P -wave fully charm and fully bottom tetraquark states

We have studied the mass spectra of the <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>P</a:mi></a:math>-wave fully charm and fully bottom tetraquark states in the framework of quantum chromodynamics (QCD) sum rules. We construct the interpolating currents by inserting the covariant derivative operator <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:msub><c:mover accent="true"><c:mi mathvariant="script">D</c:mi><c:mo stretchy="false">↔</c:mo></c:mover><c:mi>μ</c:mi></c:msub></c:math> between the S-wave diquark and antidiquark fields. The excitation structures show that the pure <h:math xmlns:h="http://www.w3.org/1998/Math/MathML" display="inline"><h:mi>λ</h:mi></h:math>-mode excited <j:math xmlns:j="http://www.w3.org/1998/Math/MathML" display="inline"><j:mi>P</j:mi></j:math>-wave fully heavy tetraquarks exist for the quantum numbers <l:math xmlns:l="http://www.w3.org/1998/Math/MathML" display="inline"><l:msup><l:mi>J</l:mi><l:mrow><l:mi>P</l:mi><l:mi>C</l:mi></l:mrow></l:msup><l:mo>=</l:mo><l:msup><l:mn>1</l:mn><l:mrow><l:mo>−</l:mo><l:mo>−</l:mo></l:mrow></l:msup><l:mo>,</l:mo><l:msup><l:mn>1</l:mn><l:mrow><l:mo>−</l:mo><l:mo>+</l:mo></l:mrow></l:msup><l:mo>,</l:mo><l:msup><l:mn>2</l:mn><l:mrow><l:mo>−</l:mo><l:mo>−</l:mo></l:mrow></l:msup><l:mo>,</l:mo><l:msup><l:mn>2</l:mn><l:mrow><l:mo>−</l:mo><l:mo>+</l:mo></l:mrow></l:msup></l:math>, and <n:math xmlns:n="http://www.w3.org/1998/Math/MathML" display="inline"><n:msup><n:mn>3</n:mn><n:mrow><n:mo>−</n:mo><n:mo>−</n:mo></n:mrow></n:msup></n:math>, while it is difficult to separate the <p:math xmlns:p="http://www.w3.org/1998/Math/MathML" display="inline"><p:mi>λ</p:mi></p:math>-mode and <r:math xmlns:r="http://www.w3.org/1998/Math/MathML" display="inline"><r:mi>ρ</r:mi></r:math>-mode excitations in the <t:math xmlns:t="http://www.w3.org/1998/Math/MathML" display="inline"><t:msup><t:mn>0</t:mn><t:mrow><t:mo>−</t:mo><t:mo>+</t:mo></t:mrow></t:msup></t:math> channel. Within three Lorentz indices, there is no pure <v:math xmlns:v="http://www.w3.org/1998/Math/MathML" display="inline"><v:mi>λ</v:mi></v:math>-mode excited <x:math xmlns:x="http://www.w3.org/1998/Math/MathML" display="inline"><x:mi>P</x:mi></x:math>-wave fully charm/bottom tetraquark operators with <z:math xmlns:z="http://www.w3.org/1998/Math/MathML" display="inline"><z:msup><z:mi>J</z:mi><z:mrow><z:mi>P</z:mi><z:mi>C</z:mi></z:mrow></z:msup><z:mo>=</z:mo><z:msup><z:mn>0</z:mn><z:mrow><z:mo>−</z:mo><z:mo>−</z:mo></z:mrow></z:msup></z:math> and <bb:math xmlns:bb="http://www.w3.org/1998/Math/MathML" display="inline"><bb:msup><bb:mn>3</bb:mn><bb:mrow><bb:mo>−</bb:mo><bb:mo>+</bb:mo></bb:mrow></bb:msup></bb:math>. Our results support that the recent observed <db:math xmlns:db="http://www.w3.org/1998/Math/MathML" display="inline"><db:mi>X</db:mi><db:mo stretchy="false">(</db:mo><db:mn>6900</db:mn><db:mo stretchy="false">)</db:mo></db:math> and <hb:math xmlns:hb="http://www.w3.org/1998/Math/MathML" display="inline"><hb:mi>X</hb:mi><hb:mo stretchy="false">(</hb:mo><hb:mn>7200</hb:mn><hb:mo stretchy="false">)</hb:mo></hb:math> resonances could be interpreted as the <lb:math xmlns:lb="http://www.w3.org/1998/Math/MathML" display="inline"><lb:mi>P</lb:mi></lb:math>-wave fully charm <nb:math xmlns:nb="http://www.w3.org/1998/Math/MathML" display="inline"><nb:mi>c</nb:mi><nb:mi>c</nb:mi><nb:mover accent="true"><nb:mi>c</nb:mi><nb:mo stretchy="false">¯</nb:mo></nb:mover><nb:mover accent="true"><nb:mi>c</nb:mi><nb:mo stretchy="false">¯</nb:mo></nb:mover></nb:math> tetraquark states with <tb:math xmlns:tb="http://www.w3.org/1998/Math/MathML" display="inline"><tb:msup><tb:mi>J</tb:mi><tb:mrow><tb:mi>P</tb:mi><tb:mi>C</tb:mi></tb:mrow></tb:msup><tb:mo>=</tb:mo><tb:msup><tb:mn>1</tb:mn><tb:mrow><tb:mo>−</tb:mo><tb:mo>+</tb:mo></tb:mrow></tb:msup></tb:math> and <vb:math xmlns:vb="http://www.w3.org/1998/Math/MathML" display="inline"><vb:msup><vb:mn>2</vb:mn><vb:mrow><vb:mo>−</vb:mo><vb:mo>+</vb:mo></vb:mrow></vb:msup></vb:math>, respectively. Some <xb:math xmlns:xb="http://www.w3.org/1998/Math/MathML" display="inline"><xb:mi>P</xb:mi></xb:math>-wave fully bottom <zb:math xmlns:zb="http://www.w3.org/1998/Math/MathML" display="inline"><zb:mi>b</zb:mi><zb:mi>b</zb:mi><zb:mover accent="true"><zb:mi>b</zb:mi><zb:mo stretchy="false">¯</zb:mo></zb:mover><zb:mover accent="true"><zb:mi>b</zb:mi><zb:mo stretchy="false">¯</zb:mo></zb:mover></zb:math> tetraquark states are predicted to be lower than the di-<fc:math xmlns:fc="http://www.w3.org/1998/Math/MathML" display="inline"><fc:msub><fc:mi>η</fc:mi><fc:mi>b</fc:mi></fc:msub><fc:mo stretchy="false">(</fc:mo><fc:mn>1</fc:mn><fc:mi>S</fc:mi><fc:mo stretchy="false">)</fc:mo></fc:math> and di-<jc:math xmlns:jc="http://www.w3.org/1998/Math/MathML" display="inline"><jc:mi mathvariant="normal">ϒ</jc:mi><jc:mo stretchy="false">(</jc:mo><jc:mn>1</jc:mn><jc:mi>S</jc:mi><jc:mo stretchy="false">)</jc:mo></jc:math> mass thresholds. Hopefully our calculations will be useful for identifying the nature of new exotic tetraquark states.