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

Aybat, S. Mert

Rogers, Ted C.

English

arXiv

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 TMDfactorization 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

Aybat, S. M.

Rogers, T. C.

Scientific Open-access Literature Archive and Repository

DOI 10.1393/ncc/i2012-11168-xColloquia: Transversity 2011IL NUOVO CIMENTO Vol. 35 C, N. 2 Marzo-Aprile 2012Universality and evolution of TMDsS. M. Aybat(1) and T. C. Rogers(2)(1) Nikhef Theory Group - Science Park 105, 1098XG Amsterdam, The Netherlands(2) C.N. Yang Institute for Theoretical Physics, Stony Brook University - Stony BrookNY 11794, USAricevuto il 27 Ottobre 2011; approvato il 25 Novembre 2011pubblicato online il 14 Febbraio 2012Summary. — In this talk, we summarize how QCD evolution can be exploited toimprove the treatment of transverse momentum dependent (TMD) parton distribu-tion and fragmentation functions. The methods allow existing non-perturbative fitsto 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 essen-tial to the predictive power of calculations that utilize TMD parton distribution andfragmentation functions, especially TMD observables that are sensitive to transversespin.PACS 12.38.Bx – Perturbative calculations.PACS 12.39.St – Factorization.PACS 12.38.Cy – Summation of perturbation theory.1. – Collinear versus TMD FactorizationThe standard collinear QCD factorization theorems have set the standard for the useof perturbative QCD calculations to probe certain properties of the microscopic structureof matter. It will be instructive to recall the essential ingredients that lend the standardfactorization treatments their predictive power:– Unambiguous prescription for calculating perturbatively well-behaved higher-ordercorrections to the hard scattering.– Correlation functions describing the non-perturbative factors are well defined andhave universality properties so that, once measured, they can be useful for futurephenomenological studies.– Evolution equations, to allow the correlation functions to be compared at differentscales.The collinear PDFs and fragmentation functions have by now been parametrized by awide range and variety of experimental data, and have become indispensable tools in gen-eral high-energy physics. Ideally, TMD-factorization, in which the PDFs and fragmenta-tion functions also carry information about the intrinsic transverse momenta of partons,c© Societa` Italiana di Fisica 2930 S. M. AYBAT and T. C. ROGERSshould follow a very similar framework. However, it is only very recently [1-3] thatTMD-factorization has reached a level of logical completeness similar to the well-knowncollinear cases. As long there was ambiguity in the formulation of TMD-factorization,the procedure for correctly identifying TMDs in fits, as well as the correct procedure forimplementing evolution has remained unclear.In the absence of a full TMD-factorization treatment, there have been several popularbut separate approaches for dealing with TMDs:– Generalized Parton Model (GPM): A phenomenological approach is to extractTMDs from fits to data while assuming a literal parton model interpretation ofthe TMDs. One typically ignores evolution and therefore the fits correspond tospecific scales.– Resummation: Begin with a collinear treatment valid for large transverse momen-tum, and attempt to improve the treatment of lower transverse momentum byresuming logarithms of qT /Q. A severe limitation of this approach is that it isbound to fail below some qT , where many of the most interesting effects of TMDphysics are expected to become important.– Models: Non-perturbative models of TMDs can be used to study specific non-perturbative aspects of hadron structure (see A. Bacchetta’s talk for a review ofmodel calculations.). But there are ambiguities in how these TMDs are relatedto the ones used in actual perturbative QCD calculations of cross sections. Inparticular, it is unclear what hard scale they should correspond to.– Lattice Calculations: Lattice calculations of TMDs (see, e.g., [4]) describe the non-perturbative distribution of partons from first principles, but also require a cleardefinition for the TMDs, and a clear prescription for use in complete cross sectioncalculations.A useful TMD-factorization treatment should allow the advantages of each of these ap-proaches to be unified within a single, clear formalism for relating TMD studies to ob-servable cross sections. Fortunately, this is now possible following the recent work ofref. [1] (see, especially, chapts. 10 and 13). (A similar general approach was developedearlier by Ji, Ma, and Yuan [2,3], which built upon the Collins-Soper-Sterman (CSS) [5,6]formalism.) We will show how fits to TMDs can be constructed from existing work thatfollows the above tabulated approaches, but which include evolution and are consistentwith a full TMD-factorization.We apply the TMD-factorization method developed recently by Collins in ref. [1].The factorization theorem for the Drell-Yan (DY) process, for example, isWμνDY =∑f|Hf (Q2, μ)|μν∫d2k1Td2k2T Ff/P1(x1,k1T , μ, ζ1F )Ff/P2(x2,k2T , μ, ζ2F )(1)×δ(2)(k1T + k2T − qT ) + Y +O(m/Q).Note the similarity of the first term above to a GPM description; there is a hardpart Hf (Q2, μ)|μν and a convolution of two TMD PDFs, Ff/P1(x1,k1T , μ, ζ1F ) andFf/P2(x2,k2T , μ, ζ2F ). However, in the full TMD-factorization treatment, they haveacquired scale dependence through the renormalization group parameter μ and ζ1F andUNIVERSALITY AND EVOLUTION OF TMDS 31ζ2F (which obey√ζ1F ζ2F = Q2). Moreover, there is no explicit appearance of a soft fac-tor. The first term in eq. (1) is appropriate for describing the small qT region (qT  Q).The Y term corrects the large qT behavior and can be calculated in terms of normal inte-grated PDFs. Reference [1] derives very specific operator definitions for the TMD PDFs,which include the role of soft gluons and account for all spurious divergences which havehindered efforts to clearly define the TMD PDFs in the past.2. – Evolution of TMDsThe evolution of the individual TMDs in transverse coordinate space is governed bythe Collins-Soper (CS) equation [5],∂ ln F˜ (x, bT , μ, ζ)∂ ln√ζ= K˜(bT , μ),(2)and the renormalization group equations,dK˜(bT , μ)d lnμ= −γK(g(μ)), d ln F˜ (x, bT , μ, ζ)d lnμ = −γF (g(μ), ζ/μ2).(3)In eq. (2), K˜(bT , μ) is the kernel for evolution with respect the energy variable ζ, whileγF (g(μ), ζ/μ2) and γK(g(μ)) are the anomalous dimensions. These can all be calculatedin perturbation theory, though K˜(bT , μ) becomes non-perturbative at large bT and needsto be fit. It is a universal function, however, both with respect to different processes andwith regard to PDFs versus fragmentation functions.3. – Specific fitsWorking within a GPM approach, Gaussian parametrizations have been fit to var-ious TMDs in, for example, ref. [7] from low energy SIDIS measurements. The non-perturbative information corresponding to K˜(bT , μ) at large bT in eq. (2) has been ex-tracted from more traditional applications of the CSS method, such as in ref. [8]. Withthe non-perturbative parts constrained, specific tables for the TMDs in eq. (1) can be gen-erated for any arbitrary scale Q once the perturbative contributions have been calculated.We have carried this out explicitly for the unpolarized TMD PDFs and fragmenta-tion functions in ref. [9], and have made the results available at the website: https://projects.hepforge.org/tmd/.An example showing the unpolarized up quark PDF for a selection of Q values is thegraph in fig. 1.Note that, once the Y term is included in eq. (1), the factorization formula exactlyvalid over the whole range of kT from kT = 0 to Q.4. – Conclusions and future directionsA crucial element of the TMD-factorization approach is that the TMDs are well-defined and, therefore, can be taken to be genuinely universal. One consequence of thisis that different theories of the non-perturbative properties of hadron structure can becompared with one another, and with experimental results over a range of hard scales.Ultimately, in order to take full advantage of the predictive power of TMD-factori-zation, requires a concerted effort to improve upon non-perturbative fits, to calculate32 S. M. AYBAT and T. C. ROGERSFig. 1. – Unpolarized TMD PDF for an up-quark, for scales Q =√2.4, 5 and 91.19GeV.higher orders to the anomalous dimensions, K, the coefficient functions, and the hardparts, and to provide numerical implementations of evolution in transverse momentumspace similar to what has been done in ref. [9], but applied to other TMD PDFs andfragmentation functions, including interesting spin dependent ones. We have recentlybeen involved in efforts to extend evolution to the Sivers function [10], and plan tocontinue this work to generate evolved fits for most of the various spin-dependent TMDs.∗ ∗ ∗MA thanks the organizers of the Transversity 2011 workshop for the invitation to thisinteresting and fruitful workshop. Support for this work was provided by the researchprogram of the “Stichting voor Fundamenteel Onderzoek der Materie (FOM)”, which isfinancially supported by the “Nederlandse Organisatie voor Wetenschappelijk Onderzoek(NWO)”.REFERENCES[1] Collins J. C., Foundations of Perturbative QCD (Cambridge University Press, Cam-bridge) 2011.[2] Ji X. D., Ma J. P. and Yuan F., Phys. Rev. D, 71 (2005) 034005.[3] Ji X. D., Ma J. P. and Yuan F., Phys. Lett. B, 597 (2004) 299.[4] Musch B. U., Hagler P., Negele J. W. and Schafer A., arXiv:1011.1213 [hep-lat].[5] Collins J. C. and Soper D. E., Nucl. Phys. B, 193 (1981) 381; 213 (1983) 545(E).[6] Collins J. C., Soper D. E. and Sterman G., Nucl. Phys. B, 250 (1985) 199.[7] Schweitzer P., Teckentrup T. and Metz A., Phys. Rev. D, 81 (2010) 094019.[8] Landry F., Brock R., Nadolsky P. M. and Yuan C. P., Phys. Rev. D, 67 (2003)073016.[9] Aybat S. M. and Rogers T. C., Phys. Rev. D, 83 (2011) 114042.[10] Aybat S. M., Collins J. C., Qiu J. W. and Rogers T. C., to be published in Phys.Rev. D, arXiv:1110.6428 [hep-ph].

Universality and Evolution of TMDs

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