# Supersymmetry

Supersymmetry (SUSY) is one of the most compelling possible extensions of the Standard Model of particle physics, and one of the most promising theory candidate for a new principle about Nature that could be discovered at high-energy colliders such as the $\textsf{LHC}$.

The hierarchy problem represents the only indicator of the energy scale in which some effects of new physics should be seen. Fermionic loop-corrections associated with top quarks are the most important among those possible in the SM. The form of the fermionic loop-corrections differs from that in equation for the mass of the Higgs boson $m^2_H(phys)\sim m_H^2+c\Lambda^2$ just for the value of the $c$ constant and from its sign, opposite with respect to the one appearing in the scalar corrections.
Bosons and fermions, therefore, give rise to quadratic corrections of opposite signs: by adding all possible contributions a cancellation is obtainable and only logarithmic terms should remain.

An exact cancellation would take place only by admitting a symmetry that relates bosons and fermions and their couplings with the Higgs boson. Furthermore, interchanges between fermions and bosons in the Lagrangian of the theory lead alike satisfied the relativistic quantum field theory constraints, ensuring at the same time the consistency with the interactions of the SM. Thus, to a such new symmetry it is given the name of Supersymmetry, and it is possible that SUSY could ultimately clear up the origin of hierarchy problem, explaining the smallness of the electroweak scale if compared with the Planck scale.

It also provides a framework for the unification of particle physics and gravity which becomes effective at the Planck energy scale, where the gravitational interactions become comparable in strength to the gauge interactions. Farther, SUSY predicts that gauge couplings, as it was experimentally measured at the electroweak scale, unify at an energy scale $\mathcal{O}(10^{16})\ GeV$, the $\Lambda_{GUT}$ scale, near the Planck scale.

It is intriguing to consider that a weakly interacting (meta)stable supersymmetric particle might make up some or all of the Dark Matter in the Universe.

On theoretical grounds, SUSY is motivated as a generalization of the space-time symmetries of a quantum field theory that transforms fermions into bosons and vice versa giving to each particle a superpartner from which it differs in spin by half a unit; the generators of its group are the Majorana’s spinors.

The transformation can be formalized as
$\mathcal{Q}|\mathcal{B}\rangle = |\mathcal{F}\rangle$
$\mathcal{Q}^\dag|\mathcal{F}\rangle = |\mathcal{B}\rangle$,

where the $\mathcal{Q}$ operator must be endowed of an anticommutative spinorial structure, in order to be able to transform scalar functions into spinors.

It is possible to find the commutation rules for the generators of the supersymmetric transformations

$\{\mathcal{Q},\mathcal{Q}^\dag\}=P^\mu$
$\{\mathcal{Q},\mathcal{Q}\}=\{\mathcal{Q}^\dag,\mathcal{Q}^\dag\}=0$
$\left[P^\mu, \mathcal{Q}\right]=\left[P^\mu,\mathcal{Q}^\dag\right]=0$

where $P^\mu$ is the Poincaré group generator of space-time traslations. The equations above show that the SUSY is a space-time symmetry too and in particular the latter of them implies the relation below

$\left[P^0,\mathcal{Q}\right]=0$

which gives

$P^0(\mathcal{Q}|\mathcal{B}\rangle)=\mathcal{Q}P^0|\mathcal{B}\rangle=E_B\mathcal{Q}|\mathcal{B}\rangle=E_B|\mathcal{F}\rangle$.

So, bosonic and corresponding fermionic states are degenerate in energy and then they must have the same mass.

For the construction of the supersymmetric Lagrangian, scalar fields and
spinorial fields have to be considered as components of a single
superfield where the supersymmetric transformation
moves a component into the other.

In order to manage scalar fields in operations with spinorial fields,
it is necessary to multiply the spinor fields with numerical quantities
$\theta_i$ which anticommute and which behave like Majorana spinors.

A superfield will therefore be dependent by the Grassmann coordinates
$\theta_i$, in conjunction with the spatial coordinates $x^\mu$:

$\hat{S}=\hat{S}(x^\mu,\theta_i)$.

The extension of the four-dimensional spacetime to the further dimensions is called superspace. The Lagrangian for the $N$ superfields $\hat{S}_{i}$ is built starting from the Kähler function,

$\mathcal{K}(\hat{S}^\dag_{i},\hat{S}_{j})=\sum_{i=1}^{N} \hat{S}^\dag_{i} \hat{S}_{i}$,

which must have the form showed in the equation above to ensure the renormalizability of the theory. Thus, the form for a general supersymmetric Lagrangian which contains just scalars and spinors is

where the term $\mathcal{W}(\hat{\mathcal{S}}_{i})$ represents the superpotential. The third term in the SUSY Lagrangian above yields the scalar potential. The masses and Yukawa interactions of fermions are all included in the last term. The model dependence of the theory is set by the choice of the superpotential that can be an arbitrary function (at most a cubic polynomial for renormalizable theories) of left-chiral superfields.

If SUSY were an exact symmetry of Nature, particles and their superpartners would be degenerate in mass. Since superpartners have not (yet) been observed, SUSY must be a broken symmetry. Nevertheless, the stability of the gauge hierarchy can still be maintained if the breaking is soft which means that the Supersymmetry breaking masses cannot be larger than a few $TeV$.

The most interesting theories of this type are theories of low-energy (or “weak-scale”) Supersymmetry, where the effective scale of Supersymmetry breaking is tied to the scale of electroweak symmetry breaking, characterized by the Standard Model Higgs VEV $v\simeq246\ GeV$.

A fundamental theory of Supersymmetry breaking is unknown at this time. However, in order to parameterize the low-energy theory in terms of the most general set of soft SUSY-breaking normalisable operators, it is important to require that the theory contains the least possible number of new particles and interactions. The resulting theory is the Minimal Supersymmetric extension of the Standard Model (MSSM) that associates a supersymmetric partner to each gauge boson and to each chiral fermion of the SM, and it provides a realistic model of Physics at the weak scale.