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For a chiral superfield of mass dimension one, a renormalizable quantum field theory has a quadratic Kähler potential and a superpotential of degree at most three. Before canonical normalization one may writewith real $k_0$, andThe purely holomorphic terms in $K$, their conjugates and its constant have vanishing full superspace integration; they are Kähler transformations of the global theory. Rescaling the chiral superfield sets $Z=1$, so the physically general kinetic choice is $K={\Phi }^{†}\Phi$ up to these terms. The constant superpotential in global supersymmetry also has no component effect. The dimensions of $\ell ,m,g$ are respectively two, one and zero.

To eliminate the linear term, translate the complex chiral superfield as $\Phi ={\Phi }^{\prime }+s$. The shifted linear coefficient isIf $g\ne 0$, this quadratic has a root over the complex numbers; if $g=0$ but $m\ne 0$, take $s=-\ell /m$. The resulting linear term vanishes and the new quadratic coefficient is $m+2gs$. The canonical Kähler potential changes only by holomorphic, antiholomorphic and constant terms, which do not change the global kinetic action. This is the superpotential critical-point shift.

The claim needs an exception: if $m=g=0$ but $\ell \ne 0$, the superpotential is purely linear and no invertible translation can remove it. More generally an invertible regular field redefinition cannot turn a nowhere-zero derivative into a critical point. The theory with $W=\ell \Phi$ has $F=-\bar{\ell }$, a constant positive scalar potential $|\ell {|}^2$, and flat F-term breaking with a linear superpotential. Below, $m$ and $g$ denote the quadratic/cubic coefficients after the valid shift, as intended by the question.

Expand $\Phi =\varphi +\sqrt{2}\theta \psi +{\theta }^{2}F$ in chiral coordinates. The highest component of $W(\Phi )$ is $W^{\prime }(\varphi )F-\frac{1}{2}{W}^{\prime \prime }(\varphi )\psi \psi$, while the full-superspace kinetic term contains $F^{\ast }F$. Thus the auxiliary field part of the component Lagrangian isVarying $F$ and $F^{\ast }$ independently givesSubstitution gives ${\mathcal{L}}_F=-|m\varphi +g{\varphi }^2{|}^2$, and hence the F-term scalar potentialIt is nonnegative and vanishes at $\varphi =0$, and also at $\varphi =-m/g$ when $g\ne 0$. These are supersymmetric vacua, since the auxiliary field vanishes. Thus the quadratic/cubic model does not spontaneously break supersymmetry. If $m=0$ but $g\ne 0$, zero is the sole critical point; if both vanish, every $\varphi$ is a zero-energy vacuum. Nonzero $m,g$ give the usual two vacua, coinciding when $m=0$.

For the loop argument, promote the coefficients to background chiral spurions. One convenient pair of independent formal $U(1)$ charge assignments isBoth symmetries act nontrivially on all three of $\Phi ,m,g$; the second is an R-symmetry. The superspace integration measure $d^2\theta$ has R-charge $-2$, so both tree monomials give invariant actions. These are spurionic selection rules, not necessarily actual symmetries after fixing numerical couplings. The R-row is an ordinary-charge redefinition of the usual Wess–Zumino spurion charge assignment.

A local Wilsonian superpotential is a holomorphic function of the chiral fields and chiral spurions. The ratio $z=g\Phi /m$ is neutral under both charge assignments and has dimension zero, whereas $m{\Phi }^2$ has exactly the charges and dimension of $W$. The general symmetry-allowed form, on a patch with $m\ne 0$ and up to a physically irrelevant global constant, is thereforewith $f$ holomorphic where that description is regular. Symmetry by itself does not determine $f$.

For a regular perturbative local Wilsonian effective action retaining the same elementary field and an infrared cutoff, expand $f(z)={\sum }_n{c}_n{z}^n$. Its term of order $n$ is $c_n{g}^n{m}^{1-n}{\Phi }^{n+2}$. Negative $n$ are singular in $g$, while $n\ge 2$ are singular in $m$; these are excluded by perturbative holomorphic regularity at the free-coupling limits. Thus only the quadratic and cubic terms survive. Their regular holomorphic coefficients cannot acquire coupling-dependent invariant corrections. At $g=0$ the quadratic theory is Gaussian, fixing $c_0=1/2$ exactly. A contribution linear in $g$ to the cubic vertex has only one cubic interaction: three external lines exhaust that vertex, leaving no contraction to form a loop. Thus $c_1=1/3$ as well in the present normalization. This supplies the holomorphy argument for superpotential non-renormalization rather than concluding non-renormalization from symmetry alone.

The Kähler potential can still undergo wave-function renormalization. If its quadratic coefficient becomes $Z$, canonical normalization gives $m_c=m/Z$ and $g_c=g/Z^{3/2}$, so physical masses and interactions can run despite superpotential non-renormalization. More generally a positive nonsingular effective Kähler metric replaces $V$ by $K^{\varphi \bar{\varphi }}|W^{\prime }(\varphi ){|}^2$ and does not remove its zero-energy critical points. Infrared-singular nonlocal one-particle-irreducible terms and eliminating the entire massive field are outside this Wilsonian statement.

Take canonical kinetic terms, gauge coupling $g>0$, and a nonzero charge $q$. Define $e=gq$ and choose the convention in which the Fayet–Iliopoulos term contributes $\xi D$ to the Lagrangian. With one charged chiral superfield, gauge invariance permits no nonconstant polynomial superpotential; an irrelevant global constant gives $F=0$. The real auxiliary field part isIts equation of motion is $D=-(\xi +e|\varphi {|}^2)$. Eliminating it gives the D-term scalar potentialThe gaugino transformation has the form $\delta \lambda =iϵD+$ a field-strength term, with a convention-dependent overall phase. In a translationally invariant vacuum with zero field strength, a nonzero $⟨D⟩$ therefore gives an inhomogeneous shift of $\lambda$: no nonzero supersymmetry parameter leaves the vacuum invariant. The massless Goldstino is consequently the gaugino, not the matter fermion, since $F=0$.

Minimize the potential over $u=|\varphi {|}^2\ge 0$. If $q\xi <0$, the allowed value $u=-\xi /e$ sets $D=0$, so supersymmetry is unbroken while the internal gauge symmetry is Higgsed. If $\xi =0$, the origin also has $D=0$. For the broken branch, $q\xi >0$ and the formal zero of $D$ would require negative $u$. The actual minimum isThis is single charged-field D-term breaking. Nonzero FI parameter alone does not force breaking if a scalar can cancel it. The degenerate case $q=0$ is different: any $\xi \ne 0$ breaks supersymmetry in the vector sector while the neutral scalar remains flat; the sign test above assumes an actually charged field.

Expand around the broken vacuum:For $\varphi =({\varphi }_R+i{\varphi }_I)/\sqrt{2}$, both canonically normalized real scalars have squared mass $e\xi =gq\xi >0$. The matter Weyl spinor has no superpotential mass. The gauge boson remains massless because $⟨\varphi ⟩=0$. The gauge Yukawa coupling, proportional to ${\varphi }^{\ast }\lambda \psi$, gives no fermion mixing or mass at this vacuum, so the gaugino is also massless. The tree-level spectrum is thereforeand the chiral-multiplet splitting isThe vector and its gaugino happen to remain degenerate at zero mass even though supersymmetry is broken. If the sign convention for the FI contribution is reversed, the sign criterion changes accordingly; the invariant condition is that the scalar condensate cannot cancel the auxiliary source. This is the classical toy-model spectrum; a quantum completion of the single charged-fermion theory also has to cancel its gauge anomalies.

Four-dimensional N=1 supersymmetry is compatible with a chiral gauge spectrum. Each matter Weyl spinor belongs to a chiral superfield with a complex scalar in the same gauge representation, and each gauge boson belongs to a vector multiplet with an adjoint gaugino. Thus a supersymmetric extension of the Standard Model predicts squarks, sleptons and gauge-fermion partners, together with Higgs-sector partners. Exact unbroken supersymmetry would make partners mass-degenerate, so a phenomenologically useful model needs supersymmetry breaking.

The Minimal supersymmetric Standard Model uses two Higgs chiral doublets. Holomorphy of the superpotential prevents use of the conjugate of one doublet as an independent chiral field, so up-type and down-type Yukawa couplings require opposite-hypercharge doublets. Their higgsinos also cancel each other's gauge anomalies. The superpotential has the schematic formwith gauge contractions understood. Supersymmetry relates the scalar quartic interactions and Yukawa interactions to the same gauge and superpotential couplings, rather than allowing arbitrary independent choices.

A central motivation is the hierarchy problem. A nonsupersymmetric scalar mass can receive corrections proportional to a high ultraviolet cutoff squared. The paired bosonic and fermionic loops in exact supersymmetry cancel that quadratic sensitivity. Soft supersymmetry breaking introduces scalar squared masses, gaugino masses and appropriate scalar bilinear and trilinear terms without restoring the original quadratic divergence. With a typical soft scale $m_{\mathrm{soft}}$, residual corrections have the schematic sizeplus finite thresholds. This favors not excessively heavy partners if one wishes to avoid a new tuning, but does not derive their masses or guarantee that they are experimentally accessible. A hidden breaking sector can communicate soft terms through gauge or gravitational interactions.

The altered particle spectrum also changes the renormalization group evolution of the gauge couplings. The MSSM one-loop coefficients, with the usual grand-unified hypercharge normalization, are $b=(33/5,1,-3)$ in $d{\alpha }_i^{-1}/d\log \mu =-b_i/(2\pi )$. They allow approximate supersymmetric gauge coupling unification near $10^{16}$ GeV, subject to threshold corrections. Top-Yukawa effects can drive the up-type Higgs squared mass negative and help produce radiative electroweak symmetry breaking. Neither phenomenon alone specifies a full breaking mechanism.

Without an additional symmetry, gauge invariance allows renormalizable baryon- or lepton-number violating superpotential terms. Conserved R-parity, $(-1{)}^{3(B-L)+2s}$, forbids the usual such renormalizable terms, distinguishes ordinary particles from their superpartners, and makes the lightest supersymmetric particle stable. A neutral stable lightest partner can be a dark matter candidate; its abundance and interactions require a separate calculation. Pair production followed by cascades to such a particle can give missing-momentum signals. R-parity is an extra model assumption, and does not by itself exclude all higher-dimensional proton-decay operators.

Possible low-energy consequences are a partner spectrum, controlled scalar-mass sensitivity, a constrained Higgs sector, modified coupling evolution and, with conserved R-parity, a stable lightest partner. Soft parameters also affect flavor-changing and CP-violating processes. Supersymmetry provides relations and mechanisms, not a unique numerical spectrum or an automatic solution to every flavor or cosmological problem.

In four dimensions, extended supersymmetry has $\mathcal{N}>1$ independent Weyl spinor supercharges, or $4\mathcal{N}$ real generators. With the ordinary particle Super-Poincaré algebra,The central charges in supersymmetry commute with spacetime generators and the supercharges; an R-symmetry can act on their internal indices. Antisymmetry makes a scalar central charge impossible for $\mathcal{N}=1$.

At positive timelike momentum, the rest-frame supercharges act as fermionic oscillators. Without central charges, a spin-$j$ Clifford vacuum generates $(2j+1)2^{2\mathcal{N}}$ states. For a massless supermultiplet, positivity makes half the supercharges act trivially. The remaining $\mathcal{N}$ creation operators generate $2^{\mathcal{N}}$ states with helicity levels $\lambda -k/2$ and multiplicities $\left(\genfrac{}{}{0ex}{}{\mathcal{N}}{k}\right)$. A separate CPT completion of a supermultiplet may be necessary. Keeping all helicities between $-1$ and $1$ gives $\mathcal{N}\le 4$ for a theory without gravity; allowing helicity two gives $\mathcal{N}\le 8$.

A nonzero central charge in supersymmetry changes the oscillator norms. For an $\mathcal{N}=2$ central-charge block, write $Z^{12}=2z$. Suitable combinations of a charge and the adjoint of the other have anticommutators $2(M+|z|)$ and $2(M-|z|)$. Their positivity givesA BPS state saturates this BPS bound in supersymmetry. The combinations with zero anticommutator annihilate it, preserving part of supersymmetry and giving a shortened BPS supermultiplet. In this example a scalar-seeded massive multiplet has four states instead of sixteen, before any required CPT completion. More central-charge blocks allow different fractions of preserved supersymmetry. The mass-charge relation is protected as long as the short state exists; this does not guarantee stability against decay everywhere in parameter space.

The main obstruction to unbroken extended supersymmetry at low energies is the chirality constraint on extended supersymmetry. An $\mathcal{N}=2$ vector multiplet contains an additional adjoint complex scalar. A full hypermultiplet contains two $\mathcal{N}=1$ matter chiral superfields in conjugate gauge representations $R$ and $\overline{R}$. Its charged fermion spectrum is therefore vectorlike. Half-hypermultiplets in pseudoreal representations do not supply genuinely complex chirality. The $\mathcal{N}=4$ gauge multiplet is still more restrictive: it has four adjoint Weyl spinors and six adjoint real scalars. These spectra and their constrained interactions cannot directly reproduce the chiral electroweak matter and arbitrary Yukawa couplings of the Standard Model.

At most unbroken $\mathcal{N}=1$ supersymmetry can directly describe the observed chiral four-dimensional gauge spectrum. Extended supersymmetry can be useful in a higher-energy theory, in an internal sector or before compactification, but it must be reduced or broken to obtain that low-energy spectrum. This is a statement about unbroken symmetry of the relevant sector, not a prohibition on extended supersymmetric ultraviolet descriptions.

At the usual two-derivative order, the two-derivative global supersymmetric gauge action isHere ${\mathcal{W}}_{\alpha }^a$ is the chiral field-strength superfield, distinguished from the superpotential $W$. The Kähler potential $K$ is a real gauge-invariant function, defined up to a Kähler transformation; $K_{i\bar{j}}$ supplies the scalar kinetic metric. The superpotential is a gauge-invariant holomorphic function of the chiral superfields. The symmetric gauge kinetic function $f_{ab}$ is holomorphic and transforms so that its contraction with the field strengths is gauge invariant. Its real part supplies the gauge and gaugino kinetic coefficients and must be positive on physical configurations; its imaginary part supplies the topological angle. The final constants are Fayet–Iliopoulos terms, normalized so that they contribute ${\xi }_a{D}^a$ when $V^a$ has highest component ${\theta }^2{\bar{\theta }}^2{D}^a/2$.

For the simple non-Abelian gauge group specified in the question, $\xi =0$. Invariance of a proposed ${\xi }_a{D}^a$ requires ${\xi }_a{f}^a{}_{bc}=0$. A simple Lie algebra equals its commutator algebra, so no nonzero invariant covector exists. This proves that Fayet–Iliopoulos terms require an Abelian gauge factor. The last term above describes the customary Abelian extension, not an additional arbitrary coupling of the simple group.

For a renormalizable theory, after choosing matter bases the functions reduce toOnly gauge-invariant monomials are included. A nonconstant $f$ would produce interactions of dimension greater than four. Here the gauge coupling is absorbed into the connection in $e^{2V}$, while $\mathrm{Re}Y=1/g_h^2$ in the holomorphic normalization. A constant $c$ has no dynamical effect in global supersymmetry. The original arbitrary functions remain useful for a nonrenormalizable two-derivative effective theory, but the following argument restricts to the requested renormalizable setting.

To analyze perturbative corrections, use a local Wilsonian effective action with a nonzero infrared cutoff and keep the same elementary fields. Introduce nondynamical background chiral superfields $X,Y$ into its F-term part:One may additionally promote each polynomial coefficient to its own chiral spurion to track flavor charges. Treating the couplings as fields ensures that a local chiral integral depends holomorphically on $\Phi ,X,Y$: antichiral dependence would violate the chirality constraint unless accompanied by derivatives or rewritten as a D-term. This is the key use of holomorphy, rather than an assumption that all effective-action terms are holomorphic.

Give $\theta$ R-charge one, the matter fields charge zero, $X$ charge two and $Y$ charge zero. Then the chiral measure has charge minus two, $X{W}_{\mathrm{tree}}$ has charge two, and ${\mathcal{W}}_{\alpha }$ has charge one. These assignments define formal spurionic symmetries even if fixed numerical couplings do not enjoy the corresponding physical symmetry. An anomalous transformation is accompanied by the appropriate shift of the gauge-coupling spurion; nonperturbative dependence can record that anomaly.

For the superpotential non-renormalization argument, perturbation theory is insensitive to continuous shifts of the topological angle, which shifts the imaginary part of $Y$. A holomorphic coefficient invariant under every such shift is independent of $Y$ by the Cauchy-Riemann equations. This proves perturbative gauge-coupling independence of the Wilsonian superpotential. At fixed infrared cutoff the perturbative expansion is regular near zero superpotential couplings. R-charge two therefore forces an allowed correction to be linear in $X$, rather than involving $X^2$, $\bar{X}$ or inverse powers of $X$. Flavor charges impose the corresponding monomial selection rules.

The remaining coefficient can be evaluated at zero gauge coupling. Free chiral propagators join a chiral and an antichiral endpoint. A single holomorphic superpotential vertex cannot close a loop; adding antichiral vertices violates holomorphy, while adding holomorphic vertices violates the linear spurion degree. Consequently the coefficient is its tree value. This proves, rather than merely quotes, the perturbative resultThe fixed-cutoff regularity matters: integrating out a whole massive multiplet can introduce inverse masses at tree level, and infrared-singular one-particle-irreducible expressions are not this local Wilsonian superpotential.

For the gauge kinetic function, R-charge zero excludes regular positive powers of $X$ in its perturbative coefficient. Flavor spurions cannot repair this without forbidden antichiral or singular dependence. A topological-angle shift changes the tree gauge term only by its topological contribution, so the correction $f-Y$ is invariant under the shift. Holomorphy then makes it independent of $Y$. An $L$-loop correction to the inverse gauge coupling carries the weak-coupling behavior $g_h^{2L-2}$: only one loop has the required coupling-independent behavior. Hence the one-loop exactness of the holomorphic gauge kinetic function givesup to a one-loop matching constant. The coefficient follows from the usual one-loop field contributions: gauge bosons and an adjoint Weyl gaugino contribute $(11/3-2/3)C_2=3C_2$, while a chiral Weyl fermion and complex scalar contribute $-(2/3+1/3)T(R)=-T(R)$. Holomorphic logarithmic thresholds appear when massive fields are eliminated. This statement is about the holomorphic Wilsonian coupling, not a canonically normalized physical gauge coupling, whose running need not be one-loop exact.

The Kähler potential is integrated over all of superspace. It can depend on $X^{†}X$, on both $Y$ and $Y^{†}$, and on both matter chiralities. These neutral combinations allow ordinary loop corrections, such as a Yukawa-dependent correction to ${\Phi }^{†}\Phi$, and no analogous argument forbids higher loops. ThusIts wave-function renormalization changes physical superpotential couplings after canonical normalization even though holomorphic vertex coefficients do not change. For a single field with kinetic coefficient $Z$, for instance, $m_c=m/Z$ and $y_c=y/Z^{3/2}$.

Finally consider perturbative renormalization of a Fayet–Iliopoulos term in an Abelian extension. A term $\int d^4\theta H(X,X^{†},Y,Y^{†})V$ must be invariant under $V\mapsto V+\Lambda +{\Lambda }^{†}$ for arbitrary chiral $\Lambda$. Projection of the variation requires ${\bar{D}}^{2}H=D^{2}H=0$. An arbitrary algebraic function of background chiral spurions and their conjugates fails this requirement unless it is constant. Therefore a general coupling-dependent local FI counterterm is forbidden. The one-loop $D$ tadpole from charged scalars can still giveIt is independent of interaction couplings in the connection normalization used above and proportional to the trace of the Abelian charge generator. Higher-loop diagrams require interactions and cannot produce the permitted coupling-independent additive term. Thus a traceless Abelian charge generator has no additive perturbative FI renormalization in this exactly supersymmetric setting; otherwise a one-loop, regulator-dependent tadpole is possible. Vector-field normalization can change the reported canonical parameter, and soft breaking falls outside this proof. For the specified simple group, gauge invariance forbids $\xi$ before and after corrections.

These are perturbative statements. Instantons or other nonperturbative effects may produce symmetry-allowed holomorphic terms, for example dependence on $e^{-8{\pi }^{2}Y}$, so neither the superpotential result nor the one-loop gauge result should be misread as forbidding all nonperturbative corrections.

Choose canonical vector normalization, $g>0$, and the convention that the Fayet–Iliopoulos term contributes $+\xi D$. In a constant Lorentz-invariant vacuum with vanishing gauge field strength, the gaugino transformation containsA nonzero $⟨D⟩$ therefore shifts a fermion by a constant under supersymmetry: no invariant vacuum can have this expectation. With only D-term breaking, the Goldstino is the gaugino $\lambda$. If F-term expectation values were present too, the Goldstino would instead be the corresponding linear combination of matter fermions and gauginos.

For a genuinely charged single chiral superfield, $q\ne 0$, gauge invariance forbids every nonconstant polynomial superpotential in that field. The renormalizable superspace action is consequentlywhere ${\mathcal{W}}_{\alpha }=-{\bar{D}}^2{D}_{\alpha }V/4$. A constant superpotential is dynamically irrelevant in global supersymmetry. Take the standard vector expansion with highest component ${\theta }^2{\bar{\theta }}^{2}D/2$. In the matter exponential, the term $2gqV$ then contributes $gq|\varphi {|}^{2}D$. The gauge kinetic chiral integral contributes $D^2/2$, and the FI term contributes $\xi D$. ThusThe auxiliary equations are $F=0$ and $D=-(gq|\varphi {|}^2+\xi )$. Completing the square givesThis is the single charged-field D-term breaking model. Its vacuum is found by minimizing over $\rho =|\varphi {|}^2\ge 0$, not by assuming that a nonzero FI constant must break supersymmetry.

For $q\xi >0$, the square cannot be canceled by any $\rho \ge 0$. It is minimized at the origin, whereSupersymmetry breaking occurs while the internal $U(1)$ gauge symmetry remains unbroken. Expanding the scalar potential about the origin givesThere is no superpotential fermion mass and no gaugino-matter mixing at $\varphi =0$. Therefore the chiral multiplet mass splitting isBoth real components of the complex scalar have the same positive squared mass; the gaugino is the massless Goldstino. With this sign convention and $\xi >0$, the breaking condition is $q>0$. Reversing the FI sign convention reverses this quoted charge inequality, while the condition that the scalar cannot cancel the auxiliary-field shift is unchanged.

For $q\xi <0$, the minimum instead satisfiesThe nonzero charged scalar expectation breaks the continuous internal gauge symmetry and preserves supersymmetry. To see the resulting supersymmetric Higgs mechanism, choose a real representative $v>0$ and expand $\varphi =v+(h+ia)/\sqrt{2}$. The scalar kinetic term supplies a vector mass, the D-term potential supplies the radial scalar mass, and the Yukawa interaction $-\sqrt{2}gq{\varphi }^{\ast }\lambda \psi +h.c.$ pairs the two Weyl fermions. They have the common squared massThe phase field $a$ is the absorbed Goldstone mode. There is no Goldstino in this supersymmetric Higgs vacuum; the fields assemble into a massive N=1 vector multiplet. This branch makes explicit that internal gauge breaking and supersymmetry breaking are different phenomena.

If $\xi =0$ and $q\ne 0$, the minimum of $g^2{q}^2|\varphi {|}^4/2$ is $\varphi =0$. Both supersymmetry and the continuous gauge symmetry are then unbroken, and the fields have zero tree-level masses at that vacuum. For the degenerate uncharged case $q=0$ with no superpotential interactions, a nonzero $\xi$ still breaks supersymmetry in the vector sector, but the free chiral multiplet stays unsplit and cannot Higgs the gauge field. An uncharged field would allow additional superpotential terms, outside the genuinely charged model above.

The calculation is classical. A single charged Weyl fermion has nonzero Abelian gauge and mixed gravitational anomaly coefficients proportional to $q^3$ and $q$; a complete quantum gauge theory needs an anomaly-canceling completion. The tree-level branches above refer to the specified field content; extra fields in such a completion can change the vacuum structure.

Use the canonical chiral-superfield component expansion $\Phi =\varphi +\sqrt{2}\theta \psi +\theta \theta F$ in chiral coordinates, with $\psi \psi ={\psi }^{\alpha }{\psi }_{\alpha }$. Multiplication of the Grassmann variables givesFor example, the two linear-in-$\theta$ factors in ${\Phi }^2$ contribute $-\theta \theta \psi \psi$, while each of the three choices of the remaining scalar in ${\Phi }^3$ gives the same contribution. Thus the required F-term isAdding its Hermitian conjugate and the canonical $F^{\ast }F$ term eliminates the auxiliary field through $F=-\overline{m\varphi +g{\varphi }^2}$ and yields the F-term scalar potentialIt is nonnegative, and for $g\ne 0$ its absolute minima are $\varphi =0$ and $\varphi =-m/g$, each with zero energy. They coincide when $m=0$. If $g=0,m\ne 0$, only $\varphi =0$ is a minimum; if $g=m=0$, every $\varphi$ is a minimum. These exhaust the local minima as well: at any other stationary point $V_{\varphi }=W^{\prime \prime }\overline{W^{\prime }}=0$ forces $W^{\prime \prime }=0$; for $mg\ne 0$ this is $\varphi =-m/(2g)$, whose quadratic variation has opposite signs in two real directions and is a saddle.

Expand around $\varphi =0$, writing $\varphi =h=(h_1+i{h}_2)/\sqrt{2}$. The scalar potential and fermionic terms areThe canonical kinetic term is $\frac{1}{2}[(\partial h_1{)}^2+(\partial h_2{)}^2]$. Its quadratic potential is $\frac{1}{2}|m{|}^2(h_1^2+h_2^2)$, while the Weyl fermion has mass $|m|$. HenceAt the other minimum, $W^{\prime \prime }=-m$, so the same mass equality holds. The quartic and Yukawa interaction are both fixed by $g$: defining $V\supset \lambda |h{|}^4$ and $\mathcal{L}\supset -yh\psi \psi +h.c.$ givesThis is the supersymmetric relation between quartic and Yukawa couplings. The printed equality of the two couplings refers to their common supersymmetric origin; literal equality of the conventional coefficients $\lambda$ and $y$ is not true for arbitrary $g$. If instead the Yukawa term is defined as $-\frac{1}{2}yh\psi \psi$, then $y=2g$ and $\lambda =|y{|}^2/4$.

To establish perturbative superpotential non-renormalization, use a Wilsonian effective action with a fixed positive infrared cutoff, so integrating out modes does not introduce massless infrared singularities. Promote $m,g$ to nondynamical chiral superfields, or spurions. The holomorphic superpotential depends holomorphically on $\Phi ,m,g$, not on their complex conjugates. Assign an ordinary $U(1)$ charge and an R-charge byThe superspace coordinate has $R(\theta )=1$, so $d^2\theta$ has charge $-2$ and $W$ must have R-charge $2$. Both original monomials satisfy the two selection rules. A holomorphic perturbative monomial $m^a{g}^b{\Phi }^c$ must obeywhich gives $c=b+2$ and $a=1-b$. Equivalently,The nonnegative integer $b$ follows from perturbative analyticity in $g$. Regularity as $m\to 0$ at the fixed Wilsonian cutoff excludes $b\ge 2$, which would require negative powers of $m$. The remaining quadratic coefficient is fixed to $c_0=1/2$ by the free theory at $g=0$. The coefficient $c_1$ is $1/3$: a three-point contribution at first order in $g$ has just one cubic vertex, uses all three legs externally, and cannot contain a loop. A loop correction would require additional interactions, whereas the selection rules and holomorphy have already excluded the corresponding monomials. ThereforeThis proof concerns the local Wilsonian superpotential. The Kähler potential can acquire a wave-function factor $Z$, and canonical normalization changes the couplings to $m_c=m/Z$ and $g_c=g/Z^{3/2}$. Such running, and nonlocal infrared terms in a one-particle-irreducible action, do not contradict the holomorphic result.

Assume global four-dimensional N=1 supersymmetry with canonical kinetic terms and no explicit gaugino mass. Write $W_i={\partial }_{i}W$ and $W_{ij}={\partial }_i{\partial }_{j}W$. To match the displayed stationarity equation, use $F^i=\overline{W_i}$; this differs by a minus sign from the eliminated auxiliary field in the component convention of Solution 2. The real D-terms are $D^a=-g^a({\bar{\varphi }}_i(T^a{)}^i{}_j{\varphi }^j+{\xi }^a)$, with Hermitian gauge generators. A constant Fayet–Iliopoulos term is permitted only for an appropriate Abelian factor. The scalar potential isDifferentiate with respect to ${\varphi }^i$, holding $\bar{\varphi }$ independent:Thus stationary point conditions gives the first requested identity. Gauge invariance of the superpotential gives $W_i(T^a{)}^i{}_j{\varphi }^j=0$. Conjugating it and using Hermiticity gives ${\bar{\varphi }}_j(T^a{)}^j{}_i{F}^i=0$, the second condition in the form needed below.

At the vacuum, define $H_{ij}=⟨W_{ij}⟩$ and $G_{ia}=g^a⟨{\bar{\varphi }}_j⟩(T^a{)}^j{}_i$. The two conditions combine asHere the upper equation is stationarity and the lower equation is conjugated gauge invariance. The chiral-superfield fermion mass matrix contributes $-\frac{1}{2}{H}_{ij}{\psi }^i{\psi }^j$, and the gauge Yukawa interaction contributes the mixed chiral-fermion/gaugino bilinear. With canonical Weyl fermion and gaugino kinetic terms, and choosing the gaugino phase to put the mixed bilinear in real-sign convention, these terms areThe factor $\sqrt{2}$ is important: the matrix directly acting on the printed $(F,D)$ is the same mass bilinear with rescaled gauginos, rather than literally the canonical matrix. In detail, for $S=\mathrm{diag}(I,\sqrt{2}I)$,Since supersymmetry breaking makes this vector nonzero, there is at least one zero mass eigenvalue. A normalized massless fermion isWith the consistent gaugino phase, the inhomogeneous vacuum supersymmetry transformation of $(\psi ,\lambda )$ is proportional to the same vector $(F,D/\sqrt{2})$. Therefore $\eta$ shifts by a nonzero constant times the transformation parameter: it is the Goldstino, the Goldstone fermion of broken global supersymmetry. The argument guarantees at least one Goldstino zero mode, not the absence of other massless fermions.

For the second calculation, first take only canonical chiral superfields and F-term breaking. At a stationary vacuum $v$, let $h=\varphi -v$, $m_{ij}=W_{ij}(v)$ andThese follow by differentiating $V={\sum }_k|W_k{|}^2$ twice. The quadratic scalar potential isThe doubled scalar mass matrix has blocksTransforming to the $2N$ canonically normalized real components of $h$ preserves its trace, so the sum of real scalar mass squares is $\mathrm{tr}{M}_s^2=2\mathrm{tr}A$. The Weyl fermion mass squares are the eigenvalues of $m^{†}m=A$ and their sum is $\mathrm{tr}A$. Each Weyl mass eigenstate has two physical spin states. The printed supertrace has scalar sign $-1$ and fermion weight $+2$, henceThe frequently used boson-minus-fermion definition has the opposite overall sign and gives the same zero. The mixing block $B$ can split scalar masses but cannot change this trace. This proves the tree-level supertrace mass sum rule; it uses canonical kinetic terms and a renormalizable global theory, rather than a general supergravity or noncanonical effective theory.

In the MSSM, trying to make all squarks and sleptons heavy through direct canonical tree-level F-term breaking encounters this constraint. In an unbroken electric-charge and color sector, the scalar mass-squared trace is twice the trace for its ordinary fermionic partners. More explicitly, take a unit vector $u$ minimizing $u^{†}Au$. For $h=e^{i\alpha }u$, choose $\alpha$ so that $\mathrm{Re}(e^{2i\alpha }{u}^{T}Bu)\le 0$. The curvature along this real scalar direction is then at most $u^{†}Au$, so at least one scalar mass square is no larger than the smallest fermion mass square in that sector. For one isolated pair the scalar masses are $m_f^2\pm |B|$. Thus light quarks can force an unacceptably light squark, or a tachyon, under these assumptions. A hidden supersymmetry-breaking sector with mediation produces effective soft supersymmetry breaking; radiative effects, noncanonical kinetic terms, or supergravity can change the assumptions and allow a viable spectrum. The trace over an arbitrary enlarged theory alone would not locate the light state in the visible sector; the MSSM conclusion uses the conserved-charge block structure.

Use left-chiral superfields throughout, abbreviating the conjugate singlets by $U_i={\bar{u}}_i^c$, $D_i={\bar{d}}_i^c$, $E_i={\bar{e}}_i^c$ and $N_i={\bar{\nu }}_i^c$. The printed hypercharge convention is the negative of the usual one, so $Q_{\mathrm{em}}=T_3-Y$; consequently $H_1$ is the down-type and $H_2$ the up-type Higgs doublet. Write $A\cdot B={ϵ}_{\alpha \beta }{A}^{\alpha }{B}^{\beta }$ for the weak-doublet contraction. Assign baryon number $B(Q)=1/3$, $B(U)=B(D)=-1/3$, and lepton number $L(L)=1$, $L(E)=L(N)=-1$; both Higgs fields have $B=L=0$.

The baryon number and lepton number conserving homogeneous cubic superpotential isEach term has zero hypercharge, is a color and weak singlet, and has $B=L=0$. Higgs expectation values give the usual quark and charged-lepton Yukawa interaction masses and, here, Dirac neutrino masses. The complete remaining cubic part isThe first two terms have $\Delta L=+1$, the third has $\Delta B=-1$, the fourth has $\Delta L=-1$, and the last has $\Delta L=-3$. Their Hermitian conjugates reverse these signs. Commutativity of chiral superfields and the antisymmetric weak/color contractions give ${\lambda }_{ijk}=-{\lambda }_{jik}$ and ${\lambda }_{ijk}^{\prime \prime }=-{\lambda }_{ikj}^{\prime \prime }$; $\rho$ can be taken completely symmetric. All other coupling tensors are unrestricted complex tensors.

To check completeness, a cubic color singlet containing a quark doublet and an antiquark singlet must contain a colorless weak doublet; the allowed hypercharges give $Q{H}_{2}U$, $Q{H}_{1}D$ and $QLD$. Three color-antitriplet singlets can form a singlet with ${ϵ}^{abc}$, but hypercharge permits only $UDD$. Three quark doublets cannot be a weak singlet. With no colored fields, an even number of weak doublets is necessary. Two doublets and a singlet give precisely $L{H}_{1}E$, $LLE$, $L{H}_{2}N$ and $H_1{H}_{2}N$; contractions of two identical Higgs superfields vanish. With no doublets the only neutral triple is $NNN$. Thus $W_3=W_{3,\mathrm{cons}}+W_{3,\mathrm{viol}}$ is the most general homogeneous cubic superpotential for the printed fields, including their singlet neutrinos.

If “cubic” is intended to mean a polynomial of degree at most three, also addThe $\mu$ term conserves both charges; the other nonconstant terms have respectively $\Delta L=1,-2,-1$. This distinction will matter for the printed claim about R-parity.

For squark-mediated proton decay, choose ${\lambda }_{11k}^{\prime }$ and ${\lambda }_{11k}^{\prime \prime }$ with $k=2$ or $3$. The same down-type singlet squark can appear in both vertices; $k=1$ would give ${\lambda }_{111}^{\prime \prime }=0$ by antisymmetry. The first term includes an electron/up-quark/squark vertex, and the second an up-quark/down-quark/squark vertex. Exchanging a squark of mass $M$ and expanding its propagator at momenta much smaller than $M$ produces the dimension-six effective operatorThe left- and right-handed pairs are separately contracted spinor bilinears. This operator converts an up/down quark pair into a positron and an antiquark; the spectator up quark can combine with the antiquark into the neutral pion. Equivalently its hadronic matrix element between a proton and a pion is permitted, so it induces $p\to e^{+}{\pi }^0$. Having both the baryon number and lepton number violating vertices is what enables this channel.

Since $C$ has mass dimension $-2$, dimensional analysis givesThe dimensionless $c_h$ includes the hadronic matrix element and two-body phase space; the pion mass changes this factor rather than the leading dimensions. Dimensional grounds alone do not give a precise hadronic coefficient. Using the lifetime supplied in the paper, rather than substituting a different bound, gives $\Gamma <1/(2.4\times 10^{64})\mathrm{GeV}=4.17\times 10^{-65}\mathrm{GeV}$ and henceFor an illustrative $100\mathrm{GeV}$ squark this is about $6.5\times 10^{-29}{c}_h^{-1/2}$ when $m_p$ is rounded to $1\mathrm{GeV}$. The bound constrains the relevant flavour product and depends on the exchanged mass; separate smallness of each coupling does not follow without an additional assumption.

Define matter parity and R-parity byEvery matter superfield $Q,U,D,L,E,N$ has $P_M=-1$, whereas $H_1,H_2$ and the vector superfields have $P_M=+1$. For ordinary quarks and antiquarks, $3(B-L)=\pm 1$ is odd and $2s=1$, so $R_p=+1$. For charged leptons, neutrinos and their antiparticles, $3(B-L)=\mp 3$ is odd and $2s=1$, again giving $R_p=+1$; the singlet neutrino fermion obeys the same rule. Gluons, the photon, $W$ and $Z$ have $B=L=0$ and spin one, hence $R_p=+1$; Higgs bosons have $B=L=s=0$ and also $R_p=+1$. Thus every Standard Model particle is R-even, including the corresponding antiparticles. Changing spin by one half within a supermultiplet reverses $R_p$, so squarks, sleptons, gauginos and Higgsinos are R-odd.

Each conserving cubic term contains two R-odd matter superfields and one even Higgs superfield, so it is even under matter parity. Each term in $W_{3,\mathrm{viol}}$ has an odd number of matter superfields: three for $LLE,LQD,UDD,NNN$, and one for $N{H}_1{H}_2$. All are therefore forbidden. At the component level the two fermion spin factors in a Yukawa interaction multiply to $+1$, so the same selection rule follows directly from R-parity. This verifies all cubic baryon/lepton-number violating terms are forbidden while the Yukawa fermion mass terms survive.

The blanket assertion needs a qualification beyond homogeneous cubic terms. The Higgs $\mu$ term is even; $L{H}_2$ and $N$ are odd and forbidden. But $N_i{N}_j$ is even and permits a Majorana neutrino mass despite violating lepton number by two. Likewise the higher-dimensional operators $QQQL$ and $(L{H}_2{)}^2$ are even. Consequently R-parity alone does not enforce all-order conservation of baryon number and lepton number, or absolute proton stability; the stated precise exclusion holds for the violating cubic interactions above.

Two physical implications follow from multiplicative conservation of R-parity. Superpartners are produced in even numbers from an initial state consisting only of Standard Model particles, leading to pair production at the lowest threshold and parity-conserving cascades. The lightest supersymmetric particle is stable, since one R-odd particle cannot decay exclusively into R-even particles and no lighter R-odd state is available. A neutral weakly interacting lightest supersymmetric particle can therefore be a dark-matter candidate and produce missing-energy signatures; neutrality and a suitable abundance require further model assumptions.

Work with a supersymmetry-preserving regulator and a local Wilsonian effective action at a nonzero infrared cutoff, in holomorphic field variables. Its two-derivative terms have the formHere ${\mathcal{W}}_{\alpha }$ is the gauge field-strength superfield, distinct from the superpotential $W$, and the gauge kinetic function $f_{ab}$ is symmetric and holomorphic. The perturbative non-renormalization theorem statesThe same elementary light fields are retained in the first equality. It does not prohibit tree-level changes to the superpotential when an entire massive field is eliminated. The Kähler potential may be renormalized at every loop order, nonperturbative effects are separate, and canonically normalized physical couplings need not satisfy these holomorphic statements.