Nodal mechanism for the suppressed $$ D\overline{D} $$ decay of ψ(4040) in the Bethe-Salpeter framework

A bstract The strong decay ψ (4040) → $$ D\overline{D} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>D</mml:mi> <mml:mover> <mml:mi>D</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> </mml:math> is anomalously suppressed despite ample phase space, whereas the $$ D{\overline{D}}^{\ast } $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>D</mml:mi> <mml:msup> <mml:mover> <mml:mi>D</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> <mml:mo>∗</mml:mo> </mml:msup> </mml:math> and $$ {D}_s{\overline{D}}_s $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>D</mml:mi> <mml:mi>s</mml:mi> </mml:msub> <mml:msub> <mml:mover> <mml:mi>D</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> <mml:mi>s</mml:mi> </mml:msub> </mml:math> channels remain sizable. In this work, we study this suppression and the associated open-charm hierarchy in the framework of the instantaneous Bethe-Salpeter equation combined with the relativistic 3 P 0 model, with the pair-creation strength fixed independently from ψ (3770) → $$ D\overline{D} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>D</mml:mi> <mml:mover> <mml:mi>D</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> </mml:math> . Within this framework, we show that the suppressed $$ D\overline{D} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>D</mml:mi> <mml:mover> <mml:mi>D</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> </mml:math> mode can be understood as a consequence of node-induced cancellations in the relativistic decay amplitude. The $$ D\overline{D} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>D</mml:mi> <mml:mover> <mml:mi>D</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> </mml:math> amplitude is strongly reduced because the corresponding overlap integral receives comparable positive and negative contributions from different momentum regions, whereas the $$ D{\overline{D}}^{\ast } $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>D</mml:mi> <mml:msup> <mml:mover> <mml:mi>D</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> <mml:mo>∗</mml:mo> </mml:msup> </mml:math> and $$ {D}_s{\overline{D}}_s $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>D</mml:mi> <mml:mi>s</mml:mi> </mml:msub> <mml:msub> <mml:mover> <mml:mi>D</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> <mml:mi>s</mml:mi> </mml:msub> </mml:math> channels do not undergo the same strong cancellation. This interpretation is further supported by the pronounced sensitivity of the $$ D\overline{D} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>D</mml:mi> <mml:mover> <mml:mi>D</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> </mml:math> width to the initial mass, the charged-neutral D -meson mass splitting, and the dip structure in the mass dependence of the partial width. Our results provide a dynamical explanation of the suppressed $$ D\overline{D} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>D</mml:mi> <mml:mover> <mml:mi>D</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> </mml:math> mode and the core open-charm hierarchy of ψ (4040) within a conventional 3 3 S 1 charmonium picture, while the precise value of the near-vanishing $$ D\overline{D} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>D</mml:mi> <mml:mover> <mml:mi>D</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> </mml:math> width remains model dependent.