Superpartners of Standard Model Fermions

The mass eigenstates of squarks and sleptons are, in principle, mixtures of their left- and right-handed components, given by:

$$M_{\tilde{f}_L}^2 = M_{\bf 2}^2 + m^2_f + D_{\tilde{f}_L} M_{\tilde{f}_R}^2 = M_{\bf 1}^2 + m^2_f + D_{\tilde{f}_R}$$ (148)

where $M_{\bf 2}$ are soft SUSY-breaking parameters for superpartners of ${\bf SU(2)_L}$ doublets, and $M_{\bf 1}$ are parameters for singlets. The $D$-terms associated with Electroweak symmetry breaking obey $D_{\tilde{f}_L}= M_Z^2 \cos (2 \beta) (T_{3_{f}} - Q_f \sin^2\theta_W)$ and $D_{\tilde{f}_R}= M_Z^2 \cos( 2 \beta) Q_f\sin^2\theta_W$, where $T_{3_f}$ is the weak isospin eigenvalue of the fermion and $Q_f$ is the electric charge.

In most high-energy models, the soft SUSY-breaking sfermion mass parameters are taken to be equal at the high-energy scale, but, in principle, they can be different for each generation or even within a generation. However, the sfermion flavor dependence can have important effects on low-energy observables, and it is often strongly constrained. The suppression of flavor changing neutral currents (FCNC's), such as $K_L\to\pi^\circ\nu\bar\nu$, requires that either $(i)$ the squark soft SUSY-breaking mass matrix is diagonal and degenerate, or $(ii)$ the masses of the first- and second-generation sfermions are very large.

The left-right sfermion mixing is determined by the product of soft SUSY-breaking parameters and the mass of the corresponding fermion. Unless the soft SUSY-breaking parameters for the first two generations are orders of magnitude greater than for the third generation, the mixing in the first two generations can be neglected and $\tilde \mathrm{q}_{L,R}$, with $\tilde \mathrm{q}= \tilde \u ,\tilde \d ,\tilde \c ,\tilde \mathrm{s}$, and $\tilde\ell _{L,R},\tilde \nu _\ell$, with $\ell = \mathrm{e}, \mu$, are the real mass eigenstates with masses $m_{\tilde \mathrm{q}_{L,R}}$ and $m_{\tilde\ell _{L,R}}, m_{\tilde \nu _{\ell}},$ respectively. For the third generation sfermions, the left-right mixing can be nontrivial. The mass matrix for the top squarks (stops) in the ( $\tilde \mathrm{t}_L, \tilde \mathrm{t}_R$) basis is given by

$$M^2_{\tilde \mathrm{t}} = \left( \begin{array}{cc} m_{Q_3}^... ... + m_\t ^2 + D_{\tilde \mathrm{t}_R} \\ \end{array} \right),$$ (149)

where $A_\t$ is a trilinear coupling. Different sign conventions for $A_\t$ occur in the literature; PYTHIA and ISASUSY use opposite signs. Unless there is a cancellation between $A_\t$ and $\mu/\tan\beta$, left-right mixing occurs for the stop squarks because of the large top quark mass. The stop mass eigenstates are then given by

$$\tilde \mathrm{t}_1 = \cos \theta_{\tilde \mathrm{t}} \;\;\tilde \mathrm{t}_L + \sin \theta_{\tilde \mathrm{t}} \;\; \tilde \mathrm{t}_R$$

$$\tilde \mathrm{t}_2 = - \sin \theta_{\tilde \mathrm{t}} \;\;\tilde \mathrm{t}_L + \cos \theta_{\tilde \mathrm{t}} \;\; \tilde \mathrm{t}_R,$$ (150)

where the masses and mixing angle $\theta_{\tilde \mathrm{t}}$ are fixed by diagonalizing the squared-mass matrix Eq. (). Note that different conventions exist also for the mixing angle $\theta_{\tilde \mathrm{t}}$, and that PYTHIA here agrees with the ISASUSY. When translating Feynman rules from the (L,R) to (1,2) basis, we use:

$$\tilde \mathrm{t}_L = \cos \theta_{\tilde \mathrm{t}} \;\;\tilde \mathrm{t}_1 - \sin \theta_{\tilde \mathrm{t}} \;\; \tilde \mathrm{t}_2$$

$$\tilde \mathrm{t}_R = \sin \theta_{\tilde \mathrm{t}} \;\;\tilde \mathrm{t}_1 + \cos \theta_{\tilde \mathrm{t}} \;\; \tilde \mathrm{t}_2.$$ (151)

Because of the large mixing, the lightest stop $\tilde \mathrm{t}_1$ can be one of the lightest sparticles. For the sbottom, an analogous formula for the mass matrix holds with $m_{U_3}\to m_{D_3}$, $A_\t\to A_\b$, $D_{\tilde \mathrm{t}_{L,R}}\to D_{\tilde \mathrm{b}_{L,R}}$, $m_\t\to m_\b$, and $\tan\beta\to 1/\tan\beta$. For the stau, substitute $m_{Q_3}\to m_{L_3}$, $m_{U_3}\to m_{E_3}$, $A_\t\to A_\tau$, $D_{\tilde \mathrm{t}_{L,R}}\to D_{\tilde\tau _{L,R}}$, $m_t\to m_\tau$ and $\tan\beta\to$ 1/$\tan\beta$. The parameters $A_\t$, $A_\b$, and $A_\tau$ can be independent, or they might be related by some underlying principle. When $m_\b\tan\beta$ or $m_\tau\tan\beta$ is large ( ${\cal O}(m_\t ))$, left-right mixing can also become relevant for the sbottom and stau.

Most of the SUSY input parameters are needed to specify the properties of the sfermions. As mentioned earlier, the effects of mixing between the interaction and mass eigenstates are assumed negligible for the first two generations. Furthermore, sleptons and squarks are treated slightly differently. The physical slepton masses $\tilde\ell_L$ and $\tilde\ell_R$ are set by RMSS(6) and RMSS(7). By default, the $\tilde\tau$ mixing is set by the parameters RMSS(13), RMSS(14) and RMSS(17), which represent $M_{L_3}$, $M_{E_3}$ and $A_\tau$, respectively, i.e. neither $D$-terms nor $m_\tau$ is included. However, for IMSS(8)=1, the $\tilde\tau$ masses will follow the same pattern as for the first two generations. Previously, it was assumed that the soft SUSY-breaking parameters associated with the stau included $D$-terms. This is no longer the case, and is more consistent with the treatment of the stop and sbottom. For the first two generations of squarks, the parameters RMSS(8) and RMSS(9) are the mass parameters $M_{\bf 2}$ and $M_{\bf 1}$, i.e. without $D$-terms included. For more generality, the choice IMSS(9)=1 means that $M_{\bf 1}$ for $\tilde \u _R$ is set instead by RMSS(22), while $M_{\bf 1}$ for $\tilde \d _R$ is RMSS(9). Note that the left-handed squark mass parameters must have the same value since they reside in the same ${\bf SU(2)_L}$ doublet. For the third generation, the parameters RMSS(10), RMSS(11), RMSS(12), RMSS(15) and RMSS(16) represent $M_{Q_3}$, $M_{D_3}$, $M_{U_3}$, $A_{\b }$ and $A_{\t }$, respectively.

There is added flexibility in the treatment of stops, sbottoms and staus. With the flag IMSS(5)=1, the properties of the third generation sparticles can be specified by their mixing angle and mass eigenvalues (instead of being derived from the soft SUSY-breaking parameters). The parameters RMSS(26) - RMSS(28) specify the mixing angle (in radians) for the sbottom, stop, and stau. The parameters RMSS(10) - RMSS(14) specify the two stop masses, the one sbottom mass (the other being fixed by the other parameters) and the two stau masses. Note that the masses RMSS(10) and RMSS(13) correspond to the left-left entries of the diagonalized matrices, while RMSS(11), RMSS(12) and RMSS(14) correspond to the right-right entries. Note that these entries need not be ordered in mass.