First Measurement of the $D_s^+\rightarrow K^0μ^+ν_μ$ Decay

Using 7.33 fb⁻¹ of $e^+e^-$ collision data collected with the BESIII detector, this paper reports the first measurement of the $D_s^+\rightarrow K^0\mu^+\nu_\mu$ decay, determining its branching fraction and the form factor $f^{K^0}_{+}(0)$ to provide the most precise extraction of $|V_{cd}|$ from this channel and the first test of lepton flavor universality in these decays.

The Gist

Imagine the subatomic world as a bustling, high-speed train station. In this station, particles called $D_s^+$ mesons are like special delivery trucks that carry heavy cargo (quarks) and then break apart into smaller packages.

This paper is a report from the BESIII Collaboration, a team of scientists working at a massive particle collider in China (the BEPCII). They have just solved a long-standing mystery about one specific type of delivery: the $D_s^+ \to K^0 \mu^+ \nu_\mu$ decay.

Here is the story of their discovery, broken down into simple concepts:

1. The Mystery: A Missing Delivery

For a long time, scientists knew that these "delivery trucks" ($D_s^+$) could drop off their cargo in two main ways:

• The Electron Route: Dropping off a package with an electron ($e^-$). This was already well-documented.
• The Muon Route: Dropping off a package with a muon ($\mu^-$). This is a heavier cousin of the electron.

Until now, nobody had successfully counted the Muon Route. It was like knowing a bus line exists but never having counted the number of passengers who actually got on. The scientists wanted to catch this specific decay in action for the first time.

2. The Detective Work: The "Tag and Track" Method

To find these rare events, the scientists used a clever trick called "Single-Tag" and "Double-Tag".

• The Setup: They created millions of these $D_s^+$ trucks in the lab.
• The Single Tag (The Witness): In many collisions, they would spot one of the trucks breaking apart in a very specific, easy-to-recognize way. This acted as a "witness" confirming that a truck had just arrived at the station.
• The Double Tag (The Crime Scene): Because these trucks are created in pairs, if they saw one truck break apart (the witness), they knew the other truck in the pair was right there too. They then looked closely at that second truck to see if it broke apart into the rare Muon package they were hunting for.

It's like seeing a pair of twins enter a room. If you see one twin leave through the front door (the tag), you know the other twin is still inside. You then watch the room carefully to see if the second twin leaves through a secret back door (the rare decay).

3. The Big Reveal: Counting the Passengers

After sifting through 7.33 billion worth of data (an enormous amount of information), they finally found 147 instances of this rare decay.

From this, they calculated the Branching Fraction. Think of this as the "success rate." They found that about 0.29% of the time, the $D_s^+$ truck chooses the Muon route. This number is crucial because it helps physicists understand the rules of the universe.

4. The "Shape-Shifter" (Form Factors)

The paper also measured something called a Form Factor.

• The Analogy: Imagine the $D_s^+$ truck is a shape-shifting robot. As it breaks apart, it changes its shape. The "Form Factor" is a mathematical description of how it changes shape during the breakup.
• The Result: The scientists measured this shape-shifting behavior with the highest precision ever recorded. They found that the "shape" at the start of the breakup matches a prediction made by Lattice QCD (a super-computer simulation of the strong nuclear force) almost perfectly.

5. Why Does This Matter?

This discovery is a big deal for three reasons:

1. Testing the Rules (Standard Model): The Standard Model is the "rulebook" of physics. It predicts that electrons and muons should behave almost identically, except for their weight. The scientists checked if the Muon route happened at the same rate as the Electron route. Result: They matched perfectly! This confirms that the "Lepton Flavor Universality" rule holds true—nature treats these particles fairly.
2. Solving a Math Puzzle: There was a slight disagreement (a "tension") in previous measurements about a number called $|V_{cd}|$, which describes how quarks mix and change flavors. By measuring this new decay, the team calculated a new, more precise value for this number. Their result helps smooth out the previous disagreement, making the universe's math look more consistent.
3. Checking the Super-Computers: The experimental data was compared against the Lattice QCD computer simulations. The fact that they agree so well gives scientists confidence that their computer models of the strong force are accurate. This is vital for understanding even heavier particles in the future.

The Bottom Line

The BESIII team successfully caught the "Muon delivery" for the first time. They proved that the universe follows its own rules consistently, their measurements match the best computer simulations, and they have provided a sharper, more precise map of how heavy particles break apart. It's a small step for a particle, but a giant leap for understanding the fundamental forces that hold our universe together.

Technical Summary

1. Problem and Motivation

The paper addresses several critical gaps in the understanding of charm meson physics and the Standard Model (SM):

• Missing Experimental Data: While the semileptonic decay $D^+_s \to K^0 e^+ \nu_e$ had been previously measured, the corresponding muon mode, $D^+_s \to K^0 \mu^+ \nu_\mu$, had never been observed. This lack of data prevented a direct test of Lepton Flavor Universality (LFU) in this specific transition.
• Tension in CKM Elements: There was a reported $\sim 2\sigma$ tension between the Cabibbo-Kobayashi-Maskawa (CKM) matrix element $|V_{cd}|$ extracted from $D^+_s \to K^0 e^+ \nu_e$ decays and that extracted from $D \to \pi \ell \nu_\ell$ decays.
• Theoretical Discrepancies: Theoretical predictions for the hadronic form factor (FF) $f_+^{K^0}(0)$ in the $D^+_s \to K^0$ transition vary significantly (ranging from 0.57 to 0.82). Recent Lattice QCD (LQCD) calculations predict $f_+^{K^0}(0) = 0.6307(20)$, but experimental precision was insufficient to rigorously test these predictions or the assumption that FFs are insensitive to the spectator quark mass (U-spin symmetry).
• LFU Tests: Theoretical models predict the ratio of branching fractions $R_{\mu/e}^{K^0} = \mathcal{B}(D^+_s \to K^0 \mu^+ \nu_\mu) / \mathcal{B}(D^+_s \to K^0 e^+ \nu_e)$ to be very close to 1 (0.96–1.00), but precise experimental verification was lacking.

2. Methodology

The analysis utilizes data collected by the BESIII detector at the BEPCII collider.

• Dataset: An integrated luminosity of 7.33 fb$^{-1}$ collected at center-of-mass energies between 4.128 and 4.226 GeV, corresponding to the production of $D_s^+ D_s^-$ pairs via $e^+e^- \to D_s^{*+} D_s^- + \text{c.c.}$.
• Tagging Technique: The analysis employs the "Single-Tag" (ST) and "Double-Tag" (DT) method:
  • ST: One $D_s^-$ meson is fully reconstructed in one of 14 hadronic decay modes.
  • DT: The recoiling $D_s^+$ is reconstructed in the signal mode $D_s^+ \to K^0 \mu^+ \nu_\mu$. The $K^0$ is reconstructed via $K^0_S \to \pi^+ \pi^-$.
• Event Selection & Kinematics:
  • Muon ID: Combined $dE/dx$, Time-of-Flight (TOF), and Electromagnetic Calorimeter (EMC) shower properties to distinguish muons from electrons and kaons.
  • Background Suppression:
    • Rejection of $D_s^+ \to K^0 \pi^+$ background by requiring the $K^0 \mu^+$ invariant mass $< 1.70$ GeV/$c^2$.
    • Suppression of events with extra $\pi^0$s by requiring the maximum unused photon energy $E_{\gamma}^{\text{max}} < 0.15$ GeV.
    • Kinematic fit requiring energy-momentum conservation for the $D_s^+ D_s^-$ pair.
  • Signal Extraction: The signal yield is extracted using the missing mass squared ($MM^2$) distribution, defined as:
    $$MM^2 = (\sqrt{s} - E_{D_s^-} - E_\gamma - E_{SL})^2 - |\vec{p}_{D_s^-} + \vec{p}_\gamma + \vec{p}_{SL}|^2$$
    where $E_{SL}$ and $\vec{p}_{SL}$ are the energy and momentum of the $K^0 \mu^+$ system.
• Form Factor Extraction: A simultaneous fit is performed on the partial decay rates of both $D_s^+ \to K^0 \mu^+ \nu_\mu$ and $D_s^+ \to K^0 e^+ \nu_e$ across seven $q^2$ bins. Three theoretical parameterizations (Simple Pole, Modified Pole, and $z$-series expansion) are tested to extract the form factor $f_+^{K^0}(q^2)$.

3. Key Contributions

• First Observation: This is the first experimental measurement of the $D^+_s \to K^0 \mu^+ \nu_\mu$ decay channel.
• Precision Branching Fraction: Determination of the absolute branching fraction with high precision.
• Form Factor Determination: Extraction of the hadronic form factor $f_+^{K^0}(0)$ and the product $f_+^{K^0}(0)|V_{cd}|$ using the $z$-series expansion, providing the most precise determination to date for this transition.
• LFU Test: The first test of Lepton Flavor Universality in $D_s \to K^0 \ell \nu$ decays, both in the integrated rate and differentially across $q^2$ intervals.
• CKM Constraint: A new determination of $|V_{cd}|$ using this channel, which helps resolve tensions with other extraction methods.

4. Key Results

• Branching Fraction:
  $$\mathcal{B}(D^+_s \to K^0 \mu^+ \nu_\mu) = (2.89 \pm 0.27_{\text{stat}} \pm 0.12_{\text{syst}}) \times 10^{-3}$$
• Form Factor and CKM Product:
  Using the $z$-series expansion and $|V_{cd}| = 0.22486 \pm 0.00068$:
  $$f_+^{K^0}(0) = 0.623 \pm 0.036_{\text{stat}} \pm 0.009_{\text{syst}}$$
  $$f_+^{K^0}(0)|V_{cd}| = 0.140 \pm 0.008_{\text{stat}} \pm 0.002_{\text{syst}}$$
• Lepton Flavor Universality Ratio:
  Combining the new muon result with the previous electron result:
  $$R_{\mu/e}^{K^0} = \frac{\mathcal{B}(D^+_s \to K^0 \mu^+ \nu_\mu)}{\mathcal{B}(D^+_s \to K^0 e^+ \nu_e)} = 0.97 \pm 0.12_{\text{stat}} \pm 0.04_{\text{syst}}$$
  This is consistent with the SM prediction ($0.981$) and LQCD calculations. No violation of LFU is observed in any $q^2$ interval.
• CKM Element $|V_{cd}|$:
  Using the LQCD prediction $f_+^{K^0}(0) = 0.6307(20)$ as input:
  $$|V_{cd}| = 0.220 \pm 0.013_{\text{stat}} \pm 0.003_{\text{syst}} \pm 0.001_{\text{LQCD}}$$
  This result is consistent with the world average from $D \to \pi \ell \nu$ decays, effectively resolving the previous $\sim 2\sigma$ tension.
• U-Spin Symmetry:
  Comparing $f_+^{K^0}(0)$ with $f_+^{\pi^0}(0)$ yields a ratio of $0.995 \pm 0.061$, consistent with U-spin symmetry expectations.

5. Significance

• Validation of Lattice QCD: The measured form factor $f_+^{K^0}(0)$ is in excellent agreement with LQCD predictions, validating the theoretical framework used for non-perturbative QCD calculations in the charm sector.
• Resolution of Tensions: The new measurement of $|V_{cd}|$ removes the discrepancy between values derived from $D_s \to K \ell \nu$ and $D \to \pi \ell \nu$ decays, strengthening the consistency of the CKM matrix unitarity tests.
• Constraints on Theory: The results provide stringent constraints on various theoretical models (CQM, LCSR, RQM, etc.). Specifically, the measured branching fraction disfavors the central values of the CCQM and RQM models, while the form factor disfavors the CQM model.
• Future Impact: The precision achieved here sets a benchmark for future studies of semileptonic decays and supports the application of LQCD to $B$-meson decays for the precise determination of $|V_{cb}|$ and $|V_{ub}|$.
• Note on High-$q^2$ Tension: While the value at $q^2=0$ agrees with LQCD, the paper notes a $\sim 2\sigma$ deviation in the shape of the form factor at high-$q^2$ regions compared to LQCD and other theoretical models, suggesting a need for further theoretical refinement in the high-momentum transfer region.