Perturbative and non-perturbative infrared behavior of ...

7.1. The supertrace theorem
particle is a massless Weyl fermion, called the goldstino. In the basis (λ a ,ψ i ), the fermion mass
matrix has the form
mf =
0 −g〈φ †
l 〉(T a ) l i
−g〈φ †
l 〉(T b ) l j 〈Wji〉
(7.1.9)
where we defined Wi ≡ ∂W/∂φ i and so on. Now, the condition for the minimum of the scalar
potential reads
0 = ∂V
∂φi = F j ∂2W ∂φi∂φj − gDa φ †
j (T a ) j
i
while the condition that the superpotential is gauge invariant
0 = δ a ∂W
gaugeW =
∂φi δa gaugeφi = F †
i (T a ) i j φj
Using these two equations it is easy to show that the vector
˜G
〈Da 〉
=
〈Fi〉
(7.1.10)
(7.1.11)
(7.1.12)
is annihilated by the fermion mass matrix (7.1.9). Hence (7.1.9) has a vanishing eigenvalue. The
corresponding eigenvector is proportional to the goldstino wavefunction
ψG ∝
〈Fi〉ψi +
〈D a 〉λ a
(7.1.13)
i
and it is nontrivial if and only if at least one of the auxiliary fields F and D has a vacuum
expectation value, thus breaking supersymmetry. This proves that if global supersymmetry is
spontaneously broken, then there must be a massless goldstino, and that its components among
the various fermions in the theory are just proportional to the corresponding auxiliary field
vacuum expectation values.
We now move to the bosonic sector. If supersymmetry is unbroken all particles within a
supermultiplet have the same mass. If supersymmetry is broken this is no longer true, but the
mass splitting in the multiplet can be computed as a function of the supersymmetry breaking
parameters, i.e. the VEVs of the auxiliary fields. Let us introduce the notation
D a i = ∂Da
= −gφ†
∂φi j (T a ) j
i
F ij = ∂2W ∂φ †
i∂φ† j
D ia = ∂Da
∂φ †
i
a
Fij = ∂2 W
∂φ i ∂φ j
= −g(T a ) i jφi
D ai
j = −g(T a ) i j (7.1.14)
〈F †
l 〉〈F pkl 〉 + 〈Dap 〉〈Dak 〉 〈Fjl〉〈F pl 〉 + 〈Dap 〉〈Da j 〉 + 〈Da 〉D ap
i
(7.1.15)
The squared masses of the real scalar degrees of freedom are the eigenvalues of the matrix
m 2 s =
〈Fil〉〈F lk 〉 + 〈Dak 〉〈Da i 〉 + 〈Da 〉Dak i 〈F l 〉〈Fijl〉 + 〈Da i 〉〈Da j 〉
(7.1.16)
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