Ultraviolet Complete Quantum Field Theory and Gauge Invariance

Abstract: 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.