Improved measurement of the branching fraction of $D_s^+\to\mu^+\nu_\mu$

Using $e^+e^-$ collision data collected by the BESIII detector, this study presents an improved measurement of the branching fraction for the leptonic decay $D_s^+\to\mu^+\nu_\mu$, which is subsequently used to determine the $D_s^+$ decay constant $f_{D_s^+}$ and the CKM matrix element $|V_{cs}|$ with enhanced precision.

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 BESIII collaboration (a team of physicists working at a giant particle collider in Beijing) decided to take a closer look at a specific type of "race car" called the $D_s^+$ meson.

Here is the story of what they did, explained simply:

1. The Setup: Catching a Rare Escape Artist

Think of the $D_s^+$ meson as a very unstable, short-lived particle. It's like a soap bubble that pops almost instantly. Usually, when it pops, it breaks apart into a chaotic mix of other particles.

However, sometimes, very rarely, it decides to take a "shortcut." Instead of breaking into a crowd, it transforms into just two things: a muon (a heavy cousin of the electron) and a neutrino (a ghost-like particle that barely interacts with anything).

This specific "shortcut" is called a leptonic decay ($D_s^+ \to \mu^+ \nu_\mu$). It's so rare that for every 1,000 times this particle pops, it only takes this shortcut about 5 times.

2. The Challenge: Finding a Needle in a Haystack

The scientists wanted to measure exactly how often this rare shortcut happens. This number is called the Branching Fraction.

Why is this hard?

• The Haystack: They had a massive pile of data (7.33 "inverse femtobarns," which is a fancy way of saying they collected a huge amount of collision data).
• The Needle: They needed to find the specific events where the $D_s^+$ turned into a muon and a neutrino.
• The Ghost: The neutrino is invisible. It leaves no trace in the detector. It's like trying to find a thief in a room by only seeing the empty space where they stood.

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

To find the invisible neutrino, the scientists used a clever trick called the "Single-Tag" and "Double-Tag" method.

Imagine you are at a busy train station (the particle collider). You know that particles are always produced in pairs (like a $D_s^+$ and a $D_s^-$).

• Step 1 (The Tag): You spot one of the pair (the $D_s^-$) and catch it in a net. You know exactly what it is and how much energy it has. This is your "Tag."
• Step 2 (The Track): Because you know the total energy of the collision and you've caught one partner, you can calculate exactly what the other partner ($D_s^+$) should have been doing.
• Step 3 (The Missing Piece): You look at the other side of the room. You see a muon. But you don't see the neutrino. However, because you know the total energy and momentum, you can calculate the "missing" energy. If the math adds up perfectly to a neutrino, you've found your rare event!

4. The New Tools: Better Glasses

In previous experiments, the scientists used "glasses" (detectors) that were good, but not perfect. They missed some muons or got confused by other particles.

In this new study, they used:

• More Data: They looked at a much larger pile of collisions (like looking at a bigger haystack).
• Better Glasses: They upgraded their "Muon Identifier" (a special layer of the detector that acts like a metal detector for muons). This allowed them to distinguish a muon from a regular particle much better than before.

5. The Results: A Precise Measurement

After all the counting and calculating, they found:

• The Frequency: The $D_s^+$ meson takes the "muon shortcut" about 0.53% of the time.
• The Physics: This number helps them calculate two fundamental secrets of the universe:
  1. $f_{D_s}$: How "strongly" the particles inside the meson are holding hands (the decay constant).
  2. $|V_{cs}|$: How likely a "charm" quark is to turn into a "strange" quark (part of the CKM matrix, which is like the rulebook for how particles change flavors).

6. Why Does This Matter?

The scientists compared their results to the Standard Model (the current best theory of how the universe works).

• The Verdict: Their results matched the theory perfectly.
• The Big Question: Recently, other experiments found that heavy particles (like B-mesons) might be breaking the rules of "Lepton Flavor Universality" (the idea that electrons, muons, and taus should behave the same way, just with different weights).
• The Conclusion: By measuring the $D_s^+$ decay so precisely, the BESIII team showed that in this specific case, the rules are not broken. The muon and the tau behave exactly as the Standard Model predicts.

Summary Analogy

Imagine you are trying to figure out if a specific type of coin is fair. You flip it 10,000 times.

• Old experiments: You flipped it 1,000 times with a blurry camera. You guessed it was fair, but you weren't 100% sure.
• This experiment: You flipped it 10,000 times with a 4K camera and a super-computer to track every spin. You confirmed it is fair, and you measured the exact weight of the coin with incredible precision.

This paper confirms that our current understanding of the "coin" (the Standard Model) is solid, at least for this specific type of particle, and provides a new, highly precise ruler for physicists to measure the fundamental forces of nature.

Technical Summary

1. Problem and Motivation

The paper addresses the need for high-precision measurements of the leptonic decay of the charmed strange meson, $D^+_s \to \ell^+ \nu_\ell$ (where $\ell = e, \mu, \tau$).

• Physics Goals:
  • Standard Model (SM) Tests: In the SM, the decay width depends on the Fermi constant ($G_F$), the CKM matrix element $|V_{cs}|$, the $D^+_s$ mass, and the decay constant $f_{D^+_s}$. Precise experimental determination of the branching fraction (BF) allows for the extraction of the product $f_{D^+_s}|V_{cs}|$.
  • Theoretical Calibration: Experimental values of $f_{D^+_s}$ are currently less precise than Lattice Quantum Chromodynamics (LQCD) calculations. Improved measurements are necessary to calibrate these theoretical models.
  • Lepton Flavor Universality (LFU): The ratio of branching fractions $R_{\tau/\mu} = \mathcal{B}(D^+_s \to \tau^+ \nu_\tau) / \mathcal{B}(D^+_s \to \mu^+ \nu_\mu)$ is predicted to be $9.75$ in the SM with negligible uncertainty. Deviations could signal New Physics, similar to tensions observed in $B$-meson decays.
• Previous Limitations: Previous measurements by BESIII and other experiments (CLEO, BaBar, Belle) had larger statistical uncertainties or relied on smaller datasets. This work aims to supersede previous BESIII results by utilizing a significantly larger dataset and improved detector capabilities.

2. Methodology

The analysis utilizes $e^+e^-$ collision data collected by the BESIII detector at the BEPCII collider.

• Dataset:
  • Integrated luminosity: 7.33 fb$^{-1}$.
  • Center-of-mass energies ($E_{cm}$): 8 points ranging from 4.128 to 4.226 GeV. These energies correspond to the production threshold of $D_s^* D_s$ pairs.
• Production Mechanism:
  • $D_s$ mesons are produced primarily via $e^+e^- \to D_s^{*\pm} D_s^\mp \to \gamma(\pi^0) D_s^\pm D_s^\mp$.
• Tagging Strategy (Single-Tag and Double-Tag):
  • Single-Tag (ST): The experiment fully reconstructs one $D_s^-$ meson (the "tag") in 16 different hadronic decay modes (e.g., $K^+K^-\pi^-$, $K_S^0 K^-$, $\eta \pi^-$, etc.).
  • Double-Tag (DT): The experiment searches for the signal decay $D_s^+ \to \mu^+ \nu_\mu$ in the recoil system of the ST $D_s^-$.
  • Transition Reconstruction: The transition photon or pion ($\gamma/\pi^0$) from the $D_s^* \to D_s \gamma(\pi^0)$ decay is reconstructed to link the tag and signal sides.
• Signal Selection:
  • Muon Identification: The muon candidate must have opposite charge to the tag, low energy deposition in the Electromagnetic Calorimeter (EMC), and satisfy specific hit-depth requirements in the muon identifier modules (interleaved with resistive plate counters). These requirements are optimized based on momentum ($p_\mu$) and polar angle ($\cos \theta$).
  • Kinematic Variables:
    • $\Delta E$: Used to select the transition $\gamma(\pi^0)$.
    • $M_{miss}^2$ (Missing Mass Squared): Defined as $E_\nu^2/c^4 - |\vec{p}_\nu|^2/c^2$. For the signal $D_s^+ \to \mu^+ \nu_\mu$, $M_{miss}^2$ should peak at zero (representing the neutrino).
  • Background Suppression: Cuts on extra photons ($E_{extra}^\gamma < 0.3$ GeV) and extra charged tracks are applied.
• Analysis Technique:
  • A simultaneous fit to the $M_{miss}^2$ distribution across all 8 energy points is performed.
  • The signal shape is modeled using Monte Carlo (MC) simulation, convolved with a Gaussian to account for data-MC resolution differences.
  • Backgrounds (non-$D_s$ and real-$D_s$ with mis-ID) are modeled using MC shapes.
  • The branching fraction is calculated using the formula:
    $$\mathcal{B}(D_s^+ \to \mu^+ \nu_\mu) = \frac{N_{DT}}{N_{ST} \cdot \epsilon_{\gamma(\pi^0)\mu^+\nu_\mu}}$$
    where $N_{DT}$ is the signal yield, $N_{ST}$ is the total single-tag yield, and $\epsilon$ is the effective efficiency.

3. Key Contributions

• Dataset Scale: Utilization of the largest $D_s$ dataset to date (7.33 fb$^{-1}$), covering a wide energy range near the $D_s^* D_s$ threshold.
• Muon Identification Upgrade: The analysis explicitly leverages the upgraded muon identifier modules (introduced in 2015), which cover approximately 83% of the data used. This allows for more efficient and pure muon selection compared to previous BESIII analyses.
• Comprehensive Systematics: A detailed evaluation of systematic uncertainties, including:
  • ST yield modeling (0.44%).
  • Muon tracking and PID (0.43% combined).
  • Transition $\gamma/\pi^0$ reconstruction (1.00%).
  • $M_{miss}^2$ fit modeling (0.72%).
  • Total systematic uncertainty: 1.61%.

4. Results

• Branching Fraction:
  The measured branching fraction is:
  $$\mathcal{B}(D_s^+ \to \mu^+ \nu_\mu) = (0.5294 \pm 0.0108_{\text{stat}} \pm 0.0085_{\text{syst}})\%$$
  This corresponds to a signal yield of $2514.5 \pm 51.6$ events.
• Decay Constant and CKM Element:
  Using the measured BF and SM inputs ($G_F$, masses, lifetime), the product of the decay constant and CKM element is determined:
  $$f_{D^+_s} |V_{cs}| = 241.8 \pm 2.5_{\text{stat}} \pm 2.2_{\text{syst}} \text{ MeV}$$
• Derived Parameters:
  • Assuming the global SM fit value for $|V_{cs}| = 0.97349 \pm 0.00016$:
    $$f_{D^+_s} = 248.4 \pm 2.5_{\text{stat}} \pm 2.2_{\text{syst}} \text{ MeV}$$
  • Assuming the recent LQCD value for $f_{D^+_s} = 249.9 \pm 0.5$ MeV:
    $$|V_{cs}| = 0.968 \pm 0.010_{\text{stat}} \pm 0.009_{\text{syst}}$$
• Lepton Flavor Universality Test:
  Combining this result with the previous BESIII measurement of $\mathcal{B}(D_s^+ \to \tau^+ \nu_\tau) = (5.32 \pm 0.07 \pm 0.07)\%$, the ratio is:
  $$\frac{\mathcal{B}(D_s^+ \to \tau^+ \nu_\tau)}{\mathcal{B}(D_s^+ \to \mu^+ \nu_\mu)} = 10.05 \pm 0.35$$
  This is consistent with the SM prediction of $9.75$ within uncertainties, indicating no evidence of $\tau$-$\mu$ LFU violation in the $D_s$ sector.

5. Significance

• Precision Improvement: The statistical uncertainty is reduced by roughly a factor of 2 compared to the previous BESIII measurement using muon identifiers, and the total uncertainty is significantly improved.
• Theoretical Validation: The result provides a stringent test for LQCD calculations of $f_{D^+_s}$. The measured value of $f_{D^+_s} \approx 248.4$ MeV is in good agreement with recent LQCD predictions, validating the theoretical framework.
• CKM Unitarity: The determination of $|V_{cs}|$ contributes to the global fit of the CKM matrix, helping to test the unitarity of the first two rows.
• New Physics Search: By confirming the SM prediction for the LFU ratio in $D_s$ decays, the paper constrains potential New Physics models that might explain anomalies seen in $B$-meson decays, suggesting that such violations (if they exist) may not be universal across all heavy flavor sectors.