Lokhov, Alexey (2006) Non-perturbative study of QCD correlators. PhD thesis CPHT, EP - CPHT Centre de Physique Théorique, EP/X p.188.

Alternative Locations: http://www.imprimerie.polytechnique.fr/Theses/Files/Lokhov.pdf

Abstract

This PhD dissertation is devoted to a non-perturbative study of QCD correlators. The main tool that we use is lattice QCD. Lattice QCD has been successfully used in phenomenology, but it can also be used to study the fundamental parameters (like the coupling constant) and properties of the theory itself. This is the main goal of the present dissertation. We concentrated our efforts on the study of the main correlators of the pure Yang - Mills theory in the Landau gauge, namely the ghost and the gluon propagators. We are particularly interested in determining the LQCD parameter -the fundamental scale of the pure Yang-Mills theory. It is extracted by means of perturbative predictions available up to NNNLO. The related topic is the influence of non-perturbative effects that show up as appearance of power-corrections to the low-momentum behaviour of the Green functions. We shall see that these corrections are quite important up to energies of the order of 1!
0 GeV. A new method of removing these power corrections allows a better estimate of LQCD. Our result is \Lambda^{n_f=0}_{\bar{MS}} = 269(5)+12−9 MeV. Another question that we address is the infrared behaviour of Green functions, at momenta of order and below LQCD. At low energy the momentum dependence of the propagators changes considerably, and this is probably related to confinement. We try to clarify the laws that govern the infrared gluodynamics in order to understand the radical nature of the changes in the infrared behaviour. Many questions arise: the Gribov ambiguity, the impact of different non-perturbative relations at low momenta, self-consistency of the lattice approach in this domain. The lattice approach allows to check the predictions of analytical methods because it gives access to non-perturbative correlators. According to our analysis the gluon propagator is finite and non-zero at vanishing mom!
entum, and the power-law behaviour of the ghost propagator is !
the same as in the free case.

Table of content

Notations and conventions 6
General introduction 9
1 Continuum and lattice QCD 11
1.1 General features of QCD - 11
1.1.1 Definitions and symmetries - 11
1.1.2 The Gribov ambiguity - 13
1.1.3 Schwinger-Dyson equations - 15
1.1.4 Slavnov-Taylor identities - 18
1.1.5 Renormalisation group equations - 19
1.2 Lattice QCD - 20
1.2.1 Lattice QCD partition function - 20
1.2.2 Fixing the Minimal Landau gauge on the lattice - 22
2 Lattice Green functions 27
2.1 Green functions in Landau gauge - 27
2.2 Numerical calculation of the ghost Green functions in Landau gauge 30
2.2.1 Lattice implementation of the Faddeev-Popov operator - 30
2.2.2 Numerical inversion of the Faddeev-Popov operator - 31
2.2.3 Calculation of the ghost-gluon vertex - 33
2.3 Errors of the calculation - 34
2.3.1 Estimating the statistical error - 34
2.3.2 Handling the discretisation errors - 34
2.4 Gribov ambiguity and lattice Green functions - 36
2.4.1 The landscape of minima of the gauge-fixing functional - 37
2.4.2 Lattice Green functions and the Gribov ambiguity - 40
3 The ultraviolet behaviour of Green functions 45
3.1 LQCD and perturbative expressions for Green functions - 46
3.2 OPE for the Green functions and dominant power corrections - 48
3.2.1 The dominant OPE power correction for the gluon propagator 50
3.2.2 The dominant OPE power correction for the ghost propagator 51
3.2.3 Constraints on theWilson coefficients from the Slavnov-Taylor identity - 52
3.3 Data analysis - 53
3.3.1 Fitting the gluon and the ghost propagators - 54
6 Contents
3.3.2 Fit of the ratio - 56
3.3.3 Comparing the results - 59
4 The infrared behaviour of Green functions 61
4.1 Review of today’s analytical results - 62
4.1.1 Zwanziger’s prediction - 62
4.1.2 Study of truncated SD and ERG equations - 63
4.2 Constraints on the infrared exponents and the Slavnov-Taylor identity 63
4.3 Relation between the infrared exponents - 65
4.4 Lattice study of the ghost Schwinger-Dyson equation - 68
4.4.1 Complete ghost Schwinger-Dyson equation in the lattice formulation
- 68
4.4.2 Checking the validity of the tree-level approximation for the ghost-gluon vertex - 70
4.5 Direct fits of infrared exponents - 72
4.5.1 Testing the relation 2aF + aG = 0 - 73
4.5.2 Lattice fits for aF and aG - 74
5 Conclusions 79
6 Resumé 83
References 87
A Large momentum behavior of the ghost propagator in SU(3) lattice gauge theory 91
B Non-perturbative power corrections to ghost and gluon propagators 105
C The infrared behaviour of the pure Yang-Mills Green functions 121
D Scaling properties of the probability distribution of lattice Gribov copies 147
E Short comment about the lattice gluon propagator at vanishing momentum171
F Is the QCD ghost dressing function finite at zero momentum ? 179

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