The Search for $K_L \rightarrow \pi^0\pi^0\gamma\gamma$ and $K_L\rightarrow \pi^0\pi^0X$ where $X\rightarrow 2\gamma$ at the KOTO Experiment

Using 2021 data, the KOTO experiment conducted the first search for the decay $K_L \rightarrow \pi^0\pi^0\gamma\gamma$ and searched for axion-like particles in the $K_L \rightarrow \pi^0\pi^0X$ channel, observing three candidate events that resulted in upper limits on the branching ratios for both processes at the 95% confidence level.

The Gist

Imagine the universe as a giant, bustling cosmic city. Most of the traffic we see involves familiar cars like protons and electrons. But physicists suspect there are "ghost cars" driving around too—particles from a "dark sector" that we can't see directly because they don't interact with light. These ghost cars might be things like Axion-Like Particles (ALPs), which are theoretical candidates for dark matter.

The KOTO experiment in Japan is like a high-tech traffic camera station set up on a very specific, narrow road. Its main job is to catch a very rare event: a neutral kaon (a type of unstable particle) decaying into a neutral pion and two neutrinos. But while looking for that specific "ghost," the team decided to scan the area for other weird traffic patterns, too.

Here is what they did, explained simply:

1. The Setup: The Cosmic Trap

The experiment uses a powerful beam of protons (like a high-speed train) crashing into a gold target. This crash creates a spray of particles, including neutral kaons ($K_L$). These kaons are like unstable firecrackers; they fly down a long, shielded tunnel (the beamline) and eventually explode (decay) inside a giant, empty room filled with detectors.

The room is lined with CsI crystals (think of them as giant, super-sensitive eyes) that can see flashes of light (photons) when particles hit them. Surrounding the room are "veto" detectors—like motion sensors—that scream "STOP!" if any charged particle tries to sneak in. The goal is to only count events where nothing enters the room except the specific particles the team is looking for.

2. The Hunt: Two Different Mysteries

The team was looking for two specific types of "ghostly" traffic:

• Mystery A: The Invisible Passenger ($X$)
  They were looking for a decay where a kaon turns into two pions and a mysterious particle called $X$. This $X$ is a "ghost" that instantly vanishes into two photons (flashes of light).

  • The Analogy: Imagine a car (the kaon) crashing and breaking into two smaller cars (pions) and a magical, invisible balloon ($X$). The balloon pops immediately, leaving two sparks (photons). The team is trying to find the sparks and prove the balloon existed, even though they never saw the balloon itself.
  • The Range: They looked for balloons of different sizes (masses) between 160 and 220 MeV/c².

• Mystery B: The Double Flash ($\gamma\gamma$)
  They also looked for a decay where the kaon turns into two pions and two photons directly, without any invisible balloon in between.

  • The Analogy: This is like the car crashing and just leaving two sparks behind. This is a known phenomenon, but it's very rare. Measuring it helps physicists check if their "traffic laws" (theories of physics) are correct.

3. The Investigation: Sorting the Noise

The detectors see millions of flashes. Most of them are just background noise—like rain hitting the roof when you're looking for a specific bird. The main "noise" comes from a common decay called $K_L \to 3\pi^0$ (a kaon turning into three pions, which then turn into six photons).

To find the signal, the team used a digital filter:

• They built a computer model of what the "ghost" events should look like.
• They used a "goodness-of-fit" test (called $\chi^2$). Imagine trying to fit puzzle pieces together. If the pieces (energy and position of the flashes) fit the "ghost" puzzle perfectly, the score is low. If they fit the "noise" puzzle better, the score is high.
• They also used AI (Deep Learning) to look at the shape of the light flashes. Some flashes look like a clean laser beam (a photon), while others look like a messy splatter (a neutron or background noise). The AI helps tell the difference.

4. The Results: What Did They Find?

After analyzing data from 2021, here is the verdict:

• For the Double Flash ($K_L \to \pi^0\pi^0\gamma\gamma$):
  They found zero events. This is actually a success! It means they set a very strict limit on how often this happens. It's like saying, "We watched the road for a year and saw zero of these specific accidents, so they must happen less than once in a million years." This helps confirm our current theories.

• For the Invisible Passenger ($K_L \to \pi^0\pi^0X$):
  They found three suspicious events.

  • Two of these events happened right around a mass of 177 MeV/c².
  • However, the team had to be very careful. In statistics, finding a few "ghosts" in a sea of noise can sometimes just be a fluke (a random coincidence).
  • After running the numbers, they concluded that while these three events are interesting, they aren't enough to say, "We found the ghost!" with 100% certainty.
  • Instead, they set a limit: If these ghost particles do exist, they must be extremely rare (less than 1 in 10 million decays).

5. Why Does This Matter?

Think of this like searching for a needle in a haystack.

• If they had found a clear needle, it would have been a Nobel Prize-winning discovery of a new particle (Dark Matter).
• Since they didn't find a clear needle, they have still done something vital: They narrowed the haystack.

By saying, "We looked here, and the needle isn't here," they force theorists to change their maps. If a theory predicts the needle should be there, that theory is now in trouble. This helps scientists refine their search for the next generation of experiments.

In short: The KOTO team built a super-sensitive trap to catch rare particle decays. They didn't catch the "ghost particle" they were hoping for, but they proved that if it exists, it's much rarer than some theories predicted. They also confirmed that our understanding of how particles decay is working correctly. It's a successful "search and rule out" mission in the quest to understand the dark side of the universe.

Technical Summary

1. Problem Statement and Motivation

The paper addresses two primary objectives in the search for physics beyond the Standard Model (BSM) and precision tests of Chiral Perturbation Theory (ChPT):

• Search for Dark Sector Particles: The experiment searches for the decay $K_L \to \pi^0\pi^0X$, where $X$ is a hypothetical axion-like particle (ALP) that promptly decays into two photons ($X \to \gamma\gamma$). Theoretical models suggest that in certain energy ranges, three-body decay modes ($K_L \to \pi^0\pi^0X$) may dominate over two-body modes ($K_L \to \pi^0X$).
• Search for Rare Standard Model Decays: The paper presents the first search at KOTO for the rare Standard Model decay $K_L \to \pi^0\pi^0\gamma\gamma$. This process is described by ChPT up to second order; measuring its branching ratio provides a critical test of second-order ChPT calculations. Furthermore, this decay is theoretically linked to the $K_L \to \pi^0\pi^0l^-l^+$ modes via virtual photon processes ($\gamma^*\gamma^*$).

The study focuses on an $X$ mass range of 160–220 MeV/c², improving sensitivity by an order of magnitude over the previous E391a experiment and extending the search closer to the $\pi^0$ mass.

2. Methodology and Experimental Setup

Experimental Facility:
The analysis utilizes data collected in 2021 by the KOTO experiment at the J-PARC facility in Tokai, Japan.

• Beam: A 30-GeV proton beam strikes a gold target, producing a secondary neutral beam ($K_L$) with a momentum spectrum peaking at 1.4 GeV/c.
• Detector: The KOTO detector features a hermetic veto system and a central electromagnetic calorimeter (CSI) composed of 2,716 undoped CsI crystals.
• Trigger: A two-stage trigger system (L1 and L2) was used. For this analysis, events were required to have six energy clusters in the CSI and no hits in the veto detectors.

Event Reconstruction and Selection:

• Kinematic Constraints: Since all final state particles are measured (no missing energy), a constrained fit strategy was employed. Five constraints were applied:
  1. The invariant mass of all six photons equals the $K_L$ mass.
  2. & 3. The invariant masses of two photon pairs equal the $\pi^0$ mass.
  3. & 5. The $K_L$ momentum vector aligns with the line connecting the target and the Center of Energy (CoE) in the CSI.
• Fit Variables: A $\chi^2_{fit}$ was minimized to determine the best fit for photon positions and energies. Two specific $\chi^2$ variables were crucial:
  • $\chi^2_{fit}(2\pi^0\gamma\gamma)$: Assumes the signal hypothesis.
  • $\chi^2_{fit}(3\pi^0)$: Assumes the background hypothesis ($K_L \to 3\pi^0$). A low value here indicates a likely background event.
• Discriminators: A deep learning cluster shape discriminator (CSDDL) was used to distinguish electromagnetic showers from photonuclear interactions (a major source of background from $K_L \to 3\pi^0$).
• Signal Region Definition:
  • For $K_L \to \pi^0\pi^0X$: The signal region is defined by the invariant mass of the two photons not used in the $\pi^0$ reconstruction ($M_{\gamma5\gamma6}$). The search covered 13 mass points from 160 to 220 MeV/c² in 5 MeV intervals.
  • For $K_L \to \pi^0\pi^0\gamma\gamma$: The search covered the high-mass region $M_{\gamma5\gamma6} \in [160, 227.66]$ MeV/c².

Background Estimation:

• The dominant background is $K_L \to 3\pi^0$, where photonuclear interactions in the calorimeter mimic the signal topology.
• A scale factor ($F$) was derived by comparing data and Monte Carlo (MC) in a control region ($M_{\gamma5\gamma6} < 130$ MeV/c²) to correct for potential underestimation of the photonuclear cross-section in MC. This factor typically ranged from 1.5 to 2.5.
• The background expectation was validated using an "inverse selection" strategy in the high-mass region.

3. Key Contributions

• First Search at KOTO: This is the first search for $K_L \to \pi^0\pi^0\gamma\gamma$ and the first search for $K_L \to \pi^0\pi^0X$ at the KOTO experiment.
• Improved Sensitivity: The analysis achieves a sensitivity improvement of up to an order of magnitude compared to the E391a experiment in the overlapping mass range.
• Advanced Background Rejection: The implementation of the CSDDL deep learning discriminator and the specific $\chi^2$ fit strategies significantly improved the rejection of the $K_L \to 3\pi^0$ background, which is the primary challenge in this channel.
• Robust Statistical Treatment: The use of a weighted $CL_s$ method accounts for correlations between overlapping signal regions when setting upper limits.

4. Results

$K_L \to \pi^0\pi^0X$ Search:

• Observations: Three events were observed in the signal regions across the 13 mass points.
  • Two events were observed near an $X$ mass of 177 MeV/c² (specifically in the 175 and 180 MeV/c² bins).
  • One event was observed at 210 MeV/c².
• Upper Limits: No significant excess over the expected background was found. The results set 95% Confidence Level (C.L.) upper limits on the branching ratio:
  $$BR(K_L \to \pi^0\pi^0X) < (1–20) \times 10^{-7}$$
  The limits are most stringent (approx. $1 \times 10^{-7}$) at masses further from the $\pi^0$ mass (e.g., 200–220 MeV/c²) and less stringent near the $\pi^0$ mass due to background overlap.

$K_L \to \pi^0\pi^0\gamma\gamma$ Search:

• Observations: Zero events were observed in the signal region.
• Upper Limit: A 95% C.L. upper limit was set:
  $$BR(K_L \to \pi^0\pi^0\gamma\gamma) < 1.69 \times 10^{-6}$$

Systematic Uncertainties:
The total systematic uncertainty on the Single Event Sensitivity (SES) ranged from 8.9% to 11.5%. The dominant sources were veto cut selection (5.1%) and trigger effects (4.1%).

5. Significance

• Constraints on ALP Models: The results directly constrain theoretical models where ALP production from neutral three-body decays dominates over two-body modes. The exclusion of certain parameter spaces helps refine the search for dark sector particles.
• ChPT Validation: The upper limit on $K_L \to \pi^0\pi^0\gamma\gamma$ provides a new benchmark for testing second-order Chiral Perturbation Theory calculations.
• Future Prospects: The methodology developed, particularly the use of deep learning for cluster shape discrimination and the rigorous background scaling techniques, establishes a framework for future searches with higher statistics at KOTO and similar experiments. The observation of three events, while consistent with background fluctuations, highlights the need for continued data taking to distinguish potential signals from statistical noise.