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Measurements of the inclusive Branching fractions of the $D_s^+ \to \eta (\eta^\prime) X$ decays

Using 7.33 fb⁻¹ of $e^+e^-$ collision data collected by the BESIII detector, this study measures the inclusive branching fractions of $D_s^+ \to \eta X$ and $D_s^+ \to \eta^\prime X$ decays with significantly improved precision compared to previous CLEO results.

Original authors: BESIII Collaboration

Published 2026-09-30

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CC BY 4.0

Original authors: BESIII Collaboration

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the subatomic world, matter is built from a small family of fundamental particles called quarks. When these quarks pair up, they form short-lived particles known as mesons. One such particle, the D+ s meson, is a specific combination of a charm quark and a strange quark. Because it is unstable, it does not last long; it quickly breaks apart into other, lighter particles. Physicists study these breakups, called decays, to understand the rules that govern how matter transforms. By measuring exactly how often a D+ s meson turns into specific groups of particles, scientists can check if their current theories of the universe are complete. If the numbers add up perfectly, the theory holds. If there is a gap between what is predicted and what is seen, it suggests there are hidden processes or undiscovered particles waiting to be found.

A researcher using the BESIII detector at the Institute of High Energy Physics in Beijing has taken a fresh, high-precision look at how the D+ s meson decays. They focused on two specific outcomes: cases where the decay produces a particle called an eta meson, and cases where it produces a slightly heavier cousin called an eta-prime meson. These two particles are special because they appear in a large number of D+ s decays, accounting for more than a third of all such events. However, previous measurements of these decays were not precise enough to tell if the known ways the particle breaks down were the only ways. The new study uses a massive amount of data collected from electron-positron collisions to measure these decay rates with much greater accuracy than ever before.

The researcher analyzed data corresponding to an integrated luminosity of 7.33 inverse femtobarns, collected at collision energies between 4.128 and 4.226 GeV. To find the rare decays they were looking for, they used a clever technique called the "double-tag" method. In their experiment, electrons and positrons collide to create pairs of D+ s mesons moving in opposite directions. The researcher fully reconstructed one of the mesons in the pair, identifying exactly how it decayed. This act of identifying one half of the pair is called "tagging." Because the two mesons are created together, knowing the properties of the first one allows the scientist to infer the existence and properties of the second one, even if they cannot see every single particle it produces. By looking at the remaining debris from the second meson, they could count how many times it produced an eta or an eta-prime particle, regardless of what other particles were created alongside them.

The results of this counting exercise are precise. The researcher found that the D+ s meson decays into an eta particle plus other debris about 30.72 percent of the time. For the eta-prime particle, the rate is 12.30 percent. These numbers come with very small margins of error, representing a significant improvement in precision over previous measurements made by other experiments. The new data is consistent with earlier findings, confirming that the basic picture of these decays is correct, but the much tighter error bars allow for a more rigorous test of the underlying physics.

When the researcher added up all the known, specific ways the D+ s meson can decay into an eta particle, they found a total of about 25.40 percent. Their new measurement of the total rate, 30.72 percent, is higher than this sum. This difference suggests that there are likely other, unobserved ways the D+ s meson can decay into an eta particle that have not yet been identified. The gap is small but statistically meaningful, hinting at hidden decay modes that future experiments may need to hunt down. In contrast, when they performed the same calculation for the eta-prime particle, the known decay modes added up to 11.97 percent, which is almost identical to their new measurement of 12.30 percent. This agreement suggests that for the eta-prime, scientists have likely already found nearly all the ways this particle appears in D+ s decays.

These findings serve as a vital reference for the broader field of particle physics. Precise measurements of how often particles decay are essential for calibrating experiments that study heavier particles, such as those containing bottom quarks. By providing these accurate numbers, the BESIII collaboration helps reduce uncertainties in other areas of research. The study confirms that while the known decay paths for the eta-prime are well understood, the story for the eta particle is not yet finished. The small but noticeable gap in the numbers points to a frontier of discovery, inviting further investigation into the unseen behaviors of the subatomic world.

Technical Summary: Measurements of the Inclusive Branching Fractions of the $D^+_s \to \eta(\eta')X$ Decays

Problem and Motivation
The $D^+_s$ meson is the ground state of charmed-strange mesons, and precise measurements of its branching fractions (BFs) are critical for understanding decay dynamics and normalizing $B$ and $B_s$ decays. However, the sum of currently measured exclusive $D^+_s$ BFs accounts for only approximately 85.7% of the total, leaving a significant fraction of unobserved decay modes. While Cabibbo-favored transitions to final states containing $\eta$ and $\eta'$ mesons are known to dominate $D^+_s$ decays, existing inclusive measurements by the CLEO and previous BESIII collaborations suffer from substantial statistical uncertainties. Specifically, previous inclusive sums for modes containing $\eta$ or $\eta'$ showed discrepancies with the sum of known exclusive modes, motivating a high-precision inclusive measurement to resolve these uncertainties and constrain the total branching fraction budget.

Methodology
This analysis utilizes $e^+e^-$ collision data collected by the BESIII detector at center-of-mass energies between 4.128 and 4.226 GeV, corresponding to an integrated luminosity of 7.33 fb $^{-1}$ . The study employs the double-tag (DT) method, leveraging the $e^+e^- \to D^{*\pm}_s D^{\mp}_s$ process, which has a cross-section approximately twenty times larger than the $D^+_s D^-_s$ process in this energy range.

Tagging: Hadronic $D^-_s$ mesons (the "tag" side) are fully reconstructed via four specific decay channels: $K^0_S K^-$ , $K^+K^-\pi^-$ , $K^+K^-\pi^-\pi^0$ , and $K^0_S K^+\pi^-\pi^-$ . Events where a tag is found are "single-tag" (ST) events.
Signal Identification: Signal candidates ( $D^+_s \to \eta X$ or $D^+_s \to \eta' X$ ) are identified on the recoil side by analyzing remaining charged tracks and neutral particles not used in the tag reconstruction.
Yield Extraction:
- ST yields are determined by fitting the reconstructed mass ( $M_{tag}$ ) distributions.
- DT yields are extracted using a two-dimensional (2D) fit of the signal mass ( $M_\eta$ or $M_{\eta'}$ ) versus $M_{tag}$ . This approach distinguishes between signal events and three types of background components: those peaking in the signal mass but flat in the tag mass, those peaking in the tag mass but flat in the signal mass, and non-peaking backgrounds.
Efficiency and Simulation: Monte Carlo (MC) simulations based on GEANT4 are used to determine detection efficiencies and estimate backgrounds. The simulation incorporates beam energy spread, initial-state radiation (via KKMC), and particle decays (via EVTGEN).

Key Results
The inclusive branching fractions are calculated using the ratio of DT yields to the product of ST yields and efficiencies, normalized by the known BFs of the tag modes and the intermediate decays of $\eta$ and $\eta'$ . The measured values are:

$B(D^+_s \to \eta X)$ : $(30.72 \pm 0.90 \text{ (stat)} \pm 0.46 \text{ (syst)})\%$
$B(D^+_s \to \eta' X)$ : $(12.30 \pm 0.94 \text{ (stat)} \pm 0.37 \text{ (syst)})\%$

These results are consistent with previous CLEO measurements but represent a significant improvement in precision, with uncertainties reduced by factors of 2.8 for the $\eta$ mode and 1.9 for the $\eta'$ mode.

Significance and Claims
The paper claims that these precise measurements serve as essential inputs for normalizing $B$ and $B_s$ decays, constraining systematic uncertainties, and improving background modeling in future heavy-flavor physics studies.

By comparing the measured inclusive BFs with the sums of known exclusive BFs listed in the Particle Data Group (PDG) tables, the author reports:

For $\eta$ modes: A difference of $(5.33 \pm 1.04)\%$ , suggesting the existence of possible unobserved $D^+_s$ decay modes involving an $\eta$ meson.
For $\eta'$ modes: A difference of $(0.33 \pm 1.03)\%$ , indicating no significant excess over the sum of known exclusive modes.

These findings help constrain the total branching fractions of all unobserved decays in the exclusive $D^+_s \to \eta(\eta')X$ sector, contributing to a more complete understanding of $D^+_s$ decay dynamics.

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