On Gribov’s Ideas on Confinement

I comment on possible relations of Gribov’s ideas on mechanism of confinement with some phenomena in QCD and in supersymmetric gauge theories. a To be published in the Boris Ioffe Festschrift ’At the Frontier of Particle Physics / Handbook of QCD’, ed. M. Shifman (World Scientific, Singapore, 2001). Based on the talks given at workshops “Challenges in QCD”, Kfar Giladi, Israel, June 20-23, 1999 and “Gribov-70”, Orsay, France, March 27-29, 2000. 1 Gribov’s mechanism of confinement of color V.N. Gribov developed his ideas on confinement in QCD over the course of more than ten years. While I had a privilege of multiple discussions on the subject with him, it was difficult for me to follow. I tried to relate some of his points to subjects I knew from QCD and from supersymmetric gauge theories. These discussions were origin of the comments presented here. First, I will try to review briefly Gribov’s ideas following his last two papers. His starting point is that confinement is due to light quarks — the mechanism which is similar to supercritical phenomena for large Z in QED. These phenomena are related with fermion bound states in a strong field. There is no such state for bosons. For this reason Gribov believed there were no glueballs in pure gluodynamics, and the theory presents scaling behavior. To realize his picture of confinement in QCD Gribov introduced a number of new points: • Formulation of QCD with no divergences and no need for renormalization. Both perturbative and nonperturbative phenomena are described by Green functions. • In QCD ultraviolet and infrared regimes are strongly interrelated to give a specific confining solution. • Chiral symmetry and corresponding Goldstone bosons — pions, play a special role in confinement. In particular, there is a short distance component in the pion wave function — a core which is pointlike. Equations are changed because of this. What is the supercritical phenomenon in QED? A heavy nucleus with Z > Zcr ∼ 1/α makes the vacuum of light charged fermions unstable. It forms a bound state with a positron at short distances (an electron component of pair goes away). As a result an effective shift, Z → Z1 = Z − 1, occurs for distances larger than the bound state size. A similar picture is proposed for QCD based on the growth of effective charge at large distances. Only colorless states are stable. The phenomenon is described by Gribov’s equation for the fermion propagator G(q)