Static force from generalized Wilson loops on the lattice using the gradient flow

The static QCD force from the lattice can be used to extract <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:msub><a:mi mathvariant="normal">Λ</a:mi><a:mover accent="true"><a:mi>MS</a:mi><a:mo stretchy="true">¯</a:mo></a:mover></a:msub></a:math>, which determines the running of the strong coupling. Usually, this is done with a numerical derivative of the static potential. However, this introduces additional systematic uncertainties; thus, we use another observable to measure the static force directly. This observable consists of a Wilson loop with a chromoelectric field insertion. We work in the pure SU(3) gauge theory. We use gradient flow to improve the signal-to-noise ratio and to address the field insertion. We extract <f:math xmlns:f="http://www.w3.org/1998/Math/MathML" display="inline"><f:mrow><f:msubsup><f:mrow><f:mi mathvariant="normal">Λ</f:mi></f:mrow><f:mrow><f:mover accent="true"><f:mrow><f:mi>MS</f:mi></f:mrow><f:mrow><f:mo stretchy="true">¯</f:mo></f:mrow></f:mover></f:mrow><f:mrow><f:msub><f:mrow><f:mi>n</f:mi></f:mrow><f:mrow><f:mi>f</f:mi></f:mrow></f:msub><f:mo>=</f:mo><f:mn>0</f:mn></f:mrow></f:msubsup></f:mrow></f:math> from the data by exploring different methods to perform the zero-flow-time limit. We obtain the value <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:mrow><k:msqrt><k:mrow><k:mn>8</k:mn><k:msub><k:mrow><k:mi>t</k:mi></k:mrow><k:mrow><k:mn>0</k:mn></k:mrow></k:msub></k:mrow></k:msqrt><k:msubsup><k:mrow><k:mi mathvariant="normal">Λ</k:mi></k:mrow><k:mrow><k:mover accent="true"><k:mrow><k:mi>MS</k:mi></k:mrow><k:mrow><k:mo stretchy="true">¯</k:mo></k:mrow></k:mover></k:mrow><k:mrow><k:msub><k:mrow><k:mi>n</k:mi></k:mrow><k:mrow><k:mi>f</k:mi></k:mrow></k:msub><k:mo>=</k:mo><k:mn>0</k:mn></k:mrow></k:msubsup><k:mo>=</k:mo><k:mn>0.62</k:mn><k:msubsup><k:mrow><k:mn>9</k:mn></k:mrow><k:mrow><k:mo>−</k:mo><k:mn>26</k:mn></k:mrow><k:mrow><k:mo>+</k:mo><k:mn>22</k:mn></k:mrow></k:msubsup></k:mrow></k:math>, where <p:math xmlns:p="http://www.w3.org/1998/Math/MathML" display="inline"><p:msub><p:mi>t</p:mi><p:mn>0</p:mn></p:msub></p:math> is a flow-time reference scale. We also obtain precise determinations of several scales: <r:math xmlns:r="http://www.w3.org/1998/Math/MathML" display="inline"><r:msub><r:mi>r</r:mi><r:mn>0</r:mn></r:msub><r:mo>/</r:mo><r:msub><r:mi>r</r:mi><r:mn>1</r:mn></r:msub></r:math>, <t:math xmlns:t="http://www.w3.org/1998/Math/MathML" display="inline"><t:msqrt><t:mrow><t:mn>8</t:mn><t:msub><t:mi>t</t:mi><t:mn>0</t:mn></t:msub></t:mrow></t:msqrt><t:mo>/</t:mo><t:msub><t:mi>r</t:mi><t:mn>0</t:mn></t:msub></t:math>, <v:math xmlns:v="http://www.w3.org/1998/Math/MathML" display="inline"><v:msqrt><v:mrow><v:mn>8</v:mn><v:msub><v:mi>t</v:mi><v:mn>0</v:mn></v:msub></v:mrow></v:msqrt><v:mo>/</v:mo><v:msub><v:mi>r</v:mi><v:mn>1</v:mn></v:msub></v:math>, and we compare these to the literature. The gradient flow appears to be a promising method for calculations of Wilson loops with chromoelectric and chromomagnetic insertions in quenched and unquenched configurations. Published by the American Physical Society 2024