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Two categories of four-fermion SMEFT operators are semileptonic (two quarks and two leptons) and hadronic (four quarks). At tree level, an operator of a given category contributes only to processes of the same category. However, when the SMEFT Hamiltonian is evolved down from the new-physics scale to low energies using the renormalization-group equations (RGEs), due to operator mixing this same SMEFT operator can generate operators of the other category at one loop. Thus, to search for a SMEFT explanation of a low-energy anomaly, or combination of anomalies, one must: (i) identify the candidate semileptonic and hadronic SMEFT operators, (ii) run them down to low energy with the RGEs, (iii) generate the required low-energy operators with the correct Wilson coefficients, and (iv) check that all other constraints are satisfied. In this paper, we illustrate this method by finding all SMEFT operators that, by themselves, provide a combined explanation of the (semileptonic)$$ \overline{b}\to \overline{s}{\ell}^{+}{\ell}^{-} $$ $\overline{b}\to \overline{s}{\ell }^{+}{\ell }^{-}$anomalies and the (hadronic)B → πKpuzzle. 

In this Letter, we perform fits to $B\to PP$decays, where $B=\{B^0,B^{+},B_s^0\}$and the pseudoscalar $P=\{\pi ,K\}$, under the assumption of flavor SU(3) symmetry [ ${\mathrm{SU}\left(3\right)}_F$]. Although the fits to $\mathrm{\Delta }S=0$or $\mathrm{\Delta }S=1$decays individually are good, the combined fit is very poor: there is a $3.6\sigma$disagreement with the ${\mathrm{SU}\left(3\right)}_F$limit of the standard model ( ${\mathrm{SM}}_{{\mathrm{SU}\left(3\right)}_{\mathrm{F}}}$). One can remove this discrepancy by adding ${\mathrm{SU}\left(3\right)}_F$-breaking effects, but 1000% ${\mathrm{SU}\left(3\right)}_F$breaking is required. The above results are rigorous, group theoretically—no dynamical assumptions have been made. When one adds an assumption motivated by QCD factorization, the discrepancy with the ${\mathrm{SM}}_{{\mathrm{SU}\left(3\right)}_{\mathrm{F}}}$grows to $4.4\sigma$. Published by the American Physical Society2024