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Introduction

The vacuum energies of spatially confined quantum fields have been of great interest since the early days of quantum field theory [1,2]. Shortly after the advent of the non-Abelian gauge theory of strong interactions [3,4,5], the bag models of hadrons [6,7,8] required estimates for the contribution of the spherically constrained vacuum to the total energy of a hadron. In essence, two lines of approaches have been pursued in the past. The canonical vacuum energy was parametrized by means of a dimensionless quantity Z₀ to be fitted to experiment [9]. While disregarding the quadratic boundary condition of the original MIT bag model, a relation between the bag radius R and the bag constant B was established by demanding stability of the calculated hadron mass under variations of R [10]. However, the quadratic boundary condition of the fermionic MIT bag model, $B_q=\left.-\frac{1}{2}\partial _r\ (\bar{\psi}\psi)\right\vert _{r=R}$ , was introduced to restore the broken four-momentum conservation of the bag [6], and thus it should be taken seriously. For a meaningful definition of the bag constant B_q according to the quadratic boundary condition, the vacuum expectation value of this operator equation must be taken [11].

There has been a great effort to compute the Casimir effect of the MIT bag model [11,12,13,14,15,16]. The vacuum expectation values of global quantities must be regularized. Several procedures, adapted to either global or local approaches, were applied. Global techniques regularize the sum over mode energies by analytical continuation (zeta-function method) [13,14,17], while local approaches compute finite densities based on two-point functions. The space-integral of these densities is regularized by volume or temporal cutoffs [2,18]. However, different regularization schemes yield different answers which is not acceptable. Various solutions have been suggested [11,13,14,15]. For instance, the vacuum energy has been separated into a classical and a quantum part. The classical contribution was parametrized by phenomenological quantities to absorb divergences due to the quantum part by appropriate renormalizations [13,15]. This procedure relies on direct experimental information which is unsatisfactory. Interesting results were obtained in the massive case [13,14,19]. By imposing the condition that the vacuum of a very massive field should not fluctuate, a unique term in the canonical vacuum energy, attributed to quantum fluctuations, was isolated.

In this paper we propose an alternative to the above procedure. Our approach is based on a separation between the perturbative and nonperturbative regimes of QCD. As suggested by Vepstas and Jackson in the framework of a chiral bag model [20], hard fluctuations should be allowed to traverse the boundary since these fluctuations are not subject to the low-energy confinement mechanism. In contrast to the work of Ref.[20], we consider only the interior of the bag. In the simple model of the QCD vacuum, which the bag-model philosophy offers, we think of hard fluctuations to be noninteracting and unconfined when calculating nonperturbative effects, such as the ground state energy of the bag.

Next: Calculation Up: Vacuum structure of a Previous: Vacuum structure of a

Marc Schumann
2000-10-16