A Color Dual Form for Gauge-Theory Amplitudes

Recently a duality between color and kinematics has been proposed, exposing a new unexpected structure in gauge theory and gravity scattering amplitudes. Here we propose that the relation goes deeper, allowing us to reorganize amplitudes into a form reminiscent of the standard color decomposition in terms of traces over generators, but with the role of color and kinematics swapped. By imposing additional conditions similar to Kleiss-Kuijf relations between partial amplitudes, the relationship between the earlier form satisfying the duality and the current one is invertible. We comment on extensions to loop level.Comment: 5 pages, 4 figure

Generalized Unitarity and Six-Dimensional Helicity

We combine the unitarity method with the six-dimensional helicity formalism of Cheung and O'Connell to construct loop-level scattering amplitudes. As a first example, we construct dimensionally regularized QCD one-loop four-point amplitudes. As a nontrivial multiloop example, we confirm that the recently constructed four-loop four-point amplitude of N=4 super-Yang-Mills theory, including nonplanar contributions, is valid for dimensions less than or equal to six. We comment on the connection of our approach to the recently discussed Higgs infrared regulator and on dual conformal properties in six dimensions.Comment: 38 pages, 7 figures, typos correcte

A Twistor Description of Six-Dimensional N=(1,1) Super Yang-Mills Theory

We present a twistor space that describes super null-lines on six-dimensional N=(1,1) superspace. We then show that there is a one-to-one correspondence between holomorphic vector bundles over this twistor space and solutions to the field equations of N=(1,1) super Yang-Mills theory. Our constructions naturally reduce to those of the twistorial description of maximally supersymmetric Yang-Mills theory in four dimensions.Comment: 15 pages, typos fixed, published versio

An R^4 non-renormalisation theorem in N=4 supergravity

We consider the four-graviton amplitudes in CHL constructions providing four-dimensional N=4 models with various numbers of vector multiplets. We show that in these models the two-loop amplitude has a prefactor of d^2R^4. This implies a non-renormalisation theorem for the R^4 term, which forbids the appearance of a three-loop ultraviolet divergence in four dimensions in the four-graviton amplitude. We connect the special nature of the R^4 term to the U(1) anomaly of pure N=4 supergravity.Comment: v2: added comments about one-loop UV divergences. Assorted stylistic corrections. Added references. v3: Eq. III.21 corrected and assorted minor corrections and clarifications. Version to be published. v4: minor corrections. 18 pages. one figur

Systematic Reviews of Research on Online Learning: An Introductory Look and Review

In this introduction to the special issue on systematic reviews of research on online learning, we introduce the need for systematic reviews on online learning. Utilizing a three-tier lens focusing on systems, pedagogical, and people levels, we have selected nine articles for this issue. At the systems level, there are two articles that focus on research trends during COVID-19, and features of high-quality online learning. At the pedagogical level, five articles were included that address online learner collaboration, help-seeking strategies, intersubjectivity, invisible participation, and online assessment. Finally, at the people level, there are two articles. The first focuses on online learning for minoritized and first-generation students. The second examines moderators in asynchronous online discussions. This introductory article provides a short summary of the nine articles and concludes with implications for practitioners and researchers on using and conducting systematic reviews on various topics in online learning

Basics of Generalized Unitarity

We review generalized unitarity as a means for obtaining loop amplitudes from on-shell tree amplitudes. The method is generally applicable to both supersymmetric and non-supersymmetric amplitudes, including non-planar contributions. Here we focus mainly on N=4 Yang-Mills theory, in the context of on-shell superspaces. Given the need for regularization at loop level, we also review a six-dimensional helicity-based superspace formalism and its application to dimensional and massive regularizations. An important feature of the unitarity method is that it offers a means for carrying over any identified tree-level property of on-shell amplitudes to loop level, though sometimes in a modified form. We illustrate this with examples of dual conformal symmetry and a recently discovered duality between color and kinematics.Comment: 37 pages, 10 figures. Invited review for a special issue of Journal of Physics A devoted to "Scattering Amplitudes in Gauge Theories", R. Roiban(ed), M. Spradlin(ed), A. Volovich(ed

Exploring cycad foliage as an archive of the isotopic composition of atmospheric nitrogen

Funding for this work was provided by a University of Washington Royalty Research Fund Grant (R.B.), National Science Foundation Graduate Research Fellowship DGE‐1256082 (M.A.K.), and German Research Foundation (DFG) Fellowship GE2558/3‐1 (M.M.G). Cyanobiont collection was funded by grant no. 265‐605 of the Australian Biodiversity and Resources Programme (M.M.G).Molecular nitrogen (N2) constitutes the majority of Earth's modern atmosphere, contributing ~0.79 bar of partial pressure (pN2). However, fluctuations in pN2 may have occurred on 107–109 year timescales in Earth's past, perhaps altering the isotopic composition of atmospheric nitrogen. Here, we explore an archive that may record the isotopic composition of atmospheric N2 in deep time: the foliage of cycads. Cycads are ancient gymnosperms that host symbiotic N2‐fixing cyanobacteria in modified root structures known as coralloid roots. All extant species of cycads are known to host symbionts, suggesting that this N2‐fixing capacity is perhaps ancestral, reaching back to the early history of cycads in the late Paleozoic. Therefore, if the process of microbial N2 fixation records the δ15N value of atmospheric N2 in cycad foliage, the fossil record of cycads may provide an archive of atmospheric δ15N values. To explore this potential proxy, we conducted a survey of wild cycads growing in a range of modern environments to determine whether cycad foliage reliably records the isotopic composition of atmospheric N2. We find that neither biological nor environmental factors significantly influence the δ15N values of cycad foliage, suggesting that they provide a reasonably robust record of the δ15N of atmospheric N2. Application of this proxy to the record of carbonaceous cycad fossils may not only help to constrain changes in atmospheric nitrogen isotope ratios since the late Paleozoic, but also could shed light on the antiquity of the N2‐fixing symbiosis between cycads and cyanobacteria.PostprintPeer reviewe

Generic multiloop methods and application to N=4 super-Yang-Mills

We review some recent additions to the tool-chest of techniques for finding compact integrand representations of multiloop gauge-theory amplitudes - including non-planar contributions - applicable for N=4 super-Yang-Mills in four and higher dimensions, as well as for theories with less supersymmetry. We discuss a general organization of amplitudes in terms of purely cubic graphs, review the method of maximal cuts, as well as some special D-dimensional recursive cuts, and conclude by describing the efficient organization of amplitudes resulting from the conjectured duality between color and kinematic structures on constituent graphs.Comment: 42 pages, 18 figures, invited review for a special issue of Journal of Physics A devoted to "Scattering Amplitudes in Gauge Theories", v2 minor corrections, v3 added reference

Yangian symmetry of light-like Wilson loops

We show that a certain class of light-like Wilson loops exhibits a Yangian symmetry at one loop, or equivalently, in an Abelian theory. The Wilson loops we discuss are equivalent to one-loop MHV amplitudes in N=4 super Yang-Mills theory in a certain kinematical regime. The fact that we find a Yangian symmetry constraining their functional form can be thought of as the effect of the original conformal symmetry associated to the scattering amplitudes in the N=4 theory.Comment: 15 pages, 5 figure

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