Detecting the flavor content of the vacuum using the Dirac operator spectrum

We compute the overlap Dirac spectrum on three gauge ensembles generated using <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mn>2</a:mn><a:mo>+</a:mo><a:mn>1</a:mn></a:math>-flavor domain wall fermions. The three ensembles have different lattice spacings, and two of them have quark masses tuned to the physical point. The spectral density is determined up to <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mi>λ</c:mi><c:mo>∼</c:mo><c:mn>100</c:mn><c:mtext> </c:mtext><c:mtext> </c:mtext><c:mi>MeV</c:mi></c:math> with subpercentage statistical uncertainty. We find that the density is close to a constant below <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mi>λ</e:mi><e:mo>∼</e:mo><e:mn>20</e:mn><e:mtext> </e:mtext><e:mtext> </e:mtext><e:mi>MeV</e:mi></e:math> as predicted by chiral perturbative theory (<g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:mi>χ</g:mi><g:mi>PT</g:mi></g:math>) and then increases linearly due to the strange quark mass. By fitting to the next-to-leading order <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:mi>χ</i:mi><i:mi>PT</i:mi></i:math> form and using the nonperturbative renormalization using the regularization independent momentum subtraction scheme, the SU(2) (keeping the strange quark mass at the physical point) and SU(3) chiral condensates at <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:mrow><k:mover accent="true"><k:mrow><k:mi>MS</k:mi></k:mrow><k:mrow><k:mo stretchy="true">¯</k:mo></k:mrow></k:mover></k:mrow></k:math> 2 GeV are determined to be <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"><o:mi mathvariant="normal">Σ</o:mi><o:mo>=</o:mo><o:mo stretchy="false">(</o:mo><o:mn>265.4</o:mn><o:mo stretchy="false">(</o:mo><o:mn>0.5</o:mn><o:mo stretchy="false">)</o:mo><o:mo stretchy="false">(</o:mo><o:mn>4.2</o:mn><o:mo stretchy="false">)</o:mo><o:mtext> </o:mtext><o:mtext> </o:mtext><o:mi>MeV</o:mi><o:msup><o:mo stretchy="false">)</o:mo><o:mn>3</o:mn></o:msup></o:math> and <x:math xmlns:x="http://www.w3.org/1998/Math/MathML" display="inline"><x:msub><x:mi mathvariant="normal">Σ</x:mi><x:mn>0</x:mn></x:msub><x:mo>=</x:mo><x:mo stretchy="false">(</x:mo><x:mn>234.3</x:mn><x:mo stretchy="false">(</x:mo><x:mn>0.5</x:mn><x:mo stretchy="false">)</x:mo><x:mo stretchy="false">(</x:mo><x:mn>25.8</x:mn><x:mo stretchy="false">)</x:mo><x:mtext> </x:mtext><x:mtext> </x:mtext><x:mi>MeV</x:mi><x:msup><x:mo stretchy="false">)</x:mo><x:mn>3</x:mn></x:msup></x:math>, respectively. The pion decay constants are also determined to be <gb:math xmlns:gb="http://www.w3.org/1998/Math/MathML" display="inline"><gb:mi>F</gb:mi><gb:mo>=</gb:mo><gb:mn>84.1</gb:mn><gb:mo stretchy="false">(</gb:mo><gb:mn>1.9</gb:mn><gb:mo stretchy="false">)</gb:mo><gb:mo stretchy="false">(</gb:mo><gb:mn>8.0</gb:mn><gb:mo stretchy="false">)</gb:mo></gb:math> and <mb:math xmlns:mb="http://www.w3.org/1998/Math/MathML" display="inline"><mb:msub><mb:mi>F</mb:mi><mb:mn>0</mb:mn></mb:msub><mb:mo>=</mb:mo><mb:mn>58.6</mb:mn><mb:mo stretchy="false">(</mb:mo><mb:mn>0.5</mb:mn><mb:mo stretchy="false">)</mb:mo><mb:mo stretchy="false">(</mb:mo><mb:mn>10.0</mb:mn><mb:mo stretchy="false">)</mb:mo><mb:mtext> </mb:mtext><mb:mtext> </mb:mtext><mb:mi>MeV</mb:mi></mb:math>. The systematic errors are carefully estimated including the effects of fitting ranges and the uncertainty of low-energy constant <sb:math xmlns:sb="http://www.w3.org/1998/Math/MathML" display="inline"><sb:msub><sb:mi>L</sb:mi><sb:mn>6</sb:mn></sb:msub></sb:math>. We also show that one can resolve the sea flavor content of the sea quarks and constrain their masses with <ub:math xmlns:ub="http://www.w3.org/1998/Math/MathML" display="inline"><ub:mrow><ub:mo>∼</ub:mo><ub:mn>10</ub:mn><ub:mo>%</ub:mo><ub:mi>–</ub:mi><ub:mn>20</ub:mn><ub:mo>%</ub:mo></ub:mrow></ub:math> statistical uncertainties using the Dirac spectral density. Published by the American Physical Society 2024