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

Abstract

CP violation (CPV) in $D^0-\overline{D^0}$ mixing is described in terms of the dispersive and absorptive `weak phases' $\phi_f^M$ and $\phi_f^\Gamma$. 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 $\phi_f^M$ (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 intrinsic dispersive and absorptive mixing phases $\phi_2^M$ and $\phi_2^\Gamma$. The latter 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. 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 $\phi_f^{M,\Gamma}$ from $\phi_2^{M,\Gamma}$ is negligible in CF/DCS decays $D^0 \to K^\pm X$, and below $10\% $ in CF/DCS decays $D^0 \to K_{S,L} X$ (up to precisely known $O(\epsilon_K)$ corrections). In Singly Cabibbo Suppressed (SCS) decays, QCD pollution enters at $O(\epsilon)$ in $U$-spin breaking and can be significant, but is $O(\epsilon^2)$ in the average over $f=K^+K^-$, $\pi^+\pi^-$. SM estimates yield $\phi_2^M, \phi_2^\Gamma = O(0.2\%)$. 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 $\phi_2^{M}$ and $\phi_2^{\Gamma}$ in the SM may be achieved at the LHCb Phase II upgrade.