Gross-Llewellyn Smith sum rule from lattice QCD

We compute the Gross-Llewellyn Smith sum rule, i.e., the lowest odd moment of the parity-violating structure function, <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:msub><a:mi>F</a:mi><a:mn>3</a:mn></a:msub></a:math>, of the nucleon from a lattice QCD calculation of the Compton amplitude. Our calculations are performed on <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:msup><c:mn>48</c:mn><c:mn>3</c:mn></c:msup><c:mo>×</c:mo><c:mn>96</c:mn></c:math> lattices at the <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mi>S</e:mi><e:mi>U</e:mi><e:mo stretchy="false">(</e:mo><e:mn>3</e:mn><e:mo stretchy="false">)</e:mo></e:math> symmetric point for two lattice spacings. We extract the moments for several values of the current momenta in the range <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:mn>0.5</i:mn><i:mo>≲</i:mo><i:msup><i:mi>Q</i:mi><i:mn>2</i:mn></i:msup><i:mo>≲</i:mo><i:mn>10</i:mn><i:mtext> </i:mtext><i:mtext> </i:mtext><i:msup><i:mi>GeV</i:mi><i:mn>2</i:mn></i:msup></i:math>, covering both the nonperturbative and perturbative regimes. We compare our moments to the Gross-Llewellyn Smith sum rule and discuss the implications for higher-twist effects, a determination of <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:msub><k:mi>α</k:mi><k:mi>s</k:mi></k:msub><k:mo stretchy="false">(</k:mo><k:msup><k:mi>Q</k:mi><k:mn>2</k:mn></k:msup><k:mo stretchy="false">)</k:mo></k:math> from a hadronic quantity complementing the phenomenological and other lattice approaches, and electroweak box contributions crucial for Cabibbo–Kobayashi–Maskawa matrix unitarity studies.