Fine structure in non-Abelian gauge theories

Abstract

Fine-structure effects of purely non-Abelian origin are isolated and evaluated for quark-antiquark bound states in a general massless non-Abelian gauge theory. The relevant kernels are identified by studying the threshold power singularities of the quark-antiquark scattering amplitude. The non-Abelian fine structure is shown to consist of (a) an $O\left({{\alpha }_s}^3\right)$ contribution ascribable to the (spatial) momentum dependence of the effective Coulomb-gluon—quark coupling, which is larger than the $O\left({{\alpha }_s}^2\right)$ Coulomb energy for a wide range of ${\alpha }_s$, and (b) $O\left({{\alpha }_s}^4\right)$ contributions comparable to the usual Abelian fine structure. A way is suggested of improving the threshold expansion to avoid the breakdown associated with logarithmic singularities piling up in higher order: In this approach the spin-independent fine structure is reasonably given by the Breit-Fermi interaction with a phenomenological potential, taken together with the purely non-Abelian contributions computed previously. The Breit-Fermi formula cannot be taken seriously for the hyperfine structure, however: Transverse- and Coulomb-gluon exchanges seem to be associated with different effective couplings. In addition, there are Pauli-type contributions to the transverse-gluon—quark vertex which involve logarithmic singularities. The threshold expansion is expected to be directly and unambiguously applicable, however, for threshold bound states of sufficiently heavy quarks (say, $m_q\gtrsim 15$ GeV), should such exist.