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

The one-loop six-dimensional hexagon integral and its relation to MHV amplitudes in N=4 SYM

We provide an analytic formula for the (rescaled) one-loop scalar hexagon integral $\tilde\Phi_6$ with all external legs massless, in terms of classical polylogarithms. We show that this integral is closely connected to two integrals appearing in one- and two-loop amplitudes in planar $\\mathcal{N}=4$ super-Yang-Mills theory, $\Omega^{(1)}$ and $\Omega^{(2)}$. The derivative of $\Omega^{(2)}$ with respect to one of the conformal invariants yields $\tilde\Phi_6$, while another first-order differential operator applied to $\tilde\Phi_6$ yields $\Omega^{(1)}$. We also introduce some kinematic variables that rationalize the arguments of the polylogarithms, making it easy to verify the latter differential equation. We also give a further example of a six-dimensional integral relevant for amplitudes in $\\mathcal{N}=4$ super-Yang-Mills.Comment: 18 pages, 2 figure

Inferring Population Preferences via Mixtures of Spatial Voting Models

Understanding political phenomena requires measuring the political preferences of society. We introduce a model based on mixtures of spatial voting models that infers the underlying distribution of political preferences of voters with only voting records of the population and political positions of candidates in an election. Beyond offering a cost-effective alternative to surveys, this method projects the political preferences of voters and candidates into a shared latent preference space. This projection allows us to directly compare the preferences of the two groups, which is desirable for political science but difficult with traditional survey methods. After validating the aggregated-level inferences of this model against results of related work and on simple prediction tasks, we apply the model to better understand the phenomenon of political polarization in the Texas, New York, and Ohio electorates. Taken at face value, inferences drawn from our model indicate that the electorates in these states may be less bimodal than the distribution of candidates, but that the electorates are comparatively more extreme in their variance. We conclude with a discussion of limitations of our method and potential future directions for research.Comment: To be published in the 8th International Conference on Social Informatics (SocInfo) 201

An Analytic Result for the Two-Loop Hexagon Wilson Loop in N = 4 SYM

In the planar N=4 supersymmetric Yang-Mills theory, the conformal symmetry constrains multi-loop n-edged Wilson loops to be basically given in terms of the one-loop n-edged Wilson loop, augmented, for n greater than 6, by a function of conformally invariant cross ratios. We identify a class of kinematics for which the Wilson loop exhibits exact Regge factorisation and which leave invariant the analytic form of the multi-loop n-edged Wilson loop. In those kinematics, the analytic result for the Wilson loop is the same as in general kinematics, although the computation is remarkably simplified with respect to general kinematics. Using the simplest of those kinematics, we have performed the first analytic computation of the two-loop six-edged Wilson loop in general kinematics.Comment: 17 pages. Extended discussion on how the QMRK limit is taken. Version accepted by JHEP. A text file containing the Mathematica code with the analytic expression for the 6-point remainder function is include

Graviton emission in Einstein-Hilbert gravity

The five-point amplitude for the scattering of two distinct scalars with the emission of one graviton in the final state is calculated in exact kinematics for Einstein-Hilbert gravity. The result, which satisfies the Steinmann relations, is expressed in Sudakov variables, finding that it corresponds to the sum of two gauge invariant contributions written in terms of a new two scalar - two graviton effective vertex. A similar calculation is carried out in Quantum Chromodynamics (QCD) for the scattering of two distinct quarks with one extra gluon in the final state. The effective vertices which appear in both cases are then evaluated in the multi-Regge limit reproducing the well-known result obtained by Lipatov where the Einstein-Hilbert graviton emission vertex can be written as the product of two QCD gluon emission vertices, up to corrections to preserve the Steinmann relations.Comment: 28 pages, LaTeX, feynmf. v2: typos corrected, reference added. Final version to appear in Journal of High Energy Physic

The fatty acid binding protein FABP7 is required for optimal oligodendrocyte differentiation during myelination but not during remyelination.

The major constituents of the myelin sheath are lipids, which are made up of fatty acids (FAs). The hydrophilic environment inside the cells requires FAs to be bound to proteins, preventing their aggregation. Fatty acid binding proteins (FABPs) are one class of proteins known to bind FAs in a cell. Given the crucial role of FAs for myelin sheath formation we investigated the role of FABP7, the major isoform expressed in oligodendrocyte progenitor cells (OPCs), in developmental myelination and remyelination. Here, we show that the knockdown of Fabp7 resulted in a reduction of OPC differentiation in vitro. Consistent with this result, a delay in developmental myelination was observed in Fabp7 knockout animals. This delay was transient with full myelination being established before adulthood. FABP7 was dispensable for remyelination, as the knockout of Fapb7 did not alter remyelination efficiency in a focal demyelination model. In summary, while FABP7 is important in OPC differentiation in vitro, its function is not crucial for myelination and remyelination in vivo.This work was supported by grants from the UK Multiple Sclerosis Society, the Adelson Medical Research Foundation, the Japan Society for the Promotion of Science (JSPS) KAKENHI Grant and a core support grant from the Wellcome Trust and MRC to the Wellcome Trust-Medical Research Council Cambridge Stem Cell Institute. AGF was also supported by an ECTRIMS postdoctoral fellowship from July 2018. SF and TB were also supported by a Wellcome-Trust PhD studentship

Single-valued harmonic polylogarithms and the multi-Regge limit

We argue that the natural functions for describing the multi-Regge limit of six-gluon scattering in planar N=4 super Yang-Mills theory are the single-valued harmonic polylogarithmic functions introduced by Brown. These functions depend on a single complex variable and its conjugate, (w,w*). Using these functions, and formulas due to Fadin, Lipatov and Prygarin, we determine the six-gluon MHV remainder function in the leading-logarithmic approximation (LLA) in this limit through ten loops, and the next-to-LLA (NLLA) terms through nine loops. In separate work, we have determined the symbol of the four-loop remainder function for general kinematics, up to 113 constants. Taking its multi-Regge limit and matching to our four-loop LLA and NLLA results, we fix all but one of the constants that survive in this limit. The multi-Regge limit factorizes in the variables (\nu,n) which are related to (w,w*) by a Fourier-Mellin transform. We can transform the single-valued harmonic polylogarithms to functions of (\nu,n) that incorporate harmonic sums, systematically through transcendental weight six. Combining this information with the four-loop results, we determine the eigenvalues of the BFKL kernel in the adjoint representation to NNLLA accuracy, and the MHV product of impact factors to NNNLLA accuracy, up to constants representing beyond-the-symbol terms and the one symbol-level constant. Remarkably, only derivatives of the polygamma function enter these results. Finally, the LLA approximation to the six-gluon NMHV amplitude is evaluated through ten loops.Comment: 71 pages, 2 figures, plus 10 ancillary files containing analytic expressions in Mathematica format. V2: Typos corrected and references added. V3: Typos corrected; assumption about single-Reggeon exchange made explici

Symbols of One-Loop Integrals From Mixed Tate Motives

We use a result on mixed Tate motives due to Goncharov (arXiv:alg-geom/9601021) to show that the symbol of an arbitrary one-loop 2m-gon integral in 2m dimensions may be read off directly from its Feynman parameterization. The algorithm proceeds via recursion in m seeded by the well-known box integrals in four dimensions. As a simple application of this method we write down the symbol of a three-mass hexagon integral in six dimensions.Comment: 13 pages, v2: minor typos correcte

Deep Inelastic Scattering in Conformal QCD

We consider the Regge limit of a CFT correlation function of two vector and two scalar operators, as appropriate to study small-x deep inelastic scattering in N=4 SYM or in QCD assuming approximate conformal symmetry. After clarifying the nature of the Regge limit for a CFT correlator, we use its conformal partial wave expansion to obtain an impact parameter representation encoding the exchange of a spin j Reggeon for any value of the coupling constant. The CFT impact parameter space is the three-dimensional hyperbolic space H3, which is the impact parameter space for high energy scattering in the dual AdS space. We determine the small-x structure functions associated to the exchange of a Reggeon. We discuss unitarization from the point of view of scattering in AdS and comment on the validity of the eikonal approximation. We then focus on the weak coupling limit of the theory where the amplitude is dominated by the exchange of the BFKL pomeron. Conformal invariance fixes the form of the vector impact factor and its decomposition in transverse spin 0 and spin 2 components. Our formalism reproduces exactly the general results predict by the Regge theory, both for a scalar target and for gamma*-gamma* scattering. We compute current impact factors for the specific examples of N=4 SYM and QCD, obtaining very simple results. In the case of the R-current of N=4 SYM, we show that the transverse spin 2 component vanishes. We conjecture that the impact factors of all chiral primary operators of N=4 SYM only have components with 0 transverse spin.Comment: 44+16 pages, 7 figures. Some correction