The hadronic running of the electromagnetic coupling and the electroweak mixing angle from lattice QCD
Marco Cè, Antoine Gérardin, Georg von Hippel, Harvey B. Meyer, Kohtaroh Miura, Konstantin Ottnad, Andreas Risch, Teseo San José, Jonas Wilhelm, Hartmut Wittig
Abstract
A bstract We compute the hadronic running of the electromagnetic and weak couplings in lattice QCD with N f = 2 + 1 flavors of $$ \mathcal{O} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>O</mml:mi></mml:math> ( a ) improved Wilson fermions. Using two different discretizations of the vector current, we compute the quark-connected and –disconnected contributions to the hadronic vacuum polarization (HVP) functions $$ \overline{\varPi} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mover><mml:mi>Π</mml:mi><mml:mo>¯</mml:mo></mml:mover></mml:math> γγ and $$ \overline{\varPi} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mover><mml:mi>Π</mml:mi><mml:mo>¯</mml:mo></mml:mover></mml:math> γZ for Euclidean squared momenta Q 2 ≤ 7 GeV 2 . Gauge field ensembles at four values of the lattice spacing and several values of the pion mass, including its physical value, are used to extrapolate the results to the physical point. The ability to perform an exact flavor decomposition allows us to present the most precise determination to date of the SU(3)-flavor-suppressed HVP function $$ \overline{\varPi} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mover><mml:mi>Π</mml:mi><mml:mo>¯</mml:mo></mml:mover></mml:math> 08 that enters the running of sin 2 θ W . Our results for $$ \overline{\varPi} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mover><mml:mi>Π</mml:mi><mml:mo>¯</mml:mo></mml:mover></mml:math> γγ , $$ \overline{\varPi} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mover><mml:mi>Π</mml:mi><mml:mo>¯</mml:mo></mml:mover></mml:math> γZ and $$ \overline{\varPi} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mover><mml:mi>Π</mml:mi><mml:mo>¯</mml:mo></mml:mover></mml:math> 08 are presented in terms of rational functions for continuous values of Q 2 below 7 GeV 2 . We observe a tension of up to 3 . 5 standard deviation between our lattice results for $$ \Delta {\alpha}_{\mathrm{had}}^{(5)} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mtext>∆</mml:mtext><mml:msubsup><mml:mi>α</mml:mi><mml:mi>had</mml:mi><mml:mfenced><mml:mn>5</mml:mn></mml:mfenced></mml:msubsup></mml:math> (− Q 2 ) and estimates based on the R -ratio for space-like momenta in the range 3–7 GeV 2 . The tension is, however, strongly diminished when translating our result to the Z pole, by employing the Euclidean split technique and perturbative QCD, which yields $$ \Delta {\alpha}_{\mathrm{had}}^{(5)}\left({M}_Z^2\right) $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mtext>∆</mml:mtext><mml:msubsup><mml:mi>α</mml:mi><mml:mi>had</mml:mi><mml:mfenced><mml:mn>5</mml:mn></mml:mfenced></mml:msubsup><mml:mfenced><mml:msubsup><mml:mi>M</mml:mi><mml:mi>Z</mml:mi><mml:mn>2</mml:mn></mml:msubsup></mml:mfenced></mml:math> = 0 . 02773(15) and agrees with results based on the R -ratio within the quoted uncertainties.