Charmoniumlike channels <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msup><mml:mn>1</mml:mn><mml:mo>+</mml:mo></mml:msup></mml:math> with isospin 1 from lattice and effective field theory

Many exotic charmoniumlike mesons have already been discovered experimentally, of which the <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:msub><a:mi>Z</a:mi><a:mi>c</a:mi></a:msub></a:math> mesons with isospin 1 are prominent examples. We investigate <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:msup><c:mi>J</c:mi><c:mrow><c:mi>P</c:mi><c:mi>C</c:mi></c:mrow></c:msup><c:mo>=</c:mo><c:msup><c:mn>1</c:mn><c:mrow><c:mo>+</c:mo><c:mo>±</c:mo></c:mrow></c:msup></c:math> states with flavor <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mover accent="true"><e:mi>c</e:mi><e:mo stretchy="false">¯</e:mo></e:mover><e:mi>c</e:mi><e:mover accent="true"><e:mi>q</e:mi><e:mo stretchy="false">¯</e:mo></e:mover><e:mi>q</e:mi></e:math> (<k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:mrow><k:mi>q</k:mi><k:mo>=</k:mo><k:mi>u</k:mi></k:mrow></k:math>, <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"><m:mrow><m:mi>d</m:mi></m:mrow></m:math>) in isospin 1 using lattice QCD. This is the first study of these mesons employing more than one volume and involving frames with nonzero total momentum. We utilize two <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"><o:msub><o:mi>N</o:mi><o:mi>f</o:mi></o:msub><o:mo>=</o:mo><o:mn>2</o:mn><o:mo>+</o:mo><o:mn>1</o:mn></o:math> CLS ensembles with <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"><q:mrow><q:msub><q:mrow><q:mi>m</q:mi></q:mrow><q:mrow><q:mi>π</q:mi></q:mrow></q:msub><q:mo>≃</q:mo><q:mn>280</q:mn><q:mtext> </q:mtext><q:mtext> </q:mtext><q:mi>MeV</q:mi></q:mrow></q:math>. The simulations are performed with unphysical light quark masses at a single lattice spacing of <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"><s:mrow><s:mi>a</s:mi><s:mo>≃</s:mo><s:mn>0.086</s:mn><s:mtext> </s:mtext><s:mtext> </s:mtext><s:mrow><s:mi>fm</s:mi></s:mrow></s:mrow></s:math> and omit <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"><u:mi>ψ</u:mi><u:mo stretchy="false">(</u:mo><u:mn>2</u:mn><u:mi>S</u:mi><u:mo stretchy="false">)</u:mo><u:mi>π</u:mi></u:math>, <y:math xmlns:y="http://www.w3.org/1998/Math/MathML" display="inline"><y:mi>ψ</y:mi><y:mo stretchy="false">(</y:mo><y:mn>3770</y:mn><y:mo stretchy="false">)</y:mo><y:mi>π</y:mi></y:math> and three-particle decay channels, so our results provide only qualitative insights. Resulting eigenenergies are compatible or just slightly shifted down with respect to noninteracting energies, where the most significant shifts occur for certain <cb:math xmlns:cb="http://www.w3.org/1998/Math/MathML" display="inline"><cb:mi>D</cb:mi><cb:msup><cb:mover accent="true"><cb:mi>D</cb:mi><cb:mo stretchy="false">¯</cb:mo></cb:mover><cb:mo>*</cb:mo></cb:msup></cb:math> states. Both channels <gb:math xmlns:gb="http://www.w3.org/1998/Math/MathML" display="inline"><gb:msup><gb:mn>1</gb:mn><gb:mrow><gb:mo>+</gb:mo><gb:mo>±</gb:mo></gb:mrow></gb:msup></gb:math> have a virtual pole slightly below the threshold if <ib:math xmlns:ib="http://www.w3.org/1998/Math/MathML" display="inline"><ib:mi>D</ib:mi><ib:msup><ib:mover accent="true"><ib:mi>D</ib:mi><ib:mo stretchy="false">¯</ib:mo></ib:mover><ib:mo>*</ib:mo></ib:msup></ib:math> is assumed to be decoupled from other channels. In addition, we perform a coupled channel analysis of <mb:math xmlns:mb="http://www.w3.org/1998/Math/MathML" display="inline"><mb:mi>J</mb:mi><mb:mo>/</mb:mo><mb:mi>ψ</mb:mi><mb:mi>π</mb:mi></mb:math> and <ob:math xmlns:ob="http://www.w3.org/1998/Math/MathML" display="inline"><ob:mi>D</ob:mi><ob:msup><ob:mover accent="true"><ob:mi>D</ob:mi><ob:mo stretchy="false">¯</ob:mo></ob:mover><ob:mo>*</ob:mo></ob:msup></ob:math> scattering with <sb:math xmlns:sb="http://www.w3.org/1998/Math/MathML" display="inline"><sb:msup><sb:mi>J</sb:mi><sb:mrow><sb:mi>P</sb:mi><sb:mi>C</sb:mi></sb:mrow></sb:msup><sb:mo>=</sb:mo><sb:msup><sb:mn>1</sb:mn><sb:mrow><sb:mo>+</sb:mo><sb:mo>−</sb:mo></sb:mrow></sb:msup></sb:math> within an effective field theory framework. The <ub:math xmlns:ub="http://www.w3.org/1998/Math/MathML" display="inline"><ub:mi>J</ub:mi><ub:mo>/</ub:mo><ub:mi>ψ</ub:mi><ub:mi>π</ub:mi></ub:math> and <wb:math xmlns:wb="http://www.w3.org/1998/Math/MathML" display="inline"><wb:mi>D</wb:mi><wb:msup><wb:mover accent="true"><wb:mi>D</wb:mi><wb:mo stretchy="false">¯</wb:mo></wb:mover><wb:mo>*</wb:mo></wb:msup></wb:math> invariant-mass distributions from BESIII and finite-volume energies from several lattice QCD simulations, including this work, are fitted simultaneously. All fits yield two poles relatively close to the <ac:math xmlns:ac="http://www.w3.org/1998/Math/MathML" display="inline"><ac:mi>D</ac:mi><ac:msup><ac:mover accent="true"><ac:mi>D</ac:mi><ac:mo stretchy="false">¯</ac:mo></ac:mover><ac:mo>*</ac:mo></ac:msup></ac:math> threshold and reasonably reproduce the experimental <ec:math xmlns:ec="http://www.w3.org/1998/Math/MathML" display="inline"><ec:msub><ec:mi>Z</ec:mi><ec:mi>c</ec:mi></ec:msub></ec:math> peaks. They also reproduce lattice energies up to slightly above the <gc:math xmlns:gc="http://www.w3.org/1998/Math/MathML" display="inline"><gc:mi>D</gc:mi><gc:msup><gc:mover accent="true"><gc:mi>D</gc:mi><gc:mo stretchy="false">¯</gc:mo></gc:mover><gc:mo>*</gc:mo></gc:msup></gc:math> threshold, while reproduction at even higher energies is better for fits that put more weight on the lattice data. Our findings suggest that the employed effective field theory can reasonably reconcile the peaks in the experimental line shapes and the lattice energies, although those lie close to noninteracting energies. We also study <kc:math xmlns:kc="http://www.w3.org/1998/Math/MathML" display="inline"><kc:mi>J</kc:mi><kc:mo>/</kc:mo><kc:mi>ψ</kc:mi><kc:mi>π</kc:mi></kc:math> scattering in s wave and place upper bounds on the phase shift.