Ultraviolet Complete Quantum Field Theory and Gauge Invariance

The problem of gauge invariance in an ultraviolet complete quantum field theory (QFT) with nonlocal interactions is investigated. For local fields that couple through a nonlocal interaction, it is demonstrated that the quantum electrodynamic (QED) sector of the electroweak (EW) model without a Higgs particle is gauge invariant. The non-Abelian QFT with massless gluons and nonlocal interactions is shown to be gauge invariant, and a perturbative S-matrix formalism in the interaction representation is investigated. The finite self-energy of a fermion and vacuum polarization are derived to first order. The K\"all\'en-Lehmann representation for the finite QED theory for local Heisenberg field operators is considered and the Coulomb potential between two charges is shown to be non-singular at $r=0$. The high energy limit of electromagnetic form factors is investigated in relation to the finite charge renormalization constant $Z_3^{-1}$. The magnitudes of the constants $Z_1,Z_2$ and $Z_3$ for charge and mass renormalization are derived without divergences and shown to be non-perturbatively finite. The proton-proton collision cross section production of lepton pairs through the Drell-Yan process is investigated and the prediction for the cross section at high energies can be tested at the LHC.