Lattice QCD calculation of the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>η</mml:mi></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:msup><mml:mrow><mml:mi

We present a lattice calculation of the <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>η</a:mi></a:math> and <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:msup><c:mi>η</c:mi><c:mo>′</c:mo></c:msup></c:math> meson masses at the physical point and in the continuum limit, based on <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:msub><e:mi>N</e:mi><e:mi>f</e:mi></e:msub><e:mo>=</e:mo><e:mn>2</e:mn><e:mo>+</e:mo><e:mn>1</e:mn><e:mo>+</e:mo><e:mn>1</e:mn></e:math> flavors of rooted staggered quarks. Our analysis includes gauge ensembles at the physical pion and kaon masses spread over six lattice spacings in the range [0.064–0.1315] fm. Our main results read <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:msub><g:mi>m</g:mi><g:mi>η</g:mi></g:msub><g:mo>=</g:mo><g:mn>543.5</g:mn><g:mo stretchy="false">(</g:mo><g:mn>5.6</g:mn><g:mo stretchy="false">)</g:mo><g:mtext> </g:mtext><g:mtext> </g:mtext><g:mi>MeV</g:mi></g:math> and <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:msub><k:mi>m</k:mi><k:msup><k:mi>η</k:mi><k:mo>′</k:mo></k:msup></k:msub><k:mo>=</k:mo><k:mn>986</k:mn><k:mo stretchy="false">(</k:mo><k:mn>38</k:mn><k:mo stretchy="false">)</k:mo><k:mtext> </k:mtext><k:mtext> </k:mtext><k:mi>MeV</k:mi></k:math>, consistent with the experimental values. This is an important numerical test that supports the validity of the fourth root procedure used in the staggered quark formalism. This calculation was the first step toward extracting the pseudoscalar transition form factors of the <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"><o:mi>η</o:mi></o:math> and <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"><q:msup><q:mi>η</q:mi><q:mo>′</q:mo></q:msup></q:math> mesons that play a crucial role in the hadronic light-by-light contribution to the muon <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"><s:mi>g</s:mi><s:mo>−</s:mo><s:mn>2</s:mn></s:math>.