Search for $K_{\mathrm{S(L)}}^{0} \rightarrow \pi^{+}\pi^{-}\mu^{+}\mu^{-}$ decays at LHCb

Using 13 TeV proton-proton collision data from the LHCb experiment, this study reports the first search for $K_{\mathrm{S(L)}}^{0} \rightarrow \pi^{+}\pi^{-}\mu^{+}\mu^{-}$ decays, finding no evidence of signals and establishing the first upper limits on their branching fractions at the 90% confidence level.

The Gist

Imagine the universe as a giant, high-speed racetrack where tiny particles zoom around at nearly the speed of light. In this paper, scientists from the LHCb experiment at CERN (a massive particle collider in Europe) acted like ultra-sensitive detectives, searching for a very specific, incredibly rare "traffic accident" involving particles called neutral kaons.

Here is the story of their search, explained simply:

The Mystery: A Ghostly Disappearance

Neutral kaons are unstable particles that usually decay (break apart) into simpler pieces very quickly. Most of the time, they split into two pions (another type of particle). However, the scientists were looking for a "ghost" version of this event.

They wanted to catch a kaon decaying into four particles at once: two pions and two muons (which are like heavy, cousin versions of electrons).

• The Analogy: Imagine a magician (the kaon) who usually pulls out two rabbits (pions). The scientists were hoping to see the magician pull out two rabbits and two heavy bowling balls (muons) at the exact same time.

Why is this hard?

This specific "four-particle" trick is incredibly difficult to pull off for two reasons:

1. It's extremely rare: Nature doesn't like to do this. The paper suggests it happens so rarely that if you watched a billion of these kaons, you might not see it happen even once.
2. The "Space" is tight: The kaon isn't very heavy. Trying to squeeze two pions and two heavy muons out of it is like trying to fit a whole family of elephants into a compact car. There just isn't enough "room" (energy) for them to all fit comfortably, making the event very unlikely.

The Detective Work

The team used data collected between 2016 and 2018 from the Large Hadron Collider. They had a massive dataset equivalent to 5.4 "inverse femtobarns" of collisions (a unit of measurement that essentially means they watched trillions of particle crashes).

To find their "needle in the haystack," they used a few clever tricks:

• The "Normal" Reference: They knew exactly how often kaons split into just two pions. They used this common event as a ruler to measure how rare the four-particle event would be.
• The Digital Filter (BDT): They used a sophisticated computer program (a "Boosted Decision Tree") trained to spot the specific signature of their target event while ignoring the billions of "noise" collisions that happen every second. Think of it as a metal detector that only beeps for gold and ignores iron, plastic, and dirt.
• The Time Traveler: They had to distinguish between two types of kaons: the "short-lived" ones ($K_S$) and the "long-lived" ones ($K_L$). Since the detector couldn't tell them apart on a case-by-case basis, they treated them as two separate searches, being extra careful not to mix up the clues.

The Verdict: No Ghosts Found

After sifting through the data, the result was clear: They found zero evidence of this decay.

• The Outcome: No "magician" was caught pulling out the rabbits and bowling balls simultaneously.
• The New Rule: Because they didn't find it, they set a new "speed limit" for how often this could happen. They announced that if this decay does happen, it must be rarer than 1 in a billion for the short-lived kaon and 1 in 1.5 million for the long-lived one.

Why does this matter?

Even though they didn't find the particle, this is a big deal.

• Ruling out the impossible: By saying "it's rarer than this," they are testing the rules of the universe (the Standard Model). If future experiments find it happens more often than this new limit, it would mean our current understanding of physics is wrong and there is some new, unknown force at play.
• Firsts: This is the very first time anyone has ever looked for this specific four-particle decay in muons. Before this paper, nobody knew if it was even possible to see it.

In summary: The LHCb team looked for a super-rare particle breakup that the laws of physics say should almost never happen. They looked at trillions of collisions, used smart computer filters, and found nothing. They didn't find the "ghost," but they successfully drew a line in the sand, telling future physicists: "If you find this ghost, it must be even more invisible than we just proved."

Technical Summary

Technical Summary: Search for $K^0_{S(L)} \to \pi^+\pi^-\mu^+\mu^-$ Decays at LHCb

Problem and Motivation
Neutral kaon radiative decays, specifically those involving a virtual photon converting into a lepton pair ($K^0 \to \pi^+\pi^-\gamma^* \to \pi^+\pi^-\ell^+\ell^-$), serve as critical laboratories for testing chiral perturbation theories and investigating CP violation. While the electron modes ($K^0_{S(L)} \to \pi^+\pi^-e^+e^-$) have been extensively studied by collaborations such as KTeV, KEK, and NA48, the equivalent muonic modes ($K^0_{S(L)} \to \pi^+\pi^-\mu^+\mu^-$) remain unexplored experimentally. The muonic channel probes higher values of the lepton-pair invariant mass ($q$), offering enhanced sensitivity to CP-violating observables where direct emission contributions are significant compared to the electron mode. However, these decays are highly suppressed in the Standard Model due to phase-space limitations arising from the small mass difference between the parent $K^0$ and the four-body final state. Theoretical predictions for the short-lived mode ($K^0_S$) exist, dominated by long-distance contributions, while no specific prediction is provided for the long-lived mode ($K^0_L$). This paper addresses the absence of experimental data by performing the first search for both $K^0_S \to \pi^+\pi^-\mu^+\mu^-$ and $K^0_L \to \pi^+\pi^-\mu^+\mu^-$ decays.

Methodology
The analysis utilizes proton-proton collision data collected by the LHCb experiment at a center-of-mass energy of 13 TeV, corresponding to an integrated luminosity of 5.4 fb$^{-1}$ (2016–2018). The search employs a relative measurement strategy, normalizing the signal yield to the abundant $K^0_S \to \pi^+\pi^-$ decay channel.

• Event Selection: Candidates are reconstructed by combining pairs of oppositely charged muons and pions originating from a common vertex displaced from the primary interaction vertex (PV). To ensure high reconstruction quality and trigger efficiency, only decays occurring within the Vertex Locator (VELO) are retained. A kinematic mass window of $490 < m(\pi^+\pi^-\mu^+\mu^-) < 600$ MeV/$c^2$ is applied. To avoid experimenter bias, the signal region ($490 < m < 510$ MeV/$c^2$) was blinded until the analysis procedure was finalized.
• Background Suppression: A Boosted Decision Tree (BDT) classifier, implemented via XGBoost, is used to distinguish signal from combinatorial background. The BDT utilizes variables such as the impact parameter significance of tracks, the displacement of the decay vertex, and the distance of closest approach between tracks. Separate BDTs are trained for "Trigger Independent of Signal" (TIS) and "Trigger on Signal" (TOS) categories.
• Efficiency and Normalization: The branching fractions are calculated using the single-event sensitivity ($\alpha$), which depends on the known branching fraction of the normalization channel, the number of observed candidates, and the ratio of reconstruction, selection, and trigger efficiencies. Efficiencies are derived from simulation samples, corrected using data-driven control channels (e.g., $J/\psi \to \mu^+\mu^-$ and $K^0_S \to \pi^+\pi^-$) to account for tracking and particle identification (PID) discrepancies.
• Statistical Analysis: A simultaneous extended unbinned maximum-likelihood fit is performed on the four-track invariant mass distributions for both trigger categories. The background model follows a kinematic threshold function, while the signal shape is modeled by the sum of two Crystal Ball functions.

Key Contributions and Results
The analysis finds no evidence of a signal, with the data consistent with the background-only hypothesis at a level of 1.9 standard deviations. Consequently, the collaboration sets the first-ever upper limits on the branching fractions for these decays at the 90% confidence level (CL):

• $\mathcal{B}(K^0_S \to \pi^+\pi^-\mu^+\mu^-) < 1.4 \times 10^{-9}$
• $\mathcal{B}(K^0_L \to \pi^+\pi^-\mu^+\mu^-) < 6.6 \times 10^{-7}$

The limits for the $K^0_L$ mode are less stringent than those for the $K^0_S$ mode due to the significantly lower LHCb acceptance for long-lived kaons ($\sim 2 \times 10^{-3}$ times that of $K^0_S$) and the inability to distinguish between $K^0_S$ and $K^0_L$ on an event-by-event basis at the reconstruction level. The systematic uncertainties, dominated by data-simulation differences in efficiency (9%) and trigger modeling (9–13%), are incorporated as Gaussian priors in the fit.

Significance
This work represents the first experimental search for $K^0_{S(L)} \to \pi^+\pi^-\mu^+\mu^-$ decays. By establishing these upper limits, the paper provides the first experimental constraints on these rare processes, which are predicted to be extremely suppressed in the Standard Model. The results offer a new benchmark for theoretical calculations, particularly for the $K^0_L$ mode where no specific prediction previously existed, and demonstrate the capability of the LHCb experiment to probe rare kaon decays with high precision despite the challenges of phase-space suppression and background combinatorics.