8 Beyond the Standard Model
479
8.2.5.3 The Moduli Space
The classical moduli space is given by solving the D-term and F-term equations:
D a = Q
+
F T a Q F − ˜
Q F T a ˜
Q
+
F
F Q F = −m F ˜
Q
F ˜
Q F
= −m F Q
(8.81)
For N F = 0 or for N F = 0 and m F = 0, there is no moduli space. Note, the vacuum
structure is an infra-red property of the system hence having m F = 0 is equivalent
to setting N F = 0 in the deep infrared.
Consider the quantum moduli space of the case where N F = 0. One can show that
the number of zero energy states of the system is no smaller than The Witten index,
Tr(−1) F = N C i.e. the rank of the group +1. This number is larger than zero and
thus there is no supersymmetry breaking in these systems. There are 2N C Fermionic
zero modes (from the vector multiplet). These Fermionic zero modes break through
instanton effects the original U(1) R down to Z 2NC . Further breaking occurs because
the gluino two point function acquires a vacuum expectation value which breaks the
symmetry down to Z 2 . This indeed leaves N C vacua. The gluino condensate is:
< λλ >= exp
2πik
N C
Λ
3
N C
(8.82)
where NC is the dynamically generated scale of the gauge theory and
k = 1, . . . ,N C − 1 label the vacua. Chiral symmetry breaking produces a mass
gap. Note, because chiral symmetry is discrete there are no Goldstone Bosons.
Further details of quantum moduli spaces will be discussed later.
Consider the case where m F = 0 and 0 < N F < N C . The classical moduli space
is determined by the following solutions to the D-term equations:
Q = ˜
Q =
⎛
⎜
⎜
⎜
⎜
⎜
⎝
a 1 0
0.. .. 0
0 a 2
0 .. .. 0
.
0 .. .. 0
.
0 .. .. 0
a NF 0 .. .. 0
⎞
⎟
⎟
⎟
⎟
⎟
⎠
N F ×N C
(8.83)
Where the row indicates the flavor and the column indicates the colour. There are
N F diagonal non-zero real entries, a i . (To validate this classical analysis the vacuum
expectation values must be much larger than any dynamically generated scale, i.e.,
a i >> . The gauge symmetry is partially broken:
SU (N C ) → SU (N C − N F ) .
(8.84)
479
8.2.5.3 The Moduli Space
The classical moduli space is given by solving the D-term and F-term equations:
D a = Q
+
F T a Q F − ˜
Q F T a ˜
Q
+
F
F Q F = −m F ˜
Q
F ˜
Q F
= −m F Q
(8.81)
For N F = 0 or for N F = 0 and m F = 0, there is no moduli space. Note, the vacuum
structure is an infra-red property of the system hence having m F = 0 is equivalent
to setting N F = 0 in the deep infrared.
Consider the quantum moduli space of the case where N F = 0. One can show that
the number of zero energy states of the system is no smaller than The Witten index,
Tr(−1) F = N C i.e. the rank of the group +1. This number is larger than zero and
thus there is no supersymmetry breaking in these systems. There are 2N C Fermionic
zero modes (from the vector multiplet). These Fermionic zero modes break through
instanton effects the original U(1) R down to Z 2NC . Further breaking occurs because
the gluino two point function acquires a vacuum expectation value which breaks the
symmetry down to Z 2 . This indeed leaves N C vacua. The gluino condensate is:
< λλ >= exp
2πik
N C
Λ
3
N C
(8.82)
where NC is the dynamically generated scale of the gauge theory and
k = 1, . . . ,N C − 1 label the vacua. Chiral symmetry breaking produces a mass
gap. Note, because chiral symmetry is discrete there are no Goldstone Bosons.
Further details of quantum moduli spaces will be discussed later.
Consider the case where m F = 0 and 0 < N F < N C . The classical moduli space
is determined by the following solutions to the D-term equations:
Q = ˜
Q =
⎛
⎜
⎜
⎜
⎜
⎜
⎝
a 1 0
0.. .. 0
0 a 2
0 .. .. 0
.
0 .. .. 0
.
0 .. .. 0
a NF 0 .. .. 0
⎞
⎟
⎟
⎟
⎟
⎟
⎠
N F ×N C
(8.83)
Where the row indicates the flavor and the column indicates the colour. There are
N F diagonal non-zero real entries, a i . (To validate this classical analysis the vacuum
expectation values must be much larger than any dynamically generated scale, i.e.,
a i >> . The gauge symmetry is partially broken:
SU (N C ) → SU (N C − N F ) .
(8.84)