SU(3) breaking effect in the Zc and Zcs states

Based on the hadronic resonance picture, we explore a unified framework to describe the observed <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:msub><a:mi>Z</a:mi><a:mrow><a:mi>c</a:mi><a:mi>s</a:mi></a:mrow></a:msub></a:math> and <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:msub><c:mi>Z</c:mi><c:mi>c</c:mi></c:msub></c:math> states that are close to the thresholds of the <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mrow><e:msup><e:mrow><e:mi>D</e:mi></e:mrow><e:mrow><e:mo stretchy="false">(</e:mo><e:mo>*</e:mo><e:mo stretchy="false">)</e:mo></e:mrow></e:msup><e:msubsup><e:mrow><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:mrow><e:mi>s</e:mi></e:mrow><e:mrow><e:mo stretchy="false">(</e:mo><e:mo>*</e:mo><e:mo stretchy="false">)</e:mo></e:mrow></e:msubsup></e:mrow></e:math> and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"><m:mrow><m:msup><m:mrow><m:mi>D</m:mi></m:mrow><m:mrow><m:mo stretchy="false">(</m:mo><m:mo>*</m:mo><m:mo stretchy="false">)</m:mo></m:mrow></m:msup><m:msup><m:mrow><m:mover accent="true"><m:mrow><m:mi>D</m:mi></m:mrow><m:mrow><m:mo stretchy="false">¯</m:mo></m:mrow></m:mover></m:mrow><m:mrow><m:mo stretchy="false">(</m:mo><m:mo>*</m:mo><m:mo stretchy="false">)</m:mo></m:mrow></m:msup></m:mrow></m:math> systems, respectively. We assume that the <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"><u:msub><u:mi>Z</u:mi><u:mrow><u:mi>c</u:mi><u:mi>s</u:mi></u:mrow></u:msub><u:mo stretchy="false">(</u:mo><u:mn>4000</u:mn><u:mo stretchy="false">)</u:mo></u:math> and <y:math xmlns:y="http://www.w3.org/1998/Math/MathML" display="inline"><y:msub><y:mi>Z</y:mi><y:mrow><y:mi>c</y:mi><y:mi>s</y:mi></y:mrow></y:msub><y:mo stretchy="false">(</y:mo><y:mn>3985</y:mn><y:mo stretchy="false">)</y:mo></y:math> are two different states. Specifically, we assign the <cb:math xmlns:cb="http://www.w3.org/1998/Math/MathML" display="inline"><cb:msub><cb:mi>Z</cb:mi><cb:mrow><cb:mi>c</cb:mi><cb:mi>s</cb:mi></cb:mrow></cb:msub><cb:mo stretchy="false">(</cb:mo><cb:mn>4000</cb:mn><cb:mo stretchy="false">)</cb:mo></cb:math> as the SU(3) partner of the <gb:math xmlns:gb="http://www.w3.org/1998/Math/MathML" display="inline"><gb:msub><gb:mi>Z</gb:mi><gb:mi>c</gb:mi></gb:msub><gb:mo stretchy="false">(</gb:mo><gb:mn>3900</gb:mn><gb:mo stretchy="false">)</gb:mo></gb:math> (<kb:math xmlns:kb="http://www.w3.org/1998/Math/MathML" display="inline"><kb:mrow><kb:msup><kb:mrow><kb:mi>J</kb:mi></kb:mrow><kb:mrow><kb:mi>P</kb:mi><kb:mi>C</kb:mi></kb:mrow></kb:msup><kb:mo>=</kb:mo><kb:msup><kb:mrow><kb:mn>1</kb:mn></kb:mrow><kb:mrow><kb:mo>+</kb:mo><kb:mo>−</kb:mo></kb:mrow></kb:msup></kb:mrow></kb:math>) due to its observation in the <mb:math xmlns:mb="http://www.w3.org/1998/Math/MathML" display="inline"><mb:mi>J</mb:mi><mb:mo>/</mb:mo><mb:mi mathvariant="normal">Ψ</mb:mi><mb:mi>K</mb:mi></mb:math> channel, the difficulty of this assignment is the large width difference between the <pb:math xmlns:pb="http://www.w3.org/1998/Math/MathML" display="inline"><pb:msub><pb:mi>Z</pb:mi><pb:mrow><pb:mi>c</pb:mi><pb:mi>s</pb:mi></pb:mrow></pb:msub><pb:mo stretchy="false">(</pb:mo><pb:mn>4000</pb:mn><pb:mo stretchy="false">)</pb:mo></pb:math> and the <tb:math xmlns:tb="http://www.w3.org/1998/Math/MathML" display="inline"><tb:msub><tb:mi>Z</tb:mi><tb:mi>c</tb:mi></tb:msub><tb:mo stretchy="false">(</tb:mo><tb:mn>3900</tb:mn><tb:mo stretchy="false">)</tb:mo></tb:math> state. Then the <xb:math xmlns:xb="http://www.w3.org/1998/Math/MathML" display="inline"><xb:msub><xb:mi>Z</xb:mi><xb:mrow><xb:mi>c</xb:mi><xb:mi>s</xb:mi></xb:mrow></xb:msub><xb:mo stretchy="false">(</xb:mo><xb:mn>3985</xb:mn><xb:mo stretchy="false">)</xb:mo></xb:math> should corresponds to the <bc:math xmlns:bc="http://www.w3.org/1998/Math/MathML" display="inline"><bc:msub><bc:mi>Z</bc:mi><bc:mi>c</bc:mi></bc:msub></bc:math> state that is close to the threshold of <dc:math xmlns:dc="http://www.w3.org/1998/Math/MathML" display="inline"><dc:mi>D</dc:mi><dc:msup><dc:mover accent="true"><dc:mi>D</dc:mi><dc:mo stretchy="false">¯</dc:mo></dc:mover><dc:mo>*</dc:mo></dc:msup></dc:math> system with <hc:math xmlns:hc="http://www.w3.org/1998/Math/MathML" display="inline"><hc:msup><hc:mi>J</hc:mi><hc:mrow><hc:mi>P</hc:mi><hc:mi>C</hc:mi></hc:mrow></hc:msup><hc:mo>=</hc:mo><hc:msup><hc:mn>1</hc:mn><hc:mrow><hc:mo>+</hc:mo><hc:mo>+</hc:mo></hc:mrow></hc:msup></hc:math>, the difficulty of this assignment is the absence of such a <jc:math xmlns:jc="http://www.w3.org/1998/Math/MathML" display="inline"><jc:msub><jc:mi>Z</jc:mi><jc:mi>c</jc:mi></jc:msub></jc:math> state in experiment. We construct the effective potentials of the <lc:math xmlns:lc="http://www.w3.org/1998/Math/MathML" display="inline"><lc:msub><lc:mi>Z</lc:mi><lc:mrow><lc:mi>c</lc:mi><lc:mi>s</lc:mi></lc:mrow></lc:msub></lc:math> and <nc:math xmlns:nc="http://www.w3.org/1998/Math/MathML" display="inline"><nc:msub><nc:mi>Z</nc:mi><nc:mi>c</nc:mi></nc:msub></nc:math> states by analogy with the effective potentials of the leading-order and next-to-leading-order <pc:math xmlns:pc="http://www.w3.org/1998/Math/MathML" display="inline"><pc:mi>N</pc:mi><pc:mo>−</pc:mo><pc:mover accent="true"><pc:mi>N</pc:mi><pc:mo stretchy="false">¯</pc:mo></pc:mover></pc:math> interactions. Then we introduce an SU(3) breaking factor <tc:math xmlns:tc="http://www.w3.org/1998/Math/MathML" display="inline"><tc:msub><tc:mi>g</tc:mi><tc:mi>x</tc:mi></tc:msub></tc:math> to identify the differences between the effective potentials in the <vc:math xmlns:vc="http://www.w3.org/1998/Math/MathML" display="inline"><vc:msub><vc:mi>Z</vc:mi><vc:mrow><vc:mi>c</vc:mi><vc:mi>s</vc:mi></vc:mrow></vc:msub></vc:math> and <xc:math xmlns:xc="http://www.w3.org/1998/Math/MathML" display="inline"><xc:msub><xc:mi>Z</xc:mi><xc:mi>c</xc:mi></xc:msub></xc:math> states. We perform two calculations to discuss the differences and similarities of the <zc:math xmlns:zc="http://www.w3.org/1998/Math/MathML" display="inline"><zc:msub><zc:mi>Z</zc:mi><zc:mrow><zc:mi>c</zc:mi><zc:mi>s</zc:mi></zc:mrow></zc:msub></zc:math> and <bd:math xmlns:bd="http://www.w3.org/1998/Math/MathML" display="inline"><bd:msub><bd:mi>Z</bd:mi><bd:mi>c</bd:mi></bd:msub></bd:math> states. In the first calculation, we show that the above two difficulties can be explained simultaneously by the inclusion of the SU(3) breaking effect. In the second calculation, we show that this framework is promising to simultaneously describe the interactions of the observed <dd:math xmlns:dd="http://www.w3.org/1998/Math/MathML" display="inline"><dd:msub><dd:mi>Z</dd:mi><dd:mrow><dd:mi>c</dd:mi><dd:mi>s</dd:mi></dd:mrow></dd:msub></dd:math> and <fd:math xmlns:fd="http://www.w3.org/1998/Math/MathML" display="inline"><fd:msub><fd:mi>Z</fd:mi><fd:mi>c</fd:mi></fd:msub></fd:math> states that are close to the thresholds of the <hd:math xmlns:hd="http://www.w3.org/1998/Math/MathML" display="inline"><hd:msup><hd:mi>D</hd:mi><hd:mrow><hd:mo stretchy="false">(</hd:mo><hd:mo>*</hd:mo><hd:mo stretchy="false">)</hd:mo></hd:mrow></hd:msup><hd:msubsup><hd:mover accent="true"><hd:mi>D</hd:mi><hd:mo stretchy="false">¯</hd:mo></hd:mover><hd:mi>s</hd:mi><hd:mrow><hd:mo stretchy="false">(</hd:mo><hd:mo>*</hd:mo><hd:mo stretchy="false">)</hd:mo></hd:mrow></hd:msubsup></hd:math> and <pd:math xmlns:pd="http://www.w3.org/1998/Math/MathML" display="inline"><pd:msup><pd:mi>D</pd:mi><pd:mrow><pd:mo stretchy="false">(</pd:mo><pd:mo>*</pd:mo><pd:mo stretchy="false">)</pd:mo></pd:mrow></pd:msup><pd:msup><pd:mover accent="true"><pd:mi>D</pd:mi><pd:mo stretchy="false">¯</pd:mo></pd:mover><pd:mrow><pd:mo stretchy="false">(</pd:mo><pd:mo>*</pd:mo><pd:mo stretchy="false">)</pd:mo></pd:mrow></pd:msup></pd:math> systems, respectively.