Nonperturbative Faddeev-Popov formula and the infrared limit of QCD

Abstract

We show that an exact nonperturbative quantization of continuum gauge theory is provided by the Faddeev-Popov formula in the Landau gauge, ␦(‫•ץ‬A)det͓Ϫ‫•ץ‬D(A)͔exp͓ϪS YM (A)͔, restricted to the region where the Faddeev-Popov operator is positive Ϫ‫•ץ‬D(A)Ͼ0 ͑Gribov region͒. Although there are Gribov copies inside this region, they have no influence on expectation values. The starting point of the derivation is stochastic quantization which determines the Euclidean probability distribution P(A) by a method that is free of the Gribov critique. In the Landau-gauge limit the support of P(A) shrinks down to the Gribov region with Faddeev-Popov weight. The cutoff of the resulting functional integral on the boundary of the Gribov region does not change the form of the Dyson-Schwinger ͑DS͒ equations because det͓Ϫ‫•ץ‬D(A)͔ vanishes on the boundary, so there is no boundary contribution. However this cutoff does provide supplementary conditions that govern the choice of solution of the DS equations. In particular the ''horizon condition,'' though consistent with the perturbative renormalization group, puts QCD into a nonperturbative phase. The infrared asymptotic limit of the DS equations of QCD is obtained by neglecting the Yang-Mills action S YM . We sketch the extension to a BRST-invariant formulation. In the infrared asymptotic limit, the BRST-invariant action becomes BRST exact, and defines a topological quantum field theory with an infinite mass gap. Confinement of quarks is discussed briefly.

Decontracted double BRST symmetry on the lattice

“…The arguments for the Gribov-Zwanziger framework to achieve this further restriction to the absolute minima in lattice formulations [30], furthermore involve the thermodynamic or infinite-volume limit, in which the common boundary of the fundamental modular region and the first Gribov region dominate the minimal Landau-gauge configuration space [31]. To find absolute minima in lattice simulations is not feasible for large lattices as this is a nonpolynomially hard computational problem.…”

Aspects of the functional renormalisation group

“…First of all this concerns the interrelation of functional methods like the general flows studied here, Dyson-Schwinger equations [149][150][151][152][153][154][155][156][157], stochastic quantisation [158][159][160], and the use of N PI effective actions [161][162][163][164][165][166][167][168][169][170][171][172][173][174][175][176][177]. All these methods have met impressive success in the last decade, in particular if it comes to physics where a perturbative treatment inherently fails.…”

“…They have been successfully used for the description of the infrared sector of QCD formulated in Landau gauge, initiated in [149,150], for a review see [151]. This approach is also tightly linked to a similar analysis in stochastic quantisation [158][159][160].…”

The gluon and ghost propagator and the influence of Gribov copies

“…Zwanziger [3] has suggested that the behaviour of both propagators in Landau gauge results from the restriction of the gauge fields to the Gribov region, where the Faddeev-Popov operator is nonnegative. Generically, one gauge orbit has more than one intersection (Gribov copies) within the Gribov region.…”

“…In particular the infrared behaviour of the ghost propagator in the Landau gauge is related to the so-called Kugo-Ojima confinement criterion [1], which expresses the absence of coloured massless asymptotic states from the spectrum of physical states in terms of the ghost propagator at vanishing momentum. On the other hand the suppression of the gluon propagator in the infrared was argued to be related to the gluon confinement [2].Zwanziger [3] has suggested that the behaviour of both propagators in Landau gauge results from the restriction of the gauge fields to the Gribov region, where the Faddeev-Popov operator is nonnegative. Generically, one gauge orbit has more than one intersection (Gribov copies) within the Gribov region.…”