Exact SMEFT formulation and expansion to $\mathcal{O}(v^4/\Lambda^4)$

The Standard Model Effective Field Theory (SMEFT) theoretical framework is increasingly used to interpret particle physics measurements and constrain physics beyond the Standard Model. We investigate the truncation of the effective-operator expansion using the geometric formulation of the SMEFT, which allows exact solutions, up to mass-dimension eight. Using this construction, we compare the exact solution to the expansion at ${\mathcal{O}}(v^2/\Lambda^2)$, partial ${\mathcal{O}}(v^4/\Lambda^4)$ using a subset of terms with dimension-6 operators, and full ${\mathcal{O}}(v^4/\Lambda^4)$, where $v$ is the vacuum expectation value and $\Lambda$ is the scale of new physics. This comparison is performed for general values of the coefficients, and for the specific model of a heavy U(1) gauge field kinetically mixed with the Standard Model. We additionally determine the input-parameter scheme dependence at all orders in $v/\Lambda$, and show that this dependence increases at higher orders in $v/\Lambda$.