102 research outputs found

    An Overview of Transverse Momentum Dependent Factorization and Evolution

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    I review TMD factorization and evolution theorems, with an emphasis on the treatment by Collins and originating in the Collins-Soper-Sterman (CSS) formalism. I summarize basic results while attempting to trace their development over that past several decades.Comment: 14 pages, 1 figure, Submission to EPJ A topical issue on "3D Structure of the Nucleon

    Connecting Different TMD Factorization Formalisms in QCD

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    In the original Collins-Soper-Sterman (CSS) presentation of the results of transverse-momentum-dependent (TMD) factorization for the Drell-Yan process, results for perturbative coefficients can be obtained from calculations for collinear factorization. Here we show how to use these results, plus known results for the quark form factor, to obtain coefficients for TMD factorization in more recent formulations, e.g., that due to Collins, and apply them to known results at order αs2\alpha_s^2 and αs3\alpha_s^3. We also show that the "non-perturbative" functions as obtained from fits to data are equal in the two schemes. We compile the higher-order perturbative inputs needed for the updated CSS scheme by appealing to results obtained in a variety of different formalisms. In addition, we derive the connection between both versions of the CSS formalism and several formalisms based in soft-collinear effective theory (SCET). Our work uses some important new results for factorization for the quark form factor, which we derive.Comment: 30 pages, 2 Figures; Fixed typos including missing term in Eq.(60

    Universality and Evolution of TMDs

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    In this talk, we summarize how QCD evolution can be exploited to improve the treatment of transverse momentum dependent (TMD) parton distribution and fragmentation functions. The methods allow existing non-perturbative fits to be turned into fully evolved TMDs that are consistent with a complete TMD-factorization formalism over the full range of kT. We argue that evolution is essential to the predictive power of calculations that utilize TMD parton distribution and fragmentation functions, especially TMD observables that are sensitive to transverse spin.Comment: To appear in the proceedings of the Third International Workshop on Transverse Polarization Phenomena in Hard Scattering (Transversity 2011), in Veli Losinj, Croatia, 29 August - 2 September 2011. 5 pages, 1 figur

    No Generalized TMD-Factorization in the Hadro-Production of High Transverse Momentum Hadrons

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    It has by now been established that standard QCD factorization using transverse momentum dependent parton distribution functions fails in hadro-production of nearly back-to-back hadrons with high transverse momentum. The essential problem is that gauge invariant transverse momentum dependent parton distribution functions cannot be defined with process-independent Wilson line operators, thus implying a breakdown of universality. This has led naturally to proposals that a correct approach is to instead use a type of "generalized" transverse momentum dependent factorization in which the basic factorized structure is assumed to remain valid, but with transverse momentum dependent parton distribution functions that contain non-standard, process dependent Wilson line structures. In other words, to recover a factorization formula, it has become common to assume that it is sufficient to simply modify the Wilson lines in the parton correlation functions for each separate hadron. In this paper, we will illustrate by direct counter-example that this is not possible in a non-Abelian gauge theory. Since a proof of generalized transverse momentum dependent factorization should apply generally to any hard hadro-production process, a single counter-example suffices to show that a general proof does not exist. Therefore, to make the counter-argument clear and explicit, we illustrate with a specific calculation for a double spin asymmetry in a spectator model with a non-Abelian gauge field. The observed breakdown of generalized transverse momentum dependent factorization challenges the notion that the role of parton transverse momentum in such processes can be described using separate correlation functions for each external hadron.Comment: 19 pages, 11 figures, typos fixed and minor explanations added, version to appear in Physical Review

    Positivity and Renormalization of Parton Densities

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    There have been recent debates about whether MS parton densities exactly obey positivity bounds (including the Soffer bound) and whether the bounds should be applied as a constraint on global fits to parton densities and on nonperturbative calculations. A recent paper [Candido et al., Can MS parton distributions be negative?, J. High Energy Phys. 11 (2020) 129] appears to provide a proof of positivity in contradiction with earlier work by other authors. We examine their derivation and find that its primary failure is in the apparently uncontroversial statement that bare parton density (or distribution) function (pdfs) are always positive. We show that under the conditions used in the derivation, that statement fails. This is associated with the use of dimensional regularization for both UV divergences (space-time dimension n \u3c 4) and for collinear divergences, with n \u3e 4. Collinear divergences appear in massless partonic quantities convoluted with bare pdfs, in the approach used by these and other authors, which we call “track B.” Divergent UV contributions are regulated and are positive when n \u3c 4, but can and often do become negative after analytic continuation to n \u3e 4. We explore ramifications of this idea and provide some elementary calculations in a model QFT that show how this situation can generically arise in reality. We examine the connection with the origin of the track B method. Our examination pinpoints considerable difficulties with track B that render it either wrong or highly problematic and explain that a different approach, which appears in some literature and that we call track A, does not suffer from this set of problems. The issue of positivity highlights that track-B methods can lead to wrong results of phenomenological importance. From our analysis we identify the restricted situations in which positivity tends to be violated