Nonperturbative part of the plaquette in pure gauge theory

Abstract

We define the nonperturbative part of a quantity as the difference between its numerical value and the perturbative series truncated by dropping the order of minimal contribution and the higher orders. For the anharmonic oscillator, the double-well potential, and the single plaquette gauge theory, the nonperturbative part can be parametrized as $A{\ensuremath{\lambda}}^{B}{\mathrm{e}}^{\ensuremath{-}C/\ensuremath{\lambda}}$ and the coefficients can be calculated analytically. For lattice QCD in the quenched approximation, the perturbative series for the average plaquette is dominated at low order by a singularity in the complex coupling plane and the asymptotic behavior can only be reached by using extrapolations of the existing series. We discuss two extrapolations that provide a consistent description of the series up to order 20--25. These extrapolations favor the idea that the nonperturbative part scales like $(a/{r}_{0}{)}^{4}$ with $a/{r}_{0}$ defined with the force method. We discuss the large uncertainties associated with this statement. We propose a parametrization of $\mathrm{ln}(a/{r}_{0})$ as the two-loop universal terms plus a constant and exponential corrections. These corrections are consistent with ${a}_{1\mathrm{\text{\ensuremath{-}}}\mathrm{loop}}^{2}$ and play an important role when $\ensuremath{\beta}<6$. We briefly discuss the possibility of calculating them semiclassically at large $\ensuremath{\beta}$.