[Paper Review] Mass Gap in Quantum Chromodynamics

Raghunath Acharya|ArXiv.org|Nov 20, 2006

Mathematical and Theoretical Analysis3 references3 citations

TL;DR

This paper presents a heuristic argument that Quantum Chromodynamics (QCD) exhibits a mass gap if the Callan-Symanzik beta function remains negative for all positive coupling strengths. By combining Elitzur’s theorem, the Federbush-Johnson-Schroer theorem, and the trace anomaly, it shows that non-Abelian color charges must vanish identically, implying dynamical gluon mass generation and confinement—supporting Cornwall's 1982 scenario of a Higgs-like confinement phase without explicit symmetry breaking.

ABSTRACT

We present a heuristic argument in support of the assertion that QCD will exhibit a mass gap, if the Callan-Symanzik function β(g) obeys the inequality β(g) < 0, for all g > 0.

Motivation & Objective

• To provide a heuristic argument for the existence of a mass gap in QCD under specific conditions on the beta function.
• To establish that the vanishing of non-Abelian color charges implies dynamical mass generation for gluons.
• To support the scenario of infrared slavery and confinement as proposed by Cornwall (1982), without explicit Higgs mechanism.
• To clarify the role of locality and vacuum structure in non-Abelian gauge theories via Greenberg’s classification of locality.
• To reconcile the absence of massless Nambu-Goldstone bosons with the trace anomaly and scale symmetry breaking.

Proposed method

• Use of Elitzur’s theorem to enforce that local color charge operators annihilate the vacuum, preventing spontaneous gauge symmetry breaking.
• Application of the Federbush-Johnson-Schroer theorem to conclude that if a local operator annihilates the vacuum, it must vanish identically.
• Analysis of the double-commutator structure involving the dilatation charge, Hamiltonian, and vector current to derive constraints on the trace anomaly.
• Derivation of the commutator $[Q^a, eta(g)/2g F^a_{ u au}F^{ u au}] = 0$ from the trace anomaly and Jacobi identity.
• Postulation of locality in the sense of Greenberg (commutation at spacelike separation) to justify the consistency of the charge-current commutator structure.
• Deduction that $Q^a \partial^\mu D_\mu \equiv 0$, leading to two solutions: conformal (A) or vanishing charge (B), with B implying mass gap.

Experimental results

Research questions

• RQ1 Under what conditions on the Callan-Symanzik function β(g) does QCD exhibit a mass gap?
• RQ2 How does the vanishing of non-Abelian color charges imply dynamical gluon mass generation?
• RQ3 Can the Federbush-Johnson-Schroer theorem be applied to derive the absence of massless colored states in QCD?
• RQ4 What is the role of the trace anomaly and dilatation current in enforcing a mass gap?
• RQ5 How does Greenberg’s classification of locality resolve apparent nonlocality in charge-current commutators?

Key findings

• If β(g) < 0 for all g > 0, the non-Abelian color charges Q^a must vanish identically, implying that colored states cannot exist as asymptotic states.
• The vanishing of Q^a leads to the conclusion that the divergence of the dilatation current must vanish, i.e., $\partial^\mu D_\mu \equiv 0$, which is incompatible with a nontrivial infrared fixed point.
• The absence of a nontrivial infrared fixed point implies that the theory cannot be conformal in the infrared, thus excluding a continuous mass spectrum and enforcing a mass gap.
• The result supports Cornwall’s 1982 proposal that infrared slavery leads to dynamical gluon mass generation and confinement without explicit Higgs mechanism.
• The analysis confirms that the vacuum is non-degenerate and points in no particular direction in group space, consistent with Elitzur’s theorem and the absence of colored Nambu-Goldstone bosons.
• The argument establishes that the only consistent solution under the given assumptions is the Higgs (confinement) phase, where gluons acquire a dynamical mass via the trace anomaly and scale symmetry breaking.