Standard Model Benchmarks for $D^0\to K^- K^+, \pi^-\pi^+, K^0_{\rm S} K^0_{\rm S}$ Decays

The non-leptonic $D^0\to K^- K^+$ and $D^0\to \pi^-\pi^+$ decays are powerful probes of the Standard Model and are related to each other through the $U$-spin symmetry of the strong interaction. Using lattice QCD inputs we calculate the corresponding colour-allowed tree amplitudes in factorisation and demonstrate that non-factorisable contributions and $U$-spin-breaking effects at the level of 50% allow us to accommodate the measured branching ratios in the Standard Model. An exciting direct probe of such non-factorisable and $U$-spin breaking effects is provided by the $D^0\to K^0_{\rm S} K^0_{\rm S}$ channel. This decay is governed by non-factorisable exchange topologies and essentially vanishes in the $U$-spin limit, although it is experimentally well established with a prominent branching ratio. Extrapolating our $D^0\to K^- K^+$ results using the isospin symmetry, we find a consistent benchmark picture. Specifically, we can accommodate the measured $D^0\to K^0_{\rm S} K^0_{\rm S}$ branching ratio with $U$-spin-breaking effects at the 50% level and exchange amplitudes at the level of 50% of the colour-allowed $D^0\to K^- K^+$, $D^0\to \pi^-\pi^+$ tree contributions. Finally, we explore the resulting range for direct CP violation in $D^0\to K^0_{\rm S} K^0_{\rm S}$, obtaining upper bounds in our benchmark scenarios of a few per mille, offering an exciting target for future measurements.