Violation and Restoration of Probability Positivity in Perturbative Quantum Field Theory: From Quantum Mechanics to Quantum Chromodynamics
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 $α\simeq1/137$. 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 $Γ^{\mathrm{amp}}=Γ_{\mathrm{LO}}\bigl|1+\frac{α_s}π(c_1+ic_2)+\frac{α_s^2}{π^2}(c_3+ic_4)\bigr|^2$, and demonstrate, with the complete NNLO QCD corrections to $J/ψ\to3γ$, $Υ\to3γ$, $e^+e^-\to J/ψ+J/ψ$ and the eight exclusive charmonium production channels at $B$ factories, that the prescription yields positive and better-converging predictions.