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Sborlini, G. F. R., de Florian, D., & Rodrigo, G. (2015). Polarized triple-collinear splitting functions at NLO for processes with photons. J. High Energy Phys., 03(3), 021–30pp.
Abstract: We compute the polarized splitting functions in the triple collinear limit at next-to-leading order accuracy (NLO) in the strong coupling alpha(S), for the splitting processes gamma -> qq gamma, gamma -> qqg and g -> qq gamma. The divergent structure of each splitting function was compared to the predicted behaviour according to Catani's formula. The results obtained in this paper are compatible with the unpolarized splitting functions computed in a previous article. Explicit results for NLO corrections are presented in the context of conventional dimensional regularization (CDR).
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Hernandez-Pinto, R. J., Sborlini, G. F. R., & Rodrigo, G. (2016). Towards gauge theories in four dimensions. J. High Energy Phys., 02(2), 044–14pp.
Abstract: The abundance of infrared singularities in gauge theories due to unresolved emission of massless particles (soft and collinear) represents the main difficulty in perturbative calculations. They are typically regularized in dimensional regularization, and their subtraction is usually achieved independently for virtual and real corrections. In this paper, we introduce a new method based on the loop-tree duality (LTD) theorem to accomplish the summation over degenerate infrared states directly at the integrand level such that the cancellation of the infrared divergences is achieved simultaneously, and apply it to reference examples as a proof of concept. Ultraviolet divergences, which are the consequence of the point-like nature of the theory, are also reinterpreted physically in this framework. The proposed method opens the intriguing possibility of carrying out purely four-dimensional implementations of higher-order perturbative calculations at next-to-leading order (NLO) and beyond free of soft and final-state collinear subtractions.
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Sborlini, G. F. R., Driencourt-Mangin, F., Hernandez-Pinto, R. J., & Rodrigo, G. (2016). Four-dimensional unsubtraction from the loop-tree duality. J. High Energy Phys., 08(8), 160–42pp.
Abstract: We present a new algorithm to construct a purely four dimensional representation of higher-order perturbative corrections to physical cross-sections at next-to-leading order (NLO). The algorithm is based on the loop-tree duality (LTD), and it is implemented by introducing a suitable mapping between the external and loop momenta of the virtual scattering amplitudes, and the external momenta of the real emission corrections. In this way, the sum over degenerate infrared states is performed at integrand level and the cancellation of infrared divergences occurs locally without introducing subtraction counter-terms to deal with soft and final-state collinear singularities. The dual representation of ultraviolet counter-terms is also discussed in detail, in particular for self-energy contributions. The method is first illustrated with the scalar three-point function, before proceeding with the calculation of the physical cross-section for gamma* -> q (q) over bar (g), and its generalisation to multi-leg processes. The extension to next-to-next-to-leading order (NNLO) is briefly commented.
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Sborlini, G. F. R., Driencourt-Mangin, F., & Rodrigo, G. (2016). Four-dimensional unsubtraction with massive particles. J. High Energy Phys., 10(10), 162–34pp.
Abstract: We extend the four-dimensional unsubtraction method, which is based on the loop-tree duality (LTD), to deal with processes involving heavy particles. The method allows to perform the summation over degenerate IR configurations directly at integrand level in such a way that NLO corrections can be implemented directly in four space-time dimensions. We define a general momentum mapping between the real and virtual kinematics that accounts properly for the quasi-collinear configurations, and leads to an smooth massless limit. We illustrate the method first with a scalar toy example, and then analyse the case of the decay of a scalar or vector boson into a pair of massive quarks. The results presented in this paper are suitable for the application of the method to any multipartonic process.
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Catani, S., de Florian, D., & Rodrigo, G. (2012). Space-like (vs. time-like) collinear limits in QCD: is factorization violated? J. High Energy Phys., 07(7), 026–88pp.
Abstract: We consider the singular behaviour of QCD scattering amplitudes in kinematical configurations where two or more momenta of the external partons become collinear. At the tree level, this behaviour is known to be controlled by factorization formulae in which the singular collinear factor is universal (process independent). We show that this strict (process-independent) factorization is not valid at one-loop and higher-loop orders in the case of the collinear limit in space-like regions (e. g., collinear radiation from initial-state partons). We introduce a generalized version of all-order collinear factorization, in which the space-like singular factors retain some dependence on the momentum and colour charge of the non-collinear partons. We present explicit results on one-loop and two-loop amplitudes for both the two-parton and multiparton collinear limits. At the level of squared amplitudes and, more generally, cross sections in hadron-hadron collisions, the violation of strict collinear factorization has implications on the non-abelian structure of logarithmically-enhanced terms in perturbative calculations (starting from the next-to-next-to-leading order) and on various factorization issues of mass singularities (starting from the next-to-next-to-next-to-leading order).
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