Nodal filtering in open-charm decays of the $\psi (4040)$--$\psi (4160)$ system

Abstract

Hadronic decay channels can provide direct information on the momentum-space structure of confined quark systems. We investigate this possibility in the $\psi (4040)$--$\psi (4160)$ system using an instantaneous Bethe--Salpeter (BS) framework combined with a relativistic ${}^3{P}_0$ decay model. The BS solutions yield nearby bare $3{}^3{S}_1$ and $2{}^3{D}_1$ states at 4051 and 4110 MeV, respectively, whose opposite displacements from the physical vector states motivate an effective two-state level-repulsion description. By comparing the matched open-charm channels $D\bar{D}$, $D{\bar{D}}^{\ast }+c.c.$, $D_s{\bar{D}}_s$, and $D_s{\bar{D}}_s^{\ast }+c.c.$ with the same leading $P$-wave threshold behavior, we show that channel-dependent overlap kernels act as momentum-space nodal filters. At $M=4146$ MeV, the $PV/PP$ width ratio is 0.253 in the nonstrange sector but 1.63 in the strange sector, reversing the ordering expected from phase space alone. The momentum-resolved amplitudes reveal that different channels weight opposite sides of the same $2D$ nodal region differently, leading to distinct cancellation patterns and a common recoil condition, $P_f\simeq 0.775$ GeV, for charge-resolved $D{\bar{D}}^{\ast }$ amplitude zeros. These results establish open-charm decays as a probe of momentum-space wave-function structures in heavy quarkonium.