High-precision muon decay predictions for ALP searches

We present an improved theoretical prediction of the positron energy spectrum for polarised Michel decay $\mu^+\to e^+ \nu_e\bar{\nu}_\mu$. In addition to full next-to-next-to-leading order correction $\alpha^2$ in electromagnetic coupling, we include logarithmically enhanced terms at even higher orders. Logarithms due collinear emission are included next-to-leading accuracy up $\alpha^4$. At endpoint spectrum, soft photon results large logarithms that resummed logarithmic accuracy. apply our context MEG II and Mu3e experiments estimate impact theory error on branching ratio sensitivity lepton-flavour-violating X$ a muon into axion-like particle $X$.