<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>η</mml:mi> </mml:mrow> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msup> <mml:mi>η</mml:mi> <mml:mo>′</m

By fitting the two-point correlation functions of topological charge density operators calculated on two <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 gauge ensembles with physical pion mass, we determine both the <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:mi>η</c:mi> </c:math> and <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"> <e:msup> <e:mi>η</e:mi> <e:mo>′</e:mo> </e:msup> </e:math> masses and also the mixing angle to be <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>0.505</g:mn> <g:mo stretchy="false">(</g:mo> <g:mn>72</g:mn> <g:mo stretchy="false">)</g:mo> <g:mo stretchy="false">(</g:mo> <g:mn>75</g:mn> <g:mo stretchy="false">)</g:mo> <g:mtext> </g:mtext> <g:mtext> </g:mtext> <g:mi>GeV</g:mi> </g:math> , <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"> <m:msub> <m:mi>m</m:mi> <m:msup> <m:mi>η</m:mi> <m:mo>′</m:mo> </m:msup> </m:msub> <m:mo>=</m:mo> <m:mn>0.952</m:mn> <m:mo stretchy="false">(</m:mo> <m:mn>47</m:mn> <m:mo stretchy="false">)</m:mo> <m:mo stretchy="false">(</m:mo> <m:mn>40</m:mn> <m:mo stretchy="false">)</m:mo> <m:mtext> </m:mtext> <m:mtext> </m:mtext> <m:mi>GeV</m:mi> </m:math> , and <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"> <s:msub> <s:mi>θ</s:mi> <s:mn>1</s:mn> </s:msub> <s:mo>=</s:mo> <s:mo>−</s:mo> <s:mn>8.9</s:mn> <s:mo stretchy="false">(</s:mo> <s:mn>2.1</s:mn> <s:mo stretchy="false">)</s:mo> <s:mo stretchy="false">(</s:mo> <s:mn>1.8</s:mn> <s:mo stretchy="false">)</s:mo> <s:mo>°</s:mo> </s:math> , respectively, where the first error is the statistical uncertainty and the second one is the systematic uncertainty. This is the first extraction of both <y:math xmlns:y="http://www.w3.org/1998/Math/MathML" display="inline"> <y:mi>η</y:mi> <y:mo>/</y:mo> <y:msup> <y:mi>η</y:mi> <y:mo>′</y:mo> </y:msup> </y:math> masses and the mixing angle <ab:math xmlns:ab="http://www.w3.org/1998/Math/MathML" display="inline"> <ab:msub> <ab:mi>θ</ab:mi> <ab:mn>1</ab:mn> </ab:msub> </ab:math> using topological charge operators. Compared with previous studies using quark bilinear operators, the error of the <cb:math xmlns:cb="http://www.w3.org/1998/Math/MathML" display="inline"> <cb:mi>η</cb:mi> </cb:math> mass is relatively large, but the mixing angle has comparable precision. This demonstrates that the topological charge operators are well suited to study the <eb:math xmlns:eb="http://www.w3.org/1998/Math/MathML" display="inline"> <eb:mi>η</eb:mi> </eb:math> and <gb:math xmlns:gb="http://www.w3.org/1998/Math/MathML" display="inline"> <gb:msup> <gb:mi>η</gb:mi> <gb:mo>′</gb:mo> </gb:msup> </gb:math> mesons.