We define a new flavor index with DefineFlavorIndex:

The default behavior of flavor indices of couplings is that they respect Einstein summation convention. An ordinary two-index coupling

can, thus, freely have repeated indices relabeled (with RelabelIndices)

Following these rules triple-repeated indices are invalid, e.g.,

In some cases, particularly heavy masses, it is convenient to assume a diagonal structure for a coupling (meaning e.g. a mass matrix has elements only along the diagonal). In such cases it is convenient to use a single index instead of two – it is even required for heavy masses when using Match. Setting a flavor index to be diagonal with the DiagonalCoupling option means that the index does not count in Einstein summation convention. For instance, the mass parameter

If diagonal indices appear with two ordinary repeated indices, they are included in the sum and relabeled simmultaneously:

Note that diagonality of coupling indices may not be stable under RG flow of the theory unless it is protected by some (accidental) symmetry.

In advanced situation a coupling can have multiple indices, some of which are diagonal and some which are not. Consider

with two indices, the first of which is diagonal. In some sense we might think of this as a rank-three tensor that is diagonal under the first two indices. In summation situations we have, e.g.,

Thus there is only a sum over the second index, which is non-diagonal and thus repeated for summation purposes.

The option EFTOrder dictates the power-counting behavior of the coupling. A coupling

with EFTOrder 4, vanishes when truncating the EFT series at dimension 3:

Couplings can have flavor indices in theories where such have been defined. Using DefineFlavorIndex to initialize a flavor representation with dimension 3,

we may define a coupling with indices in that space. For instance, the SM lepton Yukawa coupling matrix is given by

The resulting coupling has two flavor indices, which means it must be called with two arguments in its definition:

The option NiceForm can be provided to the coupling definition to change how it displays under NiceForm output. The default is to use the label of the coupling, but it can be changed by providing a string. For instance to display the coupling with a subscript, we would use

Couplings are by default assumed to be complex; however, in many cases the form of the Lagrangian may require that they should be real. In such cases, it is important to specify this, so that the simplification algorithms can work as well as possible. To do this, we may use the option SelfConjugate. To illustrate the behavior of real and complex couplings consider

Complex conjugation with Bar gives

demonstrating that Bar acts non-trivially only on the complex coupling.

If a coupling is a tensor in flavor space, complex conjugation can also have the effect of exchanging the indices of the coupling. For instance, we may define a Hermitian coupling matrix (complex conjugation exchanges the first and second indices)

In that event

When couplings have multiple indices of the same type, it might be (anti-)symmetric under permutation of the indices. Including this information allows the simplification methods to perform better.

An example of such symmetries are from the SMEFT operator C_ee, which has four flavor indices. It is symmetric under exchanges of the two first and two last indices in addition to under the exchange of indices 2 and 4:

We can verify that the coupling is in fact initialized with symmetries under the full symmetry group generated by the elements we provided. In this case we also get symmetry under the exchange of indices 1 and 3: