⚛️ high-energy experiments

Lagrangian-based model of the $e^+e^- \to V \bar V$ process applied to the $e^+e^- \to D^{\ast+} D^{\ast-}$ data

This paper derives a Lagrangian-based Breit-Wigner formula for the $e^+e^- \to V\bar{V}$ process and validates its parameters by successfully fitting 2022 BESIII Collaboration data for the $e^+e^- \to D^{\ast+}D^{\ast-}$ channel.

Original authors: Peter Lichard

Published 2026-07-20

📖 4 min read🧠 Deep dive

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

Imagine the universe as a giant, invisible dance floor where the smallest particles in existence zoom around, bumping into each other, and sometimes disappearing to create something new. This is the world of particle physics, a place where scientists smash electrons and their antimatter twins, positrons, together at nearly the speed of light. When these two collide, they can vanish and reappear as a pair of heavy, exotic particles called mesons. It's like throwing two pebbles into a pond and watching a pair of giant, glowing jellyfish pop out of the water instead of a splash.

To understand this dance, physicists use a set of rules called the "Standard Model," which acts like a cosmic instruction manual. One of the most famous tools in this manual is the "Breit-Wigner formula." Think of this formula as a musical score that predicts how loud a specific note (a particle resonance) will sound at different pitches (energies). For a long time, scientists assumed this musical score worked perfectly for every type of particle pair they could create. However, just as a guitar sounds different from a drum, different types of particles might need different musical scores. This paper asks a simple but crucial question: Does the old, standard score work when the dancers are two heavy, spinning vector mesons, or do we need to write a new tune?

The author of this paper, Peter Lichard, decides to test the old score against a brand new set of experimental data. He starts by building a new mathematical model from the ground up, using the rules of quantum field theory. Instead of just guessing, he derives a fresh formula specifically for when an electron and positron collide to create a pair of vector mesons (particles that spin like tops). He finds that the old, standard formula behaves differently at high energies than his new derivation. It's as if the old score predicts the music should fade away very quickly as the pitch gets higher, while his new score suggests the music stays louder for longer.

To see which score is correct, Lichard takes a look at real-world data collected by the BESIII Collaboration in 2022. This team smashed electrons and positrons together at energies between 4.08 and 4.60 GeV, watching closely for the creation of a specific pair of particles: a $D^{*+}$ and a $D^{*-}$ . When Lichard tried to fit the old, standard formula to this data, the result was poor. The math didn't match the reality; the "fit" was very bad, with a statistical score (chi-squared) of 2681/25 that indicated the model was significantly off the mark.

So, he applied his new, custom-built formula. The result was a significant improvement, yielding an "acceptable" fit that matched the experimental points with a p-value of 52%. While this is a good sign in this field, the author notes that the fit is not perfect and that more sophisticated mathematical descriptions of the particle shapes might be needed in the future to get even better results. By using this new formula, he was able to identify three distinct "resonances"—or heavy particle states—hiding in the data.

The first two resonances he found correspond to known particles called $\psi(4040)$ and $\psi(4230)$ . Interestingly, the new model revealed that a "bump" in the data that looked like a single new particle at 4.1 GeV wasn't a new particle at all. Instead, it was a "ghost" created by the interference of the two real particles, $\psi(4040)$ and $\psi(4230)$ , dancing together. It's like hearing a beat that isn't a drum, but the result of two different drummers hitting their drums at slightly different times.

The third resonance he found was a bit of a mystery. The model pointed to a particle at 4411 MeV, which matches the known $\psi(4415)$ perfectly. However, the model also suggested that the $\psi(4230)$ might be decaying into the $D^{*+}D^{*-}$ pair, a process that the Particle Data Group (the official record-keepers of particle physics) says has never been seen before. While the paper doesn't claim this is a confirmed discovery, it suggests this could be the first hint of such a decay.

In the end, the paper concludes that the old, standard formula is not a universal key for all particle collisions. For this specific type of dance—creating two vector mesons—the old rules don't quite work. The new formula derived in this paper provides a much better map of the territory, allowing scientists to see the true shapes of these particles and understand how they interact. It's a reminder that in the quantum world, even the most trusted tools need to be checked against the latest evidence, because the universe might just be playing a different tune than we expected.

Technical Summary: Lagrangian-based model of the e+e−→V V̄ process applied to the e+e−→D∗+D∗− data

Problem Statement
Recent experiments at $e^+e^-$ colliders have produced open-charm-meson pairs, including vector-meson pairs like $D^{*+}D^{*-}$ and $D_s^{*+}D_s^{*-}$ , which were previously unobserved with light mesons. While standard analyses often apply the Breit-Wigner (BW) formula to these cross sections, previous work by the author demonstrated that the BW formula is not universally valid. Specifically, for the annihilation into a vector ( $V$ ) and pseudoscalar ( $P$ ) meson, the cross section behaves as $s^{-2}$ at high energies, whereas the standard BW formula predicts $s^{-3}$ . This discrepancy suggests that the standard BW formula, derived for pseudoscalar-pseudoscalar ( $P$ - $P$ ) final states, fails to correctly describe $V$ - $\bar{V}$ final states. The problem addressed in this work is the derivation of a theoretically sound cross-section formula for $e^+e^- \to V\bar{V}$ based on Quantum Field Theory (QFT) and its subsequent validation against experimental data.

Methodology
The author constructs a Vector Meson Dominance (VMD) model grounded in the Lagrangian formalism of QFT.

Theoretical Derivation:
- The model utilizes an interaction Lagrangian describing the coupling of a neutral vector meson resonance ( $V_k$ ) to a $V\bar{V}$ pair.
- The Feynman diagram involves an intermediate $V_k$ state coupling to the photon and subsequently decaying into $V\bar{V}$ .
- The amplitude is calculated using the three-vector-meson vertex, which is analogous to vertices in the Standard Electroweak Model ( $ZW^+W^-$ ) and QCD (three gluons).
- By summing over initial spins and final polarizations (setting electron mass to zero) and integrating over the Mandelstam variable $t$ , the author derives a specific cross-section formula.
- The resulting formula (Eq. 7) includes a kinematic factor dependent on the meson mass $m$ and center-of-mass energy $\sqrt{s}$ , distinct from the standard BW form. It accounts for coherent sums of multiple resonances with relative phases.
- The coefficients in the formula are related to physical observables: the partial decay widths $\Gamma_{V_k \to V\bar{V}}$ and $\Gamma_{V_k \to e^+e^-}$ .
Data Fitting:
- The derived formula is applied to the $e^+e^- \to D^{*+}D^{*-}$ Born cross-section data published by the BESIII Collaboration (2022), covering the center-of-mass energy range of 4.08 to 4.60 GeV (28 data points).
- The fitting procedure treats resonance masses ( $M_k$ ), total widths ( $\Gamma_k$ ), coefficients ( $C_k$ ), and relative phases ( $\delta_k$ ) as free parameters, optimized using the standard $\chi^2$ criterion.
- The analysis proceeds iteratively: first attempting a single-resonance fit, then two, and finally three resonances to determine the optimal model complexity.

Key Contributions

Derivation of a VMD-based Cross-Section Formula: The paper provides a rigorous derivation of the $e^+e^- \to V\bar{V}$ cross-section formula, demonstrating that it differs structurally from the standard BW formula used for $P$ - $P$ states. The derived formula exhibits an asymptotic behavior of $\sigma \propto s^{-1}$ at high energies, contrasting with $s^{-3}$ for $P$ - $P$ and $s^{-2}$ for $P$ - $V$ states.
Parameterization of Resonance Properties: The work establishes a direct link between the fit parameters and the physical decay widths of the intermediate resonances, allowing for the extraction of quantities like $\Gamma_{V_k \to D^{*+}D^{*-}} \times \Gamma_{V_k \to e^+e^-}$ .
Reinterpretation of the 4.1 GeV Structure: The analysis reveals that the prominent structure observed near 4.1 GeV in the BESIII data is not a single resonance but an interference effect between two distinct resonances, $\psi(4040)$ and $\psi(4230)$ .

Results

Fit Quality: A single-resonance fit yielded a poor result ( $\chi^2/\text{NDF} = 2681/25$ ). A two-resonance fit improved the situation but remained inadequate. A three-resonance fit (involving $\psi(4040)$ , $\psi(4230)$ , and $\psi(4415)$ ) provided an acceptable description of the data with $\chi^2/\text{NDF} = 15.0/17$ and a p-value of 52%.
Resonance Parameters:
- $\psi(4040)$ : The fitted mass ( $4042 \pm 20$ MeV) agrees well with the Particle Data Group (PDG) value, though the uncertainty is larger. The width ( $108 \pm 19$ MeV) is slightly higher than the PDG average but within $1\sigma$ .
- $\psi(4230)$ : The mass ( $4217.8 \pm 2.5$ MeV) aligns well with PDG. The fitted width ( $83.4 \pm 5.5$ MeV) is significantly higher than the PDG average ( $51 \pm 7$ MeV) but is consistent with results from four recent BESIII experiments. Notably, the PDG notes that the $D^{*+}D^{*-}$ decay of $\psi(4230)$ has not been observed; this fit provides the first indication of this decay channel.
- $\psi(4415)$ : The parameters ( $M = 4411 \pm 14$ MeV, $\Gamma = 116 \pm 15$ MeV) agree perfectly with PDG values.
Interference Effects: The analysis confirms that the peak near 4.1 GeV arises from the interference between $\psi(4040)$ and $\psi(4230)$ rather than a new resonance state.

Significance and Claims
The paper claims that the standard Breit-Wigner formula is insufficient for describing $e^+e^-$ annihilation into vector meson pairs. By employing a Lagrangian-based VMD approach, the author demonstrates that a modified formula is necessary to correctly capture the high-energy behavior and kinematic dependencies of the $V\bar{V}$ final state. The successful fit to BESIII data with a p-value of 52% validates the soundness of this theoretical framework. Furthermore, the work highlights the importance of using the correct formula for specific final states to avoid misinterpreting interference patterns as new resonances. The author modestly notes that while the current parametrization (fixed masses and constant widths) yields acceptable results, more sophisticated descriptions (e.g., energy-dependent widths, Flatté approximations, or K-matrix formalisms) may be required for higher precision in future studies.

More like this