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 $b\to s l^+ l ^-$ 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 viable combined solutions to the muon $g-2$ and flavour anomalies exist with the exchange of a vector FIP with mass larger than $4 \,\textrm{GeV}$. Interestingly, the LHC has the potential to probe this region of the parameter space by increasing the precision of the $Z\to 4\mu$ 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 $B\to K +\textrm{invisible}$ and $W$-decay precision bounds.