8 Beyond the Standard Model
471
8.2.4 Spontaneous SUSY Breaking: Perturbative Analysis
Any attempt to relate supersymmetry to nature at the level of the known particles
requires the symmetry to be broken. It could have been broken explicitly, in
this section the currently known mechanisms for its spontaneous breaking are
described. This includes the breaking at the classical level in a class of gauge
theories (by D terms) and in theories with no gauge particles (by F terms). Also
are described the dynamical breaking of Supersymmetry as well as its effective
breaking in a metastable vacuum. For SUSY not to be spontaneously broken all
the SUSY generators Q i α need to annihilate the ground state. As the Hamiltonian
is constructed out of positive pairings of the SUSY generators, SUSY preservation
occurs if and only if the energy of the ground state vanishes. Conversely, SUSY is
broken if and only if the energy of the ground state does not vanish, Q i
α | G.S. > =
0 iff E G.S. = 0. As the Hamiltonian of the SUSY system is non-negative, the nonvanishing ground state energy in the case spontaneous SUSY breaking is positive.
As for the actual mechanism for the spontaneous breaking, it turned out that the
breaking of supersymmetry requires a somewhat elaborate structure.
8.2.4.1 F-terms
Consider first a system which contains only spin zero and spin one-half particles. In
that case the condition for SUSY not to be broken is the vanishing of the potential
generated by the F terms. Super Symmetry is thus unbroken when the following
equations have a solution:
V F = 0 ⇐⇒ F i = 0 ∀i,
(8.56)
These are n (complex) equations with n (complex) unknowns. Generically, they
have a solution. Take the example of the one component WZ model, where
F 1 = −λ − mA + gA
2 .
(8.57)
This has a solution. There is no supersymmetry breaking classically. The nonrenormalization theorem for the F terms ensures this result to be correct to all orders
in perturbation theory. The solution is:
V = A
∗ A|(gA − m)|
2
(8.58)
and hence there are actually two supersymmetric vacuum: either at < A > = m/g or
at < A> = 0.
Let us examine now how supersymmetry may be spontaneously broken. The
following anecdote may be of some pedagogical value. It turns out that a short time
after supersymmetry was introduced arguments were published which claimed to
471
8.2.4 Spontaneous SUSY Breaking: Perturbative Analysis
Any attempt to relate supersymmetry to nature at the level of the known particles
requires the symmetry to be broken. It could have been broken explicitly, in
this section the currently known mechanisms for its spontaneous breaking are
described. This includes the breaking at the classical level in a class of gauge
theories (by D terms) and in theories with no gauge particles (by F terms). Also
are described the dynamical breaking of Supersymmetry as well as its effective
breaking in a metastable vacuum. For SUSY not to be spontaneously broken all
the SUSY generators Q i α need to annihilate the ground state. As the Hamiltonian
is constructed out of positive pairings of the SUSY generators, SUSY preservation
occurs if and only if the energy of the ground state vanishes. Conversely, SUSY is
broken if and only if the energy of the ground state does not vanish, Q i
α | G.S. > =
0 iff E G.S. = 0. As the Hamiltonian of the SUSY system is non-negative, the nonvanishing ground state energy in the case spontaneous SUSY breaking is positive.
As for the actual mechanism for the spontaneous breaking, it turned out that the
breaking of supersymmetry requires a somewhat elaborate structure.
8.2.4.1 F-terms
Consider first a system which contains only spin zero and spin one-half particles. In
that case the condition for SUSY not to be broken is the vanishing of the potential
generated by the F terms. Super Symmetry is thus unbroken when the following
equations have a solution:
V F = 0 ⇐⇒ F i = 0 ∀i,
(8.56)
These are n (complex) equations with n (complex) unknowns. Generically, they
have a solution. Take the example of the one component WZ model, where
F 1 = −λ − mA + gA
2 .
(8.57)
This has a solution. There is no supersymmetry breaking classically. The nonrenormalization theorem for the F terms ensures this result to be correct to all orders
in perturbation theory. The solution is:
V = A
∗ A|(gA − m)|
2
(8.58)
and hence there are actually two supersymmetric vacuum: either at < A > = m/g or
at < A> = 0.
Let us examine now how supersymmetry may be spontaneously broken. The
following anecdote may be of some pedagogical value. It turns out that a short time
after supersymmetry was introduced arguments were published which claimed to