Supported particle mixing
Mixing between gauge eigenstates to new mass eigenstate appears not only in the gauge but also in the matter sector. In general the mixing is induced via bilinear terms in the Lagrangian between gauge eigenstates. These bilinear terms can either be a consequence of gauge symmetry breaking or they can correspond to bilinear superpotential- or soft-terms. In general, four kinds of bilinear terms can show up in the matter part of the Lagrangian:
\mathfrak{L}= - m^{ij}_C \phi_i^\* \phi_j - \frac{1}{2} m^{ij}_R \varphi_i \varphi_j - \frac{1}{2} m^{ij}_M \Psi^0_i \Psi^0_j - m_D^{ij} \Psi^1_i \Psi^2_j
Here,ϕ,φ,Ψx (x=0,1,2) are vectors whose components are gauge eigenstates.ϕ are complex andφ are real scalars,Ψ0,Ψ1 andΨ2 are Weyl spinors. The rotation of complex scalarsϕ to mass eigenstatesϕ̄ happens via an unitary matrixU which diagonalizes the matrixmC. For real scalars the rotation is done via a real matrixZ which diagonalizesmR:
\bar \phi = U \phi \hspace{1cm} M_C^{diag} = U m_C U^\dagger \\ \bar \varphi = Z \varphi \hspace{1cm} M_R^{diag} = Z m_R Z^T
We have to distinguish for fermions if the bilinear terms are symmetric or not. In the symmetric case the gauge eigenstates are rotated to Majorana fermions. The mass matrixmM is then diagonalized by one unitary matrix.
\bar \Psi^0 = N \Psi^0 \\ M_M^{diag} = N^\* m_M N^{-1}
In the second case, two unitary matrices are needed to transformΨ1 andΨ2 differently. This results in Dirac fermions. Both matrices together diagonalize the mass matrixmD
\bar \Psi^1 = V \Psi^1\,,\quad \bar \Psi^2 = U \Psi^2 \\ M_D^{diag} = U^\* m_D V^{-1}
Alternatively, one can use the relations
{M^2_D}^{diag} = U m_D m^T_D U^{-1} \\ {M^2_D}^{diag} = V m_D^T m_D V^{-1}
to diagonalise the mass matrix squared.
There is no restriction in SARAH how many states do mix. The most extreme case is the one with spontaneous charge, colour and CP violation where all fermions, scalars and vector bosons mix among each other. This results in a huge mass matrix which would be derived by SARAH. Phenomenological more relevant models can still have a neutralino sector mixing seven to ten states. That’s done without any problem with SARAH. Information about the calculation of the mass matrices in SARAH are given in Tree_Masses.