Perturbative and non-perturbative infrared behavior of ...

Chapter 7
Basics and motivations
The Standard Model of particle physics contains all the information we know about the elementary
particles and their strong and electroweak interactions. From the theoretical point of view, the
Standard Model is an incomplete theory. It does not describe the gravitational interactions and
it leads to quadratically divergent quantum corrections which are at the origin of the hierarchy
problem. Supersymmetry is the most compelling new physics which can be discovered at the TeV
scale, since both it solves the hierarchy problem and it naturally arises in ultraviolet complete
theories. The non-renormalization theorems guarantee that the quadratic divergences cancel out
at all orders in perturbation theory.
Since supersymmetry relates fermionic and bosonic degrees of freedom, any Standard Model
fermion (boson) will have a bosonic (fermionic) partner with the same quantum numbers. Since
the mass operator commutes with the supersymmetry generators, all particles in a given supersymmetry
multiplet necessarily have the same mass. This is trivially excluded by experiments,
and we conclude that supersymmetry must be broken, either explicitly or spontaneously. We are
mainly interested in a spontaneous breaking mechanism, since it preserves the predictive power
of supersymmetry and it voids the quadratic divergences.
It turns out that the particle masses are constrained by a sum rule even if supersymmetry is
spontaneously broken. This is the content of the so-called supertrace theorem. In particular, it
implies that at least one of the scalar partners of the quarks is lighter than the up or down quarks.
This is obviously excluded by the current experiments. Thus, we have to violate the sum rule,
by violating one of its hypothesis. The main goal of this Chapter is to show how it is possible
to overcome the phenomenological problem imposed by the sum rule, and how supersymmetry
breaking is communicated to (some supersymmetric extension of) the Standard Model.
7.1 The supertrace theorem
The supersymmetry algebra contains the translation operator Pµ
{Qα, ¯ Q ˙α} = 2σ µ
α ˙α Pµ
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(7.1.1)