Supersymmetric Action Principles and Component Reduction

Component reduction is a controlled projection, not a transcription exercise. One expands each superfield in a fixed convention, multiplies in Grassmann order, selects the measure’s top component, records spacetime total derivatives, and only then eliminates auxiliary fields. Keeping those stages separate makes it possible to verify the result twice: once by the manifestly supersymmetric superspace construction and once by direct component transformations.

This page carries out the workflow for four-dimensional rigid $\mathcal N=1$ chiral multiplets. It keeps $F^i$ off shell through the invariance check and then derives the on-shell scalar and Yukawa interactions. Gauge fixing and vector multiplets are added on later pages.

Required background. Superspace Measures, F-Terms, D-Terms, and Component Extraction fixes the measure and projection conventions. Off-Shell Closure and Auxiliary Fields explains why $F$ must be retained for an off-shell closure test.

Helpful background. The Action Principle and Field Equations supplies the boundary-sensitive variational framework.

Freeze the local convention before expanding

Use chiral superfields

$$\Phi^i(y,\theta) =\phi^i(y)+\sqrt2\,\theta\psi^i(y)+\theta^2F^i(y), \qquad y^\mu=x^\mu+i\theta\sigma^\mu\bar\theta,$$

with $\int d^2\theta\,\theta^2=1$ and the $(+---)$ metric. The rigid two-derivative action is

$$S= \int d^4x\left[ \int d^4\theta\,K(\Phi,\Phi^\dagger) +\left( \int d^2\theta\,W(\Phi)+\text{h.c.} \right) \right].$$

$K$ is real and $W$ is holomorphic. For the canonical reduction take $K=\Phi^{\dagger\bar i}\Phi^i\delta_{i\bar i}$; the nonlinear Kähler case is derived on the next geometry page. The dimensions are

$$[\phi]=1, \qquad [\psi]=\frac32, \qquad [F]=2, \qquad [W]=3.$$

The off-shell supersymmetry transformations in this convention may be chosen as

$$\begin{aligned} \delta\phi^i&=\sqrt2\,\epsilon\psi^i,\\ \delta\psi^i_\alpha &=\sqrt2\,\epsilon_\alpha F^i +i\sqrt2\,(\sigma^\mu\bar\epsilon)_\alpha\partial_\mu\phi^i,\\ \delta F^i &=i\sqrt2\,\bar\epsilon\bar\sigma^\mu\partial_\mu\psi^i. \end{aligned}$$

Their commutator is a translation on all three fields without using any field equation. A source using the mostly-plus metric or another raising convention may display different intermediate signs; the invariant checkpoints are the translation algebra, positive kinetic energy, and the pole spectrum.

Reduce the canonical chiral action in four passes

Pass 1: select the D-term

Expanding $\Phi^\dagger\Phi$ and selecting $\theta^2\bar\theta^2$ gives, after one integration by parts,

$$\mathcal L_D =\partial_\mu\phi^{*\bar i}\partial^\mu\phi^i\delta_{i\bar i} +i\bar\psi^{\bar i}\bar\sigma^\mu\partial_\mu\psi^i\delta_{i\bar i} +F^{*\bar i}F^i\delta_{i\bar i}.$$

Before the integration by parts, the scalar coefficient can be written with $\phi^*\Box\phi$ and explicit divergences. The two forms define the same action only when the boundary term vanishes or is cancelled. Keeping a “discarded derivative” line in the calculation prevents a boundary-sensitive problem from silently inheriting a closed-manifold answer.

Pass 2: select the F-term

Taylor expansion terminates because $\theta$ has two components:

$$W(\Phi)\big|_{\theta^2} =F^iW_i(\phi) -\frac12W_{ij}(\phi)\psi^i\psi^j.$$

Therefore the complete off-shell component Lagrangian is

$$\begin{aligned} \mathcal L_{\mathrm{off}}={}& \partial_\mu\phi^{*\bar i}\partial^\mu\phi^i\delta_{i\bar i} +i\bar\psi^{\bar i}\bar\sigma^\mu\partial_\mu\psi^i\delta_{i\bar i} +F^{*\bar i}F^i\delta_{i\bar i}\\ &+F^iW_i -\frac12W_{ij}\psi^i\psi^j +F^{*\bar i}\bar W_{\bar i} -\frac12\bar W_{\bar i\bar j} \bar\psi^{\bar i}\bar\psi^{\bar j}. \end{aligned}$$

Every term has mass dimension four. The factor $1/2$ is required because $W_{ij}$ is symmetric while $\psi^i\psi^j$ is symmetric in flavor indices after the antisymmetric spinor contraction. Detailed component constructions and the comparison with superspace appear in Gates et al. 1983, chs. 4–5, Open PDF and Weinberg 2000, §§26.1–26.4, pp. 55–82.

Pass 3: verify invariance before eliminating auxiliaries

Vary $\mathcal L_{\mathrm{off}}$ using the three transformations above. Terms proportional to $F\partial\psi$ cancel between the fermion kinetic term and $F^*F$; terms proportional to $W_i\partial\psi^i$ cancel against the variation of the Yukawa term after using $W_{ij}=W_{ji}$; the remainder is

$$\delta\mathcal L_{\mathrm{off}} =\partial_\mu\mathcal K^\mu(\epsilon,\bar\epsilon).$$

No equation of motion is needed. This is the decisive independent check of the Grassmann reduction. If the cancellation requires the Dirac equation or the scalar equation, either the claimed closure status is wrong or a term has been lost.

Pass 4: eliminate $F$ algebraically

Treat $F^i$ and $F^{*\bar i}$ as independent variables in the variation. Their equations are

$$F^i=-\delta^{i\bar j}\bar W_{\bar j}, \qquad F^{*\bar i}=-\delta^{j\bar i}W_j.$$

Completing the square makes the sign transparent:

$$F^{*\bar i}F^i\delta_{i\bar i} +F^iW_i+F^{*\bar i}\bar W_{\bar i} =\left|F^i+\bar W_{\bar i}\right|^2-|W_i|^2.$$

The on-shell Lagrangian is consequently

$$\begin{aligned} \mathcal L_{\mathrm{on}}={}& \partial_\mu\phi^{*\bar i}\partial^\mu\phi^i\delta_{i\bar i} +i\bar\psi^{\bar i}\bar\sigma^\mu\partial_\mu\psi^i\delta_{i\bar i} -V_F(\phi,\phi^*)\\ &-\frac12W_{ij}\psi^i\psi^j -\frac12\bar W_{\bar i\bar j} \bar\psi^{\bar i}\bar\psi^{\bar j}, \qquad V_F=\sum_i|W_i|^2. \end{aligned}$$

$V_F\geq0$ follows from the positive canonical Kähler metric. The reduced transformations close only after the fermion equation is used, because $F=-\bar W'$ has converted an independent auxiliary into a composite field. Auxiliary elimination is exact at the classical two-derivative level; it is not a low-energy approximation. It can become more subtle if higher-derivative operators give the nominal auxiliary derivatives or multiple algebraic branches.

A polynomial reduction with all coefficients visible

For one field, choose

$$W(\Phi)=a\Phi+\frac12m\Phi^2+\frac13y\Phi^3.$$

Then

$$W'=a+m\phi+y\phi^2, \qquad W''=m+2y\phi,$$

and

$$\begin{aligned} \mathcal L_{\mathrm{off}}={}& \partial_\mu\phi^*\partial^\mu\phi +i\bar\psi\bar\sigma^\mu\partial_\mu\psi+F^*F\\ &+F(a+m\phi+y\phi^2) -\frac12(m+2y\phi)\psi\psi+\text{h.c.} \end{aligned}$$

Eliminating $F$ gives

$$V=|a+m\phi+y\phi^2|^2,$$

together with fermion mass $m$ and Yukawa coupling $-y\phi\psi\psi+ \text{h.c.}$ in the expansion around $\phi=0$. Around a supersymmetric vacuum $\phi_0$ satisfying $W'(\phi_0)=0$, the fermion mass parameter is $M=W''(\phi_0)$ and the quadratic scalar potential is $|M|^2|\delta\phi|^2$. Thus the complex scalar and Weyl fermion have the same pole mass $|M|$. That equality is an invariant check of the factor $1/2$, the auxiliary sign, and the vacuum expansion.

The reduction transcript

A reproducible calculation can be recorded as this compact table:

stage	retained data	check that must pass
superspace input	$K$, $W$, measure, reality, gauge and boundary data	dimensions, chirality, gauge invariance
Grassmann selection	every selected monomial and coefficient	derivative order and Berezin signs
component rearrangement	integrations by parts and discarded divergences	boundary compatibility
off-shell action	propagating and auxiliary fields	direct supersymmetry variation closes without equations
auxiliary equation	algebraic Hessian and branch	substitution solves the same equation and preserves reality
on-shell action	kinetic, masses, Yukawa and potential	positivity and multiplet mass degeneracy

This structured transcript is also the nonvisual content required of the chapter’s governed component-reduction diagram. A runnable symbolic check is useful evidence, but it cannot replace the declared convention and the independent component variation.

Common pitfalls

Eliminating $F$ before the closure check. This turns an off-shell realization into an on-shell one and can hide a missing term behind a fermion equation of motion.

Conjugating during Euclidean reduction. In Euclidean signature, barred and unbarred fields are generally independent on the complexified field space. A reality relation is part of the chosen integration cycle, not an automatic algebraic step.

Completing the square with the wrong kinetic sign. If $F^*F$ has been mis-signed, the apparent scalar potential may be unbounded or have the wrong sign in the Lagrangian. Check the Hamiltonian and the mass spectrum, not just the appearance of $|W'|^2$.

Exercises

1. Reproduce the Yukawa coefficient. Expand $W(\Phi)=y\Phi^3/3$ through $\theta^2$.

Solution

Using $W'=y\phi^2$ and $W''=2y\phi$ gives

$$W(\Phi)\big|_{\theta^2} =y\phi^2F-y\phi\psi\psi.$$

The two identical ways of choosing one $\theta^2F$ and two identical ways of choosing the fermion components are already encoded in the derivative formula; no additional factorial should be inserted.

2. Diagnose a closure claim. After setting $F=-\bar W_{\bar\phi}$, why can two supersymmetry transformations on $\psi$ produce a fermion equation-of-motion term?

Solution

Off shell, $\delta F$ is an independent transformation chosen so the algebra closes. After elimination, $\delta F$ must instead be computed from $F=-\bar W_{\bar\phi}$. The difference between those two variations is proportional to the fermion equation. The reduced fields therefore realize the same algebra only on shell.

Where the method is used

Wess–Zumino Models analyzes the polynomial example vacuum by vacuum. Kähler Sigma Models replaces the flat auxiliary metric by $K_{i\bar j}$, and Gauge–Matter Systems adds vector auxiliaries and moment maps.

References

• Gates, S. James, Marcus T. Grisaru, Martin Roček, and Warren Siegel. Superspace, or One Thousand and One Lessons in Supersymmetry. Reading, MA: Benjamin/Cummings, 1983, chs. 4–5. Open PDF, arXiv v5.
• Weinberg, Steven. The Quantum Theory of Fields, Volume III: Supersymmetry. Cambridge: Cambridge University Press, 2000, §§26.1–26.4, pp. 55–82. DOI.

Further reading

• Wess, Julius, and Jonathan Bagger. Supersymmetry and Supergravity. 2nd ed. Princeton, NJ: Princeton University Press, 1992, chs. 4–5.