Past exam of the mathematics course of the University of Cambridge / 2001 / iii / Paper 65 (Supersymmetry)

Three distinct motivations are stabilizing the weak scale, improving gauge-coupling unification, and supplying dark matter. Supersymmetric cancellation of quadratic divergences protects the Higgs field from the hierarchy problem; soft supersymmetry breaking near the weak scale keeps the residual corrections modest. The MSSM particle content improves gauge coupling unification under the usual assumptions about intermediate thresholds. Conserved R-parity can make a neutral lightest supersymmetric particle, such as a neutralino, stable and a possible weak-scale dark matter candidate. These are motivations, not guaranteed predictions. If supersymmetry is instead broken at much higher energies, it can still organize string theory and supergravity, and it remains a powerful tool for nonperturbative quantum field theory, as illustrated by Seiberg–Witten theory.

A BPS state saturates a supersymmetry energy bound and is annihilated by some supercharges. For a massive state with one relevant central charge in supersymmetry, normalize the supersymmetry algebra by $\{Q_{\alpha }^A,Q_{\beta }^B\}=2{ϵ}_{\alpha \beta }{ϵ}^{AB}Z$. In its rest frame, appropriate linear combinations of supercharges have anticommutators $2(M+|Z|)$ and $2(M-|Z|)$. Positivity therefore gives the BPS bound in supersymmetry $M\ge |Z|$. At equality, the latter combinations have zero norm and annihilate the state, producing a shortened supermultiplet and preserved supersymmetry. With the alternative convention omitting the factor $2$ in the central-charge anticommutator, the bound reads $M\ge |Z|/2$. The mass is tied to the charge, but this does not imply unconditional stability against decay as parameters vary.

The super-Higgs mechanism is the absorption of the goldstino by the gravitino when local supersymmetry is spontaneously broken. In rigid supersymmetry breaking, the goldstino is a physical massless fermion. In supergravity, its transformation permits a gauge choice removing it; its two degrees of freedom become the longitudinal helicity $\pm \frac{1}{2}$ states of the massive gravitino. The massless gravitino has helicities $\pm \frac{3}{2}$, whereas the massive gravitino has four spin states. This is distinct from the supersymmetric Higgs mechanism, which gives an internal gauge vector a mass and need not break supersymmetry.

Anticommuting-variable dependence alone is not the defining condition: a superfield must carry a specified Super-Poincaré group transformation law. Any appropriately graded function on superspace becomes an unconstrained scalar superfield under coordinate pullback, but an arbitrary collection of coefficients in a Grassmann variable polynomial is not automatically a prescribed supermultiplet. For example, a chiral superfield has $\Phi (y,\theta )=A(y)+\sqrt{2}\theta \psi (y)+{\theta }^{2}F(y)$, with $y^{\mu }=x^{\mu }+i\theta {\sigma }^{\mu }\bar{\theta }$ and ${\bar{D}}_{\dot{\alpha }}\Phi =0$; a vector superfield obeys $V=V^{†}$ and contains a gauge vector, gaugino and auxiliary field in Wess-Zumino gauge. A polynomial with constant component fields can be a special chiral superfield; lack of explicit spacetime dependence alone does not disqualify it. This is superfield covariance is not Grassmann dependence alone.

Specify the gauge group, its gauge group representations and gaugings, a real Kähler potential $K$, a holomorphic superpotential $W$, and a symmetric holomorphic gauge kinetic function $f_{ab}$, with positive kinetic matrices. The latter includes the gauge couplings and theta angles; allowed Abelian Fayet–Iliopoulos terms require constants ${\xi }_a$. For local supersymmetry, also specify the Planck mass; constant Fayet–Iliopoulos terms in conventional N=1 supergravity require a compatible gauged R-symmetry and transformation of $W$. $W$ and $f$ are holomorphic; $K$ generally is not. Radiative corrections generally change $K$. The Wilsonian effective action obeys superpotential non-renormalization perturbatively, although physical couplings run through field normalization and nonperturbative corrections can change $W$. The gauge kinetic function does receive corrections while remaining holomorphic; its perturbative Wilsonian running has one-loop exactness of the holomorphic gauge kinetic function, not all-order constancy of the physical gauge coupling.

The irreducible massive N=1 vector multiplet consists of one massive vector, one real scalar, and two massive spin-$\frac{1}{2}$ fermions. Thus its on-shell boson and fermion counts areThe two fermions can be written as two Majorana spinors or one Dirac spinor. All these states have the same mass when supersymmetry is unbroken; auxiliary fields are not extra propagating states.

To obtain rather than assume this spectrum, use the rest-frame supersymmetry algebra $\{Q_{\alpha },Q_{\beta }^{†}\}=2M{\delta }_{\alpha \beta }$. The two normalized fermionic creation operators form a spin-$\frac{1}{2}$ doublet. Start with a spin-$\frac{1}{2}$ fermionic Clifford vacuum annihilated by the two lowering operators. The zero- and two-creation-operator levels each have spin $\frac{1}{2}$, whereas the one-creation-operator level hasIt is bosonic, giving exactly the scalar and vector above. A higher-spin Clifford vacuum would introduce spin above $1$, while a spin-zero Clifford vacuum gives maximum spin only $\frac{1}{2}$.

For a massless supermultiplet, put the momentum along the third axis. The matrix $2\sigma \cdot p$ in the supersymmetry algebra has rank one, so only one fermionic oscillator acts nontrivially. It relates states whose helicities differ by $\frac{1}{2}$. Including the antiparticles required by CPT completion of a supermultiplet gives the following complete spectra.

A chiral multiplet contains a complex scalar $A$ and a Weyl spinor $\psi$. Its scalar particle and antiparticle have helicity $0$; its fermion particle and antiparticle have helicities $-\frac{1}{2}$ and $+\frac{1}{2}$. Therefore $n_B=2=n_F$, with maximum absolute helicity $\frac{1}{2}$.

A supersymmetric vector multiplet contains a massless gauge vector $A_{\mu }$, with helicities $+1,-1$, and a Majorana spinor gaugino $\lambda$, with helicities $+\frac{1}{2},-\frac{1}{2}$. Again $n_B=2=n_F$, now with maximum absolute helicity $1$. The chiral $F$ and vector $D$ auxiliary fields contribute no on-shell states.

The Higgs mechanism is compatible with unbroken supersymmetry. A supersymmetric vacuum may have nonzero scalar vacuum expectation values that break the internal gauge symmetry while obeying both F-flatness and D-flatness. For example, oppositely charged chiral superfields with $W=0$, no Fayet–Iliopoulos term and equal nonzero scalar magnitudes have $F=D=0$ but break the Abelian gauge group. In the supersymmetric Higgs mechanism, a massless vector multiplet combines with the chiral multiplet along the broken gauge direction. One real scalar becomes the longitudinal vector polarization; the other remains a physical real scalar, and the gaugino combines with the Higgsino into a Dirac spinor. The resulting massive vector multiplet has $3+1=4$ bosonic and $4$ fermionic states with a common mass, as in part (i). Any other uneaten chiral multiplets remain separately in the spectrum.

Take canonical kinetic terms, gauge coupling $g>0$, and define the Fayet–Iliopoulos term by $+\xi D$. If $\varphi$ is the scalar component of the charge-$q$ chiral superfield, the relevant auxiliary field Lagrangian isThe algebraic equation eliminates $D$ and gives the D-term scalar potentialSince the superpotential vanishes, $F=0$. In a homogeneous vacuum, the supersymmetry transformation of the gaugino includesThus $D_0\ne 0$ gives an inhomogeneous shift: no nonzero constant supersymmetry parameter leaves this vacuum invariant. The gaugino $\lambda$ is the goldstino, because it is the field shifting under the broken supercharge; there is no $F$ contribution here.

For $q\ne 0$, minimizing over $x=|\varphi {|}^2\ge 0$ separates three cases. If $q\xi <0$, the allowed value $x=-\xi /(gq)$ gives $D_0=V_0=0$: supersymmetry is unbroken while the internal gauge symmetry is Higgsed. If $q\xi >0$, the unconstrained zero of $D$ would have negative $x$, so the minimum is $\varphi =0$ with $D_0=-\xi$ and $V_0={\xi }^2/2$. If $\xi =0$, the minimum $\varphi =0$ is supersymmetric. Consequently, with this explicitly stated sign convention,Reversing the sign used to define the Fayet–Iliopoulos term reverses the printed sign criterion, not the physics. If $q=0$, the scalar is decoupled and flat: any $\xi \ne 0$ still breaks supersymmetry through $D_0=-\xi$, but the charged-field criterion does not apply.

On the broken branch, expand about $\varphi =0$:Writing $\varphi =({\varphi }_1+i{\varphi }_2)/\sqrt{2}$ gives both real scalar modes the squared mass $gq\xi$. There is no superpotential fermion mass and no vacuum expectation value to induce gaugino–matter-fermion mixing. The Abelian gauge symmetry is unbroken, so its vector is also massless. The complete mass spectrum of single charged-field D-term breaking isThe original chiral multiplet therefore has the squared-mass splittingThe original vector multiplet has zero mass splitting, although the nonzero $D_0$ means it is not an unbroken-supersymmetry representation. As a check, the tree-level supertrace mass sum rule gives $\mathrm{Str}{M}^2=2gq\xi$ when the two real scalar degrees of freedom are counted. On the $q=0$ broken branch all the propagating fields are massless.

There is a necessary scope qualification: one nonzero-charge Weyl fermion has an Abelian cubic gauge anomaly proportional to $q^3$ and a mixed gravitational gauge anomaly proportional to $q$. These are the classical vacuum and spectrum calculations requested by the model; a consistent quantum gauge theory needs an anomaly-cancelling completion, whose additional matter must also be included in its vacuum analysis.

Keep the PDF's hypercharge signs, which are the negatives of the frequently used convention. Write $U,D,E,N$ for the displayed $U^R,D^R,E^R,N^R$ chiral superfields. Their gauge group representations are conjugate right-handed-matter representations; their Weyl spinor components are left-chiral charge conjugates. Thus assign baryon number $B(Q)=\frac{1}{3}$, $B(U)=B(D)=-\frac{1}{3}$, and lepton number $L(L)=1$, $L(E)=L(N)=-1$, with both Higgs fields neutral under $B,L$. The corresponding electromagnetic convention is $Q_{\mathrm{em}}=T_3-Y$.

Let $X\cdot Y={ϵ}_{ab}{X}^a{Y}^b$ contract the two weak doublets, and use the color Levi-Civita symbol for three color antitriplets. The most general homogeneous cubic superpotential with right-handed-neutrino superfields separates as $W_3=W_{B,L}+W_{/B\text{or}/L}$, whereandRepeated generation and color indices are summed. The four Yukawa coupling matrices are arbitrary. The weak and color contractions imply ${\lambda }_{ijk}=-{\lambda }_{jik}$ and ${\lambda }_{ijk}^{\prime \prime }=-{\lambda }_{ikj}^{\prime \prime }$, while ${\rho }_{ijk}$ is completely symmetric. The $\lambda ,{\lambda }^{\prime }$ terms have $\Delta L=1$; ${\lambda }^{\prime \prime }$ has $\Delta B=-1$; $\kappa$ and $\rho$ have respectively $\Delta L=-1,-3$. Their conjugate interactions have the opposite changes. All four Yukawa terms preserve both numbers; all five remaining types violate at least one. In particular, the neutral $N$ fields printed in the PDF require both $N{H}_1{H}_2$ and $NNN$ terms.

For completeness of this list, a cubic color gauge-invariant can contain a $Q$–$U$ or $Q$–$D$ pair and one colorless doublet, or three color antitriplets. Vanishing hypercharge gives $Q{H}_{2}U$, $Q{H}_{1}D$, $LQD$ and $UDD$. Three $Q$ doublets have no weak singlet. Among colorless terms, two doublets and a singlet give $LLE$, $L{H}_{1}E$, $L{H}_{2}N$ and $H_1{H}_{2}N$; the apparent $H_1{H}_{1}E$ contraction vanishes since the identical bosonic superfields commute. Three singlets yield only $NNN$. This also explains why no omitted cubic monomial is available. The word cubic excludes the linear $N$ term and the quadratic $H_1{H}_2$, $L_i{H}_2$, $N_i{N}_j$ terms; these are separate possible terms in the full renormalizable superpotential.

To demonstrate squark-mediated proton decay, take ${\lambda }_{11k}^{\prime }$ and ${\lambda }_{11k}^{\prime \prime }$ with $k=2$ or $3$. Identical down-flavor indices would make ${\lambda }_{111}^{\prime \prime }$ vanish. Both interactions couple to the same scalar ${\tilde{D}}_k$ of mass $M$. Suppressing overall convention-dependent vertex signs, their relevant two-component fermion bilinears areThe color index on $A$ is contracted with that on ${\tilde{D}}_k$; Lorentz spinor indices within each bilinear are contracted too. Together with $-M^2|{\tilde{D}}_k{|}^2$, its algebraic equation at momenta small compared with $M$ gives ${\tilde{D}}_k=-A^{†}/M^2$. Substitution gives $A^{†}A/M^2$, whose cross term is the four-fermion matching for squark-mediated proton decayIts electron term contains three quark fields of flavors $uud$ and a lepton, and changes baryon number and lepton number together. The Hermitian-conjugate terms describe both orientations of the process. The hadronic matrix element of the three-quark proton-decay operator between a proton and a neutral pion has the right quantum numbers and is not forbidden by a remaining symmetry. Hence the operator permits $p\to e^{+}{\pi }^0$: the lepton field creates the positron and the three-quark matrix element connects the proton to the pion. This supplies the mechanism rather than merely naming the two violating interactions.

Each fermion field has mass dimension $\frac{3}{2}$, so this effective operator has dimension six and coefficient $C_6={\lambda }^{\prime }{\lambda }^{\prime \prime }{}^{\ast }/M^2$. By dimensional analysis its contribution to the decay rate has the formHere $c_h$ collects phase space, running and the dimensionless hadronic factor; a dimensional estimate takes it of order unity rather than pretending to calculate a strong-interaction matrix element. Assuming this exchange dominates and there is no tuned cancellation, the supplied lifetime gives the proton-lifetime bound on a product of R-parity-violating couplingsThe last estimate uses $m_p\simeq 1\mathrm{GeV}$ and ${\tau }_{\min }=2.4\times 10^{64}{\mathrm{GeV}}^{-1}$. Keeping $m_p=0.938\mathrm{GeV}$ instead changes $6.5$ to about $7.6$. A numerical limit independent of the squark mass or hadronic factor is not determined by the supplied data.

Finally, matter parity is $P_M=(-1{)}^{3(B-L)}$: all matter chiral superfields, including $N$, are odd, and the Higgs chiral superfields are even. R-parity of a component field is $R_p=P_M(-1{)}^{2s}$. In a superpotential, the spin factor is irrelevant to the allowed monomials: each of $Q{H}_{2}U,Q{H}_{1}D,L{H}_{1}E,L{H}_{2}N$ has two odd matter factors and is even. Each of $LLE,LQD,UDD,NNN$ has three odd matter factors, and $N{H}_1{H}_2$ has one, so all are odd and forbidden. Higgs vacuum expectation values are even; the allowed Yukawa interactions therefore still generate ordinary fermion masses.

R-parity forbids all the baryon/lepton-number-violating cubic terms above, not every possible violating operator. In particular, $\frac{1}{2}{M}_{ij}{N}_i{N}_j$ is even under matter parity but changes lepton number by two: matter parity allows Majorana neutrino masses. Thus the literal unrestricted claim would be false for these very fields; its valid cubic interpretation gives precisely the requested exclusion while preserving the Yukawa coupling mass terms.