Abstract
We revisit, scrutinize, improve, confirm and complete our previous results [1–4] from the scalar di-gluonium sum rules within the standard SVZ-expansion at N2LO without instantons and beyond the minimal duality ansatz: “one resonance ⊕ QCD continuum” parametrization of the spectral function which is necessary for a better understanding of the complex spectra of the <math altimg="si1.svg"><mi>I</mi><mo linebreak="goodbreak" linebreakstyle="after">=</mo><mn>0</mn></math> scalar mesons. We select different (un)subtracted sum rules (USR) moments of degree ≤ 4 for extracting the two lowest gluonia masses and couplings. We obtain: <math altimg="si2.svg"><mo stretchy="false">[</mo><msub><mrow><mi>M</mi></mrow><mrow><msub><mrow><mi>σ</mi></mrow><mrow><mi>B</mi></mrow></msub></mrow></msub><mo>,</mo><msub><mrow><mi>f</mi></mrow><mrow><msub><mrow><mi>σ</mi></mrow><mrow><mi>B</mi></mrow></msub></mrow></msub><mo stretchy="false">]</mo><mo linebreak="goodbreak" linebreakstyle="after">=</mo><mo stretchy="false">[</mo><mn>1.07</mn><mo stretchy="false">(</mo><mn>13</mn><mo stretchy="false">)</mo><mo>,</mo><mn>0.46</mn><mo stretchy="false">(</mo><mn>16</mn><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>,</mo><mspace width="0.25em"/><mo stretchy="false">[</mo><msub><mrow><mi>M</mi></mrow><mrow><msub><mrow><mi>G</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msub><mo>,</mo><msub><mrow><mi>f</mi></mrow><mrow><msub><mrow><mi>G</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msub><mo stretchy="false">]</mo><mo linebreak="goodbreak" linebreakstyle="after">=</mo><mo stretchy="false">[</mo><mn>1.55</mn><mo stretchy="false">(</mo><mn>12</mn><mo stretchy="false">)</mo><mo>,</mo><mn>0.37</mn><mo stretchy="false">(</mo><mn>11</mn><mo stretchy="false">)</mo><mo stretchy="false">]</mo></math> GeV and the corresponding masses of the radial excitations: <math altimg="si3.svg"><msub><mrow><mi>M</mi></mrow><mrow><msubsup><mrow><mi>σ</mi></mrow><mrow><mi>B</mi></mrow><mrow><mo>′</mo></mrow></msubsup></mrow></msub><mo linebreak="goodbreak" linebreakstyle="after">=</mo><mn>1.11</mn><mo stretchy="false">(</mo><mn>12</mn><mo stretchy="false">)</mo></math> and <math altimg="si4.svg"><msub><mrow><mi>M</mi></mrow><mrow><msubsup><mrow><mi>G</mi></mrow><mrow><mn>1</mn></mrow><mrow><mo>′</mo></mrow></msubsup></mrow></msub><mo linebreak="goodbreak" linebreakstyle="after">=</mo><mn>1.56</mn><mo stretchy="false">(</mo><mn>14</mn><mo stretchy="false">)</mo></math> GeV which are (unexpectedly) almost degenerated with the ground states. The 2nd radial excitation is found to have a much heavier mass: <math altimg="si5.svg"><msub><mrow><mi>M</mi></mrow><mrow><msub><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msub><mo>≃</mo></math> 2.99(22) GeV. Combining these results with some Low-Energy Vertex Sum Rules (LEV-SR), we predict some hadronic widths and classify these states into two groups: – The σ-like (<math altimg="si6.svg"><msub><mrow><mi>σ</mi></mrow><mrow><mi>B</mi></mrow></msub><mo>,</mo><msubsup><mrow><mi>σ</mi></mrow><mrow><mi>B</mi></mrow><mrow><mo>′</mo></mrow></msubsup></math>) which decay copiously to ππ from OZI-violating process and the <math altimg="si207.svg"><msubsup><mrow><mi>σ</mi></mrow><mrow><mi>B</mi></mrow><mrow><mo>′</mo></mrow></msubsup></math> to <math altimg="si8.svg"><mn>2</mn><mo stretchy="false">(</mo><mi>π</mi><mi>π</mi><mo stretchy="false">)</mo><mi>S</mi></math> through σσ. – The G-like <math altimg="si9.svg"><mo stretchy="false">(</mo><msub><mrow><mi>G</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mspace width="0.25em"/><msubsup><mrow><mi>G</mi></mrow><mrow><mn>1</mn></mrow><mrow><mo>′</mo></mrow></msubsup></math> and eventually <math altimg="si10.svg"><msub><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow></msub></math>) which decay into <math altimg="si11.svg"><msup><mrow><mi>η</mi></mrow><mrow><mo>′</mo></mrow></msup><mi>η</mi><mo>,</mo><mspace width="0.25em"/><mi>η</mi><mi>η</mi></math> through the <math altimg="si12.svg"><mi>U</mi><msub><mrow><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><mrow><mi>A</mi></mrow></msub></math> gluonic vertex. Besides some eventual mixings with quarkonia states, we may expect that the observed <math altimg="si13.svg"><mi>σ</mi><mo stretchy="false">/</mo><msub><mrow><mi>f</mi></mrow><mrow><mn>0</mn></mrow></msub><mo stretchy="false">(</mo><mn>500</mn><mo stretchy="false">)</mo></math> and <math altimg="si14.svg"><msub><mrow><mi>f</mi></mrow><mrow><mn>0</mn></mrow></msub><mo stretchy="false">(</mo><mn>1.37</mn><mo stretchy="false">)</mo></math> are σ-like while the <math altimg="si15.svg"><msub><mrow><mi>f</mi></mrow><mrow><mn>0</mn></mrow></msub><mo stretchy="false">(</mo><mn>1.5</mn><mo stretchy="false">)</mo></math> and <math altimg="si16.svg"><msub><mrow><mi>f</mi></mrow><mrow><mn>0</mn></mrow></msub><mo stretchy="false">(</mo><mn>1.7</mn><mo stretchy="false">)</mo></math> are G-like gluonia. The high mass <math altimg="si17.svg"><msub><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow></msub><mo stretchy="false">(</mo><mn>2.99</mn><mo stretchy="false">)</mo></math> can also mix with the <math altimg="si18.svg"><msub><mrow><mi>G</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mspace width="0.25em"/><msubsup><mrow><mi>G</mi></mrow><mrow><mn>1</mn></mrow><mrow><mo>′</mo></mrow></msubsup></math> to bring the gluon component of the gluonia candidates above 2 GeV. We also estimate the conformal charge <math altimg="si19.svg"><msub><mrow><mi>ψ</mi></mrow><mrow><mi>G</mi></mrow></msub><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo linebreak="goodbreak" linebreakstyle="after">=</mo><mn>2.09</mn><mo stretchy="false">(</mo><mn>29</mn><mo stretchy="false">)</mo></math> GeV<sup loc="post">4</sup> and its slope <math altimg="si20.svg"><msup><mrow><mn>10</mn></mrow><mrow><mn>2</mn></mrow></msup><mo>×</mo><msubsup><mrow><mi>ψ</mi></mrow><mrow><mi>G</mi></mrow><mrow><mo>′</mo></mrow></msubsup><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo linebreak="goodbreak" linebreakstyle="after">=</mo><mo linebreak="badbreak" linebreakstyle="after">−</mo><mn>22</mn><mo stretchy="false">(</mo><mn>29</mn><mo stretchy="false">)</mo></math> GeV<sup loc="post">2</sup>. Our results are summarized in Table 1.