Generic framework for non-perturbative QCD in light hadrons

TL;DR

This work develops a comprehensive, nonperturbative framework for light-hadron QCD based on the instanton liquid model (ILM). It leverages gradient flow to reveal the topological vacuum structure, introduces a grand canonical ILM to account for fluctuations, and derives a density-expansion with emergent $t\'Hooft$ vertices that map QCD operators and hadronic matrix elements onto effective quark interactions. The framework yields a momentum-dependent constituent quark mass $M(k)$, a calculable gluon condensate, and a topological susceptibility $\chi_t$, with form factors and distribution amplitudes calculable at a low scale $\mu \lesssim 1/ρ$. Predictions show good agreement with lattice QCD, illustrating that a small set of parameters $(ρ,n_{I+A},m)$ controls nonperturbative physics of light hadrons. Overall, the ILM provides a principled, scalable bridge between vacuum topology and hadron phenomenology, enabling systematic calculations of VEVs, hadronic matrix elements, and form factors across scales.

Abstract

This paper aims to serve as an introductory resource for disseminating the concept of instanton liquid model to individuals with interests in quantum chromodynamics (QCD) for hadrons. We discuss several topological aspects of the QCD vacuum and briefly review recent progress on this intuitive unifying framework for the lowlying hadron physics rooted in QCD by introducing the vacuum as a liquid of pseudoparticles. We develop systematic density expansion on the dilute vacuum with diagrammatical Feynman rules to calculate the vacuum expectation values (VEVs) and generalize the calculations to hadronic matrix element (charges), and hadronic form factors using the instanton liquid model (ILM). The ILM prediction are well-consistent with those of recent lattice QCD calculations. Thereby, the nonperturbative physics can be well-controlled by only a few parameters: instanton size $ρ$ and instanton density $n_{I+A}$, and current quark mass $m$.

Figures (22)

• Figure 1: Visualization of the vacuum in gluodynamics, before cooling at a resolution of about $\frac{1}{10}\,{\rm fm}$ (top), and after cooling at a resolution of about $\frac{1}{3}\,{\rm fm}$ (bottom) Moran:2008xq, where the pseudoparticles emerge.
• Figure 2: Instanton density $n$ as a function of the dimensionless cooling time $\tau$Athenodorou:2018jwu where $\tau=t/a^2$ with the lattice space $a=0.139$ fm.
• Figure 3: The relation between monopoles, dyons, and instantons. Monopoles are the endpoints of the center vortices, related to dyons (instanton-monopole) via Poisson transform at finite temperature. When the temperature cools down, dyons gets denser in the vacuum and eventually recombine to instantons at zero temperature where the full 4D vacuum degrees of freedom now is described by a dilute instanton liquid.
• Figure 4: Instanton (yellow) and anti-instanton (blue) configurations in the deep-cooled Yang-Mills vacuum, threaded by center P-vortices using center projection on lattice Biddle:2019gkeBiddle:2020eec. These topological configurations form the primordial gluon epoxy (hard glue) that underpins the origin of light hadron masses Liu:2024rdmZahed:2021fxk while the string-like center P-vortices play a key role in confinement, forming a world sheet in Euclidean time direction
• Figure 5: The constituents of instantons (calorons): BPS and KK dyons, and their anti-dyons with their electric, magnetc, and topological charges labeled