A simple and previously overlooked choice of one parameter allows the Schwinger/Symanzik Generating Functional of QCD to be re-written in a manifestly gauge-invariant fashion, without the need of Fadeev-Popov insertions. When combined with Fradkin functional representations for the Green's function, G[A], of a quark in an effective color potential A, and the vacuum loop functional L[A], all QCD correlation functions can be represented as Gaussian, functional-linkage operations connecting relevant combinations of G[A] and L[A]. And because the Fradkin representations for those functionals are Gaussian in their dependence on A, the functional-linkage operation can be done exactly, and one then sees that gauge invariance here is achieved by gauge-independence, as the gauge-dependent gluon propagators exactly cancel out everywhere. In this way, the non-perturbative sums over Feynman graphs reduce to an explicit, gauge-independent functional expression.
That new, final functional expression now displays a new property we call "Effective Locality" (EL), in which the infinite sum over infinite classes of Feynman graphs corresponds to the exchange of a well-defined "gluon bundle" from specific space-time points on interacting quarks and/or antiquarks. And one then sees that it is no longer possible to continue to treat quarks as ordinary particles, with well defined asymptotic momenta or positions, for they are bound objects whose transverse coordinates can never be measured exactly. Once this necessity of introducing realistic "transverse imprecision" is realized, and introduced into the fundamental Lagrangian, all functional sums become well-defined, and one has an analytic way of obtaining physical information.
It should be noted that such progress is possible because the Fradkin representations are Potential Theory constructs, with reasonable approximations in different physical situations; e.g., at high energies, G[A] simplifies to a Bloch-Nordsieck/eikonal form. With those simplifications, and the remarkable property of EL, we have been able to calculate eikonal amplitudes for quark-antiquark scattering, and for three-quark scattering, and to extract from these eikonal functions the form of binding potentials that produce hadrons. And, most interesting, for the first time we can exhibit a mechanism which leads to effective Yukawa scattering between nucleons, including a scattering potential which becomes negative, as needed to make a deuteron from a proton and a neutron. This, to our knowledge, and for the first time ever, is Nuclear Physics from basic, realistic QCD.
This work, by myself (HMF), French colleagues Grandou and Gabellini (of the Universite de Nice), and my ex-Brown student Ming Sheu, is barely 18 months old at this writing; and there are many problems remaining to be studied, such as non-perturbative renormalization theory, the quark-gluon plasma, and - indeed - all of Nuclear Physics. On the basis of what we have been able to derive up to this point, we believe that this approach to analytic QCD calculations will, in the future, become extremely useful. It should be noted that the above remarks are a description of "textbook" QCD, with one type of quark and the massless gluons of SU(3); flavors and electroweak effects, as well as spin and angular momentum dependences are to be added in later on.