Helicity amplitude and branching fraction measurement of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msub> <mml:mi>χ</mml:mi> <mml:mrow> <mml:mi>c</mml:mi> <mml:mi>J</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretch

Utilizing <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mn>2712.4</a:mn> <a:mo>±</a:mo> <a:mn>14.3</a:mn> <a:mtext> </a:mtext> <a:mtext> </a:mtext> <a:mi>million</a:mi> </a:math> <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:mi>ψ</c:mi> <c:mo stretchy="false">(</c:mo> <c:mn>3686</c:mn> <c:mo stretchy="false">)</c:mo> </c:math> events accumulated by the BESIII experiment, we perform a partial wave analysis of <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"> <g:mi>ψ</g:mi> <g:mo stretchy="false">(</g:mo> <g:mn>3686</g:mn> <g:mo stretchy="false">)</g:mo> <g:mo stretchy="false">→</g:mo> <g:mi>γ</g:mi> <g:msub> <g:mi>χ</g:mi> <g:mrow> <g:mi>c</g:mi> <g:mi>J</g:mi> </g:mrow> </g:msub> <g:mo stretchy="false">→</g:mo> <g:mi>γ</g:mi> <g:mi mathvariant="normal">Λ</g:mi> <g:mover accent="true"> <g:mi mathvariant="normal">Λ</g:mi> <g:mo stretchy="false">¯</g:mo> </g:mover> </g:math> decay ( <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"> <q:mi>J</q:mi> <q:mo>=</q:mo> <q:mn>0</q:mn> </q:math> , 1, 2). The ratio of the helicity amplitudes with same (++) and opposite (+-) helicity for <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"> <s:msub> <s:mi>χ</s:mi> <s:mrow> <s:mi>c</s:mi> <s:mn>2</s:mn> </s:mrow> </s:msub> <s:mo stretchy="false">→</s:mo> <s:mi mathvariant="normal">Λ</s:mi> <s:mover accent="true"> <s:mi mathvariant="normal">Λ</s:mi> <s:mo stretchy="false">¯</s:mo> </s:mover> </s:math> decay is determined for the first time to be <z:math xmlns:z="http://www.w3.org/1998/Math/MathML" display="inline"> <z:msub> <z:mi>R</z:mi> <z:msub> <z:mi>χ</z:mi> <z:mrow> <z:mi>c</z:mi> <z:mn>2</z:mn> </z:mrow> </z:msub> </z:msub> <z:mo>=</z:mo> <z:mn>0.575</z:mn> <z:mo>±</z:mo> <z:mn>0.048</z:mn> <z:mo>±</z:mo> <z:mn>0.018</z:mn> </z:math> , with a relative phase angle <bb:math xmlns:bb="http://www.w3.org/1998/Math/MathML" display="inline"> <bb:mi mathvariant="normal">Δ</bb:mi> <bb:msub> <bb:mi mathvariant="normal">Φ</bb:mi> <bb:msub> <bb:mi>χ</bb:mi> <bb:mrow> <bb:mi>c</bb:mi> <bb:mn>2</bb:mn> </bb:mrow> </bb:msub> </bb:msub> <bb:mo>=</bb:mo> <bb:mn>0.37</bb:mn> <bb:mo>±</bb:mo> <bb:mn>0.15</bb:mn> <bb:mo>±</bb:mo> <bb:mn>0.05</bb:mn> <bb:mtext> </bb:mtext> <bb:mtext> </bb:mtext> <bb:mi>rad</bb:mi> </bb:math> . The parameters of the angular distribution of <fb:math xmlns:fb="http://www.w3.org/1998/Math/MathML" display="inline"> <fb:msub> <fb:mi>χ</fb:mi> <fb:mrow> <fb:mi>c</fb:mi> <fb:mn>2</fb:mn> </fb:mrow> </fb:msub> </fb:math> are determined to be <hb:math xmlns:hb="http://www.w3.org/1998/Math/MathML" display="inline"> <hb:msub> <hb:mi>α</hb:mi> <hb:msub> <hb:mi>χ</hb:mi> <hb:mrow> <hb:mi>c</hb:mi> <hb:mn>2</hb:mn> </hb:mrow> </hb:msub> </hb:msub> <hb:mo>=</hb:mo> <hb:mo>−</hb:mo> <hb:mn>0.211</hb:mn> <hb:mo>±</hb:mo> <hb:mn>0.100</hb:mn> <hb:mo>±</hb:mo> <hb:mn>0.050</hb:mn> </hb:math> and <jb:math xmlns:jb="http://www.w3.org/1998/Math/MathML" display="inline"> <jb:msub> <jb:mi>β</jb:mi> <jb:msub> <jb:mi>χ</jb:mi> <jb:mrow> <jb:mi>c</jb:mi> <jb:mn>2</jb:mn> </jb:mrow> </jb:msub> </jb:msub> <jb:mo>=</jb:mo> <jb:mo>−</jb:mo> <jb:mn>0.039</jb:mn> <jb:mo>±</jb:mo> <jb:mn>0.089</jb:mn> <jb:mo>±</jb:mo> <jb:mn>0.033</jb:mn> </jb:math> , based on the distribution <lb:math xmlns:lb="http://www.w3.org/1998/Math/MathML" display="inline"> <lb:mi>d</lb:mi> <lb:mi>N</lb:mi> <lb:mo>/</lb:mo> <lb:mi>d</lb:mi> <lb:mi>cos</lb:mi> <lb:mi>θ</lb:mi> <lb:mo>=</lb:mo> <lb:mn>1</lb:mn> <lb:mo>+</lb:mo> <lb:msub> <lb:mi>α</lb:mi> <lb:msub> <lb:mi>χ</lb:mi> <lb:mrow> <lb:mi>c</lb:mi> <lb:mn>2</lb:mn> </lb:mrow> </lb:msub> </lb:msub> <lb:msup> <lb:mi>cos</lb:mi> <lb:mn>2</lb:mn> </lb:msup> <lb:mi>θ</lb:mi> <lb:mo>+</lb:mo> <lb:msub> <lb:mi>β</lb:mi> <lb:msub> <lb:mi>χ</lb:mi> <lb:mrow> <lb:mi>c</lb:mi> <lb:mn>2</lb:mn> </lb:mrow> </lb:msub> </lb:msub> <lb:msup> <lb:mi>cos</lb:mi> <lb:mn>4</lb:mn> </lb:msup> <lb:mi>θ</lb:mi> </lb:math> . The width of <nb:math xmlns:nb="http://www.w3.org/1998/Math/MathML" display="inline"> <nb:msub> <nb:mi>χ</nb:mi> <nb:mrow> <nb:mi>c</nb:mi> <nb:mn>0</nb:mn> </nb:mrow> </nb:msub> </nb:math> is determined to be <pb:math xmlns:pb="http://www.w3.org/1998/Math/MathML" display="inline"> <pb:mn>12.31</pb:mn> <pb:mo>±</pb:mo> <pb:mn>0.26</pb:mn> <pb:mo>±</pb:mo> <pb:mn>0.12</pb:mn> <pb:mtext> </pb:mtext> <pb:mtext> </pb:mtext> <pb:mi>MeV</pb:mi> </pb:math> . Additionally, the branching fractions for <rb:math xmlns:rb="http://www.w3.org/1998/Math/MathML" display="inline"> <rb:msub> <rb:mi>χ</rb:mi> <rb:mrow> <rb:mi>c</rb:mi> <rb:mi>J</rb:mi> </rb:mrow> </rb:msub> <rb:mo stretchy="false">→</rb:mo> <rb:mi mathvariant="normal">Λ</rb:mi> <rb:mover accent="true"> <rb:mi mathvariant