PHY–396 T: SUSY Mid-term Exam. Due Tuesday, March 28, in class.
1. Consider a supersymmetric gauge theory with G=SO(N) gauge group (assume N > 4)
and chiral superfields Ai(i= 1, . . . , N ) in the vector representation N.
FYI: For SO(N) groups, Index(vector) = 1 and Index(adjoint) = N−2.
In parts (a) through (e) of this question, assume Wtree(A) = 0.
(a) Write down the scalar potential of this theory and describe its flat directions. Specif-
ically, show that modulo SO(N) there is just one flat direction and write down the
gauge-invariant chiral modulus Mfor this direction.
(b) Describe the Higgs mechanism for hMi 6= 0.
(c) Consider the low-energy EFT below the Higgs scale. Write down the Wilsonian gauge
coupling of the EFT as a function of the modulus superfield.
(d) Let us fix the modulus field for a moment and focus on the infra-red behavior of
the gauge theory. How many vacua does it have, and what is the order parameter
distinguishing those vacua?
(e) Now un-fix the modulus field Mand consider its dynamics. Calculate the non-
perturbative effective superpotential Wn.p.(M) due to gaugino condensation, and
show that the scalar potential V(M, M ∗) leads to runaway M→ ∞.
Now let Wtree =m
2A2.
(f) For small mass mΛ, the effective superpotential for the modulus comprises
Wtree(M) + Wn.p.(M). Show that this superpotential has several SUSY vacua, and
calculate the expectation values hMiand hSifor each vacuum.
(g) For large mΛ, we may integrate out the Afields perturbatively, and then study
the IR behavior in terms of the low-energy EFT, which is pure SO(N) SYM the-
ory. Calculate the gaugino condensate hSiin this regime as a function of Λ and m
parameters of the original (UV) theory.
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