Perturbative and non-perturbative infrared behavior of ...

7. Basics and motivations
where the q’s are the supersymmetry generators. Therefore, for a supersymmetric field theory
the Hamiltonian can be written as
H ≡ P 0 = 1
{ Q1,Q1} ¯ + {
4
¯ Q2,Q2}
(7.1.2)
If supersymmetry is unbroken in the vacuum state it is annihilated by the supersymmetry generators,
hence its vacuum energy vanishes:
E = 〈0|H|0〉 = 0 (7.1.3)
Conversely, if supersymmetry is spontaneously broken in the vacuum state, then by definition it
is not invariant under supersymmetry transformations and Qα|0〉 = 0 and ¯ Q ˙α|0〉 = 0. It follows
that
E = 〈0|H|0〉 = 1
||Q1|0〉||
4
2 + || ¯ Q1|0〉|| 2 + ||Q2|0〉|| 2 + || ¯ Q2|0〉|| 2 > 0 (7.1.4)
since the Hilbert space is to have positive norm. Thus, the order parameter for global supersymmetry
breaking is the vacuum energy.
If we ask the vacuum to be a Lorentz invariant configuration, then all the spacetime derivatives
and all the non-scalar fields must vanish. This means that only scalar fields can have a vacuum
expectation value:
〈A a µ 〉 = 〈λa 〉 = 〈ψ i 〉 = ∂µ〈φi〉 = 0 (7.1.5)
where A a µ are the gauge fields, λ a their fermionic superpartners (the gauginos), and ψ i are the
fermionic partners of the scalar fields φ i . The scalar vacuum expectation value is given by the
solution to the equation
∂V ∂V
=
∂φi ∂φ †
i
= 0 (7.1.6)
In a general renormalizable supersymmetric field theory the scalar potential is given by the sum
of two contributes
where
F †
i
= ∂W
∂φ i
V = F †
i F i + 1
2 Da D a
D a = −gφ †
i (T a ) i j φj
(7.1.7)
(7.1.8)
Here, T a are the gauge group generators and W is a holomorphic function called the superpotential.
The spontaneous breaking of a global symmetry always implies a massless Goldstone mode
with the same quantum numbers as the broken symmetry generator. In the case of global
supersymmetry, the broken generator is the fermionic charge Qα. Therefore, the Goldstone
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