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

2.2. THE MINIMAL SUPERSYMMETRIC STANDARD MODEL
R β0 =
( )
( )
sin β0 cos β 0
sin β± cos β
, R
− cos β 0 sin β
β± =
±
, (2.35)
0 − cos β ± sin β ±
are chosen such that the quadratic part of the potential has diagonal squared-masses. In
the tree-level approximation one finds that β 0 = β ± = β, and that the Nambu-Goldstone
bosons G 0 , G ± have zero mass and can be set to zero in the unitary gauge. The masses of
the physical Higgs bosons are given by:
m 2 A
= 2B/ sin(2β) = 2|µ| 2 + m 2 0 u + m 2 d , (2.36)
m 2 h,H = 1 (
√
)
m 2 A
+ m 2
2
0 Z ∓ (m 2 − m
A 2 0 Z )2 + 4m 2 Z m2 sin 2 (2β) , (2.37)
A 0
m 2 H
= m 2 ± A
+ m 2 0 W . (2.38)
The mixing angle α is determined, at tree-level, by
(
sin 2α m
2
sin 2β = − H
+ m 2 )
h
m 2 H − ,
m2 h
tan 2α
tan 2β = ( m
2
A 0 + m 2 Z
m 2 A 0 − m 2 Z
)
, (2.39)
and is traditionally chosen to be negative; it follows that −π/2 < α < 0 (provided m A 0 > m Z ).
In the decoupling limit, i.e. in the case m A 0 ≫ m Z , the Higgs particles H, A 0 and H ± are
very heavy and decoupled from the low-energy phenomenology. The lightest Higgs particle h
behaves as the Standard Model Higgs boson with the mass near the Fermi scale. The mixing
angle α becomes β − π/2, which will be important in the discussion of the R-parity violating
coupling of the neutralino to the Higgs boson.
Including all loop corrections, supersymmetry provides an upper bound on the mass of the
lightest Higgs boson, which is often used in its support in the light of the recent experimental
data and electroweak fits. The bound is:
m h 135 GeV. (2.40)
The µ Problem and Giudice-Masiero Mechanism We have already noted that the µ
parameter which couples the Higgs doublets in the superpotential is the only dimensionful
parameter allowed by unbroken supersymmetry. However, it also plays an important role
during electroweak symmetry breaking, since it is obviously connected with the Higgs sector.
Writing down the squared Z boson mass in terms of the fundamental parameters:
∣ m
m 2 2
Z = d
− m 2 ∣
u
√
1 − sin 2 (2β) − m2 u − m 2 d − 2 |µ|2 , (2.41)
one discovers that, barring large cancellations, all of the parameters should have values near
the Fermi scale. Why a SUSY-preserving parameter should have a value near the SUSYbreaking
scale is completely unclear. On the other hand, this problem would be solved if one
could connect the µ term with the breaking of supersymmetry. A solution was proposed by
Giudice and Masiero [104]. They observed that the µ term is generated in supergravity models
from non-renormalizable terms in Kähler potential. One way to analyze the mechanism is
to consider the low-energy effective theory below M P involving a non-renormalizable Kähler
potential:
( )
K = HuH ∗ u + Hd ∗ H λµ
d + H u H d X ∗ + h.c. , (2.42)
M P
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