A subthreshold pole in electron-positron annihilation into the $D_s^+D_s^-$ final state

Using precise data from the BESIII collaboration, this paper presents compelling evidence for a subthreshold pole in the $e^+ e^- \rightarrow D_s^+ D_s^-$ process with a mass of approximately 3896 MeV, suggesting its identification as the $G(3900)$ state.

The Gist

Imagine you are listening to a radio station that plays music only when a specific condition is met: the volume knob must be turned past a certain point (the "threshold") for the music to start playing. In the world of particle physics, this "volume knob" is the energy of a collision between an electron and a positron (a particle of antimatter).

When these two particles smash together, they can create new particles, specifically a pair of "strange" heavy mesons called $D_s^+$ and $D_s^-$. Scientists have long known that if you turn the energy knob high enough, these particles appear. But what happens if you turn the knob just below the point where they should appear?

This paper is about a mysterious "ghost note" that the scientists found playing just below that threshold.

The Setup: The Particle Collider as a Giant Piano

Think of the BESIII collider in Beijing as a giant, ultra-precise piano. When the scientists hit the keys (collide electrons and positrons), they expect to hear specific notes (resonances) that correspond to known particles.

For a long time, the "sheet music" (the theoretical models) told them exactly which notes should ring out. However, when they played the lowest notes—just below the energy required to create the $D_s$ pair—the music didn't quite match the sheet music. There was a strange, subtle distortion in the sound that the old models couldn't explain.

The Discovery: The "Subthreshold Pole"

The authors, Peter Lichard and Josef Jurán, decided to look closer. They used a mathematical tool called the Vector Meson Dominance model. Think of this model as a sophisticated audio equalizer that tries to match the sound of the piano to the actual recording.

When they tried to fit the data using only the known "notes" (resonances), the music still sounded off. It was like trying to tune a guitar, but no matter how you turned the tuning pegs, one string was still slightly out of tune.

Then, they added a new element to their model: a Subthreshold Pole (SP).

• The Analogy: Imagine you are trying to push a heavy boulder up a hill. The "threshold" is the top of the hill. You expect the boulder to only roll over the top if you push hard enough.
• The Surprise: The scientists found evidence of a "ghost boulder" that exists just below the top of the hill. It's not heavy enough to roll over the top and become a visible particle in the experiment, but its presence is so strong that it tugs on the physics of the system, changing how the other particles behave.

This "ghost boulder" is the Subthreshold Pole. It's a state of matter that is technically "stable" (it can't decay into the particles we are looking for because it doesn't have enough energy), but it still influences the collision like a hidden magnet.

The Evidence: A 7.4-Standard-Deviation Clue

In science, you need to be very sure before you claim a discovery. The authors found that including this "ghost pole" in their math improved the match between their theory and the real data from 7.4 times the usual margin of error.

To put that in perspective: If you flipped a coin 100 times and got heads every single time, that would be suspicious. Getting a result this statistically significant is like flipping a coin and getting heads 100 times in a row, then doing it again, and again. It's almost certainly not a fluke.

They calculated the "mass" (the weight/energy) of this ghost pole to be about 3896 MeV. This is just slightly less than the energy needed to create the $D_s$ pair (which is about 3938 MeV).

The Big Reveal: Who is this Ghost?

The most exciting part of the paper is connecting this ghost to a known character.

There is a famous particle called G(3900). In other types of collisions, G(3900) acts like a loud, booming resonance (a clear note). But in this specific experiment (creating $D_s$ pairs), it seems to be hiding just below the threshold.

The authors suggest that G(3900) is actually this subthreshold pole. It's the same particle, but depending on how you look at it (which "channel" of interaction you use), it either shows up as a loud resonance or as a subtle, hidden influence just below the energy limit.

Why Does This Matter?

1. It changes the rules: For a long time, scientists mostly ignored these "subthreshold" states because they couldn't be directly observed. This paper proves they are real and that they play a crucial role in how particles interact.
2. It solves a mystery: It helps explain why the data from the BESIII collider didn't fit the old models. The "missing piece" was this hidden state.
3. It hints at new physics: The fact that this particle exists so close to the threshold suggests it might be a "bound state"—like a molecule made of two smaller particles stuck together, but just barely holding on.

Summary in One Sentence

By listening very carefully to the "music" of particle collisions, these scientists discovered a hidden "ghost note" (a subthreshold pole) just below the energy limit, which turns out to be the mysterious G(3900) particle behaving in a way we haven't seen before.

Technical Summary

1. Problem Statement

The paper addresses the theoretical challenge of identifying Subthreshold Poles (SPs) in the complex energy plane of electron-positron ($e^+e^-$) annihilation into hadronic final states.

• Context: While resonances (unstable particles) appear as poles on higher Riemann sheets above the physical threshold, subthreshold poles exist on the first (physical) Riemann sheet, on the real axis below the reaction threshold.
• The Issue: These SPs correspond to states stable against decay into the specific final state but are often overlooked in standard phenomenological fits because they are not directly reachable in experiments. However, due to analyticity, causality, and unitarity, they significantly influence the scattering amplitude in the physical region.
• Specific Goal: The authors aim to uncover the role of such a pole in the $e^+e^- \to D^+_s D^-_s$ process using high-precision data, specifically investigating whether the observed excitation function requires a subthreshold pole rather than just a standard resonance.

2. Methodology

The authors employed a rigorous data-fitting approach combining the Vector Meson Dominance (VMD) model with advanced statistical analysis.

• Data Source: They utilized precise Born cross-section data from the BESIII collaboration (collected at the BEPCII collider in Beijing), covering 138 energy points from the $D^+_s D^-_s$ threshold (3.938 GeV) up to 4.951 GeV.
• Theoretical Model:
  • The cross-section $\sigma(s)$ is modeled using a VMD-based formula (Eq. 1) representing the sum of intermediate vector mesons coupled to the photon.
  • The formula includes parameters for mass ($M_k$), width ($\Gamma_k$), coupling strength ($Q_k$), and phase ($\delta_k$) for $n$ intermediate states.
  • Key Modification: To test for subthreshold poles, the authors allowed one term in the sum to have a width $\Gamma = 0$ and a mass $M$ below the threshold ($2m_{D_s}$), effectively placing a pole on the real axis below the cut.
• Fitting Procedure:
  • They performed a stepwise $\chi^2$ minimization, starting with a single resonance and incrementally adding more resonances (up to 4).
  • Statistical and systematic errors were merged quadratically.
  • They compared models with varying numbers of resonances (3R, 4R) against models including a Subthreshold Pole (SP+3R, SP+4R).
  • Statistical significance was evaluated using changes in the likelihood function ($\delta(-2 \ln L)$) and degrees of freedom.

3. Key Contributions

• Evidence for a Subthreshold Pole: The study provides compelling statistical evidence (significance up to 7.4$\sigma$) that the $e^+e^- \to D^+_s D^-_s$ cross-section cannot be adequately described by resonances alone. The inclusion of a pole below the threshold is necessary to reproduce the data, particularly near the threshold energy.
• Refinement of Resonance Parameters: By accounting for the SP, the authors derived more accurate parameters for the known vector charmonium-like states (specifically $\psi(4040)$, $\psi(4160)$, etc.) compared to previous fits that ignored subthreshold effects.
• Identification of G(3900): The paper proposes a physical interpretation for the discovered pole, linking it to the $G(3900)$ state.

4. Results

• Discovery of the Pole:
  • The optimal fit (SP+4R model) yields a subthreshold pole with a mass of $M = (3896^{+13}_{-22})$ MeV.
  • The width is consistent with zero ($\Gamma = 0$), confirming its nature as a bound state or virtual state below the $D^+_s D^-_s$ threshold (binding energy $B \approx 42$ MeV).
  • The statistical significance of adding this pole to a 4-resonance model is 5.5$\sigma$.
• Fit Quality Improvement:
  • A standard 3-resonance fit yielded a poor $p$-value ($5 \times 10^{-8}$).
  • Adding a 4th resonance improved the fit slightly ($p=0.6\%$).
  • Replacing the 4th resonance with a Subthreshold Pole (SP+3R) significantly improved the fit ($p=0.7\%$).
  • The best fit (SP+4R, 4 resonances + 1 SP) achieved a $p$-value of 37% with $\chi^2/\text{DOF} = 124.5/120$, indicating a statistically robust description of the data.
• Comparison with PDG:
  • The mass of the $\psi(4040)$ derived in this study ($4026.4 \pm 1.6$ MeV) is consistent with recent BESIII analyses but differs from older PDG compilations based on data >18 years old.
  • The mass of the SP ($3896$ MeV) is remarkably close to the mass of the $G(3900)$ resonance ($3872.5 \pm 14.2$ MeV and $3898.4 \pm 0.9$ MeV from global coupled-channel analyses).

5. Significance

• Nature of the $G(3900)$: The paper suggests that the $G(3900)$, which appears as a resonance in $e^+e^- \to D\bar{D}$ processes, manifests as a subthreshold pole in the $e^+e^- \to D^+_s D^-_s$ channel. This dual behavior implies $G(3900)$ is a genuine hadronic state with strong coupling to the $D^+_s D^-_s$ system.
• Theoretical Implications: The existence of this pole challenges the interpretation of the state as a simple $D^+_s D^-_s$ molecule (as the quantum numbers $I^G(J^{PC}) = 0^-(1^{--})$ exclude a ground-state molecule, suggesting an excited state with $L=1$).
• Methodological Impact: The study demonstrates that ignoring subthreshold poles can lead to systematic errors in extracting resonance parameters. It highlights the necessity of including analytic constraints (poles below threshold) when fitting high-precision cross-section data near thresholds.

In conclusion, Lichard and Jurán have successfully identified a subthreshold pole in the $D^+_s D^-_s$ channel, providing a unified explanation for the $G(3900)$ state and significantly improving the description of BESIII experimental data.