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

In the precision era, CP violation in $D-\bar D$ mixing is ideally described in terms of the dispersive and absorptive phases $\phi_f^M$ and $\phi_f^\Gamma$, parametrizing CP violation (CPV) in the interference of $D^0$ decays with and without dispersive (absorptive) mixing. These are distinct and separately measurable effects. This formalism is applied to (i) Cabibbo favored/doubly Cabibbo suppressed (CF/DCS) decays $D^0 \to K^\pm X$; (ii) CF/DCS decays $D^0 \to K_{S,L} X$, including the impact of $\epsilon_K$, and (iii) singly Cabibbo suppressed (SCS) decays. Expressions for the time-dependent CP asymmetries simplify: Indirect CPV only depends on $\phi_f^M$ (dispersive CPV), whereas $\phi_f^\Gamma$ (absorptive CPV) can only be probed with non-CP eigenstate final states. Measurements of the final state dependent phases $\phi_f^M$, $\phi_f^\Gamma$ determine the phases $\phi_2^M$ and $\phi_2^\Gamma$, which are the arguments of the dispersive and absorptive mixing amplitudes $M_{12}$ and $\Gamma_{12}$, relative to their dominant ($\Delta U=2$) $U$-spin components. $\phi_2^M$ and $\phi_2^\Gamma$ are experimentally accessible due to approximate universality: in the SM, $\phi_f^M-\phi_2^M$ and $\phi_f^\Gamma-\phi_2^\Gamma$ are negligible in case (i) above; and below $10\% $ in (ii), up to precisely known $O(\epsilon_K )$ corrections. In case (iii), the pollution enters at $O(\epsilon)$ in $U$-spin breaking and can be significant, but is $O(\epsilon^2)$ in the average over $f=K^+K^-$, $\pi^+\pi^-$. U-spin based estimates yield $\phi_2^M, \phi_2^\Gamma = O(0.2\%)$ in the SM. The current fit to the data thus implies an $O(10)$ window for new physics at $2\sigma$. A fit based on naively extrapolated experimental precision at the LHCb Phase II upgrade suggests that sensitivity to $\phi_2^{M,\Gamma}$ in the SM may be achievable in the precision era.