Sborlini, G. F. R., de Florian, D., & Rodrigo, G. (2014). Double collinear splitting amplitudes at next-to-leading order. J. High Energy Phys., 01(1), 018–55pp.
Abstract: We compute the next-to-leading order (NLO) QCD corrections to the 1 -> 2 splitting amplitudes in different dimensional regularization (DREG) schemes. Besides recovering previously known results, we explore new DREG schemes and analyze their consistency by comparing the divergent structure with the expected behavior predicted by Catani's formula. Through the introduction of scalar-gluons, we show the relation among splittings matrices computed using different schemes. Also, we extended this analysis to cover the double collinear limit of scattering amplitudes in the context of QCD+QED.
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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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de Florian, D., Sborlini, G. F. R., & Rodrigo, G. (2016). Two-loop QED corrections to the Altarelli-Parisi splitting functions. J. High Energy Phys., 10(10), 056–16pp.
Abstract: We compute the two-loop QED corrections to the Altarelli-Parisi (AP) splitting functions by using a deconstructive algorithmic Abelianization of the well-known NLO QCD corrections. We present explicit results for the full set of splitting kernels in a basis that includes the leptonic distribution functions that, starting from this order in the QED coupling, couple to the partonic densities. Finally, we perform a phenomenological analysis of the impact of these corrections in the splitting functions.
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Aguilera-Verdugo, J. J., Hernandez-Pinto, R. J., Rodrigo, G., Sborlini, G. F. R., & Torres Bobadilla, W. J. (2021). Causal representation of multi-loop Feynman integrands within the loop-tree duality. J. High Energy Phys., 01(1), 69–26pp.
Abstract: The numerical evaluation of multi-loop scattering amplitudes in the Feynman representation usually requires to deal with both physical (causal) and unphysical (non-causal) singularities. The loop-tree duality (LTD) offers a powerful framework to easily characterise and distinguish these two types of singularities, and then simplify analytically the underling expressions. In this paper, we work explicitly on the dual representation of multi-loop Feynman integrals generated from three parent topologies, which we refer to as Maximal, Next-to-Maximal and Next-to-Next-to-Maximal loop topologies. In particular, we aim at expressing these dual contributions, independently of the number of loops and internal configurations, in terms of causal propagators only. Thus, providing very compact and causal integrand representations to all orders. In order to do so, we reconstruct their analytic expressions from numerical evaluation over finite fields. This procedure implicitly cancels out all unphysical singularities. We also interpret the result in terms of entangled causal thresholds. In view of the simple structure of the dual expressions, we integrate them numerically up to four loops in integer space-time dimensions, taking advantage of their smooth behaviour at integrand level.
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