Loop calculations in the three dimensional Gribov–Zwanziger Lagrangian

Abstract

The three dimensional Gribov-Zwanziger Lagrangian is analysed at one and two loops. Specifically, the two loop gap equation is evaluated and the Gribov mass is expressed in terms of the coupling constant. The one loop corrections to the propagators of all the fields are determined. It is shown that when the gap equation is satisfied the Faddeev-Popov ghost and both Bose and Grassmann localizing ghosts all enhance in the infrared limit at one loop. This verifies that the Kugo-Ojima confinement criterion holds to this order and we also show that both Grassmann ghosts are enhanced at two loops. For the Bose ghost we determine the full form of the propagator in the zero momentum limit for both the transverse and longitudinal pieces and confirm Zwanziger's recent general analysis for the low energy behaviour. We provide an alternative but equivalent version of the horizon condition expressing it as the vacuum expectation value of an operator involving only the localizing Bose ghost field. The one loop static potential is also determined.