[Paper Review] Aspects of confinement from QCD correlation functions

Christian S. Fischer, Axel Maas+1 more|ArXiv.org|Dec 15, 2008

Quantum Chromodynamics and Particle Interactions7 references3 citations

TL;DR

This paper investigates the infrared behavior of ghost and gluon propagators in Landau gauge Yang-Mills theory using functional equations (Dyson-Schwinger and functional renormalization group equations). It identifies two solutions—scaling and decoupling—distinguished by the ghost dressing function's infrared behavior. The key finding is that only the scaling solution, with infrared divergent ghost dressing and vanishing gluon propagator, is consistent with the Kugo-Ojima confinement scenario and unbroken global BRST symmetry, while the decoupling solution violates these symmetries. Notably, the functional equation results for the gluon dressing function match lattice data almost pointwise across the phenomenologically relevant mid-momentum region.

ABSTRACT

We discuss the properties of ghost and gluon propagators in Landau gauge Yang-Mills theory and their relation to the confinement problem. In general two types of infrared behavior of these functions are allowed from their functional equations: scaling and decoupling. Both solutions show positivity violations in the gluon propagator and lead to a confining Polyakov loop potential. However, only the scaling solution agrees with the Kugo-Ojima confinement scenario and the related formulation of a physical Hilbert space of Yang-Mills theory. Our numerical results for the gluon dressing function agree almost pointwise with the lattice results at all physical momenta.

Motivation & Objective

• To resolve the long-standing discrepancy between lattice gauge theory and continuum functional methods regarding the infrared behavior of ghost and gluon propagators in QCD.
• To determine which infrared solution—scaling or decoupling—aligns with the Kugo-Ojima confinement scenario and global BRST symmetry.
• To assess the phenomenological relevance of the gluon and ghost dressing functions by comparing functional equation results with lattice simulations in the mid-momentum region.
• To clarify whether the decoupling solution, observed in lattice studies, is compatible with a physical Hilbert space and confinement mechanisms.

Proposed method

• Solving the functional equations of Yang-Mills theory using Dyson-Schwinger equations (DSEs) and functional renormalization group (FRG) equations in the Landau gauge.
• Implementing two distinct infrared boundary conditions for the ghost dressing function: $ G(0) = \infty $ for the scaling solution and $ G(0) = \text{const.} $ for the decoupling solution.
• Using a truncation scheme that preserves transversality and multiplicative renormalizability in the DSE framework, and a complementary FRG truncation to minimize mid-momentum artifacts.
• Analyzing the power-law infrared behavior $ Z(p^2) \sim (p^2)^{2\kappa - d/2 + 2} $ and $ G(p^2) \sim (p^2)^{-\kappa} $, with $ \kappa \approx 0.595353 $ for the scaling solution.
• Comparing numerical solutions of the functional equations with lattice data for the gluon dressing function in minimal Landau gauge.
• Assessing confinement via the Polyakov loop potential and positivity violation in the gluon propagator.

Experimental results

Research questions

• RQ1 Does the scaling solution of the ghost and gluon propagators satisfy the Kugo-Ojima confinement criterion and preserve global BRST symmetry?
• RQ2 Why do lattice simulations typically observe a finite ghost dressing function at zero momentum, contradicting continuum functional methods?
• RQ3 Can the decoupling solution, with infrared-finite ghost and massive gluon propagator, still lead to a confining Polyakov loop potential?
• RQ4 To what extent do functional equation results for the gluon dressing function agree with lattice data in the mid-momentum region?
• RQ5 What is the physical significance of the infrared boundary condition $ G(0) $ in determining the global symmetry structure of Yang-Mills theory?

Key findings

• The scaling solution, with $ \kappa \approx 0.595353 $, is the unique solution that satisfies the Kugo-Ojima confinement scenario and preserves unbroken global BRST symmetry.
• The decoupling solution, with finite $ G(0) $, leads to a massive gluon propagator and breaks global gauge and BRST symmetries, contradicting the Kugo-Ojima criterion.
• Both scaling and decoupling solutions exhibit positivity violation in the gluon propagator and lead to a confining Polyakov loop potential.
• The functional renormalization group solution for the gluon dressing function matches lattice results from ref. [23] almost pointwise in the mid-momentum region, validating the method phenomenologically.
• The infrared behavior of the ghost and gluon propagators is determined solely by the boundary condition $ G(0) $, with $ G(0) = \infty $ yielding scaling and $ G(0) = \text{const.} $ yielding decoupling.
• The scaling solution is the only one consistent with a physical Hilbert space of color-singlet states, avoiding the 'behind-the-moon' problem in confinement.