High-Energy Behavior in Field Theory and Dilatational Symmetry

The high-energy forward-scattering amplitude in $\lambda {\phi }^4$ 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 $\frac{E^{n_0}}{{\left(\ln E\right)}^c}$. This term is caused by a branch cut in a complex four-dimensional Euclidean angular-momentum plane. A lower bound for $n_0$ is obtained from a simple diagram. Using the upper bound on $n_0$ which results from unitarity, an upper bound on the coupling constant is obtained: $\lambda <\left(3\mathrm{}\surd 8\right){\pi }^2$.