Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 307 / 1

The Kähler potential $K(\Phi ,{\Phi }^{†})$ is a real function integrated over all four fermionic coordinates, $\int d^4\theta K$. Its complex Hessian gives the scalar Kähler metric and therefore the kinetic terms. The superpotential $W(\Phi )$ is holomorphic and is integrated over chiral superspace, $\int d^2\theta W+h.c.$; its derivatives determine Yukawa couplings and the F-term scalar potential.

For the canonical Kähler potential $K={\Phi }^{†}\Phi$, extracting the ${\theta }^2{\bar{\theta }}^2$ component of the supplied chiral-superfield component expansion and integrating by parts gives the bosonic actionin the mostly-plus convention. Extracting the ${\theta }^2$ component of a holomorphic function givesso its bosonic term is $F{W}^{\prime }(\varphi )$, with the Hermitian conjugate understood in a real action. For several fields, eliminating each algebraic auxiliary field by $F_i^{\ast }=-W_i$ produces $V_F={\sum }_i|W_i{|}^2$.

For $W=X^{2}Z$, the F-term scalar potential isBoth terms vanish exactly when $X=0$, while $Z$ is arbitrary. Thus the supersymmetric vacua form one complex line,

For $W=XYZ$,The F-flatness equations require at least two of $X,Y,Z$ to vanish. The vacuum space is therefore the union of the three coordinate axes in ${\mathbb{C}}^3$,and every point on it has zero vacuum energy.

Forthe three F-terms areThe equation $W_Z=0$ forces $X=0$, after which $W_Y=\alpha \ne 0$; hence no supersymmetric vacuum exists. Minimizing over $Z$ sets $Z=-2\beta XY/\gamma$, leavingThe two real quadratic eigenvalues about $X=0$ are $|\gamma {|}^2\pm 2|\alpha \beta |$, so the stated hierarchy makes $X=0$ stable. The vacua areThis is an O'Raifeartaigh model: supersymmetry is spontaneously broken and $Y$ is a classical pseudomodulus.