Rigid four-dimensional $N=1$ Wess–Zumino sigma-model action

Use the physical action brackets $[U{]}_D=-I_D(U)\mathrm{/}4$ and $[Y{]}_F=I_F(Y)\mathrm{/}2$ from Physical superspace action brackets and canonical chiral components, with raw mathematical coefficient integrals $I_D,I_F$ unchanged. Physical chiral components obey ${\psi }_{\mathrm{r}\mathrm{a}\mathrm{w}}=\sqrt{2}\psi$ and $F_{\mathrm{r}\mathrm{a}\mathrm{w}}=2F$. All physical action and coupling formulas below use these brackets and physical component names; adding Hermitian conjugation applies only to the F action and occurs once.

Use oriented and time-oriented Minkowski spacetime $M={\mathbb{R}}^{1,3}$ with coordinates $x^0,x^1,x^2,x^3$, metric $\eta =\mathrm{diag}(-1,1,1,1)$, and natural units $\mathrm{\hslash }=c=1$. Let $n\ge 1$, let $U\subseteq {\mathbb{C}}^n$ be an open set containing $0$, let $K:U\to \mathbb{R}$ be smooth, and require the Hermitian matrix $g_{i\bar{j}}(z,\bar{z})={\mathrm{\partial }}_i{\mathrm{\partial }}_{\bar{j}}K(z,\bar{z})$ to be positive definite at every point of $U$. Let $W:U\to \mathbb{C}$ be smooth and holomorphic, where holomorphic means ${\mathrm{\partial }}_{\bar{i}}W=0$ for every $i$, using the Wirtinger derivatives of the Kähler coordinate-patch definition; write $W_i={\mathrm{\partial }}_{i}W$. These data specify the Kähler coordinate patch and holomorphic superpotential of the model.

For these classical fields, take a finite-dimensional exterior algebra $\mathcal{G}$ over $\mathbb{C}$ on ordered odd generators ${\zeta }_1,\dots ,{\zeta }_m$ for $m\ge 0$, with componentwise complex addition and multiplication modulo ${\zeta }_a^2=0$ and ${\zeta }_a{\zeta }_b=-{\zeta }_b{\zeta }_a$. Give it the conjugate-linear operation $(c{\zeta }_{i_1}\cdots {\zeta }_{i_r}{)}^{\ast }=\bar{c}{\zeta }_{i_r}\cdots {\zeta }_{i_1}$, so ${\zeta }_a^{\ast }={\zeta }_a$, $(uv{)}^{\ast }=v^{\ast }{u}^{\ast }$ and $(u^{\ast }{)}^{\ast }=u$. Reversing twice proves involutivity, and reversing the defining relations preserves them; thus this is a well-defined parity-preserving anti-automorphism. Write ${\mathcal{G}}_{\bar{0}},{\mathcal{G}}_{\bar{1}}$ for its even and odd subspaces, and use its degree-zero projection $b:\mathcal{G}\to \mathbb{C}$. Form the graded tensor product $\mathcal{B}=\mathcal{G}{\otimes }_{\mathbb{C}}{\mathrm{\Lambda }}_4$ with the superspace Grassmann algebra, and let $\mathfrak{J}$ be the ideal of elements with zero total degree-zero part. Then ${\mathfrak{J}}^{m+5}=0$. For an even superfield ${\mathrm{\Phi }}^i$, its body is its image under the augmentation $\mathcal{B}\to \mathbb{C}$ that applies $b$ and sends all superspace generators to zero; denote that ordinary scalar component by ${\varphi }^i(x)$. Write ${\bar{\mathrm{\Phi }}}^{\bar{i}}=({\mathrm{\Phi }}^i{)}^{\ast }$, ${\nu }^i={\mathrm{\Phi }}^i-{\varphi }^i$, and ${\bar{\nu }}^{\bar{i}}={\bar{\mathrm{\Phi }}}^{\bar{i}}-{\bar{\varphi }}^{\bar{i}}$; these differences lie in $\mathfrak{J}$. Define the smooth function's value on the full superfields by the finite nilpotent Taylor convention

$K(\bar{\mathrm{\Phi }},\mathrm{\Phi }):={\sum }_{\begin{array}{c}\alpha ,\beta \in {\mathbb{N}}^n\\ \mathrm{\mid }\alpha \mathrm{\mid }+\mathrm{\mid }\beta \mathrm{\mid }\le m+4\end{array}}\frac{({\mathrm{\partial }}_z^{\alpha }{\mathrm{\partial }}_{\bar{z}}^{\beta }K)(\varphi ,\bar{\varphi })}{\alpha !\beta !}(\nu {)}^{\alpha }(\bar{\nu }{)}^{\beta },$

where $\alpha !={\prod }_i{\alpha }_i!$, $\beta !={\prod }_i{\beta }_i!$, ${\nu }^{\alpha }={\prod }_i({\nu }^i{)}^{{\alpha }_i}$, $(\bar{\nu }{)}^{\beta }={\prod }_i({\bar{\nu }}^{\bar{i}}{)}^{{\beta }_i}$, ${\mathrm{\partial }}_z^{\alpha }={\prod }_i{\mathrm{\partial }}_i^{{\alpha }_i}$, ${\mathrm{\partial }}_{\bar{z}}^{\beta }={\prod }_i{\mathrm{\partial }}_{\bar{i}}^{{\beta }_i}$, and the coefficient derivatives are evaluated at $(\varphi (x),\bar{\varphi }(x))$. Define the holomorphic superfield evaluation and its derivatives by the finite Taylor rule at the scalar body,

$W(\mathrm{\Phi })(x,\theta ,\bar{\theta }):={\sum }_{\mathrm{\mid }\alpha \mathrm{\mid }\le m+4}\frac{({\mathrm{\partial }}_z^{\alpha }W)(\varphi (x))}{\alpha !}(\nu {)}^{\alpha },W_i(\mathrm{\Phi }):={\sum }_{\mathrm{\mid }\alpha \mathrm{\mid }\le m+4}\frac{({\mathrm{\partial }}_z^{\alpha }{\mathrm{\partial }}_{i}W)(\varphi (x))}{\alpha !}(\nu {)}^{\alpha }\mathrm{.}$

All terms beyond these finite sums vanish because ${\mathfrak{J}}^{m+5}=0$; the rule for $K$ is the Taylor extension of a smooth real function in its $z,\bar{z}$ variables, and the rule for $W$ uses only its holomorphic Wirtinger derivatives.

For each $i=1,\dots ,n$, the full formal physical components are smooth maps $a^i,{\mathcal{F}}^i:M\to {\mathcal{G}}_{\bar{0}}$ and ${\psi }_{\alpha }^i:M\to {\mathcal{G}}_{\bar{1}}$, with compact support in spacetime $M$: equivalently each coefficient of each exterior-basis monomial is a compactly supported smooth complex function of $x$. Require the scalar body ${\varphi }^i=b(a^i)$ to lie in $U$ jointly. The lowest chiral coefficient is $a^i$, not merely its body ${\varphi }^i$; it may have even nilpotent terms. In the physical normalization definition instantiate its scalar and auxiliary component symbols by $a^i$ and ${\mathcal{F}}^i$, respectively:

${\mathrm{\Phi }}^i(y,\theta )=a^i(y)+\sqrt{2}{\theta }^{\alpha }{\psi }_{\alpha }^i(y)+2{\theta }^1{\theta }^2{\mathcal{F}}^i(y),y=x+i\theta \sigma \bar{\theta }\mathrm{.}$

Barred component fields mean the specified coefficient involution, so ${\bar{a}}^{\bar{i}}=(a^i{)}^{\ast }$ and ${\bar{\mathcal{F}}}^{\bar{i}}=({\mathcal{F}}^i{)}^{\ast }$; barred fermions are likewise the conjugates of the coefficient maps. On $\mathcal{B}$ use the supplier's graded-tensor involution, including ${\theta }^{\ast }=\bar{\theta }$ and reversal of all factors. The formal action has values in the even fixed real subspace of $\mathcal{G}$; no order or nonnegativity is assigned to nilpotent elements.

The supersymmetry parameters ${ϵ}^{\alpha },{\overline{ϵ}}^{\dot{\alpha }}$ are constant odd coefficients, in $\mathcal{G}$ or a finite exterior extension equipped with the same involution. The adopted variation is $\delta {\mathrm{\Phi }}^i=({ϵ}^{\alpha }{Q}_{\alpha }+{\overline{ϵ}}^{\dot{\alpha }}{\bar{Q}}_{\dot{\alpha }}){\mathrm{\Phi }}^i$ with the chiral-superfield supercharges. Formal parameter components may be treated independently for the algebraic identities; a conjugation-preserving variation in this left-parameter convention has ${\overline{ϵ}}^{\dot{\alpha }}=-({ϵ}^{\alpha }{)}^{\ast }$ and $\delta \bar{\mathrm{\Phi }}=(\delta \mathrm{\Phi }{)}^{\ast }$. Indeed $\delta \theta =ϵ$ and $\delta \bar{\theta }=-\overline{ϵ}$ then commute with the chosen involution. The full even coefficient spaces for $a$ and $\mathcal{F}$ are retained under these transformations; the ordinary bosonic slice below need not be invariant. The engineering dimensions are $[x^{\mu }]=-1$, $[\theta ]=[\bar{\theta }]=-1\mathrm{/}2$, $[a^i]=1$, $[{\psi }^i]=3\mathrm{/}2$, $[{\mathcal{F}}^i]=2$, $[K]=2$, and $[W]=3$ in mass units.

Adopt the classical action

$S_{\mathrm{W}\mathrm{Z}}={\int }_M{d}^{4}x([K(\bar{\mathrm{\Phi }},\mathrm{\Phi }){]}_D+[W(\mathrm{\Phi }){]}_F+\mathrm{h}\mathrm{.}\mathrm{c}\mathrm{.}),$

where only the chiral F action has a Hermitian-conjugate partner and the D bracket is real; both physical brackets are the fixed multiples of the earlier raw ordered measures. The ordinary bosonic sector is the restriction $\psi =0$, $a=\varphi \in {\mathbb{C}}^n$ and $\mathcal{F}=F\in {\mathbb{C}}^n$ with all positive-degree coefficient-algebra components set to zero; $\varphi :M\to U$ and $F:M\to {\mathbb{C}}^n$ remain smooth and compactly supported in $M$. Thus taking this sector commutes with the degree-zero augmentation, but it is a restriction for evaluating a bosonic action, not an assertion of off-shell supersymmetry closure on a purely complex field space. The component conversion below gives its complete bosonic Lagrangian, up to the displayed spacetime divergence,

${\mathcal{L}}_{\mathrm{b}\mathrm{o}\mathrm{s}}=-g_{i\bar{j}}(\varphi ,\bar{\varphi }){\eta }^{\mu \nu }{\mathrm{\partial }}_{\mu }{\varphi }^i{\mathrm{\partial }}_{\nu }{\bar{\varphi }}^{\bar{j}}+g_{i\bar{j}}(\varphi ,\bar{\varphi })F^i{\bar{F}}^{\bar{j}}+F^i{W}_i(\varphi )+{\bar{F}}^{\bar{j}}\overline{W_j(\varphi )}+\frac{1}{4}\mathrm{\square }K(\varphi ,\bar{\varphi }),\mathrm{\square }={\eta }^{\mu \nu }{\mathrm{\partial }}_{\mu }{\mathrm{\partial }}_{\nu }\mathrm{.}$

Its bosonic zero-derivative auxiliary terms are therefore precisely

${\mathcal{L}}_F=g_{i\bar{j}}(\varphi ,\bar{\varphi })F^i{\bar{F}}^{\bar{j}}+F^i{W}_i(\varphi )+{\bar{F}}^{\bar{j}}\overline{W_j(\varphi )}\mathrm{.}$

Only this ordinary complex sector is used for the ordered real scalar-potential and minimization conclusions. An arbitrary even $\mathcal{G}$-valued auxiliary field remains valid in the full formal model, but is not a vector in an ordered positive-definite complex metric space.

The canonical Wess–Zumino model is the special case $U={\mathbb{C}}^n$, $K(z,\bar{z})={\sum }_i{\bar{z}}^{\bar{i}}{z}^i$, and hence $g_{i\bar{j}}={\delta }_{i\bar{j}}$. This is an adopted classical formulation and normalization, not a deduction that nature realizes the model.

Stephen P. Martin, A Supersymmetry Primer, §§3.1–3.2 and 4.7.

Adel Bilal, Introduction to Supersymmetry, §4.3 and §7.1.

Four-dimensional $N=1$ chiral superfield on flat superspace

Four-dimensional $N=1$ superspace Grassmann algebra and Berezin integration

Kähler coordinate-patch data for a rigid $N=1$ chiral model

Physical superspace action brackets and canonical chiral components

This finite formula specifies the complete off-shell component action, including fermionic and derivative terms, without identifying our raw measures with differently ordered textbook measures. Write $h^{\mu }=i{\theta }^{\alpha }{\sigma }_{\alpha \dot{\alpha }}^{\mu }{\bar{\theta }}^{\dot{\alpha }}$, and use $\mathcal{C}(P)$ for the coefficient of ${\theta }^1{\theta }^2{\bar{\theta }}^{\dot{1}}{\bar{\theta }}^{\dot{2}}$ in a polynomial $P$ with coefficients on the left, using the earlier signed subset multiplication. Evaluation of any derivative of $K$ or $W$ at the even coefficient $a$ means its finite Taylor extension at $b(a)$, as in the Postulate. Define the following explicit polynomials in the component maps and their spacetime derivatives:

$d^i=\sqrt{2}{\theta }^{\alpha }{\psi }_{\alpha }^i+2{\theta }^1{\theta }^2{\mathcal{F}}^i+h^{\mu }{\mathrm{\partial }}_{\mu }{a}^i+\sqrt{2}{h}^{\mu }{\theta }^{\alpha }{\mathrm{\partial }}_{\mu }{\psi }_{\alpha }^i+\frac{1}{2}{h}^{\mu }{h}^{\nu }{\mathrm{\partial }}_{\mu }{\mathrm{\partial }}_{\nu }{a}^i,{\bar{d}}^{\bar{i}}=(d^i{)}^{\ast }\mathrm{.}$

The finite Taylor shift $e^{h^{\mu }{\mathrm{\partial }}_{\mu }}$ of the chiral component expansion gives exactly ${\mathrm{\Phi }}^i=a^i+d^i$ and ${\bar{\mathrm{\Phi }}}^{\bar{i}}={\bar{a}}^{\bar{i}}+{\bar{d}}^{\bar{i}}$. The shift of ${\theta }^1{\theta }^2\mathcal{F}$ vanishes because it contains three undotted generators; its action on $\theta \psi$ stops after one shift. Every monomial in $d$ has at least one undotted generator and every monomial in $\bar{d}$ at least one dotted generator, so a product of three $d$ factors or of three $\bar{d}$ factors vanishes. With empty products equal to one, the complete D-component density is the explicit finite component expression

${\mathcal{L}}_D=-\frac{1}{4}{\sum }_{r,s=0}^2\frac{K_{i_1\cdots i_r{\bar{j}}_1\cdots {\bar{j}}_s}(a,\bar{a})}{r!s!}\mathcal{C}\text{\hspace{0.17em}⁣}(d^{i_1}\cdots d^{i_r}{\bar{d}}^{{\bar{j}}_1}\cdots {\bar{d}}^{{\bar{j}}_s})\mathrm{.}$

Here repeated indices range from $1$ to $n$, and $K_{i_1\cdots i_r{\bar{j}}_1\cdots {\bar{j}}_s}={\mathrm{\partial }}_{i_1}\cdots {\mathrm{\partial }}_{i_r}{\mathrm{\partial }}_{{\bar{j}}_1}\cdots {\mathrm{\partial }}_{{\bar{j}}_s}K$. Together with the displayed $d$, this is a finite sum solely of component fields and their derivatives: the fixed sixteen-element subset table computes each $\mathcal{C}$ by addition and multiplication in $\mathcal{G}$. It includes all two-fermion, four-fermion and derivative terms, rather than retaining only the auxiliary truncation. No unspecified superfield or superspace integration remains in this component prescription.

The chiral F-component can be expanded completely in two undotted generators. Products of the two linear odd terms give coefficient $-2{\psi }_1^i{\psi }_2^j$ after contraction with the symmetric $W_{ij}$. Thus, with $W_{ij}={\mathrm{\partial }}_i{\mathrm{\partial }}_{j}W$,

$I_F(W(\mathrm{\Phi }))=2W_i(a){\mathcal{F}}^i-2W_{ij}(a){\psi }_1^i{\psi }_2^j,{\mathcal{L}}_W=W_i(a){\mathcal{F}}^i-W_{ij}(a){\psi }_1^i{\psi }_2^j+{(W_i(a){\mathcal{F}}^i-W_{ij}(a){\psi }_1^i{\psi }_2^j)}^{\ast }\mathrm{.}$

The full component action is $S_{\mathrm{W}\mathrm{Z}}={\int }_M({\mathcal{L}}_D+{\mathcal{L}}_W)d^{4}x$, interpreted coefficientwise. In particular Hermitian conjugation is added to the extracted F-component once, with the reversed fermion order imposed by $\ast$.

For a direct check of the complete bosonic conversion, the Pauli matrices give $\mathcal{C}(h^{\mu }{h}^{\nu })=-2{\eta }^{\mu \nu }$: its diagonal is $(2,-2,-2,-2)$ and off-diagonal entries vanish. On the ordinary bosonic slice, $d=h^{\mu }{\mathrm{\partial }}_{\mu }\varphi +\frac{1}{2}{h}^{\mu }{h}^{\nu }{\mathrm{\partial }}_{\mu }{\mathrm{\partial }}_{\nu }\varphi +2{\theta }^1{\theta }^{2}F$ and its conjugate has $h\mapsto -h$ and auxiliary term $-2{\bar{\theta }}^{\dot{1}}{\bar{\theta }}^{\dot{2}}\bar{F}$. The mixed auxiliary product has top coefficient $-4F^i{\bar{F}}^{\bar{j}}$, and the scalar derivative terms give $4g_{i\bar{j}}{\mathrm{\partial }}_{\mu }{\varphi }^i{\mathrm{\partial }}^{\mu }{\bar{\varphi }}^{\bar{j}}-\mathrm{\square }K$. Multiplying by $-1\mathrm{/}4$ yields the kinetic and positive complex auxiliary terms displayed in the Postulate. The F bracket's $1\mathrm{/}2$ cancels the coefficient $2$ in $F_{\mathrm{r}\mathrm{a}\mathrm{w}}=2F$, giving $F^i{W}_i$ rather than twice that term. Compact support makes ${\int }_M\mathrm{\square }K=0$: outside a compact set $K$ is the constant $K(0,0)$, so its derivative has compact support. Setting spacetime derivatives and fermions to zero then yields exactly the auxiliary Lagrangian consumed by the F-term theorem.