R-Symmetry and Nelson–Seiberg-Type Criteria

A continuous R-symmetry can turn the F-flatness problem into a powerful counting test, but only after its hypotheses are stated. Nelson–Seiberg-type criteria concern generic, regular Wess–Zumino effective theories and specified classes of vacua; they are not unconditional equivalences between “has an R-symmetry” and “breaks supersymmetry.” This page derives the original coordinate argument, proves the useful R-charge-two versus R-neutral count near an R-symmetric locus, and gives examples that separate necessity, sufficiency, spontaneous R breaking, exceptions, and runaways.

Required background. Use the explicit rank-condition models on O’Raifeartaigh and Fayet–Iliopoulos Model Laboratories and the meaning of holomorphic parameters on Holomorphic Couplings and Background Superfields.

Helpful background. Review Multiplets, Invariants, and Selection Rules for the distinction between an exact quantum symmetry, an anomalous current, and a spurionic selection rule.

Continuous R-symmetry and F-flatness

Let the Grassmann coordinate transform as $\theta\mapsto e^{-i\alpha}\theta$. Then $\mathrm d^2\theta$ has charge $+2$, so invariance of $\int\mathrm d^2\theta\,W$ requires $R(W)=-2$ in that convention. Equivalently—and more commonly in model-building—one chooses $R(\theta)=+1$ and $R(W)=+2$. This page uses the latter convention.

For chiral fields $\Phi^i$ of R-charges $r_i$,

$$W(e^{ir_i\alpha}\Phi^i)=e^{2i\alpha}W(\Phi).$$

Differentiating at $\alpha=0$ gives the quasi-homogeneity relation

$$\sum_i r_i\Phi^iW_i=2W.$$

An R-symmetric field configuration has nonzero expectation values only for R-neutral fields. If an exact continuous R-symmetry is spontaneously broken in a stable vacuum, a massless R-axion accompanies it; a dynamical-gauge ABJ anomaly or an explicit R-breaking term can lift that scalar. Supersymmetry breaking itself does not require spontaneous R breaking: the model $W=fX$ with $R(X)=2$ breaks supersymmetry through $F_X=-f$ at the R-symmetric point $X=0$.

The symmetry must be a symmetry of the quantum effective action used in the argument. An ABJ anomaly involving a dynamical gauge field makes the putative continuous R-current nonconserved and invalidates the theorem’s premise. An ’t Hooft anomaly of an exact global symmetry is different: it records a nontrivial response to background fields but does not by itself destroy the symmetry. Likewise, assigning charges to couplings as frozen spurions describes covariance of a family of theories; one fixed theory has an exact R-symmetry only if its frozen couplings are invariant or arise from background fields that remain part of the stated system.

The original Nelson–Seiberg argument

Consider a calculable four-dimensional $\mathcal N=1$ theory whose light fields near a putative ground state admit a Wess–Zumino description with a finite, nonsingular Kähler metric. Assume that a ground state exists and that the effective superpotential is locally generic in coordinates adapted to the patch containing it. The original Nelson–Seiberg statement is then twofold: an exact continuous R-symmetry is necessary for generic F-term supersymmetry breaking, and a global minimum that spontaneously breaks that R-symmetry cannot be supersymmetric.

To see the second implication, suppose a field $\phi_n$ of nonzero charge $r_n$ has $\langle\phi_n\rangle\neq0$. On a local patch where a branch of the fractional power is regular, define

$$X=\phi_n^{2/r_n}, \qquad Y_i=\phi_i\phi_n^{-r_i/r_n}.$$

Then $R(X)=2$, the $Y_i$ are neutral, and R-invariance forces

$$W=Xf(Y_1,\ldots,Y_{N-1}).$$

At $X\neq0$, the F-flatness equations are

$$f(Y)=0, \qquad \frac{\partial f}{\partial Y_i}=0 \quad(i=1,\ldots,N-1).$$

These are $N$ holomorphic equations for $N-1$ variables. A generic $f$ has no common solution, so an R-breaking global minimum in this class cannot be supersymmetric. Conversely, without an R-symmetry, symmetry-allowed local deformations generically supply enough independent parameters to restore an F-flat solution. This is the content and scope of the original argument Nelson and Seiberg 1994, §2, arXiv PDF pp. 3–8.

Every condition matters. A runaway may leave no ground state; the coordinate transformation or Kähler metric can be singular; $f$ can be nongeneric even when the original polynomial looked generic; and the superpotential can be nonpolynomial on the relevant domain. Light gauge fields and D-terms are outside this Wess–Zumino proof. The statement is therefore a theorem within a declared theory class and field patch, not a classification of every supersymmetric QFT.

The R-charge-two versus neutral count

A more directly usable test concerns R-symmetric vacua. Partition the chiral fields into

• $X_i$, $i=1,\ldots,N_X$, with R-charge $2$;
• $Y_a$, $a=1,\ldots,N_Y$, with R-charge $0$;
• $A_\alpha$ with all other R-charges.

Near the R-symmetric locus $X_i=A_\alpha=0$, a regular R-symmetric superpotential has

$$W=\sum_{i=1}^{N_X}X_i f_i(Y)+W_{\geq2}(X,A,Y),$$

where every term in $W_{\geq2}$ contains enough non-neutral fields to make its first derivatives vanish on that locus. There,

$$F_{X_i}^*=-f_i(Y), \qquad F_{Y_a}=F_{A_\alpha}=0.$$

Thus an R-symmetric supersymmetric vacuum is exactly a common zero of

$$f_1(Y)=\cdots=f_{N_X}(Y)=0.$$

For generic holomorphic $f_i$:

• if $N_X>N_Y$, an R-symmetric solution is overdetermined and generically absent;
• if $N_X=N_Y$, isolated R-symmetric solutions are expected when they lie in the field domain;
• if $N_X<N_Y$, a solution set of expected complex dimension $N_Y-N_X$ is possible.

This is first a local algebraic statement about the R-symmetric locus. At a regular common zero, the expected complex dimension is $N_Y-N_X$. Even when $N_X\leq N_Y$, dimension counting does not prove that a common zero exists in the physical field domain, nor that it is stable, within the EFT, or compatible with D-flatness.

Kang, Li, and Sun proposed a stronger global formulation: a generic perturbative Wess–Zumino model with a ground state breaks supersymmetry at its global minimum if and only if it has an R-symmetry and $N_X>N_Y$ for every consistent R-charge assignment Kang, Li, and Sun 2013, §2, arXiv PDF pp. 2–3. That formulation additionally assumes a polynomial perturbative superpotential and is not reliable without restrictions on R-breaking branches. Counterexamples contain oppositely charged fields whose product condenses and produces a supersymmetric branch despite the naive count Amariti and Sauro 2020, §§3–5, pp. 4–9. In coordinates adapted to a nonzero charged field, the corresponding $f$ is nongeneric, so these examples do not contradict the original local-coordinate argument Amariti and Sauro 2020, §6, pp. 9–12.

Accordingly, use $N_X>N_Y$ as a local obstruction unless alternative R assignments, nonstandard-charge condensates, singular loci, infinity, and genericity in the adapted coordinates have all been checked. Fields $A_\alpha$ can acquire expectation values, the R-symmetry can break, a singular branch can appear, or the potential can run away.

The O’Raifeartaigh count

For

$$W=fX+\frac h2X\phi_1^2+m\phi_1\phi_2,$$

the R-charge-two fields are $X$ and $\phi_2$, while $\phi_1$ is neutral. Hence $N_X=2>N_Y=1$. On the R-symmetric locus,

$$f_X(\phi_1)=f+\frac h2\phi_1^2, \qquad f_{\phi_2}(\phi_1)=m\phi_1.$$

The second equation sets $\phi_1=0$, where the first is $f\neq0$. The abstract count reproduces the explicit rank condition.

Examples that separate the claims

R-symmetric supersymmetric vacua

Let

$$W=X(f+\lambda Y^2), \qquad R(X)=2, \qquad R(Y)=0.$$

Here $N_X=N_Y=1$. The equations have solutions

$$X=0, \qquad Y^2=-\frac f\lambda,$$

so supersymmetry and the R-symmetry are both preserved. The mere presence of an R-symmetry therefore does not imply breaking.

Supersymmetry breaking without spontaneous R breaking

For $W=fX$, $N_X=1$ and $N_Y=0$. There is no F-flat point, but choosing $\langle X\rangle=0$ preserves the R-symmetry. At tree level every $X$ is degenerate, so other points on the valley break R; quantum or higher-dimensional terms decide which state is selected. “R-symmetry is necessary” in the generic-theory argument does not mean “R-symmetry must be spontaneously broken.”

Explicit R breaking can restore a distant vacuum

Add

$$\Delta W=\frac\epsilon2X^2$$

to $W=fX$. This term explicitly violates the original R-symmetry and gives a supersymmetric solution

$$X_{\rm susy}=-\frac f\epsilon.$$

For canonical Kähler potential the full scalar potential is simply

$$V=|f+\epsilon X|^2,$$

so it slopes directly to $X_{\rm susy}$ and has no nonsupersymmetric local minimum near the origin. This one-field model demonstrates only that the supersymmetric solution recedes to infinity as $\epsilon\to0$. If the same deformation is added to an extended model in which loops or controlled Kähler terms already stabilize $X$, a shifted local minimum may survive for sufficiently small $\epsilon$ while the supersymmetric vacuum remains far away. Its stability and lifetime must then be calculated separately. In either case, the limit is nonuniform in field space and cannot be inferred from a fixed-radius expansion.

Spontaneous R breaking requires a vacuum calculation

The field count locates possible F-flat solutions but does not determine the minimum along a pseudomodulus. Consider the renormalizable single-pseudomodulus form

$$W=fX+\frac12(M+XN)_{ij}\phi_i\phi_j$$

with canonical Kähler potential, nonsingular $M$, and a locally tachyon-free pseudomoduli space. If every field admits an R-charge assignment using only $0$ and $2$, the one-loop curvature at the R-symmetric point is nonnegative and is positive when $X$ couples nontrivially. Fields with other charges are therefore necessary for the familiar one-loop spontaneous R-breaking mechanism under these hypotheses Shih 2008, §2.2, pp. 7–8. They are not sufficient: the sign still depends on masses and couplings, and tachyons or runaways can intervene.

What each supersymmetry-breaking diagnostic can establish

Use this comparison to keep exact criteria, conditional theorems, model calculations, and deformation arguments at their proper logical strength. Read each row from its hypotheses to the controlled conclusion, then check the two limitation columns before reusing the result in another model or limit. The adversarial example is a failure test, while the final column records what must remain controlled when soft terms vanish or heavy states decouple.

What each supersymmetry-breaking diagnostic can establish.

Diagnostic/result	Logical status	Required hypotheses	Controlled conclusion	What remains uncontrolled	Adversarial failure/counterexample	Zero-soft/decoupling behavior	Primary source
Incompatible F- and D-flatness equations	Exact equation-level criterion	Exact rigid supersymmetry; complete physical field domain and gauge quotient; positive nonsingular kinetic metrics; every F- and D-equation included; finite normalizable configurations and boundary conditions fixed.	If the complete physical domain contains no simultaneous solution of all F = 0 and D = 0 equations, it contains no zero-energy supersymmetric vacuum. At a stationary physical configuration, a nonzero auxiliary expectation value is a local supersymmetry-breaking order parameter.	Existence or stability of a nonsupersymmetric ground state, runaways or singular branches outside the scanned domain, tunneling rates, and the supergravity or boundary problem.	Patch-only search: for W = fX + εX²/2 with nonzero ε, checking only X = 0 reports F_X = f, but the omitted point X = −f/ε solves F_X = 0.	Frozen nonsupersymmetric soft terms invalidate the zero-auxiliary equivalence. Sending every explicit soft coefficient to zero restores the criterion only if the field domain and limit are uniform; integrating out a heavy field requires matched effective F- and D-equations.	O’Raifeartaigh 1975, pp. 331–352; Fayet–Iliopoulos 1974, pp. 461–464 The logical criterion also follows directly from the positive auxiliary-field potential under the stated kinetic and boundary assumptions.
Stable positive-energy vacuum	Sufficient algebraic criterion	Exact global supersymmetry; a homogeneous Lorentz-invariant stable ground-state phase with finite vacuum-energy density; well-defined regulated supercharges with vanishing surface terms; the rigid positive-energy algebra; no supergravity gauging.	Strictly positive vacuum-energy density implies that at least one supercharge fails to annihilate the vacuum, so global supersymmetry is spontaneously broken.	A positive-energy saddle or local minimum is not thereby the ground state. The statement does not determine the goldstino composition, vacuum lifetime, or the supergravity cosmological constant.	Positive stationary point: W = fX − gX³/3 with positive real f and g has V(0) = f², but X = 0 is unstable and the supersymmetric minima X = ±√(f/g) have V = 0.	With explicit soft breaking, positive vacuum energy is not an order parameter. In a uniform zero-soft limit the rigid algebraic implication returns; a phase boundary or gap closing can make the limiting vacuum discontinuous.	Witten 1982, §§ 1–2, pp. 253–271
Massless pole in the supercurrent channel	Goldstone-theorem diagnostic	Exact conserved global supercurrent and charge; local relativistic QFT with a positive spectral representation; infinite-volume spontaneous breaking; vanishing improvement flux; no explicit breaking, boundary leakage, or supergravity.	A nonzero vacuum-to-fermion supercurrent matrix element gives a massless 1/p² pole with nonzero residue: the goldstino required by spontaneous global supersymmetry breaking.	Whether the goldstino is elementary or composite, the ultraviolet completion, global vacuum stability, and any decay rate. A generic massless fermion is not identified without the current residue.	Accidental zero mode: append a decoupled free massless supersymmetric chiral multiplet to an unbroken theory. Its fermion is massless, but the vacuum-to-one-fermion goldstino residue of the supercurrent vanishes.	Frozen soft breaking makes the visible-sector supercurrent nonconserved, so no exact pole is required. A uniform zero-soft limit can restore the Ward identity; a dynamical hidden sector must be restored before calling its fermion a goldstino.	Salam–Strathdee 1974, pp. 465–467
Nonzero Witten index	One-way obstruction	A well-defined graded trace or Fredholm replacement; trace-class heat kernel or controlled continuum subtraction; normalizable states; stable asymptotics and no gap-closing contribution during the deformation.	A nonzero index requires at least one unpaired zero-energy supersymmetric state, so complete spontaneous supersymmetry breaking is impossible under the stated regulator and domain.	A zero index does not imply breaking, and a nonzero index does not locate vacua, count them without signs, determine a mass gap, or prove a condensate.	Zero but unbroken: supersymmetric quantum mechanics on S¹ has one even and one odd harmonic state, so the index is 1 − 1 = 0 although supersymmetry is unbroken.	The index is an exact-supersymmetry invariant, not an observable of a softly broken Hamiltonian. Its zero-soft or heavy-threshold transport requires the regulated trace, gap, and asymptotic domain to remain controlled.	Witten 1982, § 2, pp. 261–271
Original Nelson–Seiberg criterion	Conditional genericity theorem	A calculable low-energy Wess–Zumino description without light gauge fields; a generic locally holomorphic superpotential consistent with the exact symmetries; finite nonsingular Kähler metric; for the sufficient direction, a stable ground state with a finite nonzero R-charged coordinate that spontaneously breaks an exact continuous R-symmetry.	Within that class, an R-symmetry is necessary for generic F-term breaking; a ground state that spontaneously breaks the exact R-symmetry is generically nonsupersymmetric because f = 0 and all derivatives of f = 0 overconstrain the adapted coordinates.	R-symmetric vacua, nongeneric or nonperturbative superpotentials, singular coordinate patches, fields at zero or infinity, light gauge dynamics, anomalous or spurionic R assignments, and whether a stable ground state exists.	Runaway rather than vacuum: for real C > 0 and canonical Kähler potential K = S†S, W_eff = exp(−CS) has nonzero F_S at every finite S while V = C² exp(−2C Re S) tends to zero as Re S tends to infinity. More generally, the inference requires explicit asymptotic control of the inverse Kähler metric so K^(S S̄)|∂_S W|² tends to zero; finite-S regularity alone does not suffice.	An explicit soft term removes the exact-symmetry hypothesis. Taking it to zero recovers the theorem only if the ground state and adapted coordinate patch converge uniformly rather than escaping to infinity or crossing a singularity.	Nelson–Seiberg 1994, § 2, arXiv PDF pp. 3–8, especially Eqs. (2.2)–(2.5); § 2, arXiv PDF pp. 8–9, Eqs. (2.14)–(2.16)
N_X > N_Y at an R-symmetric locus	Local generic absence count	A perturbative polynomial Wess–Zumino superpotential with a fixed consistent R assignment; X_i have charge 2, Y_j charge 0, and A_k other charges; all symmetry-allowed terms retained; a regular locus X = A = 0; generic functions f_i(Y).	At that R-symmetric locus, N_X equations f_i(Y) = 0 in N_Y variables have no generic common solution when N_X > N_Y. This excludes a generic R-symmetric supersymmetric vacuum on that locus only.	Supersymmetric vacua with other-charge condensates and broken R symmetry, alternative consistent R assignments, special coefficient loci, singular or nonpolynomial branches, runaways, and global-minimum existence.	Opposite-charge condensate: W = μ²z + az²φ₁ + bzφ₂φ₃ + cφ₁²φ₂ with R(z,φ₁,φ₂,φ₃) = (2,−2,6,−6) has N_X = 1 > 0 = N_Y, yet z = φ₁ = 0 and φ₂φ₃ = −μ²/b solve every F-term.	The count concerns exact F-flatness and does not survive a frozen soft deformation as a theorem. In a zero-soft limit, solutions can enter from a broken-R branch or infinity unless the full field domain is followed.	Kang–Li–Sun 2013, § 2, arXiv PDF pp. 2–3, Eqs. (5)–(6) and the stated theorem; Amariti–Sauro 2020, §§ 3–5, arXiv PDF pp. 4–9, especially Eqs. (3.1)–(3.2) and Theorem 3; § 6, arXiv PDF pp. 9–12 Only the local R-symmetric-locus conclusion survives; the cited counterexamples rule out an unconditional global if-and-only-if statement.
Positive Coleman–Weinberg curvature	One-loop local model evidence	The full field-dependent boson and fermion mass matrices; a stated regulator and renormalization prescription; canonical normalization of the tested direction; a tachyon-free neighborhood; perturbative loop hierarchy; all equally light modes retained.	A positive renormalized one-loop curvature establishes local stabilization of the specified pseudomodulus to that order, provided the transverse physical Hessian is also nonnegative in the same neighborhood.	Global minimality, a lower basin, a tunneling path or lifetime, stability outside the local patch, and the sign after uncontrolled higher loops or thresholds.	Tachyonic expansion point: in W = fX + mφ₁φ₂ + hXφ₁²/2, y = hf/m² > 1 gives a negative tree-level scalar mass at X = φ = 0; a formal pseudomodulus curvature there does not create a vacuum.	Removing explicit soft terms returns the unsoftened supertrace only when masses and counterterms converge uniformly. A heavy multiplet may be omitted only after matching; supersymmetric degeneracy cancels its field-independent supertrace but not every threshold derivative.	Shih 2008, § 2.2, arXiv PDF pp. 7–8, Eqs. (2.17)–(2.21); Appendix A.1, arXiv PDF pp. 13–14, Eqs. (A.1)–(A.3)
Lower basin plus a bounce with B ≫ 1	Semiclassical lifetime evidence	A locally stable false vacuum and specified lower basin; the leading finite-action zero-temperature bounce; the correct single negative mode and translational zero modes; B ≫ 1; determinant control; EFT validity along the whole path; gravity and thermal effects negligible.	The decay rate per volume is semiclassically Γ/V = A exp(−B) times controlled corrections. A parametrically large B supports a long-lived metastable vacuum within the stated EFT and channel.	Absolute stability, omitted faster channels, a cosmological survival probability without spacetime volume and history, prefactor sensitivity when B is not large, and paths that leave the EFT.	Out-of-domain bounce: a reported B = 25 follows a path reaching |φ| = 2Λ even though the EFT is declared valid only for |φ| < Λ. The formal exponential is not a controlled lifetime prediction.	A soft coefficient sent to zero can remove the barrier, send the bounce radius to infinity, or change the leading channel, so the limit must be recomputed. Heavy modes decouple from the bounce only when gradients and field-dependent masses stay below their thresholds along the entire trajectory.	Coleman 1977, pp. 2929–2936; Callan–Coleman 1977, pp. 1762–1768; Coleman 1988, pp. 178–186 Coleman’s original derivation is single-field; a multifield application additionally has to establish the relevant saddle and fluctuation spectrum rather than assume a straight path.
Nilpotent X on an F ≠ 0 branch	Low-energy representation	Exact spontaneously broken global supersymmetry; a nonzero auxiliary expectation value F on the chosen branch; energies and gradients below the sgoldstino and every removed partner; a valid derivative expansion and matching; no supergravity eating.	At leading infrared order, the goldstino can be packaged as X² = 0 with scalar component A = G²/(2F); the nonlinear symmetry and its leading interactions are reproduced below the heavy thresholds.	The ultraviolet completion, finite-heavy-mass corrections, Wilson coefficients beyond symmetry relations, a branch crossing F = 0, light partners, and the supergravity spectrum.	Singular branch: for X = A + √2θG + θ²F, setting F = 0 makes 2AF − G² = 0 unable to solve A = G²/(2F); generic G is incompatible with that coordinate chart.	The constraint sharpens as the sgoldstino and other partners decouple at fixed nonzero breaking scale. If a zero-soft limit also sends F to zero, the nilpotent chart degenerates and the linear multiplet or another EFT must be restored.	Komargodski–Seiberg 2009, § 2.2, arXiv PDF pp. 6–8, Eqs. (2.15)–(2.18); § 3.2, arXiv PDF pp. 13–15, Eqs. (3.12)–(3.23)
Girardello–Grisaru softness	Ultraviolet power-counting theorem	Four-dimensional renormalizable global supersymmetry before deformation; local gauge-invariant operators; the stated field content and singlets; a standard Girardello–Grisaru operator or an explicit divergence analysis for any extended nonholomorphic term.	The admitted soft operators do not reintroduce the forbidden quadratic scalar-mass divergences under the classification’s hypotheses; logarithmic running and finite thresholds remain.	Numerical smallness, technical naturalness of every finite threshold, flavor or CP safety, vacuum stability, a dynamical goldstino, phenomenological viability, and continuity to a large-soft phase.	Dimension alone is insufficient: a nonholomorphic cubic involving a gauge singlet, such as C S φ†φ, can generate a singlet tadpole or power-sensitive scalar term unless its actual field content and symmetries pass a separate divergence analysis.	Sending all soft coefficients to zero restores the undeformed action and its ultraviolet cancellations when the limit is uniform. Making a technically soft coefficient large does not make it a small perturbation or prove phase continuity.	Girardello–Grisaru 1982, pp. 65–76
Small-soft implicit-function continuation	Local continuation theorem	A smooth potential V(φ,ε); an isolated stationary point at ε = 0 after gauge quotient; an invertible physical Hessian with a positive gap for a local minimum; sufficiently small ε; uniform EFT validity and fixed boundary data.	There is a unique nearby stationary branch φ(ε), with first shift −H⁻¹ times the mixed ε-gradient. A strictly positive Hessian remains positive for sufficiently small ε.	Global minimality, distant vacua, tunneling, phase boundaries, flat directions, thermodynamic nonuniformity, and continuation to coefficients outside the local neighborhood.	Singular Hessian: V₀(φ) = φ⁴ has H(0) = 0. Adding −εφ² makes φ = 0 unstable and creates minima φ = ±√(ε/2), so no unique analytic nearby minimum follows from the implicit-function theorem.	On the theorem’s branch, φ(ε) tends to φ(0) as ε tends to zero. This is a local zero-soft statement only; it gives no large-soft decoupling or phase-continuity result.	Krantz–Parks 2013, § 3.3, Theorem 3.3.1, pp. 43–45
Large-soft decoupling	Conditional decoupling theorem	External energies E much smaller than a heavy mass M that remains below the ultraviolet completion scale; a renormalizable matching problem; fixed renormalized low-energy inputs; threshold matching and higher operators retained; no singular large coupling, gap closing, or intervening phase transition.	Heavy-particle effects in low-energy amplitudes are absorbed into matched renormalized parameters plus power-suppressed operators. The heavy superpartner can be removed from the low-energy field content under those assumptions.	Thresholds need not vanish; vacuum phases, anomalies and global data need separate matching; naturalness is not proved; the path does not establish continuity to nonsupersymmetric QCD or any other strongly coupled endpoint.	Unsuppressed matching term: V(H,φ) = M²H²/2 + κMHφ². Eliminating H gives V_eff ⊃ −κ²φ⁴/2, independent of M; the heavy field decouples as a propagating state but its renormalization of a light coupling does not vanish.	M_soft tending to zero restores supermultiplet degeneracy, whereas M_soft tending to infinity removes selected partners after matching. These are different limits with different field content and neither implies the other.	Appelquist–Carazzone 1975, pp. 2856–2861

Exceptions and boundaries of validity

Nongeneric coefficients. Extra zeros, factorizations, or omitted symmetry-allowed terms can create or remove F-flat solutions. Such models may be perfectly consistent, but the generic theorem no longer predicts them.

Multiple charge assignments. Accidental symmetries can permit more than one R-charge assignment. A robust counting claim must state which exact quantum R-symmetry is used and inspect alternative assignments when the theorem requires it.

Nonstandard-charge condensates. Supersymmetric vacua can occur with $A_\alpha\neq0$ and broken R-symmetry. Counterexample analyses trace apparent violations of simplified counts to precisely such branches Amariti and Sauro 2020, §§3–5, pp. 4–9 and show how adapted-coordinate genericity resolves their relation to the original theorem Amariti and Sauro 2020, §6, pp. 9–12.

Singular or nonpolynomial superpotentials. Fractional powers, poles, logarithms, and dynamically generated terms change the algebraic geometry at the excluded locus. One must analyze the actual holomorphic domain rather than polynomial equation counting.

Gauge theories and D-terms. D-flatness, gauge quotients, anomalies, and strong dynamics add equations and identifications absent from the Wess–Zumino proof. A UV gauge theory may still produce a calculable Wess–Zumino description, but matching and its regime must be shown.

Runaways. If $F_i\to0$ only at infinite field distance, finite-field counting may be correct while the theory has no normalizable vacuum. The conclusion is a runaway, not automatically a stable broken phase.

Common pitfalls

“R-symmetry implies breaking.” It does not. The equations $f_i(Y)=0$ may have solutions, as the $X(f+\lambda Y^2)$ example shows.

“Broken supersymmetry implies broken R-symmetry.” It does not. $W=fX$ breaks supersymmetry at the R-symmetric point $X=0$.

Counting fields before choosing the vacuum class. $N_X>N_Y$ directly rules out only the R-symmetric locus used in its derivation. Other branches require separate analysis.

Exercises

1. Count the canonical model. Apply the R-charge-two versus neutral test to the O’Raifeartaigh superpotential and recover its incompatible equations.

Solution

$X$ and $\phi_2$ have charge $2$, while $\phi_1$ has charge $0$, so $N_X=2>N_Y=1$. The two equations are $f+h\phi_1^2/2=0$ and $m\phi_1=0$. For nonzero $f,m$, they have no common solution.

2. A count that allows supersymmetry. For $W=X_1(a+bY)+X_2(c+dY)$, count $N_X,N_Y$ and state the generic conclusion. Under what tuning does a supersymmetric solution appear?

Solution

$N_X=2$ and $N_Y=1$, so the two equations $a+bY=0$ and $c+dY=0$ are generically incompatible. They share a solution when $ad-bc=0$ (with the obvious qualifications when $b$ or $d$ vanishes). That relation is a codimension-one tuning, illustrating why “generic” is part of the theorem.

3. A nonuniform symmetry-restoring limit. In $W=fX+\epsilon X^2/2$, show why expanding at fixed $|X|<R$ can miss the supersymmetric vacuum as $\epsilon\to0$.

Solution

The F-flat solution is $X=-f/\epsilon$. For any fixed radius $R$, it lies outside the expansion domain once $|\epsilon|<|f|/R$. The local theory near the origin approaches the R-symmetric broken model, while the supersymmetric vacuum recedes to infinity. The limits $\epsilon\to0$ and $|X|\to\infty$ do not commute.

4. An adversarial R-breaking branch. Let

$$W=fX+\lambda XAB, \qquad R(X)=2, \qquad R(A)=q, \qquad R(B)=-q.$$

For nonzero $q$, the naive assignment has $N_X=1>N_Y=0$. Find a supersymmetric branch and explain why it does not contradict the careful local count.

Solution

The F-flatness equations are

$$f+\lambda AB=0, \qquad \lambda XB=0, \qquad \lambda XA=0.$$

They are solved by

$$X=0, \qquad AB=-\frac f\lambda.$$

For $q\neq0$, nonzero $A$ and $B$ spontaneously break that R-symmetry, so the branch lies outside the R-symmetric locus used in the naive $N_X>N_Y$ count. Moreover, $q=0$ is another consistent R assignment: then $A$ and $B$ are neutral and $N_Y=2$. Equivalently, the neutral composite coordinate $Y=AB$ rewrites the relevant equations as $W=X(f+\lambda Y)$, whose single equation has a zero. The adversarial branch therefore exposes the missing vacuum-class and charge-assignment checks rather than violating the local argument.

References

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