Diagonalising the LEFT

We organise the four-fermion vector current interactions below the weak scale – i.e., in the low energy effective field theory (LEFT) – into irreps of definite parity and $SU(N)$ flavour symmetry. Their coefficients are thus arranged into small subsets with distinct phenomenology, which are significantly smaller than traditional groupings of operators by individual fermion number. As these small subsets only mix among themselves, we show that the renormalisation group evolution is soluble semi-analytically, and examine the resulting eigenvalues and eigenvectors of the one- and two-loop running. This offers phenomenological insights, for example into the radiative stability of lepton flavour non-universality. We use these to study model-independent implications for $b\to s\tau \tau$ decays, as well as setting indirect bounds on flavour changing four-quark interactions.