Born Level Bound States

Abstract

Bound state poles in the $S$-matrix of perturbative QED are generated by the {\em divergence} of the expansion in $\alpha$. The perturbative corrections are necessarily singular when expanding around free, \order{\alpha^0} $in$ and $out$ states that have no overlap with finite-sized atomic wave functions. Nevertheless, measurables such as binding energies do have well-behaved expansions in powers of $\alpha$ (and $\log\alpha$). It is desirable to formulate the concept of "lowest order" for gauge theory bound states such that higher order corrections vanish in the $\alpha \to 0$ limit. This may allow to determine a lowest order term for QCD hadrons which incorporates essential features such as confinement and chiral symmetry breaking, and thus can serve as the starting point of a useful perturbative expansion. I discuss a "Born" (no loop, lowest order in $\hbar$) approximation. Born level states are bound by gauge fields which satisfy the classical field equations. Gauss' law determines a distinct field $A^0(\xv)$ for each instantaneous position of the charges. A Poincar\'e covariant boundary condition for the gluon field leads to a confining potential for $q\bar q$ and $qqq$ states. In frames where the bound state is in motion the classical gauge field is obtained by a Lorentz boost of the rest frame field.