  This work presents some results about Wick polynomials of a vector field
renormalization in locally covariant algebraic quantum field theory in curved
spacetime. General vector fields are pictured as sections of natural vector
bundles over globally hyperbolic spacetimes and quantized through the known
functorial machinery in terms of local $^*$-algebras. These quantized fields
may be defined on spacetimes with given classical background fields, also
sections of natural vector bundles, in addition to the Lorentzian metric. The
mass and the coupling constants are in particular viewed as background fields.
Wick powers of the quantized vector field are axiomatically defined imposing in
particular local covariance, scaling properties and smooth dependence on smooth
perturbation of the background fields. A general classification theorem is
established for finite renormalization terms (or counterterms) arising when
comparing different solutions satisfying the defining axioms of Wick powers.
The result is specialized to the case of general tensor fields. In particular,
the case of a vector Klein-Gordon field and the case of a scalar field
renormalized together with its derivatives are discussed as examples. In each
case, a more precise statement about the structure of the counterterms is
proved. The finite renormalization terms turn out to be finite-order
polynomials tensorially and locally constructed with the backgrounds fields and
their covariant derivatives whose coefficients are locally smooth functions of
polynomial scalar invariants constructed from the so-called marginal subset of
the background fields. The notion of local smooth dependence on polynomial
scalar invariants is made precise in the text.
