Subleading Sudakov logarithms in electroweak high energy processes to all orders

In future collider experiments at the TeV scale, large logarithmic corrections originating from massive boson exchange can lead to significant corrections to observable cross sections. Recently double logarithms of the Sudakov type were resummed for broken gauge theories and found to exponentiate. In this paper we use the virtual contributions to the Altarelli-Parisi splitting functions to obtain the next to leading order kernel of the infrared evolution equation in the fixed angle scattering regime at high energies where particle masses can be neglected. In this regime the virtual corrections can be described by a generalized renormalization group equation with infrared singular anomalous dimensions. The results are valid for virtual electroweak corrections to fermions and transversely polarized vector bosons with an arbitrary number of external lines. The subleading terms are found to exponentiate as well and are related to external lines, allowing for a probabilistic interpretation in the massless limit. For Z-boson and $\gamma$ final states our approach leads to exponentiation with respect to each amplitude containing the fields of the unbroken theory. For longitudinal degrees of freedom it is shown that the equivalence theorem can be used to obtain the correct double logarithmic asymptotics. At the subleading level, logarithmic corrections to the would-be Goldstone bosons contribute which should be considered separately. Explicit comparisons with existing one loop calculations are made.