, they canceled out all the messy errors from the detector and the computer triggers.

The Result: The Rules Still Hold!

After analyzing data from 9 years of running the Large Hadron Collider (LHC), the team counted the results:

• The Prediction: The Standard Model says the ratio of Muons to Electrons should be 1.00 (perfectly equal).
• The Measurement: They found the ratio to be 1.08.
• The Catch: The number 1.08 is very close to 1.00. The "error bars" (the margin of uncertainty) are big enough that 1.08 is statistically indistinguishable from 1.00.

The Verdict: The universe is still playing fair. The heavy muons and the light electrons are still being treated exactly the same by the laws of physics in this high-energy zone.

Why Does This Matter?

1. It's a "First": This is the first time anyone has checked this specific high-energy zone using a hadron collider (like the LHC). Before this, only the Belle experiment (in Japan) had looked here, but with less precision.
2. It's the Most Precise: This measurement is three times more precise than previous attempts.
3. No "New Physics" Found (Yet): While scientists love finding new physics (it would mean rewriting the rulebook), finding that the rules are still correct is also a huge victory. It tells us that if there is a cheat code, it's hiding somewhere else, or it's even more subtle than we thought.

The Bottom Line

Think of this paper as the LHCb team running a very high-stakes, high-speed audit of the universe's rulebook. They checked a section of the code that was previously too messy to read. They found that, despite the noise and the complexity, the code is still running exactly as the original designers intended. The "Lepton Flavor Universality" rule is still safe, for now.

Technical Summary

1. Problem and Motivation

Lepton Flavor Universality (LFU) is a fundamental prediction of the Standard Model (SM), stating that electroweak interactions couple identically to electrons, muons, and taus, differing only by mass effects. The flavor-changing neutral current (FCNC) transition $b \to s\ell^+\ell^-$ is a sensitive probe for New Physics (NP) because it occurs only via loop diagrams in the SM and is suppressed, making it susceptible to contributions from heavy NP particles.

Recent measurements of $b \to s\mu^+\mu^-$ processes have shown deviations from SM predictions in branching ratios and angular distributions (ranging from 2 to 4$\sigma$). However, interpreting these anomalies is complicated by theoretical uncertainties in hadronic form factors and non-perturbative charm-loop effects.

• The Gap: While LFU tests in the low-$q^2$ region ($q^2 < 6 \text{ GeV}^2/c^4$) have been performed with high precision, the high-$q^2$ region (above the $\psi(2S)$ resonance) has been less explored, particularly at hadron colliders.
• The Goal: This paper presents the first measurement of the LFU ratio $R_{K^{*0}}$ in the high-$q^2$ region ($14 < q^2 < 22 \text{ GeV}^2/c^4$) using a hadron collider (LHCb), aiming to provide a theoretically cleaner test of LFU where hadronic uncertainties largely cancel.

2. Methodology

Data Sample:

• Source: Proton-proton ($pp$) collisions recorded by the LHCb detector.
• Periods: Run 1 (2011–2012) and Run 2 (2015–2018).
• Integrated Luminosity: $9 \text{ fb}^{-1}$ at center-of-mass energies of 7, 8, and 13 TeV.
• Decay Channels: $B^0 \to K^{*0}\ell^+\ell^-$ where $\ell = e, \mu$. The $K^{*0}$ is reconstructed via $K^{*0} \to K^+\pi^-$.

Analysis Strategy:
The analysis measures the ratio of branching fractions, $R_{K^{*0}}$, defined as:
$$R_{K^{*0}} = \frac{\int_{q^2_{min}}^{q^2_{max}} \frac{d\mathcal{B}(B^0 \to K^{*0}\mu^+\mu^-)}{dq^2} dq^2}{\int_{q^2_{min}}^{q^2_{max}} \frac{d\mathcal{B}(B^0 \to K^{*0}e^+e^-)}{dq^2} dq^2}$$
where $q^2_{min} = 14.0 \text{ GeV}^2/c^4$ and $q^2_{max} = 22.0 \text{ GeV}^2/c^4$.

Key Technical Components:

1. Double Ratio Normalization: To cancel detector and trigger effects, $R_{K^{*0}}$ is measured as a double ratio normalized by the resonant control mode $B^0 \to K^{*0}J/\psi (\to \ell^+\ell^-)$:
   $$R_{K^{*0}} = \frac{N/\epsilon (B^0 \to K^{*0}\mu^+\mu^-)}{N/\epsilon (B^0 \to K^{*0}J/\psi \to \mu^+\mu^-)} \Bigg/ \frac{N/\epsilon (B^0 \to K^{*0}e^+e^-)}{N/\epsilon (B^0 \to K^{*0}J/\psi \to e^+e^-)}$$
   This minimizes systematic uncertainties related to reconstruction efficiency differences between electrons and muons.

2. Bremsstrahlung Handling:

   • Electrons suffer significant energy loss due to bremsstrahlung in the detector material, degrading the $q^2$ resolution.
   • Correction: The electron momentum is corrected by adding energy from reconstructed electromagnetic calorimeter (ECAL) clusters matching the electron track.
   • Smearing: A smearing procedure is applied to the simulated dilepton invariant mass distribution to match data, accounting for residual discrepancies in $q^2$ resolution.
   • Background Suppression: To prevent resonant $B^0 \to K^{*0}\psi(2S) \to e^+e^-$ decays from leaking into the signal region due to misidentified bremsstrahlung, the $q^2$ for the electron channel is calculated before bremsstrahlung recovery for the selection, while the corrected momentum is used only for the final $B^0$ mass reconstruction.

3. Fit Model:

   • Signal: Extracted via simultaneous unbinned maximum-likelihood fits to the $B^0$ invariant mass distributions for both electron and muon channels.
   • Shapes: The muon signal is modeled with a double-sided Crystal Ball function. The electron signal is modeled with separate double-sided Crystal Ball functions for candidates with 0, 1, or $\ge 2$ associated bremsstrahlung photons.
   • Backgrounds:
     • Combinatorial: Modeled with an empirical function.
     • Partially Reconstructed: $B^+ \to K^+\pi^+\pi^-\ell^+\ell^-$ (missing pion) modeled with weighted simulation.
     • Misidentified: Hadrons misidentified as electrons (dominant in electron channel) modeled using data-driven transfer functions derived from $D^{*+}$ calibration samples.

4. Efficiency Corrections:

   • Weights ($w_{eff}(q^2)$) are applied to the muon channel data to match the $q^2$-dependent selection efficiency of the electron channel, reducing bias from imperfect form-factor modeling in simulation.

3. Key Contributions

• First Hadron Collider Measurement: This is the first measurement of $R_{K^{*0}}$ in the high-$q^2$ region performed at a hadron collider, complementing previous $e^+e^-$ collider (Belle) results.
• Precision Improvement: The result achieves a threefold improvement in precision compared to the previous Belle measurement in the same kinematic region.
• Robust Systematic Control: The analysis introduces sophisticated techniques to handle electron bremsstrahlung and misidentification backgrounds, which are the dominant challenges in high-$q^2$ electron measurements at hadron colliders.
• Validation: The analysis strategy was validated using control modes ($J/\psi$ and $\psi(2S)$ resonances), yielding results consistent with unity ($r_{J/\psi} = 1.035 \pm 0.013 \pm 0.035$ and $R_{\psi(2S)} = 1.034 \pm 0.033 \pm 0.012$).

4. Results

The measured value of the LFU ratio in the high-$q^2$ region is:
$$R_{K^{*0}} = 1.08 \, ^{+0.14}_{-0.12} \text{ (stat)} \pm 0.07 \text{ (syst)}$$

• Statistical Uncertainty: Derived from the profile likelihood scan.
• Systematic Uncertainty: The dominant systematic source is the data-driven estimation of the misidentified background in the electron channel (5.2%). Other contributions include model dependence (2.0%) and efficiency corrections (1.2%).
• Significance: The result is consistent with the Standard Model prediction (which expects $R_{K^{*0}} \approx 1$) within approximately 0.5$\sigma$.

5. Significance

• Confirmation of SM: The result reinforces the Standard Model's prediction of lepton flavor universality in the high-$q^2$ region, providing a stringent constraint on New Physics models that predict LFU violation.
• Complementarity: By covering the high-$q^2$ region, this measurement complements low-$q^2$ results. The consistency across the entire $q^2$ spectrum suggests that any potential New Physics effects are either absent or highly constrained in this specific decay channel.
• Future Prospects: The paper highlights that with the larger datasets expected from the LHCb Upgrade II, the precision can be further improved, potentially reaching the level required to definitively confirm or rule out subtle deviations in FCNC processes.

In summary, this paper represents a milestone in flavor physics, demonstrating the LHCb detector's capability to perform precision LFU tests with electrons at high invariant masses, effectively closing a significant gap in the experimental landscape of $b \to s\ell^+\ell^-$ transitions.