The Two-Loop Six-Gluon MHV Amplitude in Maximally Supersymmetric Yang-Mills Theory

We give a representation of the parity-even part of the planar two-loop six-gluon MHV amplitude of N=4 super-Yang-Mills theory, in terms of loop-momentum integrals with simple dual conformal properties. We evaluate the integrals numerically in order to test directly the ABDK/BDS all-loop ansatz for planar MHV amplitudes. We find that the ansatz requires an additive remainder function, in accord with previous indications from strong-coupling and Regge limits. The planar six-gluon amplitude can also be compared with the hexagonal Wilson loop computed by Drummond, Henn, Korchemsky and Sokatchev in arXiv:0803.1466 [hep-th]. After accounting for differing singularities and other constants independent of the kinematics, we find that the Wilson loop and MHV-amplitude remainders are identical, to within our numerical precision. This result provides non-trivial confirmation of a proposed n-point equivalence between Wilson loops and planar MHV amplitudes, and suggests that an additional mechanism besides dual conformal symmetry fixes their form at six points and beyond.Comment: 49 pages, RevTex, 2 figure file, v2 minor correction

Analytic result for the two-loop six-point NMHV amplitude in N=4 super Yang-Mills theory

We provide a simple analytic formula for the two-loop six-point ratio function of planar N = 4 super Yang-Mills theory. This result extends the analytic knowledge of multi-loop six-point amplitudes beyond those with maximal helicity violation. We make a natural ansatz for the symbols of the relevant functions appearing in the two-loop amplitude, and impose various consistency conditions, including symmetry, the absence of spurious poles, the correct collinear behaviour, and agreement with the operator product expansion for light-like (super) Wilson loops. This information reduces the ansatz to a small number of relatively simple functions. In order to fix these parameters uniquely, we utilize an explicit representation of the amplitude in terms of loop integrals that can be evaluated analytically in various kinematic limits. The final compact analytic result is expressed in terms of classical polylogarithms, whose arguments are rational functions of the dual conformal cross-ratios, plus precisely two functions that are not of this type. One of the functions, the loop integral \Omega^{(2)}, also plays a key role in a new representation of the remainder function R_6^{(2)} in the maximally helicity violating sector. Another interesting feature at two loops is the appearance of a new (parity odd) \times (parity odd) sector of the amplitude, which is absent at one loop, and which is uniquely determined in a natural way in terms of the more familiar (parity even) \times (parity even) part. The second non-polylogarithmic function, the loop integral \tilde{\Omega}^{(2)}, characterizes this sector. Both \Omega^{(2)} and tilde{\Omega}^{(2)} can be expressed as one-dimensional integrals over classical polylogarithms with rational arguments.Comment: 51 pages, 4 figures, one auxiliary file with symbols; v2 minor typo correction

One-loop corrections to AdS_5 x S^5 superstring partition function via Pohlmeyer reduction

We discuss semiclassical expansions around a class of classical string configurations lying in AdS_3 x S^1 using the Pohlmeyer-reduced from of the AdS_5 x S^5 superstring theory. The Pohlmeyer reduction of the AdS_5 x S^5 superstring theory is a gauged Wess-Zumino-Witten model with an integrable potential and two-dimensional fermionic fields. It was recently conjectured that the quantum string partition function is equal to the quantum reduced theory partition function. Continuing the previous paper (arXiv:0906.3800) where arbitrary solutions in AdS_2 x S^2 and homogeneous solutions were considered, we provide explicit demonstration of this conjecture at the one-loop level for several string solutions in AdS_3 x S^1 embedded into AdS_5 x S^5. Quadratic fluctuations derived in the reduced theory for inhomogeneous strings are equivalent to respective fluctuations found from the Nambu action in the original string theory. We also show the equivalence of fluctuation frequencies for homogeneous strings with both the orbital momentum and the winding on a big circle of S^5.Comment: 45 pages, references added, minor correction

Integrable spin chains and scattering amplitudes

In this review we show that the multi-particle scattering amplitudes in N=4 SYM at large Nc and in the multi-Regge kinematics for some physical regions have the high energy behavior appearing from the contribution of the Mandelstam cuts in the complex angular momentum plane of the corresponding t-channel partial waves. These Mandelstam cuts or Regge cuts are resulting from gluon composite states in the adjoint representation of the gauge group SU(Nc). In the leading logarithmic approximation (LLA) their contribution to the six point amplitude is in full agreement with the known two-loop result. The Hamiltonian for the Mandelstam states constructed from n gluons in LLA coincides with the local Hamiltonian of an integrable open spin chain. We construct the corresponding wave functions using the integrals of motion and the Baxter-Sklyanin approach.Comment: 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

The State of the Art in Language Workbenches. Conclusions from the Language Workbench Challenge

Language workbenches are tools that provide high-level mechanisms for the implementation of (domain-specific) languages. Language workbenches are an active area of research that also receives many contributions from industry. To compare and discuss existing language workbenches, the annual Language Workbench Challenge was launched in 2011. Each year, participants are challenged to realize a given domain-specific language with their workbenches as a basis for discussion and comparison. In this paper, we describe the state of the art of language workbenches as observed in the previous editions of the Language Workbench Challenge. In particular, we capture the design space of language workbenches in a feature model and show where in this design space the participants of the 2013 Language Workbench Challenge reside. We compare these workbenches based on a DSL for questionnaires that was realized in all workbenches

Hidden Simplicity of Gauge Theory Amplitudes

These notes were given as lectures at the CERN Winter School on Supergravity, Strings and Gauge Theory 2010. We describe the structure of scattering amplitudes in gauge theories, focussing on the maximally supersymmetric theory to highlight the hidden symmetries which appear. Using the BCFW recursion relations we solve for the tree-level S-matrix in N=4 super Yang-Mills theory, and describe how it produces a sum of invariants of a large symmetry algebra. We review amplitudes in the planar theory beyond tree-level, describing the connection between amplitudes and Wilson loops, and discuss the implications of the hidden symmetries.Comment: 46 pages, 15 figures. v2 ref added, typos fixe

Does the Effectiveness of Control Measures Depend on the Influenza Pandemic Profile?

BACKGROUND: Although strategies to contain influenza pandemics are well studied, the characterization and the implications of different geographical and temporal diffusion patterns of the pandemic have been given less attention. METHODOLOGY/MAIN FINDINGS: Using a well-documented metapopulation model incorporating air travel between 52 major world cities, we identified potential influenza pandemic diffusion profiles and examined how the impact of interventions might be affected by this heterogeneity. Clustering methods applied to a set of pandemic simulations, characterized by seven parameters related to the conditions of emergence that were varied following Latin hypercube sampling, were used to identify six pandemic profiles exhibiting different characteristics notably in terms of global burden (from 415 to >160 million of cases) and duration (from 26 to 360 days). A multivariate sensitivity analysis showed that the transmission rate and proportion of susceptibles have a strong impact on the pandemic diffusion. The correlation between interventions and pandemic outcomes were analyzed for two specific profiles: a fast, massive pandemic and a slow building, long-lasting one. In both cases, the date of introduction for five control measures (masks, isolation, prophylactic or therapeutic use of antivirals, vaccination) correlated strongly with pandemic outcomes. Conversely, the coverage and efficacy of these interventions only moderately correlated with pandemic outcomes in the case of a massive pandemic. Pre-pandemic vaccination influenced pandemic outcomes in both profiles, while travel restriction was the only measure without any measurable effect in either. CONCLUSIONS: our study highlights: (i) the great heterogeneity in possible profiles of a future influenza pandemic; (ii) the value of being well prepared in every country since a pandemic may have heavy consequences wherever and whenever it starts; (iii) the need to quickly implement control measures and even to anticipate pandemic emergence through pre-pandemic vaccination; and (iv) the value of combining all available control measures except perhaps travel restrictions

One-loop derivation of the Wilson polygon - MHV amplitude duality

We discuss the origin of the Wilson polygon - MHV amplitude duality at the perturbative level. It is shown that the duality for the MHV amplitudes at one-loop level can be proven upon the peculiar change of variables in Feynman parametrization and the use of the relation between Feynman integrals at the different space-time dimensions. Some generalization of the duality which implies the insertion of the particular vertex operator at the Wilson triangle is found for the 3-point function. We discuss analytical structure of Wilson loop diagrams and present the corresponding Landau equations. The geometrical interpretation of the loop diagram in terms of the hyperbolic geometry is discussed.Comment: 29 page

Influenza A Gradual and Epochal Evolution: Insights from Simple Models

The recurrence of influenza A epidemics has originally been explained by a “continuous antigenic drift” scenario. Recently, it has been shown that if genetic drift is gradual, the evolution of influenza A main antigen, the haemagglutinin, is punctuated. As a consequence, it has been suggested that influenza A dynamics at the population level should be approximated by a serial model. Here, simple models are used to test whether a serial model requires gradual antigenic drift within groups of strains with the same antigenic properties (antigenic clusters). We compare the effect of status based and history based frameworks and the influence of reduced susceptibility and infectivity assumptions on the transient dynamics of antigenic clusters. Our results reveal that the replacement of a resident antigenic cluster by a mutant cluster, as observed in data, is reproduced only by the status based model integrating the reduced infectivity assumption. This combination of assumptions is useful to overcome the otherwise extremely high model dimensionality of models incorporating many strains, but relies on a biological hypothesis not obviously satisfied. Our findings finally suggest the dynamical importance of gradual antigenic drift even in the presence of punctuated immune escape. A more regular renewal of susceptible pool than the one implemented in a serial model should be part of a minimal theory for influenza at the population level

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