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' ${\varphi }_f^M$ and ${\varphi }_f^{\Gamma }$. 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 ${\varphi }_f^M$ (dispersive CPV), whereas ${\varphi }_f^{\Gamma }$ (absorptive CPV) can only be probed with non-CP eigenstate final states. Measurements of the final state dependent phases ${\varphi }_f^M$, ${\varphi }_f^{\Gamma }$ determine the intrinsic dispersive and absorptive mixing phases ${\varphi }_2^M$ and ${\varphi }_2^{\Gamma }$. 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 ${\varphi }_f^{M,\Gamma }$ from ${\varphi }_2^{M,\Gamma }$ 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 $O({ϵ}_K)$ 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 ${\varphi }_2^M,{\varphi }_2^{\Gamma }=O(0.2\%)$. 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 ${\varphi }_2^M$ and ${\varphi }_2^{\Gamma }$ in the SM may be achieved at the LHCb Phase II upgrade.

Comments: 28 pages, 3 figures. v2: Modified the discussion of SM estimates for ${\varphi }_2^M$, ${\varphi }_2^{\Gamma }$ in Section IVB, cf. Eqs 110-118, includes a new upper bound on ${\varphi }_2^{\Gamma }$; typos corrected in Eqs. 73, 77, 151, 152; added Eqs 155, 156, modified Eq A6, writing clarified in a few places. v3: version to appear in Phys Rev D. Intro shortened, improved Sec. III.B.3, typos in the Appendix corrected