Abstract
We show that direct Feynman-parametric loop integration is possible for a large class of planar multi-loop integrals. Much of this follows from the existence of manifestly dual-conformal Feynman-parametric representations of planar loop integrals, and the fact that many of the algebraic roots associated with (e.g. Landau) leading singularities are automatically rationalized in momentum-twistor space — facilitating direct integration via partial fractioning. We describe how momentum twistors may be chosen non-redundantly to parameterize particular integrals, and how strategic choices of coordinates can be used to expose kinematic limits of interest. We illustrate the power of these ideas with many concrete cases studied through four loops and involving as many as eight particles. Detailed examples are included as supplementary material.
Highlights
What motivates this instinctual aversion? The naıve answer is that the problem of direct integration is open-ended — that, unlike differentiation, there is no algorithm for integrating arbitrarily complicated expressions
We show that direct Feynman-parametric loop integration is possible for a large class of planar multi-loop integrals
Much of this follows from the existence of manifestly dual-conformal Feynman-parametric representations of planar loop integrals, and the fact that many of the algebraic roots associated with (e.g. Landau) leading singularities are automatically rationalized in momentum-twistor space — facilitating direct integration via partial fractioning
Summary
We are interested in planar Feynman integrals involving massless external particles. In order to trivialize momentum conservation, we introduce dual-momentum coordinates, associating the momentum pa of the ath external particle with the difference pa ≡ (xa+1−xa) (with cyclic labeling understood). Letting GAB denote the sub-matrix of G involving rows A and columns B, det These identities can always be normalized (by dividing by the leading term, say) so that they encode relations among (multiplicatively independent) dual-conformal cross-ratios. We merely note that these relations among crossratios are at least quadratic, and their solutions involve algebraic roots depending on the set of cross-ratios chosen to be independent. Because such algebraic roots complicate much of the computational machinery involved in integration (or even analysis), it is worth enumerating at least one relevant class of these roots. We find that parametrizing external momenta in twistor space (which naturally realizes the embedding formalism) rationalizes all Gramian roots of the form (2.8). Before reviewing how to parametrize our external kinematics in momentum-twistor space, we will take a slight detour to describe how some of these roots can arise in the process of loop integration via partial fractioning
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