The Quark Model via an hbar expansion of QCD

Abstract

I discuss the possibility that the quark model emerges as the lowest order of anh expansion of QCD bound states. In a hamiltonian approach the instantaneous A 0 potential is determined by the field equations separately for each Fock component. These equations allow also a linear potential as a homogeneous solution. Stationarity of the action sets the direction of the ensuing constant electric field to be along the fermion pair separation. States bound by this non-perturbative, linear A 0 potential are analogous to the Born term of standard perturbative expansions of scattering amplitudes, in that they represent the dominant contribution at lowest order inh (no loops) and at O (g) in the potential. The Dirac equation for relativistic fermions bound by an external A 0 potential is most easily derived using retarded boundary conditions. I demonstrate why this boundary condition does not affect the bound state energies at lowest order inh. Translated to physical Feynman boundary conditions the Dirac bound states are a superposition of Fock states with any number of fermionantifermion pairs. Applying this approach to relativistic quark-antiquark states in QCD results in a bound state equation which was previously proposed without derivation and shown to provide a reasonable description of the meson spectrum, including linear Regge trajectories. The equal-time wave functions have unique Lorentz transformation properties, which ensure the correct dependence of the bound state energy on the center-of-mass momentum. This indicates that the solution is exact at the Born level, i.e., at lowest order inh and at O (g) in the QCD interaction.

Boosting equal time bound states

“…It appears possible to apply the methods presented above also in D = 3 + 1 dimensions by imposing a non-vanishing boundary condition on the solution of Gauss' law for A 0 . This gives rise to a linear instantaneous potential in all frames, which is of O (g) in the coupling and thus leading compared to the O g 2 transverse gluon exchange [2]. The frame dependence of the wave function turns out to be similar to the D = 1 + 1 case when the quark positions are aligned with the CM momentum [7].…”

“…In Appendix B we discuss the Poincaré generators of QCD 2 . The non-abelian gauge invariance brings some new features, but the formulation of QCD bound states and their frame dependence is similar to the abelian case [2].…”

“…As mentioned in the Introduction and further explained in [2], a relativistic gauge theory bound state may be described by valence-like Dirac-type wave functions provided instantaneous Coulomb exchange dominates and one uses retarded propagators. At the Born level (lowest order in ) the bound state energies will agree with the result using Feynman propagators, even though the wave functions obtained with the two types of propagator are very different.…”

“…We show how (Born level) two-body eigenstates of the translation generators P 0 and P 1 may be found in QED, making use of retarded propagation in analogy to the Dirac case mentioned above [2]. We then apply the boost generator to obtain the bound state in another frame.…”

“…Using retarded propagators there are no Z-contributions and one obtains the standard Dirac wave function describing a single particle with both positive and negative energy components. Remarkably, the same relativistic bound state can thus, at Born level, be equivalently described using two quite different wave functions [2]. The iε-prescription invariance ensures that the bound state energies are independent of the choice of wave function.…”