One-loop amplitudes in gauge theories

We propose a new “universal expansion” for one-loop amplitudes with arbitrary number of gluons in <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mrow><a:mi>D</a:mi></a:mrow></a:math> dimensions, which holds for general gauge theories with gluons, fermions, and scalars in the loop, including pure and supersymmetric Yang-Mills theories. It expresses the <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mi>n</c:mi></c:math>-gluon amplitudes as a linear combination of universal scalar-loop amplitudes with <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mi>n</e:mi><e:mo>−</e:mo><e:mi>m</e:mi></e:math> gluons and <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:mi>m</g:mi></g:math> scalars, multiplied by gauge-invariant building blocks (defined for general gauge theories); the integrands of these scalar-loop amplitudes are given in terms of tree-level objects attached to the scalar loop, or by differential operators acting on the most important part which is proportional to <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:mi>D</i:mi></i:math> (with <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:mi>m</k:mi><k:mo>=</k:mo><k:mn>0</k:mn></k:math>). We present closed-formula for these one-loop integrands and prove them by showing that the single cuts are correctly reproduced by the gluing of an additional pair of gluons (fermions, scalars) in the forward limit, plus <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"><m:mi>n</m:mi></m:math> gluons in a tree amplitude.