Generic Loop Effects of New Scalars and Fermions in $b\to s\ell^+\ell^-$ and a Vector-like $4^{\rm th}$ Generation

In this article we investigate in detail the possibility of accounting for the $b\to s\ell^+\ell^-$ anomalies via box contributions involving with new scalars and fermions. For this purpose, we first write down the most general Lagrangian which can generate the desired effects and then calculate the generic expressions for all relevant $b\to s$ Wilson coefficients. Here we extend previous analysis by allowing that the new particles can also couple to right-handed Standard Model (SM) fermions as preferred by recent $b\to s\ell^+\ell^-$ data and the anomalous magnetic moment of the muon. In the second part of this article we illustrate this generic approach for a UV complete model in which we supplement the Standard Model by a $4^{\rm th}$ generation of vector-like fermions and a real scalar field. This model allows one to coherently address the observed anomalies in $b\to s\ell^+\ell^-$ transitions and in $a_\mu$ without violating the bounds from other observables (in particular $B_s -\bar B_s$ mixing) or LHC searches. In fact, we find that our global fit to this model, after the recent experimental updates, is very good and prefers couplings to right-handed SM fermions, showing the importance of our generic setup and calculation performed in the first part of the article.