Quark masses and low-energy constants in the continuum from the tadpole-improved clover ensembles

Abstract

We present the light-flavor quark masses and low-energy constants using the <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:mrow></a:math> flavor full-QCD ensembles with stout smeared-clover fermion action and Symanzik gauge action. Both the fermion and gauge actions are tadpole improved self-consistently. The simulations are performed on 11 ensembles at three lattice spacings <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mi>a</c:mi><c:mo>∈</c:mo><c:mo stretchy="false">[</c:mo><c:mn>0.05</c:mn><c:mo>,</c:mo><c:mn>0.11</c:mn><c:mo stretchy="false">]</c:mo><c:mtext> </c:mtext><c:mtext> </c:mtext><c:mi>fm</c:mi></c:math>, four spatial sizes <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:mi>L</g:mi><g:mo>∈</g:mo><g:mo stretchy="false">[</g:mo><g:mn>2.5</g:mn><g:mo>,</g:mo><g:mn>5.1</g:mn><g:mo stretchy="false">]</g:mo><g:mtext> </g:mtext><g:mtext> </g:mtext><g:mi>fm</g:mi></g:math>, seven pion masses <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:msub><k:mi>m</k:mi><k:mi>π</k:mi></k:msub><k:mo>∈</k:mo><k:mo stretchy="false">[</k:mo><k:mn>135</k:mn><k:mo>,</k:mo><k:mn>350</k:mn><k:mo stretchy="false">]</k:mo><k:mtext> </k:mtext><k:mtext> </k:mtext><k:mi>MeV</k:mi></k:math>, and several values of the strange quark mass. The quark mass is defined through the partially conserved axial current relation and renormalized to <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"><o:mrow><o:mover accent="true"><o:mrow><o:mi>MS</o:mi></o:mrow><o:mrow><o:mo stretchy="true">¯</o:mo></o:mrow></o:mover><o:mo stretchy="false">(</o:mo><o:mn>2</o:mn><o:mtext> </o:mtext><o:mi>GeV</o:mi><o:mo stretchy="false">)</o:mo></o:mrow></o:math> through the intermediate regularization independent momentum subtraction scheme. The systematic uncertainty of using the symmetric momentum subtraction scheme is also included. Eventually, we predict <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"><u:msub><u:mi>m</u:mi><u:mi>u</u:mi></u:msub><u:mo>=</u:mo><u:mn>2.45</u:mn><u:mo stretchy="false">(</u:mo><u:mn>22</u:mn><u:mo stretchy="false">)</u:mo><u:mo stretchy="false">(</u:mo><u:mn>20</u:mn><u:mo stretchy="false">)</u:mo><u:mtext> </u:mtext><u:mtext> </u:mtext><u:mi>MeV</u:mi></u:math>, <ab:math xmlns:ab="http://www.w3.org/1998/Math/MathML" display="inline"><ab:msub><ab:mi>m</ab:mi><ab:mi>d</ab:mi></ab:msub><ab:mo>=</ab:mo><ab:mn>4.74</ab:mn><ab:mo stretchy="false">(</ab:mo><ab:mn>11</ab:mn><ab:mo stretchy="false">)</ab:mo><ab:mo stretchy="false">(</ab:mo><ab:mn>09</ab:mn><ab:mo stretchy="false">)</ab:mo><ab:mtext> </ab:mtext><ab:mtext> </ab:mtext><ab:mi>MeV</ab:mi></ab:math>, and <gb:math xmlns:gb="http://www.w3.org/1998/Math/MathML" display="inline"><gb:msub><gb:mi>m</gb:mi><gb:mi>s</gb:mi></gb:msub><gb:mo>=</gb:mo><gb:mn>98.8</gb:mn><gb:mo stretchy="false">(</gb:mo><gb:mn>2.9</gb:mn><gb:mo stretchy="false">)</gb:mo><gb:mo stretchy="false">(</gb:mo><gb:mn>4.7</gb:mn><gb:mo stretchy="false">)</gb:mo><gb:mtext> </gb:mtext><gb:mtext> </gb:mtext><gb:mi>MeV</gb:mi></gb:math> with the systematic uncertainties from lattice spacing determination, continuum extrapolation and renormalization constant included. We also obtain the chiral condensate <mb:math xmlns:mb="http://www.w3.org/1998/Math/MathML" display="inline"><mb:msup><mb:mi mathvariant="normal">Σ</mb:mi><mb:mrow><mb:mn>1</mb:mn><mb:mo>/</mb:mo><mb:mn>3</mb:mn></mb:mrow></mb:msup><mb:mo>=</mb:mo><mb:mn>268.6</mb:mn><mb:mo stretchy="false">(</mb:mo><mb:mn>3.6</mb:mn><mb:mo stretchy="false">)</mb:mo><mb:mo stretchy="false">(</mb:mo><mb:mn>0.7</mb:mn><mb:mo stretchy="false">)</mb:mo><mb:mtext> </mb:mtext><mb:mtext> </mb:mtext><mb:mi>MeV</mb:mi></mb:math> and the pion decay constant <tb:math xmlns:tb="http://www.w3.org/1998/Math/MathML" display="inline"><tb:mi>F</tb:mi><tb:mo>=</tb:mo><tb:mn>86.6</tb:mn><tb:mo stretchy="false">(</tb:mo><tb:mn>7</tb:mn><tb:mo stretchy="false">)</tb:mo><tb:mo stretchy="false">(</tb:mo><tb:mn>1.4</tb:mn><tb:mo stretchy="false">)</tb:mo><tb:mtext> </tb:mtext><tb:mtext> </tb:mtext><tb:mi>MeV</tb:mi></tb:math> in the <zb:math xmlns:zb="http://www.w3.org/1998/Math/MathML" display="inline"><zb:msub><zb:mi>N</zb:mi><zb:mi>f</zb:mi></zb:msub><zb:mo>=</zb:mo><zb:mn>2</zb:mn></zb:math> chiral limit, and the next-to-leading order low-energy constants <bc:math xmlns:bc="http://www.w3.org/1998/Math/MathML" display="inline"><bc:msub><bc:mo>ℓ</bc:mo><bc:mn>3</bc:mn></bc:msub><bc:mo>=</bc:mo><bc:mn>2.43</bc:mn><bc:mo stretchy="false">(</bc:mo><bc:mn>54</bc:mn><bc:mo stretchy="false">)</bc:mo><bc:mo stretchy="false">(</bc:mo><bc:mn>05</bc:mn><bc:mo stretchy="false">)</bc:mo></bc:math> and <hc:math xmlns:hc="http://www.w3.org/1998/Math/MathML" display="inline"><hc:msub><hc:mo>ℓ</hc:mo><hc:mn>4</hc:mn></hc:msub><hc:mo>=</hc:mo><hc:mn>4.322</hc:mn><hc:mo stretchy="false">(</hc:mo><hc:mn>75</hc:mn><hc:mo stretchy="false">)</hc:mo><hc:mo stretchy="false">(</hc:mo><hc:mn>96</hc:mn><hc:mo stretchy="false">)</hc:mo></hc:math>. Published by the American Physical Society 2024

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