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Supersymmetry Demystified
The standard model is based on the gauge group SU(3)
C
⊗ SU(2)
L
⊗U (1)
Y
.
This means that all the terms of the lagrangian must be invariant under this
symmetry. The symmetry is broken spontaneously via the Higgs mechanism
to the subgroup U(1)
em
⊗ SU(3)
C
. The fermions of the theory, as well as the
scalar Higgs field, transform in the fundamental representations of the unbroken
group (more about this below), whereas the gauge bosons transform in the adjoint
representations.
The terms in the lagrangian can be grouped into the following five categories:
•
The kinetic terms of the fermions
•
The kinetic terms of the gauge bosons
•
The kinetic term of the Higgs boson
•
The Higgs potential
•
The Yukawa terms coupling the Higgs field to the fermions
Here, by kinetic terms, we mean the expressions made gauge-invariant by replacing
the partial derivatives with the proper gauge-covariant derivatives (we will be more
specific soon).
This is it. There are no mass terms; all masses are generated through the Higgs
mechanism. The Higgs mechanism is a complex doublet, so it has four degrees of
freedom. On spontaneous symmetry breaking, three of these degrees of freedom
become the longitudinal modes of the W
±
and Z, which become massive, leaving
a single Higgs scalar particle. The masses of the fermions are also generated by
spontaneous symmetry breaking (SSB) via the Yukawa terms. The standard model
does not include neutrino masses, so we already know that it must be modified
to accommodate such terms, but this can be done in a straightforward manner
(although it depends on whether the neutrinos are Majorana or Dirac particles).
The MSSM is also written without neutrino masses, and again, these can be added
easily. We will not include neutrino masses in our presentation of the MSSM.
Before writing down the standard model (SM) lagrangian, one must first specify
the quantum numbers of the fermions and Higgs bosons. This then determines how
these fields can be coupled together and what their kinetic terms will look like.
The standard model is a chiral theory in the sense that part of the theory, the weak
interaction to be more precise, treats left- and right-chiral states of the fermions
differently. In other words, the SU(2)
L
quantum numbers of the left- and right-chiral
states of the fermions are different.
The Fermions: Quantum Numbers and Kinetic Energy Terms
Let’s be more specific. The rest of this section is not only useful as a review of
the standard model, but it is also a good preparation for construction of the MSSM