The Gaugino/Higgsino Sector

The superpartners of the charged gauge bosons and the Higgs bosons mix and form two mass eigenstates called charginos, usually denoted $\XPMj (j=1,2)$, which are linear combinations of the $\widetilde W$ and $\widetilde H^\pm$ interaction eigenstates. The mass matrix in the $(\widetilde W^\pm,\widetilde H^\pm)$ base is given by

$$M_{\tilde \chi^\pm}= \left(\begin{array}{cc} M_2 & \sqrt{2}\... ...in \beta \\ \sqrt{2}\mW \cos \beta & - \mu \end{array}\right)$$ (7)

giving the following mass relations

$$m^2_{\small {\XPMI ,\XPMII }} = \half \Biggl\{\vert\mu\vert^2+\ve... ...rt^2+2\mW ^2\mp \Biggl[ \left(\vert\mu\vert^2+\vert M_2\vert^2+2\mW ^2\right)^2$$

$$-4\vert\mu\vert^2\vert M_2\vert^2-4\mW ^4\sin^2 2\beta +8\mW ^2\sin 2\beta\,{\rm Re}(\mu M_2) \Biggr]^{1/2}\Biggr\}$$ (8)

where by definition the $\XPMI$ is the lightest chargino. A similar mixing occurs for the neutral gauginos and higgsinos, in this case they form four mass eigenstates that are usually called neutralinos. The mass matrix in the basis $(\widetilde B,\widetilde W^3,\widetilde H^0_1 , \widetilde H^0_2)$ is

$$m_{\tilde \chi_i^0} = \left(\begin{array}{cccc} M_1 & 0& -\... ...Z s_\beta s_W & -\mZ s_\beta c_W & \mu & 0 \end{array}\right)$$ (9)

where $c_\beta$, $s_\beta$, c_W, and s_W is $\cos\beta$, $\sin\beta$, $\cos\theta_W$ and $\sin\theta_W$ respectively. The different mass eigenstates are ordered according to the convention $m_{\XN {1}}<m_{\XN {2}}<m_{\XN {3}}<m_{\XN {4}}$. Figure 11 illustrates the masses of the different neutralino mass eigenstates for different values of the MSSM parameters. From Equ. 7 and Equ. 9 one can note that the masses of the charginos and neutralinos depends on four parameters, however from the GUT relation Equ. 3, and using the renormalization group equations (RGE), one gets the following relation at the electroweak scale

$${ 3M_1(\mZ ) \over 5{\alpha_1} } = { M_2(\mZ ) \over {\alpha_2} } = { M_3(\mZ ) \over {\alpha_3} }$$ (10)

where the first equality can be approximated by

$$M_1(\mZ ) = {5 \over 3} \tan^2 \theta _W M_2(\mZ ) \simeq 0.5 M_2(\mZ )$$ (11)

Thus the number of parameters, describing masses and the mixing of the chargino and neutralino mass eigenstates can be reduced to three:M₂, $\mu$ and $\tan{\beta}$. The field composition of charginos and neutralinos are of great importance of the phenomenology, since it affects production cross-sections as well as decay branching ratios. This effect is of special importance in the case of neutralinos since the bino component does not couple to the $\z$. Charginos can be produced in the s-channel process $\eeto {}$ $\Zstar$ with the subsequent decay $\Zstar \to \XPI \XMI$ , or in the t-channel through an exchange of a sneutrino. In a similar fashion neutralinos can be produced both in the s-channel and t-channel, see Fig. 6, but in the latter case through the exchange of a selectron. An important difference between the chargino and the neutralino cases, is that while for charginos the s-channel and the t-channel contributions interfere destructively, it is usually the opposite in the case of neutralino production. As a consequence of this, the production cross-sections in both cases have a crucial dependence on the slepton masses, i.e. the m₀ parameter.

Figure: The different production processes for $\eeto \XN {i}\XN {j}$.

Since all supersymmetric particles are unstable, with the exception of the LSP in case of R-parity conservation, it is important to know their decay modes. Assuming that $\XN {1}$ is the LSP, the decay modes of $\XN {2}$ are described in detail in Ref. [21].

• $\XN {2} \to \XN {1}$ $f\bar{f}$ , f= e, $\mu$, $\tau$, $\nu$ and q
• $\XN {2} \to \XN {1} \gamma$

The first mode is the most important channel in neutralino searches at LEP2. The three-body decay into a fermion pair goes either via a $\Z$ or corresponding sfermion. This decay mode is most important if both $\XN {1}$ and $\XN {2}$ have large higgsino components. When the mass difference $m_{\XN {2}} - m_{\XN {1}}$ is close to $\mZ$, the decay is mediated by a real $\Z$ and the branching ratios are the same as in the case of $\Z$ decays, but for low slepton masses the leptonic branching ratio can be considerable larger. The radiative decay, $\XN {2} \to \XN {1} \gamma$, plays an important rôle in putting a limit on $m_{\XN {1}}$. However due to the low cross-section in the region where this decay dominates, the integrated luminosity collected so far at LEP2 is not enough to extend this limit, so searches at LEP1 have to be taken into account. Apart from the decay modes mentioned above, there is another possibility in the case of sufficiently light sfermions. Then the two-body decay mode, where the neutralino decays into a real sfermion and a fermion (e.g., a selectron and an electron or a Higgs and a lighter neutralino), will tend to dominate.

Figure: Running of masses from the GUT scale down to the electroweak scale. Note that the mass of H_U, becomes negative as it approaches $\mZ$, thus breaking the $\su$ symmetry. Figure from Ref. [18].