Solving the gluon Dyson—Schwinger equation in the Mandelstam approximation

• Published in: Computer physics communications Vol. 112; no. 2; pp. 149 - 165
• Main Authors: Hauck, A., von Smekal, L., Alkofer, R.
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 01.08.1998; Elsevier Science
• Subjects: Constrained iterative solution; Dyson—Schwinger equations; Exact sciences and technology; Function theory, analysis; Gauge bosons; Gauge field theories; General properties of qcd (dynamics, confinement, etc.); General theory of fields and particles; Gluon propagator; Gluons; Infrared asymptotic series; Integral equations; Landau gauge; Mandelstam approximation; Mathematical methods in physics; Nonlinear integral equations; Nonperturbative QCD; Other nonperturbative techniques; Physics; Properties of specific particles; Quantum chromodynamics; Specific theories and interaction models; particle systematics; The physics of elementary particles and fields; Mandelstam approximation; Nonperturbative QCD; Gluon propagator; Nonlinear integral equations; Constrained iterative solution; 11.15.Tk; Landau gauge; 02.30.Rz; 12.38.Aw; 14.70.Dj; Infrared asymptotic series; Dyson—Schwinger equations; Nonlinear equations; Landau theory; Asymptotic expansion; Gluons; Integral equations; Propagator; Theoretical study; Mandelstam representation; Green function
• ISSN: 0010-4655; 1879-2944
• DOI: 10.1016/S0010-4655(98)00046-0

Truncated Dyson—Schwinger equations represent finite subsets of the equations of motion for Green's functions. Solutions to these nonlinear integral equations can account for nonperturbative correlations. We describe the solution to the Dyson—Schwinger equation for the gluon propagator of Landau gauge QCD in the Mandelstam approximation. This involves a combination of numerical and analytic methods: an asymptotic infrared expansion of the solution is calculated recursively. In the ultraviolet, the problem reduces to an analytically solvable differential equation. The iterative solution is then obtained numerically by matching it to the analytic results at appropriate points. Matching point independence is obtained for sufficiently wide ranges. The solution is used to extract a nonperturbative β-function. The scaling behavior is in good agreement with perturbative QCD. No further fixed point for positive values of the coupling is found which thus increases without bound in the infrared. The nonperturbative result implies an infrared singular quark interaction relating the scale A of the subtraction scheme to the string tension σ.