Four loop renormalization of QCD in the Curci-Ferrari gauge

We renormalize quantum chromodynamics (QCD) when gauge fixed in the nonlinear Curci-Ferrari gauge to four loops in the modified minimal subtraction (<a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mover accent="true"><a:mi>MS</a:mi><a:mo stretchy="true">¯</a:mo></a:mover></a:math>) scheme. We reproduce the four loop QCD <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mover accent="true"><e:mi>MS</e:mi><e:mo stretchy="true">¯</e:mo></e:mover></e:math> <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:mi>β</i:mi></i:math>-function from the Slavnov-Taylor identity for this gauge which relates the coupling constant renormalization to the gluon, Faddeev-Popov ghost and gauge parameter anomalous dimensions. This is carried out for a nonzero gauge parameter, without having to evaluate a vertex function. The anomalous dimension of the BRST invariant dimension two gluon and ghost mass term is deduced from a similar Slavnov-Taylor identity for this gauge. Consequently we construct the renormalization group functions in the minimal momentum subtraction scheme to four loops. As a corollary we deduce the five loop <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:mi>β</k:mi></k:math>-function and quark mass anomalous dimensions in the same scheme. We also outline the pros and cons of employing the Curci-Ferrari gauge to access the six loop QCD <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"><m:mi>β</m:mi></m:math>-function in the <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"><o:mover accent="true"><o:mi>MS</o:mi><o:mo stretchy="true">¯</o:mo></o:mover></o:math> scheme.