Wave functions and energy for the vacuum and heavy quarks from the WKB Schrödinger equation for massive gauge theories with instantons

The physics of vacuum tunneling for gauge field theories is studied following methods developed in a series of earlier papers by explicitly constructing the eigenstates of the Hamiltonian which correspond to the ground state in the presence of an arbitrary external static charge distribution. This is achieved in the $A_0=0\mathrm{}$ canonical formalism where this charge distribution was shown before to be a cyclic degree of freedom. The corresponding wave functionals represent the $\theta$ vacuum and very-heavy-quark states. We work in the Schrödinger picture where the field configurations are functions only of the space coordinates. Hence we have one less space dimension than the usual Euclidean approach, and instantons are paths in configuration space which connect different classical vacuums through classically forbidden regions. There we solve the Schrödinger equation exactly to the first two orders in $\hslash$ using the one-instanton solution. The resulting wave functionals are concentrated around the instanton paths which are the locus of maximum tunneling probability. Around classical vacuums the wave functional is shown to possess, besides the usual perturbative Gaussian, a part which is an exponentially increasing function of the fluctuation. The energy and eigenfunctional are determined by matching this wave functional to the WKB expression in an overlap region where both are valid. Special care must be taken concerning the continuous symmetries of the problem since the instantons and the classical vacuum break different subgroups of the symmetry group. Although the formalism applies to any field theory with instantons, the present discussion of matching strictly holds if there are no massless particles. It has thus the same limitations as the dilute-gas approximation (DGA) in the Euclidean approach, the results of which are comprised in the outcome of the present discussion. For the massless case the DGA result for the energy also comes out if one ignores the fact that the matching problem differes a priori in an essential way from the massive case. In field-theoretical language, one should notice that we match two different perturbative expansions, namely the expansions around the instanton and around the classical vacuum. This is rather new in quantum field theory and may have other applications, for instance in connection with the Gribov phenomenon, as discussed in an earlier paper.