Ratios of the light quark masses: Cubic curve versus ellipse

The bounds on the light quark masses are obtained by fitting the squares of pseudoscalar meson masses <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:msubsup><a:mi>m</a:mi><a:mi>π</a:mi><a:mn>2</a:mn></a:msubsup></a:math> and <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:msubsup><c:mi>m</c:mi><c:mi>K</c:mi><c:mn>2</c:mn></c:msubsup></c:math> to second order in <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mn>1</e:mn><e:mo>/</e:mo><e:msub><e:mi>N</e:mi><e:mi>c</e:mi></e:msub></e:math> expansion. The result is an algebraic cubic curve whose coefficients are the known Weinberg values for the quark mass ratios <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:msub><g:mi>m</g:mi><g:mi>u</g:mi></g:msub><g:mo>/</g:mo><g:msub><g:mi>m</g:mi><g:mi>d</g:mi></g:msub></g:math> and <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:msub><i:mi>m</i:mi><i:mi>s</i:mi></i:msub><i:mo>/</i:mo><i:msub><i:mi>m</i:mi><i:mi>d</i:mi></i:msub></i:math>. Additional restrictions arise when using the ratio <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:msub><k:mi>m</k:mi><k:mi>s</k:mi></k:msub><k:mo>/</k:mo><k:msub><k:mi>m</k:mi><k:mrow><k:mi>u</k:mi><k:mi>d</k:mi></k:mrow></k:msub><k:mo>=</k:mo><k:mn>27.23</k:mn><k:mo stretchy="false">(</k:mo><k:mn>10</k:mn><k:mo stretchy="false">)</k:mo></k:math> quoted by FLAG for lattice simulations with four quark flavors. This provides a tight constraint on the ratio <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"><o:msub><o:mi>m</o:mi><o:mi>u</o:mi></o:msub><o:mo>/</o:mo><o:msub><o:mi>m</o:mi><o:mi>d</o:mi></o:msub><o:mo>=</o:mo><o:mn>0.455</o:mn><o:mo stretchy="false">(</o:mo><o:mn>8</o:mn><o:mo stretchy="false">)</o:mo></o:math>. Published by the American Physical Society 2024