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

2.1. GLOBAL SUPERSYMMETRY
research will probably shift to another scenarios. Therefore, it is important to look beyond
standard scenarios and investigate different realizations of supersymmetric models, which
can be hidden from the LHC. This is the main topic of the present work. In the following
we will introduce our notations, some aspects of the formalism, and the phenomenological
implications of supersymmetry which will be needed in the following chapters. The discussion
follows references [78,89,102,103] which contain a comprehensive introduction to these topics.
2.1 Global Supersymmetry
We investigate the properties of N = 1 supersymmetry, which is the only phenomenologically
interesting realization of supersymmetry in four dimensions. The supersymmetric extension
of the Poincaré algebra reads:
{ }
Q α , Q †˙α
= 2σ µ α ˙α P µ, {Q α , Q β } =
{ }
[ ]
Q
†˙α
, = 0, [P Q†˙β
µ , Q α ] = P µ , Q †˙α = 0. (2.2)
Representations Irreducible representations of the supersymmetry algebra are called supermultiplets,
they unify fermionic and bosonic degrees of freedom. There are two such
(massless) supermultiplets in our case, the chiral and the vector supermultiplet.
The chiral multiplet contains one complex scalar φ and one two-component fermion ψ
transforming in the defining representation of SL(2, C), see Appendix A for details on the
two-component notation. In order for the SUSY algebra to close off-shell, the chiral multiplet
is augmented with an auxiliary complex scalar F , which has no kinetic term in the full theory.
The vector supermultiplet contains one massless vector field A µ , one two-component
fermion λ α transforming also as ( 1 2
, 0), and off-shell a real auxiliary scalar D. Since the
supersymmetry generators commute with the generators of the gauge transformations, the
whole supermultiplet transforms in the same representation of the gauge group, in particular
the fermion λ transforms in the adjoint representation.
Superfields We have seen that supersymmetry has a peculiar feature being on the one
hand an internal symmetry, and on the other hand an extension of the external Poincaré
group. Both notions can be conciliated if one introduces superspace as natural stage for
supersymmetry. Superspace is obtained by adding four fermionic coordinates to the usual
bosonic space-time coordinates. Points in superspace are labeled by coordinates:
x µ , θ α , θ
†˙α
. (2.3)
Here θ α and θ
†˙α
are constant complex anticommuting two-component spinors with dimension
[mass] − 1/2
. The components of θ are anticommuting Grassmann numbers. The objects
living in superspace are superfields, functions of the superspace coordinates, containing the
component fields of the supermultiplets. They embody linear representation of the SUSY
transformations, since global supersymmetry transformations are represented as infinitesimal
translations in the superspace. Consequently, the dichotomy between the internal and
external nature of the SUSY transformations is resolved: they are external physical transformations
in superspace perceived as internal symmetries from the viewpoint of ordinary space.
This discussion suggests that superspace should be taken as real and not as a pure heuristic
instrument, a view surely open for debate.
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