Gluon condensate via Dirac spectral density: IR phase, scale anomaly, and IR decoupling

Quark and gluon scalar densities, <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mo stretchy="false">⟨</a:mo> <a:mover accent="true"> <a:mi>ψ</a:mi> <a:mo stretchy="false">¯</a:mo> </a:mover> <a:mi>ψ</a:mi> <a:mo stretchy="false">⟩</a:mo> </a:math> and <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"> <g:mo stretchy="false">⟨</g:mo> <g:msup> <g:mi>F</g:mi> <g:mn>2</g:mn> </g:msup> <g:mo stretchy="false">⟩</g:mo> </g:math> , reflect the degree of scale-invariance violations in <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"> <k:mrow> <k:mi>SU</k:mi> <k:mo stretchy="false">(</k:mo> <k:mi>N</k:mi> <k:mo stretchy="false">)</k:mo> </k:mrow> </k:math> gauge theories with fundamental quarks. It is known that <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"> <o:mo stretchy="false">⟨</o:mo> <o:mover accent="true"> <o:mi>ψ</o:mi> <o:mo stretchy="false">¯</o:mo> </o:mover> <o:mi>ψ</o:mi> <o:mo stretchy="false">⟩</o:mo> </o:math> can be usefully scale-decomposed via the spectral density <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"> <u:mi>ρ</u:mi> <u:mo stretchy="false">(</u:mo> <u:mi>λ</u:mi> <u:mo stretchy="false">)</u:mo> </u:math> of Dirac modes. Here, we give such formula for <y:math xmlns:y="http://www.w3.org/1998/Math/MathML" display="inline"> <y:mo stretchy="false">⟨</y:mo> <y:msup> <y:mi>F</y:mi> <y:mn>2</y:mn> </y:msup> <y:mo stretchy="false">⟩</y:mo> </y:math> , which reveals that the 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 <cb:math xmlns:cb="http://www.w3.org/1998/Math/MathML" display="inline"> <cb:mo stretchy="false">⟨</cb:mo> <cb:msup> <cb:mi>F</cb:mi> <cb:mn>2</cb:mn> </cb:msup> <cb:mo stretchy="false">⟩</cb:mo> </cb:math> due to this IR part vanishes. Its glue, thus, does not contribute to the scale anomaly of the entire system and is, in this sense, scale invariant consistently with the original claim. Associated formulas are used to define the 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.