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Violation and Restoration of Probability Positivity in Perturbative Quantum Field Theory: From Quantum Mechanics to Quantum Chromodynamics

This paper resolves the issue of negative decay and production rates in perturbative Quantum Chromodynamics by proposing that for exclusive processes, probability should be defined as the modulus square of the truncated amplitude rather than its truncated square, a prescription that restores positivity and improves convergence in NNLO calculations.

Original authors: Bin Gong, Xu-Dong Huang, Jian-Xiong Wang, Chao Zeng

Published 2026-10-06

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Bin Gong, Xu-Dong Huang, Jian-Xiong Wang, Chao Zeng

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 microscopic world of subatomic particles, scientists rely on a fundamental rule to make sense of chaos: probability must always be positive. Just as you cannot have a negative chance of rain or a negative likelihood of a coin landing on heads, the mathematical tools used to predict particle behavior must yield results that are greater than zero. This rule is the bedrock of quantum mechanics, where the likelihood of an event is calculated by squaring a value derived from a wave function. If the math produces a negative number, it signals a breakdown in the theory, suggesting that the method used to calculate the answer has gone astray. For decades, physicists have used a specific shortcut to handle the complex interactions of particles, a method that worked perfectly well for simpler forces but began to show cracks when applied to the strong force that binds the heart of atoms together.

This paper addresses a long-standing crisis in the study of heavy particles called quarkonia, which are made of a heavy quark and its antimatter partner. When researchers tried to calculate how these particles decay or are produced using the standard mathematical shortcuts, they frequently arrived at impossible answers: negative probabilities. A negative probability is physically meaningless; it is the equivalent of saying there is a minus fifty percent chance of an event happening. The authors of this study, working at the Institute of High Energy Physics and Chongqing Normal University, argue that this problem is not a failure of nature, but a failure of the traditional calculation method. They propose a simple but profound correction: instead of breaking the calculation into pieces and discarding some of them, scientists should calculate the full strength of the interaction first and then square the result. By doing so, they restore the positivity of the probability and find that their predictions match experimental data much more closely than before.

The story begins with how physicists have historically handled these calculations. In the early days of quantum field theory, scientists developed a way to approximate complex interactions by adding up a series of terms, much like building a tower block by block. For processes involving the electromagnetic force, this method was safe because the force is weak, and the discarded parts of the calculation were so tiny they didn't matter. However, when this same method was applied to the strong force, which is much more powerful, the discarded parts became significant. In the traditional approach, the calculation involves squaring a sum of terms and then throwing away the higher-order pieces to keep the math manageable. For exclusive processes—where the initial and final states are clean and isolated, with no hidden particles to account for—this practice forces the removal of positive contributions. When the strong force is involved, these removed contributions are large enough that their absence flips the sign of the final answer, turning a positive probability into a negative one. This has plagued calculations for nearly fifty years, affecting predictions for how particles like the J/psi and the Upsilon decay into photons or how they are created in particle colliders.

The researchers in this paper demonstrate that the solution lies in changing the order of operations. Instead of expanding the square and discarding terms, they suggest truncating the sum of the interaction strengths first and then squaring the entire result. This approach ensures that the final number is always positive, because the square of any real number is positive. To test this idea, they applied it to a wide range of specific scenarios, including the decay of the J/psi and Upsilon particles into three photons, and the production of charmonium particles in electron-positron collisions at B factories. In every case, the traditional method produced results that oscillated wildly, sometimes becoming negative at the second or third level of precision. In contrast, the new method produced smooth, positive results that converged steadily toward a stable value.

The data presented in the study reveals a stark difference between the two methods. For the decay of the J/psi into three photons, the traditional calculation predicted a negative rate at the next-to-leading order, a result that is physically nonsensical. The new method, however, yielded a positive rate that was consistent with the expected physical behavior. This pattern held true across all eight exclusive production channels examined at the B factories. The authors show that the series of numbers generated by their method behaves like a converging geometric series, where each step gets closer to the true answer without overshooting or flipping signs. The traditional method, by discarding essential positive terms, creates a series that is unstable and prone to producing negative values, especially when the force being studied is strong.

One of the most compelling aspects of this work is that it does not require new physics or new particles to explain the discrepancy. It simply requires a more faithful application of the rules of probability to the existing theory. The authors argue that the burden of proof now lies with the traditional method, which relies on discarding terms that are no longer negligible in the context of the strong force. By keeping the full amplitude intact before squaring it, the calculation respects the fundamental requirement that probability must be positive. The results are not just theoretically cleaner; they are practically superior. The predictions made using this amplitude-level prescription align better with experimental measurements and show less sensitivity to the arbitrary choices of scale that often plague theoretical calculations.

The study concludes that for these specific types of particle interactions, the old way of cutting corners in the math is no longer acceptable. The negative probabilities that have confused physicists for decades are a direct consequence of an outdated approximation that works for weak forces but fails for strong ones. By returning to the basic principle that the probability of an event is the square of the total amplitude, the researchers have restored order to the calculations. Their findings suggest that the path forward for understanding these heavy particles is not to invent new theories, but to calculate the existing ones more carefully, ensuring that the final numbers always reflect the reality that probabilities cannot be negative. This shift in perspective offers a clearer, more reliable window into the behavior of the subatomic world, resolving a crisis that has persisted since the early days of quantum chromodynamics.

Technical Summary: Violation and Restoration of Probability Positivity in Perturbative Quantum Field Theory

Problem Statement
In standard perturbative Quantum Field Theory (QFT), the probability of a process is traditionally calculated by expanding the scattering amplitude ( $M$ ) in powers of the coupling constant, squaring the expansion, and truncating the result at a specific order. While this "squared-amplitude-level truncation" is mathematically consistent for inclusive processes in Quantum Electrodynamics (QED) due to the Kinoshita-Lee-Nauenberg (KLN) theorem and the smallness of the fine-structure constant ( $\alpha \approx 1/137$ ), it leads to a "sign crisis" in exclusive processes within Quantum Chromodynamics (QCD).

For exclusive processes—where initial and final states are colorless and electrically neutral—infrared divergences cancel amplitude-by-amplitude rather than at the level of the squared matrix element. Consequently, the traditional practice of discarding higher-order positive terms from the exact modulus square ( $|M|^2$ ) is not forced by infrared constraints. When the strong coupling constant ( $\alpha_s$ ) is large, discarding these terms results in negative decay widths and production rates at Next-to-Leading Order (NLO) and Next-to-Next-to-Leading Order (NNLO). The paper highlights that this issue has plagued exclusive quarkonium calculations for decades, affecting processes such as $J/\psi \to 3\gamma$ , $\Upsilon \to 3\gamma$ , and $e^+e^- \to J/\psi + J/\psi$ .

Methodology: The Amplitude-Level Prescription
The authors propose a reformulation of the perturbative calculation for exclusive processes, termed the "amplitude-level prescription." This approach adheres strictly to the quantum mechanical definition of probability:

Truncation at the Amplitude Level: The perturbative expansion of the scattering amplitude is truncated at a fixed order $M$ (e.g., $M = M_0 + M_1 + \dots + M_M$ ).
Complete Modulus Square: The probability is defined as the complete modulus square of this truncated amplitude: $\Gamma \propto \int d\Phi \sum_{\text{pol}} |M_0 + M_1 + \dots + M_M|^2$ .

Unlike the traditional method, which expands the square and discards terms beyond the working order (e.g., discarding $|M_1|^2$ at NLO), this prescription retains all positive contributions generated by the truncated amplitude. Within the framework of Non-Relativistic QCD (NRQCD) factorization, this is implemented by factorizing at the amplitude level, ensuring that each renormalized amplitude $A^{(l)}$ is infrared finite on its own.

The authors demonstrate that any amplitude-level truncated observable can be cast into an effective representation:
$\Gamma_{\text{amp}} = \Gamma_{\text{LO}} \left| 1 + \frac{\alpha_s}{\pi}(c_1 + ic_2) + \frac{\alpha_s^2}{\pi^2}(c_3 + ic_4) \right|^2$
where $c_i$ are effective real numbers derived from the ratios of loop amplitudes. This form explicitly guarantees positivity ( $\Gamma_{\text{amp}} \geq 0$ ) at every order.

Key Contributions and Results
The paper provides a systematic comparison between the traditional squared-amplitude truncation and the amplitude-level prescription across eleven exclusive processes. The analysis includes:

Decays: $J/\psi \to 3\gamma$ and $\Upsilon \to 3\gamma$ (complete NNLO QCD corrections).
Production: $e^+e^- \to J/\psi + J/\psi$ (NNLO corrections).
B-Factory Channels: Eight exclusive charmonium production channels ( $e^+e^- \to J/\psi + \eta_c$ , $J/\psi + \chi_{cJ}$ , $\eta_c + \gamma$ , and $\chi_{cJ} + \gamma$ for $J=0,1,2$ ).

Numerical Findings:

Restoration of Positivity: The amplitude-level prescription yields positive decay widths and cross sections at NLO and NNLO for all processes considered. In contrast, the traditional method produces negative rates for several channels (e.g., $J/\psi \to 3\gamma$ at NLO and NNLO, $\Upsilon \to 3\gamma$ at NNLO).
Improved Convergence: The series inside the modulus square (the terms $1, c_1, c_2$ , etc.) exhibits geometric convergence, with the modulus of higher-order terms decreasing. Conversely, the traditional series (expanding the square) shows oscillating behavior where the second term often exceeds the leading term in magnitude and is negative, leading to poor convergence.
Reduced Scale Dependence: The NNLO results obtained via the amplitude-level prescription demonstrate reduced dependence on the renormalization scale compared to the traditional approach.

Significance and Claims
The paper argues that the "sign crisis" in exclusive QCD processes is not a failure of QCD but a failure of the traditional truncation method, which was historically justified only for inclusive QED processes where discarded terms are negligible. The authors assert that for exclusive processes, the KLN constraint does not apply, and therefore, the truncation of the squared amplitude is unjustified when higher-order terms are large.

The primary significance claimed is that the amplitude-level prescription is the "first-principles definition of the probability" for exclusive processes. It requires no further justification beyond the correspondence between the wave function and the matrix element in quantum mechanics. The burden of justification, the authors argue, lies with the traditional method. By adopting the amplitude-level prescription, the paper demonstrates that negative probabilities are an artifact of an inappropriate truncation scheme, and that a systematic, positive, and better-converging perturbative expansion is achievable for exclusive quarkonium processes.

Technical Summary: Violation and Restoration of Probability Positivity in Perturbative Quantum Field Theory

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