Study on the properties of hidden-charm pentaquarks with double strangeness

Since the LHCb collaboration successively reported the hidden-charm exotic states <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:msub><a:mi>P</a:mi><a:mi>c</a:mi></a:msub></a:math> and <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:msub><c:mi>P</c:mi><c:mrow><c:mi>c</c:mi><c:mi>s</c:mi></c:mrow></c:msub></c:math>, these exotic states have rightfully attracted significant attention from theoretical perspectives. Motivated by these exotic states, this study systematically investigates the hidden-charm double-strange pentaquark system with the quark configuration <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mi>n</e:mi><e:mi>s</e:mi><e:mi>s</e:mi><e:mi>c</e:mi><e:mover accent="true"><e:mi>c</e:mi><e:mo stretchy="false">¯</e:mo></e:mover></e:math> using the resonating group method within the framework of the quark delocalization color screening model. The dynamical estimations, incorporating channel coupling, identify five resonance states in the hidden-charm double-strange systems, denoted as <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:msubsup><i:mi mathvariant="normal">Ξ</i:mi><i:mi>c</i:mi><i:mo>′</i:mo></i:msubsup><i:msubsup><i:mover accent="true"><i:mi>D</i:mi><i:mo stretchy="false">¯</i:mo></i:mover><i:mi>s</i:mi><i:mo>*</i:mo></i:msubsup></i:math> and <n:math xmlns:n="http://www.w3.org/1998/Math/MathML" display="inline"><n:msubsup><n:mi mathvariant="normal">Ξ</n:mi><n:mi>c</n:mi><n:mo>*</n:mo></n:msubsup><n:msubsup><n:mover accent="true"><n:mi>D</n:mi><n:mo stretchy="false">¯</n:mo></n:mover><n:mi>s</n:mi><n:mo>*</n:mo></n:msubsup></n:math> with <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"><s:msup><s:mi>J</s:mi><s:mi>P</s:mi></s:msup><s:mo>=</s:mo><s:mn>1</s:mn><s:mo>/</s:mo><s:msup><s:mn>2</s:mn><s:mo>−</s:mo></s:msup></s:math>, <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"><u:msubsup><u:mi mathvariant="normal">Ξ</u:mi><u:mi>c</u:mi><u:mo>*</u:mo></u:msubsup><u:msub><u:mover accent="true"><u:mi>D</u:mi><u:mo stretchy="false">¯</u:mo></u:mover><u:mi>s</u:mi></u:msub></u:math>, <z:math xmlns:z="http://www.w3.org/1998/Math/MathML" display="inline"><z:msubsup><z:mi mathvariant="normal">Ξ</z:mi><z:mi>c</z:mi><z:mo>*</z:mo></z:msubsup><z:msubsup><z:mover accent="true"><z:mi>D</z:mi><z:mo stretchy="false">¯</z:mo></z:mover><z:mi>s</z:mi><z:mo>*</z:mo></z:msubsup></z:math>, and <eb:math xmlns:eb="http://www.w3.org/1998/Math/MathML" display="inline"><eb:msup><eb:mi mathvariant="normal">Ω</eb:mi><eb:mo>*</eb:mo></eb:msup><eb:msup><eb:mover accent="true"><eb:mi>D</eb:mi><eb:mo stretchy="false">¯</eb:mo></eb:mover><eb:mo>*</eb:mo></eb:msup></eb:math> with <jb:math xmlns:jb="http://www.w3.org/1998/Math/MathML" display="inline"><jb:msup><jb:mi>J</jb:mi><jb:mi>P</jb:mi></jb:msup><jb:mo>=</jb:mo><jb:mn>3</jb:mn><jb:mo>/</jb:mo><jb:msup><jb:mn>2</jb:mn><jb:mo>−</jb:mo></jb:msup></jb:math>, through scattering phase shifts processes of various channels. <lb:math xmlns:lb="http://www.w3.org/1998/Math/MathML" display="inline"><lb:msubsup><lb:mi mathvariant="normal">Ξ</lb:mi><lb:mi>c</lb:mi><lb:mo>′</lb:mo></lb:msubsup><lb:msubsup><lb:mover accent="true"><lb:mi>D</lb:mi><lb:mo stretchy="false">¯</lb:mo></lb:mover><lb:mi>s</lb:mi><lb:mo>*</lb:mo></lb:msubsup></lb:math> and <qb:math xmlns:qb="http://www.w3.org/1998/Math/MathML" display="inline"><qb:msubsup><qb:mi mathvariant="normal">Ξ</qb:mi><qb:mi>c</qb:mi><qb:mo>*</qb:mo></qb:msubsup><qb:msubsup><qb:mover accent="true"><qb:mi>D</qb:mi><qb:mo stretchy="false">¯</qb:mo></qb:mover><qb:mi>s</qb:mi><qb:mo>*</qb:mo></qb:msubsup></qb:math> with <vb:math xmlns:vb="http://www.w3.org/1998/Math/MathML" display="inline"><vb:msup><vb:mi>J</vb:mi><vb:mi>P</vb:mi></vb:msup><vb:mo>=</vb:mo><vb:mn>1</vb:mn><vb:mo>/</vb:mo><vb:msup><vb:mn>2</vb:mn><vb:mo>−</vb:mo></vb:msup></vb:math>, with mass range and decay widths of (4684 MeV–4688 MeV, 6.4 MeV–24.2 MeV) and (4751 MeV–4756 MeV, 3.2 MeV–18.7 MeV), can be derived from the scattering processes of channels <xb:math xmlns:xb="http://www.w3.org/1998/Math/MathML" display="inline"><xb:mi mathvariant="normal">Ξ</xb:mi><xb:msub><xb:mi>η</xb:mi><xb:mi>c</xb:mi></xb:msub></xb:math>, <ac:math xmlns:ac="http://www.w3.org/1998/Math/MathML" display="inline"><ac:mi mathvariant="normal">Ξ</ac:mi><ac:mi>J</ac:mi><ac:mo>/</ac:mo><ac:mi>ψ</ac:mi></ac:math>, and <dc:math xmlns:dc="http://www.w3.org/1998/Math/MathML" display="inline"><dc:msub><dc:mi mathvariant="normal">Ξ</dc:mi><dc:mi>c</dc:mi></dc:msub><dc:msub><dc:mover accent="true"><dc:mi>D</dc:mi><dc:mo stretchy="false">¯</dc:mo></dc:mover><dc:mi>s</dc:mi></dc:msub></dc:math>, it should be noted that resonance state <ic:math xmlns:ic="http://www.w3.org/1998/Math/MathML" display="inline"><ic:msubsup><ic:mi mathvariant="normal">Ξ</ic:mi><ic:mi>c</ic:mi><ic:mo>′</ic:mo></ic:msubsup><ic:msubsup><ic:mover accent="true"><ic:mi>D</ic:mi><ic:mo stretchy="false">¯</ic:mo></ic:mover><ic:mi>s</ic:mi><ic:mo>*</ic:mo></ic:msubsup></ic:math>, in addition to the channels mentioned above, can also be observed in <nc:math xmlns:nc="http://www.w3.org/1998/Math/MathML" display="inline"><nc:msup><nc:mi mathvariant="normal">Ξ</nc:mi><nc:mo>*</nc:mo></nc:msup><nc:mi>J</nc:mi><nc:mo>/</nc:mo><nc:mi>ψ</nc:mi></nc:math> and <qc:math xmlns:qc="http://www.w3.org/1998/Math/MathML" display="inline"><qc:msub><qc:mi mathvariant="normal">Ξ</qc:mi><qc:mi>c</qc:mi></qc:msub><qc:msubsup><qc:mover accent="true"><qc:mi>D</qc:mi><qc:mo stretchy="false">¯</qc:mo></qc:mover><qc:mi>s</qc:mi><qc:mo>*</qc:mo></qc:msubsup></qc:math>. For the quantum number <vc:math xmlns:vc="http://www.w3.org/1998/Math/MathML" display="inline"><vc:msup><vc:mi>J</vc:mi><vc:mi>P</vc:mi></vc:msup><vc:mo>=</vc:mo><vc:mn>3</vc:mn><vc:mo>/</vc:mo><vc:msup><vc:mn>2</vc:mn><vc:mo>−</vc:mo></vc:msup></vc:math>, <xc:math xmlns:xc="http://www.w3.org/1998/Math/MathML" display="inline"><xc:msubsup><xc:mi mathvariant="normal">Ξ</xc:mi><xc:mi>c</xc:mi><xc:mo>*</xc:mo></xc:msubsup><xc:msubsup><xc:mover accent="true"><xc:mi>D</xc:mi><xc:mo stretchy="false">¯</xc:mo></xc:mover><xc:mi>s</xc:mi><xc:mo>*</xc:mo></xc:msubsup></xc:math> and <cd:math xmlns:cd="http://www.w3.org/1998/Math/MathML" display="inline"><cd:msup><cd:mi mathvariant="normal">Ω</cd:mi><cd:mo>*</cd:mo></cd:msup><cd:msup><cd:mover accent="true"><cd:mi>D</cd:mi><cd:mo stretchy="false">¯</cd:mo></cd:mover><cd:mo>*</cd:mo></cd:msup></cd:math> can be identified in channels <hd:math xmlns:hd="http://www.w3.org/1998/Math/MathML" display="inline"><hd:mi mathvariant="normal">Ξ</hd:mi><hd:mi>J</hd:mi><hd:mo>/</hd:mo><hd:mi>ψ</hd:mi></hd:math>, <kd:math xmlns:kd="http://www.w3.org/1998/Math/MathML" display="inline"><kd:msup><kd:mi mathvariant="normal">Ξ</kd:mi><kd:mo>*</kd:mo></kd:msup><kd:mi>J</kd:mi><kd:mo>/</kd:mo><kd:mi>ψ</kd:mi></kd:math> and <nd:math xmlns:nd="http://www.w3.org/1998/Math/MathML" display="inline"><nd:msub><nd:mi mathvariant="normal">Ξ</nd:mi><nd:mi>c</nd:mi></nd:msub><nd:msubsup><nd:mover accent="true"><nd:mi>D</nd:mi><nd:mo stretchy="false">¯</nd:mo></nd:mover><nd:mi>s</nd:mi><nd:mo>*</nd:mo></nd:msubsup></nd:math>, and <sd:math xmlns:sd="http://www.w3.org/1998/Math/MathML" display="inline"><sd:msup><sd:mi mathvariant="normal">Ξ</sd:mi><sd:mo>*</sd:mo></sd:msup><sd:mi>J</sd:mi><sd:mo>/</sd:mo><sd:mi>ψ</sd:mi></sd:math> and <vd:math xmlns:vd="http://www.w3.org/1998/Math/MathML" display="inline"><vd:msubsup><vd:mi mathvariant="normal">Ξ</vd:mi><vd:mi>c</vd:mi><vd:mo>′</vd:mo></vd:msubsup><vd:msubsup><vd:mover accent="true"><vd:mi>D</vd:mi><vd:mo stretchy="false">¯</vd:mo></vd:mover><vd:mi>s</vd:mi><vd:mo>*</vd:mo></vd:msubsup></vd:math>, respectively, the estimated masses and decay widths of these resonance states range from approximately (4749 MeV–4755 MeV, 16.4 MeV) and (4771 MeV-4772 MeV, 18.8 MeV). However, the residual resonance state <ae:math xmlns:ae="http://www.w3.org/1998/Math/MathML" display="inline"><ae:msubsup><ae:mi mathvariant="normal">Ξ</ae:mi><ae:mi>c</ae:mi><ae:mo>*</ae:mo></ae:msubsup><ae:msub><ae:mover accent="true"><ae:mi>D</ae:mi><ae:mo stretchy="false">¯</ae:mo></ae:mover><ae:mi>s</ae:mi></ae:msub></ae:math>, for which the estimated mass and decay width are about 4600 MeV and 21.7 MeV, is expected to be found only in channel <fe:math xmlns:fe="http://www.w3.org/1998/Math/MathML" display="inline"><fe:msub><fe:mi mathvariant="normal">Ξ</fe:mi><fe:mi>c</fe:mi></fe:msub><fe:msubsup><fe:mover accent="true"><fe:mi>D</fe:mi><fe:mo stretchy="false">¯</fe:mo></fe:mover><fe:mi>s</fe:mi><fe:mo>*</fe:mo></fe:msubsup></fe:math>. Given the current theoretical estimations, experiments are anticipated to further explore hidden-charm double-strange pentaquark states through the possible decay channels, thereby offering a strong theoretical foundation for their experimental investigation.