Study of the $f_{0}(980)$ and $f_{0}(500)$ Scalar Mesons through the Decay $D_{s}^{+} \to π^{+} π^{-} e^{+} ν_{e}$

Using $e^+e^-$ collision data from the BESIII detector, this study measures the branching fraction and form factor parameters for the decay $D_{s}^{+} \to f_{0}(980)e^{+}\nu_{e}$, while setting the first upper limit on the previously unobserved $D_{s}^{+} \to f_{0}(500)e^{+}\nu_{e}$ decay.

The Gist

Imagine the universe is a giant, bustling kitchen where particles are the ingredients. For decades, physicists have been trying to figure out the recipes for the most mysterious ingredients: light scalar mesons. These are tiny, short-lived particles that act like the "glue" holding the universe's structure together, but they are notoriously difficult to understand.

Two of these mysterious ingredients are named $f_0(980)$ and $f_0(500)$. Think of them as two very strange, shape-shifting ghosts. Scientists have been arguing for years about what they are actually made of. Are they simple pairs of a quark and an anti-quark (like a classic sandwich)? Or are they complex "tetraquarks" (four quarks stuck together like a club sandwich)? Or maybe they are molecules made of two other particles stuck together?

To solve this mystery, the BESIII Collaboration (a team of scientists using a giant particle detector in China) decided to cook up a specific experiment. They used a massive accelerator to smash electrons and positrons together, creating a shower of new particles. Specifically, they looked for a rare event where a heavy particle called a $D_s^+$ meson decays (breaks apart) into a specific set of ingredients: two pions ($\pi^+\pi^-$), an electron, and a neutrino.

Here is the breakdown of their findings using simple analogies:

1. The "Tagging" Strategy: Finding a Needle in a Haystack

Imagine you are at a crowded party (the particle collision). You are looking for a specific person (the $D_s^+$ meson) who is wearing a red hat. But the party is chaotic, and everyone looks similar.

The scientists used a clever trick called "Tagging."

• They knew that these particles are always created in pairs. If they see one particle (the "tag") breaking apart in a very specific, recognizable way, they know the other particle (the "signal") must be right there, too.
• By identifying the "tag" partner, they could isolate the "signal" particle and study its decay without the noise of the rest of the party. This is like seeing someone's twin walk into the room; you immediately know the twin is nearby.

2. The Main Discovery: Catching the $f_0(980)$ Ghost

The team successfully caught the $f_0(980)$ ghost in the act.

• What they saw: The $D_s^+$ meson decayed into an electron, a neutrino, and a pair of pions. When they looked at the energy of those two pions, they saw a distinct "bump" or resonance at a specific mass (980 MeV). This confirmed the $f_0(980)$ was there.
• The Recipe: They measured exactly how often this happens (the "branching fraction"). It's a rare event, happening about 1.7 times out of every 1,000 decays.
• The Big Clue: By analyzing the math of this decay, they found that the $f_0(980)$ is mostly made of strange quarks ($s\bar{s}$). It's like finding out that a "mystery smoothie" is actually 90% strawberry, even though it looked like a fruit salad. This helps settle the debate: it's likely a standard quark-antiquark pair, not a complex tetraquark.

3. The Missing Ghost: The Search for $f_0(500)$

Next, they tried to find the $f_0(500)$, which is even more elusive and controversial.

• The Hunt: They looked for the same decay pattern but focused on a lower energy range where the $f_0(500)$ should appear.
• The Result: They found nothing. No ghost, no signal.
• The Conclusion: They set a strict "limit" on how often this could happen. It's like saying, "If this ghost exists, it must be so shy that it only shows up less than once in 3,000 tries." This result actually favors the idea that these particles might be tetraquarks (four-quark states), because some theories predicted the $f_0(500)$ would be much easier to see if it were a simple quark pair.

4. The "Speedometer" of the Strong Force

Finally, the scientists measured something called a form factor.

• The Analogy: Imagine the strong force (which holds quarks together) as a spring. When the $D_s^+$ breaks apart, it stretches this spring. The "form factor" measures how stiff that spring is.
• The Result: They measured the stiffness of this spring for the first time. This is a crucial piece of data for theorists who try to write the "laws of physics" for how quarks behave. Their measurement matches some theories but disagrees with others, helping to refine our understanding of the universe's fundamental rules.

Why Does This Matter?

Think of the Standard Model of physics as a giant instruction manual for the universe. We know the ingredients (quarks), but we don't fully understand the recipes for how they combine to make the "light scalar mesons."

• If they are simple pairs: It means our basic understanding of quarks is solid.
• If they are complex tetraquarks: It means there is a whole new layer of complexity in nature that we are just beginning to discover.

By measuring these decays with high precision, the BESIII team has provided a new, clearer map. They've confirmed the nature of the $f_0(980)$ and placed a tight leash on the elusive $f_0(500)$, helping physicists finally solve the mystery of these shape-shifting particles.

Technical Summary

1. Problem Statement

The nature of light scalar mesons, specifically the $f_0(500)$ (also known as $\sigma$) and $f_0(980)$, remains one of the most controversial topics in hadron physics. While the constituent quark model predicts a simple $q\bar{q}$ structure, the observed mass ordering and properties of these states cannot be fully explained by this naive picture. Alternative interpretations include:

• Tetraquark states: Diquark-antidiquark configurations ($qq\bar{q}\bar{q}$).
• Molecular states: Bound states of meson-meson pairs.
• Mixtures: Superpositions of $q\bar{q}$ states with different flavor contents (e.g., mixing between $s\bar{s}$ and non-strange $n\bar{n} = \frac{1}{\sqrt{2}}(u\bar{u} + d\bar{d})$).

Previous experimental data on these states were limited in precision or statistics. Semileptonic decays of charm mesons ($D_s$) offer a unique "clean" platform to probe these structures because the leptons interact only weakly, and the hadronic transition is governed by the strong interaction between the final-state quarks. This paper aims to:

1. Precisely measure the branching fraction (BF) of $D_s^+ \to f_0(980)e^+\nu_e$ with $f_0(980) \to \pi^+\pi^-$.
2. Determine the transition form factor and the product $f_+^{f_0}(0)|V_{cs}|$.
3. Search for the decay $D_s^+ \to f_0(500)e^+\nu_e$ with $f_0(500) \to \pi^+\pi^-$, which had not been previously observed in this channel.

2. Methodology

Data Sample and Experiment:

• Experiment: BESIII detector at the BEPCII collider.
• Data: $e^+e^-$ collision data corresponding to an integrated luminosity of 7.33 fb$^{-1}$.
• Energy: Center-of-mass energies between 4.128 and 4.226 GeV, specifically tuned to produce $D_s^{*\pm} D_s^{\mp}$ pairs.
• Process: $e^+e^- \to D_s^{*\pm} D_s^{\mp} \to \gamma D_s^{\pm} D_s^{\mp}$. The $D_s^-$ acts as a "tag" to fully reconstruct the event, allowing the inference of the neutrino in the signal $D_s^+$ decay.

Event Selection and Reconstruction:

• Tagging: $D_s^-$ candidates were reconstructed in 12 hadronic decay modes (e.g., $K^+K^-\pi^-$, $K_S^0 K^-$, etc.).
• Signal Reconstruction: The signal side ($D_s^+ \to \pi^+\pi^- e^+ \nu_e$) was identified by requiring a $\pi^+\pi^-$ pair and a positron ($e^+$) recoiling against the tag.
• Kinematic Constraints:
  • The missing mass squared ($M_{miss}^2$) was calculated to infer the neutrino. Events were required to have $|M_{miss}^2| < 0.06$ GeV$^2/c^4$.
  • The recoil mass against the tag and transition photon was constrained to the $D_s^+$ mass.
• Background Suppression:
  • Inclusive Monte Carlo (MC) samples were used to model backgrounds.
  • A sideband analysis of the tag mass distribution verified the non-peaking background.
  • Specific cuts were applied to separate $f_0(980)$ and $f_0(500)$ regions.

Analysis Techniques:

• Branching Fraction Measurement: An unbinned maximum likelihood fit was performed on the $\pi^+\pi^-$ invariant mass ($M_{\pi^+\pi^-}$) distribution. The signal shape was modeled using MC simulation convolved with a Gaussian resolution function.
• Form Factor Extraction: The differential decay rate was analyzed by dividing events into four $q^2$ (four-momentum transfer squared) intervals.
  • The dynamics were modeled using a simple pole parametrization for the form factor: $f_+^{f_0}(q^2) = \frac{f_+^{f_0}(0)}{1 - q^2/M_{pole}^2}$.
  • The $f_0(980)$ line shape was described by the Flatté formula to account for the open $K\bar{K}$ channel.
• **Search for $f_0(500):** A search was conducted in the low mass region ($M_{\pi^+\pi^-} < 0.45$ GeV/$c^2$) to avoid contamination from $f_0(980)$ and $\rho^0$.

3. Key Contributions

1. Precision Measurement of $D_s^+ \to f_0(980)e^+\nu_e$:

   • Provided a measurement with significantly improved statistical precision (data sample >10$\times$ larger than previous CLEO measurements).
   • Determined the branching fraction with high accuracy, reducing uncertainties compared to prior results.

2. First Measurement of Transition Form Factor:

   • This is the first determination of the product $f_+^{f_0}(0)|V_{cs}|$ for the $D_s^+ \to f_0(980)e^+\nu_e$ decay.
   • Extracted the form factor at zero momentum transfer, $f_+^{f_0}(0)$, providing a critical test for theoretical models of non-perturbative QCD.

3. First Search for $D_s^+ \to f_0(500)e^+\nu_e$:

   • Conducted the first dedicated search for the $f_0(500)$ in this specific semileptonic decay channel.
   • Established a stringent upper limit on its branching fraction.

4. Insight into Scalar Meson Structure:

   • Used the measured branching fraction to constrain the mixing angle $\phi$ in the $q\bar{q}$ mixture picture, suggesting the $s\bar{s}$ component dominates the $f_0(980)$.
   • Compared results with theoretical predictions (Tetraquark vs. $q\bar{q}$ models) to assess the nature of light scalars.

4. Results

Branching Fractions:

• $D_s^+ \to f_0(980)e^+\nu_e$ ($f_0(980) \to \pi^+\pi^-$):
  $$\mathcal{B} = (1.72 \pm 0.13_{\text{stat}} \pm 0.10_{\text{syst}}) \times 10^{-3}$$
  This result is 2.6 times more accurate than the previous measurement by CLEO.
• $D_s^+ \to f_0(500)e^+\nu_e$ ($f_0(500) \to \pi^+\pi^-$):
  No significant signal was observed. The 90% confidence level upper limit is set to:
  $$\mathcal{B} < 3.3 \times 10^{-4}$$

Form Factor and Mixing Angle:

• Form Factor Product:
  $$f_+^{f_0}(0)|V_{cs}| = 0.504 \pm 0.017_{\text{stat}} \pm 0.035_{\text{syst}}$$
• Derived Form Factor: Using $|V_{cs}| = 0.97349 \pm 0.00016$:
  $$f_+^{f_0}(0) = 0.518 \pm 0.018_{\text{stat}} \pm 0.036_{\text{syst}}$$
• Mixing Angle ($\phi$): Assuming a $q\bar{q}$ mixture model ($\sin\phi \frac{u\bar{u}+d\bar{d}}{\sqrt{2}} + \cos\phi s\bar{s}$), the measured BF implies:
  $$\phi = (19.7 \pm 12.8)^\circ$$
  This suggests the $s\bar{s}$ component dominates the $f_0(980)$.

Theoretical Comparison:

• The measured form factor $f_+^{f_0}(0) \approx 0.52$ agrees well with predictions from Covariant Light-Front Dynamics (CLFD) and QCD Sum Rules (QCDSR).
• It is significantly higher (by 2.8$\sigma$ to 4.3$\sigma$) than predictions from Light-Cone QCD Sum Rules (LCSR) and Light-Front Quark Models (LFQM) found in literature.
• The non-observation of $f_0(500)$ and the specific BF values favor theoretical models treating $f_0(980)$ and $f_0(500)$ as tetraquark states over simple $q\bar{q}$ mixture models, although the $s\bar{s}$ dominance in the mixing angle supports a specific $q\bar{q}$ interpretation.

5. Significance

• QCD Non-Perturbative Dynamics: The results provide crucial experimental input for understanding the strong interaction in the non-perturbative regime, specifically how light scalar mesons are formed.
• Model Discrimination: The precise measurement of the form factor and branching fraction serves as a powerful discriminator between competing theoretical models (e.g., tetraquark vs. molecular vs. $q\bar{q}$). The data challenges models that predict significantly lower form factors.
• Standard Model Tests: The determination of $f_+^{f_0}(0)|V_{cs}|$ contributes to the precision testing of the Cabibbo-Kobayashi-Maskawa (CKM) matrix element $|V_{cs}|$ in the context of heavy-to-light transitions.
• Future Directions: The lack of a signal for $f_0(500)$ in this channel sets a benchmark for future experiments and theoretical calculations regarding the existence and coupling strength of the $\sigma$ meson in semileptonic decays.

In conclusion, this study represents a major advancement in the experimental characterization of light scalar mesons, utilizing a large dataset from BESIII to provide the most precise measurements to date for $D_s^+ \to f_0(980)e^+\nu_e$ and the first constraints on the $f_0(500)$ channel.