Past exam of the groups and representations course of the University of Oxford / 2021 / 2 / d / Solution

First choose the same orthonormal basis for the left and right copies of the two-dimensional unitary representation $D$. Then the family representation is $F=D\oplus \epsilon$. Because the barred fermion transforms with $F(g{)}^{†}$ and the scalar field $H$ is a trivial representation, invariance of the Yukawa interaction requiresWrite $\lambda$ in a $2+1$ block form. The off-diagonal blocks would be intertwiners between inequivalent irreducible representations $D$ and $\epsilon$, so the Schur lemma makes them zero. The doublet block must be a scalar matrix, while the singlet block is arbitrary. ThereforeEquivalently, the allowed Yukawa interaction is $H[a({\bar{\psi }}_L^1{\psi }_R^1+{\bar{\psi }}_L^2{\psi }_R^2)+b{\bar{\psi }}_L^3{\psi }_R^3]$, together with its Hermitian conjugate when constructing a real Lagrangian. The tensor product rules explain the two free parameters: both $D^{\ast }\otimes D$ and ${\epsilon }^{\ast }\otimes \epsilon$ contain one trivial representation, while the mixed products contain none.

There is a useful basis qualification. The basis used in solution is not orthonormal: its Gram matrix is $K=\left(\begin{array}{cc}2& 1\\ 1& 2\end{array}\right)$, and $D(g{)}^{†}KD(g)=K$. If the fields are expressed in exactly those coordinates, with the barred components defined by ordinary conjugate transpose, the invariant doublet coefficient is $aK$, rather than $a{I}_2$. Canonically normalizing the kinetic term returns the boxed form. Independently chosen left and right orthonormal bases replace $a{I}_2$ by $aS$, where $S$ is their fixed unitary intertwiner; the physical degeneracy is unchanged.