Role of electromagnetic corrections in the $\pi\pi$ distributions of $\psi^\prime \to J/\psi \pi \pi$

Using nonrelativistic effective field theory, this paper demonstrates that including electromagnetic Coulomb interactions in the analysis of $\psi^\prime \to J/\psi \pi \pi$ decays enhances the prominence of the cusp structure near the $\pi^+\pi^-$ threshold and alters its magnitude by 2–3%, highlighting the necessity of these corrections for precision determinations of $\pi\pi$ scattering lengths.

The Gist

Imagine you are a detective trying to solve a mystery about the fundamental rules of the universe. Specifically, you want to know how two tiny particles called pions (which are like the "atoms" of the strong nuclear force) bounce off each other.

To do this, scientists look at a specific event: a heavy particle called a $\psi'$ (Psi-prime) decaying into a lighter particle called a $J/\psi$ (J/Psi) and two pions. It's like a heavy bowling ball breaking apart into a smaller bowling ball and two ping-pong balls.

Here is the simple breakdown of what this paper is about, using some everyday analogies:

1. The "Cusp" on the Graph

When scientists measure the energy of the two pions produced in this crash, they plot it on a graph. Usually, these graphs are smooth curves. However, right at the specific energy where two charged pions ($\pi^+\pi^-$) could just barely exist, the graph for two neutral pions ($\pi^0\pi^0$) suddenly gets a sharp little spike or a "kink."

• The Analogy: Imagine you are driving a car on a smooth road. Suddenly, right at a specific mile marker, the road dips down into a tiny, sharp pothole before going back up. That pothole is the "cusp."
• Why it matters: The shape and depth of this pothole tell us exactly how sticky or bouncy the pions are when they hit each other. This "stickiness" is called the scattering length.

2. The Missing Ingredient: Electricity

For a long time, scientists calculated the shape of this road (the graph) assuming the pions only interacted via the "strong force" (the glue that holds atoms together). They ignored electricity.

But pions have electric charges. The positive and negative pions attract each other like magnets, while neutral pions don't care about electricity. The authors of this paper asked: "What happens if we finally turn on the electricity in our calculations?"

• The Analogy: Imagine you are trying to predict how two people will dance. You calculated their moves based only on their rhythm (the strong force). But you forgot that one of them is holding a heavy magnet (electric charge) that pulls them closer. If you ignore the magnet, your prediction of the dance is slightly off.

3. The Discovery: The "Pothole" Gets Deeper

The paper found that when you include the electromagnetic (electric) forces:

1. The "cusp" (the pothole) becomes more prominent and sharper.
2. The size of this effect changes the measurement by about 2% to 3%.

• The Analogy: It's like realizing that the pothole isn't just a small dip; it's actually a deep crater. If you are trying to measure the depth of the crater to understand the geology of the area, ignoring the extra depth means you get the wrong answer.

4. Why This Matters for Future Experiments

The paper uses computer simulations (Monte Carlo) to see how this affects real-world experiments.

• The Scenario: Imagine the BESIII experiment (a giant particle detector in China) and a future super-powerful machine called STCF. These machines will collect billions of these "bowling ball" crashes.
• The Result: If the machines collect enough data (billions of events) to measure the pions with extreme precision, ignoring the electric force will lead to a wrong answer.
• The Metaphor: If you are measuring a table's length with a ruler that has millimeter markings, a 2% error is huge. But if you are just using a rough tape measure, a 2% error doesn't matter much. The paper says: "Once we get our high-precision rulers (future experiments), we must include the electric force, or our measurements will be wrong."

Summary in One Sentence

This paper tells us that to perfectly understand how pions bounce off each other using data from heavy particle crashes, we can no longer ignore the tiny electric pull between them, because it changes the "shape" of the data just enough to matter when we have very precise measurements.

The Takeaway: In the world of particle physics, even the smallest forces (like electricity) can make a big difference when you are looking at the tiniest details with the sharpest eyes.

Technical Summary

1. Problem Statement

The extraction of $\pi\pi$ S-wave scattering lengths ($a_0$ and $a_2$) is crucial for understanding low-energy QCD dynamics and chiral symmetry breaking. While these parameters are well-studied in processes like $K \to 3\pi$ and pionium decay, heavy quarkonium dipion transitions (e.g., $\psi' \to J/\psi\pi\pi$) offer a promising alternative due to large data samples available at BESIII and future facilities like the Super Tau-Charm Facility (STCF).

A specific feature in the $\pi^0\pi^0$ invariant mass spectrum near the $\pi^+\pi^-$ threshold is the cusp effect, caused by the charge-exchange rescattering $\pi^+\pi^- \to \pi^0\pi^0$. This cusp is a sensitive probe for scattering lengths. However, previous theoretical frameworks for heavy quarkonium transitions (specifically within Non-Relativistic Effective Field Theory, NREFT) often neglected electromagnetic (Coulomb) interactions between the charged pions. Given that Coulomb interactions significantly alter the threshold behavior in kaon decays, the authors investigate whether these effects are negligible or critical for precision extractions in charmonium transitions.

2. Methodology

The authors employ a coupled-channel Non-Relativistic Effective Field Theory (NREFT) framework involving the $\pi^0\pi^0$ (Channel 1) and $\pi^+\pi^-$ (Channel 2) states.

• Theoretical Framework:

  • The scattering amplitude is constructed using a two-potential formalism separating strong and Coulomb interactions.
  • The $T$-matrix includes the resummation of infinite Coulomb photon exchanges between $\pi^+\pi^-$ pairs (represented by the Coulomb $T$-matrix $T_C$ and the Gamow factor $W_C$).
  • The strong interaction is modeled via contact terms satisfying the Lippmann-Schwinger equation (LSE), renormalized to be cutoff-independent.
  • The scattering length matrix is parameterized in terms of isospin $I=0$ and $I=2$ scattering lengths ($a_0, a_2$), including isospin-breaking effects.

• Decay Amplitude Construction:

  • The decay $\psi' \to J/\psi\pi\pi$ is treated as a short-distance vertex followed by final-state interactions (FSI).
  • The amplitudes for $\psi' \to J/\psi\pi^0\pi^0$ and $\psi' \to J/\psi\pi^+\pi^-$ are derived, explicitly incorporating the Coulomb resummation factor $W_C$ for the charged channel and the mixing between neutral and charged channels via strong interactions.
  • Parameters for the short-distance vertex are fixed by matching to Chiral Perturbation Theory (ChPT) and fitting existing BESII and ATLAS data.

• Monte Carlo (MC) Simulation:

  • To quantify the impact of neglecting Coulomb corrections, the authors generate synthetic data sets conforming to the NREFT prediction with Coulomb interactions.
  • These data sets are fitted using two models: one with Coulomb interactions and one without.
  • The simulation varies the number of events (from $1.5 \times 10^5$ to $4 \times 10^7$) and bin widths (0.1 to 2 MeV) to mimic current BESIII data and future STCF precision.

3. Key Contributions

• Inclusion of Coulomb Resummation: The work provides the first rigorous application of coupled-channel NREFT with full Coulomb resummation to the $\psi' \to J/\psi\pi\pi$ process.
• Quantification of Electromagnetic Effects: It demonstrates that Coulomb interactions are not negligible in the near-threshold region of charmonium decays, altering the cusp structure.
• Precision Benchmarking: Through MC simulations, the paper establishes the statistical precision required for experiments to necessitate the inclusion of electromagnetic corrections. It provides a "ready-to-use" theoretical framework for experimental analyses.

4. Results

• Cusp Prominence: Including Coulomb interactions makes the cusp structure at the $\pi^+\pi^-$ threshold in the $\pi^0\pi^0$ invariant mass spectrum more prominent.
• Magnitude of Correction: The electromagnetic corrections alter the magnitude of the threshold cusp by approximately 2%–3%.
• Impact on Scattering Length Extraction:
  • When fitting synthetic data generated with Coulomb effects using a model without Coulomb effects, the extracted value of $(a_0 - a_2)M_{\pi^+}$ deviates from the input value by 1%–5%.
  • For datasets with $\sim 1.5 \times 10^5$ events (current BESIII scale), the statistical uncertainty is large enough ($\sim 4-6\%$) that neglecting Coulomb corrections is acceptable.
  • For datasets with $\sim 4 \times 10^7$ events (expected at STCF), the statistical precision improves to $\sim 0.3-0.6\%$. At this level, the central value extracted without Coulomb corrections falls outside the error region of the true value, leading to a significant overestimation of the scattering lengths if electromagnetic effects are ignored.
• Pionium Structure: The inclusion of Coulomb interactions reveals peaks in the spectrum corresponding to the ground and excited states of pionium ($\pi^+\pi^-$ atom) near the threshold, which are absent in the non-Coulomb model.

5. Significance

• Future Experimental Planning: The study is critical for the analysis of upcoming high-statistics data from the Super Tau-Charm Facility (STCF). It establishes that for precision measurements aiming at sub-percent accuracy, electromagnetic corrections are mandatory.
• Theoretical Framework: The derived coupled-channel amplitude serves as a robust tool for experimentalists to analyze fine structures near $\pi\pi$ thresholds, moving beyond simple isospin-symmetric approximations.
• Physics Insight: The results reinforce the necessity of treating electromagnetic interactions on equal footing with strong interactions in precision low-energy QCD studies involving charged hadrons, even in processes dominated by heavy quarkonium transitions.

In conclusion, while Coulomb corrections are a small effect (2-3%) on the line shape, they are essential for the precision extraction of $\pi\pi$ scattering lengths in future high-statistics experiments, preventing systematic biases in the determination of fundamental QCD parameters.