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

Massive amplitudes on the Coulomb branch of N=4 SYM

We initiate a systematic study of amplitudes with massive external particles on the Coulomb-branch of N=4 super Yang Mills theory: 1) We propose that (multi-)soft-scalar limits of massless amplitudes at the origin of moduli space can be used to determine Coulomb-branch amplitudes to leading order in the mass. This is demonstrated in numerous examples. 2) We find compact explicit expressions for several towers of tree-level amplitudes, including scattering of two massive W-bosons with any number of positive helicity gluons, valid for all values of the mass. 3) We present the general structure of superamplitudes on the Coulomb branch. For example, the n-point "MHV-band" superamplitude is proportional to a Grassmann polynomial of mixed degree 4 to 12, which is uniquely determined by supersymmetry. We find explicit tree-level superamplitudes for this MHV band and for other simple sectors of the theory. 4) Dual conformal generators are constructed, and we explore the dual conformal properties of the simplest massive amplitudes. Our compact expressions for amplitudes and superamplitudes should be of both theoretical and phenomenological interest; in particular the tree-level results carry over to truncations of the theory with less supersymmetry.Comment: 29 pages, 1 figur

On BCFW shifts of integrands and integrals

In this article a first step is made towards the extension of Britto-Cachazo-Feng-Witten (BCFW) tree level on-shell recursion relations to integrands and integrals of scattering amplitudes to arbitrary loop order. Surprisingly, it is shown that the large BCFW shift limit of the integrands has the same structure as the corresponding tree level amplitude in any minimally coupled Yang-Mills theory in four or more dimensions. This implies that these integrands can be reconstructed from a subset of their `single cuts'. The main tool is powercounting Feynman graphs in a special lightcone gauge choice employed earlier at tree level by Arkani-Hamed and Kaplan. The relation between shifts of integrands and shifts of its integrals is investigated explicitly at one loop. Two particular sources of discrepancy between the integral and integrand are identified related to UV and IR divergences. This is cross-checked with known results for helicity equal amplitudes at one loop. The nature of the on-shell residue at each of the single-cut singularities of the integrand is commented upon. Several natural conjectures and opportunities for further research present themselves.Comment: 43 pages, 6 figures, v2: minor improvement in exposition, typos fixed, bibliography update

The Soft-Collinear Bootstrap: N=4 Yang-Mills Amplitudes at Six and Seven Loops

Infrared divergences in scattering amplitudes arise when a loop momentum $\ell$ becomes collinear with a massless external momentum $p$. In gauge theories, it is known that the L-loop logarithm of a planar amplitude has much softer infrared singularities than the L-loop amplitude itself. We argue that planar amplitudes in N=4 super-Yang-Mills theory enjoy softer than expected behavior as $\ell \parallel p$ already at the level of the integrand. Moreover, we conjecture that the four-point integrand can be uniquely determined, to any loop-order, by imposing the correct soft-behavior of the logarithm together with dual conformal invariance and dihedral symmetry. We use these simple criteria to determine explicit formulae for the four-point integrand through seven-loops, finding perfect agreement with previously known results through five-loops. As an input to this calculation we enumerate all four-point dual conformally invariant (DCI) integrands through seven-loops, an analysis which is aided by several graph-theoretic theorems we prove about general DCI integrands at arbitrary loop-order. The six- and seven-loop amplitudes receive non-zero contributions from 229 and 1873 individual DCI diagrams respectively.Comment: 27 pages, 48 figures, detailed results including PDF and Mathematica files available at http://goo.gl/qIKe8 v2: minor corrections v3: figure 7 corrected, Lemma 2 remove

Integrands for QCD rational terms and N=4 SYM from massive CSW rules

We use massive CSW rules to derive explicit compact expressions for integrands of rational terms in QCD with any number of external legs. Specifically, we present all-n integrands for the one-loop all-plus and one-minus gluon amplitudes in QCD. We extract the finite part of spurious external-bubble contributions systematically; this is crucial for the application of integrand-level CSW rules in theories without supersymmetry. Our approach yields integrands that are independent of the choice of CSW reference spinor even before integration. Furthermore, we present a recursive derivation of the recently proposed massive CSW-style vertex expansion for massive tree amplitudes and loop integrands on the Coulomb-branch of N=4 SYM. The derivation requires a careful study of boundary terms in all-line shift recursion relations, and provides a rigorous (albeit indirect) proof of the recently proposed construction of massive amplitudes from soft-limits of massless on-shell amplitudes. We show that the massive vertex expansion manifestly preserves all holomorphic and half of the anti-holomorphic supercharges, diagram-by-diagram, even off-shell.Comment: 30 pages, many figure

Spectral Parameters for Scattering Amplitudes in N=4 Super Yang-Mills Theory

49 pages, 20 figures; v2: typos fixedPlanar N=4 Super Yang-Mills theory appears to be a quantum integrable four-dimensional conformal theory. This has been used to find equations believed to describe its exact spectrum of anomalous dimensions. Integrability seemingly also extends to the planar space-time scattering amplitudes of the N=4 model, which show strong signs of Yangian invariance. However, in contradistinction to the spectral problem, this has not yet led to equations determining the exact amplitudes. We propose that the missing element is the spectral parameter, ubiquitous in integrable models. We show that it may indeed be included into recent on-shell approaches to scattering amplitude integrands, providing a natural deformation of the latter. Under some constraints, Yangian symmetry is preserved. Finally we speculate that the spectral parameter might also be the regulator of choice for controlling the infrared divergences appearing when integrating the integrands in exactly four dimensions.Peer reviewe

Two-loop Sudakov form factor in ABJM

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On super form factors of half-BPS operators in N=4 super Yang-Mills

Open Access, (c) The Authors. Article funded by SCOAP3. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited

Standardized metadata for human pathogen/vector genomic sequences

High throughput sequencing has accelerated the determination of genome sequences for thousands of human infectious disease pathogens and dozens of their vectors. The scale and scope of these data are enabling genotype-phenotype association studies to identify genetic determinants of pathogen virulence and drug/insecticide resistance, and phylogenetic studies to track the origin and spread of disease outbreaks. To maximize the utility of genomic sequences for these purposes, it is essential that metadata about the pathogen/vector isolate characteristics be collected and made available in organized, clear, and consistent formats. Here we report the development of the GSCID/BRC Project and Sample Application Standard, developed by representatives of the Genome Sequencing Centers for Infectious Diseases (GSCIDs), the Bioinformatics Resource Centers (BRCs) for Infectious Diseases, and the U.S. National Institute of Allergy and Infectious Diseases (NIAID), part of the National Institutes of Health (NIH), informed by interactions with numerous collaborating scientists. It includes mapping to terms from other data standards initiatives, including the Genomic Standards Consortium's minimal information (MIxS) and NCBI's BioSample/BioProjects checklists and the Ontology for Biomedical Investigations (OBI). The standard includes data fields about characteristics of the organism or environmental source of the specimen, spatial-temporal information about the specimen isolation event, phenotypic characteristics of the pathogen/vector isolated, and project leadership and support. By modeling metadata fields into an ontology-based semantic framework and reusing existing ontologies and minimum information checklists, the application standard can be extended to support additional project-specific data fields and integrated with other data represented with comparable standards. The use of this metadata standard by all ongoing and future GSCID sequencing projects will provide a consistent representation of these data in the BRC resources and other repositories that leverage these data, allowing investigators to identify relevant genomic sequences and perform comparative genomics analyses that are both statistically meaningful and biologically relevant

Lipopolysaccharide and Tumor Necrosis Factor Regulate Parkin Expression via Nuclear Factor-Kappa B

Inflammation and oxidative stress have been implicated in the pathophysiology of Parkinson's disease (PD) and inhibition of microglial activation attenuates degeneration of dopaminergic (DA) neurons in animal models of PD. Loss-of-function mutations in the parkin gene, which encodes an E3 ubiquitin ligase, cause autosomal recessive parkinsonism. While most studies on Parkin have focused on its function in neurons, here we demonstrate that Parkin mRNA and protein is detectable in brain-resident microglia and peripheral macrophages. Using pharmacologic and genetic approaches, we found that Parkin levels are regulated by inflammatory signaling. Specifically, exposure to LPS or Tumor Necrosis Factor (TNF) induced a transient and dose-dependent decrease in Parkin mRNA and protein in microglia, macrophages and neuronal cells blockable by inhibitors of Nuclear Factor-Kappa B (NF-κB) signaling and not observed in MyD88-null cells. Moreover, using luciferase reporter assays, we identified an NF-κB response element in the mouse parkin promoter responsible for mediating the transcriptional repression, which was abrogated when the consensus sequence was mutated. Functionally, activated macrophages from Parkin-null mice displayed increased levels of TNF, IL-1β, and iNOS mRNA compared to wild type macrophages but no difference in levels of Nrf2, HO-1, or NQO1. One implication of our findings is that chronic inflammatory conditions may reduce Parkin levels and phenocopy parkin loss-of-function mutations, thereby increasing the vulnerability for degeneration of the nigrostriatal pathway and development of PD