Past exam of the mathematics course of the University of Cambridge / 2005 / iii / Paper 55 / 1

For null four-momentum, $\{Q^I,{\bar{Q}}^J\}=2{\delta }^{IJ}\sigma \cdot p$ has rank one in its spinor indices. Positivity makes the null combinations act trivially; the remaining $N$ normalized pairs obey $\{a_I,a_J^{†}\}={\delta }_{IJ}$. Acting with distinct lowering operators on a highest-helicity state gives the helicity spectrum of a massless supermultiplet:A CPT completion of a supermultiplet adds the opposite helicities if needed. Interpreting the physical spin bounds as $|\lambda |\le s$, the width $N/2$ fits inside $[-s,s]$ only if $N\le 4s$. A literal one-sided bound on helicity without CPT would not give this maximum. The maximal spectra areandThus the maxima are (N=4) for spin at most one and (N=8) for spin at most two, with 16 and 256 states. Dimensional reduction of ten-dimensional super Yang-Mills theory on a six-torus gives the first: the vector yields one four-dimensional vector and six real scalar fields, and its Majorana-Weyl spinor yields four Weyl spinors. Reducing eleven-dimensional supergravity on a seven-torus gives the second: metric components supply one graviton, seven vectors and 28 scalars; the three-form supplies 21 vectors, 35 scalars and seven dualized two-form scalars; the gravitino supplies eight gravitini and 56 spin-half fields. These total 28 vectors and 70 scalars, with 128 boson and 128 fermion states.

R-parity is the multiplicative ${\mathbb{Z}}_2$ quantum numberwhere $B,L,s$ are baryon number, lepton number and spin. Ordinary Standard Model particles have $R_p=+1$ and their superpartners have $R_p=-1$. Gauge invariance alone allows MSSM superpotential terms $LL{E}^c$, $LQ{D}^c$, $U^c{D}^c{D}^c$ and $L{H}_u$. Simultaneous baryon- and lepton-number violation can induce rapid proton decay. Exact R-parity forbids these renormalizable terms, since the associated matter parity is odd; it does not forbid every higher-dimensional proton-decay operator, for example $QQQL$.

Two consequences are superpartner production in pairs from ordinary-particle initial states and stability of the lightest supersymmetric particle: a single parity-odd particle cannot decay only into parity-even particles. Cascades therefore end in the lightest supersymmetric particle; if it is neutral and weakly interacting, it produces missing-energy signatures.

A continuous R-symmetry rotates $\theta$ and the supercharges. With $R(\theta )=1$, the superpotential has R-charge two and the fermion in a chiral superfield has one less charge than its scalar. R-parity is only a discrete restriction, represented by combining matter parity with $\theta \mapsto -\theta$, and need not preserve a full continuous R-symmetry. For example, a gaugino Majorana mass term violates an unbroken continuous R-symmetry but preserves R-parity. Thus proton protection by parity and continuous R-charge selection rules are distinct requirements.

In conventional Kaluza-Klein theory, bulk fields have a Kaluza-Klein tower with masses of order $1/R$. If ordinary matter propagates in the extra dimension, the absence of accessible excited modes requires a small $R$: a TeV compactification scale corresponds to roughly $10^{-19}$ m; Planck-scale compactification would be much smaller, of order the Planck length. This is a scale estimate, not a universal radius limit independent of the fields and spectrum.

An unwarped brane-world scenario localizes ordinary fields on a brane while gravity explores a large compact bulk. The volume relationmakes gravity weak without a fundamental Planck-scale cutoff. In the original large-extra-dimension model, $M_{\ast }\sim \mathrm{TeV}$ permits approximately millimetre/submillimetre dimensions for $n=2$; more dimensions are smaller. Ordinary-field KK bounds no longer limit these gravity-only radii, although gravitational and other constraints do. This replaces the large fundamental scale hierarchy by a large compact volume needing stabilization.

A warped brane-world scenario instead uses a position-dependent metric factor. In the two-brane Randall–Sundrum model, $m_{\mathrm{IR}}=e^{-kL}{m}_0$, so $kL\sim 35$ generates a weak scale from a Planck-sized input without a millimetre proper separation; for $k$ of Planck order, $L$ is tens of inverse-Planck lengths. The single-brane version even allows infinite proper size through gravity localization in a noncompact warped dimension; it is not by itself the two-brane redshift hierarchy construction. Thus warped proper size can be unbounded, while finite hierarchy models need specified warping and radius stabilization.

Supersymmetry addresses the hierarchy problem differently: boson/fermion cancellations protect scalar masses against quadratic sensitivity to high scales, with residual corrections controlled by soft supersymmetry breaking. It stabilizes a chosen hierarchy rather than explaining its initial size. Volume dilution and the warped redshift mechanism explain scale separation geometrically; neither automatically determines its moduli or removes all tuning.