A case study of SMEFT $\mathcal O(1/Λ^4)$ effects in diboson processes: $pp \to W^\pm(\ell^\pm ν) γ$

In this paper we explore $pp \to W^\pm (\ell^\pm ν) γ$ to $\mathcal O(1/Λ^4)$ in the SMEFT expansion. Calculations to this order are necessary to properly capture SMEFT contributions that grow with energy, as the interference between energy-enhanced SMEFT effects at $\mathcal O(1/Λ^2)$ and the Standard Model is suppressed. We find that there are several dimension eight operators that interfere with the Standard Model and lead to the same energy growth, $\sim \mathcal O(E^4/Λ^4)$, as dimension six squared. While energy-enhanced SMEFT contributions are a main focus, our calculation includes the complete set of $\mathcal O(1/Λ^4)$ SMEFT effects consistent with $U(3)^5$ flavor symmetry. Additionally, we include the decay of the $W^\pm \to \ell^\pmν$, making the calculation actually $\bar q q' \to \ell^\pm νγ$. As such, we are able to study the impact of non-resonant SMEFT operators, such as $(L^†\barσ^μτ^I\, L) (Q^†\barσ^ντ^I\, Q)\, B_{μν}$, which contribute to $\bar q q' \to \ell^\pm νγ$ directly and not to $\bar q q' \to W^\pm γ$. We show several distributions to illustrate the shape differences of the different contributions.