Gluon Condensate via Dirac Spectral Density: IR Phase, Scale Anomaly and IR Decoupling

Abstract

Quark and gluon scalar densities, $\langle \barψ ψ\rangle$ and $\langle F^2 \rangle$, reflect the degree of scale-invariance violations in SU(N) gauge theories with fundamental quarks. It is known that $\langle \barψ ψ\rangle$ can be usefully scale-decomposed via spectral density $ρ(λ)$ of Dirac modes. Here I give such formula for $\langle F^2 \rangle$, which reveals that gluon condensate is a strictly UV quantity. For the recently-found IR phase [1,2], where the infrared (IR) degrees of freedom separate out and become independent of the system's bulk, it implies that $\langle F^2 \rangle$ due to this IR part vanishes. Its glue thus doesn't contribute to scale anomaly of the entire system and is, in this sense, scale invariant consistently with the original claim. Associated formulas are used to define IR decoupling of glue, which may serve as an alternative indicator of IR phase transition. Using the simplest form of coherent lattice QCD, we express the effective action of full QCD entirely via Dirac spectral density.

Figures (2)

• Figure 1: Types of thermal states for theories in ${\cal T}$ and schematics of their Dirac spectral densities $\rho(\lambda)$. Here $\lambda$ is the Dirac eigenvalue (scale) in the continuum-like notation where $D\psi_\lambda = i \lambda \psi_\lambda$. Left: B phase (standard confined phase) involves a single-component system with correlated parts. Its leading IR power behavior $\lambda^0$ includes cases when density is logarithmically divergent. Middle: IR phase involves a multi-component system with IR separated and decoupled from the bulk. $\Lambda_{{\text{\rm IR}}}$ is the energy scale of IR-bulk separation and $\lambda_{dc}$ the associated Dirac scale. Right: the hypothetical UV phase describes a single-component system of weakly interacting quarks and gluons.
• Figure 2: Left: schematic view of phases in set ${\cal T}$ based on a degree of deep-IR degrees of freedom proliferation and their scale invariance. Direction of arrows for parameters indicates the direction of possible phase changes along the chain $\text{B} \!\to\! \text{IR} \!\to\! \text{UV}$Alexandru:2019gdm. Right: the case of near-massless quarks with Banks-Zaks (B-Z) regime and the asymptotic freedom (AF) boundary indicated.