Precision lattice calculation of the hadronic contribution to the running of the electroweak gauge couplings

This is the authors' abstract. We don't add key points for this paper.

We present an update of our lattice QCD determination of the hadronic contribution to the running of the electromagnetic coupling, <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mi mathvariant="normal">Δ</a:mi> <a:msubsup> <a:mi>α</a:mi> <a:mi>had</a:mi> <a:mrow> <a:mo stretchy="false">(</a:mo> <a:mn>5</a:mn> <a:mo stretchy="false">)</a:mo> </a:mrow> </a:msubsup> <a:mo stretchy="false">(</a:mo> <a:mo>−</a:mo> <a:msup> <a:mi>Q</a:mi> <a:mn>2</a:mn> </a:msup> <a:mo stretchy="false">)</a:mo> </a:math> , and of the electroweak mixing angle in the spacelike momentum region up to <h:math xmlns:h="http://www.w3.org/1998/Math/MathML" display="inline"> <h:msup> <h:mi>Q</h:mi> <h:mn>2</h:mn> </h:msup> <h:mo>=</h:mo> <h:mn>12</h:mn> <h:mtext> </h:mtext> <h:mtext> </h:mtext> <h:msup> <h:mi>GeV</h:mi> <h:mn>2</h:mn> </h:msup> </h:math> . The calculation is based on coordinated lattice simulations ensembles with <j:math xmlns:j="http://www.w3.org/1998/Math/MathML" display="inline"> <j:msub> <j:mi>N</j:mi> <j:mi>f</j:mi> </j:msub> <j:mo>=</j:mo> <j:mn>2</j:mn> <j:mo>+</j:mo> <j:mn>1</j:mn> </j:math> flavors of <l:math xmlns:l="http://www.w3.org/1998/Math/MathML" display="inline"> <l:mi>O</l:mi> <l:mo stretchy="false">(</l:mo> <l:mi>a</l:mi> <l:mo stretchy="false">)</l:mo> </l:math> -improved Wilson fermions, covering five lattice spacings between 0.039 and 0.085 fm and a range of pion masses, including the physical point. A refined analysis employing a telescopic window strategy allows for a clean separation of systematic effects across Euclidean distance scales. Statistical precision is further enhanced through low-mode averaging, combined with a spectral reconstruction of the vector-vector correlator at long distances on the most chiral ensembles. We confirm significant tensions of up to <p:math xmlns:p="http://www.w3.org/1998/Math/MathML" display="inline"> <p:mn>7</p:mn> <p:mi>σ</p:mi> </p:math> at spacelike virtualities around <r:math xmlns:r="http://www.w3.org/1998/Math/MathML" display="inline"> <r:msup> <r:mi>Q</r:mi> <r:mn>2</r:mn> </r:msup> <r:mo>=</r:mo> <r:mn>1</r:mn> <r:mtext> </r:mtext> <r:mtext> </r:mtext> <r:msup> <r:mi>GeV</r:mi> <r:mn>2</r:mn> </r:msup> </r:math> between our lattice results for <t:math xmlns:t="http://www.w3.org/1998/Math/MathML" display="inline"> <t:mi mathvariant="normal">Δ</t:mi> <t:msubsup> <t:mi>α</t:mi> <t:mi>had</t:mi> <t:mrow> <t:mo stretchy="false">(</t:mo> <t:mn>5</t:mn> <t:mo stretchy="false">)</t:mo> </t:mrow> </t:msubsup> <t:mo stretchy="false">(</t:mo> <t:mo>−</t:mo> <t:msup> <t:mi>Q</t:mi> <t:mn>2</t:mn> </t:msup> <t:mo stretchy="false">)</t:mo> </t:math> and the corresponding data-driven estimates based on <ab:math xmlns:ab="http://www.w3.org/1998/Math/MathML" display="inline"> <ab:msup> <ab:mi>e</ab:mi> <ab:mo>+</ab:mo> </ab:msup> <ab:msup> <ab:mi>e</ab:mi> <ab:mo>−</ab:mo> </ab:msup> </ab:math> cross section data. Combining our lattice data with perturbative QCD via the Euclidean split technique, we obtain at the <cb:math xmlns:cb="http://www.w3.org/1998/Math/MathML" display="inline"> <cb:mi>Z</cb:mi> </cb:math> pole <eb:math xmlns:eb="http://www.w3.org/1998/Math/MathML" display="inline"> <eb:mi mathvariant="normal">Δ</eb:mi> <eb:msubsup> <eb:mi>α</eb:mi> <eb:mi>had</eb:mi> <eb:mrow> <eb:mo stretchy="false">(</eb:mo> <eb:mn>5</eb:mn> <eb:mo stretchy="false">)</eb:mo> </eb:mrow> </eb:msubsup> <eb:mo stretchy="false">(</eb:mo> <eb:msubsup> <eb:mi>M</eb:mi> <eb:mi>Z</eb:mi> <eb:mn>2</eb:mn> </eb:msubsup> <eb:mo stretchy="false">)</eb:mo> <eb:mo>=</eb:mo> <eb:mn>0.027821</eb:mn> <eb:mo stretchy="false">(</eb:mo> <eb:mn>34</eb:mn> <eb:msub> <eb:mo stretchy="false">)</eb:mo> <eb:mi>lat</eb:mi> </eb:msub> <eb:mo stretchy="false">(</eb:mo> <eb:mn>35</eb:mn> <eb:msub> <eb:mo stretchy="false">)</eb:mo> <eb:mtext>pQCD</eb:mtext> </eb:msub> </eb:math> , which is more than two times more precise than recent data-driven estimates. Our result deviatesslightly, by <pb:math xmlns:pb="http://www.w3.org/1998/Math/MathML" display="inline"> <pb:mn>1</pb:mn> <pb:mo>−</pb:mo> <pb:mn>2</pb:mn> <pb:mi>σ</pb:mi> </pb:math> , from the value produced by global electroweak fits. For the electroweak mixing angle, we present the hadronic contribution to its running and provide a precise determination of the octet-singlet mixing component <rb:math xmlns:rb="http://www.w3.org/1998/Math/MathML" display="inline"> <rb:msup> <rb:mover accent="true"> <rb:mi mathvariant="normal">Π</rb:mi> <rb:mo stretchy="false">¯</rb:mo> </rb:mover> <rb:mrow> <rb:mo stretchy="false">(</rb:mo> <rb:mn>0</rb:mn> <rb:mo>,</rb:mo> <rb:mn>8</rb:mn> <rb:mo stretchy="false">)</rb:mo> </rb:mrow> </rb:msup> </rb:math> , in good agreement with phenomenological models but with significantly higher precision.