Infrared Phases and the Conformal Window

The infrared regime of massless $SU(N_c)$ SQCD is governed not by one slogan but by a sequence of qualitatively different exact descriptions. The rank inequalities can be derived cleanly; the physical labels require more care. Below, “exact” means a protected holomorphic or anomaly statement, “controlled” means a weak-coupling expansion exists, and “duality-supported” means the conclusion also uses Seiberg’s infrared electric–magnetic equivalence, which is exceptionally well tested but is not a mathematical theorem.

Required background. Quantum-modified moduli and s-confinement supplies the exact low-rank boundary cases. Holomorphic and canonical couplings and the NSVZ relation supplies the beta-function conventions. Helpful background. Ultraviolet and infrared fixed points explains the evidence needed to distinguish a fixed point from slow running.

The theory whose phases are being classified

Unless stated otherwise, take four-dimensional Lorentzian $\mathcal N=1$ $SU(N_c)$ gauge theory with $N_c\geq3$, $N_f$ pairs $(Q^i,\widetilde Q_i)$, no masses, no tree superpotential, and the simply connected global form. The one-loop electric coefficient is

$$b_e=3N_c-N_f.$$

Thus the electric variables are asymptotically free only for $N_f<3N_c$. For $N_f>3N_c$ the same Lagrangian can be treated as a cutoff effective theory with an infrared-free gauge coupling, but it is not a UV-complete asymptotically free theory by itself.

Fundamental matter breaks the electric center one-form symmetry and screens a fundamental probe. Consequently “confinement” below never means an exact fundamental Wilson-loop area law. It means that the appropriate massless infrared variables are gauge-invariant composites and no gauge boson remains in the infrared description. Chiral symmetry realization, massless spectrum, and line-operator behavior are separate observables.

Exact special regimes

Nf = 0: pure super-Yang–Mills

Anomaly and holomorphy give $N_c$ supersymmetric vacua and a gaugino condensate with phases $e^{2\pi i k/N_c}$. The vacuum count and condensate are protected. A mass gap and flux-tube dynamics are standard, strongly supported dynamical conclusions, not consequences of holomorphy alone; see Veneziano and Yankielowicz 1982, pp. 233–235.

0 < Nf < Nc: ADS runaway

The exact superpotential

$$W=(N_c-N_f) \left(\frac{\Lambda^{3N_c-N_f}}{\det M}\right)^{1/(N_c-N_f)}$$

has no stationary point at finite $M$ in the massless theory. The correct statement is therefore “runaway,” not a vacuum phase. Adding generic masses stabilizes the fields and produces the expected pure-theory vacua after holomorphic decoupling.

Nf = Nc: quantum-modified moduli

The exact constraint $\det M-B\widetilde B=\Lambda^{2N_c}$ removes the classical origin. There are supersymmetric vacua on a smooth moduli space, with flavor symmetry broken differently on different branches. This is sometimes called confinement with chiral symmetry breaking, but the phrase suppresses the branch dependence and the presence of massless moduli.

Nf = Nc + 1: s-confinement

The fields $M,B,\widetilde B$ and their exact confining superpotential give a smooth description also at the origin. The origin preserves the nonanomalous continuous flavor symmetry and contains massless composites. This is the cleanest s-confining SQCD example; it is not a gapped phase.

These four statements use holomorphy, anomalies, moduli counting, and mass deformations but do not use the full electric–magnetic duality conjecture. Their unified derivation appears in Intriligator and Seiberg 1996, §§ 4.1–4.3, pp. 39–49.

Magnetic rank and the two open intervals

For $N_f\geq N_c+2$, Seiberg’s proposed infrared description has gauge group

$$SU(\widetilde N_c), \qquad \widetilde N_c=N_f-N_c,$$

magnetic quarks $q,\widetilde q$, a singlet meson $M$, and $W=yM q\widetilde q$. Its one-loop coefficient is

$$b_m=3\widetilde N_c-N_f=2N_f-3N_c.$$

The signs of $b_e$ and $b_m$ divide the remaining range.

Nc + 2 ≤ Nf < 3Nc/2: free magnetic phase

Here $b_e>0$ but $b_m<0$. The electric theory becomes strong while the magnetic gauge and Yukawa couplings run to zero in the infrared. The magnetic quarks, gauge multiplet, and meson therefore provide a weakly coupled long-distance description. The rank inequality is perturbative; identifying this description with the electric infrared theory is duality-supported. It is called “free magnetic,” not confining, because an infrared gauge field survives.

The interval contains integers only when the ranks permit them. For example, at $N_c=3$ the inequality $5\leq N_f<4.5$ has no solution. Writing a phase name without first checking that the integer interval is nonempty is a common source of false examples.

3Nc/2 < Nf < 3Nc: interacting non-Abelian Coulomb phase

Both descriptions are asymptotically free, so each becomes strongly coupled toward the infrared. The duality proposal and superconformal constraints support flow to one interacting fixed point described by either theory. The anomaly-free candidate $R$-charge gives

$$R(Q)=R(\widetilde Q)=1-\frac{N_c}{N_f}, \qquad \Delta(M)=\frac32R(M)=3\left(1-\frac{N_c}{N_f}\right).$$

At $N_f=3N_c/2$, $\Delta(M)=1$, the scalar unitarity bound. This reproduces the lower boundary and is independent of the one-loop magnetic sign. Within the open interval the fixed point is perturbatively controlled near $N_f=3N_c$ in electric variables and near $N_f=3N_c/2$ in magnetic variables; away from both edges, the existence and dictionary are duality-supported. The full fixed-point and accidental-symmetry analysis belongs with the duality chapter; here the formula is used only to test the phase boundary. The foundational evidence is Seiberg 1995, §§ 3–5, pp. 135–144.

Endpoints are not interior points

When $3N_c/2$ is an integer, the lower endpoint has $b_m=0$, and the meson and magnetic variables approach free-field dimensions. The cubic magnetic coupling is marginally irrelevant; the endpoint is most precisely described as a logarithmically free magnetic boundary, not as a generic interacting member of the open conformal window.

At $N_f=3N_c$, $b_e=0$. The electric gauge coupling is marginally irrelevant near the origin, giving a free infrared endpoint; the theory is not asymptotically free. For $N_f>3N_c$, $b_e<0$ and the electric theory is infrared free but has a Landau pole in the ultraviolet. Thus “free electric” is an infrared statement plus a UV-cutoff qualification.

For $N_c=2$, pseudoreality enhances the flavor symmetry to $SU(2N_f)$ and reorganizes mesons and baryons. The numerical inequalities remain useful, but anomaly tables and operator dictionaries must be recomputed in the enhanced symmetry. The $N_c=1$ notation does not describe a non-Abelian gauge theory.

How strong is each conclusion?

Three logically independent checks should be kept visible:

1. Exact structural checks: nonanomalous symmetries, ‘t Hooft anomalies, holomorphic decoupling, quantum constraints, and protected chiral operators.
2. Controlled dynamical checks: a small electric or magnetic coupling near a boundary, so beta functions and operator dimensions can be computed.
3. Duality checks: matching moduli spaces, deformations, anomalies, baryons, and flows between different ranks. These strongly constrain the proposed equivalence but are mutually connected consequences of one dictionary, not dozens of independent proofs.

The regime map, including the free-magnetic and non-Abelian Coulomb terminology, was developed in Seiberg 1995, pp. 134–146 and organized with explicit evidence limits in Intriligator and Seiberg 1996, § 5, pp. 49–61. Its exact anomaly and holomorphy tests remain valid; the assertion of a common interacting fixed point remains an infrared-duality claim.

Nothing in this map may be extrapolated mechanically to nonsupersymmetric QCD. The superpotential, chiral ring, holomorphy, anomaly-free $R$ symmetry, and weakly coupled magnetic description do essential work. Removing the gaugino or squarks removes those arguments even if the symbols $N_c,N_f$ remain.

Common pitfalls

Treating every rank range as a vacuum phase. Massless SQCD with $0<N_f<N_c$ runs away and has no finite vacuum. A mass deformation creates vacua, but that is a different theory.

Including endpoints silently. Both $N_f=3N_c/2$ and $N_f=3N_c$ have vanishing one-loop coefficients on one side and logarithmic qualifications. Use open intervals for the interacting conformal window.

Using “confinement” as one universal observable. Composite infrared variables, a mass gap, center symmetry, Wilson-loop behavior, and chiral symmetry realization can disagree. State which one is meant.

Exercises

1. Classify three examples

Classify massless $SU(4)$ SQCD with $N_f=5,6,9$ and state the evidence type for each label.

Solution

$N_f=5=N_c+1$ is s-confining, an exact composite description. $N_f=6=3N_c/2$ is the logarithmically free magnetic endpoint, supported by duality plus a weakly coupled magnetic limit; it is not inside the open interacting window. $N_f=9$ lies in $6<N_f<12$, the duality-supported interacting non-Abelian Coulomb phase. It is not parametrically close to the electric Banks–Zaks edge at $N_f=12$, so neither electric nor magnetic variables are necessarily very weakly coupled.

2. Recover the lower boundary twice

Derive the lower edge of the candidate conformal window from (a) the magnetic one-loop coefficient and (b) the meson unitarity bound.

Solution

(a) $b_m=2N_f-3N_c$ changes sign at $N_f=3N_c/2$. Below it the magnetic theory is infrared free. (b) $\Delta(M)=3(1-N_c/N_f)\geq1$ gives $N_f\geq3N_c/2$. Equality means $M$ reaches the free-scalar bound, so it marks a boundary rather than a generic interacting point.

References

• Intriligator, Kenneth, and Nathan Seiberg. “Lectures on Supersymmetric Gauge Theories and Electric–Magnetic Duality.” Nuclear Physics B Proceedings Supplements 45BC (1996): 1–28. DOI; arXiv PDF.
• Seiberg, Nathan. “Electric–Magnetic Duality in Supersymmetric Non-Abelian Gauge Theories.” Nuclear Physics B 435 (1995): 129–146. DOI; arXiv.
• Veneziano, Gabriele, and Shimon Yankielowicz. “An Effective Lagrangian for the Pure $N=1$ Supersymmetric Yang–Mills Theory.” Physics Letters B 113 (1982): 231–236. DOI.