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Bierenbaum, I., Catani, S., Draggiotis, P., & Rodrigo, G. (2010). A tree-loop duality relation at two loops and beyond. J. High Energy Phys., 10(10), 073–22pp.
Abstract: The duality relation between one-loop integrals and phase-space integrals, developed in a previous work, is extended to higher-order loops. The duality relation is realized by a modification of the customary +i0 prescription of the Feynman propagators, which compensates for the absence of the multiple-cut contributions that appear in the Feynman tree theorem. We rederive the duality theorem at one-loop order in a form that is more suitable for its iterative extension to higher-loop orders. We explicitly show its application to two-and three-loop scalar master integrals, and we discuss the structure of the occurring cuts and the ensuing results in detail.
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Catani, S., & Cieri, L. (2022). Multiple soft radiation at one-loop order and the emission of a soft quark-antiquark pair. Eur. Phys. J. C, 82(2), 97–27pp.
Abstract: We consider the radiation of two or more soft partons in QCD hard-scattering at one-loop order. The corresponding scattering amplitude is singular, and the singular behaviour is controlled by a process-independent soft current. Using regularization in d = 4 – 2 epsilon space-time dimensions, we explicitly evaluate the ultraviolet and infrared divergent (epsilon-pole) terms of the one-loop soft current for emission of an arbitrary number of soft partons in a generic hard-scattering process. Then we consider the specific case of soft quark-antiquark (q (q) over bar) emission and we compute the one-loop current by including the finite terms. We find that the one-loop soft-q (q) over bar current exhibits a new type of transverse-momentum singularity, which has a quantum (absorptive) origin and a purely non-abelian character. At the squared amplitude (cross section) level, this transverse-momentum singularity produces contributions to multijet production processes in hadron collisions. The one-loop squared current also leads to charge asymmetry terms, which are a distinctive features of soft-q (q) over bar radiation. We also extend these results to the cases of QED and mixed QCDxQED radiative corrections for soft fermion-antifermion emission.
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Catani, S., Cieri, L., Colferai, D., & Coradeschi, F. (2023). Soft gluon-quark-antiquark emission in QCD hard scattering. Eur. Phys. J. C, 83(1), 38–18pp.
Abstract: We consider the radiation of a soft gluon (g) and a soft quark-antiquark (qq & macr;) pair in QCD hard scattering. In the soft limit the scattering amplitude has a singular behaviour that is factorized and controlled by a soft current, which has a process-independent structure in colour space. We evaluate the soft gqq & macr; current at the tree level for an arbitrary multiparton scattering process. The irreducible correlation component of the current includes strictly nonabelian terms and also terms with an abelian character. Analogous abelian correlations appear for soft photon-lepton- antilepton emission in QED. The squared current for soft gqq & macr; emission produces colour dipole and colourtripole interactions between the hard-scattering partons. The colour tripole interactions are odd under charge conjugation and lead to charge asymmetry effects. We consider the specific applications to processes with two and three hard partons, and we discuss the structure of the corresponding charge asymmetry contributions. We also generalize our QCD results to the cases of QED and mixed QCD x QED radiative corrections.
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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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