Abstract
In this paper, we attempt to discuss the topics mentioned in the title by scrutinizing and improving the <math altimg="si1.svg"><msup><mrow><mn>0</mn></mrow><mrow><mo linebreak="badbreak" linebreakstyle="after">−</mo><mo linebreak="badbreak" linebreakstyle="after">+</mo></mrow></msup></math> pseudoscalar di-gluonia/glueballs sum rules within the standard SVZ-expansion at N2LO without instantons. First, we reconsider the estimate of the slope of the topological charge <math altimg="si2.svg"><msqrt><mrow><msup><mrow><mi>χ</mi></mrow><mrow><mo>′</mo></mrow></msup><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></msqrt><mo stretchy="false">(</mo><msup><mrow><mi>Q</mi></mrow><mrow><mn>2</mn></mrow></msup><mo linebreak="badbreak" linebreakstyle="after">=</mo><mn>2</mn><mspace width="0.25em"/><msup><mrow><mtext>GeV</mtext></mrow><mrow><mn>2</mn></mrow></msup><mo stretchy="false">)</mo><mo linebreak="goodbreak" linebreakstyle="after">=</mo><mn>24.3</mn><mo stretchy="false">(</mo><mn>3.4</mn><mo stretchy="false">)</mo></math> MeV from low degree moments which imply <math altimg="si3.svg"><msubsup><mrow><mo>∫</mo></mrow><mrow><mn>0</mn></mrow><mrow><mn>1</mn></mrow></msubsup><mspace width="0.2em"/><mi>d</mi><mi>x</mi><mspace width="0.2em"/><msubsup><mrow><mi>g</mi></mrow><mrow><mn>1</mn></mrow><mrow><mi>P</mi></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo linebreak="goodbreak" linebreakstyle="after">=</mo><mn>0.144</mn><mo stretchy="false">(</mo><mn>5</mn><mo stretchy="false">)</mo><mspace width="0.25em"/><mo stretchy="false">[</mo><mrow><mi mathvariant="normal">data</mi><mo>:</mo><mspace width="0.25em"/><mn mathvariant="normal">0.145</mn><mo stretchy="false">(</mo><mn mathvariant="normal">14</mn><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mrow></math> for the first moment of the polarized proton structure function (proton spin) and <math altimg="si4.svg"><msubsup><mrow><mi>G</mi></mrow><mrow><mi>A</mi></mrow><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></msubsup><mo stretchy="false">(</mo><mn>10</mn><mspace width="0.25em"/><msup><mrow><mtext>GeV</mtext></mrow><mrow><mn>2</mn></mrow></msup><mo stretchy="false">)</mo><mo>≡</mo><mrow><mi mathvariant="normal">Δ</mi><mi mathvariant="normal">u</mi><mo linebreak="goodbreak" linebreakstyle="after">+</mo><mi mathvariant="normal">Δ</mi><mi mathvariant="normal">d</mi><mo linebreak="goodbreak" linebreakstyle="after">+</mo><mi mathvariant="normal">Δ</mi><mi mathvariant="normal">s</mi></mrow><mo linebreak="goodbreak" linebreakstyle="after">=</mo><mn>0.340</mn><mo stretchy="false">(</mo><mn>50</mn><mo stretchy="false">)</mo><mspace width="0.25em"/><mo stretchy="false">[</mo><mrow><mi mathvariant="normal">data</mi><mo linebreak="goodbreak" linebreakstyle="after">=</mo><mn mathvariant="normal">0.330</mn><mo stretchy="false">(</mo><mn mathvariant="normal">39</mn><mo stretchy="false">)</mo></mrow><mo stretchy="false">]</mo></math> for the singlet form factor of the axial current. Second, we work with high degree moments and parametrize the spectral function beyond the minimal duality ansatz: “One resonance ⊕ QCD continuum” to get the <math altimg="si1.svg"><msup><mrow><mn>0</mn></mrow><mrow><mo linebreak="badbreak" linebreakstyle="after">−</mo><mo linebreak="badbreak" linebreakstyle="after">+</mo></mrow></msup></math> pseudoscalar di-gluonia spectra. Then, we obtain three groups of gluonia: The familiar light<math altimg="si5.svg"><msub><mrow><mi mathvariant="italic">η</mi></mrow><mrow><mn>1</mn></mrow></msub></math> [singlet gluon component of the <math altimg="si6.svg"><msup><mrow><mi>η</mi></mrow><mrow><mo>′</mo></mrow></msup><mo stretchy="false">(</mo><mn>958</mn><mo stretchy="false">)</mo></math>] with <math altimg="si7.svg"><mo stretchy="false">[</mo><msub><mrow><mi>M</mi></mrow><mrow><msub><mrow><mi>η</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msub><mo>,</mo><msub><mrow><mi>f</mi></mrow><mrow><msub><mrow><mi>η</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>825</mn><mo stretchy="false">(</mo><mn>45</mn><mo stretchy="false">)</mo><mo>,</mo><mn>905</mn><mo stretchy="false">(</mo><mn>72</mn><mo stretchy="false">)</mo><mo stretchy="false">]</mo></math> MeV which is important for understanding the <math altimg="si8.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> anomaly; The two new medium gluonia with <math altimg="si9.svg"><msub><mrow><mi>M</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mn>1</mn><mi>a</mi></mrow></msub></mrow></msub><mo linebreak="goodbreak" linebreakstyle="after">=</mo><mn>1338</mn><mo stretchy="false">(</mo><mn>112</mn><mo stretchy="false">)</mo></math> MeV and <math altimg="si10.svg"><mo stretchy="false">[</mo><msub><mrow><mi>M</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mn>1</mn><mi>b</mi></mrow></msub></mrow></msub><mo linebreak="goodbreak" linebreakstyle="after">=</mo><mn>1462</mn><mo stretchy="false">(</mo><mn>117</mn><mo stretchy="false">)</mo></math> MeV or their mean <math altimg="si11.svg"><mo stretchy="false">[</mo><msub><mrow><mi>M</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msub><mo>,</mo><msub><mrow><mi>f</mi></mrow><mrow><msub><mrow><mi>P</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>1397</mn><mo stretchy="false">(</mo><mn>81</mn><mo stretchy="false">)</mo><mo>,</mo><mn>594</mn><mo stretchy="false">(</mo><mn>144</mn><mo stretchy="false">)</mo><mo stretchy="false">]</mo></math> MeV which support the gluonium nature of the excellent experimental candidate <math altimg="si12.svg"><mi>η</mi><mo stretchy="false">(</mo><mn>1405</mn><mo stretchy="false">)</mo></math> and may bring a small gluon piece to the <math altimg="si13.svg"><mi>η</mi><mo stretchy="false">(</mo><mn>1295</mn><mo stretchy="false">)</mo></math>; their corresponding 1st radial excitations: <math altimg="si14.svg"><msub><mrow><mi>M</mi></mrow><mrow><msubsup><mrow><mi>P</mi></mrow><mrow><mn>1</mn><mi>a</mi></mrow><mrow><mo>′</mo></mrow></msubsup></mrow></msub><mo linebreak="goodbreak" linebreakstyle="after">=</mo><mn>1508</mn><mo stretchy="false">(</mo><mn>226</mn><mo stretchy="false">)</mo></math> MeV and <math altimg="si15.svg"><msub><mrow><mi>M</mi></mrow><mrow><msubsup><mrow><mi>P</mi></mrow><mrow><mn>1</mn><mi>b</mi></mrow><mrow><mo>′</mo></mrow></msubsup></mrow></msub><mo linebreak="goodbreak" linebreakstyle="after">=</mo><mn>1553</mn><mo stretchy="false">(</mo><mn>139</mn><mo stretchy="false">)</mo></math> MeV with their mean: <math altimg="si16.svg"><mo stretchy="false">[</mo><msub><mrow><mi>M</mi></mrow><mrow><msubsup><mrow><mi>P</mi></mrow><mrow><mn>1</mn></mrow><mrow><mo>′</mo></mrow></msubsup></mrow></msub><mo>,</mo><msub><mrow><mi>f</mi></mrow><mrow><msubsup><mrow><mi>P</mi></mrow><mrow><mn>1</mn></mrow><mrow><mo>′</mo></mrow></msubsup></mrow></msub><mo stretchy="false">]</mo><mo linebreak="goodbreak" linebreakstyle="after">=</mo><mo stretchy="false">[</mo><mn>1541</mn><mo stretchy="false">(</mo><mn>118</mn><mo stretchy="false">)</mo><mo>,</mo><mn>205</mn><mo stretchy="false">(</mo><mn>282</mn><mo stretchy="false">)</mo><mo stretchy="false">]</mo></math> MeV which may be identified (up to some eventual mixings with <math altimg="si17.svg"><mover accent="true"><mrow><mi>q</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mi>q</mi></math> states) with the observed <math altimg="si18.svg"><mi>η</mi><mo stretchy="false">(</mo><mn>1475</mn><mo>,</mo><mn>1700</mn><mo stretchy="false">)</mo></math> states; The heavy gluonium with the mean mass: <math altimg="si19.svg"><mo stretchy="false">[</mo><msub><mrow><mi>M</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msub><mo>,</mo><msub><mrow><mi>f</mi></mrow><mrow><msub><mrow><mi>P</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msub><mo stretchy="false">]</mo><mo linebreak="goodbreak" linebreakstyle="after">=</mo><mo stretchy="false">[</mo><mn>2751</mn><mo stretchy="false">(</mo><mn>140</mn><mo stretchy="false">)</mo><mo>,</mo><mn>500</mn><mo stretchy="false">(</mo><mn>43</mn><mo stretchy="false">)</mo><mo stretchy="false">]</mo></math> MeV which can be compared with the lattice results. One can remark the (natural) one to one correspondence between the pseudoscalar gluonia and their chiral scalar analogue from Ref. [1]: <math altimg="si20.svg"><mi>σ</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">→</mo><msub><mrow><mi>η</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>;</mo><mspace width="0.25em"/><msub><mrow><mi>G</mi></mrow><mrow><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><mn>1.55</mn><mo stretchy="false">)</mo><mo stretchy="false">→</mo><msub><mrow><mi>P</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>;</mo><mspace width="0.25em"/><mo stretchy="false">[</mo><msup><mrow><mi>σ</mi></mrow><mrow><mo>′</mo></mrow></msup><mo stretchy="false">(</mo><mn>1.1</mn><mo stretchy="false">)</mo><mo>,</mo><msup><mrow><mi>G</mi></mrow><mrow><mo>′</mo></mrow></msup><mo stretchy="false">(</mo><mn>1.56</mn><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo stretchy="false">→</mo><msubsup><mrow><mi>P</mi></mrow><mrow><mn>1</mn><mi>a</mi><mo>,</mo><mn>1</mn><mi>b</mi></mrow><mrow><mo>′</mo></mrow></msubsup><mo>;</mo><mspace width="0.25em"/><msub><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow></msub><mo stretchy="false">(</mo><mn>3</mn><mo stretchy="false">)</mo><mo stretchy="false">→</mo><msub><mrow><mi>P</mi></mrow><mrow><mn>2</mn></mrow></msub></math> which is mainly due to the importance of the QCD PT contributions in the sum rule analysis that are almost equal in these two channels.