A case study of SMEFT \( \mathcal{O}\left(1/{\Lambda}^4\right) \) effects in diboson processes: pp → W^±(ℓ^±ν)γ

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