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:

\mathcal{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, \phi, \varphi, \Psi^x (x=0,1,2) are vectors whose components are gauge eigenstates. \phi are complex and \varphi are real scalars, \Psi^0, \Psi^1, and \Psi^2 are Weyl spinors. The rotation of complex scalars \phi to mass eigenstates \bar{\phi} happens via an unitary matrix U which diagonalizes the matrix m_C. For real scalars the rotation is done via a real matrix Z which diagonalizes m_R:

\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 matrix m_M 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 transform \Psi^1 and \Psi^2 differently. This results in Dirac fermions. Both matrices together diagonalize the mass matrixm_D

\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.