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.

The distinction between scalar and fermion charges is important. In the displayed basis the complete ultraviolet fermion card is

$$\begin{array}{c|c|rr} \text{left-handed fermion}&\text{multiplicity}&X&R\\ \hline \psi_A&10&1&-2\\ \psi_{\overline F}&5&-3&-4\\ \lambda&24&0&1 \end{array}$$

where the last row is the $SU(5)$ gaugino. For global-form bookkeeping it is sometimes cleaner to use $R_0=R+X$. Its scalar charges are $(0,-6)$, so the half-turn $e^{i\pi R_0}$ acts as fermion parity. We will quote the faithful ordinary flavor group as $U(1)_X/\mathbb Z_5$ and keep the spin-$R$ extension separate. The anomaly-free phase-rotation kernel is connected in this model, so there is no additional independent ordinary discrete factor hidden by the continuous notation.

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. Define the Pfaffian-adjugate vector

$$v_e=\epsilon_{abcde}A^{ab}A^{cd}.$$

Antisymmetry in five dimensions gives $v_eA^{ef}=0$, so the displayed contraction is zero. Higher candidate matter singlets factor through the same identities. Equivalently, the holomorphic invariant ring generated by $A$ and $\overline F$ is just the constants.

Why does an algebraic statement settle the $D$ terms? For a reductive gauge group, every $D$-flat configuration represents a closed orbit of the complexified gauge group, and distinct closed orbits are separated by holomorphic invariants. A nonzero $D$-flat orbit would therefore give some invariant a nonzero value. Since no such invariant exists, 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 invariant-orbit correspondence and its use in supersymmetric gauge theories are reviewed in Luty and Taylor 1996, §§ 2–3, arXiv PDF.

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.

The full Abelian anomaly record

Using the fermion charges above, the purely flavor and gravitational coefficients are

$$\begin{aligned} \operatorname{Tr}X&=-5,& \operatorname{Tr}X^3&=-125,\\ \operatorname{Tr}R&=24+10(-2)+5(-4)=-16,& \operatorname{Tr}R^3&=24+10(-2)^3+5(-4)^3=-376,\\ \operatorname{Tr}X^2R&=10(1)^2(-2)+5(-3)^2(-4)=-200,& \operatorname{Tr}XR^2&=10(1)(-2)^2+5(-3)(-4)^2=-200. \end{aligned}$$

Together with $SU(5)^3=SU(5)^2X=SU(5)^2R=0$, these are all continuous local and ‘t Hooft anomaly coefficients in the chosen Abelian basis. Under $R\mapsto R+tX$, the coefficients containing $R$ transform by the corresponding polynomial substitution; this basis dependence is not an ambiguity in the anomaly polynomial.

Anomalies constrain candidate infrared phases

If $U(1)_X$ remained unbroken and the infrared were a trivially gapped theory with no topological sector, these anomalies could not match. A symmetry-preserving proposal with no massless degrees of freedom is therefore inconsistent unless a suitable nontrivial infrared sector is supplied. Spontaneous breaking is a separate option and necessarily introduces a Goldstone mode. Numerically, the two $X$ anomalies equal those of one Weyl fermion of charge $-5$. In an unbroken supersymmetric description that fermion would need a chiral-multiplet partner, yet the ultraviolet matter invariant ring contains no nonconstant coordinate of charge $-5$. This excludes a naive one-composite description, not an emergent interacting sector. Gauge-field-strength operators such as $\operatorname{Tr}W^\alpha W_\alpha$ also exist, but they carry $X=0$ and do not solve this match. The 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. If a continuous global symmetry is known independently to break, then unbroken supersymmetry would place its Goldstone boson in a chiral multiplet with another massless real scalar. That partner parametrizes a noncompact flat direction, contradicting the no-flat-direction result. This is the conditional “broken continuous symmetry plus no flat directions” argument explained in Skiba 1997, § 2, pp. 2–4, arXiv PDF. The missing step is proving that symmetry breaking actually occurs and that every quantum escape route, including infinity or an emergent branch, has been excluded.

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 vectorlike extensions analyzed by Poppitz and Trivedi 1996, §§ 2–4, arXiv PDF and the antisymmetric-tensor dual descriptions of Pouliot 1996, §§ 2–4, arXiv PDF; Skiba 1997, §§ 3–4, arXiv PDF reviews their role in DSB arguments.

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.

3. Complete the Abelian anomaly polynomial

Using the fermion table, compute $\operatorname{Tr}R$, $\operatorname{Tr}R^3$, $\operatorname{Tr}X^2R$, and $\operatorname{Tr}XR^2$. Then explain why the numbers change under $R\mapsto R+tX$ without changing the physical anomaly.

Solution

The gaugino contributes $24$ to both $\operatorname{Tr}R$ and $\operatorname{Tr}R^3$. The matter contributions use multiplicities 10 and 5 and fermion charges $R=(-2,-4)$, giving

$$\operatorname{Tr}R=-16, \quad \operatorname{Tr}R^3=-376, \quad \operatorname{Tr}X^2R=-200, \quad \operatorname{Tr}XR^2=-200.$$

Changing $R$ by $tX$ changes the coordinate basis on the same two-dimensional Abelian symmetry group. Substituting $R+tX$ into the anomaly polynomial mixes the displayed coefficients but does not change the underlying multilinear anomaly functional.

References

• Affleck, Ian, Michael Dine, and Nathan Seiberg. “Dynamical Supersymmetry Breaking in Chiral Theories.” Physics Letters B 137 (1984): 187–192. DOI.
• Luty, Markus A., and Washington Taylor IV. “Varieties of Vacua in Classical Supersymmetric Gauge Theories.” Physical Review D 53 (1996): 3399–3405. DOI; arXiv.
• Poppitz, Erich, and Sandip P. Trivedi. “Some Examples of Chiral Moduli Spaces and Dynamical Supersymmetry Breaking.” Physics Letters B 365 (1996): 125–131. DOI; arXiv.
• Pouliot, Philippe. “Duality in SUSY $SU(N)$ with an Antisymmetric Tensor.” Physics Letters B 367 (1996): 151–156. DOI; arXiv.
• Skiba, Witold. “Dynamical Supersymmetry Breaking.” Modern Physics Letters A 12 (1997): 737–750. DOI; arXiv.