8 Beyond the Standard Model
475
U(1) R . Inverting this argument leads to the conclusion that supersymmetric breaking
in nature is not easy to obtain since we do not observe even such a particle. This
argument was revisited in recent years.
So far only systems with U(1) vector fields were discussed. To have a nonvanishing FayetIliopoulos term there needs to be a U(1) factor in the gauge group.
To obtain a breaking of SUSY in a gauge system with does not have a U(1) factor
one needs to consider dynamical SUSY breaking. This is a more complex problem
and new tools were developed to explore this possibility. The functional form of
such a breaking allows to obtain a scale of SUSY breaking which is naturally much
smaller than the relevant cutoff of the problem. This is reflected by the relation:
M SUSY−breaking = M cutoff exp (−c/g (M cutoff )) .
(8.71)
8.2.5 Dynamics of SUSY Gauge Theories and SUSY Breaking
The description of the mechanism of dynamical breaking of SUSY is preceded by
a survey of the general possible phase structure of gauge theories as well as its
concrete realizations in SUSY gauge theories.
8.2.5.1 Phases of Gauge Theories
The phase structure of gauge theories can be introduced by analyzing them as
statistical mechanical systems regulated by a finite lattice. The basics can be
illustrated by considering D = 4 lattice gauge theories, in particular those for which
the gauge fields which are Z N valued. The system has a coupling g.
The effective “temperature” of the system is given by, T =
Ng 2
2π .
For a given theory there is a lattice of electric and magnetically charged operators.
The electric charge is denoted by n and the magnetic charge by m. An operator with
charges (n,m) is perturbative, i.e., it is an irrelevant operator and weakly coupled to
system, so long as the free energy, F > 0, i.e.,
n
2 T +
m 2
T
>
C
N
,
(8.72)
however, when the free energy is negative for the operator (n,m), it condenses
indicating the presence of a relevant operator and hence an infra-red instability, this
occur when,
n
2 T +
m 2
T
<
C
N
.
(8.73)
Keeping N,C fixed and vary T.
475
U(1) R . Inverting this argument leads to the conclusion that supersymmetric breaking
in nature is not easy to obtain since we do not observe even such a particle. This
argument was revisited in recent years.
So far only systems with U(1) vector fields were discussed. To have a nonvanishing FayetIliopoulos term there needs to be a U(1) factor in the gauge group.
To obtain a breaking of SUSY in a gauge system with does not have a U(1) factor
one needs to consider dynamical SUSY breaking. This is a more complex problem
and new tools were developed to explore this possibility. The functional form of
such a breaking allows to obtain a scale of SUSY breaking which is naturally much
smaller than the relevant cutoff of the problem. This is reflected by the relation:
M SUSY−breaking = M cutoff exp (−c/g (M cutoff )) .
(8.71)
8.2.5 Dynamics of SUSY Gauge Theories and SUSY Breaking
The description of the mechanism of dynamical breaking of SUSY is preceded by
a survey of the general possible phase structure of gauge theories as well as its
concrete realizations in SUSY gauge theories.
8.2.5.1 Phases of Gauge Theories
The phase structure of gauge theories can be introduced by analyzing them as
statistical mechanical systems regulated by a finite lattice. The basics can be
illustrated by considering D = 4 lattice gauge theories, in particular those for which
the gauge fields which are Z N valued. The system has a coupling g.
The effective “temperature” of the system is given by, T =
Ng 2
2π .
For a given theory there is a lattice of electric and magnetically charged operators.
The electric charge is denoted by n and the magnetic charge by m. An operator with
charges (n,m) is perturbative, i.e., it is an irrelevant operator and weakly coupled to
system, so long as the free energy, F > 0, i.e.,
n
2 T +
m 2
T
>
C
N
,
(8.72)
however, when the free energy is negative for the operator (n,m), it condenses
indicating the presence of a relevant operator and hence an infra-red instability, this
occur when,
n
2 T +
m 2
T
<
C
N
.
(8.73)
Keeping N,C fixed and vary T.