Connected sequential Bargmann invariants and $$\mathcal CP~$$-sensitive geometric correlation structures in neutral meson systems

Abstract We investigate connected sequential geometric structures in correlated neutral meson systems within the framework of Bargmann invariants. Building upon previously developed third- and fourth-order rephasing-invariant geometric structures involving decay-projected conditional states, we introduce a connected sequential fourth-order Bargmann invariant in which the conditional meson states associated with two decay channels are directly linked through their geometric overlap. This construction modifies the connectivity of the cyclic overlap sequence by introducing an explicit overlap between the conditional meson states prepared by different decay channels. The connected sequential invariant encodes the geometric relation between decay-projected states, thereby extending the geometric framework developed for correlated neutral meson systems. To characterize the resulting geometric structures, we define rephasing-invariant ratios that quantify connected sequential correlations and provide a direct comparison with the previously studied disconnected geometric correlations. The behavior of these quantities is analyzed in the regime of small $$\\mathcal CP~$$ C P violation using the standard rephasing-invariant interference parameters together with a small-asymmetry expansion. We show that the connected sequential ratios exhibit characteristic scaling behaviors governed by both mixing asymmetry and relative interference-phase alignment, leading to geometric scaling properties distinct from those of the disconnected structures. We further discuss the geometric interpretation of the connected sequential invariants and their relevance for correlated neutral meson systems, where mixing and decay interference are experimentally studied. The resulting framework extends the hierarchy of Bargmann invariant geometric correlations associated with neutral meson mixing and decay, providing a complementary geometric perspective on $$\\mathcal CP~$$ C P -sensitive interference phenomena.