8 Beyond the Standard Model
483
One examines the charges of W under the various symmetries. Automatically, the
charges of W for the flavor symmetries, SU(N F ) L × SU(N F ) R and the U(1) V vanish.
If one requires the U(1) A charge to vanish then this implies a = −b. Requiring
the U(1) R charge to vanish implies that b =
1
N F −N C
. These restrictions fix:
W = c
Λ 3N C −N F
det M
1
N C −N F
(8.97)
For non-vanishing c, all the moduli are now lifted and there is no ground state.
What is the value of c? This is a difficult to calculate directly unless there is
complete higgsing. For N F = N C − 1 there is complete symmetry breaking and one
can turn to weak coupling. From instant on calculations one calculates that c = 0
and the prepotential for the matter fields is
W ∼
Λ 2N C +1
det M
.
(8.98)
One may now go to N F < N C − 1 by adding masses and integrating out the heavy
degrees of freedom. This produces
< M
i
j > min =
m
−1
i
j
Λ
3N C −N F det m
1
N C .
(8.99)
Thus in this region the dynamical effects remove the supersymmetric vaccum. In
fact the system has no ground state and supersymmetry is broken dynamically. One
can go one step further and consider the possibility that the effective potential above
is modified so as to have a local minimum with positive vacuum energy. It is possible
to construct many such examples and consider that supersymmetry seems broken
as a result of the universe being for the (long) time being in the metastable state.
Eventually the system may tunnel to another lower energy vacuum and perhaps
eventually will reach a SUSY vacuum. In the analysis of such systems once needs
also to take into account gravity. In the cases which follow supersymmetry is not
broken, they are described to illustrate the rich structure which emerges once one
can treat the system when it strongly interacts. Perhaps also structures like those of
the dyonic condensates will yet play a role somewhere in nature (Fig. 8.4).
We now turn to study the dynamics and quantum moduli space for the N F ≥ N C
case. In this situation there is a surviving moduli space. In the presence of a mass
matrix, m ij for matter one obtains
< M
i
j >=
m
−1
i
j
Λ
3N C −N F det m
1
N C .
(8.100)
483
One examines the charges of W under the various symmetries. Automatically, the
charges of W for the flavor symmetries, SU(N F ) L × SU(N F ) R and the U(1) V vanish.
If one requires the U(1) A charge to vanish then this implies a = −b. Requiring
the U(1) R charge to vanish implies that b =
1
N F −N C
. These restrictions fix:
W = c
Λ 3N C −N F
det M
1
N C −N F
(8.97)
For non-vanishing c, all the moduli are now lifted and there is no ground state.
What is the value of c? This is a difficult to calculate directly unless there is
complete higgsing. For N F = N C − 1 there is complete symmetry breaking and one
can turn to weak coupling. From instant on calculations one calculates that c = 0
and the prepotential for the matter fields is
W ∼
Λ 2N C +1
det M
.
(8.98)
One may now go to N F < N C − 1 by adding masses and integrating out the heavy
degrees of freedom. This produces
< M
i
j > min =
m
−1
i
j
Λ
3N C −N F det m
1
N C .
(8.99)
Thus in this region the dynamical effects remove the supersymmetric vaccum. In
fact the system has no ground state and supersymmetry is broken dynamically. One
can go one step further and consider the possibility that the effective potential above
is modified so as to have a local minimum with positive vacuum energy. It is possible
to construct many such examples and consider that supersymmetry seems broken
as a result of the universe being for the (long) time being in the metastable state.
Eventually the system may tunnel to another lower energy vacuum and perhaps
eventually will reach a SUSY vacuum. In the analysis of such systems once needs
also to take into account gravity. In the cases which follow supersymmetry is not
broken, they are described to illustrate the rich structure which emerges once one
can treat the system when it strongly interacts. Perhaps also structures like those of
the dyonic condensates will yet play a role somewhere in nature (Fig. 8.4).
We now turn to study the dynamics and quantum moduli space for the N F ≥ N C
case. In this situation there is a surviving moduli space. In the presence of a mass
matrix, m ij for matter one obtains
< M
i
j >=
m
−1
i
j
Λ
3N C −N F det m
1
N C .
(8.100)