The discovery of the Higgs boson in 2012 at the Large Hadron Collider completed the Standard
Model field content. Many questions though remain unanswered by the Standard Model triggering
a search for new physics. New physics could manifest itself at the Large Hadron Collider by the
discovery of new particles. However, the lack of new resonances might suggest that these new
particles are still out of reach which leaves us with few options. Two possibilities are explored
in this thesis. The first is to study precision measurements which might indicate new physics as
small deviations from the Standard Model predictions. The second is to construct an ultraviolet
completion of the Standard Model which allows for a naturally light Higgs while maintaining a
sizable hierarchy between the Higgs mass and the cutoff scale of the theory.
A general approach to parametrize small deviations from the Standard Model is by the use
of so-called effective operators. We consider all operators of higher dimensions made of the
Standard Model’s fields while respecting its symmetries. The Standard Model effective field
theory is obtained by adding these operators to the Standard Model. In this thesis, we use
electroweak precision data to set constraints on a subset of the parameters of the Standard Model
effective field theory. We give detailed explanations on the calculation of the Standard Model
effective field theory predictions for the observables. Furthermore, we develop a novel statistical
method to fit for the parameters of the Standard Model effective field theory which accounts for
the missing higher orders of these parameters in the theoretical predictions of the observables.
The statistical method developed is not limited to effective field theories, it can be applied to any
theory where predictions are expressed as power series with missing higher order terms. We also
show how to connect ultraviolet models of new physics to the Standard Model effective field
theory and calculate bounds on them using the Standard Model effective field theory fit results.
Finally, we study a nonrelativistic ultraviolet completion of the Standard Model which restores
technical naturalness while keeping a hierarchy between the Higgs mass and the cutoff of the
Standard Model. More precisely, the theory allows for a naturally light Higgs while preserving
a hierarchy of seven orders of magnitude and reasonable magnitudes for the gauge coupling,
the Yukawa couplings and the Higgs quartic self-interaction. Achieving such a hierarchy while
maintaining the rest of the parameters at a reasonable order necessitates a cancelation of the
gauge field one-loop corrections to the squared Higgs mass. We check by explicit calculations
that these one-loop corrections indeed cancel and show furthermore that the gauge field one-loop
corrections to the squared Higgs speed of light cannot be canceled which prevents us from
improving the hierarchy further.