A new method to measure the time-integrated $CP$ asymmetry in singly-Cabibbo suppressed $D^0$ $\rightarrow$ $K^+K^-$ and $D$$^{0}$ $\rightarrow$$\pi^+\pi^-$ decays at LHCb.

A new method to measure time integrated ${C\!P}$ asymmetry of $D^0$ singly Cabibbo-suppressed decays to the ${C\!P}$-even final states $K^+K^-$ and $\pi^+\pi^-$ is developed. The method is applied to decays produced in $pp$ collision and collected by the LHCb experiment between 2016 and 2018 at the center-of-mass energy of 13$\,$TeV and corresponding to an integrated luminosity of 5.7$\,$fb$^{-1}$. The strong-interaction decay $D^{*+}\rightarrow D^0\pi^+$ is used to infer the flavour of the $D^0$ meson at production. The measured asymmetries are expressed as the difference between the raw asymmetries of the decays of interest and the raw asymmetry of the $D^0\rightarrow K^0_S\pi^+\pi^-$ calibration channel. The final results are: $$ \Delta A_{C\!P}(K^+K^-) = A_{C\!P}(K^+K^-)−A_{C\!P}(K^0_S\pi^+\pi^-)=0.194 \pm 0.049\%$$ $$ \Delta A_{C\!P}(\pi^+\pi^-) = A_{C\!P}(\pi^+\pi^-)−A_{C\!P}(K^0_S\pi^+\pi^-)=0.402 \pm 0.057\%$$ where the uncertainties are statistical and the central values are blinded. The final uncertainty improves the world average statistical uncertainty, currently dominated by LHCb, by a factor of about 2.5.