Collinear and Soft Limits of Multi-Loop Integrands in N=4 Yang-Mills

It has been argued in arXiv:1112.6432 that the planar four-point integrand in N=4 super Yang-Mills theory is uniquely determined by dual conformal invariance together with the absence of a double pole in the integrand of the logarithm in the limit as a loop integration variable becomes collinear with an external momentum. In this paper we reformulate this condition in a simple way in terms of the amplitude itself, rather than its logarithm, and verify that it holds for two- and three-loop MHV integrands for n>4. We investigate the extent to which this collinear constraint and a constraint on the soft behavior of integrands can be used to determine integrands. We find an interesting complementarity whereby the soft constraint becomes stronger while the collinear constraint becomes weaker at larger n. For certain reasonable choices of basis at two and three loops the two constraints in unison appear strong enough to determine MHV integrands uniquely for all n.Comment: 27 pages, 14 figures; v2: very minor change

Multi-Regge kinematics and the moduli space of Riemann spheres with marked points

We show that scattering amplitudes in planar N = 4 Super Yang-Mills in multi-Regge kinematics can naturally be expressed in terms of single-valued iterated integrals on the moduli space of Riemann spheres with marked points. As a consequence, scattering amplitudes in this limit can be expressed as convolutions that can easily be computed using Stokes' theorem. We apply this framework to MHV amplitudes to leading-logarithmic accuracy (LLA), and we prove that at L loops all MHV amplitudes are determined by amplitudes with up to L + 4 external legs. We also investigate non-MHV amplitudes, and we show that they can be obtained by convoluting the MHV results with a certain helicity flip kernel. We classify all leading singularities that appear at LLA in the Regge limit for arbitrary helicity configurations and any number of external legs. Finally, we use our new framework to obtain explicit analytic results at LLA for all MHV amplitudes up to five loops and all non-MHV amplitudes with up to eight external legs and four loops.Comment: 104 pages, six awesome figures and ancillary files containing the results in Mathematica forma

All-mass n-gon integrals in n dimensions

We explore the correspondence between one-loop Feynman integrals and (hyperbolic) simplicial geometry to describe the "all-mass" case: integrals with generic external and internal masses. Specifically, we focus on $n$-particle integrals in exactly $n$ space-time dimensions, as these integrals have particularly nice geometric properties and respect a dual conformal symmetry. In four dimensions, we leverage this geometric connection to give a concise dilogarithmic expression for the all-mass box in terms of the Murakami-Yano formula. In five dimensions, we use a generalized Gauss-Bonnet theorem to derive a similar dilogarithmic expression for the all-mass pentagon. We also use the Schl\"afli formula to write down the symbol of these integrals for all $n$. Finally, we discuss how the geometry behind these formulas depends on space-time signature, and we gather together many results related to these integrals from the mathematics and physics literature.Comment: 49 pages, 8 figure

Depression, Anxiety and Glucose Metabolism in the General Dutch Population: The New Hoorn Study

BACKGROUND: There is a well recognized association between depression and diabetes. However, there is little empirical data about the prevalence of depressive symptoms and anxiety among different groups of glucose metabolism in population based samples. The aim of this study was to determine whether the prevalence of increased levels of depression and anxiety is different between patients with type 2 diabetes and subjects with impaired glucose metabolism (IGM) and normal glucose metabolism (NGM). METHODOLOGY/PRINCIPAL FINDINGS: Cross-sectional data from a population-based cohort study of 2667 residents, 1261 men and 1406 women aged 40-65 years from the Hoorn region, the Netherlands. Depressive symptoms and anxiety were measured using the Centre for Epidemiologic Studies Depression Scale (CES-D, score >or=16) and the Hospital Anxiety and Depression Scale--Anxiety Subscale (HADS-A, score >or=8), respectively. Glucose metabolism status was determined by oral glucose tolerance test. In the total study population the prevalence of depressive symptoms and anxiety for the NGM, IGM and type 2 diabetes were 12.5, 12.2 and 21.0% (P = 0.004) and 15.0, 15.3 and 19.9% (p = 0.216), respectively. In men, the prevalence of depressive symptoms was 7.7, 9.5 and 19.6% (p<0.001), and in women 16.4, 15.8 and 22.6 (p = 0.318), for participants with NGM, IGM and type 2 diabetes, respectively. Anxiety was not associated with glucose metabolism when stratified for sex. Intergroup differences (NGM vs. IGM and IGM vs. type 2 diabetes) revealed that higher prevalences of depressive symptoms are mainly manifested in participants with type 2 diabetes, and not in participants with IGM. CONCLUSIONS: Depressive symptoms, but not anxiety are associated with glucose metabolism. This association is mainly determined by a higher prevalence of depressive symptoms in participants with type 2 diabetes and not in participants with IGM

Generating All Tree Amplitudes in N=4 SYM by Inverse Soft Limit

The idea of adding particles to construct amplitudes has been utilized in various ways in exploring the structure of scattering amplitudes. This idea is often called Inverse Soft Limit, namely it is the reverse mechanism of taking particles to be soft. We apply the Inverse Soft Limit to the tree-level amplitudes in $\mathcal{N}=4$ super Yang-Mills theory, which allows us to generate full tree-level superamplitudes by adding "soft" particles in a certain way. With the help from Britto-Cachazo-Feng-Witten recursion relations, a systematic and concrete way of adding particles is determined recursively. The amplitudes constructed solely by adding particles not only have manifest Yangian symmetry, but also make the soft limit transparent. The method of generating amplitudes by Inverse Soft Limit can also be generalized for constructing form factors.Comment: 33 pages, 10 figures. v2: figures corrected, JHEP versio