Résumé
We present a review of our results for the masses and couplings of the scalar fully heavy four-quarks and <math altimg="si2.svg"><msub><mrow><mi>T</mi></mrow><mrow><mi>Q</mi><mi>Q</mi><mover accent="true"><mrow><mi>q</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><msup><mrow><mover accent="true"><mrow><mi>q</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mrow><mrow><mo>′</mo></mrow></msup></mrow></msub><mspace width="0.2em"/><mo stretchy="false">(</mo><msup><mrow><mi>J</mi></mrow><mrow><mi>P</mi></mrow></msup><mo linebreak="badbreak" linebreakstyle="after">=</mo><msup><mrow><mn>0</mn></mrow><mrow><mo>±</mo></mrow></msup><mo>,</mo><msup><mrow><mn>1</mn></mrow><mrow><mo>±</mo></mrow></msup><mo stretchy="false">)</mo></math> tetraquarks states from (inverse) QCD Laplace sum rule (LSR), their ratios <math altimg="si3.svg"><mi mathvariant="script">R</mi></math> and double ratio of sum rules (DRSR) within stability criteria and including Factorized Next-to-Leading Order (FNLO) Perturbative (PT) corrections. As the Operator Product Expansion (OPE) usually converges for <math altimg="si4.svg"><mi>d</mi><mo>⩽</mo><mn>6</mn><mo linebreak="goodbreak" linebreakstyle="after">−</mo><mn>8</mn></math>, we evaluated the QCD spectral functions at Lowest Order (LO) of PT QCD and up to <math altimg="si5.svg"><mo stretchy="false">〈</mo><msup><mrow><mi>G</mi></mrow><mrow><mn>3</mn></mrow></msup><mo stretchy="false">〉</mo></math>. Our results for the <math altimg="si6.svg"><msup><mrow><mn>0</mn></mrow><mrow><mo linebreak="badbreak" linebreakstyle="after">+</mo><mo linebreak="badbreak" linebreakstyle="after">+</mo></mrow></msup></math> fully heavy four-quark states may explain the LHCb broad structure around <math altimg="si7.svg"><mo stretchy="false">(</mo><mn>6.2</mn><mo linebreak="badbreak" linebreakstyle="after">−</mo><mn>6.7</mn><mo stretchy="false">)</mo><mspace width="0.2em"/><mtext>GeV</mtext></math> which can be due to <math altimg="si8.svg"><mover accent="true"><mrow><msub><mrow><mi>η</mi></mrow><mrow><mi>c</mi></mrow></msub></mrow><mo>‾</mo></mover><msub><mrow><mi>η</mi></mrow><mrow><mi>c</mi></mrow></msub><mo>,</mo><mspace width="0.25em"/><mspace width="0.25em"/><mover accent="true"><mrow><msub><mrow><mi>χ</mi></mrow><mrow><mi>c</mi><mn>1</mn></mrow></msub></mrow><mo>‾</mo></mover><msub><mrow><mi>χ</mi></mrow><mrow><mi>c</mi><mn>1</mn></mrow></msub></math> and <math altimg="si9.svg"><mover accent="true"><mrow><mi>J</mi><mo stretchy="false">/</mo><mi>ψ</mi></mrow><mo>‾</mo></mover><mi>J</mi><mo stretchy="false">/</mo><mi>ψ</mi></math> molecules or/and their analogue <math altimg="si10.svg"><msub><mrow><mi>S</mi></mrow><mrow><mi>c</mi></mrow></msub><msub><mrow><mi>S</mi></mrow><mrow><mi>c</mi></mrow></msub><mo>,</mo><mspace width="0.25em"/><mspace width="0.25em"/><msub><mrow><mi>A</mi></mrow><mrow><mi>c</mi></mrow></msub><msub><mrow><mi>A</mi></mrow><mrow><mi>c</mi></mrow></msub></math> and <math altimg="si11.svg"><msub><mrow><mi>V</mi></mrow><mrow><mi>c</mi></mrow></msub><msub><mrow><mi>V</mi></mrow><mrow><mi>c</mi></mrow></msub></math> tetraquarks. The peak at <math altimg="si12.svg"><mo stretchy="false">(</mo><mn>6.8</mn><mo linebreak="badbreak" linebreakstyle="after">−</mo><mn>6.9</mn><mo stretchy="false">)</mo><mspace width="0.2em"/><mtext>GeV</mtext></math> can be identified to the <math altimg="si13.svg"><mover accent="true"><mrow><msub><mrow><mi>χ</mi></mrow><mrow><mi>c</mi><mn>0</mn></mrow></msub></mrow><mo>‾</mo></mover><msub><mrow><mi>χ</mi></mrow><mrow><mi>c</mi><mn>0</mn></mrow></msub></math> molecule or/and the <math altimg="si14.svg"><msub><mrow><mi>P</mi></mrow><mrow><mi>c</mi></mrow></msub><msub><mrow><mi>P</mi></mrow><mrow><mi>c</mi></mrow></msub></math> tetraquark state. Then, combining <math altimg="si3.svg"><mi mathvariant="script">R</mi></math> and DRSR we focus on the analysis of the four-quark nature of <math altimg="si15.svg"><msub><mrow><mi>T</mi></mrow><mrow><mi>c</mi><mi>c</mi><mover accent="true"><mrow><mi>q</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><msup><mrow><mover accent="true"><mrow><mi>q</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mrow><mrow><mo>′</mo></mrow></msup></mrow></msub></math><math altimg="si16.svg"><msup><mrow><mn>1</mn></mrow><mrow><mo>±</mo></mrow></msup></math> and <math altimg="si17.svg"><msup><mrow><mn>0</mn></mrow><mrow><mo>±</mo></mrow></msup></math> states. We show that combining <math altimg="si3.svg"><mi mathvariant="script">R</mi></math> and DRSR can provide more precise results: <math altimg="si18.svg"><msub><mrow><mi>M</mi></mrow><mrow><msubsup><mrow><mi>T</mi></mrow><mrow><mi>c</mi><mi>c</mi></mrow><mrow><msup><mrow><mn>1</mn></mrow><mrow><mo linebreak="badbreak" linebreakstyle="after">+</mo></mrow></msup></mrow></msubsup></mrow></msub><mo linebreak="goodbreak" linebreakstyle="after">=</mo><mn>3886</mn><mo stretchy="false">(</mo><mn>6</mn><mo stretchy="false">)</mo><mspace width="0.2em"/><mtext>MeV</mtext></math> and <math altimg="si19.svg"><msub><mrow><mi>M</mi></mrow><mrow><msubsup><mrow><mi>T</mi></mrow><mrow><mi>c</mi><mi>c</mi></mrow><mrow><msup><mrow><mn>0</mn></mrow><mrow><mo linebreak="badbreak" linebreakstyle="after">+</mo></mrow></msup></mrow></msubsup></mrow></msub><mo linebreak="goodbreak" linebreakstyle="after">=</mo><mn>3883</mn><mo stretchy="false">(</mo><mn>3</mn><mo stretchy="false">)</mo><mspace width="0.2em"/><mtext>MeV</mtext></math>. In a similar way, one obtain the improved mass predictions: <math altimg="si20.svg"><msub><mrow><mi>M</mi></mrow><mrow><msubsup><mrow><mi>T</mi></mrow><mrow><mi>c</mi><mi>c</mi><mover accent="true"><mrow><mi>s</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mover accent="true"><mrow><mi>u</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mrow><mrow><msup><mrow><mn>1</mn></mrow><mrow><mo linebreak="badbreak" linebreakstyle="after">+</mo></mrow></msup></mrow></msubsup></mrow></msub><mo linebreak="goodbreak" linebreakstyle="after">=</mo><mn>3931</mn><mo stretchy="false">(</mo><mn>7</mn><mo stretchy="false">)</mo><mspace width="0.2em"/><mtext>MeV</mtext></math>, <math altimg="si21.svg"><msub><mrow><mi>M</mi></mrow><mrow><msubsup><mrow><mi>T</mi></mrow><mrow><mi>c</mi><mi>c</mi><mover accent="true"><mrow><mi>s</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mover accent="true"><mrow><mi>u</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mrow><mrow><msup><mrow><mn>0</mn></mrow><mrow><mo linebreak="badbreak" linebreakstyle="after">+</mo></mrow></msup></mrow></msubsup></mrow></msub><mo linebreak="goodbreak" linebreakstyle="after">=</mo><mn>3983</mn><mo stretchy="false">(</mo><mn>7</mn><mo stretchy="false">)</mo><mspace width="0.2em"/><mtext>MeV</mtext></math> and <math altimg="si22.svg"><msub><mrow><mi>M</mi></mrow><mrow><msubsup><mrow><mi>T</mi></mrow><mrow><mi>c</mi><mi>c</mi><mover accent="true"><mrow><mi>s</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mover accent="true"><mrow><mi>s</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mrow><mrow><msup><mrow><mn>0</mn></mrow><mrow><mo linebreak="badbreak" linebreakstyle="after">+</mo></mrow></msup></mrow></msubsup></mrow></msub><mo linebreak="goodbreak" linebreakstyle="after">=</mo><mn>3993</mn><mo stretchy="false">(</mo><mn>11</mn><mo stretchy="false">)</mo><mspace width="0.2em"/><mtext>MeV</mtext></math>. From our estimates of the masses of the Pseudoscalar and Vector <math altimg="si15.svg"><msub><mrow><mi>T</mi></mrow><mrow><mi>c</mi><mi>c</mi><mover accent="true"><mrow><mi>q</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><msup><mrow><mover accent="true"><mrow><mi>q</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mrow><mrow><mo>′</mo></mrow></msup></mrow></msub></math> states, we observe that the interpolating currents lead to two classes: Class H (Heavy) states with masses around <math altimg="si23.svg"><mn>6</mn><mspace width="0.2em"/><mtext>GeV</mtext></math> and Class L (Light) states around <math altimg="si24.svg"><mo stretchy="false">(</mo><mn>3.8</mn><mo linebreak="badbreak" linebreakstyle="after">−</mo><mn>4.4</mn><mo stretchy="false">)</mo><mspace width="0.2em"/><mtext>GeV</mtext></math> where the pseudoscalar (resp. all vector states) are below the <math altimg="si25.svg"><mover accent="true"><mrow><mi>D</mi></mrow><mo>‾</mo></mover><msub><mrow><mi>D</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>,</mo><mspace width="0.25em"/><mspace width="0.25em"/><msub><mrow><mover accent="true"><mrow><mi>D</mi></mrow><mo>‾</mo></mover></mrow><mrow><mi>s</mi></mrow></msub><msub><mrow><mi>D</mi></mrow><mrow><mi>s</mi><mn>0</mn></mrow></msub></math> (resp. <math altimg="si26.svg"><mover accent="true"><mrow><mi>D</mi></mrow><mo>‾</mo></mover><msub><mrow><mi>D</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mspace width="0.25em"/><mspace width="0.25em"/><msub><mrow><mover accent="true"><mrow><mi>D</mi></mrow><mo>‾</mo></mover></mrow><mrow><mi>s</mi></mrow></msub><msub><mrow><mi>D</mi></mrow><mrow><mi>s</mi><mn>1</mn></mrow></msub></math>) open charm thresholds. Finally, we extend the whole study to the bottom sector and confront our results with the ones from different LSR predictions and some other approaches (lattice, potential models, ⋯) in the literature.