Bootstrapping pentagon functions

Abstract

In Phys. Rev. Lett. 116 (2016) 062001, the space of planar pentagon functions that describes all two-loop on-shell five-particle scattering amplitudes was introduced. In the present paper we present a natural extension of this space to non-planar pentagon functions. This provides the basis for our pentagon bootstrap program. We classify the relevant functions up to weight four, which is relevant for two-loop scattering amplitudes. We constrain the first entry of the symbol of the functions using information on branch cuts. Drawing on an analogy from the planar case, we introduce a conjectural second-entry condition on the symbol. We then show that the information on the function space, when complemented with some additional insights, can be used to efficiently bootstrap individual Feynman integrals. The extra information is read off of Mellin-Barnes representations of the integrals, either by evaluating simple asymptotic limits, or by taking discontinuities in the kinematic variables. We use this method to evaluate the symbols of two non-trivial non-planar five-particle integrals, up to and including the finite part.

Prescriptive unitarity for non-planar six-particle amplitudes at two loops

“…This will not be easy, but may lead to the discovery of new simplicities such as that hinted by the 'non-planar dual-conformal symmetry' described in refs. [73][74][75], as well as being a natural testing ground for questions about the restrictions on symbol entries [76], for example. Integrating the amplitudes as we have represented them, however, may require upgrading them to 4 − 2 dimensions-in order to exploit the loop-integration technology available in dimensional regularization.…”

“…Our six-particle results represent a natural step forward in complexity, following recent developments at two loops for five particles [60][61][62][63][64][67][68][69][70][71][72]. It furthermore opens up avenues for exploring a potential extension of dual conformal symmetry beyond the planar limit [44,55,56,[73][74][75] as well as an extension of the empirical second entry conditions for five-particle amplitudes [76]. This paper is organized as follows.…”

Integration-by-parts reductions of Feynman integrals using Singular and GPI-Space

“…In recent years, great progress has been made in multi-loop scattering amplitude calculations, for instance, in the case of 2 → 3 processes [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15]. The progress is due to modern developments of scattering amplitudes, like the integrand construction method [16,17], canonical integrals [18,19], numeric unitarity [20,21], bootstrap methods [22][23][24][25][26][27][28][29], reconstruction using finite fields [30][31][32][33] and new ideas in the integration-by-parts (IBP) reduction. The latter is the main topic to be discussed in this paper.…”

Manifestly dual-conformal loop integration

“…Indeed, in the case of six (or seven) particle amplitudes in planar SYM-the primary example studied in this work-the symbolic bootstrap approach described in refs. [19][20][21][22][23][24][25][26][27][28] has proven dramatically more powerful than integrand-tointegral strategies. This strategy eschews reference to loop integrands altogether, determining finite quantities directly from the space of functions expected to arise from loop integration-fixed by some number of globally-defined physical constraints.…”