A gauge-invariant spectral scale from covariant diffusion in d-dimensional Yang-Mills backgrounds
DOI:
https://doi.org/10.31349/RevMexFis.72.060701Keywords:
Yang–Mills theory, covariant Laplacian, heat kernel, gauge invariance, spectral diagnosticsAbstract
We introduce a simple gauge-invariant spectral diagnostic for Euclidean Yang--Mills backgrounds based on covariant diffusion. Using the gauge-covariant Laplacian $\Delta_A$, we consider heat-kernel trace observables built from the semigroup $e^{-t\Delta_A}$ and address the infinite-volume divergence through two complementary normalizations: a trace per unit volume in a finite domain followed by the thermodynamic limit, and a relative trace with respect to the trivial background $A=0$ that cancels the leading volume contribution. From these trace observables we define an effective spectral scale via a logarithmic derivative in diffusion time, yielding a mass-like parameter that characterizes the large-$t$ decay of covariant diffusion when the limit exists. In parallel, we outline a diffusion-based spectral-dimension diagnostic and discuss basic invariance and positivity properties of the construction. The framework is formulated in general dimension $d$ and is intended as a clean, continuum starting point for connecting gauge-invariant long-time diffusion behaviour with spectral information of the covariant operator, without discretization or lattice input.
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Copyright (c) 2026 Ángel Escamilla-Aké
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