Measurement of branching fractions of $D^+_s\to K^0_SK^0_S \pi^+\pi^0$ and $D^+_s\to K^0_S K^+\pi^0\pi^0$

Using 7.33 fb$^{-1}$ of $e^+e^-$ collision data collected by the BESIII detector, this study reports the first observations of the hadronic decays $D^+_s\to K^0_SK^0_S\pi^+\pi^0$ and $D^+_s\to K^0_S K^+\pi^0\pi^0$ and determines their branching fractions to be $(4.08\pm0.46_{\rm stat}\pm0.45_{\rm syst})\times 10^{-3}$ and $(3.32\pm0.64_{\rm stat}\pm0.31_{\rm syst})\times 10^{-3}$, respectively.

The Gist

The Big Picture: A Cosmic "Tag and Track" Game

Imagine a massive, high-speed particle collider as a giant, ultra-precise bowling alley. Scientists shoot tiny particles (electrons and positrons) at each other at nearly the speed of light. When they crash, they create a shower of new, short-lived particles, much like how a bowling ball hitting pins creates a chaotic scatter of debris.

The goal of this paper is to find and count two very specific, rare types of "debris" that fly out of these crashes:

1. Decay A: A particle called a $D^+_s$ breaking apart into two neutral kaons ($K^0_S$), a positive pion ($\pi^+$), and a neutral pion ($\pi^0$).
2. Decay B: A $D^+_s$ breaking apart into one neutral kaon, one charged kaon, and two neutral pions.

These specific combinations had never been seen before. It's like looking for a specific, rare color of marble in a bucket of mixed marbles that no one has ever found in that exact pattern.

The Detective Work: The "Double-Tag" Method

Finding these rare particles is hard because they are produced alongside thousands of other messy particles. To solve this, the scientists used a clever trick called the "Double-Tag" method.

Think of it like a game of "Find the Twin" at a crowded party:

1. The Setup: When the particles collide, they don't just make one $D^+_s$; they usually make a pair: a $D^+_s$ and its antimatter twin, a $D^-_s$. They are born together and fly off in opposite directions.
2. The Single Tag (Finding the Twin): The scientists first look for the $D^-_s$ (the twin). They know exactly what this twin looks like because it can decay in 16 different, well-known ways (like a twin wearing a very distinct, recognizable outfit). If they spot the twin in one of these 16 outfits, they know, "Aha! There is a $D^+_s$ hiding on the other side of the room!"
3. The Double Tag (Finding the Mystery): Once they have identified the twin ($D^-_s$), they look at the other side of the collision to see what the $D^+_s$ did. They ask: "Did it turn into the rare combination we are looking for?"

By using the twin to confirm the existence of the partner, they can ignore all the background noise and focus only on the events where they are sure a $D^+_s$ was present.

The Experiment: The BESIII Detector

The scientists used a giant camera called the BESIII detector (located at the BEPCII collider in China) to take these pictures.

• The Camera: It's a giant cylinder that wraps around the collision point, acting like a 360-degree security camera. It tracks the paths of charged particles (like pions and kaons) and measures the energy of light particles (like photons from neutral pions).
• The Data: They analyzed data equivalent to 7.33 "inverse femtobarns" of collisions. To put that in perspective, this is like taking billions of high-speed snapshots of particle crashes over several years to ensure they didn't miss a single rare event.

The Results: Two New Discoveries

After sifting through millions of events, the team found:

• 124 events of the first rare decay ($D^+_s \to K^0_S K^0_S \pi^+ \pi^0$).
• 135 events of the second rare decay ($D^+_s \to K^0_S K^+ \pi^0 \pi^0$).

They calculated the Branching Fraction for these events. In simple terms, this is the "odds" of this specific breakup happening.

• For the first decay, it happens about 4 times out of every 1,000 $D^+_s$ particles.
• For the second decay, it happens about 3.3 times out of every 1,000.

The paper states that these results are statistically significant (meaning it's highly unlikely they were just random noise) and that the two rates are very similar to each other.

Why Does This Matter?

The authors explain that studying these four-particle breakups helps physicists understand the "rules of the road" for how quarks (the building blocks of matter) stick together and break apart.

• The Mystery: They noticed that while the two decays are similar, they aren't identical. One of them might be influenced by a specific intermediate step involving a particle called the $f_0(980)$, which acts like a temporary bridge before the final pieces fly apart.
• The Goal: By measuring these rates, scientists can test theories about Symmetry Breaking. Imagine if you had a perfect mirror image of a process, but the mirror image behaved slightly differently. Understanding why it behaves differently helps us understand the fundamental forces of the universe.

Summary

In short, the BESIII collaboration used a "twin-finding" strategy to hunt for two previously unseen ways that a specific particle ($D^+_s$) can decay. They successfully found them, measured how often they happen, and provided new clues about how the subatomic world is put together. They did not claim these findings have immediate medical or technological applications; the value is purely in deepening our understanding of particle physics.

Technical Summary

Technical Summary: Measurement of Branching Fractions of $D^+_s \to K^0_S K^0_S \pi^+ \pi^0$ and $D^+_s \to K^0_S K^+ \pi^0 \pi^0$

Problem and Motivation
Experimental investigations into hadronic $D^+_s$ decays are critical for refining the understanding of CP violation and quark SU(3) symmetry breaking within the charm sector. While four-body heavy hadron weak decays offer a rich phenomenology for studying QCD hadronization, differential distributions, and testing factorization approaches, specific decay modes had remained unexplored. Prior to this work, decays such as $D^+_s \to K^+ K^- \pi^+ \pi^0$, $D^+_s \to K^0_S K^- \pi^+ \pi^+$, and $D^+_s \to K^0_S K^+ \pi^+ \pi^-$ were well-investigated. However, no experimental information existed regarding the modes $D^+_s \to K^0_S K^0_S \pi^+ \pi^0$ and $D^+_s \to K^0_S K^+ \pi^0 \pi^0$. This paper addresses this gap by reporting the first observation of these two decay channels and determining their branching fractions.

Methodology
The analysis utilizes $e^+e^-$ collision data corresponding to an integrated luminosity of 7.33 fb$^{-1}$, collected with the BESIII detector at center-of-mass energies ranging from 4.128 to 4.226 GeV. The measurement employs the double-tag (DT) method, a technique originally developed by the MARKIII Collaboration.

1. Single-Tag (ST) Reconstruction: The $D^-_s$ meson is fully reconstructed via one of sixteen hadronic decay modes (e.g., $K^+K^-\pi^-$, $\pi^-\pi^+\pi^-$, $K^0_S K^-$, etc.). Charged tracks are selected based on vertex proximity and particle identification (PID) using $dE/dx$ and time-of-flight (TOF) information. $K^0_S$ candidates are formed from $\pi^+\pi^-$ pairs with invariant masses within specific windows. Photon candidates are selected from electromagnetic calorimeter (EMC) showers, and $\pi^0$ candidates are reconstructed from $\gamma\gamma$ pairs.
2. Kinematic Constraints: To suppress backgrounds from non-$D^*_s D_s$ processes, the beam-constrained mass ($M_{BC}$) of the ST candidate is required to fall within specific ranges defined for each energy point. If multiple candidates exist, the one with a recoil mass closest to the known $D^*_s$ mass is retained.
3. Double-Tag (DT) Selection: In the recoil system against the tagged $D^-_s$, the signal $D^+_s$ decay is reconstructed. For the signal modes, the transition photon or $\pi^0$ from the $D^*_s \to \gamma(\pi^0) D^+_s$ process is identified by minimizing the energy difference $\Delta E$. Signal candidates are selected based on invariant mass constraints for intermediate resonances and final state particles.
4. Yield Extraction: The signal yields are extracted using unbinned maximum likelihood fits to the signal mass ($M_{sig}$) distributions. The signal shape is modeled using Monte Carlo (MC) simulations, while background shapes are derived from inclusive MC samples. A peaking background from $D^+_s \to \rho^+ \phi$ is accounted for in the $K^0_S K^0_S \pi^+ \pi^0$ fit.
5. Efficiency and Systematics: Detection efficiencies are determined using MC simulations generated with mixed sub-decays based on Particle Data Group (PDG) branching fractions. Systematic uncertainties are evaluated by varying selection criteria, comparing data and MC for tracking/PID efficiencies, assessing MC generator model dependence, and accounting for "tag bias" arising from differences between inclusive and signal MC environments.

Key Contributions and Results
The paper reports the first observation of the decays $D^+_s \to K^0_S K^0_S \pi^+ \pi^0$ and $D^+_s \to K^0_S K^+ \pi^0 \pi^0$. The statistical significances are estimated at $8.3\sigma$ and $5.1\sigma$, respectively.

The measured branching fractions are:

• $\mathcal{B}(D^+_s \to K^0_S K^0_S \pi^+ \pi^0) = (4.08 \pm 0.46_{\text{stat}} \pm 0.45_{\text{syst}}) \times 10^{-3}$
• $\mathcal{B}(D^+_s \to K^0_S K^+ \pi^0 \pi^0) = (3.32 \pm 0.64_{\text{stat}} \pm 0.31_{\text{syst}}) \times 10^{-3}$

The total systematic uncertainties are determined to be 11.1% and 9.2% for the respective modes. The analysis includes a detailed breakdown of systematic sources, including tracking, PID, photon/$\pi^0$ reconstruction, MC generator modeling, and tag bias.

Significance
The authors note that the measured branching fractions for the two modes are consistent with each other within uncertainties. They observe that while the decay $D^+_s \to \bar{K}^{*0} K^{*+}$ can contribute to both final states, the decay chain $D^+_s \to f_0(980)(\to K^0_S K^0_S)\rho^+(\to \pi^+ \pi^0)$ contributes exclusively to the $K^0_S K^0_S \pi^+ \pi^0$ mode. Consequently, the difference in the measured branching fractions may be attributed to the contribution from the $D^+_s \to f_0(980)\rho^+$ decay.

The paper concludes that future amplitude analyses of these newly observed decays will provide valuable insights into two-body decay modes involving scalar, vector, axial-vector, and tensor mesons, thereby aiding in the improvement of the understanding of quark-level SU(3)-flavor symmetry.