Flavour anomalies and the muon $g-2$ from feebly interacting particles

We perform a phenomenological analysis of simplified models of light, feebly interacting particles (FIPs) that can provide a combined explanation of the anomalies in $bto s l^+ l ^-$ transitions at LHCb and the anomalous magnetic moment of the muon. Different scenarios are categorised according to the explicit momentum dependence of the FIP coupling to the $b-s$ and $mu-mu$ vector currents and they are subject to several constraints from flavour and precision physics. We show that a phenomenologically viable combined solution to the muon $g-2$ and flavour anomalies always exists if a vector with mass larger than $4 ,textrm{GeV}$ is exchanged. Interestingly, the LHC has the potential to probe this region of the parameter space by increasing the precision of the $Zto 4mu$ cross-section measurement. Conversely, we find that solutions based on the exchange of a lighter vector, in the $m_V < 1,textrm{GeV}$ range, are essentially excluded by a combination of $Bto K +textrm{invisible}$ and $W$-decay precision bounds.