Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 48 / 2

A four-dimensional $\mathcal{N}=1$ chiral superfield is a complex scalar superfield satisfyingThe supersymmetric covariant derivatives anticommute with the supercharges, so the constraint is preserved by supersymmetry. Use left Grassmann derivatives and the printed convention ${\bar{D}}_{\dot{\alpha }}=-{\bar{\partial }}_{\dot{\alpha }}-i{\theta }^{\beta }{\sigma }_{\beta \dot{\alpha }}^{\mu }{\partial }_{\mu }$. The conjugate antichiral superfield satisfies $D_{\alpha }{\Phi }^{†}=0$.

Any nonsingular holomorphic function $f({\Phi }_1,\dots ,{\Phi }_n)$ of chiral superfields is itself a chiral superfield. The graded Leibniz rule and the chain rule giveThus sums, products, convergent power series and inverses on patches where the denominator is nonzero preserve chirality. An ordinary function involving ${\Phi }^{†}$ is generally not chiral, because $\bar{D}{\Phi }^{†}$ need not vanish. This is why a superpotential is holomorphic. Spacetime derivatives also commute with $\bar{D}$, so derivatives of chiral superfields remain chiral; chirality by itself does not impose renormalizability.

With left Grassmann derivatives, differentiating $\theta {\sigma }^{\mu }\bar{\theta }$ with respect to $\bar{\theta }$ introduces a minus sign. ConsequentlyAt fixed $y$, the chirality condition becomes independence of $\bar{\theta }$. There are only two independent Grassmann variables ${\theta }^{\alpha }$, so the chiral-superfield component expansion terminates:Here $\varphi$ is a complex scalar field, $\psi$ a Weyl spinor, and $F$ a complex auxiliary field. Its four real off-shell bosonic components, two in $\varphi$ and two in $F$, match the four real off-shell fermionic components. Equivalently, in ordinary coordinates the complete expansion is fixed without ambiguous contraction signs byThe exponential terminates because of the Grassmann algebra.

For ungauged chiral superfields, the two-derivative supersymmetric action is a real full-superspace integral plus a holomorphic half-superspace integral:The real Kähler potential $K$ determines the Kähler metric and kinetic terms; the superpotential $W$ determines Yukawa interactions and the F-term scalar potential. For canonical normalization, $K={\sum }_i{\Phi }_i^{†}{\Phi }_i$. Berezin integration is normalized to extract the highest component, $\int d^2\theta {\theta }^2=1$, and the conjugate measure is chosen so the canonical term is $F_i^{\ast }{F}_i$. The D-term and F-term highest components vary by spacetime total derivatives, making the action invariant under supersymmetry. Gauging would replace the canonical bilinear by its gauge-covariant version and add a gauge kinetic F-term; no gauge multiplet is needed for the chiral theory here.

In four spacetime dimensions, canonical mass dimensions areSince a Lagrangian has mass dimension four, a power-counting renormalizable quantum field theory requires $[K]=2$ and $[W]=3$, with couplings of nonnegative mass dimension. Up to a positive constant kinetic matrix, field redefinitions and Kähler transformations, the renormalizable ungauged kinetic term is canonical. Holomorphic additions to $K$ integrate to a boundary term. The general renormalizable superpotential isThe coefficients have dimensions $3,2,1,0$, respectively; $m_{ij}$ and $y_{ijk}$ may be taken symmetric. The constant $W_0$ does not affect a global chiral theory. Higher polynomial degree in $W$, nontrivial higher-degree terms in $K$, or higher-derivative interactions are nonrenormalizable interactions and can instead be retained in an effective field theory.

For canonical $K$, extract the ${\theta }^2{\bar{\theta }}^2$ component of ${\Phi }_i^{†}{\Phi }_i$. The part depending on the auxiliary fields is $F_i^{\ast }{F}_i$. To extract the F-term of the superpotential, expand around the complex scalar fields:All higher terms contain at least three identical-type Grassmann variables and vanish. In the convention $(\theta {\psi }_i)(\theta {\psi }_j)=-\frac{1}{2}{\theta }^2{\psi }_i{\psi }_j$, the highest component is $W_i{F}_i-\frac{1}{2}{W}_{ij}{\psi }_i{\psi }_j$. ThereforeHere $W_i=\partial W/\partial {\varphi }_i$ and $W_{ij}={\partial }_i{\partial }_{j}W$. The Yukawa interaction is displayed to distinguish it from the terms actually containing $F$. With signature $(+---)$, the other canonical terms are ${\partial }_{\mu }{\varphi }_i^{\ast }{\partial }^{\mu }{\varphi }_i+i{\bar{\psi }}_i{\bar{\sigma }}^{\mu }{\partial }_{\mu }{\psi }_i$, up to a total derivative.

Vary $F_i$ and $F_i^{\ast }$ independently. The auxiliary fields have algebraic, rather than differential, Euler-Lagrange equations:Alternatively complete the square:Substitution therefore yields the on-shell auxiliary fields and nonnegative scalar potentialThe sign in the Lagrangian is minus the F-term scalar potential; changing the metric convention does not change this algebraic result. A supersymmetric vacuum requires all $W_i=0$, namely F-flatness. For a noncanonical positive Kähler metric $K_{i\bar{j}}$, the purely bosonic elimination instead gives $V_F=K^{i\bar{j}}{W}_i\overline{W_j}$; fermionic connection terms must also be included in a complete nonlinear component action.