As a simple example, consider a U(3)⁵ flavor symmetry acting on the chiral fermions in the unbroken electroweak phase of the SMEFT.

First, we have to load the Warsaw basis Lagrangian and define global symmetry groups for the flavor symmetries:

Where we identified U(3)~SU(3)U(1) since Matchete only allows for defining simple Lie groups. Note that for U(3) in the SMEFT there is no difference between using U(3) or SU(3). However, the U(1) factor become important, for example, when considering U(2) since the fundamental representation of SU(2) is pseudo real and therefore allows for an anti-symmetric singlet contraction of two fundamentals, which is however forbidden by the U(1) factors.

Then we can define the U(3)⁵ flavor symmetry as an association by

where the keys are the field labels and the values are lists of rules giving the transformation properties. The left-hand side of these rules are lists of flavors and the right-hand side contain the representation under which these flavors transform. In this example, for all fields, all flavors transform under the same representation, but specifying different transformation properties for different flavors is possible. The only requirement is that every flavor can at most transform under one non-Abelian representation at a time, while an arbitrary number of U(1) factors is allowed.

Note that the list of representations provided on the right-hand side can also be replaced by TensorProduct. For example: {SU3d[fund],U1d[1]} can be replaced by SU3d[fund]U1d[1].

We can the assign this flavor symmetry to the Wilson coefficients of the Warsaw basis:

Note that the values in this association are lists of lists. The reason is that all invariants are either self-Hermitian or belong to a manifestly complex coefficient (the chirality flipping ones). In more complicated scenarios (for example when including spurions) non-self-Hermitian structures can be found which are then grouped together with their conjugate at the inner most level of the lists.
[See the documentation of FlavorInvariantCoupling for more details.]

For example, we find the Yukawas to be forbidden by the symmetry:

While for C_ℓℓ we find the two expected invariant structures

If flavorStructures is provided to DefaultParamCard or ExportUFO through the option FlavorInvariants it will incorporate a parametrization of the couplings following these flavor invariant structures and setting all elements that are not populated in the invariants to "ZERO", i.e., dropping them entirely from the UFO files.

A mixed U(3)³U(2)² can be defined similarly using:

Here, we have defined the global groups not used before. Note that the specification of Singlet as done for the third generation of the u field above is optional. If a flavor does not have a given transformation property it is automatically assumed to be a Singlet, as done for the third generation of the q field above.

The symmetry can be assigned similarly to the Lagrangian:

and we find, for example,

Note that the Lagrangian given to ImposeFlavorSymmetry must contain the fields for which the symmetry was defined. For example, in the SM(EFT) flavor symmetries are conventionally defined for the chiral fermions before electroweak symmetry breaking. Therefore, also the Lagrangian of the unbroken phase must be provided to ImposeFlavorSymmetry.

Note also that for the automatic identification to work, the flavor indices of the couplings must be directly contracted into fields (as it is usually the case in the unbroken phase) and not into other couplings.