8 Beyond the Standard Model
465
This gives:
L = i∂ψ i σ ψ i + A ∗
i ∗ A i −
1
2 m ij ψ i ψ j −
1
2 m ∗
ij ψ ∗
i ψ ∗
j
− g ij k ψ i ψ j A k − g ∗
ij k ψ i ψ j − A ∗
k − F ∗
i F i
(8.26)
where the last term leads after integration on the F i s to a potential for the fields A,A ∗ ;
these are known as the F terms, V F (A ∗ ,A). (Note, V F ≥ 0). It turns out as explained
later that at the ground state this must vanish i.e. V F (A ∗ ,A) = 0 if SUSY is not to
be spontaneously broken. This in turn implies that F i = 0 for such a symmetric
ground state. Although this is a classical analysis so far, in fact it is true to all orders
in perturbation theory as there exists a non-renormalization theorem for such an
effective potential in SUSY theories. The Lagrangian described above is called the
Wess-Zumino Lagrangian (WZ).
So far the scalar fields have been defined over simple flat manifolds. To describe
the kinetic terms of supersymmetric Lagrangians of systems containing scalar
fields spanning complicated manifolds it is convenient to introduce the following
supersymmetry invariant:
d
4 θK
Φ, Φ
.
(8.27)
K
Φ, Φ
is called the Kähler potential, unlike the potential W() but similar to
the kinetic term introduced above, the Kähler potential depends on both and Φ.
One may add any function of or Φ to the integrand since these terms will vanish
after integration. For the usual kinetic terms, K is taken to be given by K = ΦΦ
which produces the −δ ij ∂ μ A ∗ i ∂ μ A j kinetic terms for the scalars. For the case of a
sigma model with a target space whose metric is g ij ; this metric is related to the
Kähler potential by:
g ij =
∂ 2 K
∂Φ i ∂Φ j
.
(8.28)
The above supersymmetry invariant (8.27) which previously gave the usual
kinetic terms in the action, produces for general K the action of a supersymmetric
sigma model, with the target space metric given by Eq. (8.28).
8.2.2.2 Global Symmetries
It is possible to construct Lagrangians which have a global symmetry which does
not commute with supersymmetry and thus assigns different quantum numbers
to particles in the same supermultiplet. This symmetry already in its discrete
form forbids unwanted interaction terms which strongly violate baryon and lepton
conservation laws. Such interactions arise due to the bosonic superpartners to
465
This gives:
L = i∂ψ i σ ψ i + A ∗
i ∗ A i −
1
2 m ij ψ i ψ j −
1
2 m ∗
ij ψ ∗
i ψ ∗
j
− g ij k ψ i ψ j A k − g ∗
ij k ψ i ψ j − A ∗
k − F ∗
i F i
(8.26)
where the last term leads after integration on the F i s to a potential for the fields A,A ∗ ;
these are known as the F terms, V F (A ∗ ,A). (Note, V F ≥ 0). It turns out as explained
later that at the ground state this must vanish i.e. V F (A ∗ ,A) = 0 if SUSY is not to
be spontaneously broken. This in turn implies that F i = 0 for such a symmetric
ground state. Although this is a classical analysis so far, in fact it is true to all orders
in perturbation theory as there exists a non-renormalization theorem for such an
effective potential in SUSY theories. The Lagrangian described above is called the
Wess-Zumino Lagrangian (WZ).
So far the scalar fields have been defined over simple flat manifolds. To describe
the kinetic terms of supersymmetric Lagrangians of systems containing scalar
fields spanning complicated manifolds it is convenient to introduce the following
supersymmetry invariant:
d
4 θK
Φ, Φ
.
(8.27)
K
Φ, Φ
is called the Kähler potential, unlike the potential W() but similar to
the kinetic term introduced above, the Kähler potential depends on both and Φ.
One may add any function of or Φ to the integrand since these terms will vanish
after integration. For the usual kinetic terms, K is taken to be given by K = ΦΦ
which produces the −δ ij ∂ μ A ∗ i ∂ μ A j kinetic terms for the scalars. For the case of a
sigma model with a target space whose metric is g ij ; this metric is related to the
Kähler potential by:
g ij =
∂ 2 K
∂Φ i ∂Φ j
.
(8.28)
The above supersymmetry invariant (8.27) which previously gave the usual
kinetic terms in the action, produces for general K the action of a supersymmetric
sigma model, with the target space metric given by Eq. (8.28).
8.2.2.2 Global Symmetries
It is possible to construct Lagrangians which have a global symmetry which does
not commute with supersymmetry and thus assigns different quantum numbers
to particles in the same supermultiplet. This symmetry already in its discrete
form forbids unwanted interaction terms which strongly violate baryon and lepton
conservation laws. Such interactions arise due to the bosonic superpartners to