Superpartners of Gauge and Higgs Bosons

The chargino and neutralino masses and their mixing angles (that is, their gaugino and Higgsino composition) are determined by the SM gauge boson masses ($M_\mathrm{W}$ and $M_\mathrm{Z}$), $\tan\beta$, two soft SUSY-breaking parameters (the ${\bf SU(2)_L}$ gaugino mass $M_2$ and the ${\bf U(1)_Y}$ gaugino mass $M_1$), and the Higgsino mass parameter $\mu$, all evaluated at the electroweak scale $\sim M_\mathrm{Z}$. PYTHIA assumes the input parameters are evaluated at the ``correct'' scale. Obviously, more care is needed to set precise experimental limits or to make a connection to higher-order calculations. Explicit solutions of the chargino and neutralino masses and their mixing angles (which appear in Feynman rules) are found by diagonalizing the $2\times 2$ chargino $\mathbf{M_C}$ and $4\times 4$ neutralino $\mathbf{M_N}$ mass matrices:

$$\mathbf{M_{C}} = \left( \begin{array}{cc} M_2 & \sqrt{2}M_\mathrm{W}... ...hbf{M}_i & \mathbf{Z} \\ \mathbf{Z^T} & \mathbf{M}_\mu \\ \end{array} \right)$$ (147)

$$\mathbf{M}_i = \left( \begin{array}{cc} M_1 & 0 \\ 0 & M_2 \\ \end... ..._W \\ M_\mathrm{Z}c\beta c_W & -M_\mathrm{Z}s\beta c_W \\ \end{array} \right)$$

$\mathbf{M_C}$ is written in the $\widetilde W^+-\widetilde H^+$ basis, $\mathbf{M_N}$ in the $\widetilde B-\widetilde W^3-\widetilde H_1-\widetilde H_2$ basis, with the notation $s\beta=\sin\beta,c\beta=\cos\beta,s_W=\sin\theta_W$ and $c_W=\cos\theta_W$. Different sign conventions and bases sometimes appear in the literature. In particular, PYTHIA agrees with the ISASUSY [Bae93] convention for $\mu$, but uses a different basis of fields and has different-looking mixing matrices. In general, the soft SUSY-breaking parameters can be complex valued, resulting in CP violation. Presently, the consequences of arbitrary phases are only considered in the chargino and neutralino sector, though it is well known that they can have a significant impact on the Higgs sector. A generalization of the Higgs sector is among the plans for the future development of the program. The chargino and neutralino input parameters are RMSS(5) ($\tan\beta$), RMSS(1) (the modulus of $M_1$) and RMSS(30) (the phase of $M_1$), RMSS(2) and RMSS(31) (the modulus and phase of $M_2$), and RMSS(4) and RMSS(33) (the modulus and phase of $\mu$). To simulate the case of real parameters (which is CP-conserving), the phases are zeroed by default. In addition, the moduli parameters can be signed, to make a simpler connection to the CP-conserving case. (For example, RMSS(5)=-100.0 and RMSS(30)=0.0 represents $\mu=-100$ GeV.)

The expressions for the production cross sections and decay widths of neutralino and chargino pairs contain the phase dependence, but ignore possible effects of the phases in the sfermion masses. The production cross sections have been updated to include the dependence on beam polarization through the parameters PARJ(151,152) (see Sect. ). There are several approximations made for three-body decays. The numerical expressions for three-body decay widths ignore the effects of finite fermion masses in the matrix element, but include them in the phase space. No three-body decays $\chi_i^0\to \t\bar \t\chi^0_j$ are simulated, nor $\chi^+_i (\chi^0_i) \to \t\bar \b\chi^0_j (\chi^-_j)$. Finally, the effects of mixing between the third generation interaction and mass eigenstates for sfermions is ignored, except that the physical sfermion masses are used. The kinematic distributions of the decay products are spin-averaged, but include the correct matrix-element weighting. Note that for the $R$-parity violating decays (see below), both sfermion mixing effects and masses of $b$, $t$, and $\tau$ are fully included.

Since the $SU(3)_C$ symmetry of the SM is not broken, the gluinos have masses determined by the $SU(3)_C$ gaugino mass parameter $M_3$, input through the parameter RMSS(3). The physical gluino mass is shifted from the value of the gluino mass parameter $M_3$ because of radiative corrections. As a result, there is an indirect dependence on the squark masses. Nonetheless, it is sometimes convenient to input the physical gluino mass, assuming that there is some choice of $M_3$ which would be shifted to this value. This can be accomplished through the input parameter IMSS(3). A phase for the gluino mass can be set using RMSS(32), and this can influence the gluino decay width (but no effect is included in the $\tilde \mathrm{g}+\widetilde \chi$ production). Three-body decays of the gluino to $\t\bar\t$ and $\b\bar\b$ and $\t\bar \b$ plus the appropriate neutralino or chargino are allowed and include the full effects of sfermion mixing. However, they do not include the effects of phases arising from complex neutralino or chargino parameters.

The neutralinos and the gluinos are Majorana particles, and do not distinguish between states and their charged conjugate. Signatures of liked-sign lepton pairs, for example, are possible.

There is one exception to the above discussion about the input parameters to the neutralino and chargino mass matrices. In the case when $M_2$ is much smaller than other mass parameters (as occurs in models of anomaly-mediated SUSY breaking), radiative corrections are very important in keeping the lightest neutralino lighter than the lightest chargino. If ever the opposite occurs in solving the eigenvalue problem numerically, the chargino mass is set to the neutralino mass plus 2 times the charged pion mass.