Precise prediction of the decay rate for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msub> <mml:mi>η</mml:mi> <mml:mi>b</mml:mi> </mml:msub> <mml:mo stretchy="false">→</mml:mo> <mml:mi>γ</mml:mi> <mml:mi>γ</mml:mi>

We calculate the decay rate for <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:msub> <a:mi>η</a:mi> <a:mi>b</a:mi> </a:msub> <a:mo stretchy="false">→</a:mo> <a:mi>γ</a:mi> <a:mi>γ</a:mi> </a:math> in lattice QCD for the first time, providing a precise prediction for the Belle II experiment. Our calculation includes <d:math xmlns:d="http://www.w3.org/1998/Math/MathML" display="inline"> <d:mi>u</d:mi> </d:math> , <f:math xmlns:f="http://www.w3.org/1998/Math/MathML" display="inline"> <f:mi>d</f:mi> </f:math> , <h:math xmlns:h="http://www.w3.org/1998/Math/MathML" display="inline"> <h:mi>s</h:mi> </h:math> , and <j:math xmlns:j="http://www.w3.org/1998/Math/MathML" display="inline"> <j:mi>c</j:mi> </j:math> quarks in the sea, using gluon field configurations generated by the MILC collaboration, at three values of the lattice spacing from 0.06 to 0.03 fm. All quarks are treated in the highly improved staggered quark formalism, which enables us to reach the <l:math xmlns:l="http://www.w3.org/1998/Math/MathML" display="inline"> <l:mi>b</l:mi> </l:math> quark mass for our valence quarks on these fine lattices. We calculate quark-line connected correlation functions only. By working at additional heavy quark masses between those of <n:math xmlns:n="http://www.w3.org/1998/Math/MathML" display="inline"> <n:mi>c</n:mi> </n:math> and <p:math xmlns:p="http://www.w3.org/1998/Math/MathML" display="inline"> <p:mi>b</p:mi> </p:math> we map out the behavior of the ratio <r:math xmlns:r="http://www.w3.org/1998/Math/MathML" display="inline"> <r:msub> <r:mi>f</r:mi> <r:msub> <r:mi>η</r:mi> <r:mi>h</r:mi> </r:msub> </r:msub> <r:mo>/</r:mo> <r:mo stretchy="false">(</r:mo> <r:msubsup> <r:mi>M</r:mi> <r:msub> <r:mi>η</r:mi> <r:mi>h</r:mi> </r:msub> <r:mn>2</r:mn> </r:msubsup> <r:msub> <r:mi>F</r:mi> <r:msub> <r:mi>η</r:mi> <r:mi>h</r:mi> </r:msub> </r:msub> <r:mo stretchy="false">(</r:mo> <r:mn>0</r:mn> <r:mo>,</r:mo> <r:mn>0</r:mn> <r:mo stretchy="false">)</r:mo> <r:mo stretchy="false">)</r:mo> </r:math> , where <x:math xmlns:x="http://www.w3.org/1998/Math/MathML" display="inline"> <x:mi>f</x:mi> </x:math> is the decay constant, <z:math xmlns:z="http://www.w3.org/1998/Math/MathML" display="inline"> <z:mi>M</z:mi> </z:math> , the mass and <bb:math xmlns:bb="http://www.w3.org/1998/Math/MathML" display="inline"> <bb:mi>F</bb:mi> <bb:mo stretchy="false">(</bb:mo> <bb:mn>0</bb:mn> <bb:mo>,</bb:mo> <bb:mn>0</bb:mn> <bb:mo stretchy="false">)</bb:mo> </bb:math> , the form factor for decay to two on-shell photons for the pseudoscalar heavyonium meson, <fb:math xmlns:fb="http://www.w3.org/1998/Math/MathML" display="inline"> <fb:msub> <fb:mi>η</fb:mi> <fb:mi>h</fb:mi> </fb:msub> </fb:math> . This ratio takes the approximate value 0.5 in leading-order nonrelativistic QCD (NRQCD) but we are able to give a much more accurate analysis than this. Focusing on the <hb:math xmlns:hb="http://www.w3.org/1998/Math/MathML" display="inline"> <hb:mi>b</hb:mi> </hb:math> quark mass, we find a ratio of 0.467(11), giving <jb:math xmlns:jb="http://www.w3.org/1998/Math/MathML" display="inline"> <jb:mrow> <jb:mi mathvariant="normal">Γ</jb:mi> <jb:mo stretchy="false">(</jb:mo> <jb:msub> <jb:mrow> <jb:mi>η</jb:mi> </jb:mrow> <jb:mrow> <jb:mi>b</jb:mi> </jb:mrow> </jb:msub> <jb:mo stretchy="false">→</jb:mo> <jb:mi>γ</jb:mi> <jb:mi>γ</jb:mi> <jb:mo stretchy="false">)</jb:mo> <jb:mo>=</jb:mo> <jb:mn>0.559</jb:mn> <jb:mo stretchy="false">(</jb:mo> <jb:mn>32</jb:mn> <jb:msub> <jb:mrow> <jb:mo stretchy="false">)</jb:mo> </jb:mrow> <jb:mrow> <jb:mi>fit</jb:mi> </jb:mrow> </jb:msub> <jb:mo stretchy="false">(</jb:mo> <jb:mn>1</jb:mn> <jb:msub> <jb:mrow> <jb:mo stretchy="false">)</jb:mo> </jb:mrow> <jb:mrow> <jb:mi>syst</jb:mi> </jb:mrow> </jb:msub> <jb:mtext> </jb:mtext> <jb:mtext> </jb:mtext> <jb:mi>keV</jb:mi> </jb:mrow> </jb:math> . Combined with a value for the branching fraction from NRQCD, our result can be used to determine the total width of the <tb:math xmlns:tb="http://www.w3.org/1998/Math/MathML" display="inline"> <tb:msub> <tb:mi>η</tb:mi> <tb:mi>b</tb:mi> </tb:msub> </tb:math> with a 6% uncertainty.