[Paper Review] Dispersion Relations in Quantum Chromodynamics

Reinhard Oehme|ArXiv.org|May 26, 1992

Quantum optics and atomic interactions3 citations

TL;DR

This paper establishes dispersion relations for hadronic scattering in Quantum Chromodynamics (QCD) using BRST cohomology to define physical states and spectral conditions. It demonstrates that hadronic amplitudes lack ordinary and anomalous thresholds tied to quark-gluon structure, ensuring analyticity and supporting the validity of dispersion-theoretic methods in non-Abelian gauge theories.

ABSTRACT

Dispersion relations for the scattering of hadrons are considered within the framework of Quantum Chromodynamics. It is argued that the original methods of proof remain applicable. The setting and the spectral conditions are provided by an appropriate use of the BRST cohomology. Confinement arguments are used in order to exclude quarks and gluons from the physical subspace. Local, BRST-invariant hadron fields are considered as leading terms in operator product expansions for products of fundamental fields. The hadronic amplitudes have neither ordinary nor anomalous thresholds which are directly associated with the underlying quark-gluon-structure. Proofs involving the Edge of the Wedge Theorem and analytic completion are discussed briefly.

Motivation & Objective

• To extend dispersion relation techniques—previously used in quantum field theories with stable particles—to non-Abelian gauge theories like QCD.
• To address the challenge of hadronic scattering amplitudes in a theory where quarks and gluons are confined and not directly observable.
• To establish rigorous spectral conditions and physical state conditions via BRST cohomology, ensuring consistency with confinement.
• To prove that hadronic amplitudes do not exhibit thresholds directly linked to the underlying quark-gluon structure, preserving analyticity.
• To validate the use of analytic completion and the Edge of the Wedge Theorem in QCD for deriving dispersion relations.

Proposed method

• Utilizes BRST cohomology to define the physical subspace of the theory, excluding unphysical states and ensuring gauge invariance.
• Constructs local, BRST-invariant hadron fields as leading terms in operator product expansions of fundamental quark and gluon fields.
• Applies the Edge of the Wedge Theorem and analytic completion techniques to extend amplitudes analytically into complex momentum planes.
• Imposes spectral conditions derived from the physical state conditions in the BRST framework to ensure unitarity and causality.
• Demonstrates that hadronic amplitudes are analytic in the complex energy plane, avoiding thresholds from quark-gluon intermediate states.
• Relies on confinement arguments to exclude quarks and gluons from the physical spectrum, ensuring only stable hadronic states contribute.

Experimental results

Research questions

• RQ1 Can dispersion relations be rigorously applied to hadronic scattering amplitudes in QCD despite confinement?
• RQ2 Do hadronic amplitudes in QCD exhibit ordinary or anomalous thresholds associated with the underlying quark-gluon structure?
• RQ3 How can BRST cohomology be used to define physical states and spectral conditions in a non-Abelian gauge theory?
• RQ4 To what extent do analyticity theorems like the Edge of the Wedge Theorem remain valid in QCD for hadronic amplitudes?
• RQ5 Can operator product expansion techniques be used to construct local, BRST-invariant hadron fields from fundamental quark and gluon fields?

Key findings

• Hadronic amplitudes in QCD do not possess ordinary or anomalous thresholds directly tied to the quark-gluon structure, preserving analyticity.
• The use of BRST cohomology successfully isolates the physical subspace, excluding unphysical states and ensuring consistency with confinement.
• Local, BRST-invariant hadron fields emerge as leading terms in operator product expansions, providing a field-theoretic realization of hadrons.
• The Edge of the Wedge Theorem and analytic completion techniques are applicable in QCD, supporting the derivation of dispersion relations.
• The spectral conditions derived from BRST cohomology ensure unitarity and causality in the scattering amplitudes.
• The framework confirms that hadronic scattering amplitudes are analytic in the complex energy plane, consistent with dispersion theory.