Variational approach to quark confinement in field theory

Abstract

We explore the use of the variational principle in a relativistic field theory of quarks coupled linearly to real scalar gluons. Renormalization is discussed, in principle and in practice. In the actual calculation the theory is cut off at a large but fixed momentum $\ensuremath{\Lambda}$, and variations are performed before taking the limit $\ensuremath{\Lambda}\ensuremath{\rightarrow}\ensuremath{\infty}$. Trial wave functions are such that quarks find themselves in a central scalar square-well potential. The outside mass ${M}_{0}$ the inside mass ${M}_{1}$, and the range ${r}_{0}$ are variational parameters. By construction the wave functions are eigenstates of total momentum and angular momentum, and form a complete set. In the limit of infinite renormalized gluon mass the entire calculation can be done analytically, with the following results. There is a choice of model constants such that ${M}_{0}\ensuremath{\rightarrow}\ensuremath{\infty}$ in the renormalized limit. Futhermore, the model constants can be so chosen that the calculation reduces to one that bears formal resemblance to calculations in a classical field theory. We examine in greater detail this "semiclassical" limit, and, in particular, investigate two consistent possibilities that are similar respectively to the MIT bag model and the SLAC model. In either case the model still contains two free renormalized constants. Depending on their values there may or may not be a vacuum well (i.e., ${M}_{1}\ensuremath{\ne}{M}_{0}$ in the lowest state). Illustrations of physical applications are given in the MIT case. Hadronic structure and nucleon deep-inelastic structure functions are calculated, with results similar to those of the MIT bag model. In particular, the structure functions exhibit scaling.