Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 307 / 3

With canonical Kähler potential, the Wess–Zumino model action isA spurion is a nondynamical background superfield assigned charges so that a coupling transforms covariantly and the action can be treated as formally symmetric.

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Take an ordinary $U(1)$ and an $R$-symmetry, assigning $q(\Phi )=R(\Phi )=1$. The superpotential has chargesbecause $d^2\theta$ is neutral under the first symmetry and has $R$-charge $-2$. Formal invariance of each term then assigns

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The ratiois neutral under both $U(1)$ symmetries. The unique monomial with mass dimension three and charges $(0,2)$ isHolomorphy and the spurion symmetries therefore requireExpanding a term givesRegularity as $g\to 0$ requires $n\ge 2$, while regularity as $m\to 0$ requires $n\le 3$. Hence only $n=2,3$ can occur.

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Weak-coupling matching givesbecauseThus $W_{\mathrm{eff}}=W_{\mathrm{tree}}$. The holomorphic superpotential receives neither perturbative nor nonperturbative corrections compatible with the limits and symmetries. The Kähler potential and wavefunction normalization can still be renormalized, so canonically normalized couplings can run through anomalous dimensions.

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Eliminating the auxiliary field givesExplicitly,The two supersymmetric vacua for $g\ne 0$ are $\varphi =0$ and $\varphi =-m/g$.

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The component interactions include the Yukawa vertexand the quartic scalar vertex $-|g{|}^2|\varphi {|}^4$, with cubic scalar interactions fixed by the same $m$ and $g$. At one loop, the scalar propagator has a bosonic tadpole from the quartic vertex and a closed fermion loop with two Yukawa vertices. The loop's fermionic minus sign makes their quadratic cutoff dependence equal and opposite; diagrams involving the dimensionful cubic scalar vertex are at most logarithmically divergent. This is the supersymmetric cancellation of quadratic divergences, and it relies on the coupling relations imposed by supersymmetry.

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