Diagonalising the LEFT

TL;DR

The paper presents a symmetry-based decomposition of LEFT four-fermion vector current operators into definite flavour and parity irreps, yielding small, non-mixing blocks that render the RG evolution semi-analytic. Focusing on the down-type octet sector, it demonstrates a semi-analytic Magnus-expansion approach to solve the RG and examines one- and two-loop effects, including the protection of certain dd eigen-directions by full flavour symmetry. The analysis shows that lepton-flavour non-universality is radiatively unstable in a way that generically induces lepton-flavour universal effects, placing indirect constraints on tauonic new physics and four-quark SMEFT operators through RG running from the electroweak scale to low energies. The framework provides a coherent, model-independent method to connect high-scale flavour structures to low-energy observables such as $b\to s\tau\tau$, with implications for upcoming Belle II, LHCb, and FCC-ee probes. Overall, the work demonstrates how non-Abelian flavour symmetries can drastically simplify LEFT operator RGs and illuminate the interplay between LFUV, tauonic NP, and hadronic flavour processes.

Abstract

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 $b\to s ττ$ decays, as well as setting indirect bounds on flavour changing four-quark interactions.

Figures (14)

• Figure 1: The mapping of the components (labelled in red) of the irreps $\mathbf{1}$, $\mathbf{3}$, $\mathbf{8}$, $\mathbf{10}$, and $\mathbf{27}$ onto their flavour charges --- the generational analogues of isospin $\mathcal{I}$, the third component of isospin $\mathcal{I}_3$, and the hypercharge $\mathcal{Y}$ --- in the conventions of deSwart:1963pdg. 'Isospin' ($\mathcal{I}$) key: $\,=0$, $\,=\frac{1}{2}$, $\,=1$, $\,=\frac{3}{2}$, $\,=2$. The dashed hexagon intersects components that contribute to $\Delta F = 1$ processes, the dot-dashed hexagon intersects those that contribute to $\Delta F = 2$ processes. (Note that the conventions of Aebischer:2025hsx distinguish the points on the vertices of the dot-dashed hexagon from those in the middle of the edges, respectively labelling them $\Delta F = 2$ and $\Delta F = 1.5$.) The diagrams for $\overline{\mathbf{3}}$ and $\overline{\mathbf{10}}$ are obtained by inverting the diagrams for the corresponding unconjugated irrep about the origin.
• Figure 2: LEFT dimension five and six operator coefficients, arranged by the total helicity $(\sum h)$ and the number of legs ($n$) of the tree level amplitudes they generate. Conjugation reflects the graph in the $n$ axis.
• Figure 3: Four-point amplitudes built from dipole operators (drawn as black circles) and SM gauge vertices. The helicities of (ingoing) legs are shown as $+$ or $-$, and the number of legs and total helicity of the amplitudes are given in $(n,\sum h)$ form below the amplitudes.
• Figure 4: The ADM of the charged four-fermion vector operators in the basis of definite parity and flavour charges. Shaded blocks indicate non-zero entries. Diagonal blocks are ordered by decreasing dimensions to match \ref{['tab:large_blocks_table']} followed by \ref{['tab:small_blocks_table']}.
• Figure 5: The eigenvectors and eigenvalues $d_i$, defined in \ref{['eq:effectiveAnomalousDims']}, of the parity-even (top) and parity-odd (bottom) $\mathbf{d}_{d} = 8, \mathbf{d}_{u,e}=1$ blocks. Normalised eigenvectors, in terms of their components in the Wilson coefficients listed at the side, are arranged vertically with their corresponding eigenvalues at the top.