LFV in flavourful SMEFT: Dimension-Six Running versus Dimension-Eight Mixing

We study how charged-lepton flavour violation (CLFV) generated in the $\tau$ sector propagates into the $\mu \to e$ sector within the Standard Model Effective Field Theory (SMEFT). Assuming that new physics at a scale $\mathrm{\Lambda }\gg v$ produces only dimension-6 operators that mediate $\tau \to e$ and $\tau \to \mu$ transitions, we identify and compare the three mechanisms that induce $\mu \to e$ observables. All three originate from double insertions of the two $\tau \to \mathrm{\ell }$ operators and scale as $C_6^{\tau e}{C}_6^{\tau \mu }\mathrm{/}{\mathrm{\Lambda }}^4$. (T1) Renormalization-group mixing misaligns the charged-lepton mass matrix, and the rotation to the mass basis generates dimension-6 $\mu \to e$ operators. (T2) A finite, non-logarithmic remainder, which we find to be negligible. (T3) The overall divergence renormalizes dimension-8 operators, contributing after electroweak symmetry breaking. Since T1 and T3 carry no relative power of $\mathrm{\Lambda }$, neither can be neglected at any scale, and only their comparison reveals which mechanism, and hence which $\mu \to e$ observable, most strongly probes a given operator pair. Surveying the Warsaw-basis operators, we classify representative $(\tau \to e,\tau \to \mu )$ pairs into T1-dominated, T3-dominated, and T1–T3-comparable regimes, and explain the hierarchy through the underlying mixing chains. Translating current $\mu \to e\gamma$, $\mu \to eee$ and $\mu \to e$ conversion limits into bounds on products of $\tau \to \mathrm{\ell }$ Wilson coefficients, we find that they exceed the direct $\tau$-decay constraints by factors of ten to several hundred. Excluding either T1 or T3 would miss these limits by orders of magnitude for some pairs, and shift them by factors of two to three in others. Interpreting $\mu \to e$ data as a probe of $\tau$ flavour violation therefore requires both the dimension-6 RG running and the dimension-8 mixing contributions.