Flavor diagonal nucleon charges using clover fermions on MILC HISQ ensembles

We present lattice results for the flavor diagonal charges of the proton from the analysis of eight ensembles generated using <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mrow><a:mn>2</a:mn><a:mo>+</a:mo><a:mn>1</a:mn><a:mo>+</a:mo><a:mn>1</a:mn></a:mrow></a:math>-flavors of highly improved staggered quarks by the MILC Collaboration. The calculation includes all the needed connected and disconnected contributions to nucleon three-point function. For extracting matrix elements using fits to the spectral decomposition of these correlation functions, two strategies to remove excited state contributions are employed and compared. To renormalize these charges, the <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mrow><c:mn>2</c:mn><c:mo>+</c:mo><c:mn>1</c:mn></c:mrow></c:math>-flavor mixing matrix is calculated in the regularization independent symmetric momentum subtraction intermediate scheme on the lattice. The final results are presented in the <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mover accent="true"><e:mi>MS</e:mi><e:mo stretchy="true">¯</e:mo></e:mover></e:math> scheme at scale 2 GeV. The axial charges for the proton are <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:mrow><i:msubsup><i:mrow><i:mi>g</i:mi></i:mrow><i:mrow><i:mi>A</i:mi></i:mrow><i:mrow><i:mi>u</i:mi></i:mrow></i:msubsup><i:mo>=</i:mo><i:mn>0.781</i:mn><i:mo stretchy="false">(</i:mo><i:mn>25</i:mn><i:mo stretchy="false">)</i:mo></i:mrow></i:math>, <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"><m:msubsup><m:mi>g</m:mi><m:mi>A</m:mi><m:mi>d</m:mi></m:msubsup><m:mo>=</m:mo><m:mo>−</m:mo><m:mn>0.440</m:mn><m:mo stretchy="false">(</m:mo><m:mn>39</m:mn><m:mo stretchy="false">)</m:mo></m:math>, and <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"><q:msubsup><q:mi>g</q:mi><q:mi>A</q:mi><q:mi>s</q:mi></q:msubsup><q:mo>=</q:mo><q:mo>−</q:mo><q:mn>0.055</q:mn><q:mo stretchy="false">(</q:mo><q:mn>9</q:mn><q:mo stretchy="false">)</q:mo></q:math>; the tensor charges are <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"><u:msubsup><u:mi>g</u:mi><u:mi>T</u:mi><u:mi>u</u:mi></u:msubsup><u:mo>=</u:mo><u:mn>0.782</u:mn><u:mo stretchy="false">(</u:mo><u:mn>28</u:mn><u:mo stretchy="false">)</u:mo></u:math>, <y:math xmlns:y="http://www.w3.org/1998/Math/MathML" display="inline"><y:msubsup><y:mi>g</y:mi><y:mi>T</y:mi><y:mi>d</y:mi></y:msubsup><y:mo>=</y:mo><y:mo>−</y:mo><y:mn>0.195</y:mn><y:mo stretchy="false">(</y:mo><y:mn>16</y:mn><y:mo stretchy="false">)</y:mo></y:math>, and <cb:math xmlns:cb="http://www.w3.org/1998/Math/MathML" display="inline"><cb:msubsup><cb:mi>g</cb:mi><cb:mi>T</cb:mi><cb:mi>s</cb:mi></cb:msubsup><cb:mo>=</cb:mo><cb:mo>−</cb:mo><cb:mn>0.0016</cb:mn><cb:mo stretchy="false">(</cb:mo><cb:mn>12</cb:mn><cb:mo stretchy="false">)</cb:mo></cb:math>; and the scalar charges are <gb:math xmlns:gb="http://www.w3.org/1998/Math/MathML" display="inline"><gb:msubsup><gb:mi>g</gb:mi><gb:mi>S</gb:mi><gb:mi>u</gb:mi></gb:msubsup><gb:mo>=</gb:mo><gb:mn>9.39</gb:mn><gb:mo stretchy="false">(</gb:mo><gb:mn>88</gb:mn><gb:mo stretchy="false">)</gb:mo></gb:math>, <kb:math xmlns:kb="http://www.w3.org/1998/Math/MathML" display="inline"><kb:msubsup><kb:mi>g</kb:mi><kb:mi>S</kb:mi><kb:mi>d</kb:mi></kb:msubsup><kb:mo>=</kb:mo><kb:mn>8.84</kb:mn><kb:mo stretchy="false">(</kb:mo><kb:mn>93</kb:mn><kb:mo stretchy="false">)</kb:mo></kb:math>, and <ob:math xmlns:ob="http://www.w3.org/1998/Math/MathML" display="inline"><ob:msubsup><ob:mi>g</ob:mi><ob:mi>S</ob:mi><ob:mi>s</ob:mi></ob:msubsup><ob:mo>=</ob:mo><ob:mn>0.37</ob:mn><ob:mo stretchy="false">(</ob:mo><ob:mn>14</ob:mn><ob:mo stretchy="false">)</ob:mo></ob:math>. Results for the neutron are given by the <sb:math xmlns:sb="http://www.w3.org/1998/Math/MathML" display="inline"><sb:mi>u</sb:mi><sb:mo stretchy="false">↔</sb:mo><sb:mi>d</sb:mi></sb:math> interchange. Results for the sigma terms are <vb:math xmlns:vb="http://www.w3.org/1998/Math/MathML" display="inline"><vb:mrow><vb:msub><vb:mrow><vb:mi>σ</vb:mi></vb:mrow><vb:mrow><vb:mi>π</vb:mi><vb:mi>N</vb:mi></vb:mrow></vb:msub><vb:msub><vb:mrow><vb:mo stretchy="false">|</vb:mo></vb:mrow><vb:mrow><vb:mtext>standard</vb:mtext></vb:mrow></vb:msub><vb:mo>=</vb:mo><vb:mn>42</vb:mn><vb:mo stretchy="false">(</vb:mo><vb:mn>6</vb:mn><vb:mo stretchy="false">)</vb:mo><vb:mtext> </vb:mtext><vb:mtext> </vb:mtext><vb:mi>MeV</vb:mi></vb:mrow></vb:math> from a “standard” analysis and <ac:math xmlns:ac="http://www.w3.org/1998/Math/MathML" display="inline"><ac:mrow><ac:msub><ac:mrow><ac:mi>σ</ac:mi></ac:mrow><ac:mrow><ac:mi>π</ac:mi><ac:mi>N</ac:mi></ac:mrow></ac:msub><ac:msub><ac:mrow><ac:mo stretchy="false">|</ac:mo></ac:mrow><ac:mrow><ac:mi>N</ac:mi><ac:mi>π</ac:mi></ac:mrow></ac:msub><ac:mo>=</ac:mo><ac:mn>61</ac:mn><ac:mo stretchy="false">(</ac:mo><ac:mn>6</ac:mn><ac:mo stretchy="false">)</ac:mo><ac:mtext> </ac:mtext><ac:mtext> </ac:mtext><ac:mi>MeV</ac:mi></ac:mrow></ac:math> from an “<fc:math xmlns:fc="http://www.w3.org/1998/Math/MathML" display="inline"><fc:mi>N</fc:mi><fc:mi>π</fc:mi></fc:math>” analysis that includes the contributions of multihadron <hc:math xmlns:hc="http://www.w3.org/1998/Math/MathML" display="inline"><hc:mi>N</hc:mi><hc:mi>π</hc:mi></hc:math> excited states as motivated by chiral perturbation theory. Our preferred value <jc:math xmlns:jc="http://www.w3.org/1998/Math/MathML" display="inline"><jc:msub><jc:mi>σ</jc:mi><jc:mrow><jc:mi>π</jc:mi><jc:mi>N</jc:mi></jc:mrow></jc:msub><jc:msub><jc:mo stretchy="false">|</jc:mo><jc:mrow><jc:mi>N</jc:mi><jc:mi>π</jc:mi></jc:mrow></jc:msub></jc:math> is consistent with the phenomenological extraction from <mc:math xmlns:mc="http://www.w3.org/1998/Math/MathML" display="inline"><mc:mi>π</mc:mi><mc:mo>−</mc:mo><mc:mi>N</mc:mi></mc:math> scattering data. The strangeness content of the proton, for which the standard analysis is appropriate, is <oc:math xmlns:oc="http://www.w3.org/1998/Math/MathML" display="inline"><oc:mrow><oc:msub><oc:mrow><oc:mi>σ</oc:mi></oc:mrow><oc:mrow><oc:mi>s</oc:mi></oc:mrow></oc:msub><oc:msub><oc:mrow><oc:mo stretchy="false">|</oc:mo></oc:mrow><oc:mrow><oc:mtext>standard</oc:mtext></oc:mrow></oc:msub><oc:mo>=</oc:mo><oc:mn>35</oc:mn><oc:mo stretchy="false">(</oc:mo><oc:mn>13</oc:mn><oc:mo stretchy="false">)</oc:mo><oc:mtext> </oc:mtext><oc:mtext> </oc:mtext><oc:mi>MeV</oc:mi></oc:mrow></oc:math>.