Supersymmetric QCD

Supersymmetric QCD is a family of four-dimensional $\mathcal N=1$ gauge theories used to study strongly coupled vacua, confinement, and infrared duality. Changing the number of flavors changes the infrared problem. Despite its name, massless SQCD with unbroken supersymmetry is not ordinary QCD with experimentally established superpartners.

Gauge group, matter, and interactions

Take gauge group $SU(N_c)$ with $N_c\ge3$ and $N_f$ massless pairs of chiral multiplets $Q_i,\widetilde Q^{\,j}$ in the fundamental and antifundamental representations. Each contains a scalar and a Weyl fermion. The vector multiplet contains the gauge boson and an adjoint Weyl gaugino.

Use canonical gauge-covariant kinetic terms, the supersymmetric Yang–Mills gauge action, and zero tree-level superpotential. The gauge coupling and theta angle specify the complex gauge parameter; a renormalization prescription fixes its associated strong scale $\Lambda$. These data, rather than $(N_c,N_f)$ alone, define the model. A quark superpotential mass or soft supersymmetry breaking changes it.

Gauge-invariant chiral coordinates include mesons $M_i{}^j=\widetilde Q^{\,j}Q_i$ and, when enough flavors exist, baryons formed by antisymmetrizing $N_c$ quarks. Classical vacua obey the gauge D-term conditions modulo gauge transformations. Quantum effects can change their relations and dynamics.

An anomaly and running check

With the fundamental index normalized to $1/2$, the one-loop gauge coefficient is

$$b_0=3N_c-N_f.$$

The electric description is asymptotically free for $N_f<3N_c$. For $N_f>0$, a candidate anomaly-free R symmetry assigns the scalar quarks charge $R_Q=1-N_c/N_f$. Their Weyl fermions have charge $R_Q-1$. The gauge–gauge–R anomaly is then

$$N_c+N_f(R_Q-1)=0.$$

The adjoint gaugino supplies $N_c$; the $N_f$ fundamental–antifundamental pairs supply the second term. This check constrains possible infrared descriptions. It does not determine every correlation function.

Flavor-dependent infrared behavior

Consider the origin of the moduli space, $M=B=\widetilde B=0$. Generic scalar expectation values can Higgs the gauge group and introduce additional scales; the following phase statements do not describe every such vacuum.

For $N_c+2\le N_f<3N_c$, Seiberg’s proposed magnetic description has gauge group $SU(N_f-N_c)$, magnetic quarks, a singlet meson, and a meson–quark superpotential. The two descriptions are proposed to agree in the infrared, with nontrivial checks from anomalies, operator maps, and deformations Seiberg 1995, §§2–4.

The interacting conformal window is $3N_c/2<N_f<3N_c$ in this framework. The interval $N_c+2\le N_f\le3N_c/2$, when nonempty, has a free magnetic description in the infrared; at its upper endpoint the approach to zero coupling is logarithmic. At $N_f=N_c$ or $N_c+1$, the special confining/moduli descriptions need separate formulas. A single generic “confining SQCD” label therefore loses essential information.

Protected holomorphic quantities can be controlled beyond ordinary perturbation theory, while unprotected Kähler data and generic real-time observables remain difficult.

Develop the calculation

SQCD Fields, Symmetries, Global Data, and Classical Moduli supplies the full defining data. Seiberg Duality in SQCD gives the magnetic dictionary and rank qualifications. Compare the greater supersymmetry of N=4 Super Yang–Mills.

References

• Seiberg, Nathan. “Electric-Magnetic Duality in Supersymmetric Non-Abelian Gauge Theories.” Nuclear Physics B 435 (1995): 129–146. DOI. Open text.