Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 307 (Supersymmetry)

Choose $P_{\mu }=-i{\partial }_{\mu }$ and left Grassmann derivatives. A consistent differential realization of four-dimensional N=1 supersymmetry isThe signs can change with the definitions of $P_{\mu }$, Grassmann differentiation and the finite transformation, but these operators obey the convention-independent content of the Super-Poincaré algebra,They follow by expanding the finite transformation to first order and requiring a supersymmetry translation of the superspace coordinates.

A vector superfield is a real superfield,For a non-Abelian gauge theory it is also Lie-algebra valued, $V=V^a{T}_a$. The reality condition permits the component expansion to be reduced to Wess-Zumino gauge, where the physical fields are a gauge field and gaugino together with a real auxiliary field.

A chiral superfield is annihilated by the antichiral supersymmetric covariant derivative:Equivalently, it depends on $(y^{\mu },\theta )$ with $y^{\mu }=x^{\mu }+i\theta {\sigma }^{\mu }\bar{\theta }$, and has the finite expansion $\Phi =\varphi +\sqrt{2}\theta \psi +{\theta }^{2}F$.

The gauge-transformation superfield $\Lambda ={\Lambda }^a{T}_a$ is a dimensionless chiral superfield:It takes values in the complexified $\mathfrak{su}(3)$ Lie algebra. It need not be real; its Hermitian conjugate is antichiral. This complexification is necessary because a general superspace gauge transformation must preserve chirality.

With canonical normalization, the renormalizable supersymmetric gauge theory hasThe first term is the gauge-covariant Kähler potential and the second is the Supersymmetric Yang-Mills action. A holomorphic superpotential would also be integrated as $\int d^2\theta \mathcal{W}(\Phi )+h.c.$, but one commuting field in the fundamental representation of $SU(3)$ admits no nonconstant renormalizable gauge-invariant superpotential: the apparent cubic ${ϵ}_{ijk}{\Phi }^i{\Phi }^j{\Phi }^k$ vanishes.

Under ${\Phi }^{\prime }=e^{-2i\Lambda }\Phi$, the conjugate field transforms as ${\Phi }^{\prime }{}^{†}={\Phi }^{†}{e}^{2i{\Lambda }^{†}}$. Invariance of the canonical Kähler term therefore requiresorThis is the finite non-Abelian supergauge transformation of the vector superfield.

The relation in part vi givesso the full-superspace term has gauge invariance. The chiral field-strength superfield transforms covariantly,Cyclicity of the matrix trace then gives $\mathrm{Tr}(W^{\prime }{}^{\alpha }{W}_{\alpha }^{\prime })=\mathrm{Tr}(W^{\alpha }{W}_{\alpha })$. Both superspace integrals, and hence the entire Lagrangian density, are invariant.

The hierarchy problem is visible in the one-loop top-quark contribution to the Higgs boson mass parameter,In a supersymmetric theory the corresponding stop loop has the opposite quadratic term. If supersymmetry is softly broken, the remaining correction is roughlyThe two relevant one-loop Feynman diagrams are a Higgs two-point function with a closed top-quark loop and the analogous stop loop. Requiring the residual correction not to exceed the electroweak scale by many orders of magnitude gives the naturalness expectation $m_{\tilde{t}}$, and broadly other sparticle masses, at a few TeV or below. This is an order-of-magnitude argument rather than a mass bound.

R-parity iswhere $B$ is baryon number, $L$ is lepton number and $s$ is spin angular momentum. Standard Model particles have $R_p=+1$, while their superpartners have $R_p=-1$. The renormalizable R-parity-violating superpotential of the Minimal supersymmetric Standard Model iswith ${\lambda }_{ijk}=-{\lambda }_{jik}$ and ${\lambda }_{ijk}^{\prime \prime }=-{\lambda }_{ikj}^{\prime \prime }$.

Combining ${\lambda }_{i1k}^{\prime }{L}_i{Q}_1{D}_k^c$ with ${\lambda }_{11k}^{\prime \prime }{U}_1^c{D}_1^c{D}_k^c$ produces proton decay through exchange of a virtual right-handed down-type squark. The tree diagram has $u$ and $d$ entering the baryon-number-violating vertex, an internal ${\tilde{d}}_{Rk}$ line, and the lepton-number-violating vertex emitting ${\bar{\nu }}_i$ and a light quark. Hadronization givesThe invisible particle is an antineutrino of any family $i=1,2,3$, and electric-charge conservation requires a positively charged pion. The antisymmetry of ${\lambda }_{11k}^{\prime \prime }$ requires $k=2$ or $3$.

Integrating out a squark of mass $m_{\tilde{q}}$ gives a dimension-six interaction with coefficient $C\sim {\lambda }^{\prime }{\lambda }^{\prime \prime }/m_{\tilde{q}}^2$. Ignoring order-one hadronic and phase-space factors,For $m_p\sim 1\mathrm{GeV}$ and $m_{\tilde{q}}\sim 10^4\mathrm{GeV}$, the factor before the couplings is about $10^{-18}$--$10^{-17}\mathrm{GeV}$. A lifetime above $10^{34}$ years is about $10^{65}{\mathrm{GeV}}^{-1}$, so ${\Gamma }_p\lesssim 10^{-65}\mathrm{GeV}$ andat order-of-magnitude accuracy. Hadronic matrix elements and flavour choices alter the numerical coefficient, not the conclusion that simultaneous baryon- and lepton-number violation is extraordinarily constrained.

The matter-parity factor $(-1{)}^{3(B-L)}$ is odd for every quark or lepton superfield and even for either Higgs superfield. Each term in $W_{\mathrm{RPV}}$ contains an odd number of matter superfields, so R-parity forbids all four types of term. Standard MSSM Yukawa interactions contain two matter superfields and remain allowed.

Finally, the requested four-point interaction contains a squark, an antisquark, a $W$ boson and a gluon. It comes from the squark gauge-covariant kinetic termThe cross term isbecause the colour and weak generators commute. With all fields incoming, its Feynman rule iswith the conventional $1/\sqrt{2}$ conversion when $W^I$ is replaced by $W^{\pm }$.

With ${ϵ}_{0123}=+1$, define the Pauli-Lubanski pseudovector byFor a translation $a^{\mu }$ and a Lorentz transformation $\Lambda =\exp (\omega )$, a left-handed Weyl spinor transforms asEquivalently, the active field at a fixed point uses ${\psi }^{\prime }(x)=S(\Lambda )\psi ({\Lambda }^{-1}(x-a))$.

The Clebsch-Gordan decomposition isIndeed,The symmetric spinor is the $(1,0)$ representation, while its antisymmetric part is the Lorentz scalar $(0,0)$.

In the conventions used below, the nonzero brackets involving the supercharges areand $\{Q_{\alpha },Q_{\beta }\}=\{{\bar{Q}}_{\dot{\alpha }},{\bar{Q}}_{\dot{\beta }}\}=0$. These equations, together with the Poincare algebra, are the four-dimensional $\mathcal{N}=1$ Super-Poincaré algebra without central charges.

Both terms in $B_{\mu }=W_{\mu }-\frac{1}{4}\bar{Q}{\bar{\sigma }}_{\mu }Q$ commute with momentum. For the first,because the two resulting products of momenta are symmetric in indices contracted with the Levi-Civita symbol. For the second, $[P_{\rho },Q]=[P_{\rho },\bar{Q}]=0$. Therefore

Using part i and $[P_{\mu },P_{\rho }]=0$,Thus

Move $Q_{\alpha }$ through the two odd supercharges and use $\{Q_{\alpha },Q_{\beta }\}=0$:Consequently,This unsimplified form keeps all signs transparent; changing the placement convention for the conjugate supercharge changes the displayed intermediate sign consistently with the definition of $B_{\mu }$.

The Pauli-Lubanski pseudovector and the spinor Lorentz transformation giveCombining this with part iii and the Pauli matrix identity ${\sigma }^{\nu }{\bar{\sigma }}_{\mu }={\eta }^{\nu }{}_{\mu }1+2i{\sigma }^{\nu }{}_{\mu }$ cancels the term containing ${\sigma }_{\mu \nu }{P}^{\nu }$. In these conventions,Simultaneously reversing the convention for $W_{\mu }$ reverses the final sign, but the proportionality to $P_{\mu }{Q}_{\alpha }$ is invariant and is what the next part needs.

Since momentum commutes with the supercharge, part iv givesThereforeHermitian conjugation also gives $[C_{\mu \nu },{\bar{Q}}_{\dot{\alpha }}]=0$.

The antisymmetric $C_{\mu \nu }$ transforms as a Lorentz tensor, so its complete contractionis a Lorentz scalar. HenceParts ii and v give $[P_{\rho },{\tilde{C}}_2]=[Q_{\alpha },{\tilde{C}}_2]=[{\bar{Q}}_{\dot{\alpha }},{\tilde{C}}_2]=0$. It therefore commutes with every generator of the Super-Poincaré algebra and is a Casimir element, called the superspin Casimir.