Assessment of ϒ -states above BB¯ -threshold using a constituent-quark-model based meson-meson coupled-channels framework

The <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi mathvariant="normal">ϒ</a:mi><a:mo stretchy="false">(</a:mo><a:mn>10753</a:mn><a:mo stretchy="false">)</a:mo></a:math> state has been recently observed by the Belle and Belle II collaborations with enough global significance to motivate an assessment of the high-energy spectrum usually predicted by any reasonable quark model. In the framework of a constituent quark model which satisfactorily describes a wide range of properties of conventional hadrons containing heavy quarks, the quark-antiquark and meson-meson degrees of freedom have been incorporated with the goal of elucidating the influence of open-bottom meson-meson thresholds into the <f:math xmlns:f="http://www.w3.org/1998/Math/MathML" display="inline"><f:mi mathvariant="normal">ϒ</f:mi></f:math> states whose masses are within the energy range of the <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:mi mathvariant="normal">ϒ</i:mi><i:mo stretchy="false">(</i:mo><i:mn>10753</i:mn><i:mo stretchy="false">)</i:mo></i:math>’s mass. It is well known that such effects could be relevant enough as to dynamically generate new states and thus provide a plausible explanation of the nature of the <n:math xmlns:n="http://www.w3.org/1998/Math/MathML" display="inline"><n:mi mathvariant="normal">ϒ</n:mi><n:mo stretchy="false">(</n:mo><n:mn>10753</n:mn><n:mo stretchy="false">)</n:mo></n:math> state. In particular, we have performed a coupled-channels calculation in which the bare states <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"><s:mi mathvariant="normal">ϒ</s:mi><s:mo stretchy="false">(</s:mo><s:mn>4</s:mn><s:mi>S</s:mi><s:mo stretchy="false">)</s:mo></s:math>, <x:math xmlns:x="http://www.w3.org/1998/Math/MathML" display="inline"><x:mi mathvariant="normal">ϒ</x:mi><x:mo stretchy="false">(</x:mo><x:mn>3</x:mn><x:mi>D</x:mi><x:mo stretchy="false">)</x:mo></x:math>, <cb:math xmlns:cb="http://www.w3.org/1998/Math/MathML" display="inline"><cb:mi mathvariant="normal">ϒ</cb:mi><cb:mo stretchy="false">(</cb:mo><cb:mn>5</cb:mn><cb:mi>S</cb:mi><cb:mo stretchy="false">)</cb:mo></cb:math>, and <hb:math xmlns:hb="http://www.w3.org/1998/Math/MathML" display="inline"><hb:mi mathvariant="normal">ϒ</hb:mi><hb:mo stretchy="false">(</hb:mo><hb:mn>4</hb:mn><hb:mi>D</hb:mi><hb:mo stretchy="false">)</hb:mo></hb:math> are considered together with the threshold channels <mb:math xmlns:mb="http://www.w3.org/1998/Math/MathML" display="inline"><mb:mi>B</mb:mi><mb:mover accent="true"><mb:mi>B</mb:mi><mb:mo stretchy="false">¯</mb:mo></mb:mover></mb:math>, <qb:math xmlns:qb="http://www.w3.org/1998/Math/MathML" display="inline"><qb:mi>B</qb:mi><qb:msup><qb:mover accent="true"><qb:mi>B</qb:mi><qb:mo stretchy="false">¯</qb:mo></qb:mover><qb:mo>*</qb:mo></qb:msup></qb:math>, <ub:math xmlns:ub="http://www.w3.org/1998/Math/MathML" display="inline"><ub:msup><ub:mi>B</ub:mi><ub:mo>*</ub:mo></ub:msup><ub:msup><ub:mover accent="true"><ub:mi>B</ub:mi><ub:mo stretchy="false">¯</ub:mo></ub:mover><ub:mo>*</ub:mo></ub:msup></ub:math>, <yb:math xmlns:yb="http://www.w3.org/1998/Math/MathML" display="inline"><yb:msub><yb:mi>B</yb:mi><yb:mi>s</yb:mi></yb:msub><yb:msub><yb:mover accent="true"><yb:mi>B</yb:mi><yb:mo stretchy="false">¯</yb:mo></yb:mover><yb:mi>s</yb:mi></yb:msub></yb:math>, <cc:math xmlns:cc="http://www.w3.org/1998/Math/MathML" display="inline"><cc:msub><cc:mi>B</cc:mi><cc:mi>s</cc:mi></cc:msub><cc:msubsup><cc:mover accent="true"><cc:mi>B</cc:mi><cc:mo stretchy="false">¯</cc:mo></cc:mover><cc:mi>s</cc:mi><cc:mo>*</cc:mo></cc:msubsup></cc:math>, and <gc:math xmlns:gc="http://www.w3.org/1998/Math/MathML" display="inline"><gc:msubsup><gc:mi>B</gc:mi><gc:mi>s</gc:mi><gc:mo>*</gc:mo></gc:msubsup><gc:msubsup><gc:mover accent="true"><gc:mi>B</gc:mi><gc:mo stretchy="false">¯</gc:mo></gc:mover><gc:mi>s</gc:mi><gc:mo>*</gc:mo></gc:msubsup></gc:math>. Among the results we have described, the following are of particular interest: (i) a richer complex spectrum is gained when thresholds are present and bare bound states are sufficiently nonrelativistic; (ii) those poles obtained in the complex energy plane do not have to appear as simple peaks in the relevant cross sections; and (iii) the <kc:math xmlns:kc="http://www.w3.org/1998/Math/MathML" display="inline"><kc:mi mathvariant="normal">ϒ</kc:mi><kc:mo stretchy="false">(</kc:mo><kc:mn>10753</kc:mn><kc:mo stretchy="false">)</kc:mo></kc:math> candidate is interpreted as a dressed hadronic resonance whose structure is an equally mixture of a conventional <pc:math xmlns:pc="http://www.w3.org/1998/Math/MathML" display="inline"><pc:mi>b</pc:mi><pc:mover accent="true"><pc:mi>b</pc:mi><pc:mo stretchy="false">¯</pc:mo></pc:mover></pc:math> state and <tc:math xmlns:tc="http://www.w3.org/1998/Math/MathML" display="inline"><tc:msup><tc:mi>B</tc:mi><tc:mo>*</tc:mo></tc:msup><tc:msup><tc:mover accent="true"><tc:mi>B</tc:mi><tc:mo stretchy="false">¯</tc:mo></tc:mover><tc:mo>*</tc:mo></tc:msup></tc:math> molecule. Published by the American Physical Society 2024