Nonlinear Goldstino Dynamics and Constrained Effective Theory

When global supersymmetry is spontaneously broken and every non-goldstino excitation is heavy, the infrared theory still carries the symmetry—but nonlinearly. The goldstino shifts by the breaking scale, its interactions are derivative and fixed at leading order by current algebra, and the same dynamics can be packaged in a nilpotent chiral superfield. This page derives that equivalence, solves the constraint in components, explains how constrained matter multiplets remove heavy partners, and makes the auxiliary branch, heavy thresholds, and nonlinear cutoff part of every validity claim.

Required background. The order parameter and supercurrent residue are derived on F- and D-Term Breaking, Vacuum Energy, and the Goldstino. Use the component logic of Constrained and On-Shell Superfields when solving algebraic superfield constraints.

Helpful background. Power Counting and Predictive Order supplies the derivative-expansion test used below.

Nonlinear realization from the supersymmetry algebra

Let $G_\alpha$ be the canonically normalized goldstino and let $f>0$ have mass dimension two, with $V_{\rm vac}=f^2$ in rigid supersymmetry. A convenient Volkov–Akulov normalization is

$$\delta_\epsilon G_\alpha =\sqrt2f\,\epsilon_\alpha -\frac{i}{\sqrt2f} \left(G\sigma^\mu\bar\epsilon-\epsilon\sigma^\mu\bar G\right) \partial_\mu G_\alpha.$$

The second term is a field-dependent translation. Two transformations close on an ordinary spacetime translation without introducing an independent auxiliary field. Overall signs and factors of $\sqrt2$ vary in the literature; the invariant normalization is the supercurrent matrix element and the leading shift magnitude.

Introduce the induced vierbein

$$A_\mu{}^a =\delta_\mu{}^a +\frac{i}{2f^2} \left( G\sigma^a\partial_\mu\bar G -\partial_\mu G\sigma^a\bar G \right).$$

Its determinant transforms by a total derivative, so

$$\mathcal L_{\rm VA}=-f^2\det A$$

realizes supersymmetry nonlinearly. Expanding and making local field redefinitions gives

$$\mathcal L_{\rm VA} =-f^2-iG\sigma^\mu\partial_\mu\bar G +\frac{1}{4f^2}\bar G^2\Box G^2 +O\!\left(\frac{G^2\bar G^2\Box G^2\Box\bar G^2}{f^6}\right).$$

Terms proportional to the leading equation of motion move between equivalent bases, so individual higher-order operators should not be compared without specifying the goldstino field definition. The determinant construction and nonlinear fermionic shift originate in Akulov and Volkov 1974, pp. 28–35.

The current-algebra low-energy theorem gives the same leading couplings. If $S^\mu_\alpha$ is the supercurrent of the light-plus-heavy theory, then below the breaking scale one may write schematically

$$\mathcal L_{\rm int} =-\frac{1}{f}\,\partial_\mu G^\alpha S^\mu_\alpha +\text{h.c.}$$

up to improvements and terms proportional to equations of motion. Integrating by parts connects goldstino amplitudes to the supersymmetry variation of external states. For a light scalar–fermion pair with splitting $m_b^2-m_f^2$, this produces couplings of order $(m_b^2-m_f^2)/f$. The coupling vanishes as the multiplet becomes degenerate, a useful soft-limit check.

The nilpotent chiral superfield

Let

$$X_{\rm NL}=A+\sqrt2\theta G+\theta^2F$$

be chiral and impose

$$X_{\rm NL}^2=0.$$

The $\theta^2$ component of this equation is

$$2AF-G^2=0.$$

On the branch where $F$ is invertible as a low-energy expansion, the unique nontrivial solution is

$$A=\frac{G^2}{2F}.$$

The remaining components of $X^2$ then vanish by Grassmann nilpotence. The scalar is not set to zero; it is replaced by a goldstino bilinear. Division by $F$ is the key hypothesis: the solution is singular at a point where the supersymmetry-breaking auxiliary expectation value vanishes.

Consider the constrained superspace action

$$\mathcal L =\int\!\mathrm d^4\theta\,X_{\rm NL}^\dagger X_{\rm NL} +\left( f\int\!\mathrm d^2\theta\,X_{\rm NL}+\text{h.c.} \right).$$

At zero goldstino, the auxiliary equation gives $F=-f$ up to phase and $V=f^2$. Substituting $A=G^2/(2F)$ and then eliminating $F$ reproduces the Volkov–Akulov action through local field redefinitions. Roček first exhibited this constrained linear representation Roček 1978, pp. 451–453; its modern infrared formulation and relation to the supercurrent multiplet are developed in Komargodski and Seiberg 2009, §§2–3.

How nilpotency emerges from a heavy sgoldstino

A simple linear parent theory makes the approximation visible:

$$K=X^\dagger X-\frac{1}{\Lambda^2}(X^\dagger X)^2, \qquad W=fX.$$

Near $X=0$, the scalar partner has

$$m_s^2=\frac{4f^2}{\Lambda^2}$$

in these conventions. At momenta $E\ll m_s$, its equation of motion gives

$$A=\frac{G^2}{2F}+\Delta A, \qquad \frac{\Delta A}{G^2/(2F)} =O\!\left(\frac{E^2}{m_s^2}\right).$$

For matrix elements in which the leading bilinear is nonzero, the displayed ratio makes the suppression explicit; more generally $\Delta A$ is a sum of local operators with the same quantum numbers and coefficients suppressed by heavy scales. Thus $X^2=0$ is the leading infrared relation obtained after the sgoldstino is removed. At finite $m_s$, derivative corrections remember the linear parent theory. The sign of the Kähler correction was chosen to make $m_s^2>0$; reversing it makes the origin unstable rather than producing a valid constrained EFT.

This example also prevents a common circular argument. One may not impose nilpotency to discard a scalar and then cite the absence of that scalar as evidence that it was heavy. A UV mass, a strong-dynamics gap, or an independently justified decoupling limit must come first.

Constraining matter multiplets

Additional constrained superfields encode which partner of a light state has been integrated out. Let

$$Q=q+\sqrt2\theta\chi+\theta^2F_Q$$

be chiral. The constraint

$$X_{\rm NL}Q=0$$

removes the independent scalar. Solving its lowest components gives

$$q=\frac{G\chi}{F} -\frac{G^2}{2F^2}F_Q.$$

The matter fermion $\chi$ remains. Different constraints remove different components:

Constraint	Independent low-energy content	Necessary UV fact
$X^2=0$	goldstino, auxiliary field	sgoldstino is heavy
$XQ=0$	matter fermion, no independent scalar	scalar partner is heavy
$X\bar D_{\dot\alpha}\bar Q=0$	scalar, no independent matter fermion	fermion partner is heavy
$XW_\alpha=0$	gauge boson, no independent gaugino	gaugino is heavy without removing the gauge field

The table is a low-energy map, not a menu of identities one may impose arbitrarily. Constraints must respect gauge transformations and any remaining global symmetries. They can become mutually inconsistent if two eliminated components are required by a light multiplet or if integrating out one field generates a threshold of the same order as the retained terms.

For finite superpartner masses, the component solutions receive corrections suppressed by $E/m_{\rm heavy}$ and by additional supersymmetry-breaking ratios. The appropriate constraint can also change across parameter space when a nominally heavy field becomes light.

Power counting and the cutoff

The leading four-goldstino operator has coefficient $1/f^2$. At fixed angle its scattering amplitude scales as $E^4/f^2$, so perturbative unitarity fails at an energy of order $\sqrt f$, up to convention-dependent $4\pi$ factors. A conservative cutoff is therefore

$$\Lambda_{\rm EFT} \lesssim \min\!\left\{ m_s,\,m_{\rm other\ partners},\,\sqrt{4\pi f},\,\Lambda_{\rm UV} \right\}.$$

The factor $\sqrt{4\pi}$ is an estimate from naive dimensional analysis, not a universal threshold. A weakly coupled parent theory can introduce a heavy state below it; a strongly coupled completion can change the numerical coefficient. The actual validity statement should compare every process energy and background gradient with the smallest relevant scale.

For an operator with $n_\partial$ derivatives and $n_G$ goldstini, write its coefficient in powers of $f$ and the heavy scale so the Lagrangian has dimension four. Predictive truncation requires both

$$\frac{E}{m_{\rm heavy}}\ll1, \qquad \frac{E^2}{f}\ll1,$$

along with small background-field invariants. A process can satisfy one inequality and violate the other.

Matching, not only symmetry

Nonlinear supersymmetry fixes relations among operators, but Wilson coefficients still require matching. For example, the coefficient of a goldstino–matter interaction is tied to the measured or calculated superpartner splitting only after kinetic terms are canonical and the correct supercurrent is used. Integrating out a mediator can generate additional symmetry-invariant contact operators at the same order.

A reproducible constrained-EFT claim should state

1. the order parameter $f$ and its normalization;
2. the heavy components and their masses;
3. the branch with $F\neq0$;
4. the constraints and their component solutions;
5. the matching scale and retained operators;
6. the energy and background range;
7. the leading omitted corrections.

Boundaries of validity

Supergravity. The goldstino is eaten by the gravitino through the super-Higgs mechanism. Nilpotent superfields remain useful in supergravity, but the spectrum, auxiliary equations, and cutoff are different; the rigid VA action is not the complete theory.

Several breaking sectors. Only the linear combination aligned with the total order parameter is the true goldstino. Orthogonal “goldstini” generally acquire masses from interactions, supergravity, or mixing and require a multi-sector EFT.

Explicit breaking. A theory with only explicit soft terms has no conserved supercurrent and no exact massless goldstino. A dynamical hidden sector can restore the interpretation, but its fields and decoupling must be specified.

Massless partners. If a sgoldstino, gaugino, or matter scalar is as light as the process, the corresponding constraint removes a physical pole and violates unitarity or analyticity. Keep the full multiplet instead.

Crossing $F=0$. The component solution $A=G^2/(2F)$ fails on a branch where $F$ vanishes. A constrained chart cannot be continued through that point without changing variables or restoring degrees of freedom.

Common pitfalls

Treating $X^2=0$ as $A=0$. The correct solution is $A=G^2/(2F)$. Dropping the bilinear destroys the nonlinear transformations and the required contact interactions.

Quoting only $E\ll\sqrt f$. A partner mass can be much lower than the nonlinear unitarity scale. The smallest heavy threshold is also a cutoff.

Imposing every available constraint. Each constraint encodes a particular decoupling pattern. Removing a component without a UV mass hierarchy changes the theory rather than approximating it.

Exercises

1. Solve nilpotency. Square $X=A+\sqrt2\theta G+\theta^2F$ and show that $A=G^2/(2F)$ solves every component of $X^2=0$ when $F\neq0$.

Solution

Using two-component Grassmann algebra,

$$X^2=A^2+2\sqrt2\theta AG+\theta^2(2AF-G^2).$$

The $\theta^2$ coefficient gives $A=G^2/(2F)$. Then $A^2\propto(G^2)^2=0$ and $AG_\alpha\propto G^2G_\alpha=0$, so the lower components vanish as well.

2. Remove a scalar. Derive the lowest-component solution of $XQ=0$ for chiral $X$ and $Q$.

Solution

The $\theta^2$ component of $XQ$ is $AF_Q+Fq-G\chi=0$. Substituting $A=G^2/(2F)$ and solving for $q$ gives

$$q=\frac{G\chi}{F}-\frac{G^2F_Q}{2F^2}.$$

The lower components then vanish by the same Grassmann identities as in $X^2=0$.

3. Compare two cutoffs. Let $\sqrt f=10\,\mathrm{TeV}$, $m_s=3\,\mathrm{TeV}$, and every other partner be heavier. At what energies is a nilpotent goldstino-only theory parametrically justified?

Solution

The first physical threshold is the sgoldstino mass, not the nonlinear estimate. One needs $E\ll3\,\mathrm{TeV}$, together with small background gradients. The condition $E\ll10\,\mathrm{TeV}$ is weaker and does not justify integrating out the sgoldstino by itself.

4. Diagnose a branch failure. A background solution has $F(x_0)=0$ at one point but uses $A=G^2/(2F)$ everywhere. What has gone wrong?

Solution

The constrained coordinate chart is singular at $x_0$. Near that point the presumed heavy scalar or another degree of freedom must generally be restored, or a different low-energy description must be matched. The nilpotent solution cannot be divided through $F=0$.

References

• Akulov, V. P., and D. V. Volkov. “Goldstone Fields with Spin 1/2.” Theoretical and Mathematical Physics 18 (1974): 28–35. DOI.
• Komargodski, Z., and N. Seiberg. “From Linear SUSY to Constrained Superfields.” Journal of High Energy Physics 2009, no. 09 (2009): 066. DOI. Open preprint.
• Roček, M. “Linearizing the Volkov–Akulov Model.” Physical Review Letters 41 (1978): 451–453. DOI.