$D$ meson mixing as an inverse problem

Abstract

We calculate the parameters $x$ and $y$ for the $D$ meson mixing in the Standard Model by considering a dispersion relation between them. The dispersion relation for a fictitious charm quark of arbitrary mass squared $s$ is turned into an inverse problem, via which the mixing parameters at low $s$ are solved with the perturbative inputs $x(s)$ and $y(s)$ from large $s$. It is shown that nontrivial solutions for $x$ and $y$ exist, whose values around the physical charm scale agree with the data in both CP-conserving and CP-violating cases. We then predict the observables $|q/p|-1\approx 2\times 10^{-4}$ and $Arg(q/p)\approx 6\times 10^{-3}$ degrees associated with the coefficient ratio for the $D$ meson mixing, which can be confronted with more precise future measurements. Our work represents the first successful quantitative attempt to explain the $D$ meson mixing parameters in the Standard Model.