Charmonia in an unquenched quark model

In this work, we study the charmonium spectrum within an unquenched quark model including coupled-channel effects. In couple-channel calculations, we include all of the opened charmed meson channels with the once-subtracted method, meanwhile we adopt a suppressed factor to soften the hard vertices given by the <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mrow><a:mmultiscripts><a:mrow><a:msub><a:mrow><a:mi>P</a:mi></a:mrow><a:mrow><a:mn>0</a:mn></a:mrow></a:msub></a:mrow><a:mprescripts/><a:none/><a:mrow><a:mn>3</a:mn></a:mrow></a:mmultiscripts></a:mrow></a:math> model in the high momentum region. We obtain a good description of both the masses and widths for the well-established states in the charmonium spectrum. Furthermore, we give predictions for the higher <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mi>S</c:mi></c:math>-, <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mi>P</e:mi></e:math>-, and <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:mi>D</g:mi></g:math>-wave charmonium states up to mass region of <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:mo>∼</i:mo><i:mn>5.0</i:mn><i:mtext> </i:mtext><i:mtext> </i:mtext><i:mi>GeV</i:mi></i:math>. The magnitude of mass shifts due to the coupled-channel effects is estimated to be about <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:mrow><k:mn>10</k:mn><k:mi>s</k:mi></k:mrow></k:math> MeV. Although many decay channels are opened for the higher charmonium states, they are relatively narrow states. Their widths scatter in the range of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"><m:mrow><m:mo>∼</m:mo><m:mn>10</m:mn><m:mi>s</m:mi><m:mi>–</m:mi><m:mn>100</m:mn><m:mtext> </m:mtext><m:mtext> </m:mtext><m:mi>MeV</m:mi></m:mrow></m:math>. Many charmoniumlike states, such as <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"><o:msub><o:mi>χ</o:mi><o:mrow><o:mi>c</o:mi><o:mn>1</o:mn></o:mrow></o:msub><o:mo stretchy="false">(</o:mo><o:mn>4274</o:mn><o:mo stretchy="false">)</o:mo></o:math>, <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>0</s:mn></s:mrow></s:msub><s:mo stretchy="false">(</s:mo><s:mn>3915</s:mn><s:mo stretchy="false">)</s:mo></s:math>, <w:math xmlns:w="http://www.w3.org/1998/Math/MathML" display="inline"><w:msub><w:mi>χ</w:mi><w:mrow><w:mi>c</w:mi><w:mn>0</w:mn></w:mrow></w:msub><w:mo stretchy="false">(</w:mo><w:mn>4500</w:mn><w:mo stretchy="false">)</w:mo></w:math>, <ab:math xmlns:ab="http://www.w3.org/1998/Math/MathML" display="inline"><ab:msub><ab:mi>χ</ab:mi><ab:mrow><ab:mi>c</ab:mi><ab:mn>0</ab:mn></ab:mrow></ab:msub><ab:mo stretchy="false">(</ab:mo><ab:mn>4700</ab:mn><ab:mo stretchy="false">)</ab:mo></ab:math>, <eb:math xmlns:eb="http://www.w3.org/1998/Math/MathML" display="inline"><eb:mi>X</eb:mi><eb:mo stretchy="false">(</eb:mo><eb:mn>4160</eb:mn><eb:mo stretchy="false">)</eb:mo></eb:math>, <ib:math xmlns:ib="http://www.w3.org/1998/Math/MathML" display="inline"><ib:mi>X</ib:mi><ib:mo stretchy="false">(</ib:mo><ib:mn>4350</ib:mn><ib:mo stretchy="false">)</ib:mo></ib:math>, <mb:math xmlns:mb="http://www.w3.org/1998/Math/MathML" display="inline"><mb:mi>Y</mb:mi><mb:mo stretchy="false">(</mb:mo><mb:mn>4500</mb:mn><mb:mo stretchy="false">)</mb:mo></mb:math>, and <qb:math xmlns:qb="http://www.w3.org/1998/Math/MathML" display="inline"><qb:mi>ψ</qb:mi><qb:mo stretchy="false">(</qb:mo><qb:mn>4660</qb:mn><qb:mo stretchy="false">)</qb:mo><qb:mo>/</qb:mo><qb:mi>Y</qb:mi><qb:mo stretchy="false">(</qb:mo><qb:mn>4710</qb:mn><qb:mo stretchy="false">)</qb:mo></qb:math>, can be accommodated by the charmonium spectrum when the unquenched coupled-channel effects are carefully considered. Published by the American Physical Society 2024