3.3.3.5 Full Reconstruction of the Chargino Mass Matrix

The analysis presented above can be worked out further[29] to fully reconstruct the chargino mass matrix given by Eq. (3.10), which we repeat here for convenience:

$${\cal M}_C = \pmatrix{ M_2 & \sqrt{2}m_W\cos\beta \cr \sqrt{2}m_W\sin\beta & \mu }$$

Notice that the mass matrix, being asymmetric, requires two unitary matrices for diagonalization. We thus have two (real) mixing angles, $\phi_L$ and $\phi_R$, which are given in terms of M₂, $\mu$, and $\tan{\beta}$. These SUSY parameters are, in general, complex, implying a possible non-trivial CP phase which cannot be removed by field redefinitions. Without loss of generality, we can attribute this phase to the $\mu$ parameter.

As mentioned above, only the s-channel B exchange diagram contribute to $e^+e^-_R \to \tilde{\chi}^+_1 \tilde{\chi}^-_1$, thereby singling out the higgsino component of the chargino. Because of this the corresponding cross section ( $\sigma _R\{11\}$) is symmetric with respect to $\cos2\phi_L$ and $\cos2\phi_R$ and is independent of the mass of the sneutrino that could be exchanged in the t-channel if the electron beam were left-handed⁴ On the other hand, the left-handed ( $\sigma _L\{11\}$) and the transverse ( $\sigma _T\{11\}$) cross sections vary with the sneutrino mass and the $e\tilde{\nu}\tilde{W}$Yukawa coupling ( $g_{e\tilde{\nu}\tilde{W}}$). When mapped into the $\cos2\phi_L$- $\cos2\phi_R$ plane, these cross sections comprise elliptic or parabolic contours, which may cross each other up to four times. Remember that $\sigma _R\{11\}$ is invariant against any change of $m_{\tilde{\nu}_e}$ or $g_{e\tilde{\nu}\tilde{W}}$, while $\sigma _L\{11\}$ and $\sigma _T\{11\}$ move with them. Since the three cross section measurements meet at a single point in the $\cos2\phi_L$- $\cos2\phi_R$ plane only for the correct solution, we can thus decide the mixing angles by changing $m_{\tilde{\nu}_e}$ provided that the $e\tilde{\nu}\tilde{W}$Yukawa coupling is identified with the gauge coupling as dictated by the supersymmetry. Fig. 3.13 demonstrate this.

Instead of using beam polarizations, we can also use the final-state polarization of charginos, which can be extracted by measuring the distributions of their decay daughters. It has been shown that we can reconstruct the polarization vectors of the charginos and their spin-spin correlation tensor dynamics-independently, and use them to decide the mixing angles[29]. Note also that these additional measurements, including the differential cross section measurements, can be used, in combination with the beam polarizations, to check the equality of the $e\tilde{\nu}\tilde{W}$Yukawa coupling to the gauge coupling (a quantitative test of supersymmetry)[31].

When the machine energy reaches the pair production threshold for the heavier chargino, we will be able to study all of the three combinations:

$$\to \tilde{\chi}^+_1 \tilde{\chi}^-_1$$ e⁺e^-

$$\to \tilde{\chi}^+_1 \tilde{\chi}^-_2$$ (3.10)

$$\to \tilde{\chi}^+_2 \tilde{\chi}^-_2.$$ (3.11)

This will allow us to measure the masses of the two charginos, the production cross sections for the left- and right-handed electron beams for the three processes, and various decay distributions. These measurements can then be use to unambiguously determine all of the chargino mass matrix parameters, including CP phase. Figure 3.14

  

Figure 3.14: Contours of the cross sections (a) $\sigma _{L/R}\{11\}$, (b) $\sigma _{L/R}\{12\}$, and (c) $\sigma _{L/R}\{22\}$in the $\cos2\phi_L$- $\cos2\phi_R$ plane for $\tan\beta=3$, $m_0=100~{\rm GeV}$, $M_{1/2}=200~{\rm GeV}$ at $\sqrt{s} = 800~$GeV[29].

shows the cross section contours in the $\cos2\phi_L$- $\cos2\phi_R$ plane for the three processes with different beam polarizations. We can see that the three figures uniquely select a single point $(\cos\theta_L,\cos\theta_R) = (0.645,0.844)$in the mixing angle plane. In this way, we can thus decide the mixing angles with a statistical error of a percent level or better, given an integrated luminosity of $1~{\rm ab}^{-1}$. Notice also that the mass matrix is clearly over-constrained by the measurements, we can relax the supersymmetry condition on the $e\tilde{\nu}\tilde{W}$ Yukawa coupling, and test it instead of assuming it. The ratio of the $e\tilde{\nu}\tilde{W}$ Yukawa coupling to the gauge coupling can thus be tested at per mil level. Table 3.2 compares the input and output values of the masses, mixing angles, and the Yukawa coupling for $1~{\rm ab}^{-1}$[29].

 

Table: 3.2 Comparison of the input and output values of the mixing angles in the chargino sector extracted from different chargino measurements for $\int {\cal L} dt = 1 {\rm ab}^{-1}$[29].

Input	Extracted
$m_{\tilde{\chi}_{1}^{\pm}}$ = 128 GeV,	$m_{\tilde{\chi}_{1}^{\pm}}$ = 128 $\pm 0.04$GeV,
$m_{\tilde{\chi}_2^{\pm}}$ = 346 GeV.	$m_{\tilde{\chi}_2^{\pm}} = 346$$\pm 0.25$ GeV.
$\cos 2 \phi_{L} = 0.645$,	$\cos 2 \phi_{L} = 0.645 \pm 0.02,$
$\cos 2 \phi_{R} = 0.844.$	$\cos 2 \phi_{R} = 0.844 \pm 0.005$.
$g_{e \tilde {\nu} \tilde {W}} / g_{e \nu W} = 1$	$g_{e \tilde {\nu} \tilde {W}} / g_{e \nu W} = 1 \pm 0.01$

On the other hand Table 3.3 summarizes the input and extracted values of the fundamental parameters of the chargino mass matrix.

 

Table: 3.3 The input and output values of $M_2,\mu$ and $\tan{\beta}$ for two input points for an integrated luminosity 1 ab^-1[29].

parameter	Input	Extracted	Input	Extracted
M2	152	$152 \pm 1.75$	150	$150 \pm 1.2$
$\mu$	316	$316 \pm 0.87$	263	$263 \pm 0.7$
$\tan{\beta}$	3	3 $\pm$ 0.69	30	> 20.2

Note that the expected precisions are remarkable except for the large $\tan{\beta}$ case. This is easy to understand since all the chargino variables are proportional to $\cos 2\beta$ whose dependence on $\beta$ becomes flat as $2\beta$ goes to $\pi$ (i.e. $\tan{\beta}$ increases).