  Extensions of the Standard Model (SM) often contain new particles with masses
far above the electroweak scale. Due to the presence of a mass hierarchy,
effective field theory (EFT) is a suitable tool for the study of such
extensions. In this thesis, we provide a solution to the matching problem at
the one-loop level in the form of the Bispinor Universal One-Loop Effective
Action (BSUOLEA), which is a pre-computed expression for the effective action,
parameterised in terms of derivatives of the Lagrangian of the ultraviolet (UV)
model. The BSUOLEA is derived using functional methods and is sufficiently
general to match any renormalisable, Lorentz invariant UV model to an EFT,
containing operators of mass dimension up to and including six at the one-loop
level. This includes supersymmetric models regularised in dimensional
reduction. Since the pre-computed expression contains many operators, it is
implemented into a currently private Mathematica package, which automatically
performs the matching based on the BSUOLEA. We illustrate the use of this
package, which will be made publicly available in the near future, by applying
it to the matching of the Singlet Extended SM to the SM EFT. We then expand the
framework in order to apply it to a non-renormalisable extension of the SM
containing a vector resonance, which is assumed to arise from the confinement
of a gauge group at high energies. This model raises some interesting
theoretical questions since such a vector resonance comprises a system with
second-class constraints. We perform the constraint analysis of the model and
derive the generating functional in order to apply functional methods to the
matching. We finally match the model to the SM EFT and present the full
one-loop result including operators up to mass dimension six.
