  The gauge-invariant operators up to dimension six in the low-energy effective
field theory below the electroweak scale are classified. There are 70 Hermitian
dimension-five and 3631 Hermitian dimension-six operators that conserve baryon
and lepton number, as well as $\Delta B= \pm \Delta L = \pm 1$, $\Delta L=\pm
2$, and $\Delta L=\pm 4$ operators. The matching onto these operators from the
Standard Model Effective Field Theory (SMEFT) up to order $1/\Lambda^2$ is
computed at tree level. SMEFT imposes constraints on the coefficients of the
low-energy effective theory, which can be checked experimentally to determine
whether the electroweak gauge symmetry is broken by a single fundamental scalar
doublet as in SMEFT. Our results, when combined with the one-loop anomalous
dimensions of the low-energy theory and the one-loop anomalous dimensions of
SMEFT, allow one to compute the low-energy implications of new physics to
leading-log accuracy, and combine them consistently with high-energy LHC
constraints.
