Sifting out the neutral and charged components of X ( 3872 ) from a spin observable

In this work, we propose a new method to detect the composition of the <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mi>X</a:mi> <a:mo stretchy="false">(</a:mo> <a:mn>3872</a:mn> <a:mo stretchy="false">)</a:mo> </a:math> state. Based on a widely accepted interpretation that <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"> <e:mi>X</e:mi> <e:mo stretchy="false">(</e:mo> <e:mn>3872</e:mn> <e:mo stretchy="false">)</e:mo> </e:math> is a weakly bound <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"> <i:mi>S</i:mi> </i:math> -wave molecule of the <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"> <k:msup> <k:mi>D</k:mi> <k:mrow> <k:mo>*</k:mo> <k:mn>0</k:mn> </k:mrow> </k:msup> <k:msup> <k:mover accent="true"> <k:mi>D</k:mi> <k:mo stretchy="false">¯</k:mo> </k:mover> <k:mn>0</k:mn> </k:msup> </k:math> (neutral) and <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"> <o:msup> <o:mi>D</o:mi> <o:mrow> <o:mo>*</o:mo> <o:mo>+</o:mo> </o:mrow> </o:msup> <o:msup> <o:mi>D</o:mi> <o:mo>−</o:mo> </o:msup> </o:math> (charged) configurations, the process <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"> <q:mi>X</q:mi> <q:mo stretchy="false">(</q:mo> <q:mn>3872</q:mn> <q:mo stretchy="false">)</q:mo> <q:mo stretchy="false">→</q:mo> <q:msup> <q:mover accent="true"> <q:mi>D</q:mi> <q:mo stretchy="false">¯</q:mo> </q:mover> <q:mrow> <q:mo>*</q:mo> <q:mn>0</q:mn> </q:mrow> </q:msup> <q:msup> <q:mi>D</q:mi> <q:mn>0</q:mn> </q:msup> </q:math> is described by one-loop triangle diagrams with the corresponding components as intermediate particles, and with the parameters constrained by the experimental branching ratio. Unlike the mild behavior of the neutral component, the charged component manifests itself as a distinct structure near the <x:math xmlns:x="http://www.w3.org/1998/Math/MathML" display="inline"> <x:msup> <x:mi>D</x:mi> <x:mrow> <x:mo>*</x:mo> <x:mo>+</x:mo> </x:mrow> </x:msup> <x:msup> <x:mi>D</x:mi> <x:mo>−</x:mo> </x:msup> </x:math> threshold at 3.880 GeV, in the <z:math xmlns:z="http://www.w3.org/1998/Math/MathML" display="inline"> <z:msub> <z:mi>M</z:mi> <z:mrow> <z:msup> <z:mover accent="true"> <z:mi>D</z:mi> <z:mo stretchy="false">¯</z:mo> </z:mover> <z:mrow> <z:mo>*</z:mo> <z:mn>0</z:mn> </z:mrow> </z:msup> <z:msup> <z:mi>D</z:mi> <z:mn>0</z:mn> </z:msup> </z:mrow> </z:msub> </z:math> invariant mass distribution of the spin density matrix element <db:math xmlns:db="http://www.w3.org/1998/Math/MathML" display="inline"> <db:msubsup> <db:mi>ρ</db:mi> <db:mn>00</db:mn> <db:msup> <db:mi>D</db:mi> <db:mo>*</db:mo> </db:msup> </db:msubsup> </db:math> . Hence, the line shape of <fb:math xmlns:fb="http://www.w3.org/1998/Math/MathML" display="inline"> <fb:msubsup> <fb:mi>ρ</fb:mi> <fb:mn>00</fb:mn> <fb:msup> <fb:mi>D</fb:mi> <fb:mo>*</fb:mo> </fb:msup> </fb:msubsup> </fb:math> near this structure depends sensitively on the proportions of the two configurations. We propose high-precision measurements of this matrix element by future BESIII and Belle experiments to elucidate how a molecular <hb:math xmlns:hb="http://www.w3.org/1998/Math/MathML" display="inline"> <hb:mi>X</hb:mi> <hb:mo stretchy="false">(</hb:mo> <hb:mn>3872</hb:mn> <hb:mo stretchy="false">)</hb:mo> </hb:math> state is composed.