Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 307 / 1

In the chiral coordinate $y^{\mu }=x^{\mu }+i\bar{\theta }{\bar{\sigma }}^{\mu }\theta$, a chiral superfield isHere $\varphi$ is a complex scalar, ${\psi }_{\alpha }$ is a two-component Weyl spinor, and $F$ is a complex auxiliary field. Off shell, $\varphi$ and $F$ contribute two real bosonic components each, while $\psi$ has two complex Grassmann components. Thus there are four real bosonic and four real fermionic degrees of freedom.

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The eight Gell-Mann matrices generate $G=SU(3)$, and the three superfields form one fundamental triplet. A renormalizable superpotential may have degree at most three. There is no invariant vector or symmetric rank-two tensor in the fundamental. The only apparent cubic singlet isbut chiral superfields are Grassmann-even and commute, so this contraction of a symmetric product with ${ϵ}^{ijk}$ vanishes identically. Consequently the most general invariant superpotential as the question is written is only a constant,which has no effect in global supersymmetry. In particular, $W=\lambda {\Phi }_1{\Phi }_2{\Phi }_3$ is not invariant under a general $SU(3)$ rotation.

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The canonical Kähler potential isUnder $\Phi \mapsto U\Phi$ with $U\in SU(3)$, it becomes ${\Phi }^{†}{U}^{†}U\Phi =K$, so it respects $G$.

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The F-term scalar potential isSince the invariant superpotential is constant, all derivatives vanish andThere are no gauge fields in the stated model, so there is no D-term potential.

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Soft supersymmetry breaking means adding supersymmetry-breaking operators with positive-mass-dimension coefficients that do not create new ultraviolet quadratic divergences. Standard examples are scalar masses, holomorphic bilinear and trilinear scalar couplings, gaugino masses, and singlet tadpoles when the field content and symmetries allow them.

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For one $SU(3)$ triplet and no gauge multiplet, the only nonzero $G$-invariant soft term is the universal scalar mass,An invariant bilinear does not exist, and the formal trilinear $A{ϵ}^{ijk}{\varphi }_i{\varphi }_j{\varphi }_k$ vanishes because the scalar fields commute. Thus the phrase “general soft terms” does not generate further interactions for the field content stated in the question.

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There are no scalar interaction vertices in the theory as written. The canonical Kähler term supplies kinetic terms, the constant superpotential supplies none, and the soft term is only a two-point mass insertion with Feynman rule $-i{m}_s^2$ in Minkowski conventions. Hence no interaction diagram involving only scalar components exists.

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The Yukawa matrix is $W_{ij}={\partial }_i{\partial }_{j}W$, which vanishes for constant $W$. Therefore there is no scalar-fermion interaction vertex and no corresponding Feynman diagram. The requests in parts (vi)(a) and (vi)(b) would become nontrivial only after changing the field content or reducing the symmetry so that a nonzero invariant superpotential exists.

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The three Weyl components form one fundamental representation of the global $SU(3)$, so coupling $G$ to background gauge fields reveals a nonzero $SU(3{)}^3$ 't Hooft anomaly. This does not destroy the global symmetry in flat space or make the theory inconsistent; it constrains any infrared description through 't Hooft anomaly matching. If $G$ were instead intended as a dynamical gauge symmetry, the same uncancelled gauge anomaly would make the model inconsistent and additional chiral matter would be required. Ordinary quantum corrections cannot generate the forbidden superpotential, and the non-renormalization theorem protects its absence.

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