How higher charmonia shape the puzzling data of the e+e−→ηJ/ψ cross section

Recently, the BESIII collaboration performed a precise measurement of the <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:msup><a:mi>e</a:mi><a:mo>+</a:mo></a:msup><a:msup><a:mi>e</a:mi><a:mo>−</a:mo></a:msup><a:mo stretchy="false">→</a:mo><a:mi>η</a:mi><a:mi>J</a:mi><a:mo>/</a:mo><a:mi>ψ</a:mi></a:math> cross section. It is puzzling that the resonance parameters of the reported <d:math xmlns:d="http://www.w3.org/1998/Math/MathML" display="inline"><d:mi>Y</d:mi><d:mo stretchy="false">(</d:mo><d:mn>4230</d:mn><d:mo stretchy="false">)</d:mo></d:math> show a substantial divergence from the previously measured results in both the open-charmed and hidden-charmed decay channels, and the line shape asymmetry of the data approaching 4.2 GeV also suggests that it might be difficult to characterize the details of the structure around 4.2 GeV by a single resonance. This has motivated our great curiosity about how the charmonium states are distributed in the measured energy range and how they shape the puzzling data of the <h:math xmlns:h="http://www.w3.org/1998/Math/MathML" display="inline"><h:msup><h:mi>e</h:mi><h:mo>+</h:mo></h:msup><h:msup><h:mi>e</h:mi><h:mo>−</h:mo></h:msup><h:mo stretchy="false">→</h:mo><h:mi>η</h:mi><h:mi>J</h:mi><h:mo>/</h:mo><h:mi>ψ</h:mi></h:math> cross section. In this work, we use five theoretically constructed charmonia in the range of 4.0–4.5 GeV, i.e., <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:mi>ψ</k:mi><k:mo stretchy="false">(</k:mo><k:mn>4040</k:mn><k:mo stretchy="false">)</k:mo></k:math>, <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>4160</o:mn><o:mo stretchy="false">)</o:mo></o:math>, <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>4220</s:mn><s:mo stretchy="false">)</s:mo></s:math>, <w:math xmlns:w="http://www.w3.org/1998/Math/MathML" display="inline"><w:mi>ψ</w:mi><w:mo stretchy="false">(</w:mo><w:mn>4380</w:mn><w:mo stretchy="false">)</w:mo></w:math>, and <ab:math xmlns:ab="http://www.w3.org/1998/Math/MathML" display="inline"><ab:mi>ψ</ab:mi><ab:mo stretchy="false">(</ab:mo><ab:mn>4415</ab:mn><ab:mo stretchy="false">)</ab:mo></ab:math>, to apply a combined fit to the data, in which their calculated decay ratios into <eb:math xmlns:eb="http://www.w3.org/1998/Math/MathML" display="inline"><eb:mi>η</eb:mi><eb:mi>J</eb:mi><eb:mo>/</eb:mo><eb:mi>ψ</eb:mi></eb:math> via hadronic loop mechanism are taken as input. The fit results can reproduce the measured cross section data well, especially for the subtle line shape around 4.2 GeV, showing that the structure around 4.2 GeV is possible from the contribution of both <gb:math xmlns:gb="http://www.w3.org/1998/Math/MathML" display="inline"><gb:mi>ψ</gb:mi><gb:mo stretchy="false">(</gb:mo><gb:mn>4160</gb:mn><gb:mo stretchy="false">)</gb:mo></gb:math> and <kb:math xmlns:kb="http://www.w3.org/1998/Math/MathML" display="inline"><kb:mi>ψ</kb:mi><kb:mo stretchy="false">(</kb:mo><kb:mn>4220</kb:mn><kb:mo stretchy="false">)</kb:mo></kb:math>.