Coarse graining in effective theories of lattice QCD in 1+1d and 2+1d

TL;DR

This paper applies coarse graining techniques to effective lattice QCD theories in 1+1d and 2+1d, analyzing phase transitions and fixed points, and comparing results with Monte Carlo simulations.

Contribution

It introduces a coarse graining approach to effective lattice QCD theories in low dimensions, providing analytical and numerical insights into phase transitions and fixed points.

Findings

Analytical solutions for recursion relations in 1d

Numerical continuum extrapolation results

Approximate agreement with Monte Carlo data (~12%)

Abstract

In the strong coupling and heavy quark mass regime, lattice QCD dimensionally reduces to effective theories of Polyakov loops depending on the parameters of the original Wilson action $\beta, \kappa$ and $N_\tau$. We apply coarse graining techniques to such theories in 1d and 2d, corresponding to lattice QCD at finite temperature and non-zero chemical potential in 1+1d and 2+1d, respectively. In 1d the method is applied to the effective theories up to $\mathcal{O}(\kappa^4)$. Using the transfer matrix, the recursion relations are solved analytically. The thermodynamic limit is taken for some observables. Afterwards, continuum extrapolation is performed numerically and results are discussed. In 2d the coarse graining method is applied in the pure gauge and static quark limit. Running couplings are obtained and the fixed points of the transformations are discussed. Finally, the critical coupling of the deconfinement transition is determined in both limits. Agreement to about 12% with Monte Carlo results of 2+1d Yang-Mills theory from the literature is observed.