Effect of $D^0$-$\bar{D}{}^0$ Mixing upon Cabibbo-favored $D^0$ Decays

The parameter $y_{CP}$ is used to characterize mixing in the $D^0$-$\bar{D}{}^0$ meson system. To determine $y_{CP}$, one measures the effective decay width of a $D^0$ or $\bar{D}{}^0$ decaying to a $CP$ eigenstate, relative to the effective width for a decay to a Cabibbo-favored, flavor-specific final state. When using $y_{CP}$ to extract information about $D^0$-$\bar{D}{}^0$ mixing and $CP$ violation, the decay width of the Cabibbo-favored decay is usually assumed to equal $1\mathrm{/}\tau$, the reciprocal of the $D^0$ lifetime. However, there is a small correction to this that should be taken into account when $y_{CP}$ is measured with sufficiently high precision. We calculate this correction in terms of charm mixing and $CP$ violation parameters $x$, $y$, $\mathrm{\mid }q\mathrm{/}p\mathrm{\mid }$, and $\phi$.