Abstract
We present a systematic procedure to compute complete, analytic form factors of gauge-invariant operators at loop level in pure Yang-Mills. We consider applications to operators of the form $\mathrm{Tr}\, F^n$ where $F$ is the gluon field strength. Our approach is based on an extension to form factors of the dimensional reconstruction technique, in conjunction with the six-dimensional spinor-helicity formalism and generalised unitarity. For form factors this technique requires the introduction of additional scalar operators, for which we provide a systematic prescription. We also discuss a generalisation of dimensional reconstruction to any number of loops, both for amplitudes and form factors. Several novel results for one-loop minimal and non-minimal form factors of $\mathrm{Tr}\, F^n$ with $n>2$ are presented. Finally, we describe the \texttt{Mathematica} package \texttt{SpinorHelicity6D}, which is tailored to handle six-dimensional quantities written in the spinor-helicity formalism.
Highlights
The aim of this paper is to construct complete, analytic form factors of gauge-invariant operators at one loop.In supersymmetric theories, four-dimensional unitarity [1,2] is sufficient to obtain complete answers for amplitudes at one loop
The key point of this work is that we extend dimensional reconstruction to any form factor of operators involving vector fields, which requires the subtraction of form factors of an appropriate class of scalar operators that we identify
In the first three we review the spinor-helicity formalism in four and six dimensions, as well as the structure of six-dimensional tree amplitudes
Summary
The aim of this paper is to construct complete, analytic form factors of gauge-invariant operators at one loop. The dimensional reconstruction approach can be effectively combined with the spinor-helicity formalism in six dimensions of [17], which allows for compact expressions of the on-shell building blocks At higher loops, these techniques were used in [18] to derive the five-point all-plus gluon amplitude integrand in pure Yang-Mills, while a generalization to incorporate fermions was carried out in [19]. III we study tree-level form factors for a wide class of operators involving field strengths in four and six dimensions These quantities are needed in the one-loop unitarity-based calculations of Sec. IV. Appendix F contains a short description of the SpinorHelicity6D Mathematica package we have used in our numerical calculations, focusing on the functions required to replicate our results
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