Light single-gluon hybrid states with various exotic quantum numbers

We apply the QCD sum rule method to study the light single-gluon hybrid states with various (exotic) quantum numbers. We construct 24 single-gluon hybrid currents and use 18 of them to calculate the masses of 44 single-gluon hybrid states with the quark-gluon contents <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mover accent="true"><a:mi>q</a:mi><a:mo stretchy="false">¯</a:mo></a:mover><a:mi>q</a:mi><a:mi>g</a:mi></a:math> (<e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mi>q</e:mi><e:mo>=</e:mo><e:mi>u</e:mi><e:mo>/</e:mo><e:mi>d</e:mi></e:math>) and <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:mover accent="true"><g:mi>s</g:mi><g:mo stretchy="false">¯</g:mo></g:mover><g:mi>s</g:mi><g:mi>g</g:mi></g:math>. We concentrate on the hybrid states with the exotic quantum number <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:msup><k:mi>J</k:mi><k:mrow><k:mi>P</k:mi><k:mi>C</k:mi></k:mrow></k:msup><k:mo>=</k:mo><k:msup><k:mn>1</k:mn><k:mrow><k:mo>−</k:mo><k:mo>+</k:mo></k:mrow></k:msup></k:math>, whose masses and widths are calculated to be <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"><m:msub><m:mi>M</m:mi><m:mrow><m:mo stretchy="false">|</m:mo><m:mover accent="true"><m:mi>q</m:mi><m:mo stretchy="false">¯</m:mo></m:mover><m:mi>q</m:mi><m:mi>g</m:mi><m:mo>;</m:mo><m:msup><m:mn>1</m:mn><m:mo>−</m:mo></m:msup><m:msup><m:mn>1</m:mn><m:mrow><m:mo>−</m:mo><m:mo>+</m:mo></m:mrow></m:msup><m:mo stretchy="false">⟩</m:mo></m:mrow></m:msub><m:mo>=</m:mo><m:mn>1.6</m:mn><m:msubsup><m:mn>7</m:mn><m:mrow><m:mo>−</m:mo><m:mn>0.17</m:mn></m:mrow><m:mrow><m:mo>+</m:mo><m:mn>0.15</m:mn></m:mrow></m:msubsup><m:mtext> </m:mtext><m:mtext> </m:mtext><m:mi>GeV</m:mi></m:math>, <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"><s:msub><s:mi mathvariant="normal">Γ</s:mi><s:mrow><s:mo stretchy="false">|</s:mo><s:mover accent="true"><s:mi>q</s:mi><s:mo stretchy="false">¯</s:mo></s:mover><s:mi>q</s:mi><s:mi>g</s:mi><s:mo>;</s:mo><s:msup><s:mn>1</s:mn><s:mo>−</s:mo></s:msup><s:msup><s:mn>1</s:mn><s:mrow><s:mo>−</s:mo><s:mo>+</s:mo></s:mrow></s:msup><s:mo stretchy="false">⟩</s:mo></s:mrow></s:msub><s:mo>=</s:mo><s:mn>53</s:mn><s:msubsup><s:mn>0</s:mn><s:mrow><s:mo>−</s:mo><s:mn>330</s:mn></s:mrow><s:mrow><s:mo>+</s:mo><s:mn>540</s:mn></s:mrow></s:msubsup><s:mtext> </s:mtext><s:mtext> </s:mtext><s:mi>MeV</s:mi></s:math>, <z:math xmlns:z="http://www.w3.org/1998/Math/MathML" display="inline"><z:msub><z:mi>M</z:mi><z:mrow><z:mo stretchy="false">|</z:mo><z:mover accent="true"><z:mi>q</z:mi><z:mo stretchy="false">¯</z:mo></z:mover><z:mi>q</z:mi><z:mi>g</z:mi><z:mo>;</z:mo><z:msup><z:mn>0</z:mn><z:mo>+</z:mo></z:msup><z:msup><z:mn>1</z:mn><z:mrow><z:mo>−</z:mo><z:mo>+</z:mo></z:mrow></z:msup><z:mo stretchy="false">⟩</z:mo></z:mrow></z:msub><z:mo>=</z:mo><z:mn>1.6</z:mn><z:msubsup><z:mn>7</z:mn><z:mrow><z:mo>−</z:mo><z:mn>0.17</z:mn></z:mrow><z:mrow><z:mo>+</z:mo><z:mn>0.15</z:mn></z:mrow></z:msubsup><z:mtext> </z:mtext><z:mtext> </z:mtext><z:mi>GeV</z:mi></z:math>, <fb:math xmlns:fb="http://www.w3.org/1998/Math/MathML" display="inline"><fb:msub><fb:mi mathvariant="normal">Γ</fb:mi><fb:mrow><fb:mo stretchy="false">|</fb:mo><fb:mover accent="true"><fb:mi>q</fb:mi><fb:mo stretchy="false">¯</fb:mo></fb:mover><fb:mi>q</fb:mi><fb:mi>g</fb:mi><fb:mo>;</fb:mo><fb:msup><fb:mn>0</fb:mn><fb:mo>+</fb:mo></fb:msup><fb:msup><fb:mn>1</fb:mn><fb:mrow><fb:mo>−</fb:mo><fb:mo>+</fb:mo></fb:mrow></fb:msup><fb:mo stretchy="false">⟩</fb:mo></fb:mrow></fb:msub><fb:mo>=</fb:mo><fb:mspace linebreak="goodbreak"/><fb:mn>12</fb:mn><fb:msubsup><fb:mn>0</fb:mn><fb:mrow><fb:mo>−</fb:mo><fb:mn>110</fb:mn></fb:mrow><fb:mrow><fb:mo>+</fb:mo><fb:mn>160</fb:mn></fb:mrow></fb:msubsup><fb:mtext> </fb:mtext><fb:mtext> </fb:mtext><fb:mi>MeV</fb:mi></fb:math>, <nb:math xmlns:nb="http://www.w3.org/1998/Math/MathML" display="inline"><nb:msub><nb:mi>M</nb:mi><nb:mrow><nb:mo stretchy="false">|</nb:mo><nb:mover accent="true"><nb:mi>s</nb:mi><nb:mo stretchy="false">¯</nb:mo></nb:mover><nb:mi>s</nb:mi><nb:mi>g</nb:mi><nb:mo>;</nb:mo><nb:msup><nb:mn>0</nb:mn><nb:mo>+</nb:mo></nb:msup><nb:msup><nb:mn>1</nb:mn><nb:mrow><nb:mo>−</nb:mo><nb:mo>+</nb:mo></nb:mrow></nb:msup><nb:mo stretchy="false">⟩</nb:mo></nb:mrow></nb:msub><nb:mo>=</nb:mo><nb:mn>1.8</nb:mn><nb:msubsup><nb:mn>4</nb:mn><nb:mrow><nb:mo>−</nb:mo><nb:mn>0.15</nb:mn></nb:mrow><nb:mrow><nb:mo>+</nb:mo><nb:mn>0.14</nb:mn></nb:mrow></nb:msubsup><nb:mtext> </nb:mtext><nb:mtext> </nb:mtext><nb:mi>GeV</nb:mi></nb:math>, and <tb:math xmlns:tb="http://www.w3.org/1998/Math/MathML" display="inline"><tb:msub><tb:mi mathvariant="normal">Γ</tb:mi><tb:mrow><tb:mo stretchy="false">|</tb:mo><tb:mover accent="true"><tb:mi>s</tb:mi><tb:mo stretchy="false">¯</tb:mo></tb:mover><tb:mi>s</tb:mi><tb:mi>g</tb:mi><tb:mo>;</tb:mo><tb:msup><tb:mn>0</tb:mn><tb:mo>+</tb:mo></tb:msup><tb:msup><tb:mn>1</tb:mn><tb:mrow><tb:mo>−</tb:mo><tb:mo>+</tb:mo></tb:mrow></tb:msup><tb:mo stretchy="false">⟩</tb:mo></tb:mrow></tb:msub><tb:mo>=</tb:mo><tb:mn>10</tb:mn><tb:msubsup><tb:mn>0</tb:mn><tb:mrow><tb:mo>−</tb:mo><tb:mn>80</tb:mn></tb:mrow><tb:mrow><tb:mo>+</tb:mo><tb:mn>110</tb:mn></tb:mrow></tb:msubsup><tb:mtext> </tb:mtext><tb:mtext> </tb:mtext><tb:mi>MeV</tb:mi></tb:math>. Our results support the interpretations of the <ac:math xmlns:ac="http://www.w3.org/1998/Math/MathML" display="inline"><ac:msub><ac:mi>π</ac:mi><ac:mn>1</ac:mn></ac:msub><ac:mo stretchy="false">(</ac:mo><ac:mn>1600</ac:mn><ac:mo stretchy="false">)</ac:mo></ac:math> and <ec:math xmlns:ec="http://www.w3.org/1998/Math/MathML" display="inline"><ec:msub><ec:mi>η</ec:mi><ec:mn>1</ec:mn></ec:msub><ec:mo stretchy="false">(</ec:mo><ec:mn>1855</ec:mn><ec:mo stretchy="false">)</ec:mo></ec:math> as the hybrid states <ic:math xmlns:ic="http://www.w3.org/1998/Math/MathML" display="inline"><ic:mo stretchy="false">|</ic:mo><ic:mover accent="true"><ic:mi>q</ic:mi><ic:mo stretchy="false">¯</ic:mo></ic:mover><ic:mi>q</ic:mi><ic:mi>g</ic:mi><ic:mo>;</ic:mo><ic:msup><ic:mn>1</ic:mn><ic:mo>−</ic:mo></ic:msup><ic:msup><ic:mn>1</ic:mn><ic:mrow><ic:mo>−</ic:mo><ic:mo>+</ic:mo></ic:mrow></ic:msup><ic:mo stretchy="false">⟩</ic:mo></ic:math> and <oc:math xmlns:oc="http://www.w3.org/1998/Math/MathML" display="inline"><oc:mo stretchy="false">|</oc:mo><oc:mover accent="true"><oc:mi>s</oc:mi><oc:mo stretchy="false">¯</oc:mo></oc:mover><oc:mi>s</oc:mi><oc:mi>g</oc:mi><oc:mo>;</oc:mo><oc:msup><oc:mn>0</oc:mn><oc:mo>+</oc:mo></oc:msup><oc:msup><oc:mn>1</oc:mn><oc:mrow><oc:mo>−</oc:mo><oc:mo>+</oc:mo></oc:mrow></oc:msup><oc:mo stretchy="false">⟩</oc:mo></oc:math>, respectively. Considering the uncertainties, our results suggest that the <uc:math xmlns:uc="http://www.w3.org/1998/Math/MathML" display="inline"><uc:msub><uc:mi>π</uc:mi><uc:mn>1</uc:mn></uc:msub><uc:mo stretchy="false">(</uc:mo><uc:mn>1600</uc:mn><uc:mo stretchy="false">)</uc:mo></uc:math> and <yc:math xmlns:yc="http://www.w3.org/1998/Math/MathML" display="inline"><yc:msub><yc:mi>η</yc:mi><yc:mn>1</yc:mn></yc:msub><yc:mo stretchy="false">(</yc:mo><yc:mn>1855</yc:mn><yc:mo stretchy="false">)</yc:mo></yc:math> may also be interpreted as the hybrid states <cd:math xmlns:cd="http://www.w3.org/1998/Math/MathML" display="inline"><cd:mo stretchy="false">|</cd:mo><cd:mover accent="true"><cd:mi>q</cd:mi><cd:mo stretchy="false">¯</cd:mo></cd:mover><cd:mi>q</cd:mi><cd:mi>g</cd:mi><cd:mo>;</cd:mo><cd:msup><cd:mn>1</cd:mn><cd:mo>−</cd:mo></cd:msup><cd:msup><cd:mn>1</cd:mn><cd:mrow><cd:mo>−</cd:mo><cd:mo>+</cd:mo></cd:mrow></cd:msup><cd:mo stretchy="false">⟩</cd:mo></cd:math> and <id:math xmlns:id="http://www.w3.org/1998/Math/MathML" display="inline"><id:mo stretchy="false">|</id:mo><id:mover accent="true"><id:mi>q</id:mi><id:mo stretchy="false">¯</id:mo></id:mover><id:mi>q</id:mi><id:mi>g</id:mi><id:mo>;</id:mo><id:msup><id:mn>0</id:mn><id:mo>+</id:mo></id:msup><id:msup><id:mn>1</id:mn><id:mrow><id:mo>−</id:mo><id:mo>+</id:mo></id:mrow></id:msup><id:mo stretchy="false">⟩</id:mo></id:math>, respectively. To differentiate these two assignments and to verify whether they are hybrid states or not, we propose to examine the <od:math xmlns:od="http://www.w3.org/1998/Math/MathML" display="inline"><od:msub><od:mi>a</od:mi><od:mn>1</od:mn></od:msub><od:mo stretchy="false">(</od:mo><od:mn>1260</od:mn><od:mo stretchy="false">)</od:mo><od:mi>π</od:mi></od:math> decay channel in future experiments. Published by the American Physical Society 2024