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

After the scalar field obtains a vacuum expectation value $h$, the fermion mass matrix in a canonically normalized basis is $M=h\lambda$. Physical masses are its singular values, not arbitrary entries or signed eigenvalues. ThusThe independent Yukawa coupling $b$ can fit the tau lepton mass relative to the doublet scale, but the electron and muon are predicted to have equal masses. The values supplied by the paper instead give $m_{\mu }/m_e\simeq 105.7/0.511\simeq 207$. Consequently exact unbroken $S_3$ with this representation assignment cannot reproduce all three masses. Complex phases of $a,b$, or changes of basis, cannot split the two equal singular values. Additional breaking of the symmetry or a different field content would be needed; the discrepancy cannot be removed merely by choosing different invariant Yukawa couplings.