Renormalization of beta decay at three loops and beyond

The anomalous dimension for heavy-heavy-light effective theory operators describing nuclear beta decay is computed through three-loop order in the static limit. The result at order <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:msup><a:mi>Z</a:mi><a:mn>2</a:mn></a:msup><a:msup><a:mi>α</a:mi><a:mn>3</a:mn></a:msup></a:math> corrects a previous result in the literature. An all-orders symmetry is shown to relate the anomalous dimensions at leading and subleading powers of <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mi>Z</c:mi></c:math> at a given order of <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mi>α</e:mi></e:math>. The first unknown coefficient for the anomalous dimension now appears at <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:mi>O</g:mi><g:mo stretchy="false">(</g:mo><g:msup><g:mi>Z</g:mi><g:mn>2</g:mn></g:msup><g:msup><g:mi>α</g:mi><g:mn>4</g:mn></g:msup><g:mo stretchy="false">)</g:mo></g:math>.