Light-Cone Operator Expansions in Perturbation Theory. II

Abstract

A systematic investigation, in perturbation theory, is presented of the light-cone behavior of multiparticle matrix elements of time-ordered products of local fields: $\ensuremath{\int}{d}^{4}x{e}^{\mathrm{idx}}〈\ensuremath{\beta}|T{\ensuremath{\psi}}_{1}(x){\ensuremath{\psi}}_{2}(0)|\ensuremath{\alpha}〉$. In the limit ${q}_{\ensuremath{-}}\ensuremath{\rightarrow}\ensuremath{\infty}$, the contribution of any single Feynman graph is of the form ${{q}_{\ensuremath{-}}}^{\ensuremath{\beta}}{(\mathrm{ln}{q}_{\ensuremath{-}})}^{\ensuremath{\gamma}}$. The main result here is a rule by means of which the integers $\ensuremath{\beta}$ and $\ensuremath{\gamma}$ can be read off from the topology of the graph. The implications of this investigation for local field theories are organized and discussed in operator language in a companion paper. A by-product of the special methods here developed to obtain asymptotic estimates in perturbation theory is a refinement of Weinberg's theorem for the Euclidean region: the determination of the logarithmic factors in the asymptotic form of Feynman amplitudes when a set of external momenta ${q}_{1}, {q}_{2}, \dots{}$ is allowed to approach infinity according to ${q}_{i}=\ensuremath{\eta}{{q}^{\ensuremath{'}}}_{i}, \ensuremath{\eta}\ensuremath{\rightarrow}\ensuremath{\infty}$.