Blockspin renormalization-group study of color confinement due to violation of the non-Abelian Bianchi identity

TL;DR

This study applies block-spin transformations to topological defects in lattice gauge theory, revealing a scaling behavior of the effective monopole action that approaches the continuum limit and allows analytical estimates of physical quantities like string tension and glueball mass.

Contribution

It introduces a novel smoothing technique for lattice vacuum, demonstrates a universal scaling of the effective monopole action, and analytically connects the infrared monopole action to the string model for confinement.

Findings

Effective action exhibits scaling as a function of the product of blocking step and lattice spacing.

The almost perfect action is independent of the gauge fixing used.

Analytical estimates of string tension and glueball mass are consistent with numerical results.

Abstract

Block-spin transformation of topological defects is applied to the violation of the non-Abelian Bianchi identity (VMABI) on lattice defined as Abelian monopoles. To get rid of lattice artifacts, we introduce various techniques smoothing the vacuum. The effective action can be determined by adopting the inverse Monte-Carlo method. The coupling constants $F(i)$ of the effective action depend on the coupling of the lattice action $\beta$ and the number of the blocking step $n$. But it is found that $F(i)$ satisfy a beautiful scaling, that is, they are a function of the product $b=na(\beta)$ alone for lattice coupling constants $3.0\le\beta\le3.9$ and the steps of blocking $1\le n\le 12$. The effective action showing the scaling behavior can be regarded as an almost perfect action corresponding to the continuum limit, since $a\to 0$ as $n\to\infty$ for fixed $b$. The almost perfect action showing the scaling is found to be independent of the smooth gauges adopted here. Then we compare the results with those obtained by the analytic blocking method of topological defects from the continuum. The infrared monopole action can be transformed into that of the string model. The physical string tension and the lowest glueball mass can be evaluated \textit{analytically} by the strong-coupling expansion of the string model. We get $\sqrt{\sigma}\simeq 1.3\sqrt{\sigma_{phys}}$ for $b\ge 1.0\ \ (\sigma_{phys}^{-1/2})$, whereas the scalar glueball mass is kept to be near $M(0^{++})\sim 3.7\sqrt{\sigma_{phys}}$. Also we can almost reproduce \textit{analytically} the scaling function of the squared monopole density determined numerically for large $b$ region $b>1.2\ (\sigma_{phys}^{-1/2})$.