Is the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mn>3</mml:mn> <mml:mi>S</mml:mi> <mml:mtext>−</mml:mtext> <mml:mn>2</mml:mn> <mml:mi>D</mml:mi> </mml:mrow> </mml:math> mixing strong for the charmonia

In this work, we revisit the <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mrow> <a:mn>3</a:mn> <a:mi>S</a:mi> <a:mtext>−</a:mtext> <a:mn>2</a:mn> <a:mi>D</a:mi> </a:mrow> </a:math> mixing scheme for the charmonia <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>4040</c:mn> <c:mo stretchy="false">)</c:mo> </c:math> and <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>4160</g:mn> <g:mo stretchy="false">)</g:mo> </g:math> . We introduce a coupled-channel mechanism—distinct from the tensor-force contribution in potential models, which alone is insufficient to induce significant mixing—to describe the mixing between these states. Our analysis yields mixing angles of <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"> <k:msub> <k:mi>θ</k:mi> <k:mn>1</k:mn> </k:msub> <k:mo>=</k:mo> <k:mn>7</k:mn> <k:mi>°</k:mi> </k:math> and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"> <m:msub> <m:mi>θ</m:mi> <m:mn>2</m:mn> </m:msub> <m:mo>=</m:mo> <m:mn>1</m:mn> <m:mn>0</m:mn> <m:mi>°</m:mi> </m:math> , inconsistent with the larger angle inferred from experimental data, such as the dielectronic widths of the <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"> <o:mi>ψ</o:mi> <o:mo stretchy="false">(</o:mo> <o:mn>4040</o:mn> <o:mo stretchy="false">)</o:mo> </o:math> and <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"> <s:mi>ψ</s:mi> <s:mo stretchy="false">(</s:mo> <s:mn>4160</s:mn> <s:mo stretchy="false">)</s:mo> </s:math> . We discuss possible origins of this discrepancy and emphasize the need for future experiments to resolve it. Precise measurements of the resonance parameters and dielectronic decay widths, via both inclusive and exclusive processes, will be crucial in clarifying this issue.