  This note presents an analysis of lepton-flavour-violating muon decays within
the framework of a low-energy effective field theory that contains
higher-dimensional operators allowed by QED and QCD symmetries. The decay modes
$\mu\to e \gamma$ and $\mu\to 3e$ are investigated below the electroweak
symmetry-breaking scale, down to energies at which such processes occur, i.e.
the muon mass scale. The complete class of dimension-5 and dimension-6
operators is studied systematically at the tree level, and one-loop
contributions to the renormalisation group equations are fully taken into
account. Current experimental limits are used to extract bounds on the Wilson
coefficients of some of the operators and, ultimately, on the effective
couplings at any energy level below the electroweak symmetry-breaking scale.
Correlations between two couplings relevant to both processes illustrate the
complementarity of searches planned for the MEG II and Mu3e experiments.
