1,721,009 research outputs found
On the singularities of scattering amplitudes: from cluster adjacency tropical geometry
No full textPlanar maximally supersymmetric Yang-Mills theory (N = 4 SYM) is a unique quantum field theory. Its interesting properties include conformal symmetry at the quantum level, expected integrability, and being a prime example of the AdS/CFT correspondence. Witten’s twistor string theory [1] reignited an interest in this theory, leading to enormous progress in utilising its special features to uncover new mathematical structures, such as the Grassmannian [2], and to better understand gauge theory in general. One such feature is the hidden dual conformal symmetry in SYM [3, 4] which led to the introduction of momentum twistors [5] as useful variables when describing scattering amplitudes in SYM. It was then shown in [6] that the kinematic space parametrised by these momentum twistors, itself a Grassmannian, comes equipped with a particular mathematical structure called a cluster algebra. It was shown that scattering amplitudes in SYM depend on cluster coordinates and, at least for six and seven-points, the cluster algebra provided all the singularities, or alphabet, for all known amplitudes in SYM. The ultimate hope is that by understanding all the analytic structure of the functions scattering amplitudes consist of, one could simply write down the form of an amplitude without the need for an explicit calculation. The overarching theme of this thesis is to better understand the analytic properties of scattering amplitudes, mainly (but not exclusively) in SYM. Although this thesis has two parts, the main focus is on cluster algebras; extracting the information they contain about amplitudes and exploring their relationship with other areas of mathematics, as well as what those relationships imply for the singularity properties of amplitudes. In the first part, we introduce and develop the notion of cluster adjacency and how it controls the poles and branch cuts an amplitude is allowed to have. We also provide an example of how cluster adjacency can aid in computations in calculating the seven-point, four-loop, NMHV scattering amplitude. The second part expands on the discussion started in [7] linking tropical geometry to scattering amplitudes in the biadjoint φ 3 theory. We demonstrate how the connection between tropical Grassmannians and cluster algebras allows for straightforward calculation of amplitudes in this theory. We go on to utilise this connection to generalise the notion of cluster adjacency and conjecture how different helicity configurations (MHV, NMHV, etc.) may each have their own cluster adjacency rules. Finally, tropical geometry allows us to approach certain issues with eight-point scattering in SYM. Namely, the infinite set of cluster coordinates provided by cluster algebra associated with eight-point scattering, as well as generating the algebraic singularities known to appear even at one-loop for N2MHV amplitudes. This thesis is based on [8–13] with considerable overlap with these papers
Tropical Grassmannians, cluster algebras and scattering amplitudes
No full textWe provide a cluster-algebraic approach to the computation of the recently introduced generalised biadjoint scalar amplitudes related to Grassmannians Gr(k, n). A finite cluster algebra provides a natural triangulation for the tropical Grassmannian whose volume computes the scattering amplitudes. Using this method one can construct the entire colour-ordered amplitude via mutations starting from a single term
Algebraic singularities of scattering amplitudes from tropical geometry
No full textWe address the appearance of algebraic singularities in the symbol alphabet of scattering amplitudes in the context of planar = 4 super Yang-Mills theory. We argue that connections between cluster algebras and tropical geometry provide a natural language for postulating a finite alphabet for scattering amplitudes beyond six and seven points where the corresponding Grassmannian cluster algebras are finite. As well as generating natural finite sets of letters, the tropical fans we discuss provide letters containing square roots. Remarkably, the minimal fan we consider provides all the square root letters recently discovered in an explicit two-loop eight-point NMHV calculation
Cluster adjacency properties of scattering amplitudes in N=4 supersymmetric Yang-Mills Theory
Get PDFWe conjecture a new set of analytic relations for scattering amplitudes in planar N=4 super Yang-Mills theory. They generalize the Steinmann relations and are expressed in terms of the cluster algebras associated to Gr(4,n). In terms of the symbol, they dictate which letters can appear consecutively. We study heptagon amplitudes and integrals in detail and present symbols for previously unknown integrals at two and three loops which support our conjecture
Going Beyond Counting First Authors in Author Co-citation Analysis
Get PDFThe present study examines one of the fundamental aspects of author co-citation analysis (ACA) - the way co-citation
counts are defined. Co-citation counting provides the data on which all subsequent statistical analyses and mappings
are based, and we compare ACA results based on two different types of co-citation counting - the traditional type that
only counts the first one among a cited work's authors on the one hand and a non-traditional type that takes into
account the first 5 authors of a cited work on the other hand. Results indicate that the picture produced through this non-traditional author co-citation counting contains more coherent author groups and is therefore considerably clearer. However, this picture represents fewer specialties in the research field being studied than that produced through the traditional first-author co-citation counting when the same number of top-ranked authors is selected and analyzed. Reasons for these effects are discussed
Cluster adjacency beyond MHV
Get PDFWe explore further the notion of cluster adjacency, focussing on non-MHV amplitudes. We extend the notion of adjacency to the BCFW decomposition of tree-level amplitudes. Adjacency controls the appearance of poles, both physical and spurious, in individual BCFW terms. We then discuss how this notion of adjacency is connected to the adjacency already observed at the level of symbols of scattering amplitudes which controls the appearance of branch cut singularities. Poles and symbols become intertwined by cluster adjacency and we discuss the relation of this property to the Q¯ -equation which imposes constraints on the derivatives of the transcendental functions appearing in loop amplitudes.</p
Tropical fans, scattering equations and amplitudes
No full textAbstract We describe a family of tropical fans related to Grassmannian cluster algebras. These fans are related to the kinematic space of massless scattering processes in a number of ways. For each fan associated to the Grassmannian Gr(k, n) there is a notion of a generalised ϕ 3 amplitude and an associated set of scattering equations which further generalise the Gr(k, n) scattering equations that have been recently introduced. Here we focus mostly on the cases related to finite Grassmannian cluster algebras and we explain how face variables for the cluster polytopes are simply related to the scattering equations. For the Grassmannians Gr(4, n) the tropical fans we describe are related to the singularities (or symbol letters) of loop amplitudes in planar N = 4 super Yang-Mills theory. We show how each choice of tropical fan leads to a natural class of polylogarithms, generalising the notion of cluster adjacency and we describe how the currently known loop data fit into this classification
Variations on the Author
Get PDF“Variations on the Author” discusses two of Eduardo Coutinho’s recent films (Um Dia na Vida, from 2010, and Últimas Conversas, posthumously released in 2015) and their contribution to the general question of documentary authorship. The director’s filmography is characterized by a consistent yet self-effacing form of authorial self-inscription: Coutinho often features as an interviewer that rather than express opinions propels discourses; an interviewer that is good at listening. This mode of self-inscription characterizes him as an author who is not expressive but who is nonetheless markedly present on the screen. In Um Dia na Vida, however, Coutinho is completely absent form the image, while Últimas Conversas, on the contrary, includes a confessional prologue that moves the director from the margins to the center of his films. This article examines the ways in which these works stand out in the filmography of a director who offers new insights into the notion of cinematic authorship
Appropriate Similarity Measures for Author Cocitation Analysis
Get PDFWe provide a number of new insights into the methodological discussion about author cocitation analysis. We first argue that the use of the Pearson correlation for measuring the similarity between authors’ cocitation profiles is not very satisfactory. We then discuss what kind of similarity measures may be used as an alternative to the Pearson correlation. We consider three similarity measures in particular. One is the well-known cosine. The other two similarity measures have not been used before in the bibliometric literature. Finally, we show by means of an example that our findings have a high practical relevance.information science;Pearson correlation;cosine;similarity measure;author cocitation analysis
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