Top-down and bottom-up: Studying the SMEFT beyond leading order in $1/\Lambda^2$

In order to assess the relevance of higher order terms in the Standard Model effective field theory (SMEFT) expansion we consider four new physics models and their impact on the Drell Yan cross section. Of these four, one scalar model has no effect on Drell Yan, a model of fermions while appearing to generate a momentum expansion actually belongs to the vacuum expectation value expansion and so has a nominal effect on the process. The remaining two, a leptoquark and a $Z'$ model exhibit a momentum expansion. After matching these models to dimension-ten we study the how the inclusion of dimension-eight and dimension-ten operators in hypothetical effective field theory fits to the full ultraviolet models impacts fits. We do this both in the top-down approach, and in a very limited approximation to the bottom up approach of the SMEFT to infer the impact of a fully general fit to the SMEFT. We find that for the more weakly coupled models a strictly dimension-six fit is sufficient. In contrast when stronger interactions or lighter masses are considered the inclusion of dimension-eight operators becomes necessary. However, their Wilson coefficients perform the role of nuisance parameters with best fit values which can differ statistically from the theory prediction. In the most strongly coupled theories considered (which are already ruled out by data) the inclusion of dimension-ten operators allows for the measurement of dimension-eight operator coefficients consistent with theory predictions and the dimension-ten operator coefficients then behave as nuisance parameters. We also study the impact of the inclusion of partial next order results, such as dimension-six squared contributions, and find that in some cases they improve the convergence of the series while in others they hinder it.