First energy scan measurement of $e^{+}e^{-}\to K^{+}K^{-}$ around the $\psi(2S)$ resonance

Using the energy scan method with BESIII data, this paper presents the first measurement of the $e^{+}e^{-}\to K^{+}K^{-}$ cross section around the $\psi(2S)$ resonance, revealing two distinct solutions for the branching fraction and relative phase while establishing the critical role of interference effects and reporting the first results for the $\psi(2S)$ strong form factor and the charged kaon's electromagnetic form factor in this energy region.

The Gist

Imagine you are a detective trying to figure out how two very different types of musicians are playing together in a band. One musician plays a loud, electric guitar (representing the Electromagnetic force), and the other plays a deep, resonant cello (representing the Strong nuclear force).

In the world of subatomic particles, specifically when an electron and a positron smash together to create a pair of charged particles called Kaons ($K^+K^-$), these two "musicians" are both trying to play the same song. Sometimes they play in perfect harmony (constructive interference), making the music louder. Other times, they play out of sync (destructive interference), canceling each other out and making the music quieter.

The big mystery in physics has been: What is the exact timing difference (phase) between these two musicians?

This paper, written by the massive BESIII Collaboration (a team of hundreds of scientists from around the world), solves this mystery for a specific "note" in the universe called the $\psi(2S)$ resonance. Think of the $\psi(2S)$ as a very heavy, short-lived particle that acts like a temporary bridge between the electron-positron collision and the final Kaon pair.

Here is the breakdown of their discovery in simple terms:

1. The Experiment: Tuning the Radio

The scientists didn't just listen to one radio station; they scanned a whole range of frequencies. They used a giant particle accelerator (BEPCII) to smash electrons and positrons together at nine slightly different energy levels around the mass of the $\psi(2S)$ particle.

It's like tuning a radio dial slowly across a specific station. As they tuned, they counted how many Kaon pairs were produced at each frequency. This created a "line shape"—a graph showing how the music volume (the number of particles) changed as they tuned the dial.

2. The Big Reveal: Two Possible Stories

When they analyzed the shape of that graph, they found something fascinating. The data didn't point to just one answer. Instead, it fit two distinct stories equally well:

• Story A (The "Loud" Version): The two musicians are playing in a way that creates constructive interference. The electric guitar and the cello boost each other. In this scenario, the branching fraction (the probability of this specific decay happening) is about 7.5 out of 100,000.
• Story B (The "Quiet" Version): The musicians are playing in a way that creates destructive interference. They are slightly out of step, canceling each other out. To get the same total volume of sound (the data we see), the musicians must actually be playing much harder individually. In this scenario, the probability is higher, about 11 out of 100,000.

The Twist: Both stories fit the data perfectly! The scientists cannot yet say which one is the "true" reality without more data, but they have proven that interference matters. You cannot just add the two forces together; you have to account for how they dance with each other.

3. Why This Matters

For decades, physicists have been trying to understand the "Phase" (the timing) between these forces.

• The "Universal Phase" Question: Some theories suggest that this timing difference should be the same for all heavy particles (like a universal rule of the universe). This paper provides the most precise measurement yet to test that rule.
• The "12% Rule": There is a famous rule in physics that says the $\psi(2S)$ particle should be much weaker than its cousin, the $J/\psi$. This experiment confirms that the interference effects are indeed visible and significant, helping us understand why some particles decay the way they do.

4. The "Form Factors": Measuring the Shape

The paper also measured something called Form Factors. Imagine the Kaon isn't a point, but a fuzzy cloud.

• The Electromagnetic Form Factor tells us how the electric charge is distributed in that cloud.
• The Strong Form Factor tells us how the strong force binds the cloud together.

The scientists measured these for the first time in this specific energy region. It's like taking a high-resolution photo of the "fuzziness" of the Kaon particle at a specific moment in time. Their results match previous measurements but are more precise, helping to refine our map of the subatomic world.

Summary

In short, this paper is a precision measurement of a cosmic dance.

• The Players: Electrons, Positrons, and Kaons.
• The Stage: The $\psi(2S)$ resonance.
• The Dance: A complex tango between the Electromagnetic and Strong forces.
• The Result: We now know the dance has two possible rhythms (phases), and we have measured the steps with unprecedented precision.

This helps physicists solve long-standing puzzles about why particles decay the way they do and brings us closer to understanding the fundamental rules that govern the universe's smallest building blocks.

Technical Summary

1. Problem Statement and Motivation

The paper addresses a long-standing ambiguity in charmonium physics regarding the relative phase ($\Phi$) between the strong (three-gluon mediated, $A_g$) and electromagnetic (virtual-photon mediated, $A_\gamma$) amplitudes in the decay of vector charmonium states ($J^{PC}=1^{--}$) into pseudoscalar meson pairs ($P\bar{P}$).

• The Puzzle: While the phase $\Phi$ in $J/\psi$ decays is known to be approximately $\pm 90^\circ$, measurements for the $\psi(2S)$ have yielded conflicting results and large uncertainties in previous studies (BES, CLEO). These discrepancies hinder the understanding of the "universal" nature of this phase and the "12% rule" (which suggests $\psi(2S)$ amplitudes are suppressed relative to $J/\psi$).
• The Interference Effect: The decay $\psi(2S) \to K^+K^-$ proceeds via both strong and electromagnetic mechanisms. The interference between these amplitudes significantly affects the measured branching fraction. Previous measurements often neglected or poorly constrained this interference, leading to systematic errors in extracted branching fractions.
• Goal: To perform a direct, model-independent measurement of the cross-section line-shape of $e^+e^- \to K^+K^-$ around the $\psi(2S)$ resonance to extract $\Phi$, the branching fraction ($B$), and the relevant form factors.

2. Methodology

Data Collection

• Experiment: BESIII detector at the BEPCII collider.
• Method: Energy scan technique.
• Dataset: Data collected at nine center-of-mass energies ranging from 3.58 GeV to 3.71 GeV, covering the $\psi(2S)$ resonance mass ($\approx 3.686$ GeV).
• Integrated Luminosity: Total of 495 pb$^{-1}$. Luminosity was measured using di-photon events to avoid resonance-continuum interference.

Event Selection and Analysis

• Signal: Events with exactly two charged tracks identified as kaons ($K^+K^-$).
• Selection Criteria:
  • Tracks within polar angle $|\cos \theta| < 0.93$.
  • Particle Identification (PID) using $dE/dx$ (MDC) and Time-of-Flight (TOF) with likelihood $L(K) > L(\pi)$.
  • Rejection of Bhabha events ($E/p < 0.8$) and cosmic rays ($\Delta T < 3$ ns).
  • Momentum difference between tracks $< 0.07$ GeV/c.
• Background Suppression: The dominant background is $e^+e^- \to \mu^+\mu^-$. Other backgrounds ($\pi^+\pi^-$, $p\bar{p}$) were found negligible via Monte Carlo (MC) simulation.
• Yield Extraction: An unbinned maximum likelihood fit was performed on the $K^+K^-$ invariant mass ($M_{K^+K^-}$) distribution. Signal and background shapes were modeled using MC templates convolved with Gaussian functions to account for resolution differences.

Theoretical Framework and Fitting

The observed cross-section ($\sigma_{obs}$) was modeled using a coherent sum of amplitudes:

1. Continuum ($A_{cont}$): Purely electromagnetic $e^+e^- \to \gamma^* \to K^+K^-$.
2. Resonant Electromagnetic ($A_\gamma$): $e^+e^- \to \gamma^* \to \psi(2S) \to \gamma^* \to K^+K^-$.
3. Resonant Strong ($A_g$): $e^+e^- \to \gamma^* \to \psi(2S) \to 3g \to K^+K^-$.

The total amplitude is parameterized as:
$$A = A_{cont} + A_\gamma + A_g$$
The fit parameters included:

• $\Phi$: The relative phase between $A_g$ and $A_\gamma$.
• $C$: The ratio of magnitudes $|A_g|/|A_\gamma|$.
• $F$: A parameter describing the energy-dependent electromagnetic form factor of the charged kaon ($|F_K(s)| \propto 1/s$).
• Systematics: Initial State Radiation (ISR) and beam energy spread were included via convolution. Systematic uncertainties (tracking, PID, luminosity, MC modeling) were incorporated into the $\chi^2$ minimization.

3. Key Contributions

1. First Direct Line-Shape Measurement: This is the first measurement of the $e^+e^- \to K^+K^-$ cross-section using an energy scan specifically around the $\psi(2S)$, allowing for a direct extraction of interference parameters without relying solely on SU(3) flavor symmetry assumptions.
2. Resolution of Two Distinct Solutions: The analysis revealed two physically distinct, non-overlapping solutions for the phase and branching fraction, corresponding to constructive and destructive interference scenarios.
3. Form Factor Extraction: The paper reports the first measurement of the $\psi(2S)$ strong form factor ($f_K$) and provides updated results for the charged kaon electromagnetic form factor ($F_K$) in this energy region.
4. Correlation Analysis: It establishes a strong correlation between the branching fraction ($B$) and the phase ($\Phi$), demonstrating that ignoring interference leads to significant errors in determining $B$.

4. Results

The fit yielded two equally likely solutions:

Parameter	Constructive Interference Solution	Destructive Interference Solution
Phase ($\Phi$)	$110.1 \pm 6.7^\circ$	$-106.8 \pm 5.7^\circ$
Branching Fraction ($B$)	$(7.49 \pm 0.41) \times 10^{-5}$	$(10.94 \pm 0.48) \times 10^{-5}$
Ratio ($C = |A_g|/|A_\gamma|$)	$3.18 \pm 0.16$	$3.77 \pm 0.15$

• Form Factors:

  • EM Form Factor: At the $\psi(2S)$ mass, $|F_K(M^2)| = 0.0687 \pm 0.0013$. The energy dependence follows a power law $|F_K| \propto \alpha_s(\sqrt{s})/s^n$ with $n = 0.965 \pm 0.024$, consistent with perturbative QCD predictions but differing from BaBar results.
  • Strong Form Factor: The modulus of the strong form factor was measured for the first time:
    • $|f_K|_+ = (3.53 \pm 0.15) \times 10^{-3}$ (Constructive)
    • $|f_K|_- = (4.18 \pm 0.13) \times 10^{-3}$ (Destructive)

• Comparison with Previous Work: The results improve the precision of $\Phi$ and $C$ significantly compared to previous BES and CLEO analyses. The measured branching fractions differ from older values that did not account for the specific interference phase constraints.

5. Significance

• Charmonium Decay Dynamics: The results provide crucial input for understanding the "12% rule" and the nature of the $\rho\pi$ puzzle. The phase values ($\sim \pm 110^\circ$) are close to the $\pm 90^\circ$ observed in $J/\psi$, supporting the hypothesis of a universal phase structure in charmonium decays, though the large uncertainty on the sign remains.
• Interference Necessity: The study conclusively demonstrates that interference effects between strong and electromagnetic amplitudes cannot be ignored in precision measurements of $\psi(2S)$ branching fractions. The branching fraction varies drastically depending on the phase assumption.
• Form Factor Physics: The measurement of the strong form factor $f_K$ offers a new observable for testing QCD models of hadronization. The updated EM form factor data helps refine the understanding of the charged kaon's internal structure at intermediate energy scales.
• Future Impact: These results serve as a benchmark for future high-precision experiments and theoretical calculations aiming to resolve the remaining ambiguity between the two interference solutions (constructive vs. destructive).