The landscape in the search for new physics is shifting from dedicated searches to a focus on finding small deviations from the Standard Model (SM) via indirect methods. To this end, there is a need for our knowledge of SM backgrounds to be as precise as possible, through precise knowledge of partonic scattering amplitudes, proton structure and SM parameters. Alongside this, the frameworks that are available for characterising deviations, such as the SM Effective Field Theory (SMEFT), must be utilised effectively such that in the future it would be possible to confidently characterise a deviation that may be due to new physics.

In this thesis we will present work that is a part of both of these complementary programmes. In the field of SM proton structure we will present work that aims to increase both the accuracy and precision of the parton distribution functions (PDFs) of the proton, and thereby increase the accuracy and precision of Large Hadron Collider (LHC) cross section predictions. This work takes the form of projects that seek to systematically include (i) scale uncertainties and (ii) data from single top production in PDFs for the first time. While, on the topic of improving our ability to characterise new physics, we will present work that assesses the importance of accounting for new physics effects in PDF determinations when constraining the very same new physics contributions with the SMEFT. This work is an important step in working towards the development of a new framework that can be used to accurately constrain new physics at the LHC via statistical analyses that treat all ingredients entering theoretical predictions consistently.