Abstract
We use generalized unitarity at the integrand-level to directly construct local, manifestly dual-conformally invariant formulae for all two-loop scattering amplitudes in planar, maximally supersymmetric Yang-Mills theory (SYM). This representation separates contributions into manifestly finite and divergent terms---in a way that makes manifest the exponentiation of infrared divergences at the integrand-level. These results perfectly match the all-loop BCFW recursion relations, to which we provide a closed-form solution valid through two-loop-order. Finally, we describe and document a Mathematica package which implements these results, available as part of this work's source files on the arXiv.
Highlights
All scattering amplitudes in planar supersymmetric Yang-Mills theory (SYM), [22]
We describe the generalization of this approach to two-loop amplitudes in planar SYM
Our confidence in the correctness of our local representations of two-loop amplitudes follows in part from direct comparison with the all-loop recursion relations, [19], to which we provide a closed-form solution which is valid through twoloop-order in appendix B
Summary
Not all on-shell diagrams are independent as functions; they satisfy many identities referred to as residue theorems. We have shown that dressing each on-shell diagram in (2.12) by its corresponding integrand according to (2.13) must result in an integrand which is free of any dependence on X and must match field theory everywhere: A(nk),1 = This was the form of one-loop amplitude integrands derived in ref. Let us briefly note that because the representation (2.19) is independent of X, while each term involves an X-dependent factor of the form (X, Yq)/( , X), it must be the case that sum of terms in the numerator (put over a common denominator) factorizes to become proportional to ( , X) directly
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