Supersymmetry Breaking and Controlled Deformations

Supersymmetry can fail in several logically different ways. A rigid theory may have no state annihilated by all supercharges, a calculable model may possess only a locally stable nonsupersymmetric vacuum, an effective theory may realize the symmetry nonlinearly after heavy partners are removed, or the Lagrangian may be deformed explicitly by soft operators. This chapter separates those statements and supplies the calculation appropriate to each one: order parameters and the goldstino, incompatible auxiliary-field equations, symmetry criteria, loop-lifted pseudomoduli, vacuum decay, constrained-superfield power counting, and controlled soft limits.

The scope is four-dimensional rigid $\mathcal N=1$ field theory unless a page explicitly says otherwise. Supergravity changes two central conclusions—the goldstino is eaten by the gravitino, and vacuum energy is not simply a sum of auxiliary-field squares—so it is treated only as a boundary of validity here.

Helpful background. Review false-vacuum bounces before assigning a lifetime to a local minimum; effective-theory power counting before using a constrained superfield; and ultraviolet and infrared fixed points before interpreting a soft deformation across a long renormalization-group flow.

Enter this chapter

A sound diagnosis begins by asking what has actually been established.

Evidence in hand	What it establishes	What it does not establish
No simultaneous solution of $F_i=0$ and $D^a=0$	No supersymmetric field configuration in the stated field domain	Existence of a stable vacuum
A stationary point with $V>0$ and positive physical Hessian	A locally stable, spontaneously broken rigid-SUSY vacuum	Its lifetime or global dominance
A nonzero goldstino matrix element of the supercurrent	Spontaneous breaking and the massless fermionic pole under the Goldstone hypotheses	A fundamental goldstino in supergravity or at energies above compositeness
A Nelson–Seiberg-type field count	A generic F-term conclusion under its symmetry, regularity, and vacuum assumptions	A theorem about arbitrary gauge dynamics, singular loci, or nongeneric couplings
Positive one-loop pseudomodulus curvature	Local quantum stabilization in a controlled perturbative regime	Metastability without a lower state and a suppressed bounce
A nilpotent chiral superfield	A low-energy nonlinear realization on a branch with nonzero auxiliary field	A UV completion or validity beyond the heavy-state threshold
Soft masses taken large	A path through explicitly broken theories, with thresholds	A proof that a remote nonsupersymmetric phase is continuously connected

These distinctions are already visible in the rigid supersymmetry algebra. For a translationally invariant state with well-defined global charges,

$$\sum_{\alpha=1}^{2}\left( \lVert Q_\alpha|\Omega\rangle\rVert^2 +\lVert Q_\alpha^\dagger|\Omega\rangle\rVert^2 \right) =4\,\langle\Omega|H|\Omega\rangle.$$

Thus zero energy and invariance under all supercharges are equivalent after the Hamiltonian normalization is fixed by the algebra. In a homogeneous infinite-volume phase, the statement is made for the energy density and local order parameters; surface terms, ill-defined charges, or a boundary can invalidate the displayed step. The component and supercurrent versions of this argument are developed on F- and D-term breaking, following the careful rigid-theory treatment in Weinberg 2000, §§29.1–29.2, pp. 248–263.

A calculation-first route

1. F- and D-Term Breaking, Vacuum Energy, and the Goldstino derives the positive auxiliary-field potential, the fermion zero mode, and the supercurrent pole. It also states the stationarity, gauge, boundary, and local-versus-global qualifications.
2. O’Raifeartaigh and Fayet–Iliopoulos Model Laboratories solves one F-term rank-condition model and one anomaly-free Abelian D-term model, including spectra, flat directions, gauge breaking, and tachyon boundaries.
3. R-Symmetry and Nelson–Seiberg-Type Criteria turns a continuous R-symmetry into a field-counting test, while keeping genericity, regularity, and vacuum-location hypotheses visible.
4. Pseudomoduli, Quantum Lifting, and Metastability computes the field-dependent supertrace and Coleman–Weinberg potential, then separates local curvature from a semiclassical decay claim.
5. Nonlinear Goldstino Dynamics and Constrained Effective Theory derives $X_{\rm NL}^2=0$, solves its components, and identifies the derivative expansion and heavy-state cutoff.
6. Soft Breaking, Spurions, and Controlled Decoupling classifies soft operators, derives them from spurions, follows their running and thresholds, and states what the restoration and large-soft limits can and cannot prove.

The first four leaves concern spontaneous breaking. The fifth is the infrared realization of that breaking. The sixth concerns explicit breaking and should not be used as evidence for a spontaneous order parameter.

The common diagnostic chain

For chiral multiplets $\Phi^i$ and vector multiplets $V^a$, the canonical tree-level potential has the schematic form

$$V(\phi,\bar\phi) =\sum_i |F_i|^2+\frac12\sum_a(D^a)^2.$$

A reliable analysis then passes through six gates:

1. Field domain and equations. Specify the Kähler metric, gauge quotient, superpotential, Fayet–Iliopoulos data if present, and whether points at infinity or singular loci belong to the problem.
2. Order parameter. Show that every candidate vacuum has at least one nonzero $F_i$ or $D^a$, or use an equivalent current-algebra matrix element.
3. Stationarity and physical stability. Solve first derivatives, remove gauge directions, and diagonalize the physical quadratic form. Incompatible $F$ equations alone do not guarantee a vacuum.
4. Quantum control. If a flat direction is lifted, display the field-dependent masses, loop parameter, subtraction scheme, and range with no tachyons.
5. Global structure and lifetime. Locate a lower vacuum or runaway and identify the interpolating path. Only then does a bounce exponent turn a local minimum into a metastability claim.
6. Infrared description. Integrate out states only below their masses and below the nonlinear strong-coupling scale. Keep the auxiliary branch and corrections visible.

O’Raifeartaigh’s original chiral model and the Fayet–Iliopoulos Abelian construction supply the two canonical tree-level mechanisms O’Raifeartaigh 1975, pp. 331–352, Fayet and Iliopoulos 1974, pp. 461–464. Their value is not that every realistic theory resembles them, but that every logical step can be inspected.

Criteria are conditional statements

An R-symmetry criterion is not a substitute for solving a model. In a generic, calculable Wess–Zumino effective theory, Nelson and Seiberg related F-term breaking to continuous R-symmetry and its spontaneous breaking Nelson and Seiberg 1994, pp. 46–62. A useful refined test counts R-charge-two fields against R-neutral fields near an R-symmetric locus, but it still assumes a regular polynomial superpotential and a specified class of vacua Kang, Li, and Sun 2013, §§2–3. Gauge fields, D-terms, nonpolynomial terms, runaways, accidental symmetries, and special coefficients require a direct analysis.

The Witten index is likewise one-way evidence. A nonzero, well-defined index obstructs spontaneous breaking; a vanishing index permits either breaking or paired supersymmetric vacua. A continuous spectrum or a changing asymptotic potential can also invalidate the naive trace. The operator-level qualifications are developed in the Witten-index chapter; they are not repeated as a shortcut here.

Local minima, metastability, and effective descriptions

The one-loop effective potential along a pseudomodulus $X$ is determined by the field-dependent spectrum,

$$V_1(X)=\frac{1}{64\pi^2} \operatorname{STr}\!\left[ \mathcal M^4(X) \left(\log\frac{\mathcal M^2(X)}{\mu^2}-\frac32\right) \right].$$

This formula can establish a local mass and a tachyon-free neighborhood. It does not supply the destination, path, or Euclidean action needed for a lifetime. Coleman’s bounce formalism supplies those extra data Coleman 1977, pp. 2929–2936, while the ISS construction is a controlled example in which a small parameter separates a nonsupersymmetric local vacuum from supersymmetric vacua Intriligator, Seiberg, and Shih 2006, §§2–7.

At energies well below every non-goldstino mass, the same broken phase admits a nonlinear description. The goldstino translation is encoded either by the Volkov–Akulov determinant or by a chiral field satisfying $X_{\rm NL}^2=0$ Akulov and Volkov 1974, pp. 28–35, Komargodski and Seiberg 2009, §§2–3. This is an effective equivalence, not permission to remove a light sgoldstino or to cross a point where the auxiliary expectation value vanishes.

Review the chapter

Diagnosis. For a proposed vacuum, list the auxiliary expectation values, the physical Hessian eigenvalues, the goldstino direction, and the global alternative vacua or runaways. If any entry is unknown, qualify the conclusion accordingly.

Comparison. Explain why $F_X=f$ in a Wess–Zumino model is spontaneous breaking, while a scalar mass term $m_s^2|\phi|^2$ inserted without a dynamical spurion is explicit breaking. The first changes the state in a symmetric theory; the second changes the Lagrangian.

Failure analysis. A paper finds $m_X^2>0$ at one loop and calls the point a long-lived metastable vacuum. Identify the missing evidence: a lower endpoint, a barrier, a controlled bounce, its negative mode, and a hierarchy making the exponent large.

Transfer. Suppose a nilpotent description is proposed at energy $E$. Ask for the nonzero $F$ branch, the sgoldstino and other partner masses, the suppression of higher-derivative operators, and the nonlinear unitarity scale. The effective theory is controlled only if $E$ lies below all relevant bounds.

Where to go next

• Continue to Dynamical Supersymmetry Breaking and Calculability for strongly coupled mechanisms whose order parameters are inferred through exact dynamics and duality.
• Use supergraphs and quantum effective actions for Wilsonian versus 1PI distinctions that underlie the loop analysis.
• Return to false-vacuum decay for determinant prefactors, multiple fields, finite temperature, and gravitational corrections.
• Use matching and decoupling when the soft spectrum crosses thresholds.

References

• Akulov, V. P., and D. V. Volkov. “Goldstone Fields with Spin 1/2.” Theoretical and Mathematical Physics 18 (1974): 28–35. DOI.
• Coleman, S. “The Fate of the False Vacuum. I. Semiclassical Theory.” Physical Review D 15 (1977): 2929–2936; erratum 16 (1977): 1248. DOI.
• Fayet, P., and J. Iliopoulos. “Spontaneously Broken Supergauge Symmetries and Goldstone Spinors.” Physics Letters B 51 (1974): 461–464. DOI.
• Intriligator, K., N. Seiberg, and D. Shih. “Dynamical SUSY Breaking in Meta-Stable Vacua.” Journal of High Energy Physics 2006, no. 04 (2006): 021. DOI. Open preprint.
• Kang, Z., T. Li, and Z. Sun. “The Nelson–Seiberg Theorem Revised.” Journal of High Energy Physics 2013, no. 12 (2013): 093. DOI. Open preprint.
• Komargodski, Z., and N. Seiberg. “From Linear SUSY to Constrained Superfields.” Journal of High Energy Physics 2009, no. 09 (2009): 066. DOI. Open preprint.
• Nelson, A. E., and N. Seiberg. “R Symmetry Breaking versus Supersymmetry Breaking.” Nuclear Physics B 416 (1994): 46–62. DOI. Open preprint.
• O’Raifeartaigh, L. “Spontaneous Symmetry Breaking for Chiral Scalar Superfields.” Nuclear Physics B 96 (1975): 331–352. DOI.
• Weinberg, S. The Quantum Theory of Fields, Volume III: Supersymmetry. Cambridge: Cambridge University Press, 2000. DOI.