Scale Invariance, Conformal Invariance, and the High-Energy Behavior of Scattering Amplitudes

Abstract

The constraints placed by scale invariance upon the asymptotic behavior of scattering amplitudes, in theories with no dimensional coupling constants, are discussed. It is proved that for a wide class of Larangian theories, which include all renormalizable interactions except for ${\ensuremath{\varphi}}^{3}$ coupling, scale invariance implies invariance under conformal transformations. The equations that scattering amplitudes should satisfy in theories where the breaking of scale invariance is due solely to nonzero masses are derived, under the assumption of large energies compared to the masses. These equations are derived by two methods, first by direct scale and conformal transformations of the Green's functions and second by considering the low-energy theorem for the emission of the divergence of the currents which generate scale and conformal invariance. These divergences are essentially given by the trace of the symmetric energy-momentum tensor. A low-energy theorem is proved for the emission of an energy-momentum tensor (or graviton) from an arbitrary process up to quadratic terms in the graviton momenta. The equations of scale and conformal invariance are applied to the scattering of scalars off spinors, and photons. It is argued that scale invariance leads to asymptotic behavior that is governed by fixed cuts and not by Regge poles in the angular momentum. Although the strong interactions seem to be manifestly non-scale invariant, scale and conformal invariances may prove useful in discussing asymptotic behavior in quantum electrodynamics, in model field theories, and in high-energy inelastic electroproduction.