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Use the canonical Kähler potential and normalize superspace integration by $\int d^2\theta {\theta }^2=1$. The superspace Lagrangian density isThe full superspace integration produces a D-term, while the chiral superfield integral produces an F-term. Work in chiral coordinates for the chiral-superfield component expansion, so the omitted spacetime-derivative terms do not enter the extraction of $F$.

Set $\delta \Phi =\sqrt{2}\theta \psi +{\theta }^{2}F$. A Taylor expansion of the superpotential givesHigher terms vanish because there are only two Grassmann variables ${\theta }^{\alpha }$. With $\theta \psi ={\theta }^{\alpha }{\psi }_{\alpha }$ and $\psi \psi ={\psi }^{\alpha }{\psi }_{\alpha }$, their anticommutation gives $(\theta \psi {)}^2=-\frac{1}{2}{\theta }^2\psi \psi$. ThusThe canonical Kähler potential contributes $F^{\ast }F$. Consequently the part of the Lagrangian density containing the auxiliary field isThe fermion terms coming from the superpotential areThe Weyl spinor kinetic term and the complex scalar field kinetic term are canonical because $K={\Phi }^{†}\Phi$.

Treat $F$ and $F^{\ast }$ as independent variables in their algebraic Euler-Lagrange equations. They giveEquivalently,After eliminating the auxiliary field, the Lagrangian density contains $-V$, so the F-term scalar potential isIt is nonnegative for arbitrary complex $m,g,\phi$, since it is a modulus squared. A supersymmetric vacuum satisfies F-flatness, so its zeros are:If $m=0$ and $g\ne 0$ these coincide; if $g=0$ and $m\ne 0$ only the origin remains; if $m=g=0$ every constant $\phi$ is a supersymmetric vacuum.

To compare masses, expand around a supersymmetric vacuum $v$ satisfying $W^{\prime }(v)=0$, writing $\phi =v+\eta$. With $M=W^{\prime \prime }(v)=m+2gv$,Writing $\eta =(A+iB)/\sqrt{2}$ gives the scalar field mass term $-\frac{1}{2}|M{|}^2(A^2+B^2)$; the Weyl spinor has physical mass $|M|$. Therefore the supersymmetric mass degeneracy isIn particular, at $v=0$ the boson and fermion masses are both $|m|$. At the second supersymmetric vacuum $v=-m/g$ the sign of $M$ reverses, but the physical mass is again $|m|$. Equality here concerns fluctuations about a supersymmetric vacuum, not arbitrary backgrounds with $W^{\prime }(v)\ne 0$.

In the convention $V\supset \lambda |\phi {|}^4$ and ${\mathcal{L}}_{\mathrm{Yukawa}}=-y\phi \psi \psi +h.c.$, the same superpotential coefficient gives $\lambda =|g{|}^2$ and $y=g$. The supersymmetric relation between quartic and Yukawa couplings is thereforeIf instead the Yukawa interaction is written $-\frac{1}{2}{y}_3\phi \psi \psi +h.c.$, then $y_3=2g$ and the same physical relation is $\lambda =|y_3{|}^2/4$. The numerical factor depends on the definition of the Yukawa coupling.

The perturbative non-renormalization theorem applies to the local holomorphic superpotential in the Wilsonian effective action, using regularization in quantum field theory that preserves supersymmetry. There are two complementary ways to see why it applies here.

First, in supergraph calculations in perturbative quantum field theory, the algebra of supersymmetric covariant derivatives puts any candidate local correction to the superpotential into a full $d^4\theta$ superspace integration. It therefore corrects a D-term, such as the Kähler potential, rather than a local chiral $d^2\theta$ F-term. Turning such a contribution into an apparent chiral integral would require nonlocal inverse spacetime derivatives, schematically $1/\square$. These are excluded from the local Wilsonian effective action with a fixed nonzero momentum cutoff. This gives the diagrammatic non-renormalization theorem.

Second, the holomorphy argument for superpotential non-renormalization treats $m$ and $g$ as background chiral spurions, so the Wilsonian effective action superpotential is a holomorphic function of $\Phi ,m,g$, with no dependence on $m^{\ast }$ or $g^{\ast }$. Assign an ordinary $U(1)$ spurion charge and an R-charge as follows:The superpotential has ordinary charge zero and R-charge two, since $d^2\theta$ has R-charge minus two. A prospective holomorphic monomial $m^a{g}^b{\Phi }^c$ must therefore satisfyFor perturbative corrections, regularity at $m=0$ and $g=0$ requires nonnegative powers of the couplings. Regularity in $m$ is justified by keeping a nonzero Wilsonian momentum cutoff and retaining $\Phi$ as a light field. Hence only $c=2$ and $c=3$ survive: the existing $m{\Phi }^2$ and $g{\Phi }^3$ structures. The quadratic coefficient is fixed by the free theory at $g=0$, where there are no interaction loops. The cubic coefficient is fixed at first order in $g$, where a connected three-field interaction has only its tree-level Feynman diagram; loop corrections would require further powers of the interaction couplings, which the holomorphy argument for superpotential non-renormalization charge constraints forbid. ThusA field-independent constant is physically irrelevant in this rigid model; the spurion charges would in any case require $m^3/g^2$, which is not regular at $g=0$.

The Kähler potential can still receive wave-function renormalization. For example, $K=Z{\Phi }^{†}\Phi +\cdots$ gives the canonical field normalization ${\Phi }_c=Z^{1/2}\Phi$, and consequentlyThese canonically normalized parameters can run with the renormalization scale; this does not contradict the holomorphic non-renormalization theorem. When massless modes are included in the full quantum effective action, nonlocal terms associated with infrared divergences can also imitate an F-term. Distinguishing that action from the local Wilsonian effective action is part of the theorem's convention.

Treating derivatives with respect to the Grassmann variables as graded derivatives and using $\{{\partial }_{\alpha },{\theta }^{\beta }\}={\delta }_{\alpha }{}^{\beta }$ givesThe terms in $\{{\mathcal{D}}_{\alpha },{\mathcal{D}}_{\beta }\}$ and $\{{\overline{\mathcal{D}}}_{\dot{\alpha }},{\overline{\mathcal{D}}}_{\dot{\beta }}\}$ cancel pairwise, so both vanish.

A chiral superfield obeys ${\overline{\mathcal{D}}}_{\dot{\alpha }}\Phi =0$, while an antichiral superfield obeys ${\mathcal{D}}_{\alpha }{\Phi }^{†}=0$. Since ${\overline{\mathcal{D}}}_{\dot{\alpha }}{y}^{\mu }=0$ for $y^{\mu }=x^{\mu }+i\theta {\sigma }^{\mu }\bar{\theta }$, the general chiral expansion isIn the original coordinates this iswith signs following the conventions of the question.

A holomorphic function $W(\Phi )$ is chiral. Its highest component transforms into a spacetime divergence, so the F-term actionis supersymmetric even though it integrates over only chiral half of superspace.

The non-renormalization theorem says that perturbative loop corrections are full-superspace D-terms and cannot generate a new local superpotential. Holomorphy and spurion symmetries therefore preserveThe Kähler potential is renormalized, however. If its kinetic term is $Z{\Phi }^{†}\Phi$, canonical normalization ${\Phi }_c=Z^{1/2}\Phi$ givesup to scheme and scale conventions. Thus superpotential parameters are holomorphic invariants while physical masses and couplings still run through wave-function renormalization.

Choose $P_{\mu }=-i{\partial }_{\mu }$ and left Grassmann derivatives. A consistent differential realization of four-dimensional N=1 supersymmetry isThe signs can change with the definitions of $P_{\mu }$, Grassmann differentiation and the finite transformation, but these operators obey the convention-independent content of the Super-Poincaré algebra,They follow by expanding the finite transformation to first order and requiring a supersymmetry translation of the superspace coordinates.

The gauge-transformation superfield $\Lambda ={\Lambda }^a{T}_a$ is a dimensionless chiral superfield:It takes values in the complexified $\mathfrak{su}(3)$ Lie algebra. It need not be real; its Hermitian conjugate is antichiral. This complexification is necessary because a general superspace gauge transformation must preserve chirality.

A product of scalar superfields is a scalar superfield: the pullback of a product is the product of the pullbacks at the same transformed superspace point. Infinitesimally, the even supersymmetry variation obeys $\delta (ST)=(\delta S)T+S(\delta T)$. This follows from the graded Leibniz rule for the odd supercharges, after including their odd parameters.

With left Grassmann derivatives, let $Q_{\alpha }={\partial }_{\alpha }-i{\sigma }_{\alpha \dot{\beta }}^{\mu }{\bar{\theta }}^{\dot{\beta }}{\partial }_{\mu }$ and ${\bar{Q}}_{\dot{\alpha }}={\bar{\partial }}_{\dot{\alpha }}-i{\theta }^{\beta }{\sigma }_{\beta \dot{\alpha }}^{\mu }{\partial }_{\mu }$. Left-placed odd parameters generate $\delta \theta =ϵ$, $\delta \bar{\theta }=\overline{ϵ}$ and $\delta x^{\mu }=i\theta {\sigma }^{\mu }\overline{ϵ}-iϵ{\sigma }^{\mu }\bar{\theta }$. A scalar superfield transforms by pullback under this superspace translation. The parameter-weighted variation is an even derivation, although each supercharge is odd.

A superfield is a field on superspace with a specified Super-Poincaré group transformation law. Being a polynomial in anticommuting coordinates alone does not specify this law or make coefficients a supermultiplet. Conversely, a suitably graded function on full superspace, acted on by coordinate pullback, is an unconstrained scalar superfield. A polynomial with spacetime-independent coefficients can be a special constant-component chiral superfield; dependence on spacetime is not itself the defining criterion. Chirality and reality impose additional conditions.

A supergraph is a Feynman diagram formulated with superfield propagators and vertices in superspace. Performing its spinor-derivative algebra organizes perturbative corrections into superspace integration terms and provides a proof of the local superpotential non-renormalization theorem.

A gauge connection on superspace has both vector and spinor components. Subtracting the flat-frame torsion from the graded commutator of its covariant derivatives defines its supercurvature. The conventional four-dimensional $\mathcal{N}=1$ spinor-curvature constraints leave one real vector superfield as a prepotential. Its chiral field-strength superfield contains the ordinary gauge curvature, gaugino and real auxiliary field.

Supersymmetry extends spacetime symmetry by odd supercharges that interchange boson and fermion states. Their Super-Poincaré algebra pairs positive-energy states into supermultiplets and gives energy positivity in global supersymmetry. Superspace and superfields package the component fields into representations of this symmetry.

A supersymmetry transformation is the infinitesimal action generated by an odd supercharge, with constant anticommuting spinor parameter in a global theory. In superspace it acts as $\delta \mathcal{S}=(ϵQ+\overline{ϵ}\bar{Q})\mathcal{S}$, with possible conventional overall factors of $i$. On the fermion of a chiral superfield, the transformation includes $\delta \psi =\sqrt{2}ϵF_{\mathrm{aux}}+i\sqrt{2}{\sigma }^{\mu }\overline{ϵ}{\partial }_{\mu }\varphi$. A nonzero vacuum auxiliary field therefore gives an inhomogeneous fermion shift, identifying the Goldstino direction when global supersymmetry is spontaneously broken.

A supertranslation is generated by translations and supercharges, excluding the Lorentz transformations. For flat Majorana superspace with ${\Pi }^m=d{X}^m+i\bar{\theta }{\gamma }^{m}d\theta$, the rigid odd transformation $\delta \theta =ϵ$, $\delta X^m=-i\overline{ϵ}{\gamma }^m\theta$ leaves ${\Pi }^m$ invariant. Its commutator gives an ordinary translation.

The displayed one-forms give an invertible triangular change from the coordinate coframe on flat Majorana superspace. A rigid supertranslation has $dϵ=0$, and the variation of $d{X}^m$ cancels that of $i\bar{\theta }{\gamma }^{m}d\theta$. Their Maurer-Cartan equation is $d{\Pi }^m=id\bar{\theta }{\gamma }^{m}d\theta$. The bosonic frame transforms as a Lorentz vector and the odd frame as a Majorana spinor.