## Sean Solari - PHS3350 Project 2020## Crossing Relations for Double-Gluon Emission Antenna functions |

It is known that charged particles emit radiation when they are accelerated up to high energies. This is observed in abundance from those particles accelerated within the Large-Hadron Collider, as well as from the by-products of collisions between these accelerated particles. As a consequence of this, a series of cascaded emissions or bremsstrahlung is observed in the detectors around these high-energy collisions. Due to the large number of particles involved in these so-called parton showers, direct matrix element calculations are not a viable option in trying to model these events. Instead, the known singularity structure of matrix elements in general is exploited and used to factor out a splitting function, thereby isolating the effect of individual bremsstrahlung and enabling an iterative approach to modelling parton showers. We explore structure within the antenna function for double-gluon emission from a quark-antiquark pair in the initial state, before they annihilate to form a Z-boson. Within this antenna function, there is physically interesting structure related to the different configurations that the emitted gluons can take with respect to the quarks. Specifically, we explore the double-soft limit, wherein both gluons are emitted with negligible momenta, as well as the triple-collinear limit, where both gluons become collinear with the quark from which they were emitted. To validate the antenna function, we employ a uniform sampling over its domain and compare the resulting values to direct matrix element calculations. In a similar fashion, cuts are introduced into the domain of the antenna function to distinguish those points in phase-space corresponding to double-soft or triple-collinear configurations, thereby enabling a similar sampling of the derived limits. Ultimately, having the structure of the antenna function in these singular limits is useful as they are likely candidates in trying to find a simple overestimate of the full matrix element to enable Monte Carlo simulations of these events. Current parton shower models view double-gluon emission as iterated single- emission, and so are not able to account for the interference effects of Feynman diagrams containing two consecutive gluon emissions. Exploring the double-gluon emission splitting function will therefore aid in understanding the role of these coherent emissions. |

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