The two-loop five-particle amplitude in N=8\mathcal{N}=8 supergravity

We compute for the first time the two-loop five-particle amplitude in $\mathcal{N}=8$ supergravity. Starting from the known integrand, we perform an integration-by-parts reduction and express the answer in terms of uniform weight master integrals. The latter are known to evaluate to non-planar pentagon functions, described by a 31-letter symbol alphabet. We express the final result for the amplitude in terms of uniform weight four symbols, multiplied by a small set of rational factors. The amplitude satisfies the expected factorization properties when one external graviton becomes soft, and when two external gravitons become collinear. We verify that the soft divergences of the amplitude exponentiate, and extract the finite remainder function. The latter depends on fewer rational factors, and is independent of one of the symbol letters. By analyzing identities involving rational factors and symbols we find a remarkably compact representation in terms of a single seed function, summed over all permutations of external particles. Finally, we work out the multi-Regge limit, and present explicitly the leading logarithmic terms in the limit. The full symbol of the IR-subtracted hard function is provided as an ancillary file.Comment: 22 pages, 1 figure, 8 ancillary file

A first look at the function space for planar two-loop six-particle Feynman integrals

Two-loop corrections to scattering amplitudes are crucial theoretical input for collider physics. Recent years have seen tremendous advances in computing Feynman integrals, scattering amplitudes, and cross sections for five-particle processes. In this paper, we initiate the study of the function space for planar two-loop six-particle processes. We study all genuine six-particle Feynman integrals, and derive the differential equations they satisfy on maximal cuts. Performing a leading singularity analysis in momentum space, and in Baikov representation, we find an integral basis that puts the differential equations into canonical form. The corresponding differential equation in the eight independent kinematic variables is derived with the finite-field reconstruction method and the symbol letters are identified. We identify the dual conformally invariant hexagon alphabet known from maximally supersymmetric Yang-Mills theory as a subset of our alphabet. This paper constitutes an important step in the analytic calculation of planar two-loop six-particle Feynman integrals

New differential equations for on-shell loop integrals

We present a novel type of differential equations for on-shell loop integrals. The equations are second-order and importantly, they reduce the loop level by one, so that they can be solved iteratively in the loop order. We present several infinite series of integrals satisfying such iterative differential equations. The differential operators we use are best written using momentum twistor space. The use of the latter was advocated in recent papers discussing loop integrals in N=4 super Yang-Mills. One of our motivations is to provide a tool for deriving analytical results for scattering amplitudes in this theory. We show that the integrals needed for planar MHV amplitudes up to two loops can be thought of as deriving from a single master topology. The master integral satisfies our differential equations, and so do most of the reduced integrals. A consequence of the differential equations is that the integrals we discuss are not arbitrarily complicated transcendental functions. For two specific two-loop integrals we give the full analytic solution. The simplicity of the integrals appearing in the scattering amplitudes in planar N=4 super Yang-Mills is strongly suggestive of a relation to the conjectured underlying integrability of the theory. We expect these differential equations to be relevant for all planar MHV and non-MHV amplitudes. We also discuss possible extensions of our method to more general classes of integrals.Comment: 39 pages, 8 figures; v2: typos corrected, definition of harmonic polylogarithms adde

The global integrated world ocean assessment: linking observations to science and policy across multiple scales

In 2004, the United Nations (UN) General Assembly approved a Regular Process to report on the environmental, economic and social aspects of the world's ocean. The Regular Process for Global Reporting and Assessment of the State of the Marine Environment, including Socioeconomic Aspects produced the first global integrated assessment of the marine environment in December 2016 (known as the first World Ocean Assessment). The second assessment, to be delivered in December 2020, will build on the baselines included in the first assessment, with a focus on establishing trends in the marine environment with relevance to global reporting needs such as those associated with the UN Sustainable Development Goals. Central to the assessment process and its outputs are two components. First, is the utilization of ocean observation and monitoring outputs and research to temporally assess physical, chemical, biological, social, economic and cultural components of coastal and marine environments to establish their current state, impacts currently affecting coastal and marine environments, responses to those impacts and associated ongoing trends. Second, is the knowledge brokering of ocean observations and associated research to provide key information that can be utilized and applied to address management and policy needs at local, regional and global scales. Through identifying both knowledge gaps and capacity needs, the assessment process also provides direction to policy makers for the future development and deployment of sustained observation systems that are required for enhancing knowledge and supporting national aspirations associated with the sustainable development of coastal and marine ecosystems. Input from the ocean observation community, managers and policy makers is critical for ensuring that the vital information required for supporting the science policy interface objectives of the Regular Process is included in the assessment. This community white paper discusses developments in linking ocean observations and science with policy achieved as part of the assessment process, and those required for providing strategic linkages into the future.Agência financiadora - United Nations Division for Ocean Affairs and the Law of the Seainfo:eu-repo/semantics/publishedVersio