Perturbative determination of O(am) improvement on the QCD running coupling

We present the perturbative results of the discretization errors proportional to the quark mass [<a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi mathvariant="script">O</a:mi><a:mo stretchy="false">(</a:mo><a:mi>a</a:mi><a:mi>m</a:mi><a:mo stretchy="false">)</a:mo></a:math>] on the QCD running coupling within lattice perturbation theory. Our analysis involves calculating the two-loop renormalization factor <f:math xmlns:f="http://www.w3.org/1998/Math/MathML" display="inline"><f:msub><f:mi>Z</f:mi><f:mi>g</f:mi></f:msub></f:math> using improved lattice actions for the <h:math xmlns:h="http://www.w3.org/1998/Math/MathML" display="inline"><h:mi>S</h:mi><h:mi>U</h:mi><h:mo stretchy="false">(</h:mo><h:msub><h:mi>N</h:mi><h:mi>c</h:mi></h:msub><h:mo stretchy="false">)</h:mo></h:math> gauge group and <l:math xmlns:l="http://www.w3.org/1998/Math/MathML" display="inline"><l:msub><l:mi>N</l:mi><l:mi>f</l:mi></l:msub></l:math> multiplets of fermions with a finite quark mass. We employ the background field method to compute <n:math xmlns:n="http://www.w3.org/1998/Math/MathML" display="inline"><n:msub><n:mi>Z</n:mi><n:mi>g</n:mi></n:msub></n:math>, by calculating quantum corrections on both the background and quantum gluon propagator, respecting the <p:math xmlns:p="http://www.w3.org/1998/Math/MathML" display="inline"><p:mi mathvariant="script">O</p:mi><p:mo stretchy="false">(</p:mo><p:mi>a</p:mi><p:mo stretchy="false">)</p:mo></p:math> improvement. This allows us to evaluate the perturbative <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"><u:mi mathvariant="script">O</u:mi><u:mo stretchy="false">(</u:mo><u:mi>a</u:mi><u:mi>m</u:mi><u:mo stretchy="false">)</u:mo></u:math> lattice errors which affect the determination of the running coupling. Eliminating these <z:math xmlns:z="http://www.w3.org/1998/Math/MathML" display="inline"><z:mi mathvariant="script">O</z:mi><z:mo stretchy="false">(</z:mo><z:mi>a</z:mi><z:mi>m</z:mi><z:mo stretchy="false">)</z:mo></z:math> effects is crucial for the nonperturbative studies of precision determinations of the strong coupling constant using lattice field theory.