Violation and Restoration of Probability Positivity in Perturbative Quantum Field Theory: From Quantum Mechanics to Quantum Chromodynamics
Abstract
In quantum mechanics the modulus square of the wave function is the probability of an event --- positivity is self-evident. Quantum field theory corresponds to quantum mechanics one-to-one: the wave function corresponds to the matrix element, so the probability of a process should be the modulus square of the matrix element, positive by construction. In the founding era of perturbative quantum field theory, however, Feynman, Schwinger and Tomonaga treated inclusive processes in quantum electrodynamics, where the truncation of the squared amplitude was forced by infrared cancellations and rendered harmless by the small coupling α≃1/137. Carried over to exclusive processes in quantum chromodynamics, where the coupling is large, this practice produced negative decay and production rates already at NLO, persisting at NNLO. We point out that for exclusive processes, whose initial and final states are colorless for QCD and electrically neutral for QED, the infrared divergences cancel amplitude by amplitude and the KLN constraint is absent, so the complete modulus square of the truncated amplitude is the proper, manifestly positive definition of the probability. We formulate amplitude-level NRQCD factorization as the systematic prescription that restores positivity, show that every amplitude-level truncated observable admits the effective representation Γamp=ΓLO|1+αs/π(c1+ic2)+αs2/π2(c3+ic4)|2, and demonstrate, with the complete NNLO QCD corrections to J/ψ→3γ, Υ→3γ, e+e-→ J/ψ+J/ψ and the eight exclusive charmonium production channels at B factories, that the prescription yields positive and better-converging predictions.
Related
- Analytic Methods for Perturbative Quantum Chromodynamics
- Unitarity violation and restoration in radiative bound-state formation
- No violation on a generalisation of Leggett-Garg inequality and Bell-CHSH inequality with extended probability
- Quantum kinetic theory for quantum chromodynamics
- Instanton liquid framework for non-perturbative QCD in light hadrons
BibTeX
@misc{gong2026violation,
title = {{Violation and Restoration of Probability Positivity in Perturbative Quantum Field Theory: From Quantum Mechanics to Quantum Chromodynamics}},
author = {Bin Gong and Xu-Dong Huang and Jian-Xiong Wang and Chao Zeng},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.04557},
eprint = {2610.04557},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.04557},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Violation and Restoration of Probability Positivity in Perturbative Quantum Field Theory: From Quantum Mechanics to Quantum Chromodynamics
AU - Bin Gong
AU - Xu-Dong Huang
AU - Jian-Xiong Wang
AU - Chao Zeng
PY - 2026
DA - 2026/10/03
PB - arXiv
DO - 10.48550/arxiv.2610.04557
UR - https://arxiv.org/abs/2610.04557
AB - In quantum mechanics the modulus square of the wave function is the probability of an event --- positivity is self-evident. Quantum field theory corresponds to quantum mechanics one-to-one: the wave function corresponds to the matrix element, so the probability of a process should be the modulus square of the matrix element, positive by construction. In the founding era of perturbative quantum field theory, however, Feynman, Schwinger and Tomonaga treated inclusive processes in quantum electrodynamics, where the truncation of the squared amplitude was forced by infrared cancellations and rendered harmless by the small coupling α≃1/137. Carried over to exclusive processes in quantum chromodynamics, where the coupling is large, this practice produced negative decay and production rates already at NLO, persisting at NNLO. We point out that for exclusive processes, whose initial and final states are colorless for QCD and electrically neutral for QED, the infrared divergences cancel amplitude by amplitude and the KLN constraint is absent, so the complete modulus square of the truncated amplitude is the proper, manifestly positive definition of the probability. We formulate amplitude-level NRQCD factorization as the systematic prescription that restores positivity, show that every amplitude-level truncated observable admits the effective representation Γ^amp=Γ_LO|1+α_s/π(c_1+ic_2)+α_s^2/π^2(c_3+ic_4)|^2, and demonstrate, with the complete NNLO QCD corrections to J/ψ→3γ, Υ→3γ, e^+e^-→ J/ψ+J/ψ and the eight exclusive charmonium production channels at B factories, that the prescription yields positive and better-converging predictions.
ER -