Asymptotic behaviour of total cross sections in quantum field theory

We examine the issue of high energy scattering in generic d-dimensional quantum field theories admitting an ultraviolet fixed point. We find that the total cross section behaves asymptotically as s1−d/2(a + b/sδ), with δ ≥ (d − X)/2, X being the scaling dimension of the least relevant operator responsible for the renormalisation group flow. PACS numbers: 11.10.Jj The set of ideas which commonly goes under the name of renormalisation group (RG) provided us with a new global insight into quantum field theory. It shows that a quantum field theory is associated with a trajectory in a space of coupling constants, different points along the trajectory being related by a sequence of RG transformations and corresponding to the description of the theory at different distance scales [1]. In the extreme ultraviolet limit in which distances are infinitely smaller than any intrinsic length scale, the theory is expected to become scale invariant, and the corresponding trajectory in the coupling constant space to end into a fixed point of the RG. Quite generally, this description can be formalised by writing the action of the theory in d dimensions in the form A = AFP + ∑