Résumé
In this talk, I review the results obtained recently in Ref. [1]. First, we estimate the LO hadronic vacuum polarization contribution to the muon and τ anomalous magnetic moments to be: <math altimg="si2.svg"><msub><mrow><mi>a</mi></mrow><mrow><mi>μ</mi></mrow></msub><msubsup><mrow><mo stretchy="false">|</mo></mrow><mrow><mi>l</mi><mo>.</mo><mi>o</mi></mrow><mrow><mi>h</mi><mi>v</mi><mi>p</mi></mrow></msubsup><mo linebreak="goodbreak" linebreakstyle="after">=</mo><mo stretchy="false">(</mo><mn>7036.5</mn><mo>±</mo><mn>38.9</mn><mo stretchy="false">)</mo><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mo linebreak="badbreak" linebreakstyle="after">−</mo><mn>11</mn></mrow></msup></math>, <math altimg="si3.svg"><msub><mrow><mi>a</mi></mrow><mrow><mi>τ</mi></mrow></msub><msubsup><mrow><mo stretchy="false">|</mo></mrow><mrow><mi>l</mi><mo>.</mo><mi>o</mi></mrow><mrow><mi>h</mi><mi>v</mi><mi>p</mi></mrow></msubsup><mo linebreak="goodbreak" linebreakstyle="after">=</mo><mo stretchy="false">(</mo><mn>3494.8</mn><mo>±</mo><mn>24.7</mn><mo stretchy="false">)</mo><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mo linebreak="badbreak" linebreakstyle="after">−</mo><mn>9</mn></mrow></msup></math> (see Table 1) leading to: <math altimg="si4.svg"><mi mathvariant="normal">Δ</mi><msub><mrow><mi>a</mi></mrow><mrow><mi>μ</mi></mrow></msub><mo>≡</mo><msubsup><mrow><mi>a</mi></mrow><mrow><mi>μ</mi></mrow><mrow><mi>e</mi><mi>x</mi><mi>p</mi></mrow></msubsup><mo linebreak="goodbreak" linebreakstyle="after">−</mo><msubsup><mrow><mi>a</mi></mrow><mrow><mi>μ</mi></mrow><mrow><mi>S</mi><mi>M</mi></mrow></msubsup><mo linebreak="goodbreak" linebreakstyle="after">=</mo><mo stretchy="false">(</mo><mn>143</mn><mo>±</mo><msub><mrow><mn>42</mn></mrow><mrow><mi>t</mi><mi>h</mi></mrow></msub><mo>±</mo><msub><mrow><mn>22</mn></mrow><mrow><mi>e</mi><mi>x</mi><mi>p</mi></mrow></msub><mo stretchy="false">)</mo><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mo linebreak="badbreak" linebreakstyle="after">−</mo><mn>11</mn></mrow></msup></math> which is about 3σ discrepancy between the SM predictions and experiment. One also finds: <math altimg="si5.svg"><msup><mrow><mi>α</mi></mrow><mrow><mo stretchy="false">(</mo><mn>5</mn><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">(</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>Z</mi></mrow></msub><mo stretchy="false">)</mo><msub><mrow><mo stretchy="false">|</mo></mrow><mrow><mi>h</mi><mi>a</mi><mi>d</mi></mrow></msub><mo linebreak="goodbreak" linebreakstyle="after">=</mo><mo stretchy="false">(</mo><mn>2766.3</mn><mo>±</mo><mn>4.5</mn><mo stretchy="false">)</mo><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mo linebreak="badbreak" linebreakstyle="after">−</mo><mn>5</mn></mrow></msup></math>. Second, we estimate the QCD power corrections up to dimension 20 from the ratio of Laplace sum rule and from τ-like decay high moments (see Table 3). We do not observe any exponential growth of their size which may not favour a duality violation of the spectral function. We obtain <math altimg="si6.svg"><mo stretchy="false">〈</mo><msub><mrow><mi>α</mi></mrow><mrow><mi>s</mi></mrow></msub><msup><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow></msup><mo stretchy="false">〉</mo><mo linebreak="goodbreak" linebreakstyle="after">=</mo><mo stretchy="false">(</mo><mn>7.8</mn><mo>±</mo><mn>3.5</mn><mo stretchy="false">)</mo><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mo linebreak="badbreak" linebreakstyle="after">−</mo><mn>2</mn></mrow></msup><mspace width="0.25em"/><msup><mrow><mtext>GeV</mtext></mrow><mrow><mn>4</mn></mrow></msup></math> in agreement with the more precise one from heavy quark sum rules, while <math altimg="si7.svg"><mi>ρ</mi><msub><mrow><mi>α</mi></mrow><mrow><mi>s</mi></mrow></msub><msup><mrow><mo stretchy="false">〈</mo><mover accent="true"><mrow><mi>ψ</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mi>ψ</mi><mo stretchy="false">〉</mo></mrow><mrow><mn>2</mn></mrow></msup><mo linebreak="goodbreak" linebreakstyle="after">=</mo><mo stretchy="false">(</mo><mn>5.98</mn><mo>±</mo><mn>0.64</mn><mo stretchy="false">)</mo><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mo linebreak="badbreak" linebreakstyle="after">−</mo><mn>4</mn></mrow></msup><mspace width="0.25em"/><msup><mrow><mtext>GeV</mtext></mrow><mrow><mn>6</mn></mrow></msup></math> confirms a violation of the four-quark condensate factorization by a factor <math altimg="si8.svg"><mi>ρ</mi><mo>≃</mo><mn>6</mn></math>. Third, using the previous values of the condensates, we re-extract <math altimg="si9.svg"><msub><mrow><mi>α</mi></mrow><mrow><mi>s</mi></mrow></msub></math> from the lowest τ-decay Braaten-SN-Pich (BNP) moment and find to order <math altimg="si10.svg"><msubsup><mrow><mi>α</mi></mrow><mrow><mi>s</mi></mrow><mrow><mn>4</mn></mrow></msubsup><mo>:</mo><msub><mrow><mi>α</mi></mrow><mrow><mi>s</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>τ</mi></mrow></msub><mo stretchy="false">)</mo><mo linebreak="goodbreak" linebreakstyle="after">=</mo><mn>0.3081</mn><mo stretchy="false">(</mo><mn>86</mn><mo stretchy="false">)</mo><mspace width="0.25em"/><mo stretchy="false">[</mo><mrow><mi mathvariant="normal">resp</mi><mo>.</mo></mrow><mspace width="0.25em"/><mn>0.3260</mn><mo stretchy="false">(</mo><mn>79</mn><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo stretchy="false">⟶</mo><mspace width="-0.33cm"/><mo>∘</mo><mspace width="0.25em"/><mspace width="0.2em"/><msub><mrow><mi>α</mi></mrow><mrow><mi>s</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>Z</mi></mrow></msub><mo stretchy="false">)</mo><mo linebreak="goodbreak" linebreakstyle="after">=</mo><mn>0.1170</mn><mo stretchy="false">(</mo><mn>7</mn><mo stretchy="false">)</mo><mspace width="0.25em"/><mo stretchy="false">[</mo><mrow><mi mathvariant="normal">resp</mi><mo>.</mo></mrow><mspace width="0.25em"/><mn>0.1192</mn><mo stretchy="false">(</mo><mn>7</mn><mo stretchy="false">)</mo><mo stretchy="false">]</mo></math> for Fixed Order (FO) [resp. Contour Improved (CI)] PT series. We also show that the contributions beyond the Shifman-Vainshtein-Zakharov (SVZ)-expansion are negligible.