Perturbative and non-perturbative infrared behavior of ...

9.2. The ISS model
and it gives the leading quantum corrections to the effective action because the superpotential
coupling h is marginally irrelevant. Substituting (9.2.32) into (9.2.35) and expanding up to
quadratic order in the pseudo-moduli fields we find
V (1)
eff = |h4 µ 2 |(log 4 − 1)
8π 2
1
2 (Nf − Ñ)Trˆχ2 + ÑTrˆ Φ †ˆ Φ
+ ... (9.2.36)
The kinetic terms for the pseudo-moduli fields are inherited from the tree-level kinetic terms
of the full theory, so they are canonical and diagonal at leading order. The quadratic effective
Lagrangian
Leff = Tr ∂µ ˆ
Φ
2
+ 1
2 Tr (∂µ ˆχ) 2 − V (1)
eff + ... (9.2.37)
shows that the pseudo-moduli are stabilized with positive mass squared and we conclude that
the vacua (9.2.28) are stable, without any tachyonic directions.
Let us resume the spectrum of the theory. We find that the nonsupersymmetric vacuum has
a hierarchy of mass scales dictated by the coupling h. There are fields with tree-level masses
of order |hµ|. The pseudo-moduli have masses of order |h2 µ|; note that they are suppressed by
a loop factor. The massless spectrum contains the Goldstone bosons and the exactly massless
Goldstino from supersymmetry breaking.
We now turn to the inclusion of the gauge fields. We gauge the SU( Ñ) symmetry. Because
we are restricting our analysis to the case Nc + 1 < Nf < 3
2Nc, the SU( Ñ) gauge theory has
Nf > 3Ñ and it is IR free instead of asymptotically free: above its dynamical scale ˜ Λ it is strongly
coupled. For energies of order ˜ Λ and above, the weakly coupled electric description is much more
accurate.
Having gauged SU( Ñ), the D-term contribution should be added to the full potential
VD = g2
2
A
Trq † TAq − Tr˜qTA˜q † 2
(9.2.38)
The D-term potential vanishes in the nonsupersymmetric vacuum, so it remains a minimum of
the tree-level potential. The SU( Ñ) gauge group is completely Higgsed in this vacuum. The
SU( Ñ) gauge fields acquire mass gµ through the super-Higgs mechanism. The traceless parts
of the would-be Goldstone bosons Imµ ∗χ−/|µ| are eaten and the traceless parts of the pseudomoduli
ˆχ get a positive tree-level mass gµ from the coupling with the gauge fields. Then ˆ Φ
and Trˆχ only remain as classical pseudo-moduli. Along the lines of our previous discussion, we
should compute the quantum effective potential for them to determine if they are stabilized or
get tachyonic masses. It turns out that the effect of the gauge fields drops out in the leading
order effective potential for the pseudo-moduli. The reason is that the tree-level spectrum of the
massive SU( Ñ) fields is supersymmetric, so the supertrace vanishes. This is due to the fact that
the SU( Ñ) gauge fields do not directly couple to the supersymmetry breaking fields: the D-terms
vanish on the pseudo-flat space, and the non-zero expectation values of q and ˜q, which give the
SU( Ñ) gauge fields their masses, do not couple directly to any non-zero F-terms. Thus, the
leading order effective potential we already computed stabilizes the pseudo-moduli with positive
squared masses. The supersymmetry breaking vacuum (9.2.28) survives to the SU( Ñ) gauging.
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