Unlocking time-dependent $CP$ violation without signal vertexing at $B$ factories

We present a method to measure time-dependent $CP$ violation in $B^0$ decays produced in $\Upsilon(4S) \to B^0\overline{B}^0$ events at $B$ factories without reconstructing the signal decay vertex. The method exploits the sensitivity of the tag $\overline{B}^0$ decay time to $CP$ violation in the $B^0$ signal. It relies on a compact $e^+e^-$ interaction region and excellent vertex resolution, which enable precise measurement of the displacement between production and decay vertices of the tag meson. We study an application to $B^0\to\pi^0\pi^0$ decays using a simplified simulation that approximates the conditions of the Belle II experiment at the SuperKEKB collider. Our approach achieves a sensitivity on mixing-induced $CP$ violation that would require 20 times larger samples with standard approaches. When incorporated into the $B \to \pi\pi$ isospin analysis, the expected results would reduce the degeneracy of $\phi_2$ solutions and significantly increase precision. This work opens a path to previously inaccessible $CP$-violation studies enhancing and accelerating the reach of Belle II and future flavor physics programs.