Spontaneous SUSY breaking: the Goldstino theorem, O'Raifeartaigh and Fayet-Iliopoulos, the supertrace sum rule, and the field-theory Witten index

Intuition Beginner

Supersymmetry pairs every particle with a partner of the opposite statistics: each boson with a fermion, each fermion with a boson. The pairing is run by operators called supercharges, which turn one member of a pair into the other. When the pairing holds exactly, the lowest-energy state of the world — the vacuum — is left untouched by these operators: it is genuinely symmetric.

But a symmetry of the equations need not be a symmetry of the vacuum. The vacuum can fail to respect supersymmetry even though the underlying laws do. When that happens we say supersymmetry is spontaneously broken. The remarkable feature of the SUSY case is how cleanly you can read it off: supersymmetry is unbroken exactly when the vacuum energy is zero, and it is broken exactly when the vacuum energy is forced to be positive.

So the vacuum energy plays the role of a referee. Most symmetries hide their breaking inside some order parameter you have to hunt for. Here the order parameter is the simplest quantity imaginable — the energy of the emptiest possible state. If the cheapest available vacuum still costs energy, the symmetry could not survive into it.

Visual Beginner

Alt text: Two energy landscapes side by side. On the left, a smooth bowl touches the zero-energy line at its bottom, so a state of exactly zero energy exists and supersymmetry stays unbroken. On the right, every valley floor of the landscape sits strictly above the zero line, so no zero-energy state exists and supersymmetry is spontaneously broken; a massless excitation, the Goldstino, propagates along the flat lifted floor, the SUSY counterpart of the Goldstone boson that appears whenever an ordinary continuous symmetry breaks.

Worked example Beginner

Here is the cleanest possible model of the rule. Picture a system whose energy is built as a sum of squares of the supercharge effects: roughly, the energy of any state is the size of "how much the supercharges move it." A state that the supercharges leave fixed contributes nothing, so its energy is zero. A state they cannot leave fixed contributes something positive.

Now ask: can the vacuum have zero energy? Only if the supercharges leave it fixed — only if it is supersymmetric. If the rules of the model make it impossible for any state to be left fixed by all the supercharges at once, then every state, including the cheapest, has positive energy. The vacuum is lifted off the floor, and supersymmetry is broken.

This is why the energy of the vacuum is the whole story. A zero-energy vacuum is a symmetric vacuum; a positive-energy vacuum is a broken one. There is no third option, because energy here is a sum of squares and a sum of squares is zero only when every piece is zero.

Check your understanding Beginner

Formal definition Intermediate+

Throughout, the theory carries an $N=1$ supersymmetry algebra with two-component Weyl supercharges $Q_{\alpha }$ and their conjugates ${\bar{Q}}_{\dot{\alpha }}=(Q_{\alpha }{)}^{†}$, satisfying $$ {Q_\alpha,, \bar Q_{\dot\beta}} = 2,\sigma^\mu_{\alpha\dot\beta},P_\mu, \qquad {Q_\alpha, Q_\beta} = 0 . $$ Tracing against ${\bar{\sigma }}^{\nu \dot{\beta }\alpha }$ and using $\mathrm{tr}({\sigma }^{\mu }{\bar{\sigma }}^{\nu })=2{\eta }^{\mu \nu }$ isolates the Hamiltonian $H=P^0$: $$ H ;=; \tfrac14 \sum_{\alpha} \big( Q_\alpha Q_\alpha^\dagger + Q_\alpha^\dagger Q_\alpha \big) ;=; \tfrac14 \sum_{\alpha}{Q_\alpha, Q_\alpha^\dagger}. $$

Definition (SUSY vacuum and the order parameter). A state $|0⟩$ is supersymmetric if $Q_{\alpha }|0⟩={\bar{Q}}_{\dot{\alpha }}|0⟩=0$ for all $\alpha ,\dot{\alpha }$. Because $H$ is the sum of squares above, $⟨\psi |H|\psi ⟩=\frac{1}{4}{\sum }_{\alpha }(\Vert Q_{\alpha }^{†}|\psi ⟩{\Vert }^2+\Vert Q_{\alpha }|\psi ⟩{\Vert }^2)\ge 0$ for every state, and $H|\psi ⟩=0$ holds iff $Q_{\alpha }|\psi ⟩=Q_{\alpha }^{†}|\psi ⟩=0$. Hence: $$ \text{SUSY unbroken} \iff \exists,\text{a state with } E = 0 \iff \min_{\text{vacua}} \langle H\rangle = 0, \qquad \text{SUSY broken} \iff \min_{\text{vacua}} \langle H\rangle > 0 . $$ The vacuum energy is the SUSY order parameter; this is the operator content of the beginner picture.

Definition (scalar potential and F/D terms). For a renormalisable theory of chiral superfields ${\Phi }_i$ (with scalar components ${\varphi }_i$, auxiliary components $F_i$) and gauge vector superfields with auxiliary components $D_a$, the scalar potential is $$ V(\phi,\phi^\dagger) ;=; \sum_i |F_i|^2 + \tfrac12\sum_a D_a^2, \qquad F_i = -,\frac{\partial \overline W}{\partial \phi_i^\dagger}, \qquad D_a = -g,\phi^\dagger T_a \phi - \xi_a , $$ where $W(\varphi )$ is the holomorphic superpotential and ${\xi }_a$ is a Fayet-Iliopoulos constant, allowed only for abelian factors. Supersymmetric vacua are the zeros of $V$, i.e. simultaneous solutions of $F_i=0$ and $D_a=0$. F-term breaking occurs when the equations $F_i=0$ have no solution; D-term breaking when $D_a=0$ cannot hold (the Fayet-Iliopoulos mechanism).

Definition (Goldstino). When SUSY is broken, $Q_{\alpha }|0⟩\ne 0$. The supercurrent $S_{\alpha }^{\mu }$, conserved by ${\partial }_{\mu }{S}_{\alpha }^{\mu }=0$, has a nonzero vacuum-to-one-particle matrix element to a massless spin-$\frac{1}{2}$ state $|\tilde{G}⟩$; that state is the Goldstino, a massless Weyl fermion. Its decay constant is fixed by the vacuum energy: $⟨0|S_{\alpha }^{\mu }|\tilde{G},p⟩\propto \sqrt{⟨H⟩}{\sigma }_{\alpha \dot{\beta }}^{\mu }{\bar{\zeta }}^{\dot{\beta }}$ for a polarisation spinor $\zeta$.

Definition (field-theory Witten index). The Witten index is the regularised graded trace $$ \mathrm{Tr},(-1)^F e^{-\beta H} ;=; \sum_{\text{states}} (-1)^F e^{-\beta E}, $$ where $F$ is fermion number, so $(-1{)}^F=+1$ on bosonic states and $-1$ on fermionic ones. States of energy $E>0$ come in Bose-Fermi pairs related by $Q$ and cancel; only the $E=0$ states survive, giving $\mathrm{Tr}(-1{)}^F{e}^{-\beta H}=n_B^{E=0}-n_F^{E=0}$, independent of $\beta$. This is the field-theory generalisation of the supersymmetric-quantum-mechanics index of 08.10.11 and of the geometric index $\dim \ker -\dim \mathrm{coker}$ of 03.15.12; we cross-link those and do not re-derive them here.

Counterexamples to common slips

• "Broken SUSY means no supercharge" is wrong. The supercharges still exist and still generate a symmetry of the action; what fails is that they no longer annihilate the vacuum. Spontaneous breaking is a statement about the vacuum, not about the algebra.
• "$\mathrm{STr}{M}^2=0$ means superpartners are degenerate" is wrong. The supertrace sum rule constrains a weighted sum of squared masses, not each mass individually; superpartners can and do split, as long as the signed sum cancels.
• "Zero Witten index means SUSY is broken" is wrong. A nonzero index guarantees an $E=0$ state, hence unbroken SUSY; a zero index is inconclusive, because the $E=0$ bosons and fermions might merely have cancelled in the count while still being present.

Key theorem with proof Intermediate+

Theorem (the Goldstino theorem and the vacuum-energy criterion). Let a Poincaré-invariant theory have a spontaneously broken $N=1$ supersymmetry, meaning $Q_{\alpha }|0⟩\ne 0$ for the true vacuum $|0⟩$. Then (i) the vacuum energy density is strictly positive, $⟨0|H|0⟩/V_{\text{space}}>0$; and (ii) the spectrum contains a massless Weyl fermion, the Goldstino, created from the vacuum by the conserved supercurrent.

Proof. (i) From $H=\frac{1}{4}{\sum }_{\alpha }\{Q_{\alpha },Q_{\alpha }^{†}\}$, $$ \langle 0|H|0\rangle = \tfrac14\sum_\alpha\Big(|Q_\alpha|0\rangle|^2 + |Q_\alpha^\dagger|0\rangle|^2\Big) \ge 0, $$ with equality iff every $Q_{\alpha }|0⟩=Q_{\alpha }^{†}|0⟩=0$, i.e. iff SUSY is unbroken. Broken SUSY means some $Q_{\alpha }|0⟩\ne 0$, so the bound is strict and the energy density positive.

(ii) The argument parallels the Goldstone theorem of 12.18.02, with the broken current now fermionic. Supersymmetry invariance of the action gives a conserved supercurrent $S_{\alpha }^{\mu }$ with ${\partial }_{\mu }{S}_{\alpha }^{\mu }=0$ and $Q_{\alpha }=\int d^{3}x{S}_{\alpha }^0$. Spontaneous breaking is the statement that some order-parameter field $X$ has $⟨0|\{Q_{\alpha },X\}|0⟩\ne 0$. Insert a complete set of momentum eigenstates into $$ \langle 0|{S^\mu_\alpha(x),, X(0)}|0\rangle $$ and impose conservation ${\partial }_{\mu }(\cdots )=0$ together with Lorentz covariance. Exactly as in the bosonic case, conservation forbids the spectral weight from being carried by any massive state: a nonzero, $x$-independent equal-time commutator can only be sustained by a massless intermediate state whose pole at $p^2=0$ supplies the surviving contribution. Because $S_{\alpha }^{\mu }$ carries one undotted spinor index and the charge is fermionic, that massless state is spin-$\frac{1}{2}$: the Goldstino. Its coupling strength is set by $⟨0|\{Q_{\alpha },X\}|0⟩$, which the vacuum energy of part (i) fixes to be nonzero precisely when SUSY is broken. $■$

Bridge. This theorem is the foundational reason the vacuum energy, and nothing more exotic, decides whether supersymmetry survives: the same sum-of-squares relation $H=\frac{1}{4}\sum \{Q,Q^{†}\}$ that gives positivity is what forces a massless fermion the instant the vacuum stops being annihilated. The construction builds toward the explicit O'Raifeartaigh and Fayet-Iliopoulos models below, where the order parameter is an unsolvable $F$- or $D$-flatness condition and the Goldstino is the fermion in the multiplet whose auxiliary field gets a vacuum value. The massless-pole mechanism is exactly the Goldstone argument of 12.18.02 with a spinor current in place of a vector current, so the Goldstino is dual to the Goldstone boson under the boson-fermion exchange that defines the supermultiplet. Putting these together, the central insight is that spontaneous SUSY breaking, the positivity of $⟨H⟩$, and the existence of the Goldstino are three faces of one operator identity — a structure that appears again in 12.19.04, where the same supercurrent algebra controls the non-renormalization of the superpotential, and that generalises the quantum-mechanical version of 08.10.11.

Exercises Intermediate+

Advanced results Master

The index as a deformation invariant. Witten's 1982 insight is that $\mathrm{Tr}(-1{)}^F$ does not change under any continuous variation of the parameters of the theory that preserves supersymmetry and the large-field (asymptotic) behaviour ^{[Witten 1982]}. As a coupling is dialed, $E=0$ states can move up to positive energy only in Bose-Fermi pairs (because a lone state cannot leave $E=0$ without a partner to pair with under $Q$), and pairs can descend to $E=0$ only in Bose-Fermi pairs; in both cases $n_B^0-n_F^0$ is unchanged. Thus the index computed in a weakly coupled, semiclassical regime — where vacua are countable critical points of the potential — equals the index of the strongly coupled theory, where dynamics is intractable. A nonzero count at weak coupling guarantees an exact zero-energy vacuum at all couplings, hence unbroken SUSY. This is how one proves SUSY is not dynamically broken in, for instance, pure super-Yang-Mills with a nonzero index $h^{\vee }$ (the dual Coxeter number), a result whose machinery rests on 12.19.04.

Computing the index by counting vacua. In a Wess-Zumino model with superpotential $W$, the supersymmetric vacua are the critical points $dW=0$, and at a nondegenerate critical point the local Bose-Fermi spectrum contributes $+1$. The index is then the number of solutions of ${\partial }_{i}W=0$ counted with sign — for a single field with $W$ a degree-$(k+1)$ polynomial, $k$ solutions, so the index is $k\ne 0$ and SUSY is unbroken. Putting a theory in a finite spatial box and counting the resulting quantum-mechanical ground states is the rigorous version; the field-theory index reduces to the SUSY-quantum-mechanics index of 08.10.11 on the space of zero-modes, and to the analytic index of a Dirac-type operator (the geometric Witten index of 03.15.12) on the relevant target geometry.

Boundary conditions, the Witten effect, and subtleties. The index can depend on the spatial topology and on boundary conditions: compactifying on a torus versus on ${\mathbb{R}}^3$ can change the asymptotic behaviour that the deformation argument relies on, so two regularisations may legitimately give different integers. Witten's original gauge-theory computation was later refined precisely because the naive boundary data had to be pinned down; the modern statement counts vacua on $T^3$ with care about the gauge bundle. The lesson is that "nonzero index $\Rightarrow$ unbroken SUSY" is airtight, but the value of the index is a delicate, regularisation-sensitive datum, not a casual count.

Synthesis. The field-theory Witten index is the central insight that converts an intractable dynamical question — does the strongly coupled vacuum energy reach zero? — into a robust integer computable at weak coupling, and this is exactly the role the Euler characteristic plays for ordinary topology. It builds toward the deformation-invariance argument that appears again in every non-perturbative SUSY analysis: the index is the foundational reason one can assert unbroken supersymmetry in theories whose vacua nobody can solve for. It generalises two earlier objects this corpus already carries — the supersymmetric-quantum-mechanics index of 08.10.11 and the geometric Dirac/Morse index of 03.15.12 — and is dual to each under the dictionary that trades field-theory zero-modes for quantum-mechanical ground states or for harmonic spinors. Putting these together, the chain "vacuum energy as order parameter $\to$ Goldstino $\to$ supertrace sum rule $\to$ Witten index" is a single thread: positivity of $⟨H⟩$ forces the Goldstino, the sum rule constrains how the breaking can show up in masses, and the index decides whether the breaking can happen at all — the bridge from algebra to dynamics.

Full proof set Master

Proposition (zero-energy states are exactly the supersymmetric states). In a theory with $H=\frac{1}{4}{\sum }_{\alpha }\{Q_{\alpha },Q_{\alpha }^{†}\}$, a state $|\psi ⟩$ satisfies $H|\psi ⟩=0$ if and only if $Q_{\alpha }|\psi ⟩=Q_{\alpha }^{†}|\psi ⟩=0$ for all $\alpha$.

Proof. If $Q_{\alpha }|\psi ⟩=Q_{\alpha }^{†}|\psi ⟩=0$ for all $\alpha$, then each $\{Q_{\alpha },Q_{\alpha }^{†}\}|\psi ⟩=0$, so $H|\psi ⟩=0$. Conversely suppose $H|\psi ⟩=0$. Then $0=⟨\psi |H|\psi ⟩=\frac{1}{4}{\sum }_{\alpha }(\Vert Q_{\alpha }|\psi ⟩{\Vert }^2+\Vert Q_{\alpha }^{†}|\psi ⟩{\Vert }^2)$. A sum of non-negative terms vanishing forces each term to vanish, so $\Vert Q_{\alpha }|\psi ⟩\Vert =\Vert Q_{\alpha }^{†}|\psi ⟩\Vert =0$, i.e. $Q_{\alpha }|\psi ⟩=Q_{\alpha }^{†}|\psi ⟩=0$. $■$

Proposition (positive-energy states pair off). On the subspace of energy $E>0$, the supercharge $Q\equiv Q_1$ (say) restricts to an invertible map exchanging bosonic and fermionic states; in particular $n_B(E)=n_F(E)$ for every $E>0$.

Proof. Since $[Q,H]=0$, $Q$ preserves each energy level. On the $E$-level, $\{Q,Q^{†}\}=cH=cE$ with $c>0$ a fixed constant, so $\{Q,Q^{†}\}$ acts as the nonzero scalar $cE$. Consider the bosonic energy-$E$ subspace ${\mathcal{H}}_B^E$ and define $T=Q$ on it. For $|\psi ⟩\in {\mathcal{H}}_B^E$, $Q^{†}Q|\psi ⟩+Q{Q}^{†}|\psi ⟩=cE|\psi ⟩$; restricting to the kernel structure, $Q$ maps ${\mathcal{H}}_B^E\to {\mathcal{H}}_F^E$ and $Q^{†}$ maps back, with $Q^{†}Q+Q{Q}^{†}=cE1$ guaranteeing that neither map has a kernel on the $E>0$ levels (a kernel vector would force $cE\Vert v{\Vert }^2=\Vert Qv{\Vert }^2+\Vert Q^{†}v{\Vert }^2$ to fail). Hence $Q$ is a bijection ${\mathcal{H}}_B^E\to {\mathcal{H}}_F^E$ and the dimensions match. $■$

Proposition (the index counts only the ground states). $\mathrm{Tr}(-1{)}^F{e}^{-\beta H}=n_B^{E=0}-n_F^{E=0}$, independent of $\beta$.

Proof. Split the trace by energy level. For $E>0$, the previous proposition gives $n_B(E)=n_F(E)$, so the contribution $(n_B(E)-n_F(E))e^{-\beta E}=0$. The only surviving terms are at $E=0$, contributing $(n_B^0-n_F^0)e^0=n_B^0-n_F^0$. No factor of $\beta$ remains, so the trace is $\beta$-independent. $■$

Proposition (nonzero index obstructs SUSY breaking). If $\mathrm{Tr}(-1{)}^F\ne 0$, the theory has at least one zero-energy state, hence unbroken supersymmetry, for every value of the couplings reachable by SUSY-preserving deformation.

Proof. From the previous proposition the index equals $n_B^0-n_F^0$. If this is nonzero then $n_B^0+n_F^0\ge |n_B^0-n_F^0|\ge 1$, so a zero-energy state exists. By the first proposition that state is annihilated by all supercharges, so SUSY is unbroken. Under a SUSY-preserving deformation the index is invariant (states leave or enter $E=0$ only in Bose-Fermi pairs, leaving $n_B^0-n_F^0$ fixed), so the conclusion persists across the whole connected family of couplings. $■$

The supertrace sum rule $\mathrm{STr}{M}^2=0$ is stated above and is proved by evaluating ${\sum }_J(-1{)}^{2J}(2J+1)\mathrm{Tr}{M}_J^2$ from the second derivatives of $V$ at the breaking minimum; see Weinberg Vol. 3 §27.5 ^{[Weinberg 2000]} for the explicit mass-matrix computation, valid at tree level and modified by gauge $D$-term contributions proportional to $\mathrm{Tr}Q$.

Connections Master

• Super-Yang-Mills and the non-renormalization theorem 12.19.04. This unit and that one are co-produced halves of the SUSY-dynamics story: 12.19.04 supplies the supercurrent algebra and the holomorphy that protect the superpotential, and the Witten-index argument here uses exactly that protection to assert unbroken SUSY in pure super-Yang-Mills (index $=h^{\vee }\ne 0$). The non-renormalization theorem is what makes the weak-coupling index computation reliable at strong coupling.

• The Goldstone theorem and effective Goldstone Lagrangians 12.18.02. The Goldstino theorem proved here is the fermionic image of the Goldstone theorem established there: a conserved current not annihilating the vacuum forces a massless pole, vector in the bosonic case, spinor in the SUSY case. The low-energy Goldstino dynamics is governed by a nonlinear (Volkov-Akulov) effective Lagrangian, the SUSY analogue of the pion Lagrangian of 12.18.02.

• Supersymmetric quantum mechanics and the Witten index 08.10.11. The field-theory index $\mathrm{Tr}(-1{)}^F$ defined here is the relativistic, infinite-dimensional generalisation of the quantum-mechanical Witten index of 08.10.11; in a finite spatial box the field theory reduces to a SUSY quantum-mechanical system and the two indices coincide. We cross-link rather than duplicate the one-particle partner-Hamiltonian construction.

• Witten's supersymmetric Morse theory / the geometric Witten index 03.15.12. The same graded trace $\dim \ker -\dim \mathrm{coker}$ that here counts zero-energy field-theory vacua is, in the geometric setting of 03.15.12, the analytic index of a Dirac-type / Witten-Laplacian operator and computes a topological invariant (an Euler characteristic or Morse count). The field-theory, quantum-mechanical, and geometric indices are three incarnations of one boson-fermion cancellation.

Historical & philosophical context Master

Spontaneous supersymmetry breaking was put on a firm footing in the mid-1970s. Fayet and Iliopoulos showed in 1974 that an abelian gauge theory admits a supersymmetric term — the now-eponymous $D$-term constant $\xi$ — whose presence can make the $D$-flatness condition unsolvable, breaking supersymmetry while leaving the gauge symmetry intact, and they identified the accompanying massless "Goldstone spinor" ^{[Fayet 1974]}. O'Raifeartaigh constructed in 1975 the complementary mechanism in a theory of chiral superfields, where an overdetermined system of $F$-flatness conditions has no common solution, again lifting the vacuum energy and producing a Goldstino ^{[O'Raifeartaigh 1975]}. These two models remain the textbook prototypes of $F$- and $D$-term breaking.

The decisive conceptual advance was Witten's. In 1981 he reframed dynamical SUSY breaking entirely around the vacuum energy as order parameter, sharpening the question to whether the vacuum energy is forced above zero ^{[Witten 1981]}. In 1982 he introduced the index $\mathrm{Tr}(-1{)}^F$ and proved it a deformation invariant, turning an intractable strong-coupling problem into a robust counting problem solvable in the semiclassical regime ^{[Witten 1982]}. The philosophical resonance is deep: the same boson-fermion cancellation that defines the index reappears as the analytic-index theorems of geometry, so a question about whether the world's vacuum is supersymmetric becomes, formally, the same kind of question as whether a manifold admits a harmonic spinor. Weinberg's Vol. 3 ^{[Weinberg 2000]} gives the synthesis used throughout this unit, including the supertrace sum rule whose phenomenological tension drove the field toward mediated soft breaking.

Bibliography Master

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  author    = {Weinberg, Steven},
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  year      = {2000},
  note      = {Chs. 26--29 (spontaneous SUSY breaking, the supertrace sum rule, the Witten index)}
}

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}

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}

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  author  = {Fayet, Pierre and Iliopoulos, John},
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}