MILICEVIC L.,Wilberforce Road
Combinatorics Probability and Computing | Year: 2017
Erdős asked the following question: given n points in the plane in almost general position (no four collinear), how large a set can we guarantee to find that is in general position (no three collinear)? Füredi constructed a set of n points in almost general position with no more than o(n) points in general position. Cardinal, Tóth and Wood extended this result to ℝ3, finding sets of n points with no five in a plane whose subsets with no four points in a plane have size o(n), and asked the question for higher dimensions: for given n, is it still true that the largest subset in general position we can guarantee to find has size o(n)? We answer their question for all d and derive improved bounds for certain dimensions. Copyright © Cambridge University Press 2017
Jack I.,University of Liverpool |
Osborn H.,Wilberforce Road
Nuclear Physics B | Year: 2014
The response of four dimensional quantum field theories to a Weyl rescaling of the metric in the presence of local couplings and which involve a, the coefficient of the Euler density in the energy momentum tensor trace on curved space, is reconsidered. Previous consistency conditions for the anomalous terms, which implicitly define a metric G on the space of couplings and give rise to gradient flow like equations for a, are derived taking into account the role of lower dimension operators. The results for infinitesimal Weyl rescaling are integrated to finite rescalings e2σ to a form which involves running couplings gσ and which interpolates between IR and UV fixed points. The results are also restricted to flat space where they give rise to broken conformal Ward identities.Expressions for the three loop Yukawa β-functions for a general scalar/fermion theory are obtained and the three loop contribution to the metric G for this theory is also calculated. These results are used to check the gradient flow equations to higher order than previously. It is shown that these are only valid when β→. B, a modified β-function, and that the equations provide strong constraints on the detailed form of the three loop Yukawa β-function. N=1 supersymmetric Wess-Zumino theories are also considered as a special case. It is shown that the metric for the complex couplings in such theories may be restricted to a hermitian form. © 2014 The Authors.
Mason L.,Mathematical Institute andrew Wiles Building |
Skinner D.,Wilberforce Road
Journal of High Energy Physics | Year: 2014
We show that string theories admit chiral infinite tension analogues in which only the massless parts of the spectrum survive. Geometrically they describe holomorphic maps to spaces of complex null geodesics, known as ambitwistor spaces. They have the standard critical space-time dimensions of string theory (26 in the bosonic case and 10 for the superstring). Quantization leads to the formulae for tree-level scattering amplitudes of massless particles found recently by Cachazo, He and Yuan. These representations localize the vertex operators to solutions of the same equations found by Gross and Mende to govern the behaviour of strings in the limit of high energy, fixed angle scattering. Here, localization to the scattering equations emerges naturally as a consequence of working on ambitwistor space. The worldsheet theory suggests a way to extend these amplitudes to spinor fields and to loop level. We argue that this family of string theories is a natural extension of the existing twistor string theories. © 2014 The Author(s).
Bjornsson J.,Wilberforce Road
Journal of High Energy Physics | Year: 2011
This paper provides a more detailed background concerning the previously obtained properties of multi-loop amplitudes in a pure spinor formulation of field theories with maximal supersymmetry. This involves the development of a first quantised field theory version of the non-minimal pure spinor formalism originally designed for describing superstring amplitudes. In addition to superspace world-line fields, the formalism involves a set of world-line ghost fields that are required for implementing BRST invariance with a composite b ghost. In particular, we show that BRST invariance requires the presence of certain contact terms. For four-point amplitudes, these are important beyond two loops. We also present an alternative proof of the "no-triangle hypothesis" and the vanishing of amplitudes with fewer than four external particles. © SISSA 2011.
Casali E.,Wilberforce Road |
Tourkine P.,Wilberforce Road
Journal of High Energy Physics | Year: 2015
Abstract: The recently introduced ambitwistor string led to a striking proposal for one-loop maximal supergravity amplitudes, localised on the solutions of the ambitwistor one-loop scattering equations. However, these amplitudes have not been explicitly analysed, due to the apparent complexity of the equations that determine the localisation. In this paper we propose an analytic solution to the four-point one-loop scattering equations in the infrared (IR) regime of the amplitude. Using this solution, we compute the ambitwistor integrand and demonstrate that it correctly reproduces the four-graviton integrand in the IR regime. This solution qualitatively extends to n points. To conclude, we explain that the ambitwistor one-loop scattering equations actually correspond to the standard Gross & Mende saddle point. © 2015, The Author(s).
Casali E.,Wilberforce Road
Journal of High Energy Physics | Year: 2014
In this short note I show that the soft limit for colour-ordered tree-level YangMills amplitudes contains a sub-leading divergent term. This term is analogous to those found recently by Cachazo and Strominger for tree-level gravity amplitudes. © 2014 The Author(s).
Petridis G.,Wilberforce Road
Combinatorics Probability and Computing | Year: 2011
Plünnecke's inequality is a standard tool for obtaining estimates on the cardinality of sumsets and has many applications in additive combinatorics. We present a new proof. The main novelty is that the proof is completed with no reference to Menger's theorem or Cartesian products of graphs. We also investigate the sharpness of the inequality and show that it can be sharp for arbitrarily long, but not for infinite commutative graphs. A key step in our investigation is the construction of arbitrarily long regular commutative graphs. Lastly we prove a necessary condition for the inequality to be attained. © 2011 Cambridge University Press.
Bjornsson J.,Wilberforce Road |
Green M.B.,Wilberforce Road
Journal of High Energy Physics | Year: 2010
A first quantised approach to loop amplitudes based on the pure spinor particle is applied to the systematics of four-particle amplitudes in maximally supersymmetric field theories. Counting of fermionic zero modes allows the identification of momentum factors multiplying R4 in the case of supergravity (and F4 in the Yang-Mills case) thereby making manifest their ultraviolet properties as a function of dimension, D. For L = 2; 3; 4 loops the leading supergravity divergence is in D = 4+6/L dimensions and proportional to ∂2L R4, in line with earlier field theory calculations. However, at five loops there is a radical change in the systematics, suggesting the presence of a contribution with an explicit L = 5 logarithmic ultraviolet divergence when D = 24/5 that is proportional to ∂8R4. We further argue that ∂8R 4 should receive contributions from all loops, which would imply that N = 8 supergravity (with D = 4) is not protected by supersymmetry from a seven-loop divergence. © 2010 SISSA.
Osborn H.,Wilberforce Road
Physics Letters, Section B: Nuclear, Elementary Particle and High-Energy Physics | Year: 2012
Conformal blocks for the finite dimension conformal group SO(2,2) for four point functions for fields with arbitrary spins in two dimensions are obtained by evaluating an appropriate integral. The results are just products of hypergeometric functions of the conformally invariant cross ratios formed from the four complex coordinates. Results for scalars previously obtained are a special case. Applications to four point functions involving the energy momentum tensor are discussed. © 2012 Elsevier B.V.
Bahr B.,Wilberforce Road
Journal of Physics: Conference Series | Year: 2012
A dual holonomy version of operator spin foam models is presented, which is particularly adapted to the notion of coarse graining. We discuss how this leads to a natural way of comparing models on different discretization scales, and a notion of renormalization group flow on the partially ordered set of 2-complexes. © Published under licence by IOP Publishing Ltd.