Bound states and QCD

Abstract

The similarities of hadrons and atoms motivate a study of the principles of QED bound states and of their applicability to QCD. The power series in $\alpha$ and $\log\alpha$ of the binding energy is reflected in the Fock expansion of the bound state in temporal gauge ($A^0=0$). Gauss' constraint on physical states fixes the gauge for time independent transformations and determines the instantaneous interaction within each Fock state. Positronium atoms generate a classical (dipole) electric field, whereas there can be no color octet gluon field for color singlet hadrons. Hence the gluon field generated by each color component of a hadron need not vanish at spatial infinity. Gauss' constraint has a homogeneous solution with a single parameter $\Lambda$ that is compatible with Poincar\'e invariance. The corresponding potential is linear for $q\bar q$ and $gg$ Fock states, and confining also for other states ($q\bar qg,\,qqq$). This approach is consistent with the quarkonium phenomenology based on the Cornell potential at lowest order. The relativistic meson and glueball eigenstates of the QCD Hamiltonian with the $O(\alpha_s^0)$ linear potential are determined. The states lie on linear Regge trajectories and their daughters. There are also massless bound states which allow to include a $J^{PC}=0^{++}$ condensate in the perturbative vacuum, thus breaking chiral symmetry spontaneously.