Abstract
We investigate the asymptotically large loop-momentum behavior of multi-loop amplitudes in maximally supersymmetric quantum field theories in four dimensions. We check residue-theorem identities among color-dressed leading singularities in $\mathcal{N}=4$ supersymmetric Yang-Mills theory to demonstrate the absence of poles at infinity of all MHV amplitudes through three loops. Considering the same test for $\mathcal{N}=8$ supergravity leads us to discover that this theory does support non-vanishing residues at infinity starting at two loops, and the degree of these poles grow arbitrarily with multiplicity. This causes a tension between simultaneously manifesting ultraviolet finiteness---which would be automatic in a representation obtained by color-kinematic duality---and gauge invariance---which would follow from unitarity-based methods.
Highlights
Much of the remarkable recent progress in our understanding of scattering amplitudes in perturbative quantum field theory has followed from the investigation of the singularity structure of scattering amplitudes—especially when viewed at the integrand level, as rational differential forms on the space of internal loop momenta
We have found that scattering amplitude integrands in N 1⁄4 8 sugra behave worse at infinity than anticipated from the point of view of UV divergences of integrated amplitudes
The fact that the asymptotic behavior of the cut (12) does not signal any UV divergences in the theory is easy to see from color-kinematic duality [31–33]—provided, that is, that such a representation exists for all two-loop amplitude integrands in N 1⁄4 8 sugra
Summary
Much of the remarkable recent progress in our understanding of scattering amplitudes in perturbative quantum field theory has followed from the investigation of the singularity structure of scattering amplitudes—especially when viewed at the integrand level, as rational differential forms on the space of internal loop momenta (see e.g. [1–6]). We demonstrate that integrands in this theory support poles at infinity starting at two loops, and that the degree of these poles grows arbitrarily with multiplicity This suggests an unboundedly bad, term-wise UV behavior with increasing multiplicity, and that the theory becomes non-cut-constructible [28] (in four dimensions) at sufficient multiplicity. To be clear, this does not imply that there are UV divergences in N 1⁄4 8 at two loops; these are known to be absent from general arguments Rather, this is a term-wise behavior that seems necessary within any local, gauge-invariant integrand representation of amplitudes in the theory, analogous to what was seen for four-particle amplitudes in the planar limit of SYM starting at eight loops [29,30]. At the end of this Letter, we discuss color-kinematics duality [31–33], and a tension between gauge-invariance and cut-constructibility
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