Dispersive and Absorptive CP Violation in $D^0-\overline{D^0}$ Mixing

CP violation (CPV) in $D^0-\overline{D^0}$ mixing is described in terms of the dispersive and absorptive `weak phases’ ${\varphi }_f^M$ and ${\varphi }_f^{\mathrm{\Gamma }}$. They parametrize CPV originating from the interference of $D^0$ decays with and without dispersive mixing, and with and without absorptive mixing, respectively, for CP conjugate hadronic final states $f$, $\bar{f}$. These are distinct and separately measurable effects. For CP eigenstate final states, indirect CPV only depends on ${\varphi }_f^M$ (dispersive CPV), whereas ${\varphi }_f^{\mathrm{\Gamma }}$ (absorptive CPV) can only be probed with non-CP eigenstate final states. Measurements of the final state dependent phases ${\varphi }_f^M$, ${\varphi }_f^{\mathrm{\Gamma }}$ determine the intrinsic dispersive and absorptive mixing phases ${\varphi }_2^M$ and ${\varphi }_2^{\mathrm{\Gamma }}$. The latter are the arguments of the dispersive and absorptive mixing amplitudes $M_{12}$ and ${\mathrm{\Gamma }}_{12}$, relative to their dominant ($\mathrm{\Delta }U=2$) $U$-spin components. The intrinsic phases are experimentally accessible due to approximate universality: in the SM, and in extensions with negligible new CPV phases in Cabibbo favored/doubly Cabibbo suppressed (CF/DCS) decays, the deviation of ${\varphi }_f^{M,\mathrm{\Gamma }}$ from ${\varphi }_2^{M,\mathrm{\Gamma }}$ is negligible in CF/DCS decays $D^0\to K^{\pm }X$, and below $10\mathrm{\%}$ in CF/DCS decays $D^0\to K_{S,L}X$ (up to precisely known $O({ϵ}_K)$ corrections). In Singly Cabibbo Suppressed (SCS) decays, QCD pollution enters at $O(ϵ)$ in $U$-spin breaking and can be significant, but is $O({ϵ}^2)$ in the average over $f=K^{+}{K}^{-}$, ${\pi }^{+}{\pi }^{-}$. SM estimates yield ${\varphi }_2^M,{\varphi }_2^{\mathrm{\Gamma }}=O(0.2\mathrm{\%})$. A fit to current data allows $O(10)$ larger phases at $2\sigma$, from new physics. A fit based on naively extrapolated experimental precision suggests that sensitivity to ${\varphi }_2^M$ and ${\varphi }_2^{\mathrm{\Gamma }}$ in the SM may be achieved at the LHCb Phase II upgrade.