Constrained, On-Shell, and Nonlinear Superfields

A superfield constraint can do several inequivalent jobs. It may define an irreducible off-shell multiplet, express a gauge field strength in terms of a prepotential, impose an equation of motion, or eliminate heavy components nonlinearly in an effective theory. The algebraic form alone does not reveal which job it performs. One must solve the constraint in components, check integrability and branches, and then identify whether dynamics entered.

Required background. Chiral, Vector, Linear, and Field-Strength Superfields supplies the standard linear constraints and component projections. Off-Shell Closure and Auxiliary Fields supplies the difference between algebraic elimination and off-shell completion.

Helpful background. Local versus Integrated Operator Redundancies explains how equations of motion and total derivatives are treated in operator bases without turning them into kinematic identities.

Four jobs for a constraint

For a superfield $\mathscr U$, classify a condition $\mathcal C(\mathscr U)=0$ by its input and consequence:

Class	Example	What it changes	Dynamics?
Kinematic irreducibility	$\bar D_{\dot\alpha}\Phi=0$	Removes independent $\bar\theta$ coefficients	No
Field-strength or gauge	$W_\alpha=-\tfrac14\bar D^2D_\alpha V$ with $V\sim V+i(\Lambda-\Lambda^\dagger)$	Replaces redundant prepotential data by curvature data	No
On-shell differential	$-\tfrac14\bar D^2\bar\Phi=0$	Imposes component field equations	Yes
Nonlinear algebraic	$X^2=0$	Expresses one component as a composite on a branch	Not by itself

Every row is supersymmetry covariant. Only the third is intrinsically an equation of motion, and the last becomes a dynamical EFT statement only after an action, scale hierarchy, and branch have been supplied.

Linear constraints: irreducible versus on shell

A chiral superfield

$$\bar D_{\dot\alpha}\Phi=0$$

has the unconstrained chiral-coordinate expansion

$$\Phi=A+\sqrt2\,\theta\psi+\theta\theta F.$$

No spacetime field equation follows. The constraint defines an off-shell $4+4$ multiplet.

For the free canonical action, variation with respect to the chiral field instead gives

$$-\frac14\bar D^2\bar\Phi=0,$$

with conjugate equation

$$-\frac14D^2\Phi=0.$$

Successive component projections yield

$$F=0, \qquad \bar\sigma^{\mu\dot\alpha\alpha} \partial_\mu\psi_\alpha=0, \qquad \Box A=0.$$

The same superfield is now on shell. The difference is not that one equation contains more $D$ operators; it is that the second condition is the Euler–Lagrange equation of a specified action. Gates, Grisaru, Roček, and Siegel separate constrained off-shell superfields from on-shell representations in Gates et al. 1983, §§ 3.5 and 3.12, pp. 89–92 and 138–146.

The real-linear conditions

$$D^2L=\bar D^2L=0, \qquad L=L^\dagger,$$

are likewise kinematic when $L$ describes a tensor or current multiplet. Adding a model-dependent equation such as a first-order relation between $L$ and another field can put it on shell. Constraint names do not determine closure class without the full system.

Solving the nilpotent chiral constraint

Let

$$X(y,\theta) =x(y)+\sqrt2\,\theta G(y)+\theta\theta F(y)$$

be chiral. Its square is

$$X^2 =x^2+2\sqrt2\,\theta xG +\theta\theta(2xF-GG).$$

The constraint

$$X^2=0$$

therefore implies

$$x^2=0, \qquad xG_\alpha=0, \qquad 2xF-GG=0.$$

On the branch where $F$ has a nonzero invertible body,

$$x=\frac{GG}{2F}.$$

The other two component equations then follow from the nilpotence of a two-component Grassmann spinor: $(GG)G_\alpha=0$ and $(GG)^2=0$. The independent fields are $G_\alpha$ and $F$; the scalar is composite.

This solution is local on field space. It is singular when $F$ cannot be inverted, so $X^2=0$ must not be used across a branch where the auxiliary expectation value passes through zero. A different branch can have different component content.

Roček’s constrained linear realization already exhibited the relation to the Volkov–Akulov goldstino Roček 1978, pp. 451–453. Komargodski and Seiberg identify the nilpotent chiral field as the infrared goldstino multiplet and derive its controlled low-energy couplings in Komargodski and Seiberg 2009, §§ 1–3, pp. 1–13.

Nilpotence is not the goldstino equation of motion

The condition $X^2=0$ eliminates $x$ but leaves $F$ independent. To obtain a particular goldstino dynamics one must also choose an action, for example

$$S = \int\mathrm d^4x\,\mathrm d^4\theta\, X^\dagger X +\left( f\int\mathrm d^4x\,\mathrm d^2\theta\,X +\text{c.c.} \right),$$

subject to $X^2=0$. Eliminating $F$ then generates the nonlinear goldstino action. For real $f$ with the signs displayed above, the auxiliary equation begins $F=-f+\cdots$. The inhomogeneous leading transformation is therefore

$$\delta G_\alpha =-\sqrt2\,f\,\epsilon_\alpha+\cdots$$

and signals spontaneous supersymmetry breaking; the omitted terms are field-dependent and enforce the nonlinear algebra. Reversing the sign of the linear superpotential reverses this conventional overall sign without changing the physics.

The separation is essential:

• $X^2=0$ is an algebraic constraint on superfield coordinates;
• the $F$ equation follows from the chosen constrained action;
• identifying $f$ with an order parameter requires a vacuum and normalization;
• the Volkov–Akulov equivalence can involve a nonlinear field redefinition.

One sometimes adds a differential condition such as

$$X\bar D^2\bar X=4fX$$

in a compatible convention. That stronger relation incorporates an auxiliary equation and should not be described as equivalent to nilpotence alone.

Constraints from decoupling

Suppose a linear ultraviolet theory contains a heavy scalar partner with mass $m_s$ and a light goldstino. At energies

$$E\ll m_s,$$

solving the heavy-field equation in a derivative expansion can lead to a constrained superfield. The resulting relation is valid only with:

• the hierarchy and retained order in $E/m_s$ stated;
• the branch of the auxiliary background fixed;
• singular denominators excluded;
• gauge covariance preserved;
• higher-derivative and loop corrections bounded.

An exact algebraic constraint in the low-energy variables need not be an exact operator identity of the microscopic theory at arbitrary momentum. Composite operators such as $GG/F$ require a renormalized definition in quantum correlation functions. The constrained-superfield formalism is powerful precisely because it packages a controlled infrared realization; it does not erase its EFT cutoff.

Other constraints can remove selected components. For example, with nilpotent $X$, a condition of the schematic form

$$X\,Y=0$$

can solve the lowest component of another chiral field $Y$ in terms of its fermion, auxiliary, and the goldstino. A condition involving $X\bar D_{\dot\alpha}\bar Y$ can remove a fermion instead. Each case must be solved separately: multiplying by $X$ can create additional branches, and dividing by $F$ repeats the nonzero-background assumption.

Gauge covariance and integrability

For a charged constrained field, replace flat derivatives by gauge-covariant ones. A constraint written with $D$ may fail to transform covariantly even if its component solution looks plausible in one gauge. The consistency test is

$$[\nabla_r,\nabla_s\}\mathscr U \in \langle\mathcal C(\mathscr U)\rangle,$$

up to declared gauge transformations. Curvature components that survive outside this ideal are obstructions.

Nilpotence is algebraically integrable because supersymmetry acts as a derivation:

$$\delta(X^2)=2X\,\delta X.$$

If $X^2=0$, its supersymmetry variation vanishes on the constrained space. This does not establish that the component solution is nonsingular, gauge covariant, or quantum exact; those are independent checks.

Constraint checks

For every constrained superfield, record:

1. dimension, signature, supersymmetry algebra, and reality;
2. whether the constraint is algebraic or differential;
3. its integrability conditions;
4. all component equations by Grassmann degree;
5. which components are removed, composite, gauge, auxiliary, or dynamical;
6. branch assumptions and every denominator;
7. whether an action or equation of motion was used;
8. closure before and after solving the constraint;
9. the EFT regime and quantum meaning of composite operators.

If any step uses an Euler–Lagrange equation, the resulting multiplet is on shell even when the final constraint is written compactly in superspace.

Common pitfalls

“Constrained” does not mean “on shell.” Chirality and linearity can define off-shell irreducible multiplets. The closure class follows from the component equations, not the adjective.

Nilpotence does not justify division by zero. The solution $x=GG/(2F)$ is valid only on the $F\ne0$ branch. It cannot describe a supersymmetric locus with vanishing auxiliary background without a separate analysis.

A low-energy constraint is not automatically ultraviolet exact. State the decoupled modes, scale hierarchy, derivative order, and possible operator-mixing corrections.

Exercises

1. Solve $X^2=0$

Derive the scalar solution and verify the remaining component equations.

Solution

The $\theta\theta$ coefficient gives $2xF-GG=0$, hence

$$x=\frac{GG}{2F}$$

when $F$ is invertible. Then $xG_\alpha$ contains three factors of the two-component spinor $G$ and vanishes, while $x^2$ contains four factors and also vanishes.

2. Classify two constraints

Classify $\bar D_{\dot\alpha}\Phi=0$ and $-\tfrac14\bar D^2\bar\Phi=0$ for the free canonical chiral theory.

Solution

The first is a kinematic irreducibility condition and leaves $(A,\psi,F)$ off shell. The second is the superfield Euler–Lagrange equation; its components set $F=0$ and impose the Weyl and Klein–Gordon equations, so it puts the multiplet on shell.

3. Find the hidden assumption

A derivation removes the sgoldstino using $x=GG/(2F)$ and then studies a configuration with $F=0$. What failed?

Solution

The component solution was obtained by dividing by $F$. It is not defined on the proposed configuration. One must return to the unsolved component equations of $X^2=0$ and analyze the $F=0$ branch separately.

Continue

F- and D-Term Breaking, Vacuum Energy, and the Goldstino supplies the dynamical breaking analysis. Nonlinear Goldstino and Constrained-Superfield EFT develops the effective theory beyond this constraint-classification page.

References

• Gates, S. James, Jr., Marcus T. Grisaru, Martin Roček, and Warren Siegel. Superspace, or One Thousand and One Lessons in Supersymmetry. Frontiers in Physics 58. Reading, MA: Benjamin/Cummings, 1983. Corrected open edition, 2001. arXiv:hep-th/0108200.

• Komargodski, Zohar, and Nathan Seiberg. “From Linear SUSY to Constrained Superfields.” Journal of High Energy Physics 2009, no. 9 (2009): 066. DOI. arXiv:0907.2441.

• Roček, Martin. “Linearizing the Volkov–Akulov Model.” Physical Review Letters 41 (1978): 451–453. DOI.