Gravitinos and hidden Supersymmetry at the LHC - Universität ...

2.2. THE MINIMAL SUPERSYMMETRIC STANDARD MODEL
not very illuminating. However, in all cases considered in the present work, the electroweak
symmetry breaking effects are only small perturbations on the neutralino mass matrix and
there is a hierarchy between the gaugino and higgsino mass terms:
m Z < |µ ± M 1 | , |µ ± M 2 | . (2.47)
In this case it is possible to perturbatively diagonalize the mass matrix and the neutralino
mass eigenstates are very nearly “bino-like”, “wino-like” and “higgsino-like”. The ordering of
the masses depends on the ordering of gaugino and higgsino masses. If m Z < M 1 < M 2 < µ
we diagonalize the mass matrix to the second order in m Z /µ and find:
( (
m χ 0
1
= M 1 − m2 Z s2 w(M 1 + µs 2β ) m
2
))
(µ 2 − M1 2) 1 + O Z
µ 2
( (
m χ 0
2
= M 2 − m2 Z c2 w(M 2 + µs 2β ) m
2
))
(µ 2 − M2 2) 1 + O Z
µ 2
m χ 0
3
= µ + m2 Z (1 − s 2β)(µ + M 1 c 2 w + M 2 s 2 w)
2(µ + M 1 )(µ + M 2 )
m χ 0
4
= µ + m2 Z (1 + s 2β)(µ − M 1 c 2 w − M 2 s 2 w)
2(µ − M 1 )(µ − M 2 )
,
,
( ( m
2
))
1 + O Z
µ 2
( ( m
2
))
1 + O Z
µ 2
,
, (2.48)
where we have defined s 2β = sin(2β), and have assumed that sign (µ) = +1. The lightest neutralino
is bino-like, as expected. The perturbative diagonalization technique will be essential
in the case of R-parity violation. It is summarized in Appendix B.
The chargino mass term in the gauge eigenstate basis ψ − = (w − , h − d ), ψ+ = (w + , h + u ) T
reads:
− L chargino mass = ψ − M C ψ + + h.c. , (2.49)
where
(
√ )
M C M
=
2 2mZ √2mZ
s β c w
. (2.50)
c β c w µ
The chargino mass matrix is an arbitrary complex matrix, therefore one has to use its singular
value decomposition (also described in Appendix B), in order to obtain the physical masses:
U (c)† M C Ũ (c) = M C diag , (2.51)
where U (c) and Ũ c are unitary. The chargino masses can be easily given in analytical form, but
we are again interested in the limit of eq. (2.47), in which case the chargino mass eigenstates
consist of a wino-like χ ± 1 and a higgsino-like χ± 2 , with masses
( (
m χ
± = M 2 − m2 Z c2 w(M 2 + µs 2β ) m
2
))
1
(µ 2 − M2 2) 1 + O Z
µ 2
m χ
±
2
( (
= µ + m2 Z c2 w(µ + M 2 s 2β ) m
2
))
(µ 2 − M2 2) 1 + O Z
µ 2 . (2.52)
The lightest chargino is degenerate with the neutralino χ 0 2 up to the higher orders. The mass
degeneracies between the charginos and neutralinos will have important phenomenological
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