Handling of non-fundamental representations

In the handling of non-fundamental fields under a symmetry, SARAH distinguishes if the corresponding symmetry gets broken or not: for unbroken symmetries it is convenient to work with fields which transform as vector under the symmetry with the appropriate length. For instance, a 6 under SU(3)_c is taken to be

\phi_\alpha \hspace{1cm} \alpha=1,2, \dots 6

I.e. it carries one charge index. In contrast, non-fundamental fields under a broken gauge symmetry are represented by tensor products of the fundamental representation. For instance, a 3 under S**U(2)_L is taken to be

\phi_{ab} \hspace{1cm} a,b=1,2

Thus, the triplet can be given as usual as 2 × 2 matrix. For Abelian gauge groups one can not only define charges for superfields which are real numbers, but also variables can be used for that. All interactions are then expressed keeping these charges as free parameter. For all chiral superfield SARAH adds the soft-breaking masses. For fields appearing in N generations, these are treated as hermitian N × N matrices. As written above, also soft-terms mixing two scalars are included if allowed by all symmetries. Hence, the soft-breaking mass terms read in general

𝔏_S**B, ϕ = ∑_i**jδ̃_i**jϕ_i^†m_i**j²ϕ_j + h.c.

Note, i, j label different scalar fields, generation indices are not shown. δ̃_i**j is 1, if fields ϕ_i and ϕ_j have exactly the same transformation properties under all local and global symmetries, and otherwise 0.

See also