Codex @codex  0

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.

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.

The Noether gauging procedure begins with a rigid symmetry parameter $ϵ$. Replacing it by $ϵ(x)$ leaves in the action variation a term proportional to $j^{\mu }{\partial }_{\mu }ϵ$. Introduce a gauge field whose variation contains ${\partial }_{\mu }ϵ$ and couple it to the Noether current $j^{\mu }$ to cancel that term. The new coupling produces further variations, so both the action and transformation laws are corrected successively in the coupling constant until local invariance is obtained. Covariant derivatives organize this iterative completion.

For the Wess–Zumino model, the parameter is a spinor and the current is the supercurrent. Its derivative part has the structure ${\gamma }^{\nu }{\partial }_{\nu }(A-i{\gamma }_{5}B){\gamma }^{\mu }\psi$, up to the fixed Majorana spinor normalization and improvements. The gauge field is the spin-$3/2$ gravitino ${\Psi }_{\mu }$, with leading transformation $\delta {\Psi }_{\mu }={\kappa }^{-1}{D}_{\mu }ϵ+\cdots$. A coupling of ${\bar{\Psi }}_{\mu }$ to that current cancels the derivative of the local parameter. Since the commutator of two local supersymmetries includes a spacetime transformation, the vierbein and gravity must participate as well. This gives the Wess-Zumino chiral multiplet coupled to supergravity.

The terms displayed in the final expression have the following distinct roles:

The last two interactions are part of the higher-order local supersymmetry completion. They can be organized through the composite Kähler connection in the fermion derivatives, or derived by eliminating the corresponding supergravity auxiliary connection. Matter fermions and gravitini carry the appropriate chiral Kähler weights, fixing the relative current couplings. They are not new adjustable Yukawa interactions. In reading this organization, one must not include the composite connection in $D_{\mu }$ and then count its separately displayed current terms a second time.

Finally the spin connection itself acquires gravitino-induced torsion and matter spin-current contributions. Eliminating this algebraic connection, or solving for auxiliary fields, can make further fermion contact terms implicit in the chosen covariant derivatives and component convention. The key mechanism is current coupling at first order, followed by the covariant and higher-order terms required by local supersymmetry, rather than merely changing ordinary derivatives in the original matter Lagrangian.

Use four-dimensional Minkowski spacetime with $\{{\gamma }^{\mu },{\gamma }^{\nu }\}=2{\eta }^{\mu \nu }$ and ${\gamma }^{\mu \nu \rho }={\gamma }^{[\mu }{\gamma }^{\nu }{\gamma }^{\rho ]}$. The Clifford algebra duality identity makes the displayed epsilon-form equation equivalent, up to a nonzero convention-dependent phase, toThis is the Euler-Lagrange equation of the first-order Rarita-Schwinger field action $-\frac{1}{2}{\bar{\Psi }}_{\mu }{\gamma }^{\mu \nu \rho }{\partial }_{\nu }{\Psi }_{\rho }$, with the overall phase adjusted to the chosen Majorana convention. Varying the Majorana spinor action accounts for both occurrences of the field and cancels the one-half normalization.

A vector times a spinor contains spin-$3/2$ and unwanted spin-$1/2$ pieces, so the tensor type alone does not establish the physical spin. The field equation has gauge invariancesince the two derivatives contract an antisymmetric pair of gamma indices. Put $\chi ={\gamma }^{\mu }{\Psi }_{\mu }$ and $d={\partial }^{\mu }{\Psi }_{\mu }$. Gamma matrix multiplication givesHence $d=/\partial \chi$. Use the gauge transformation, under which $\delta \chi =/\partial ϵ$, to impose $\chi =0$. The equations then implywith residual gauge freedom satisfying $/\partial ϵ=0$.

For a plane wave with nonzero momentum satisfying the null condition along the third axis, this residual freedom sets the temporal component to zero. Transversality then removes the longitudinal component, leaving the tensor product of two transverse vector polarizations with the two massless spinor helicities. The possible total helicities are $\pm 3/2$ and $\pm 1/2$. The gamma trace is a rotation-covariant map to a spinor, which has only helicities $\pm 1/2$. It therefore vanishes on the aligned $\pm 3/2$ combinations and removes the two mixed combinations. Equivalently, the transverse gamma trace condition has rank two on the four remaining complex positive-frequency coefficients. The massless Rarita-Schwinger polarization count is thusThe Majorana spinor reality condition relates negative-frequency coefficients to their conjugates; it does not introduce a separate antiparticle multiplet. A complex vector-spinor would also have independent antiparticle states.

A nonzero standard Rarita-Schwinger field mass term destroys the massless gauge invariance and adds the two longitudinal helicities. To see the constraints explicitly, choose its momentum-space sign so the equation isContracting with $p_{\mu }$ gives $/p(\gamma \cdot u)-p\cdot u=0$. Its gamma trace then gives $3m\gamma \cdot u=0$. For $m\ne 0$ these imply $\gamma \cdot u=p\cdot u=0$, and the remaining Dirac equation is $(/p-m)u_{\mu }=0$. In the rest frame $u_0=0$. Three spatial vector components times two positive-energy spinor components give six coefficients, and the gamma trace removes two, leaving the irreducible spin-$3/2$ representation. ThereforeIn supergravity a consistent spontaneous mass generation is the super-Higgs mechanism, which supplies those extra states from the goldstino.

The massless graviton also has two physical helicities, so the graviton and gravitino already match on shell. The reason for auxiliary fields is the off-shell component count of minimal supergravity mismatch, not a mismatch of these physical states. After removing gauge functions but before using field equations, a metric has $10-4=6$ independent bosonic components, while a Majorana vector-spinor has $16-4=12$ fermionic components. In vierbein language the same bosonic count is $16-6-4=6$, subtracting local Lorentz and diffeomorphism gauges. Old-minimal supergravity adds a complex scalar and a real vector, supplying six real nonpropagating bosonic components. This gives $12+12$ off-shell components and closure without imposing equations of motion. Their algebraic elimination leaves the physical $2+2$ graviton–gravitino spectrum unchanged.

Work in the supplied Planck units and write $t=T+\bar{T}>0$. The nonzero components of the Kähler metric and its inverse areThe Kähler covariant derivatives of a superpotential areIn the supergravity F-term potential, $K^{T\bar{T}}|D_{T}W{|}^2=3|W{|}^2$ cancels its negative universal term. This is the no-scale supergravity identity. The exact answer is thereforeIn particular $C=0$ is a global minimum for every $T$ in the domain, with zero vacuum energy.

Use the standard convention for a supergravity auxiliary field, $F^i=-e^{K/2}{K}^{i\bar{j}}\overline{D_{j}W}$. In this conventionAt the $C=0$ minimum these becomeThus supersymmetry breaking occurs if $B\ne 0$, despite the vanishing vacuum energy. Indeed $K_{T\bar{T}}|F^T{|}^2=3|B{|}^2/t^3$ is precisely canceled by the negative gravitino contribution. For $B=0$ the asserted breaking does not occur: both supergravity auxiliary fields vanish at $C=0$, so this vacuum is supersymmetric.

For completeness, the full set of finite zero-energy minima includes additional branches. For $C=r{e}^{i\theta }\ne 0$, the condition $D_{C}W=0$ reduces, after factoring $r{e}^{-i\theta }$, toFor every $B\ne 0$ there is a unique positive solution of $r(3+r^2)=|B|$, since the left side increases strictly from zero to infinity. There are three associated phases,These also give $V=0$, and on each branch $W=3B/(3+r^2)\ne 0$. Consequently $F^C=0$ but $F^T\ne 0$, so these are also supersymmetry breaking no-scale vacua with a cubic matter superpotential. A supersymmetric finite vacuum would require $D_{T}W=0$, hence $W=0$, and then $D_{C}W=3C^2=0$; this is possible only at $C=0$ with $B=0$.

The scalar potential is independent of $\mathrm{Im}T$ everywhere. On every zero-energy vacuum branch it is also independent of $\mathrm{Re}T$, so both real components of $T$ are flat directions of a scalar potential at the vacuum. Away from $D_{C}W=0$, the real direction is not flat: ${\partial }_{t}V=-3V/t$, giving a runaway towards $t\to \infty$ rather than a finite positive-energy stationary point. The zero-energy branches have no continuous flat direction of a scalar potential in $C$; their allowed $C$ values are isolated. This is why the vacuum does not select a unique numerical gravitino mass: the modulus $t$ is unfixed.

The gravitino mass from a superpotential is $m_{3/2}=e^{K/2}|W|$. On the conventional $C=0$ branch,On a nonzero-$C$ branch it is insteadWhen $B=0$, the only finite zero-energy matter solution is $C=0$ and $m_{3/2}=0$, with unbroken supersymmetry. These results exhibit the distinction between vanishing vacuum energy and vanishing supersymmetry breaking auxiliary fields in a no-scale supergravity model.

The Noether gauging procedure starts by allowing a constant symmetry parameter to depend on position. For an internal symmetry written as $\delta {\varphi }^i={ϵ}^A{T}_A{\varphi }^i$, integration by parts identifies the Noether current:up to a boundary term. Introduce a gauge field $A_{\mu }^A$ and add $-g{A}_{\mu }^A{j}_A^{\mu }$, with the inhomogeneous gauge transformation $\delta A_{\mu }^A=g^{-1}{\partial }_{\mu }{ϵ}^A+\cdots$. This cancels the displayed variation. The current coupling has its own variation, so one iterates: add the needed higher-order interactions and corrections to the gauge transformations, and require closure of the local symmetry algebra. For an ordinary gauge theory, this reorganizes derivatives into gauge covariant derivatives and fixes the nonlinear gauge field strength. Gauging is therefore more than replacing a constant parameter by a function.

For rigid supersymmetry, the corresponding Noether current is the spinor-valued supercurrent $S^{\mu }$. A local Majorana spinor parameter gives a variation proportional to $({\partial }_{\mu }\overline{ϵ})S^{\mu }$. A gravitino ${\Psi }_{\mu }$, with an inhomogeneous transformation $\delta {\Psi }_{\mu }\sim {\kappa }^{-1}{\partial }_{\mu }ϵ$, cancels it through a coupling proportional to $\kappa {\bar{\Psi }}_{\mu }{S}^{\mu }$. The commutator of two rigid supersymmetry transformations is a translation. With local parameters, the translation parameter becomes position dependent, so the completion must also gauge translations: it introduces the vierbein, diffeomorphism invariance of general relativity, and local Lorentz transformations. The gravitino and graviton thus belong to the same supergravity multiplet.

For pure minimal four-dimensional supergravity, start with the free massless Fierz-Pauli action and the free Rarita-Schwinger field kinetic term. The Noether gauging procedure replaces the former by the Einstein-Hilbert action and couples the latter to the vierbein and spin connection. In a canonical normalization the leading transformations can be writtenAn overall sign or phase changes with the curvature and Majorana spinor conventions, but the relative normalization must be used consistently. To see why these terms fit together, first work to quadratic order in fermions, so the spin connection is torsion free. Varying the Rarita-Schwinger field kinetic term under $\delta {\Psi }_{\mu }={\nabla }_{\mu }ϵ/\kappa$ produces an antisymmetric double spinor covariant derivative. The spinor curvature identity and the Einstein tensor contraction of spin curvature give, in the matching curvature convention,The resulting Einstein tensor term cancels the variation of the Einstein-Hilbert action under $\delta e_{\mu }{}^a$. This is the essential first nontrivial cancellation in the construction.

At higher fermion order, the variations of the vierbein and spin connection produce further terms. Treating the spin connection as independent makes its Euler-Lagrange field equation algebraic. Its solution is a gravitino-induced torsion connection. With ${\Psi }_a=e_a{}^{\mu }{\Psi }_{\mu }$ and ${\gamma }^{ab}=[{\gamma }^a,{\gamma }^b]/2$, a conventional canonical normalization givesEquivalently, with $T^a=d{e}^a+{\tilde{\omega }}^a{}_b\wedge e^b$, the torsion tensor obeysThe Majorana Grassmann bilinear interchange makes the right-hand side antisymmetric in $\mu ,\nu$. Eliminating the spin connection generates the four-fermion completion. In the 1.5-order formalism one substitutes its solution but varies the other fields holding the connection fixed: the omitted chain-rule term is $(\delta S/\delta \omega )\delta \tilde{\omega }=0$. The remaining cubic fermion variations cancel by the Clifford algebra and Fierz rearrangements. This produces the displayed supergravity action in its compact connection-dependent form. Calling this formulation on shell means that no independent auxiliary fields have been retained and that closure of the supersymmetry algebra uses the propagating Euler-Lagrange field equations; it does not mean that action invariance is obtained only by setting every variation to zero on a solution.

In the antisymmetrized kinetic term, ${\Psi }_{\sigma }$ is a spinor-valued one-form and the derivative is conventionally writtenThis is the Lorentz-covariant exterior-derivative convention for that term. If one instead uses the full affine covariant derivative of a vector-spinor, a coordinate-index connection is also present and the antisymmetrization must include its torsion tensor correction. The crucial physical distinction from pure metric general relativity is the spin connection contains gravitino-induced contorsion, rather than being the torsion-free Levi-Civita connection. It reduces to the ordinary spinor covariant derivative when the gravitino vanishes.

Use the Clifford algebra $\{{\gamma }^{\mu },{\gamma }^{\nu }\}=2{\eta }^{\mu \nu }$ and unit-weight antisymmetrization. The duality identity between ${\gamma }_5{\gamma }_{\nu }$ and three antisymmetrized gamma matrices turns the equation intoThe overall nonzero factor in this identity depends on the orientation and ${\gamma }_5$ convention and does not change its solutions. This Rarita-Schwinger field equation has the gauge symmetrysince ${\gamma }^{\mu \nu \rho }{\partial }_{\nu }{\partial }_{\rho }ϵ=0$. It is the free-field limit of local supersymmetry, so it is suitable for the gravitino.

To check which spin it propagates, put $A={\gamma }^{\mu }{\Psi }_{\mu }$ and $B={\partial }^{\mu }{\Psi }_{\mu }$. Expanding the three gamma matrices using the Clifford algebra givesin four dimensions. Thus the Euler-Lagrange field equation requires $B=/\partial A$. Under the gauge transformation, $A\mapsto A+/\partial ϵ$; locally one may choose gamma-trace gauge fixing, $A=0$, by solving $/\partial ϵ=-A$. The Euler-Lagrange field equation then yieldsThese are the massless Dirac equation, transversality and gamma-trace constraints that remove the lower-spin part of a vector-spinor.

For a more explicit massless Rarita-Schwinger polarization count, take a positive-frequency plane wave with null momentum along the third axis, in signature $(-,+,+,+)$. Transversality relates its temporal and longitudinal components. Residual gauge transformations with $/pϵ=0$ remove both. Write the surviving components as $u_1,u_2$; their gamma-trace condition reads ${\gamma }^1{u}_1+{\gamma }^2{u}_2=0$, henceThe massless Dirac equation leaves two complex positive-frequency amplitudes in $u_1$. If $i{\gamma }^1{\gamma }^2{u}_1=s{u}_1$, $s=\pm 1$, then $(u_1,u_2)$ is proportional to $(1,-is)$ in the transverse vector index. With spatial rotation generators $J_3^{\mathrm{vec}}{}_{12}=-i$ and $J_3^{\mathrm{spin}}=-i{\gamma }^1{\gamma }^2/2$, the vector has helicity $-s$ and the Majorana spinor has helicity $-s/2$, giving total helicity $-3s/2$. The Majorana spinor condition fixes the negative-frequency coefficients from these amplitudes. Therefore a massless gravitino has spin $3/2$, with the two physical helicities $\pm 3/2$. A complex vector-spinor would also have independent antiparticle modes.

The requested count is instead the off-shell component count of minimal supergravity, before imposing any Euler-Lagrange field equations. A four-dimensional Majorana spinor has four real components. Its vector index supplies four copies, and local supersymmetry supplies four real gauge functions:One must not additionally subtract transversality and the massless Dirac equation in this count: those are on shell conditions. The gamma-trace gauge fixing already uses the same four gauge functions subtracted above. For the graviton, a symmetric metric tensor has ten real components, with four diffeomorphism gauge functions. Equivalently, a vierbein has sixteen real components, minus six local Lorentz transformation and four diffeomorphism functions:The gravitino and graviton each have two propagating states on shell, but their off-shell counts disagree by six.

An auxiliary field has an algebraic Euler-Lagrange field equation and no propagating kinetic term. In old-minimal supergravity, a complex scalar $M$ contributes two real bosonic components and a real vector $b_{\mu }$ contributes four, so the bosonic count becomes $6+2+4=12$, matching the gravitino. Their supersymmetry transformations also absorb the terms that would otherwise require the propagating Euler-Lagrange field equations to close the local algebra. Thus auxiliary fields allow off-shell supersymmetry closure and matching component counts without adding physical particles. Eliminating them recovers an on shell formulation.

For a single chiral superfield, the supergravity F-term potential, with $\kappa =m_p^{-1}$ restored, isThe canonical Kähler potential has $K_{z\bar{z}}=K^{z\bar{z}}=1$, so the Kähler covariant derivative of a superpotential isConsequently the Polonyi model has the scalar potentialThe negative term is essential; retaining only the square of the supergravity auxiliary field would give the global F-term scalar potential rather than the supergravity F-term potential.

Set $x=\kappa z=a+iy$ and $b=\kappa \beta =2-\sqrt{3}$. In this notation $V=|\mu {|}^2{m}_p^2{e}^{|x{|}^2}U$, whereAt $a_0=\sqrt{3}-1$, $y=0$, one has $a_0+b=1$ and $1+a_0(a_0+b)=\sqrt{3}$, so $U=0$. To prove that this is a local minimum in both real directions, rather than checking only the real axis, put $u=a-a_0$. Direct expansion gives the stable zero-energy Polonyi vacuum identityEvery coefficient outside the square is strictly positive, and the exponential factor is positive. Thus $V\ge 0$ everywhere and equality requires $u=y=0$, provided $\mu \ne 0$. This proves the stronger conclusiona unique global minimum of this Polonyi model. If $\mu =0$, the scalar potential vanishes identically and this uniqueness and supersymmetry breaking conclusion do not hold.

For an independent local stability check, the Hessian matrix of $U$ at this global minimum is diagonal:Since $U$ and its first derivatives vanish there, differentiating the exponential introduces no additional Hessian matrix terms at the global minimum. For canonically normalized real fluctuations $z=z_0+(s+it)/\sqrt{2}$, the squared scalar masses are thereforewhich are both positive.

Although the vacuum energy is zero, the supergravity auxiliary field is not. In the convention $F^z=-e^{{\kappa }^{2}K/2}\overline{D_{z}W}$, its vacuum expectation value obeysThe matter fermion transformation contains a term proportional to $F^zϵ$, so no nonzero constant supersymmetry parameter leaves this vacuum invariant. Hence supersymmetry is spontaneously broken even though the vacuum energy vanishes: the positive auxiliary field contribution cancels the negative gravitational term. The matter fermion supplies the goldstino, which is absorbed by the gravitino in the super-Higgs mechanism.

Using the gravitino mass from a superpotential, and $z_0+\beta =m_p$, givesAn auxiliary field $F^z$ has mass dimension two. Defining the supersymmetry breaking scale by ${\Lambda }_{\mathrm{SUSY}}=\sqrt{|F^z|}$, one obtainsThus the gravitino mass is small compared with the supersymmetry breaking scale when that scale is sub-Planckian, as in the usual hierarchy $|\mu |\ll m_p$. This smallness is gravitational suppression; it is not a conclusion valid for arbitrary choices of the mass parameter.