Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 307 (Supersymmetry)

The Coleman–Mandula theorem says that, under its assumptions on a nontrivial analytic relativistic S-matrix and the particle spectrum, every continuous bosonic symmetry algebra is a direct sum of the Poincare algebra and an internal symmetry algebra. Supersymmetry becomes possible by weakening the assumption that the symmetry algebra is an ordinary Lie algebra: a Z2-graded Lie superalgebra admits odd generators whose bracket is an anticommutator. The Haag–Łopuszański–Sohnius theorem then classifies the allowed extension and leads to the Super-Poincaré algebra.

An internal generator $T_a$ is a Lorentz scalar. The Coleman–Mandula theorem and the graded extension allow it to act nontrivially on supercharges only as an R-symmetry. Since R-symmetries are excluded,

Translation invariance of a conserved global supercharge and the graded Jacobi identities giveLorentz covariance requires the supercharges to transform as Weyl spinors,with the complex-conjugate dotted-spinor relation for ${\bar{Q}}_{\dot{\alpha }}$, up to the sign convention used for the action of generators.

The anticommutator $\{Q_{\alpha },{\bar{Q}}_{\dot{\beta }}\}$ transforms as a Lorentz vector because $(1/2,0)\otimes (0,1/2)=(1/2,1/2)$. The only translation generator with that transformation law is $P_{\mu }$, and a normalization of $Q$ fixesAn equal-chirality anticommutator could only contain an antisymmetric spinor contraction times a central charge, but the anticommutator is symmetric under exchange of the complete supercharges. For one supercharge and no central extension this forcesTogether with the stated Poincare brackets, these are the four-dimensional $\mathcal{N}=1$ Super-Poincare relations.

Apply parity to the mixed anticommutator. The transformed left-hand side isbecause $|{\eta }_P|=1$. The Pauli-matrix identity represented by this product leaves the temporal matrix unchanged and reverses the three spatial matrices, so it equalsBut a parity transformation acts on momentum as $P^0\mapsto P^0$ and $P^i\mapsto -P^i$. This is exactly the parity transform ofso $\{Q_{\alpha },{\bar{Q}}_{\dot{\beta }}\}=2{\sigma }_{\alpha \dot{\beta }}^{\mu }{P}_{\mu }$ is consistent with the stated transformation law. The arbitrary intrinsic phase cancels.

Fermion parity is defined byIt anticommutes with $Q_{\alpha }$ and ${\bar{Q}}_{\dot{\alpha }}$. At fixed nonzero energy and momentum, choose a supercharge combination $S$ for which $\{S,S^{†}\}=cI$ with $c>0$. Taking the finite-dimensional trace over one supermultiplet givesbecause cyclicity of the trace and anticommutation of $(-1{)}^F$ with $S$ make the two terms cancel. ThereforeThis is boson-fermion degeneracy in a supermultiplet.

If explicit or soft supersymmetry breaking terms are added, the supercharge is no longer a conserved symmetry of the full Hamiltonian and states need not form representations of the supersymmetry algebra at equal energy. The positive anticommutator cannot be replaced by a constant on a purported multiplet, so the supertrace proof and mass degeneracy fail.

A chiral superfield obeysExamples in the Minimal supersymmetric Standard Model include the quark, lepton and Higgs chiral superfields. Such a multiplet contains a complex scalar $\varphi$, a two-component Weyl spinor $\psi$, and a complex auxiliary field $F$; only the scalar and fermion propagate on shell.

The chiral coordinate $y^{\mu }=x^{\mu }+i\theta {\sigma }^{\mu }\bar{\theta }$ is annihilated in the required combination by ${\bar{D}}_{\dot{\alpha }}$, so the solution has the simple formTaylor-expand each component about $x$. Nilpotence truncates the series, and the Grassmann identitygives the chiral-superfield component expansionTherefore

Only the Weyl fermions in the charged chiral multiplets contribute to the Abelian triangle gauge anomaly. Their cubic coefficient isand the mixed gauge-gravitational coefficient is $2+0-2=0$. Diagrammatically, the oppositely charged $X_{2+}$ and $X_{2-}$ fermion triangles have equal magnitudes and opposite signs. The neutral gaugino also contributes nothing, so the theory is anomaly free.

Gauge invariance and renormalizability permit the most general superpotentialAfter setting the linear and quadratic parameters to zero, the supersymmetric non-renormalization theorem ensures that perturbative quantum corrections do not regenerate them in the Wilsonian superpotential.

Write the scalar components as $x_{+},x_0,x_{-}$. For generic nonzero $\lambda ,y$, the F-term scalar potential and D-term scalar potential, including a Fayet–Iliopoulos term, areEvery term is nonnegative. The F-flat equations force $x_0=0$ and $x_{+}{x}_{-}=0$. If $\xi >0$, the vacuumalso makes $D=0$; if $\xi <0$, interchange $x_{+}$ and $x_{-}$ and use $|x_{+}{|}^2=-\xi /(2g)$. Thus for nonzero $\xi$ the charged vacuum expectation value spontaneously breaks the gauged $U(1)$ through the Higgs mechanism, but $F_i=D=0$ means supersymmetry remains unbroken. For $\xi =0$, the origin preserves both the gauge symmetry and supersymmetry.

At one loop, plot each inverse squared coupling $g_a^{-2}$ against $\log \mu$. The three Standard Model lines have slopes that do not pass through one common point. Above the superpartner threshold, additional scalar and fermion vacuum-polarization diagrams change the slopes: for example, an $SU(2)$ gauge-boson two-point function receives a loop from a left-handed Standard Model fermion doublet and an additional loop from its scalar superpartner, while the $SU(2)$ gauge multiplet adds a gaugino loop alongside the gauge-boson and ghost loops. With the Minimal supersymmetric Standard Model field content, the three resulting straight lines meet to good accuracy. This is supersymmetric gauge coupling unification.

The differential equationimpliesHenceor

Let $L=\log (M_{GUT}/M_Z)$. Equality of $g_1$ and $g_2$ at the unification scale givessoSimilarly,whileEliminating $L$ yieldswithFor the GUT-normalized MSSM one-loop coefficients proportional to $(33/5,1,-3)$, this is $\boxed{A=5/7}$.

Current coupling measurements approximately satisfy this MSSM relation and point to $M_{GUT}$ of order $10^{16}$ GeV, much more accurately than nonsupersymmetric one-loop running. Exact equality is not expected because two-loop evolution and threshold corrections from split superpartner and GUT-scale masses shift the lines; the absence so far of directly observed superpartners also prevents the threshold spectrum from being fixed experimentally.