Muon decays stand as highly sensitive probes in the ongoing quest to find physics beyond the Standard Model (BSM). As generic BSM models do not necessarily conserve flavour, charged-lepton-flavour-violating decay modes of anti-muon, such as the positron-photon and the positron-electron-positron channels, play a dominant role in the search for indirect evidence induced by virtual effects of heavy New Physics (NP). The current best limits of these processes will be improved further with upcoming new results from the MEG-II and Mu3e experiments. An alternative scenario where NP triggers mu-e transitions is that of a low-mass axion-like particle (ALP) with non-zero flavour-violating couplings to leptons. Stringent limits have been obtained by the SINDRUM collaboration for a promptly decaying ALP, and it has been shown that in the nearly-massless regime, where the ALP escapes undetected, it proves advantageous to consider highly polarised anti-muons and search in the forward directions of the positron distribution, where the standard background all but vanishes. A new state-of-the-art computation of the positron energy spectrum in the SM polarised muon decay is presented and we ascertain the impact of a reduced theory error on the sensitivity to ALP signals at MEG-II, Mu3e, and a hypothetical forward detector.
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