Renormalization of determinant lines in quantum field theory

On a compact manifold $M$, we consider the affine space $A$ of non self-adjoint perturbations some invertible elliptic operator acting on sections Hermitian bundle, by differential lower order. We construct and classify all complex analytic functions Fr\'echet vanishing exactly over elements, having minimal order which are obtained local renormalizations, concept coming from quantum field theory, called renormalized determinants. The additive group polynomial functionals finite degrees acts freely transitively provide different representations determinants in terms spectral zeta determinants, Gaussian Free Fields, infinite product Feynman amplitudes perturbation theory position \`a la Epstein-Glaser. Specializing to case Dirac operators coupled vector potentials reformulating our results determinant line bundles, prove define trivializations holomorphic bundle relating conjectural picture unpublished notes Quillen [52] April 1989.