High-precision scale setting with the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi mathvariant="normal">Ω</mml:mi> </mml:math> -baryon mass and gradient flow

The gradient-flow scale <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:msub> <a:mi>w</a:mi> <a:mn>0</a:mn> </a:msub> </a:math> in lattice QCD is determined using the mass of the <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:msup> <c:mi mathvariant="normal">Ω</c:mi> <c:mo>−</c:mo> </c:msup> </c:math> baryon to set the physical scale. Nine ensembles using the highly improved staggered quark (HISQ) action with lattice spacings of 0.15 fm down to 0.04 fm are used, seven of which have nearly physical light-quark masses. Electromagnetic corrections to the <f:math xmlns:f="http://www.w3.org/1998/Math/MathML" display="inline"> <f:msup> <f:mi mathvariant="normal">Ω</f:mi> <f:mo>−</f:mo> </f:msup> </f:math> mass are defined in order to compute a pure-QCD <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"> <i:mi mathvariant="normal">Ω</i:mi> </i:math> mass. The final result is <l:math xmlns:l="http://www.w3.org/1998/Math/MathML" display="inline"> <l:msub> <l:mi>w</l:mi> <l:mn>0</l:mn> </l:msub> <l:mo>=</l:mo> <l:mn>0.17187</l:mn> <l:mo stretchy="false">(</l:mo> <l:mn>68</l:mn> <l:mo stretchy="false">)</l:mo> <l:mtext> </l:mtext> <l:mtext> </l:mtext> <l:mi>fm</l:mi> </l:math> , corresponding to a relative uncertainty of 0.40% and a central value in good agreement with previous calculations in the literature.