8 Beyond the Standard Model
467
produces
V = |λ|
2
|A|
4
+ λAψψ.
(8.37)
This term is not R-invariant with the R-charges given by (8.35). To restore Rinvariance requires λ is assigned an R-charge of −1. This can be viewed as simply
a book keeping device or more physically one can view the coupling as the vacuum
expectation value of some field. The expectation value inherits the quantum numbers
of the field. This is how one treats for example the mass parameters of Fermions
in the standard model. There is also one other global U(1) symmetry, one that
commutes with the supersymmetry. All component fields are charged the same with
respect to this U(1) symmetry. Demanding that the terms in the action maintain this
symmetry requires an assignment of U(1) charges to λ, and m.
The charges are summarized in the following table:
U(1) U(1) R
Φ 1
1
m −2
0
λ −3 −1
W 0
2
These symmetries can be used to prove important nonrenormalisation theorems.
In particular it can be shown that the potential:
W =
1
2
m Φ
2
+
1
3
λΦ
3 .
(8.39)
does not change under renormalization.
These non-renormalization theorems play an important role in analyzing the
dynamics of supersymmetric systems and in addressing the so called hierarchy
problem.
8.2.2.3 Lagrangians for SUSY Gauge Theories
A vector superfield contains spin 1 and spin
1
2 component fields. It obeys a reality
condition V = V .
V = B + θχθ χ + +θ 2 C + θ
2 C − θσ μ θA μ
+ iθ 2 θ
λ +
1
2 σ
μ ∂ μ χ
− iθ
2 θ
λ −
1
2 σ μ ∂ μ χ
+
1
2 θ 2 θ
2
D 2 + ∂ 2 B
(8.40)
467
produces
V = |λ|
2
|A|
4
+ λAψψ.
(8.37)
This term is not R-invariant with the R-charges given by (8.35). To restore Rinvariance requires λ is assigned an R-charge of −1. This can be viewed as simply
a book keeping device or more physically one can view the coupling as the vacuum
expectation value of some field. The expectation value inherits the quantum numbers
of the field. This is how one treats for example the mass parameters of Fermions
in the standard model. There is also one other global U(1) symmetry, one that
commutes with the supersymmetry. All component fields are charged the same with
respect to this U(1) symmetry. Demanding that the terms in the action maintain this
symmetry requires an assignment of U(1) charges to λ, and m.
The charges are summarized in the following table:
U(1) U(1) R
Φ 1
1
m −2
0
λ −3 −1
W 0
2
These symmetries can be used to prove important nonrenormalisation theorems.
In particular it can be shown that the potential:
W =
1
2
m Φ
2
+
1
3
λΦ
3 .
(8.39)
does not change under renormalization.
These non-renormalization theorems play an important role in analyzing the
dynamics of supersymmetric systems and in addressing the so called hierarchy
problem.
8.2.2.3 Lagrangians for SUSY Gauge Theories
A vector superfield contains spin 1 and spin
1
2 component fields. It obeys a reality
condition V = V .
V = B + θχθ χ + +θ 2 C + θ
2 C − θσ μ θA μ
+ iθ 2 θ
λ +
1
2 σ
μ ∂ μ χ
− iθ
2 θ
λ −
1
2 σ μ ∂ μ χ
+
1
2 θ 2 θ
2
D 2 + ∂ 2 B
(8.40)