Four loop renormalization in six dimensions using forcer

We employ the algorithm to renormalize a variety of six dimensional field theories to four loops. In order to achieve this we construct the master integrals in six dimensions from their four dimensional counterparts by using the Tarasov method. The <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>ε</a:mi></a:math> expansion of the six dimensional masters are determined up to weight 9 where <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mi>d</c:mi><c:mo>=</c:mo><c:mn>6</c:mn><c:mo>−</c:mo><c:mn>2</c:mn><c:mi>ε</c:mi></c:math>. By applying the routine the four loop <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> renormalization of <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:msup><i:mi>ϕ</i:mi><i:mn>3</i:mn></i:msup></i:math> theory is reproduced before gauge theories are considered. The renormalization of these theories is also determined in the <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:mover accent="true"><k:mrow><k:mrow><k:mi>MOM</k:mi></k:mrow></k:mrow><k:mo stretchy="true">˜</k:mo></k:mover></k:math> scheme. For instance the absence of <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"><o:msub><o:mi>ζ</o:mi><o:mn>4</o:mn></o:msub></o:math> and <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"><q:msub><q:mi>ζ</q:mi><q:mn>6</q:mn></q:msub></q:math> is confirmed to five loops in the <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"><s:mover accent="true"><s:mrow><s:mrow><s:mi>MOM</s:mi></s:mrow></s:mrow><s:mo stretchy="true">˜</s:mo></s:mover></s:math> renormalization of <w:math xmlns:w="http://www.w3.org/1998/Math/MathML" display="inline"><w:msup><w:mi>ϕ</w:mi><w:mn>3</w:mn></w:msup></w:math> theory. We also evaluate the three loop <y:math xmlns:y="http://www.w3.org/1998/Math/MathML" display="inline"><y:mi>β</y:mi></y:math>-function of the gauge coupling in six dimensional QCD.