The Gribov problem and QCD dynamics N. Vandersickel 1 a, Daniel Zwanziger 2 b a Ghent University, Department of Mathematical Physics and Astronomy Krijgslaan 281-S9, 9000 Gent,Belgium b New York University, New York, NY 10003, USA arXiv:1202.1491v2 [hep-th] 25 Apr 2012 [email protected] [email protected] Abstract In 1967, Faddeev and Popov were able to quantize the Yang-Mills theory by introducing new particles called ghost through the introduction of a gauge. Ever since, this quantization has become a standard textbook item. Some years later, Gribov discovered that the gauge fixing was not complete, gauge copies called Gribov copies were still present and could affect the infrared region of quantities like the gauge dependent gluon and ghost propagator. This feature was often in literature related to con- finement. Some years later, the semi-classical approach of Gribov was generalized to all orders and the GZ action was born. Ever since, many related articles were published. This review tends to give a pedagogic review of the ideas of Gribov and the subsequent construction of the GZ action, including many other toipics related to the Gribov region. It is shown how the GZ action can be viewed as a non-perturbative tool which has relations with other approaches towards confinement. Many different features related to the GZ action shall be discussed in detail, such as BRST breaking, the KO cri- terion, the propagators, etc. We shall also compare with the lattice data and other non-perturbative approaches, including stochastic quantization. Contents 1 Introduction 5 1.1 Patchwork quilt of QCD . .5 1.2 Gribov’s gluon confinement scenario . .6 1.3 Approaches to the confinement problem . .8 1.4 Outline of the article . .8 1.5 Limitations of the present approach . .9 1.6 The Yang-Mills theory: definitions and conventions . .9 2 From Gribov to the local action 12 2.1 The Faddeev-Popov quantization . 12 2.1.1 Zero modes . 12 2.1.2 A two dimensional example . 13 2.1.3 The Yang-Mills case . 15 2.1.4 Two important remarks concerning the Faddeev-Popov derivation . 18 2.1.5 Other gauges . 18 2.1.6 The BRST symmetry . 19 2.2 The Gribov problem . 20 2.2.1 The Gribov region: a possible solution to the Gribov problem? . 21 2.2.2 The fundamental modular region (FMR) . 25 2.2.3 Other solutions to the Gribov problem . 26 2.2.4 Summary . 27 2.3 The relation of the Gribov horizon to Abelian and center-vortex dominance . 27 2.4 Semi classical solution of Gribov . 30 2.4.1 The no-pole condition . 30 2.4.2 The gluon and the ghost propagator . 33 2.5 The local renormalizable action . 36 2.5.1 A toy model . 36 2.5.2 The non-local GZ action . 39 2.5.3 The local GZ action . 51 2.5.4 The gluon and the ghost propagator . 53 2.6 Algebraic renormalization of the GZ action . 56 2.6.1 The starting action and BRST . 56 2.6.2 The Ward identities . 59 2.6.3 The counter-term . 61 2.6.4 The renormalization factors . 63 2.7 Relation between Gribov no-pole condition and the GZ action . 64 2 3 Features of the GZ action 65 3.1 The propagators of the GZ action . 65 3.2 The transversality of the gluon propagator . 70 3.3 The hermiticity of the GZ action . 71 3.4 The soft breaking of the BRST symmetry . 71 3.4.1 The breaking . 71 3.4.2 The BRST breaking as a tool to prove that the Gribov parameter is a physical parameter . 72 3.4.3 The Maggiore-Schaden construction and spontaneous breaking of BRST sym- metry . 73 3.4.4 Local version of Maggiore-Schaden construction . 75 3.5 Restoring the BRST . 80 3.5.1 Adapting the BRST symmetry s is not possible . 80 3.5.2 Restoring the BRST by the introduction of new fields . 81 3.5.3 Two different phases of the restored BRST symmetry . 83 3.6 The GZ action and its relation to the Kugo-Ojima confinement criterion . 86 3.6.1 Introduction: the Kugo-Ojima criterion . 86 3.6.2 Remarks on the KO criteria . 87 3.6.3 Imposing u(0) = -1 as a boundary condition in the Faddeev-Popov action . 87 3.6.4 Conclusion . 87 3.7 The relation of the GZ action to the lattice data . 88 3.7.1 The lattice data . 88 3.7.2 Reflection positivity . 89 3.8 Color confinement . 90 3.8.1 Exact bounds on free energy . 90 3.8.2 Simple model . 93 4 Overview of various approaches to the Gribov problem 94 4.1 GZ action . 94 4.2 Refined GZ action . 95 4.3 Coulomb gauge . 95 4.4 Dyson-Schwinger equation . 95 4.5 Stochastic quantization . 97 4.5.1 Background . 97 4.5.2 Time-dependent approach . 99 4.5.3 Time-independent approach . 99 A Some formulae and extra calculations 102 A.1 Gaussian integrals . 102 A.1.1 Gaussian integral for scalar variables . 102 A.1.2 Gaussian integral for complex conjugated scalar variables . 102 A.1.3 Gaussian integral for Grassmann variables . 102 A.2 Dµ(A)w = 0 is a gauge invariant equation. 102 A.3 s decreases with increasing k2............................. 103 A.4 Determinant of Kµn ................................... 104 3 B Cohomologies and the doublet theorem 105 B.1 Cohomology . 105 B.2 Doublet theorem . 105 C Positivity and uniqueness of solution of time-independent Fokker-Planck equation 108 C.1 Positivity . 108 C.2 Uniqueness . 109 4 Chapter 1 Introduction 1.1 Patchwork quilt of QCD Quantum Chromodynamics (QCD) is the theory which describes the strong interaction, one of the four fundamental forces in our universe. This force describes the interactions between quarks and gluons, which are fundamental building blocks of our universe. At very high energies, QCD is asymptotically free, meaning that quarks and gluons behave like free particles1. However at low energies, i.e our daily world, due to the strong force, quarks and gluons interact and form bound states called hadrons. A well known example of these bound states are the proton and the neutron, but a whole zoo of hadrons has been observed in particle detectors. In fact, the only way to obtain information about the strong force is through bound states, as no free quark or gluon has even been detected. We call this phenomenon confinement. Although 40 years of intensive research have passed since the formulation of the standard model (which includes QCD), no good answer has been found to probably one of the most fundamental questions in QCD. Even the formulation of what confinement really is, is under discussion [1]. The difficulty for solving confinement lies in the fact that the standard techniques which have been so successful in QED, are not applicable in QCD. In QED the coupling constant is small enough2, so one can apply perturbation theory, which amounts to writing down a series in the coupling constant. In QCD, the coupling constant increases with decreasing energy, a phenomenon know as “infrared slavery”, and at low energy the coupling constant becomes too large and perturbation theory alone can never give a good description of the theory. Therefore, other techniques are required which we call non-perturbative methods. There exists a wide range of non-perturbative methods, which all try to approach QCD from one way or another. One should not see these different techniques as com- peting, but as patchwork trying to cover all aspects of QCD. Many different approaches have been developed to describe confinement, e.g. Abelian dominance [2, 3, 4], center-vortex dominance [5], light-cone dominance [6], the Kugo-Ojima confinement mechanism [7, 8], Wilson’s lattice gauge the- ory approach [9], and the approach initiated by Gribov [10]. The last approach has been elaborated in widely scattered articles, but it has not been reviewed, and the relation of different approaches is not well known. Let us mention that even if one omits quarks in QCD, one can still call the remaining theory con- 1E.g. in deep inelastic scattering (DIS) experiments, quarks can be treated as free particles. 2This depends of course on the energy range of interest. 5 fining. Although there is no real experimental evidence, because the theory of gluons without quarks is a gedankentheorie, lattice simulations have shown that gluons form bound states which we call glueballs, and no free gluon can occur. Therefore, it is of interest to investigate pure QCD without quarks, and try to find out what happens. One could say that confinement is hidden in the behavior of the gluons. 1.2 Gribov’s gluon confinement scenario The gauge concept was introduced into physics by Hermann Weyl to describe electromagnetism, by analogy with Einstein’s geometrical theory of gravity. According to Weyl, the vector potential Aµ(x) n is the electromagnetic analog of the transporter or Christoffel symbol Glµ of general relativity, and serves to transport the phase factor of a charged field. It is remarkable that the theories of the weak and strong interaction which have been developed since Weyl’s time and which, with electromag- netism, form the standard model, have also proven to be gauge theories. The group is extended from U(1) to SU(2) and, for QCD, SU(3), but these theories embody the geometrical gauge character intro- duced by Weyl. In particular they respect the principle of local gauge invariance in the same way that Einstein’s theory of gravity respects coordinate invariance.