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

In this paper we explore $pp \to W^\pm (\ell^\pm \nu) \gamma$ to $\mathcal O(1/\Lambda^4)$ 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 $\mathcal O(1/\Lambda^2)$ 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, $\sim \mathcal O(E^4/\Lambda^4)$, as dimension six squared. While energy-enhanced SMEFT contributions are a main focus, our calculation includes the complete set of $\mathcal O(1/\Lambda^4)$ SMEFT effects consistent with $U(3)^5$ flavor symmetry. Additionally, we include the decay of the $W^\pm \to \ell^\pm\nu$, making the calculation actually $\bar q q' \to \ell^\pm \nu \gamma$. As such, we are able to study the impact of non-resonant SMEFT operators, such as $(L^\dag\bar\sigma^\mu \tau^I\, L) (Q^\dag\bar\sigma^\nu \tau^I\, Q)\, B_{\mu\nu}$, which contribute to $\bar q q' \to \ell^\pm \nu \gamma$ directly and not to $\bar q q' \to W^\pm \gamma$. We show several distributions to illustrate the shape differences of the different contributions.