Dispersive analysis of neutral meson mixing

Abstract

We analyze the neutral meson mixing by directly solving the dispersion relation obeyed by the mass and width differences of the two meson mass eigenstates. We solve for the parameters $x$ and $y$, proportional to the mass and width differences in the charm mixing, respectively, taking the box-diagram contributions to $x(s)$ and $y(s)$ at large mass squared $s$ of a fictitious $D$ meson as inputs. The SU(3) symmetry breaking is introduced through physical thresholds of different $D$ meson decay channels for $y(s)$. These threshold-dependent effects, acting like nonperturbative power corrections in QCD sum rules, stabilize the solutions of $y(s=m_D^2)$ with the $D$ meson mass $m_D$. We then calculate $x(s)$ through the dispersive integration of $y(s)$, and show that our predictions $x(m_D^2)\approx 0.21\%$ and $y(m_D^2)\approx 0.52\%$ are close to the data in both $CP$-conserving and $CP$-violating cases. It is observed that the channel containing di-kaon states provides the major source of SU(3) breaking, which enhances $x(m_D^2)$ and $y(m_D^2)$ by four orders of magnitude relative to the perturbative results. We also predict the coefficient ratio $q/p$ involved in the charm mixing with $|q/p|-1\approx 2\times 10^{-4}$ and $Arg(q/p)\approx 6\times 10^{-3}$ degrees, which can be scrutinized by precise future measurements. The formalism is extended to studies of the $B_{s(d)}$ meson mixing and the kaon mixing, and the small deviations of the obtained width differences from the perturbative inputs explain why the above mixing can be understood via short-distance dynamics. We claim that the puzzling charm mixing is attributed to the strong Glashow-Iliopoulos-Maiani suppression on perturbative contributions, instead of to breakdown of the quark-hadron duality, which occurs only at 15\% level.