First bounds on effective muon interactions using the NA64$\mu$ experiment at CERN

We analyze how NA64$\mu$ can contribute to the global SMEFT program demonstrating that it can probe two effective four lepton operators completely unbounded so far and break one of the current flat directions. Furthermore, we also study an extension of SMEFT that includes fermion singlets of the SM gauge group in the low energy field content. This effective field theory, usually dubbed $\nu$SMEFT, is well motivated by the observation of light neutrino masses and leptonic mixing. We find that NA64$\mu$ can constrain three unbounded four fermion operators of the $\nu$SMEFT. We derive the current leading bounds on these operators and compute the future sensitivity. Our results fill the gap between the current experimental program and a possible future muon collider able to probe this type of New Physics.