In this example we determine the flavor invariants for a four-fermion operator such as [Q_ℓℓ]_prst=( γ_μℓ_r)( γ^μℓ_t) under the assumption of as U(2)⁵ flavor symmetry.

First, we define the symmetry as a global symmetry group, by specifying the simple Lie Groups, i.e., we have to identify U(2) ~ SU(2)U(1) since Matchete does not allow to directly define a U(2) group.

We can then provide the flavor symmetry as a list where the nth entry contains the flavor transformation properties of the nth index (or field). The transformation properties are given as a list of rules, where the left hand side is a list of the flavors which transform under the representation on the right. The righthand side is allowed to contain at most one non-Abelian representation per flavor and an arbitrary number of U(1) factors. The right-hand side can be provided as a list (as done below) or as a TensorProduct ({SU2l[fund],U1l[+1]} can be replaced by SU2l[fund]U1l[+1])

The flavor invariants are returned in the form of SparseArray. They can be inspected, for example, using

In the example of a U(2)⁵ flavor symmetry for [Q_ℓℓ]_prst given above, we can now also include one insertion of a spurion V_ℓ transforming in the fundamental of SU(2)

where we consider both the insertion of the spurion and its conjugate, representing the fifth index of the resulting SparseArray.

Parametrizing the spurion as V_ℓ=(0,ϵ) we find

The flavor invariants determined in the example above can now be associated to the C_ℓℓ coefficient of the corresponding Warsaw basis operator.

First, the Warsaw basis has to be loaded:

Then we can use FlavorInvariantCoupling to assign the flavor structure to the coupling:

The result can be inspected using:

Note that the symmetries of the coupling (pr↔st) have been employed to symmetrize the flavor structures obtained by FlavorInvariants.