⚛️ phenomenology

The $\psi$ (2S) state as a subthreshold pole in electron positron annihilation into $D\bar D$ final states

By analyzing merged BES and BESIII data on $e^+e^-\to D\bar D$ cross-sections, this study demonstrates with 8 $\sigma$ statistical significance that the subthreshold $\psi(2\mathrm S)$ pole markedly influences the process by altering the shapes, positions, and widths of seven resonances through interference with the continuum.

Original authors: Peter Lichard

Published 2026-09-30

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CC BY 4.0

Original authors: Peter Lichard

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the subatomic world, particles do not merely bounce off one another like billiard balls; they interact through a complex web of forces that can be described by mathematical maps called amplitudes. These maps reveal how likely a specific reaction is to occur at different energy levels. Sometimes, these maps contain hidden features known as poles. A familiar example is a resonance, which appears as a sharp peak in the data, signaling the brief creation of a new, unstable particle. However, there is another type of feature called a subthreshold pole. This is a point on the map that lies below the energy required to create a specific pair of particles. Even though the energy is too low to actually produce these particles, the existence of this hidden point still exerts a powerful influence on the reaction, shaping the outcome in ways that are visible to observers. Understanding these hidden influences is crucial for physicists trying to decode the fundamental structure of matter, as ignoring them can lead to a distorted view of how particles behave.

A researcher has recently turned their attention to a specific collision process where an electron and a positron smash together to produce a pair of particles known as D mesons. By analyzing a massive collection of experimental data, they discovered that a well-known particle, the psi-2S, is acting as one of these hidden subthreshold poles in this specific reaction. The psi-2S is a particle that usually appears as a distinct peak when enough energy is available to create it, but in this particular experiment, the energy is just slightly too low for it to be formed directly. Despite this, the researcher found that the presence of this particle, lurking just below the threshold, dramatically alters the behavior of the collision. When they included this hidden influence in their calculations, the fit to the experimental data improved so significantly that the result reached a statistical certainty of eight standard deviations, a level of confidence that in physics is considered a definitive discovery.

The study relied on merging two large sets of data collected by the BES and BESIII collaborations in Beijing. These experiments measured the rate at which electron-positron collisions produced D meson pairs across a wide range of energies. When the researcher first tried to explain this data using only the standard model of seven known resonances, the results were unsatisfactory. The mathematical model failed to capture the true shape of the data, leaving a large gap between the theory and the observation. The researcher then introduced a new element into their model: a subthreshold pole. As they adjusted the parameters, a clear signal emerged. A new feature appeared in the fit with a mass of approximately 3691 MeV and a width of zero, sitting just below the energy threshold of 3729.68 MeV required to create the D meson pairs. This feature was identified as the psi-2S state.

The discovery of this subthreshold pole changed the entire picture of the reaction. The researcher found that the presence of this hidden pole interferes with the other resonances in the system, effectively flipping their shapes. In the data, peaks that should have appeared as bumps turned into dips, and dips turned into peaks, a phenomenon caused by the complex interplay between the visible resonances and the invisible subthreshold pole. This interference was so strong that previous attempts to fit the data without accounting for the psi-2S pole were bound to fail, leading to incorrect conclusions about the properties of the other particles involved. The researcher demonstrated that without including this subthreshold pole, any model attempting to describe the data would produce false parameters for the other resonances, misidentifying their masses and widths.

To confirm their findings, the researcher performed a final, comprehensive fit that included the psi-2S subthreshold pole along with seven other resonances. This model matched the experimental data with a quality that corresponds to a 97 percent probability of being correct, a result that stands in stark contrast to the poor performance of models that ignored the subthreshold pole. The analysis revealed that the psi-2S, while stable in other contexts, behaves uniquely in this specific reaction because it sits just below the energy line where D mesons can be created. This position allows it to influence the reaction strongly without being produced directly. The study concludes that the widely used formulas for describing these collisions are insufficient because they cannot detect this type of hidden influence, and that a complete understanding of electron-positron annihilation into D mesons requires acknowledging the profound role of these subthreshold poles.

Technical Summary: The $\psi(2S)$ State as a Subthreshold Pole in $e^+e^- \to D\bar{D}$

Problem Statement
The paper addresses the challenge of accurately modeling the cross-section for electron-positron annihilation into $D\bar{D}$ final states ( $e^+e^- \to D^0\bar{D}^0$ and $e^+e^- \to D^+D^-$ ). While Vector Meson Dominance (VMD) models typically parametrize such processes using Breit-Wigner resonances accessible above the reaction threshold, the author hypothesizes that subthreshold poles (SPs)—stable states lying on the real axis of the complex $s$ -plane below the physical threshold—significantly influence the amplitude. Previous analyses, including those by the BESIII collaboration, assumed only resonances above the threshold, potentially leading to discrepancies in fit quality and resonance parameter extraction. The specific region of interest lies between the $D\bar{D}$ threshold and 3.8 GeV, a gap in the recent BESIII data that is expected to be most sensitive to subthreshold effects.

Methodology
The author performs a global fit to a merged dataset comprising 328 data points from the BES Collaboration (1990s, BEPC collider) and the BESIII Collaboration (2018–2024, BEPCII collider). The dataset includes "dressed" cross sections for both $D^0\bar{D}^0$ and $D^+D^-$ channels.

The theoretical framework utilizes a VMD model based on a specific interaction Lagrangian. The cross-section formula (Eq. 2) sums contributions from $n$ resonances and a subthreshold pole. Key methodological choices include:

Parameterization: The model treats resonance masses ( $M_k$ ), widths ( $\Gamma_k$ ), coupling factors ( $Q_k$ ), and phase shifts ( $\delta_k$ ) as free parameters, with separate coupling/phase parameters for the neutral and charged $D$ meson channels to account for physical and experimental differences.
Fit Strategy: The fitting process began with a minimal set of resonances and incrementally added them. The author systematically tested models with varying numbers of resonances (up to seven) and the inclusion of a subthreshold pole.
Statistical Analysis: The quality of fits is evaluated using the $\chi^2$ per degree of freedom (DOF) and $p$ -values. The statistical significance of adding a new component (resonance or SP) is estimated via changes in the likelihood function, $\delta(-2 \ln L)$ .

Key Contributions and Results

Identification of the Subthreshold Pole:
The author found that a fit including only resonances yielded poor statistical quality ( $p$ -value = 0.16% for six resonances). Introducing a new component with unconstrained parameters revealed a "resonance" with a mass of $3691 \pm 56$ MeV and a width consistent with zero ( $0 \pm 160$ MeV). This entity lies below the $D\bar{D}$ threshold (3729.68 MeV), identifying it as a subthreshold pole.
Association with $\psi(2S)$ :
The mass of the discovered subthreshold pole is compatible with the known mass of the $\psi(2S)$ state ($3686.097$ MeV). The author fixes the pole's mass to the $\psi(2S)$ value with a vanishing width for subsequent analysis.
Optimal Fit Configuration:
The best fit is achieved with one subthreshold pole ( $\psi(2S)$ ) and seven resonances. This configuration yields:
- $\chi^2/\text{DOF} = 240.4/284$
- $p$ -value = 97%
- Statistical significance of the $\psi(2S)$ as a subthreshold pole: 8 $\sigma$ .
Resonance Interference and "Flip-Over":
A striking result is the interference between the subthreshold pole and the resonances. This interference causes an inversion ("flip-over") of the resonance shapes in the cross-section curves for all resonances except the dominant $\psi(3770)$ . Peaks in the resonance-only curve become dips in the full curve, and vice versa. This explains why previous fits assuming fixed PDG resonance parameters failed to reproduce the data.
Resonance Parameters:
The fitted parameters for the seven resonances differ from standard PDG values in several cases. For instance, the resonance corresponding to $\psi(3770)$ agrees well with PDG, while others appear where $\psi(4160)$ or $\psi(4230)$ were expected, suggesting the interference effects significantly alter the apparent mass and width of states in this channel.

Significance and Claims
The paper claims that the $\psi(2S)$ state, while a resonance in other processes, acts as a subthreshold pole in the $e^+e^- \to D\bar{D}$ process. The author asserts that:

Ignoring this subthreshold pole leads to untrustworthy fits and incorrect extraction of resonance parameters (masses, widths, and couplings).
The high goodness-of-fit (97% $p$ -value) is contingent on the simultaneous inclusion of the $\psi(2S)$ subthreshold pole and the freedom to adjust resonance parameters.
Standard Breit-Wigner parametrizations using the product of branching ratios and electronic widths are insufficient to detect such poles, as they may encounter mathematical inconsistencies (negative values under square roots) in this context.

The study concludes that achieving a reliable description of $e^+e^- \to D\bar{D}$ data requires acknowledging the analytic properties of the amplitude, specifically the presence of subthreshold poles, rather than relying solely on a sum of resonances above the threshold.

Technical Summary: The ψ(2S)\psi(2S)ψ(2S) State as a Subthreshold Pole in e+e−→DDˉe^+e^- \to D\bar{D}e+e−→DDˉ

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Technical Summary: The $\psi(2S)$ State as a Subthreshold Pole in $e^+e^- \to D\bar{D}$