High-Energy Behavior in Field Theory and Dilatational Symmetry

The high-energy forward-scattering amplitude in λφ⁴ theory is investigated by means of the Bethe-Salpeter equation. The class of irreducible diagrams which have dilatational symmetry at high energies is considered as kernels for the Bethe-Salpeter equation. For these kernels, the equation is solved using a Mellin transform. The solution is found to contain a term which, at high energies, behaves like E^n₀(lnE)ᶜ. This term is caused by a branch cut in a complex four-dimensional Euclidean angular-momentum plane. A lower bound for n₀ is obtained from a simple diagram. Using the upper bound on n₀ which results from unitarity, an upper bound on the coupling constant is obtained: λ<(38)π².