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

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

From polygons and symbols to polylogarithmic functions

We present a review of the symbol map, a mathematical tool that can be useful in simplifying expressions among multiple polylogarithms, and recall its main properties. A recipe is given for how to obtain the symbol of a multiple polylogarithm in terms of the combinatorial properties of an associated rooted decorated polygon. We also outline a systematic approach to constructing a function corresponding to a given symbol, and illustrate it in the particular case of harmonic polylogarithms up to weight four. Furthermore, part of the ambiguity of this process is highlighted by exhibiting a family of non-trivial elements in the kernel of the symbol map for arbitrary weight.Comment: 75 pages. Mathematica files with the expression of all HPLs up to weight 4 in terms of the spanning set are include

Two-parton scattering in the high-energy limit

Considering $2\to 2$ gauge-theory scattering with general colour in the high-energy limit, we compute the Regge-cut contribution to three loops through next-to-next-to-leading high-energy logarithms (NNLL) in the signature-odd sector. Our formalism is based on using the non-linear Balitsky-JIMWLK rapidity evolution equation to derive an effective Hamiltonian acting on states with a fixed number of Reggeized gluons. A new effect occurring first at NNLL is mixing between states with $k$ and $k+2$ Reggeized gluons due non-diagonal terms in this Hamiltonian. Our results are consistent with a recent determination of the infrared structure of scattering amplitudes at three loops, as well as a computation of $2\to 2$ gluon scattering in ${\cal N}=4$ super Yang-Mills theory. Combining the latter with our Regge-cut calculation we extract the three-loop Regge trajectory in this theory. Our results open the way to predict high-energy logarithms through NNLL at higher-loop orders.Comment: 62 pages, 7 figure

Planossolos e Gleissolos Utilizados na Fabricação de Cerâmica Artesanal no Semiárido de Minas Gerais

O conhecimento etnopedológico tem fornecido informações importantes sobre o modo de vida das populações rurais a respeito de suas tradições ancestrais, como a arte de elaborar peças artesanais a partir do barro advindo de solos com características próprias a esse uso. O objetivo deste trabalho foi avaliar física, química e mineralogicamente Planossolos e Gleissolos explorados para a produção de artefatos de cerâmica artesanal em Minas Gerais. Nos barreiros, foram coletados dois perfis de Planossolos (P1 e P2) e um Gleissolo (P3) usados como matéria-prima na produção artesanal de cerâmica. Foram realizadas análises físicas e químicas, limites de liquidez (LL) e plasticidade (LP), índice de plasticidade (IP) e de atividade coloidal (IA), além da mineralogia da fração argila. Os horizontes selecionados pelos ceramistas para a fabricação de cerâmica artesanal (BA, Btg e BCg, do P1; Btg1 e Btg2, do P2; e C2g e C3g, do P3) apresentaram os maiores teores de argila e silte, IP e IA, importantes para a qualidade final da cerâmica. O horizonte Cg do perfil P1 possui potencial de ser utilizado para a produção artesanal, em virtude do seu IP, superior aos dos horizontes normalmente usados, além dos teores de argila, silte e areia fina e suas características mineralógicas. A proporção ideal das frações areia, silte e argila e a porcentagem de matéria orgânica na definição de um bom material para cerâmica são difíceis de estabelecer e variam principalmente em razão de aspectos quantitativos e qualitativos da argila nos solos