Shallow Bound States and Hints for Broad Resonances with Quark Content b¯c¯ud in B−</mml:…

We present the first determination of the energy dependence of the <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mrow><a:mi>B</a:mi><a:mtext>−</a:mtext><a:mover accent="true"><a:mrow><a:mi>D</a:mi></a:mrow><a:mrow><a:mo stretchy="false">¯</a:mo></a:mrow></a:mover></a:mrow></a:math> and <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mrow><e:msup><e:mrow><e:mi>B</e:mi></e:mrow><e:mrow><e:mo>*</e:mo></e:mrow></e:msup><e:mtext>−</e:mtext><e:mover accent="true"><e:mrow><e:mi>D</e:mi></e:mrow><e:mrow><e:mo stretchy="false">¯</e:mo></e:mrow></e:mover></e:mrow></e:math> isospin-0, <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:mi>S</i:mi></i:math>-wave scattering amplitudes both below and above the thresholds using lattice QCD, which allows us to investigate rigorously whether mixed bottom-charm <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:mrow><k:mover accent="true"><k:mrow><k:mi>b</k:mi></k:mrow><k:mrow><k:mo stretchy="false">¯</k:mo></k:mrow></k:mover><k:mover accent="true"><k:mrow><k:mi>c</k:mi></k:mrow><k:mrow><k:mo stretchy="false">¯</k:mo></k:mrow></k:mover><k:mi>u</k:mi><k:mi>d</k:mi></k:mrow></k:math> tetraquarks exist as bound states or resonances. The scattering phase shifts are obtained using Lüscher’s method from the energy spectra in two different volumes. To ensure that no relevant energy level is missed, we use large, symmetric <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"><q:mn>7</q:mn><q:mo>×</q:mo><q:mn>7</q:mn></q:math> and <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"><s:mn>8</s:mn><s:mo>×</s:mo><s:mn>8</s:mn></s:math> correlation matrices that include, at both source and sink, <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"><u:mrow><u:msup><u:mrow><u:mi>B</u:mi></u:mrow><u:mrow><u:mo stretchy="false">(</u:mo><u:mo>*</u:mo><u:mo stretchy="false">)</u:mo></u:mrow></u:msup><u:mtext>−</u:mtext><u:mover accent="true"><u:mrow><u:mi>D</u:mi></u:mrow><u:mrow><u:mo stretchy="false">¯</u:mo></u:mrow></u:mover></u:mrow></u:math> scattering operators with the lowest three or four possible back-to-back momenta in addition to local <ab:math xmlns:ab="http://www.w3.org/1998/Math/MathML" display="inline"><ab:mover accent="true"><ab:mi>b</ab:mi><ab:mo stretchy="false">¯</ab:mo></ab:mover><ab:mover accent="true"><ab:mi>c</ab:mi><ab:mo stretchy="false">¯</ab:mo></ab:mover><ab:mi>u</ab:mi><ab:mi>d</ab:mi></ab:math> operators. We fit the energy dependence of the extracted scattering phase shifts using effective-range expansions. We observe sharp peaks in the <gb:math xmlns:gb="http://www.w3.org/1998/Math/MathML" display="inline"><gb:mrow><gb:msup><gb:mrow><gb:mi>B</gb:mi></gb:mrow><gb:mrow><gb:mo stretchy="false">(</gb:mo><gb:mo>*</gb:mo><gb:mo stretchy="false">)</gb:mo></gb:mrow></gb:msup><gb:mtext>−</gb:mtext><gb:mover accent="true"><gb:mrow><gb:mi>D</gb:mi></gb:mrow><gb:mrow><gb:mo stretchy="false">¯</gb:mo></gb:mrow></gb:mover></gb:mrow></gb:math> scattering rates close to the thresholds, which are associated with shallow bound states, either genuine or virtual, a few MeV or less below the <mb:math xmlns:mb="http://www.w3.org/1998/Math/MathML" display="inline"><mb:mrow><mb:msup><mb:mrow><mb:mi>B</mb:mi></mb:mrow><mb:mrow><mb:mo stretchy="false">(</mb:mo><mb:mo>*</mb:mo><mb:mo stretchy="false">)</mb:mo></mb:mrow></mb:msup><mb:mtext>−</mb:mtext><mb:mover accent="true"><mb:mrow><mb:mi>D</mb:mi></mb:mrow><mb:mrow><mb:mo stretchy="false">¯</mb:mo></mb:mrow></mb:mover></mb:mrow></mb:math> thresholds. In addition, we find hints for resonances with masses of order 100 MeV above the thresholds and decay widths of order 200 MeV.