No Generalized TMD-Factorization in the Hadro-Production of High Transverse Momentum Hadrons arXiv:1001.2977v2 [hep-ph] 10 May 2010 Ted C. Rogers ∗ and Piet J. Mulders † Department of Physics and Astronomy, Vrije Universiteit Amsterdam, NL-1081 HV Amsterdam, The Netherlands (Dated: October 7, 2018) It has by now been established that standard QCD factorization using transverse momentum de- pendent 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 op- erators, 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 fac- torization 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 momen- tum 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. Keywords: perturbative QCD, factorization There has been much recent activity devoted to the study of parton transverse momentum in high energy hadronic collisions. Observables that are sensitive to par- ton transverse momentum can potentially provide new insight into the structure of hadrons. However, to in- terpret data it is important to understand the extent to which familiar perturbative QCD (pQCD) factorization approaches can be extended to situations that involve parton transverse momentum. In this paper, we specif- ically address recent efforts to apply the usual general framework of pQCD factorization (with some modifica- tions) to describe parton transverse momentum in colli- sions between high energy hadrons with production of a pair of nearly back-to-back high transverse momentum hadrons or jets in the final state: For a discussion of the relevant issues, it will be impor- tant to first clearly define our terminology. Very gener- ally, a QCD factorization theorem [3] is said to be valid if an observable (such as a cross section) can be written as a convolution product of factors that describe different regions of parton momentum. Schematically, one expects for process (1): Here, the symbol ⊗ denotes all relevant convolution in- tegrals and traces, and a sum over different types of par- tons and subprocesses is understood. The hard factor H describes the short range behavior in the hard collision between partons, while Φ$_{H}$$_{1}$ and Φ$_{H}$$_{2}$ are the parton dis- tribution functions (PDFs) for initial state hadrons H$_{1}$ and H$_{2}$ . The Δ$_{H}$$_{3}$ and Δ$_{H}$$_{4}$ are fragmentation functions for final state hadrons H$_{3}$ and H$_{4}$ . In general, a soft factor may also be needed, though we will not write it explicitly. The symbol “p.s.c.” indicates that power sup- pressed corrections are neglected. The distribution of transverse momentum inside the col- liding hadrons can be probed, for example, by measuring the small imbalance in the distribution of transverse mo- mentum between the final state pair (see, e.g., [1, 2]). Parton transverse momentum also plays a central role in the generation of spin asymmetries. A transverse momentum dependent (TMD) factoriza- tion theorem (also called unintegrated factorization or k$_{T}$ -factorization) is said to be valid if the role of parton transverse momentum can be taken into account by us- ing PDFs and FFs in Eq. (2) which depend explicitly on parton transverse momentum. In a TMD-factorization formula, one therefore refers to TMD PDFs and TMD FFs (or “unintegrated” PDFs and FFs). TMD-factorization should be contrasted with the more common collinear factorization theorems, applica- H$_{1}$ + H$_{2}$ → H$_{3}$ + H$_{4}$ + X. (1) dσ = H ⊗ Φ$_{H}$$_{1}$ ⊗ Φ$_{H}$$_{2}$ ⊗ Δ$_{H}$$_{3}$ ⊗ Δ$_{H}$$_{4}$ + p . s . c . (2) I. INTRODUCTION $^{∗}$Electronic address: trogers@few.vu.nl $^{†}$Electronic address: mulders@few.vu.nl