Next-to-Leading-Order QCD Matching for $\mathrm{\Delta }F=2$ Processes in Scalar Leptoquark Models

Leptoquarks provide viable solutions to the flavour anomalies, i.e. they can explain the tensions between the measurements and the Standard Model predictions of the anomalous magnetic moment of the muon as well as $b\to s\ell^+\ell^-$and$b\to c\tau\nu$ processes. However, leptoquarks also contribute to other flavour observables, such as $\mathrm{\Delta }F=2$ processes, at the loop-level. In particular, $B_s-{\bar{B}}_s$ mixing provides a crucial bound in setups addressing $b\to c\tau \nu$ data, often excluding a big portion of the parameter space that could otherwise account for it. In this article, we first derive the complete leading order matching, including all five scalar LQ representations, for $D^0-{\bar{D}}^0$, $K^0-{\bar{K}}^0$ and $B_{s,d}-{\bar{B}}_{s,d}$ mixing (at the dimension-six level). We then calculate the next-to-leading order ${\alpha }_s$ matching corrections to these $\mathrm{\Delta }F=2$ processes in generic scalar leptoquark models. We find that the two-loop corrections increase the effects in $\mathrm{\Delta }F=2$ processes by $\sim 5$-$10\mathrm{\%}$ and significantly reduce the matching scale uncertainty.