Browsing by Author "Cardona, Carlos A."
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Item CHY-graphs on a torus(Springer Verlag, 2016-10-12) Cardona, Carlos A.; Gómez, HumbertoRecently, we proposed a new approach using a punctured Elliptic curve in the CHY framework in order to compute one-loop scattering amplitudes. In this note, we further develop this approach by introducing a set of connectors, which become the main ingredient to build integrands on M 1 , n , the moduli space of n-punctured Elliptic curves. As a particular application, we study the Φ 3 bi-adjoint scalar theory. We propose a set of rules to construct integrands on M 1 , n from Φ 3 integrands on M 0 , n , the moduli space of n-punctured spheres. We illustrate these rules by computing a variety of Φ 3 one-loop Feynman diagrams. Conversely, we also provide another set of rules to compute the corresponding CHY-integrand on M 1 , n by starting instead from a given Φ 3 one-loop Feynman diagram. In addition, our results can easily be extended to higher loops. © 2016, The Author(s).Item Cross-ratio identities and higher-order poles of CHY-integrand(Springer Verlag, 2016-09-01) Cardona, Carlos A.; Feng, Bo; Gomez, Humberto; Huang, RijunThe evaluation of generic Cachazo-He-Yuan(CHY)-integrands is a big challenge and efficient computational methods are in demand for practical evaluation. In this paper, we propose a systematic decomposition algorithm by using cross-ratio identities, which provides an analytic and easy to implement method for the evaluation of any CHY-integrand. This algorithm aims to decompose a given CHY-integrand containing higher-order poles as a linear combination of CHY-integrands with only simple poles in a finite number of steps, which ultimately can be trivially evaluated by integration rules of simple poles. To make the method even more efficient for CHY-integrands with large number of particles and complicated higher-order pole structures, we combine the Λ-algorithm and the cross-ratio identities, and as a by-product it provides us a way to deal with CHY-integrands where the Λ-algorithm was not applicable in its original formulation. © 2016, The Author(s).Item Elliptic scattering equations(Springer Verlag, 2016-06-01) Cardona, Carlos A.; Gomez, HumbertoRecently the CHY approach has been extended to one loop level using elliptic functions and modular forms over a Jacobian variety. Due to the difficulty in manipulating these kind of functions, we propose an alternative prescription that is totally algebraic. This new proposal is based on an elliptic algebraic curve embedded in a ℂP 2 space. We show that for the simplest integrand, namely the n − gon, our proposal indeed reproduces the expected result. By using the recently formulated Λ−algorithm, we found a novel recurrence relation expansion in terms of tree level off-shell amplitudes. Our results connect nicely with recent results on the one-loop formulation of the scattering equations. In addition, this new proposal can be easily stretched out to hyperelliptic curves in order to compute higher genus. © 2016, The Author(s).