Abstract
Threshold logarithms become dominant in partonic cross sections when the selected final state forces gluon radiation to be soft or collinear. Such radiation factorizes at the level of scattering amplitudes, and this leads to the resummation of threshold logarithms which appear at leading power in the threshold variable. In this paper, we consider the extension of this factorization to include effects suppressed by a single power of the threshold variable. Building upon the Low-Burnett-Kroll-Del Duca (LBKD) theorem, we propose a decomposition of radiative amplitudes into universal building blocks, which contain all effects ultimately responsible for next-to-leading power (NLP) threshold logarithms in hadronic cross sections for electroweak annihilation processes. In particular, we provide a NLO evaluation of the "radiative jet function", responsible for the interference of next-to-soft and collinear effects in these cross sections. As a test, using our expression for the amplitude, we reproduce all abelian-like NLP threshold logarithms in the NNLO Drell-Yan cross section, including the interplay of real and virtual emissions. Our results are a significant step towards developing a generally applicable resummation formalism for NLP threshold effects, and illustrate the breakdown of next-to-soft theorems for gauge theory amplitudes at loop level.
Highlights
In the literature, such as diagrammatic techniques based on factorization theorems [1,2,3], approximations using Wilson lines [4, 5], renormalization group arguments [6], dedicated effective field theories [7,8,9,10] and path integral techniques [11]
Building upon the Low-Burnett-Kroll-Del Duca (LBKD) theorem, we propose a decomposition of radiative amplitudes into universal building blocks, which contain all effects responsible for next-to-leading-power (NLP) threshold logarithms in hadronic cross sections for electroweak annihilation processes
We have considered the generalization of the soft-collinear factorization theorem for gauge-theory scattering amplitudes to include corrections which are responsible for next-to-leading power threshold logarithms in high-energy cross sections
Summary
We discuss in detail the construction of an expression, at the amplitude level, that directly leads to organizing threshold logarithms up to next-to-leading power. In order not to introduce in the jet functions Ji spurious collinear singularities not associated with emissions from the i-th hard parton, it is customary in factorization studies [13] to take n2i = 0. Where Kiνμ is the K tensor appropriate to the i-th jet, we have again Taylor expanded in k, and we have recognized the non-radiative amplitude A in the second line Upon combining this result with the emission from the hard function, as given in eq (2.22), one obtains. In the presence of final-state hadrons (for example in the case of Deep Inelastic Scattering (DIS), NLP threshold logarithms may in principle be associated with hard collinear emission These are potentially not taken into account in eq (2.29), which relies upon the soft expansion. Having presented our general framework, we turn to the calculation of the radiative jet function defined by eq (2.14), focusing in the present case on the abelian-like contributions generated by the current in eq (2.15), up to one loop
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