Chiral Gauge Theories and Anomaly Constraints

Chiral gauge theories do not admit the matrix-of-mesons shortcut familiar from vectorlike SQCD. A reliable analysis begins with the exact gauge and global data, proves that the microscopic gauge theory is consistent, determines the complete invariant ring and its branches, and only then asks which nonperturbative terms or infrared spectra are possible. The minimal $SU(5)$ theory with one ${\bf 10}$ and one $\overline{\bf 5}$ is a compact example: every elementary check is explicit, yet those checks constrain rather than uniquely determine its strongly coupled infrared physics.

Required background. SQCD fields, symmetries, and moduli supplies the theory-card method, while nonperturbative superpotentials supplies the selection-rule logic. Helpful background. Global and torsion anomalies covers nonperturbative gauge consistency, and Nelson–Seiberg-type criteria explains why an $R$ symmetry is a diagnostic rather than a proof of supersymmetry breaking.

A complete chiral-theory test

For a proposed four-dimensional $\mathcal N=1$ theory, the following order prevents circular reasoning.

1. Declare the gauge algebra and global form, every chiral representation, the tree superpotential, and line-operator data.
2. Cancel perturbative cubic gauge anomalies and check global gauge anomalies. A failure here means there is no quantum gauge theory to analyze.
3. Compute the nonanomalous continuous and discrete global symmetry, including finite quotients shared with the gauge center.
4. Construct the holomorphic invariant ring and its relations. Use it to find $D$-flat branches and unbroken gauge subgroups.
5. Assign spurion charges to each holomorphic scale, list every allowed exact term, and test zero-mode counting and weakly Higgsed limits where available.
6. Propose an infrared spectrum only after these steps, then match all ‘t Hooft anomalies and deform the theory to neighboring regimes.

An anomaly match is necessary for an unbroken global symmetry. It is not sufficient to prove confinement, completeness of a proposed composite spectrum, or supersymmetry breaking.

The minimal chiral SU(5) theory

Consider simply connected $SU(5)$ with no tree superpotential and chiral multiplets

$$A^{ab}=-A^{ba}\in{\bf 10}, \qquad \overline F_a\in\overline{\bf 5}.$$

These representations do not descend to a nontrivial quotient of $SU(5)$, and the antifundamental screens every center charge. There is therefore no electric one-form symmetry. With $T({\bf 5})=1/2$ and $T({\bf 10})=3/2$, the holomorphic one-loop coefficient is

$$b_0=3T(\mathrm{adj})-T({\bf 10})-T(\overline{\bf 5}) =15-\frac32-\frac12=13,$$

so the instanton factor is proportional to $\Lambda^{13}$.

Gauge consistency

Normalize the cubic $SU(5)$ anomaly of a left-handed ${\bf 5}$ to $+1$. The two-index antisymmetric has coefficient $A({\bf 10})=5-4=+1$, whereas $A(\overline{\bf 5})=-1$. Hence

$$\mathcal A[SU(5)^3]=1-1=0.$$

There is no four-dimensional Witten-type global gauge anomaly because $\pi_4(SU(5))=0$. These are logically separate checks: the first is a local triangle anomaly, the second tests large gauge transformations.

Continuous global symmetry

An anomaly-free flavor symmetry $U(1)_X$ can be normalized by

$$X(A)=1, \qquad X(\overline F)=-3,$$

because $\frac32(1)+\frac12(-3)=0$. Its order-five subgroup overlaps the gauge center: an $X$ rotation by $2\pi/5$ acts like the gauge-center element $z^3$. Thus the faithful ordinary flavor action is $U(1)_X/\mathbb Z_5$; although this quotient is abstractly another circle, it matters for allowed background bundles and charge normalization.

One convenient anomaly-free $R$ assignment for scalar components is

$$R(A)=-1, \qquad R(\overline F)=-3.$$

Indeed the mixed gauge anomaly, including the charge-one gaugino, is

$$\mathcal A[SU(5)^2U(1)_R] =5+\frac32(-1-1)+\frac12(-3-1)=0.$$

Mixing $R\mapsto R+tX$ produces the expected one-parameter family. No preferred superconformal combination follows from this ultraviolet calculation.

Why there is no classical flat direction

The tempting cubic $A^{ab}\overline F_a\overline F_b$ vanishes because chiral superfields commute, so $\overline F_a\overline F_b$ is symmetric while $A^{ab}$ is antisymmetric. A second-looking contraction,

$$\epsilon_{abcde}A^{ab}A^{cd}A^{ef}\overline F_f,$$

also vanishes identically: for any antisymmetric $5\times5$ matrix, its Pfaffian-adjugate vector is a null vector of that matrix. Higher candidate singlets factor through the same identities. Equivalently, the holomorphic invariant ring is just the constants.

For a reductive gauge group, nonzero supersymmetric $D$-flat orbits are separated from the origin by holomorphic invariants. Since there is no nonconstant invariant, the only classical $D$-flat configuration is the origin. This is more informative than a component-by-component attempt to solve 24 $D$-term equations: there is no Higgs regime in which the original theory becomes weakly coupled along a modulus.

The absence of an invariant also forbids an ordinary holomorphic superpotential built solely from gauge-invariant chiral coordinates. It does not imply that strong dynamics is absent. A quantum theory may break a global symmetry, contain an interacting sector, or generate nonlocal infrared variables not visible as polynomial ultraviolet invariants.

Anomalies constrain candidate infrared phases

The microscopic fermions give simple nonzero $U(1)_X$ anomalies:

$$\operatorname{Tr}X=10(1)+5(-3)=-5, \qquad \operatorname{Tr}X^3=10(1)^3+5(-3)^3=-125.$$

If $U(1)_X$ remained unbroken and the infrared were a trivially gapped theory with no topological sector, these anomalies could not match. A proposal with no massless composites is therefore inconsistent unless the symmetry is broken or a suitable nontrivial infrared sector is supplied. Conversely, inventing a single composite is impossible here because no gauge-invariant chiral polynomial exists. This sharp obstruction was central to the original strong-dynamics analysis of Affleck, Dine, and Seiberg 1984, pp. 188–191.

What does this establish? It excludes a symmetric, trivially gapped vacuum. It does not alone choose between spontaneous global-symmetry breaking, an interacting massless phase, or dynamical supersymmetry breaking. The last claim needs an argument that every supersymmetric escape route is absent, including quantum branches not visible classically.

Deformations make stronger tests possible

A useful strategy embeds the theory in a nearby model with extra vectorlike matter or weak interactions, where a Higgsed or dual description is calculable. One then determines the exact superpotential, follows all vacua as the deformation is removed, and checks whether any run to infinity. Holomorphy controls the continuation only away from singularities; a vacuum escaping to infinite field distance invalidates a naive index or continuity argument. This deformation logic, rather than SQCD analogy, underlies the classic chiral examples reviewed in Skiba 1997, §§ 3–4.

What can and cannot be concluded

For the one-family $SU(5)$ model the following statements have different status:

• Gauge-anomaly cancellation, $b_0=13$, the continuous anomaly-free symmetries, the absence of polynomial invariants, and the displayed ‘t Hooft anomalies are exact algebraic results.
• Excluding a symmetric trivial gap follows exactly if the stated global symmetry is unbroken and no anomaly-carrying topological sector is present.
• Dynamical supersymmetry breaking is the standard strong-coupling interpretation, supported by deformations and consistency arguments, but the undeformed theory has no parametrically weak regime in which its Kähler potential and vacuum energy can be computed.

This last distinction motivates the calculable product-group example on the dynamical supersymmetry-breaking page.

Common pitfalls

Checking only the cubic gauge anomaly. A vanishing triangle anomaly does not remove possible global gauge anomalies or specify the faithful global group. Both must be checked before infrared reasoning.

Treating an allowed term as a generated term. Symmetry and dimension provide a candidate list. Zero modes, branches, decoupling, and a controlled limit decide whether a coefficient is nonzero.

Inferring supersymmetry breaking from an $R$ symmetry. An $R$ symmetry is an important necessary ingredient in generic F-term models, subject to hypotheses. In a strongly coupled gauge theory one must still exclude runaway vacua, emergent fields, and alternative branches.

Exercises

1. Reconstruct the anomaly-free charges

Let $X(A)=x$ and $X(\overline F)=f$. Solve the $SU(5)^2U(1)_X$ condition and verify the quoted faithful $\mathbb Z_5$ overlap.

Solution

The mixed anomaly is $\frac32x+\frac12f$, so $f=-3x$; choose $x=1$. Under $\alpha=2\pi/5$, the fields acquire phases $(z,z^{-3})=(z,z^2)$. The gauge-center element $z^3$ acts on the ${\bf 10}$ as $z^6=z$ and on the $\overline{\bf 5}$ as $z^{-3}=z^2$. Hence that flavor transformation is gauge-equivalent to the identity and must be quotiented.

2. Why anomaly matching is not a DSB proof

List three infrared possibilities compatible with the nonzero $U(1)_X$ anomalies, and state which extra fact a DSB proof must establish.

Solution

The symmetry may be spontaneously broken, massless interacting degrees of freedom may reproduce the anomalies, or a nontrivial topological sector may carry them. A DSB proof must additionally show that no zero-energy supersymmetric state exists anywhere, including at infinity or on an emergent quantum branch. The anomaly calculation alone excludes only the symmetric trivial gap.

References

• Affleck, Ian, Michael Dine, and Nathan Seiberg. “Dynamical Supersymmetry Breaking in Chiral Theories.” Physics Letters B 137 (1984): 187–192. DOI.
• Skiba, Witold. “Dynamical Supersymmetry Breaking.” Modern Physics Letters A 12 (1997): 737–750. DOI; arXiv.