  The standard model effective field theory (SMEFT) provides systematic
parameterization of all possible new physics above the electroweak scale.
According to the amplitude-operator correspondence, an effective operator can
be decomposed into a linear combination of several j-basis operators, which
correspond to local amplitudes carrying certain spin and gauge quantum numbers
in a particular scattering channel. Based on the Poincare and gauge symmetries
of scattering amplitude, we construct the j-basis using the Casimir method for
both the Lorentz and gauge sectors. The quantum numbers of the j-basis
operators fix the quantum numbers of any intermediate state in the
corresponding amplitudes, such as a UV resonance. This can be re-interpreted as
the j-basis/UV correspondence, thus obtaining the j-bases in all partitions of
fields for an operator amounts to finding all of its UV origins at tree level,
constituting the central part of the bottom-up EFT framework. Applying the
j-basis analysis to SMEFT, we obtain a complete list of possible tree-level UV
origins of the effective operators at the dimension 5, 6, 7, and all the
bosonic operators at the dimension 8.
