Three-body final state interactions in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msup> <mml:mi>B</mml:mi> <mml:mo>+</mml:mo> </mml:msup> <mml:mo stretchy="false">→</mml:mo> <mml:mi>D</mml:mi> <mml:mover accent="t

This paper presents a detailed analysis of the three-body final state interactions in the <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:msup> <a:mi>B</a:mi> <a:mo>+</a:mo> </a:msup> <a:mo stretchy="false">→</a:mo> <a:mi>D</a:mi> <a:mover accent="true"> <a:mi>D</a:mi> <a:mo stretchy="false">¯</a:mo> </a:mover> <a:msup> <a:mi>K</a:mi> <a:mo>+</a:mo> </a:msup> </a:math> process, whose phase space is sufficient small. To precisely extract the resonance parameters, for instance the <f:math xmlns:f="http://www.w3.org/1998/Math/MathML" display="inline"> <f:msub> <f:mi>χ</f:mi> <f:mrow> <f:mi>c</f:mi> <f:mn>0</f:mn> <f:mo>/</f:mo> <f:mn>2</f:mn> </f:mrow> </f:msub> <f:mo stretchy="false">(</f:mo> <f:mn>3930</f:mn> <f:mo stretchy="false">)</f:mo> </f:math> in the <j:math xmlns:j="http://www.w3.org/1998/Math/MathML" display="inline"> <j:mi>D</j:mi> <j:mover accent="true"> <j:mi>D</j:mi> <j:mo stretchy="false">¯</j:mo> </j:mover> </j:math> invariant mass distributions, in this process, one has to take into account final state interaction, especially the three-body final state interaction. We employ the dispersive Khuri-Treiman formalism, combined with a parametrization of the <n:math xmlns:n="http://www.w3.org/1998/Math/MathML" display="inline"> <n:mi>D</n:mi> <n:mover accent="true"> <n:mi>D</n:mi> <n:mo stretchy="false">¯</n:mo> </n:mover> </n:math> interaction based on heavy quark spin symmetry. By performing a simultaneous fit to the experimental data from LHCb, , and Belle collaborations, the scheme with three-body interaction successfully describes the invariant mass distributions of the three two-body subsystems. We precisely extract the pole structures of <r:math xmlns:r="http://www.w3.org/1998/Math/MathML" display="inline"> <r:msub> <r:mi>χ</r:mi> <r:mrow> <r:mi>c</r:mi> <r:mn>0</r:mn> </r:mrow> </r:msub> <r:mo stretchy="false">(</r:mo> <r:mn>3930</r:mn> <r:mo stretchy="false">)</r:mo> </r:math> and <v:math xmlns:v="http://www.w3.org/1998/Math/MathML" display="inline"> <v:mi>ψ</v:mi> <v:mo stretchy="false">(</v:mo> <v:mn>3770</v:mn> <v:mo stretchy="false">)</v:mo> </v:math> as <z:math xmlns:z="http://www.w3.org/1998/Math/MathML" display="inline"> <z:mrow> <z:mn>3.913</z:mn> <z:mo>−</z:mo> <z:mn>0.016</z:mn> <z:mi>i</z:mi> <z:mtext> </z:mtext> <z:mi>GeV</z:mi> </z:mrow> </z:math> and <bb:math xmlns:bb="http://www.w3.org/1998/Math/MathML" display="inline"> <bb:mrow> <bb:mn>3.761</bb:mn> <bb:mo>−</bb:mo> <bb:mn>0.006</bb:mn> <bb:mi>i</bb:mi> <bb:mtext> </bb:mtext> <bb:mi>GeV</bb:mi> </bb:mrow> </bb:math> in <db:math xmlns:db="http://www.w3.org/1998/Math/MathML" display="inline"> <db:msup> <db:mi>B</db:mi> <db:mo>+</db:mo> </db:msup> </db:math> decay. By performing the pole trajectory analysis on a uniformized complex plane, we find that both of them stem from the input bare state.