Colour-Kinematics Duality for One-Loop Rational Amplitudes

Colour-kinematics duality is the conjecture of a group theory-like structure for the kinematic dependence of scattering amplitudes in gauge theory and gravity. This structure has been verified at tree level in various ways, but similar progress has been lacking at loop level, where the power of the duality would be most significant. Here we explore colour-kinematics duality at one loop using the self-dual sector as a starting point. The duality is shown to exist in pure Yang-Mills theory for two infinite classes of amplitudes: amplitudes with any number of particles either all of the same helicity or with one particle helicity opposite the rest. We provide a simple Lagrangian-based argument in favour of the double copy relation between gauge theory and gravity amplitudes in these classes, and provide some explicit examples. We further discuss aspects of the duality which persist after integration, leading to relations among partial amplitudes. Finally, we describe form factors in the self-dual theory at tree level which also satisfy the duality.Comment: 36 pages, 5 figures; v2: published versio

New relations for scattering amplitudes in Yang-Mills theory at loop level

The calculation of scattering amplitudes in Yang-Mills theory at loop level is important for the analysis of background processes at particle colliders as well as our understanding of perturbation theory at the quantum level. We present tools to derive relations for especially one loop amplitudes, as well as several explicit examples for gauge theory coupled to a wide variety of matter. These tools originate in certain scaling behavior of permutation and cyclic sums of Yang-Mills tree amplitudes and loop integrands. In the latter case evidence exists for relations at all loop orders.Comment: 12 pages, 4 figures. v3: typos corrected, figures and clarifications adde

Yang-Mills amplitude relations at loop level from non-adjacent BCFW shifts

This article studies methods to obtain relations for scattering amplitudes at the loop level, with concrete examples at one loop. These methods originate in the analysis of large so-called Britto-Cachazo-Feng-Witten shifts of tree level amplitudes and loop level integrands. In particular BCFW shifts for particles which are not color adjacent and some particular generalizations of this situation are analyzed in some detail in four and higher dimensions. For generic non-adjacent shifts our results are independent of loop order for integrands and hold for generic minimally coupled gauge theories with possible scalar potential and Yukawa terms. By a standard argument this result indicates a generalization of the Bern-Carrasco-Johansson relations for tree level amplitudes exists to the integrand at all loop levels. A concrete relation is presented at one loop. Furthermore, inspired by results in QED it is shown that the results on generalized BCFW shifts of tree level amplitudes imply relations for the so-called rational, bubble and triangle terms of one loop amplitudes in pure Yang-Mills theory. Bubble and triangle terms for instance are shown to obey a five photon decoupling identity, while a three photon decoupling identity is demonstrated for the rational terms. Along the same lines recently conjectured relations for helicity equal amplitudes at one loop are shown to generalize to helicity independent relations for the massive box coefficient of the rational terms.Comment: 69 pages, 27 figure

Of Gluons and Gravitons: Exploring Color-Kinematics Duality

In this thesis color-kinematics duality will be investigated. This duality is a statement aboutthe kinematical dependence of a scattering amplitude in Yang-Mills gauge theories obeyinggroup theoretical relations similar to that of the color gauge group. The major consequenceof this duality is that gravity amplitudes can be related to a certain double copy of gaugetheory amplitudes. The main focus of this thesis will be on exploring the foundations ofcolor-kinematics duality and its consequences. It will be shown how color-kinematics dualitycan be made manifest at the one-loop level for rational amplitudes. A Lagrangian-basedargument will be given for the validity of the double copy construction for these amplitudesincluding explicit examples at four points. Secondly, it will be studied how color-kinematicsduality can be used to improve powercounting in gravity theories. To this end the dualitywill be reformulated in terms of linear maps. It will be shown as an example how this canbe used to derive the large BCFW shift behavior of a gravity integrand constructed throughthe duality to any loop order up to subtleties inherent to the duality that will be addressed.As will become clear the duality implies massive cancellations with respect to the usualpowercounting of Feynman graphs indicating that gravity theories are much better behavedthan navely expected. As another example the linear map approach will be used to investigatethe question of UV-niteness of N = 8 supergravity and it will be seen that the amount ofcancellations depends on the exact implementation of the duality at loop level. Lastly, colorkinematicsduality will be considered from a Feynman-graph perspective reproducing some ofthe results of the earlier chapters thus giving non-trivial evidence for the duality at the looplevel from a dierent perspective