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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:
Notice that the mass matrix, being asymmetric, requires
two unitary matrices for diagonalization.
We thus have two (real) mixing angles,
and ,
which are given in terms of M_{2}, ,
and .
These SUSY parameters are, in general,
complex, implying a possible nontrivial CP phase
which cannot be removed by field redefinitions.
Without loss of generality, we can attribute this phase
to the
parameter.
As mentioned above, only the schannel
B exchange diagram contribute to
,
thereby singling out the higgsino component of
the chargino.
Because of this the corresponding cross section
(
)
is symmetric with respect to
and
and is
independent of the mass
of the sneutrino that could be exchanged in the
tchannel if the electron beam were lefthanded^{4}
On the other hand, the lefthanded (
)
and the transverse (
)
cross sections
vary with the sneutrino mass and the
Yukawa coupling (
).
When mapped into the

plane,
these cross sections comprise elliptic or parabolic contours,
which may cross each other up to four times.
Remember that
is invariant
against any change of
or
,
while
and
move with them.
Since the three cross section measurements meet at a
single point in the

plane
only for the correct solution,
we can thus decide the mixing angles by changing
provided that the
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 finalstate 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 spinspin correlation tensor
dynamicsindependently, 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
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:
This will allow us to measure the masses of the two charginos,
the production cross sections for the left and righthanded
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)
, (b)
, and (c)
in the

plane for
,
,
at
GeV[29].

shows the cross section contours in the

plane
for the three processes with different beam polarizations.
We can see that the three figures uniquely select a single
point
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
.
Notice also that the mass matrix is clearly overconstrained
by the measurements, we can relax the supersymmetry
condition on the
Yukawa coupling,
and test it instead of assuming it.
The ratio of the
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
[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
[29].
Input 
Extracted 
= 128 GeV, 
= 128 GeV, 
= 346 GeV. 
GeV. 
, 


. 


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 and
for two input points for an integrated luminosity
1 ab^{1}[29].
parameter 
Input 
Extracted 
Input 
Extracted 
M_{2} 
152 

150 


316 

263 


3 
3
0.69 
30 
> 20.2 
Note that the expected precisions are remarkable except
for the large
case.
This is easy to understand since
all the chargino variables are proportional
to
whose dependence on
becomes flat
as
goes to
(i.e.
increases).
Next: 3.3.3.6 Extraction of Sneutrino
Up: 3.3.3 Study of
Previous: 3.3.3.4 Test of SUGRAGUT
ACFA Linear Collider Working Group
EMail:acfareport@acfahep.kek.jp