Dipionic transitions of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>Y</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mn>4500</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:math> to <mml:math xmlns:mml="ht

We, using an effective Lagrangian approach, investigated the dipionic transition of the <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mi>Y</a:mi> <a:mo stretchy="false">(</a:mo> <a:mn>4500</a:mn> <a:mo stretchy="false">)</a:mo> </a:math> newly well established in the process <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"> <e:msup> <e:mi>e</e:mi> <e:mo>+</e:mo> </e:msup> <e:msup> <e:mi>e</e:mi> <e:mo>−</e:mo> </e:msup> <e:mo stretchy="false">→</e:mo> <e:msup> <e:mi>K</e:mi> <e:mo>+</e:mo> </e:msup> <e:msup> <e:mi>K</e:mi> <e:mo>−</e:mo> </e:msup> <e:mi>J</e:mi> <e:mo>/</e:mo> <e:mi>ψ</e:mi> </e:math> . In this study, the <h:math xmlns:h="http://www.w3.org/1998/Math/MathML" display="inline"> <h:mi>Y</h:mi> <h:mo stretchy="false">(</h:mo> <h:mn>4500</h:mn> <h:mo stretchy="false">)</h:mo> </h:math> was considered as a mixture of the <l:math xmlns:l="http://www.w3.org/1998/Math/MathML" display="inline"> <l:mi>ψ</l:mi> <l:mo stretchy="false">(</l:mo> <l:mn>5</l:mn> <l:mi>S</l:mi> <l:mo stretchy="false">)</l:mo> </l:math> and <p:math xmlns:p="http://www.w3.org/1998/Math/MathML" display="inline"> <p:mrow> <p:mi>ψ</p:mi> <p:mo stretchy="false">(</p:mo> <p:mn>4</p:mn> <p:msub> <p:mrow> <p:mmultiscripts> <p:mrow> <p:mi>D</p:mi> </p:mrow> <p:mprescripts/> <p:none/> <p:mrow> <p:mn>3</p:mn> </p:mrow> </p:mmultiscripts> </p:mrow> <p:mrow> <p:mn>1</p:mn> </p:mrow> </p:msub> <p:mo stretchy="false">)</p:mo> </p:mrow> </p:math> , and its dipionic decay to the <t:math xmlns:t="http://www.w3.org/1998/Math/MathML" display="inline"> <t:mi>J</t:mi> <t:mo>/</t:mo> <t:mi>ψ</t:mi> </t:math> was assumed to occur via charmed meson loops including box and two kinds of triangle loops. The calculated invariant mass spectra due to the different loop mechanisms exhibit distinct differences. These spectrum patterns are found to be nearly independent of the cutoff parameter, especially for the box and <v:math xmlns:v="http://www.w3.org/1998/Math/MathML" display="inline"> <v:mi mathvariant="script">R</v:mi> </v:math> loops. By means of comparing our calculated results with the future experimental measurements of the <y:math xmlns:y="http://www.w3.org/1998/Math/MathML" display="inline"> <y:msup> <y:mi>π</y:mi> <y:mo>+</y:mo> </y:msup> <y:msup> <y:mi>π</y:mi> <y:mo>−</y:mo> </y:msup> </y:math> and <ab:math xmlns:ab="http://www.w3.org/1998/Math/MathML" display="inline"> <ab:msup> <ab:mi>π</ab:mi> <ab:mo>±</ab:mo> </ab:msup> <ab:mi>J</ab:mi> <ab:mo>/</ab:mo> <ab:mi>ψ</ab:mi> </ab:math> invariant mass distributions, we could judge which loop mechanism is of more importance in this dipionic transition. The interference among the different kinds of loops is also exhibited. Despite the unknown phase angles between different kinds of loop diagrams, the partial decay width for the process <cb:math xmlns:cb="http://www.w3.org/1998/Math/MathML" display="inline"> <cb:mi>Y</cb:mi> <cb:mo stretchy="false">(</cb:mo> <cb:mn>4500</cb:mn> <cb:mo stretchy="false">)</cb:mo> <cb:mo stretchy="false">→</cb:mo> <cb:msup> <cb:mi>π</cb:mi> <cb:mo>+</cb:mo> </cb:msup> <cb:msup> <cb:mi>π</cb:mi> <cb:mo>−</cb:mo> </cb:msup> <cb:mi>J</cb:mi> <cb:mo>/</cb:mo> <cb:mi>ψ</cb:mi> </cb:math> is estimated to be 250–700 keV, aligning with the estimation obtained by combining BESIII Collaboration measurements and theoretical predictions for <hb:math xmlns:hb="http://www.w3.org/1998/Math/MathML" display="inline"> <hb:mi>Y</hb:mi> <hb:mo stretchy="false">(</hb:mo> <hb:mn>4500</hb:mn> <hb:mo stretchy="false">)</hb:mo> <hb:mo stretchy="false">→</hb:mo> <hb:msup> <hb:mi>K</hb:mi> <hb:mo>+</hb:mo> </hb:msup> <hb:msup> <hb:mi>K</hb:mi> <hb:mo>−</hb:mo> </hb:msup> <hb:mi>J</hb:mi> <hb:mo>/</hb:mo> <hb:mi>ψ</hb:mi> </hb:math> . We hope that the present calculations would be tested by the future BESIII or Belle II experiments.