On-Shell Supermultiplets and Supersymmetric Ward Identities

On-shell superspace packages the physical helicities of a massless supermultiplet into a polynomial in Grassmann variables. Supercharges then act by multiplication and differentiation, so the many component supersymmetric Ward identities become two linear equations on one superamplitude. Those equations relate components and expose forbidden sectors, but they do not determine the remaining kinematic function or replace amplitude dynamics.

Required background. Massive and massless unitary supermultiplets supplies the physical helicity pairs. On-shell states and little-group scaling supplies spinor-helicity weights and the all-outgoing convention.

Helpful background. Spinor-helicity variables supplies brackets and complex three-point kinematics. Localized transformations and Ward–Takahashi identities supplies the current-to-amplitude logic.

One-particle on-shell superspace

Work in four-dimensional Lorentzian kinematics with massless external momenta written

$$p_{\alpha\dot\alpha}=\lambda_\alpha\widetilde\lambda_{\dot\alpha}.$$

Under the little group,

$$\lambda\mapsto t\lambda, \qquad \widetilde\lambda\mapsto t^{-1}\widetilde\lambda.$$

Introduce one Grassmann coordinate $\eta$ for $\mathcal N=1$ and assign $\eta\mapsto t^{-1}\eta$. A helicity pair can then be written

$$\Omega_h(p,\eta)=|h\rangle+\eta|h-\tfrac12\rangle.$$

Both terms scale as $t^{-2h}$. This is a generating function for physical states, not an off-shell superfield on spacetime. Its two components are exactly the rank-one massless representation constructed on the preceding page.

The CPT-conjugate pair has helicities $-h+\tfrac12$ and $-h$. It can be represented by another polynomial,

$$\widetilde\Omega_{-h+1/2}(p,\eta) =|-h+\tfrac12\rangle+\eta|-h\rangle,$$

or by a Grassmann Fourier transform of the first description. CPT is therefore not ordinary complex conjugation at fixed $\eta$.

Representative choices are

Physical multiplet	First on-shell polynomial	CPT-conjugate polynomial
chiral	$\Psi^++\eta\phi$	$\bar\phi+\eta\Psi^-$
vector	$g^++\eta\lambda^+$	$\lambda^-+\eta g^-$

The superscripts label helicity signs, not electric charge. An amplitude must record which polynomial is assigned to each external leg; otherwise a Grassmann coefficient has no unambiguous particle interpretation.

Supercharges as multiplication and differentiation

On one leg define normalized kinematic operators

$$q_\alpha=\lambda_\alpha\eta, \qquad \bar q_{\dot\alpha} =\widetilde\lambda_{\dot\alpha}\frac{\partial}{\partial\eta}.$$

They obey

$$\{q_\alpha,\bar q_{\dot\alpha}\} =p_{\alpha\dot\alpha}.$$

The physical supercharges of the four-dimensional N=1 convention are $Q=\sqrt2\,q$ and $\bar Q=\sqrt2\,\bar q$, which restores $\{Q,\bar Q\}=2p$. The little-group weights cancel in both $q$ and $\bar q$, an immediate check that they are Lorentz spinors rather than helicity-dependent artifacts.

For $n$ all-outgoing legs,

$$\mathsf q_\alpha=\sum_{i=1}^n\lambda_{i\alpha}\eta_i, \qquad \bar{\mathsf q}_{\dot\alpha} =\sum_{i=1}^n\widetilde\lambda_{i\dot\alpha} \frac{\partial}{\partial\eta_i},$$

and hence

$$\{\mathsf q_\alpha,\bar{\mathsf q}_{\dot\alpha}\} =\sum_i p_{i\alpha\dot\alpha}=0$$

on the momentum-conserving support of an amplitude. For $\mathcal N$-extended supersymmetry one introduces $\eta_i^A$ and repeats the construction for each $A$.

Ward identities for the S-matrix

Suppose the vacuum is supersymmetric, the asymptotic charges are defined, and the quantum symmetry has no anomaly. Then $[Q,S]=0$. Inserting complete external one-particle multiplets turns this operator equation into

$$\mathsf q_\alpha\,\mathcal A_n(\lambda,\widetilde\lambda,\eta)=0, \qquad \bar{\mathsf q}_{\dot\alpha}\,\mathcal A_n(\lambda,\widetilde\lambda,\eta)=0.$$

These are the supersymmetric Ward identities in generating-function form. The component version was developed for helicity amplitudes in Grisaru and Pendleton 1977, pp. 81–92; on-shell superspace makes the same relations manifest, with the maximally supersymmetric construction pioneered in Nair 1988, pp. 215–218.

The Grassmann delta polynomial

$$\delta^{(2)}(\mathsf q) \equiv\frac12\epsilon^{\alpha\beta} \mathsf q_\alpha\mathsf q_\beta =\sum_{i<j}\langle ij\rangle\eta_i\eta_j$$

is annihilated by multiplication with either component of $\mathsf q$. Moreover,

$$\bar{\mathsf q}_{\dot\alpha}\delta^{(2)}(\mathsf q) \propto \left(\sum_i\lambda_i^\alpha\widetilde\lambda_{i\dot\alpha}\right) \mathsf q_\alpha=0$$

by total momentum conservation. Therefore any expression of the form

$$\mathcal A_n=\delta^{(2)}(\mathsf q)\,F(\lambda,\widetilde\lambda)$$

solves both Ward identities, provided $F$ has the required little-group weights, permutation properties, mass dimension, locality or factorization behavior, and internal charges. More general Grassmann sectors can have additional invariant polynomials; the Ward identities still reduce them to a basis rather than fixing their coefficient functions. Systematic treatments are given in Elvang, Freedman, and Kiermaier 2010, §§ 2–4 and Elvang and Huang 2015, Chapter 4.

Extracting component relations

Expand

$$\mathcal A_n(\eta) =\sum_{S\subseteq\{1,\ldots,n\}} \eta_S\,A_S, \qquad \eta_S=\prod_{i\in S}\eta_i$$

in a fixed increasing order. A component amplitude is obtained by differentiating with respect to the $\eta_i$ belonging to its lower-helicity external states and then setting all $\eta$ to zero.

For the simple Ward solution above,

$$A_{ij}=\langle ij\rangle F.$$

Thus any three nonzero degree-two components obey, with the signs fixed by the chosen Grassmann ordering,

$$\frac{A_{ij}}{\langle ij\rangle} =\frac{A_{k\ell}}{\langle k\ell\rangle} =F.$$

These are genuine component-amplitude relations. They say that supersymmetry reduces several amplitudes to one dynamical function. They do not give the value of $F$. Couplings, color factors, poles, contact terms, loop integrals, and factorization data must still be computed or otherwise constrained.

Component extraction also catches an important bookkeeping error. If leg $i$ uses the CPT-conjugate polynomial, differentiating by $\eta_i$ selects a different particle than it would in the original polynomial. A correct Ward identity can appear false when external-state conventions are mixed.

Scope and failure conditions

This construction assumes:

• four-dimensional massless asymptotic states;
• an all-outgoing momentum convention;
• a fixed normalization of $Q$ and $\eta$;
• little-group-covariant external states;
• a CPT completion stated leg by leg;
• a supersymmetric vacuum and well-defined asymptotic charges; and
• no anomaly or regulator violation of the Ward identity.

Massive on-shell superspace needs extra little-group indices. Spontaneously broken supersymmetry gives Goldstino Ward identities rather than the unbroken relations used here. Infrared-divergent gauge-theory amplitudes may require regulated, inclusive, or dressed observables. At loop level, the algebraic Ward identities remain constraints when the regulator and renormalization preserve supersymmetry, but they do not fix branch cuts, rational terms, or subtraction data.

The next scientific step is three-point amplitudes, which develops Lorentzian or complex kinematics, locality, factorization seeds, and dynamical amplitude construction. This page supplies the supersymmetric state representation and Ward constraints only.

Common pitfalls

Treating $\eta$ as an ordinary scalar. Its little-group weight is required for the supercharge and every term in the on-shell polynomial to transform consistently.

Forgetting CPT completion. A two-state $\mathcal N=1$ helicity pair is generally not a full local spectrum. The conjugate polynomial must be included or deliberately separated as another charge sector.

Calling $\delta^{(2)}(\mathsf q)$ an amplitude. It solves the supersymmetry constraints. The function $F$ still contains the theory’s dynamics and must pass dimension, little-group, locality, unitarity, and factorization tests.

Check your understanding

Verify that $\bar{\mathsf q}_{\dot\alpha}\delta^{(2)}(\mathsf q)=0$ on momentum-conserving support.

Answer

Differentiating the quadratic Grassmann polynomial removes one $\eta_i$ and produces a factor $\widetilde\lambda_{i\dot\alpha}\lambda_i^\alpha$. Summing over $i$ gives the total momentum matrix $P^{\alpha}{}_{\dot\alpha}$ multiplying $\mathsf q_\alpha$. Since all momenta are outgoing, $\sum_i p_i=0$, so the result vanishes.

References

• Henriette Elvang, Daniel Z. Freedman, and Michael Kiermaier, “Solution to the Ward Identities for Superamplitudes,” Journal of High Energy Physics 2010 (2010), 103, DOI.
• Henriette Elvang and Yu-tin Huang, Scattering Amplitudes in Gauge Theory and Gravity, Cambridge University Press (2015), Chapter 4, DOI.
• Marcus T. Grisaru and Hugh N. Pendleton, “Some Properties of Scattering Amplitudes in Supersymmetric Theories,” Nuclear Physics B 124 (1977), 81–92, DOI.
• V. P. Nair, “A Current Algebra for Some Gauge Theory Amplitudes,” Physics Letters B 214 (1988), 215–218, DOI.