<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>X</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>3872</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:math> and hidden charmed tetraquarks

In a constituent quark model, a hidden charmed tetraquark is assumed consisting of a <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>c</a:mi><a:mi>q</a:mi></a:math> diquark and an <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mover accent="true"><c:mi>c</c:mi><c:mo stretchy="false">¯</c:mo></c:mover><c:mover accent="true"><c:mi>q</c:mi><c:mo stretchy="false">¯</c:mo></c:mover></c:math> antidiquark or vice versa. The Semay-Silvestre-Brac potentials are employed to calculate the masses of <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:mi>c</i:mi><i:mi>q</i:mi></i:math> (<k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:mrow><k:mi mathvariant="normal">q</k:mi><k:mo>=</k:mo><k:mi mathvariant="normal">u</k:mi></k:mrow></k:math>, d) diquarks. The mass of the <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"><o:mi>c</o:mi><o:mi>q</o:mi></o:math> diquark or <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"><q:mover accent="true"><q:mi>c</q:mi><q:mo stretchy="false">¯</q:mo></q:mover><q:mover accent="true"><q:mi>q</q:mi><q:mo stretchy="false">¯</q:mo></q:mover></q:math> antidiquark with spin-0 is predicted with <w:math xmlns:w="http://www.w3.org/1998/Math/MathML" display="inline"><w:mo>∼</w:mo><w:mn>2175</w:mn><w:mtext> </w:mtext><w:mtext> </w:mtext><w:mi>MeV</w:mi></w:math>, and the spin-1 one is predicted with <y:math xmlns:y="http://www.w3.org/1998/Math/MathML" display="inline"><y:mo>∼</y:mo><y:mn>2220</y:mn><y:mtext> </y:mtext><y:mtext> </y:mtext><y:mi>MeV</y:mi></y:math>. The masses of hidden charmed tetraquarks from <ab:math xmlns:ab="http://www.w3.org/1998/Math/MathML" display="inline"><ab:mn>1</ab:mn><ab:mi>S</ab:mi></ab:math> to <cb:math xmlns:cb="http://www.w3.org/1998/Math/MathML" display="inline"><cb:mn>2</cb:mn><cb:mi>P</cb:mi></cb:math> excitations are systematically calculated in terms of the same potentials. It is found that the mass of hidden charmed tetraquarks without radial excitation grows higher in <eb:math xmlns:eb="http://www.w3.org/1998/Math/MathML" display="inline"><eb:msup><eb:mn>1</eb:mn><eb:mrow><eb:mo>+</eb:mo><eb:mo>−</eb:mo></eb:mrow></eb:msup><eb:mo>,</eb:mo><eb:msup><eb:mn>1</eb:mn><eb:mrow><eb:mo>+</eb:mo><eb:mo>+</eb:mo></eb:mrow></eb:msup><eb:mo>,</eb:mo><eb:msup><eb:mn>1</eb:mn><eb:mrow><eb:mo>−</eb:mo><eb:mo>−</eb:mo></eb:mrow></eb:msup><eb:mo>,</eb:mo><eb:msup><eb:mn>0</eb:mn><eb:mrow><eb:mo>−</eb:mo><eb:mo>+</eb:mo></eb:mrow></eb:msup><eb:mo>,</eb:mo><eb:msup><eb:mn>0</eb:mn><eb:mrow><eb:mo>−</eb:mo><eb:mo>−</eb:mo></eb:mrow></eb:msup><eb:mo>,</eb:mo><eb:msup><eb:mn>1</eb:mn><eb:mrow><eb:mo>−</eb:mo><eb:mo>+</eb:mo></eb:mrow></eb:msup><eb:mo>,</eb:mo><eb:mo>…</eb:mo></eb:math> sequence, and the tetraquarks with exotic <gb:math xmlns:gb="http://www.w3.org/1998/Math/MathML" display="inline"><gb:msup><gb:mi>J</gb:mi><gb:mrow><gb:mi>P</gb:mi><gb:mi>C</gb:mi></gb:mrow></gb:msup><gb:mo>=</gb:mo><gb:msup><gb:mn>0</gb:mn><gb:mrow><gb:mo>−</gb:mo><gb:mo>−</gb:mo></gb:mrow></gb:msup><gb:mo>,</gb:mo><gb:msup><gb:mn>1</gb:mn><gb:mrow><gb:mo>−</gb:mo><gb:mo>+</gb:mo></gb:mrow></gb:msup></gb:math> have higher masses. The hidden charmed tetraquarks with radial excitations have masses greater than 4300 MeV. The <ib:math xmlns:ib="http://www.w3.org/1998/Math/MathML" display="inline"><ib:mn>1</ib:mn><ib:mi>S</ib:mi><ib:mo>−</ib:mo><ib:mn>1</ib:mn><ib:mi>P</ib:mi></ib:math> and <kb:math xmlns:kb="http://www.w3.org/1998/Math/MathML" display="inline"><kb:mn>1</kb:mn><kb:mi>S</kb:mi><kb:mo>−</kb:mo><kb:mn>2</kb:mn><kb:mi>S</kb:mi></kb:math> mass splittings of the hidden charmed tetraquarks are about 390–400 and 550–570 MeV, respectively, which are about 50 and 40 MeV smaller than those of normal charmoniums. The <mb:math xmlns:mb="http://www.w3.org/1998/Math/MathML" display="inline"><mb:mn>1</mb:mn><mb:mi>P</mb:mi><mb:mo>−</mb:mo><mb:mn>2</mb:mn><mb:mi>P</mb:mi></mb:math> and <ob:math xmlns:ob="http://www.w3.org/1998/Math/MathML" display="inline"><ob:mn>2</ob:mn><ob:mi>S</ob:mi><ob:mo>−</ob:mo><ob:mn>2</ob:mn><ob:mi>P</ob:mi></ob:math> mass splittings are similar to those for conventional <qb:math xmlns:qb="http://www.w3.org/1998/Math/MathML" display="inline"><qb:mi>c</qb:mi><qb:mover accent="true"><qb:mi>c</qb:mi><qb:mo stretchy="false">¯</qb:mo></qb:mover></qb:math> charmonium mesons. Based on our predicted masses for hidden charmed tetraquarks, some XYZ exotics are analyzed and tentatively assigned. <ub:math xmlns:ub="http://www.w3.org/1998/Math/MathML" display="inline"><ub:msup><ub:mi>X</ub:mi><ub:mo>*</ub:mo></ub:msup><ub:mo stretchy="false">(</ub:mo><ub:mn>3860</ub:mn><ub:mo stretchy="false">)</ub:mo></ub:math> is possibly the <yb:math xmlns:yb="http://www.w3.org/1998/Math/MathML" display="inline"><yb:msup><yb:mn>0</yb:mn><yb:mrow><yb:mo>+</yb:mo><yb:mo>+</yb:mo></yb:mrow></yb:msup></yb:math> tetraquark. <ac:math xmlns:ac="http://www.w3.org/1998/Math/MathML" display="inline"><ac:msub><ac:mi>Z</ac:mi><ac:mi>c</ac:mi></ac:msub><ac:mo stretchy="false">(</ac:mo><ac:mn>3900</ac:mn><ac:mo stretchy="false">)</ac:mo></ac:math> and <ec:math xmlns:ec="http://www.w3.org/1998/Math/MathML" display="inline"><ec:mi>X</ec:mi><ec:mo stretchy="false">(</ec:mo><ec:mn>3940</ec:mn><ec:mo stretchy="false">)</ec:mo></ec:math> are possibly the <ic:math xmlns:ic="http://www.w3.org/1998/Math/MathML" display="inline"><ic:msup><ic:mn>1</ic:mn><ic:mrow><ic:mo>+</ic:mo><ic:mo>−</ic:mo></ic:mrow></ic:msup></ic:math> tetraquarks, and <kc:math xmlns:kc="http://www.w3.org/1998/Math/MathML" display="inline"><kc:mi>X</kc:mi><kc:mo stretchy="false">(</kc:mo><kc:mn>3872</kc:mn><kc:mo stretchy="false">)</kc:mo></kc:math> is possibly a <oc:math xmlns:oc="http://www.w3.org/1998/Math/MathML" display="inline"><oc:msup><oc:mn>1</oc:mn><oc:mrow><oc:mo>+</oc:mo><oc:mo>+</oc:mo></oc:mrow></oc:msup></oc:math> tetraquark. <qc:math xmlns:qc="http://www.w3.org/1998/Math/MathML" display="inline"><qc:mrow><qc:mi>X</qc:mi><qc:mo stretchy="false">(</qc:mo><qc:mn>4250</qc:mn><qc:mo stretchy="false">)</qc:mo></qc:mrow></qc:math> may be a <uc:math xmlns:uc="http://www.w3.org/1998/Math/MathML" display="inline"><uc:msup><uc:mn>0</uc:mn><uc:mrow><uc:mo>−</uc:mo><uc:mo>+</uc:mo></uc:mrow></uc:msup></uc:math>, <wc:math xmlns:wc="http://www.w3.org/1998/Math/MathML" display="inline"><wc:msup><wc:mn>0</wc:mn><wc:mrow><wc:mo>+</wc:mo><wc:mo>+</wc:mo></wc:mrow></wc:msup></wc:math>, or <yc:math xmlns:yc="http://www.w3.org/1998/Math/MathML" display="inline"><yc:msup><yc:mn>1</yc:mn><yc:mrow><yc:mo>−</yc:mo><yc:mo>+</yc:mo></yc:mrow></yc:msup></yc:math> tetraquark, <ad:math xmlns:ad="http://www.w3.org/1998/Math/MathML" display="inline"><ad:mi>X</ad:mi><ad:mo stretchy="false">(</ad:mo><ad:mn>4240</ad:mn><ad:mo stretchy="false">)</ad:mo></ad:math> may be a <ed:math xmlns:ed="http://www.w3.org/1998/Math/MathML" display="inline"><ed:msup><ed:mn>0</ed:mn><ed:mrow><ed:mo>−</ed:mo><ed:mo>−</ed:mo></ed:mrow></ed:msup></ed:math> tetraquark. With radial excitations, <gd:math xmlns:gd="http://www.w3.org/1998/Math/MathML" display="inline"><gd:mi>X</gd:mi><gd:mo stretchy="false">(</gd:mo><gd:mn>4350</gd:mn><gd:mo stretchy="false">)</gd:mo></gd:math> may be a <kd:math xmlns:kd="http://www.w3.org/1998/Math/MathML" display="inline"><kd:msup><kd:mn>0</kd:mn><kd:mrow><kd:mo>+</kd:mo><kd:mo>+</kd:mo></kd:mrow></kd:msup></kd:math> tetraquark, <md:math xmlns:md="http://www.w3.org/1998/Math/MathML" display="inline"><md:msub><md:mi>Z</md:mi><md:mi>c</md:mi></md:msub><md:mo stretchy="false">(</md:mo><md:mn>4430</md:mn><md:mo stretchy="false">)</md:mo></md:math> may be a <qd:math xmlns:qd="http://www.w3.org/1998/Math/MathML" display="inline"><qd:msup><qd:mn>1</qd:mn><qd:mrow><qd:mo>+</qd:mo><qd:mo>−</qd:mo></qd:mrow></qd:msup></qd:math> tetraquark, <sd:math xmlns:sd="http://www.w3.org/1998/Math/MathML" display="inline"><sd:mi>X</sd:mi><sd:mo stretchy="false">(</sd:mo><sd:mn>4630</sd:mn><sd:mo stretchy="false">)</sd:mo></sd:math> may be a <wd:math xmlns:wd="http://www.w3.org/1998/Math/MathML" display="inline"><wd:msup><wd:mn>0</wd:mn><wd:mrow><wd:mo>−</wd:mo><wd:mo>+</wd:mo></wd:mrow></wd:msup></wd:math> or <yd:math xmlns:yd="http://www.w3.org/1998/Math/MathML" display="inline"><yd:msup><yd:mn>1</yd:mn><yd:mrow><yd:mo>−</yd:mo><yd:mo>+</yd:mo></yd:mrow></yd:msup></yd:math> tetraquark, and <ae:math xmlns:ae="http://www.w3.org/1998/Math/MathML" display="inline"><ae:mi>X</ae:mi><ae:mo stretchy="false">(</ae:mo><ae:mi>Y</ae:mi><ae:mo stretchy="false">)</ae:mo><ae:mo stretchy="false">(</ae:mo><ae:mn>4660</ae:mn><ae:mo stretchy="false">)</ae:mo></ae:math> may be the <ge:math xmlns:ge="http://www.w3.org/1998/Math/MathML" display="inline"><ge:msup><ge:mn>1</ge:mn><ge:mrow><ge:mo>−</ge:mo><ge:mo>−</ge:mo></ge:mrow></ge:msup></ge:math> tetraquark. <ie:math xmlns:ie="http://www.w3.org/1998/Math/MathML" display="inline"><ie:mrow><ie:mi>Y</ie:mi><ie:mo stretchy="false">(</ie:mo><ie:mn>4008</ie:mn><ie:mo stretchy="false">)</ie:mo></ie:mrow></ie:math> or <me:math xmlns:me="http://www.w3.org/1998/Math/MathML" display="inline"><me:mi>Y</me:mi><me:mo stretchy="false">(</me:mo><me:mn>4390</me:mn><me:mo stretchy="false">)</me:mo></me:math> seems impossible to be the <qe:math xmlns:qe="http://www.w3.org/1998/Math/MathML" display="inline"><qe:msup><qe:mn>1</qe:mn><qe:mrow><qe:mo>−</qe:mo><qe:mo>−</qe:mo></qe:mrow></qe:msup></qe:math> tetraquark. In experiments, for the identification of a normal charmonium or a charmonium-like exotic, it is also important to take into account the mixing of normal charmoniums with hidden-flavor charmonium-like exotics.