Perturbative and non-perturbative infrared behavior of ...

8. Supersymmetric QCD and Seiberg duality
interactions. This leads us to the concept of the phases of the gauge theories.
The phases of gauge theories are characterized by the potential V (r) between test charges at
a large distance r. According to how the gauge symmetry is realized, we have
• Coulomb phase: the vector bosons remain massless and mediate interactions with V (r) ∼
1/r;
• Higgs phase: the gauge group is spontaneously broken and all the vector bosons get masses.
In this case the potential is constant;
• confining phase: the sources of the gauge group are bound into singlets; V (r) ∼ r.
The Coulomb phase is also present in non-Abelian Yang-Mills theories: when the theory is not
asymptotically free the long-range potential between the charges is of the form e(r)/r, where
e(r) is the renormalized charge which decreases as the logarithm of the distance. In this case,
the theory flows to a fixed point of the renormalization group: the infrared theory is a nontrivial
conformal field theory.
The above discussion applies to electric charges. In addition, one can also have magnetic
charges in the theory. At large distances, their potential behaves as
• Coulomb phase: V (r) ∼ 1/r;
• Higgs phase: V (r) ∼ r;
• confining phase: V (r) ∼ constant.
The relation between these phases is particularly well understood in the Abelian case. The
Dirac quantization condition relates the electric charge e(r) to the magnetic charge g(r) by
e(r)g(r) ∼ 1. When an electrically charged field acquires a vacuum expectation value (Higgs
phase), the magnetically charges are confined in flux tubes. Analogously, when a magnetically
charged field acquires a vacuum expectation value, the electric charges are confined in flux tubes,
i.e., we are in the confining phase.
In the non-Abelian cases the relation we described cannot be made manifest. When there are
matter fields in the fundamental representation of the gauge group, virtual pairs can completely
screen the charges. In fact, there is no invariant distinction between the Higgs and confinement
phases: the flux tube in which the charges are confined can break. Usually, one interprets the
theory for large expectation values as being in the Higgs phase, while for small expectation values
as being in the confining phase. However, one can still smoothly interpolate between the two.
What we just described is the electric-magnetic duality, which exchanges the electric charges
with the magnetic ones. When the theory is in its Coulomb phase, the duality does not bring
the theory to another phase. In particular, if the microscopic, electric theory is strongly coupled,
its description in terms of dual magnetic charges is weakly coupled. Because our main goal
is to found low-energy models of supersymmetry breaking, and because supersymmetric gauge
theories are often strongly coupled at low energies, it is very convenient to have a consistent
description of their behavior in terms of the dual magnetic degrees of freedom. Then, in the
next sections we describe supersymmetric Yang-Mills theories with fundamental matter and how
the duality transformations allow us to perform reliable computations of dynamical quantities at
large distances.
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