Measurement of the branching fractions and longitudinal polarisations of $B^0_{(s)} \to K^{*0} \kern 0.18em \overline{\kern -0.18em K}{}^{*0}$ decays

Using 9 fb⁻¹ of LHCb collision data from 2011–2018, this study measures the branching fractions and longitudinal polarisation fractions of $B^0$ and $B^0_s$ decays to $K^{*0} \overline{K}{}^{*0}$, confirming a 4.4$\sigma$ tension between experimental results and theoretical predictions regarding longitudinal polarisation in $B \to VV$ decays.

The Gist

Imagine the universe as a giant, high-speed dance floor where tiny particles called quarks are constantly swapping partners and spinning. In this paper, the LHCb collaboration at CERN acts like a team of super-observant choreographers, watching a very specific, rare dance move performed by particles called B mesons.

Here is the story of what they found, explained simply.

The Dance: A Rare Spin

The particles they are watching are the $B^0$ and $B^0_s$ mesons. These are heavy particles that eventually decay (break apart) into two lighter, spinning particles called $K^*$ mesons (which quickly turn into a Kaon and a Pion).

Think of the $K^*$ mesons as spinning tops. When they are created, they can spin in different ways:

1. Longitudinal: Spinning like a bullet fired from a gun (aligned with their direction of travel).
2. Transverse: Spinning like a wheel rolling on the ground (sideways to their direction).

The Big Surprise: The "Polarisation Puzzle"

For a long time, physicists had a theory (based on the Standard Model of physics) that predicted how these particles should spin. The theory said: "Because of the way the universe works, these heavy particles should mostly spin like bullets (longitudinal)."

However, when the LHCb team looked at the $B^0_s$ particle, they found something weird. It wasn't spinning like a bullet at all. It was mostly spinning sideways!

• $B^0$ particle: Spins like a bullet 60% of the time.
• $B^0_s$ particle: Spins like a bullet only 16% of the time.

This huge difference is a mystery. It's like if you threw two identical-looking bowling balls, and one always rolled straight down the lane, while the other always spun wildly on its side. The paper calls this the "polarisation puzzle."

The Investigation: A Massive Data Hunt

To solve this, the team didn't just look at a few dances; they watched 9 billion collisions from the Large Hadron Collider (LHC) between 2011 and 2018. That's like watching a stadium full of people dance for eight years straight to find just a few hundred specific moves.

They used a technique called an amplitude analysis. Imagine trying to figure out the choreography of a dance by looking at a blurry, fast-moving video. The team had to build a complex mathematical model to separate the "signal" (the real dance) from the "noise" (people bumping into each other or background music).

They improved their tools significantly compared to previous studies:

• They used better "cameras" (detector simulations) to see the dance clearly.
• They modeled the "background noise" (other particles) much more accurately.
• They used a new mathematical language (covariant tensor formalism) to describe the spins, which removed some of the guesswork that plagued earlier studies.

The Results: The Puzzle Gets Bigger

After crunching the numbers, the team confirmed the mystery with much higher precision than ever before.

• They measured the exact "spin ratios" with very small margins of error.
• They calculated a specific number (called $L_{K^*0K^*0}$) that compares the two particles. Their result was 4.92.
• The best theoretical prediction for this number was 26.08.

The difference between their measurement (4.92) and the theory (26.08) is huge—about 4.4 times the size of the expected error margin. In the world of particle physics, this is a "4.4 sigma" result. It's like flipping a coin 100 times and getting heads every single time; it's so unlikely that you start to suspect the coin is rigged or your understanding of how coins work is wrong.

What Does This Mean?

The paper concludes that there is a significant tension between what we observe in the lab and what our current best theories predict.

There are two main possibilities the paper suggests:

1. New Physics: There might be a hidden force or a new particle (something beyond our current "Standard Model" of physics) influencing the dance, causing the $B^0_s$ to spin differently.
2. Hidden Complexity: Our current theory might be missing some subtle, complicated details about how these particles interact (hadronic effects) that we haven't calculated correctly yet.

The Bottom Line

This paper doesn't claim to have solved the mystery or found a new particle yet. Instead, it provides the most precise measurement to date of this strange spinning behavior. It tells the scientific community: "We have measured this very carefully, and the numbers definitely do not match the theory. We need to rethink our rules."

It's a high-precision measurement that keeps the door open for discovering something entirely new about how the universe works, or at the very least, forces us to polish our existing theories until they fit the data.

Technical Summary

Technical Summary of CERN-EP-2025-265: Measurement of the branching fractions and longitudinal polarisations of $B^0_{(s)} \to K^{*0}K^{*0}$ decays

Problem and Motivation
The decays $B^0 \to K^{*0}K^{*0}$ and $B^0_s \to K^{*0}K^{*0}$ proceed predominantly via gluonic loop (penguin) transitions ($b \to dss$ and $b \to sdd$, respectively). Due to their loop-suppressed nature, these processes serve as sensitive probes for New Physics (NP). A specific area of tension exists between experimental measurements and theoretical predictions regarding the longitudinal polarisation fractions ($f_L$) in $B \to VV$ decays. While naive factorisation predicts that the longitudinal component should dominate due to helicity suppression of transverse amplitudes, previous measurements have shown significantly lower $f_L$ values, particularly for $B^0_s$ decays. This discrepancy, known as the "polarisation puzzle," has led to various theoretical explanations involving weak-annihilation, charm-loops, and final-state interactions.

A key observable for testing these dynamics is the ratio of squared longitudinally polarised decay amplitudes, $L_{K^{*0}K^{*0}}$, defined as:
$$L_{K^{*0}K^{*0}} \equiv G \frac{\mathcal{B}(B^0_s \to K^{*0}K^{*0})}{\mathcal{B}(B^0 \to K^{*0}K^{*0})} \frac{f^s_L}{f^d_L}$$
where $G$ accounts for mass and lifetime differences. Hadronic uncertainties largely cancel in this ratio, making it a "golden channel" for NP searches. Previous LHCb measurements using 3 fb$^{-1}$ of data indicated a significant difference between $f^d_L$ and $f^s_L$, but with limited precision.

Methodology
This paper presents a time- and flavour-integrated amplitude analysis of $B^0$ and $B^0_s$ decays to the $(K^+\pi^-)(K^-\pi^+)$ final state, utilizing $pp$ collision data collected by the LHCb detector between 2011 and 2018 (Runs 1 and 2), corresponding to an integrated luminosity of 9 fb$^{-1}$.

• Event Selection: Signal candidates are selected requiring four charged tracks consistent with $K^\pm \pi^\mp$ pairs within the $K^*(892)^0$ mass region. Normalisation modes ($B^0 \to D^-\pi^+$ and $B^0_s \to D^-_s\pi^+$) are used to measure branching fractions relative to known modes. A Boosted Decision Tree (BDT) classifier is employed to suppress combinatorial background, trained separately for signal and normalisation modes across different data-taking periods.
• Amplitude Analysis: The analysis employs a covariant tensor (Zemach) formalism rather than the helicity formalism used in previous studies. This approach mitigates ambiguities associated with barrier factors at the production vertex for vector-vector states. The total amplitude is constructed as a coherent sum of intermediate components, including vector-vector ($VV$) states in $S$, $P$, and $D$-waves, and scalar-scalar ($SS$) or vector-scalar ($VS$) configurations.
• S-wave Modelling: The $K\pi$ $S$-wave contribution is modelled using a dispersive approach based on $\pi K$ scattering data, modulated by a production amplitude determined directly from the data via complex-valued polynomials.
• Efficiency and Simulation: A critical improvement over previous analyses is the inclusion of detector acceptance effects in the calculation of time-integrated quantities ($c_t$ and $s_t$). Efficiency maps are derived from large phase-space uniform simulation samples, calibrated with control samples (e.g., $D^0 \to K^-\pi^+$, $J/\psi \to \mu^+\mu^-$) to correct for differences between data and simulation in tracking, particle identification (PID), and trigger efficiency.
• Fitting Strategy: Extended unbinned maximum-likelihood fits are performed on the four-body mass distributions to extract yields. The amplitude fit utilizes the sFit method with sWeights to subtract background. A bootstrapping procedure is used to estimate statistical uncertainties due to the presence of sWeights.

Key Contributions and Results
The analysis provides the most precise measurements to date for these decay modes, superseding previous LHCb results.

1. Branching Fractions:
   The branching fractions are measured relative to normalisation modes:

   • $\mathcal{B}(B^0_s \to K^{*0}K^{*0}) = (0.938 \pm 0.025 \text{ (stat)} \pm 0.019 \text{ (syst)} \pm 0.036 \text{ (ext)}) \times 10^{-5}$
   • $\mathcal{B}(B^0 \to K^{*0}K^{*0}) = (4.73 \pm 0.30 \text{ (stat)} \pm 0.43 \text{ (syst)} \pm 0.16 \text{ (ext)}) \times 10^{-7}$
     These results represent an increase in precision by factors of 5.7 and 4.4, respectively, compared to world averages.

2. Longitudinal Polarisation Fractions:
   The measured fractions are:

   • $f^d_L = 0.600 \pm 0.022 \text{ (stat)} \pm 0.017 \text{ (syst)}$
   • $f^s_L = 0.159 \pm 0.010 \text{ (stat)} \pm 0.007 \text{ (syst)}$
     The result confirms that the longitudinal polarisation in $B^0_s \to K^{*0}K^{*0}$ is significantly lower than in $B^0 \to K^{*0}K^{*0}$, contrary to expectations from U-spin symmetry and QCD factorisation.

3. The Observable $L_{K^{*0}K^{*0}}$:
   The theory-motivated ratio is determined to be:
   $$L_{K^{*0}K^{*0}} = 4.92 \pm 0.55 \text{ (stat)} \pm 0.48 \text{ (syst)} \pm 0.02 \text{ (ext)} \pm 0.10 \text{ (}f_s/f_d\text{)}$$
   This value is in good agreement with previous LHCb Run 1 results but with considerably higher precision.

Significance and Claims
The paper claims that this measurement confirms the tension between experimental determinations and theoretical predictions (specifically QCD factorisation calculations) for longitudinal polarisation in $B \to VV$ decays at the level of 4.4 standard deviations.

The authors state that these measurements provide valuable input for refining theoretical form-factor calculations and constraining hadronic effects that impact precision tests of the Standard Model. While the observed deviations may indicate contributions from physics beyond the Standard Model, the paper modestly notes they could also result from poorly constrained Standard Model effects. The results motivate continued theoretical and experimental scrutiny of $B^0_{(s)} \to K^{*0}K^{*0}$ and related $B \to VV$ modes. The paper concludes that with the anticipated increased statistical precision from LHCb Run 3 and future reductions in theoretical uncertainties, more stringent tests of flavour symmetries will be possible.