Abstract
We present improved estimates of the couplings, masses and mass ratios of the <math altimg="si1.svg"><msub><mrow><mi>Z</mi></mrow><mrow><mi>Q</mi></mrow></msub><mo>,</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>Q</mi></mrow></msub></math> 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></math> states (<math altimg="si3.svg"><mi>Q</mi><mo>≡</mo><mi>c</mi><mo>,</mo><mi>b</mi><mspace width="0.25em"/><mo>;</mo><mspace width="0.25em"/><mi>q</mi><mo>,</mo><msup><mrow><mi>q</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>≡</mo><mi>u</mi><mo>,</mo><mi>d</mi><mo>,</mo><mi>s</mi></math>) using (inverse) QCD Laplace sum rules (LSR), their ratios <math altimg="si4.svg"><mi mathvariant="script">R</mi></math> and double ratios DRSR within stability criteria, where the NLO factorized PT QCD corrections are included which is important for giving a meaning on the running <math altimg="si5.svg"><mover accent="true"><mrow><mi>M</mi><mi>S</mi></mrow><mo>‾</mo></mover></math> heavy quark mass used in the analysis. We show that combined <math altimg="si4.svg"><mi mathvariant="script">R</mi></math> and DRSR can provide more precise results. In the 1st part of the paper, we conclude that the observed <math altimg="si6.svg"><msub><mrow><mi>X</mi></mrow><mrow><mi>c</mi></mrow></msub><mo stretchy="false">(</mo><mn>3872</mn><mo stretchy="false">)</mo></math> and <math altimg="si7.svg"><msub><mrow><mi>Z</mi></mrow><mrow><mi>c</mi></mrow></msub><mo stretchy="false">(</mo><mn>3900</mn><mo stretchy="false">)</mo></math> are tetramoles states (superposition of quasi-degenerated molecule and a tetraquark states having (almost) the same coupling to the currents) with the predicted masses: <math altimg="si8.svg"><msub><mrow><mi>M</mi></mrow><mrow><msub><mrow><mi mathvariant="script">T</mi></mrow><mrow><msub><mrow><mi>X</mi></mrow><mrow><mi>c</mi></mrow></msub></mrow></msub></mrow></msub><mo linebreak="goodbreak" linebreakstyle="after">=</mo><mn>3876</mn><mo stretchy="false">(</mo><mn>44</mn><mo stretchy="false">)</mo><mspace width="0.25em"/><mtext>MeV</mtext></math> and <math altimg="si9.svg"><msub><mrow><mi>M</mi></mrow><mrow><msub><mrow><mi mathvariant="script">T</mi></mrow><mrow><msub><mrow><mi>Z</mi></mrow><mrow><mi>c</mi></mrow></msub></mrow></msub></mrow></msub><mo linebreak="goodbreak" linebreakstyle="after">=</mo><mn>3900</mn><mo stretchy="false">(</mo><mn>42</mn><mo stretchy="false">)</mo><mspace width="0.25em"/><mtext>MeV</mtext></math>. In the 2nd part, we focus on the analysis of the four-quark nature of different <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></math><math altimg="si10.svg"><msup><mrow><mn>1</mn></mrow><mrow><mo linebreak="badbreak" linebreakstyle="after">+</mo></mrow></msup></math> and <math altimg="si11.svg"><msup><mrow><mn>0</mn></mrow><mrow><mo linebreak="badbreak" linebreakstyle="after">+</mo></mrow></msup></math> states within the <math altimg="si12.svg"><msub><mrow><mover accent="true"><mrow><mn>3</mn></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mrow><mrow><mi>c</mi></mrow></msub><msub><mrow><mn>3</mn></mrow><mrow><mi>c</mi></mrow></msub></math> interpolating currents. The final results from <math altimg="si4.svg"><mi mathvariant="script">R</mi></math> and <math altimg="si13.svg"><mi mathvariant="script">R</mi><mspace width="0.2em"/><mo>⊕</mo></math> DRSR are summarized in Table 7. Combined <math altimg="si4.svg"><mi mathvariant="script">R</mi></math> and DRSR calibrated to the observed <math altimg="si6.svg"><msub><mrow><mi>X</mi></mrow><mrow><mi>c</mi></mrow></msub><mo stretchy="false">(</mo><mn>3872</mn><mo stretchy="false">)</mo></math> lead to a precise prediction of e.g. <math altimg="si14.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="0.25em"/><mtext>MeV</mtext></math>. In a similar way, the DRSR for the <math altimg="si15.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 stretchy="false">/</mo><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></math> calibrated to <math altimg="si16.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></math> gives <math altimg="si17.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="0.25em"/><mtext>MeV</mtext></math>. The SU3 breaking ratios <math altimg="si18.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 stretchy="false">/</mo><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></math> lead to the improved mass predictions: <math altimg="si19.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>3988</mn><mo stretchy="false">(</mo><mn>12</mn><mo stretchy="false">)</mo><mspace width="0.25em"/><mtext>MeV</mtext></math>. In the 3rd part, the analysis is extended to the beauty mesons, where we find the tetramole masses: <math altimg="si20.svg"><msub><mrow><mi>M</mi></mrow><mrow><msub><mrow><mi mathvariant="script">T</mi></mrow><mrow><msub><mrow><mi>Z</mi></mrow><mrow><mi>b</mi></mrow></msub></mrow></msub></mrow></msub><mo linebreak="goodbreak" linebreakstyle="after">=</mo><mn>10579</mn><mo stretchy="false">(</mo><mn>99</mn><mo stretchy="false">)</mo><mspace width="0.25em"/><mtext>MeV</mtext></math> and <math altimg="si21.svg"><msub><mrow><mi>M</mi></mrow><mrow><msub><mrow><mi>X</mi></mrow><mrow><mi>b</mi></mrow></msub></mrow></msub><mo linebreak="goodbreak" linebreakstyle="after">=</mo><mn>10545</mn><mo stretchy="false">(</mo><mn>131</mn><mo stretchy="false">)</mo><mspace width="0.25em"/><mtext>MeV</mtext></math>. We also observe that the <math altimg="si22.svg"><msubsup><mrow><mi>T</mi></mrow><mrow><mi>b</mi><mi>b</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><mrow><msup><mrow><mn>1</mn></mrow><mrow><mo linebreak="badbreak" linebreakstyle="after">+</mo></mrow></msup><mo>,</mo><msup><mrow><mn>0</mn></mrow><mrow><mo linebreak="badbreak" linebreakstyle="after">+</mo></mrow></msup></mrow></msubsup></math> (<math altimg="si23.svg"><mi>q</mi><mo>,</mo><msup><mrow><mi>q</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>≡</mo><mi>u</mi><mo>,</mo><mi>d</mi><mo>,</mo><mi>s</mi></math>) states are (almost) stable (within the errors) against strong interactions. In the 4th part, we (critically) review and correct some recent LSR estimates of the <math altimg="si24.svg"><msubsup><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><mrow><msup><mrow><mn>1</mn></mrow><mrow><mo linebreak="badbreak" linebreakstyle="after">+</mo></mrow></msup><mo>,</mo><msup><mrow><mn>0</mn></mrow><mrow><mo linebreak="badbreak" linebreakstyle="after">+</mo></mrow></msup></mrow></msubsup></math> masses. Our combined LSR ⊕ DRSR results are confronted with the ones from some other approaches (lattices and quark models) in Fig. 26.