Past exam of the mathematics course of the University of Cambridge / 2006 / iii / Paper 54 (Supersymmetry and Extra Dimensions)

Kaluza-Klein theory explains apparently four-dimensional physics as the low-energy limit of a higher-dimensional theory with compact spatial directions. In units $\hslash =c=1$, take a circle coordinate $y\sim y+2\pi R$. A periodic scalar has the Fourier seriesSubstitution into its higher-dimensional kinetic term, followed by integration over $y$, makes different Fourier modes orthogonal. The internal derivative contributes $n^2/R^2$ to the lower-dimensional squared mass, soThe quantized internal momentum produces the Kaluza-Klein tower. A real higher-dimensional field satisfies ${\varphi }_{-n}={\varphi }_n^{\ast }$; the tower does not introduce independent complex modes for both signs. More general compact spaces replace $n^2/R^2$ by Laplacian eigenvalues, with boundary conditions determining the spectrum.

At energies $E\ll R^{-1}$, nonzero Kaluza-Klein modes cannot be produced and may be integrated out, leaving a four-dimensional effective field theory. Dimensional reduction retains the zero modes at leading order; integrating out the massive modes also generates interactions suppressed by the Kaluza-Klein compactification scale. The higher-dimensional metric decomposes into a four-dimensional metric, vectors from mixed components, and scalar fields from internal components. For a circle, these include an Abelian gauge field and a radion. Internal coordinate shifts induce the vector's gauge symmetry, while the radion describes the radius. Pure circle reduction alone does not automatically give the chiral matter spectrum of the Standard Model.

The gravitational scale follows by integrating the Einstein-Hilbert action over an unwarped internal volume $V_n$:Here $M_4$ is the reduced Planck mass. For $n$ equal circle radii, $V_n=(2\pi R{)}^n$. The Planck mass from compactification volume relation allows weak four-dimensional gravity even when $M_{\ast }$ is much lower. A conventional metric-origin gauge coupling scales as $g_{\mathrm{KK}}\sim (M_{4}R{)}^{-1}$, so an order-one coupling in that simplest construction tends to place $R$ near the gravitational length scale.

In the brane-world scenario, ordinary matter and its gauge fields can instead be localized on a three-dimensional spatial brane, while gravity explores the bulk. The absence of ordinary-matter excitations along the extra dimensions then permits larger radii than a model in which all fields propagate there. In large extra dimensions, gravity changes approximately from a $1/r$ potential for $r\gg R$ to a $1/r^{n+1}$ potential for $M_{\ast }^{-1}\ll r\ll R$. For illustration, $M_{\ast }$ of order a TeV and $n=2$ give $R$ of order $0.1$ mm; one extra dimension would require an astronomically large radius.

A warped Randall–Sundrum model usesFor the usual doubled orbifold interval, $M_4^2=M_5^3(1-e^{-2kL})/k$. The relevant lengths are the bulk curvature radius $k^{-1}$, the proper separation $L$, and the inverse warped excitation scale, of order $(k{e}^{-kL}{)}^{-1}$. Thus the first mass need not equal $1/L$. Compact volume dilutes gravity; warping redshifts local masses. Both mechanisms require consistent field localization and radius stabilization; choosing a large radius or separation is not itself a dynamical explanation of it.

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.

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.

Use ${\eta }_{\mu \nu }=\mathrm{diag}(1,-1,-1,-1)$, ${\sigma }^{\mu }=(I,\mathbf{\sigma })$ and ${\bar{\sigma }}^{\mu }=(I,-\mathbf{\sigma })$, where $\mathbf{\sigma }$ are the Pauli matrices. The translation and Lorentz generators obey the Poincare algebraOne can derive the additional relations by realizing the supercharges as differential operators on superspace. With left Grassmann derivatives and $P_{\mu }=i{\partial }_{\mu }$, takeUsing $\{{\partial }_{{\theta }^{\alpha }},{\theta }^{\beta }\}={\delta }_{\alpha }{}^{\beta }$, the two mixed derivative-coordinate terms each give $i{\sigma }_{\alpha \dot{\beta }}^{\mu }{\partial }_{\mu }$. The remaining terms cancel by Grassmann anticommutation. Equal-chirality terms cancel in the same way. HenceAll coefficients are independent of $x$, so direct differentiation gives $[P_{\mu },Q_{\alpha }]=[P_{\mu },{\bar{Q}}_{\dot{\alpha }}]=0$. This also shows that supercharges commute with translations and preserve the four-momentum of a state.

For the Lorentz generators, define the left-handed spinor matricesThe identity ${\sigma }^{\mu }{\bar{\sigma }}^{\nu }+{\sigma }^{\nu }{\bar{\sigma }}^{\mu }=2{\eta }^{\mu \nu }I$ gives their Lorentz commutation relations. Acting on the spinor superspace coordinates and their dual derivatives then gives, in a column-component convention,Here dotted index values are identified with the conjugate undotted values, and ${\bar{Q}}_{\dot{\alpha }}=Q_{\alpha }^{†}$. The second formula also follows directly by taking the adjoint of the first, using Hermitian $M_{\mu \nu }$. For example, $S_{12}={\sigma }^3/2$, so $[J_3,Q_1]=-Q_1/2$ and $[J_3,Q_2]=Q_2/2$ for $J_3=M_{12}$. This explicit component convention avoids reversing signs by mixing state matrices with operator-component matrices. Covariance of the mixed anticommutator follows by applying these matrices to ${\sigma }^{\rho }{P}_{\rho }$.

These are the minimal Super-Poincaré algebra relations. In the ordinary particle algebra a possible scalar term in $\{Q_{\alpha },Q_{\beta }\}$ would be proportional to ${ϵ}_{\alpha \beta }$, but an anticommutator is symmetric in its two labels; with only one supersymmetry such a central term vanishes. Extended algebras can instead use an antisymmetric internal index tensor. Tensor charges associated with extended objects are not part of this minimal particle algebra.

For a positive-energy massless supermultiplet, choose $p^{\mu }=(E,0,0,E)$ with $E>0$, so $p_{\mu }=(E,0,0,-E)$ andPositivity of the Hilbert-space norm implies that both $Q_1$ and $Q_1^{†}$ annihilate every state: their squared norms sum to zero. SetFor momentum along the positive third axis, helicity is $h=J_3$. The Lorentz commutators give $[h,a]=a/2$ and $[h,a^{†}]=-a^{†}/2$. Choose a highest-helicity Clifford vacuum $|\lambda ⟩$ with $a|\lambda ⟩=0$. Its partner $a^{†}|\lambda ⟩$ has unit norm if the first state does and has helicity $\lambda -1/2$. No further state is generated because $(a^{†}{)}^2=0$. ThereforeThe two states have opposite fermion parity. A CPT completion of a supermultiplet adds helicities $(-\lambda ,-\lambda +1/2)$ and conjugate internal charges when necessary. Taking $\lambda =1/2$ produces a complex scalar and a Weyl spinor after CPT completion; taking $\lambda =1$ produces the massless vector multiplet. This construction assumes ordinary finite-helicity particle representations, rather than continuous-spin representations.

Consider a physical supermultiplet at fixed four-momentum with positive energy $E$ and finitely many polarization states. Let $\Pi =(-1{)}^F$ be fermion parity, equal to $+1$ on bosons and $-1$ on fermions. Every supercharge is odd, so $\Pi Q_{\alpha }=-Q_{\alpha }\Pi$ and likewise for its adjoint.

Trace cyclicity gives, for each component,Thus the supertrace pairing at positive energy yieldswhere the middle equality uses $\mathrm{tr}{\sigma }^0=2$ and $\mathrm{tr}{\sigma }^i=0$. Since $E>0$,This proves boson-fermion degeneracy in a supermultiplet for both massive and massless particle multiplets, counting physical states rather than unphysical gauge components or auxiliary fields. The positive-energy qualification is essential: a zero-energy supersymmetric vacuum can be a one-dimensional bosonic singlet with all supercharges zero. It need not have a fermionic partner, so the printed assertion cannot literally include arbitrary zero-energy representations.

Sum the diagonal mixed anticommutators in the Super-Poincaré algebra. Their spatial terms have zero Pauli trace, givingFor every normalized state $|u⟩$ in the common operator domain,This proves energy positivity in global supersymmetry. The zero of energy is fixed by this algebra, rather than by an arbitrary independent subtraction.

A supersymmetric vacuum is invariant under every supercharge and its adjoint. Its energy therefore vanishes. Conversely, a normalized vacuum of zero energy makes every nonnegative norm on the right zero, so all supercharges annihilate it. ConsequentlyFor an infinite spatial volume the corresponding statement is about vacuum energy density, obtainable by first regulating the volume. It concerns rigid supersymmetry; the gravitational scalar potential in supergravity has additional terms and is not generally nonnegative.

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.

First complexify the Clifford algebra in $D=2n$ spacetime dimensions. Multiplying appropriate generators by $i$ converts the signature to the Euclidean form $\{{\Gamma }_a,{\Gamma }_b\}=2{\delta }_{ab}$ without changing the complex module dimension. Pair the generators intoThe Clifford algebra relations giveThe commuting occupation operators $N_j=b_j^{†}{b}_j$ have eigenvalues zero or one. Choose a nonzero state killed by every $b_j$; repeated annihilation reaches such a state in any finite irreducible module. Acting with distinct creation operators produces the $2^n$ occupation statesTheir distinct occupation patterns make them linearly independent, and their span is invariant under every Clifford generator. Irreducibility therefore makes it the entire Dirac module. This derives the even-dimensional Clifford spinor dimension:The chirality matrix is a phase-adjusted product of all $2n$ generators. It squares to one, anticommutes with each generator and acts as occupation parity up to an overall sign. Its two eigenspaces each have $2^{n-1}$ states. The even Clifford algebra, and hence the connected spin group, preserves them, giving irreducible Weyl spinors of complex dimension $2^{D/2-1}$. A Majorana spinor imposes a reality condition only in signatures where it exists; simultaneous Majorana and Weyl conditions are not available in every even dimension.

In Lorentzian ten dimensions, the complex Dirac module has dimension $32$, and a Weyl module dimension $16$. A Majorana-Weyl spinor is allowed and has $16$ real components before its equation of motion. The massless Dirac equation leaves eight physical gaugino polarizations. A ten-dimensional massless gauge field also has $10-2=8$ physical polarizations. Thus ten-dimensional super Yang-Mills theory has eight bosonic and eight fermionic states per gauge generator, with sixteen real supercharges.

For dimensional reduction, take a flat six-torus with periodic fermions and retain fields independent of its coordinates. Split $A_M$ into $A_{\mu }$ and six internal components $A_m$, $m=4,\dots ,9$. The latter are four-dimensional real adjoint scalar fields. Under the spin groups of the two factors, a ten-dimensional chiral spinor decomposes asfor $\mathrm{Spin}(1,3)\times \mathrm{Spin}(6)$, with $\mathrm{Spin}(6)\simeq SU(4)$. The Majorana condition relates these conjugate pieces, leaving four independent four-dimensional Weyl spinors. The supersymmetry parameters reduce in exactly the same fashion: sixteen real charges are four four-dimensional supersymmetries. The resulting field content isThe on-shell counts are $2+6=8$ bosonic and $4\times 2=8$ fermionic states, as required. The internal spin group supplies the $SU(4)$ R-symmetry of the reduced action.

To derive its scalar potential, use Hermitian gauge matrices and a positive trace inner product, withIf the internal volume is $V_6$, integration gives $g_4^2=g_{10}^2/V_6$. Since internal derivatives vanish,With mostly-minus spacetime signature, $F_{\mu m}{F}^{\mu m}$ has an extra minus sign and the double sum over mixed indices gives a factor two. Rescale $A_{\mu }=g_4{a}_{\mu }$ and $A_m=g_4{X}_m$ to canonical fields. The resulting bosonic Lagrangian iswhere $D_{\mu }{X}_m={\partial }_{\mu }{X}_m-i{g}_4[a_{\mu },X_m]$. A commutator of Hermitian matrices is anti-Hermitian, so the last term is minus a nonnegative scalar potential:The sum is over ordered internal pairs; a sum over $m<n$ would instead have coefficient $g_4^2/2$. Trace normalization merely changes the matched gauge-coupling convention.

The origin attains zero potential, so the global minima are exactlyFor a compact connected gauge group, a commuting tuple is simultaneously conjugate into a Cartan subalgebra. Write $X_m={\sum }_{a=1}^r{x}_m^a{H}_a$, where $r$ is the rank. Every choice of the $6r$ real numbers $x_m^a$ has zero potential. After the residual Weyl group identifications, the classical commuting-scalar vacua of four-dimensional N=4 Yang-Mills theory form $({\mathbb{R}}^6{)}^r/\mathcal{W}$ in the four-dimensional theory. For $SU(N)$ one may describe them as six simultaneous diagonal matrices, with tracelessness and permutations of their common eigenvalue labels. Varying these eigenvalues changes gauge-invariant configurations while leaving $V=0$, proving the existence of genuine flat directions of a scalar potential, not just gauge-equivalent directions. Generic values break the gauge group to its maximal torus; coincident eigenvalue vectors produce enhanced unbroken gauge symmetry. If one keeps a finite compact torus rather than the four-dimensional limit, large gauge transformations add periodic identifications of the Wilson-line coordinates.

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.