In this section, we discuss some basic features of transverse momentum dependent, or unintegrated, parton distribution functions. In particular, when these correlation functions are combined in a factorization formulae with hard processes beyond the simplest cases, there are basic problems with universality and factorization. We discuss some of these problems as well as the opportunities that they offer.
We give an overview of the current status of perturbative QCD factorization theorems in processes that involve transverse momentum dependent (TMD) parton distribution functions (PDFs) and fragmentation functions (FF). We enumerate those cases where TMD-factorization is well-established, and mention cases where it is likely to fail. We discuss recent progress in the implementation of specific TMD-factorization calculations, including the implementation of evolution. We also give examples of hard part calculations. We end by discussing future strategies for the implementation of TMD-factorizatio…
Typically, a production of a particle with a small transverse momentum in hadron-hadron collisions is described by CSS-based TMD factorization at moderate Bjorken $x_B\sim 1$ and by $k_T$-factorization at small $x_B$. A uniform description valid for all $x_B$ is provided by rapidity-only TMD factorization developed in a series of recent papers at the tree level. In this paper the rapidity-only TMD factorization for particle production by gluon fusion is extended to the one-loop level.
I show that the forms of the Collins-Soper-Sterman and renormalization-group equations for the evolution of transverse-momentum-dependent (TMD) parton densities in QCD follow from the structure of TMD factorization. A derivation does not need to directly use detailed properties of the operator definition of the TMD parton densities.
We extract the pion transverse momentum dependent (TMD) parton distribution by fitting the pion-induced Drell-Yan process within the framework of TMD factorization. The analysis is done at the next-to-next-to-leading order (NNLO) with proton TMD distribution and non-perturbative TMD evolution extracted earlier in the global fit. We observe the significant difference in the normalization of transverse momentum differential cross-section measured by E615 experiment and the theory prediction.
We examine some of the complications involved when combining (matching) TMD factorization with collinear factorization to allow accurate predictions over the whole range of measured transverse momentum in a process like Drell-Yan. Then we propose some improved methods for combining the two types of factorization. (This talk is based on work reported in arXiv:1605.00671.)
A typical factorization formula for production of a particle with a small transverse momentum in hadron-hadron collisions is given by a convolution of two leading-twist TMD parton densities with cross section of production of the final particle by the two partons. For practical applications at a given transverse momentum, though, one should estimate at what momenta the power corrections to TMD factorization formula due to higher-twist operators become essential. In this paper we calculate the first power corrections to TMD factorization formula for Z-boson production and Drell-Yan process in h…
I discuss how the rapidity evolution of gluon transverse momentum dependent distribution changes from nonlinear evolution at small $x\ll 1$ to linear evolution at moderate $x\sim 1$.
Quarkonia production in different high-energy processes has recently been proposed in order to probe gluon transverse-momentum-dependent parton distribution and fragmentation functions (TMDs in general). However, no proper factorization theorems have been derived for the discussed processes, but rather just ansatzs, whose main assumption is the factorization of the two soft mechanisms present in the process: soft-gluon radiation and the formation of the bound state. In this paper it is pointed out that, at low transverse momentum, these mechanisms are entangled and thus encoded in a new kind o…
We consider the azimuthal distribution of the final observed hadron in semi-inclusive deep-inelastic scattering and the lepton pair in the Drell-Yan process. In particular, we focus on the $\cos ϕ$ modulation of the unpolarized cross section and on its dependence upon transverse momentum. At low transverse momentum, for these observables we propose a factorized expression based on tree-level approach and conjecture that the same formula is valid in transverse-momentum dependent (TMD) factorization when written in terms of subtracted TMD parton distributions. Our formula correctly matches with …
We report on recent study of the production of forward di-jets in proton-proton and proton-lead collisions at the Large Hadron Collider with Improved Transversal Momentum Factorization \cite{vanHameren:2016ftb}. The results as compared to results obtained within High Energy Factorization show noticable effects related to detailed treatment of nonlinear effects.
Proof of transverse-momentum-dependent(TMD) factorization for hadron-hadron collision is given in this paper. We focus on processes without detected soft final hadrons or detected final hadrons that are collinear to initial hadrons. This contradicts the widely accepted viewpoint that TMD- factorization does not hold in such processes even in the generalized sense. The key point is that singular points of the type l+ = 0 can be absorbed into Wilson lines of soft gluons, where l is collinear to the plus direction. Thus one should subtract such singular points from the collinear region of l. Afte…
Quarkonium production at low transverse momentum in hadron collisions can be used to extract Transverse-Momentum-Dependent(TMD) gluon distribution functions, if TMD factorization holds there. We show that TMD factorization for the case of P-wave quarkonium with $J^{PC}=0^{++}, 2^{++}$ holds at one-loop level, but is violated beyond one-loop level. TMD factorization for other P-wave quarkonium is also violated already at one-loop.
This paper is a part of a series of works where we in detail examine the concept of Transverse Momentum Dependent (TMD), or k_T, factorization, which is frequently encountered in the literature and is widely used in the phenomenological applications of QCD at very high energies. We address the question of what exactly factorization is, as it is meant in different contexts and formalisms, and we compare the formalisms to each other. We clarify some basic concepts regarding factorization and how it exactly is applied in high energy QCD, and we make important notes on some key and fundamental poi…
With a consistent definition of transverse-momentum dependent (TMD) light-cone wave function, we show that the amplitude for the process $γ^* π^0 \toγ$ can be factorized when the virtuality of the initial photon is large. In contrast to the collinear factorization in which the amplitude is factorized as a convolution of the standard light-cone wave function and a hard part, the TMD factorization yields a convolution of a TMD light-cone wave function, a soft factor and a hard part. We explicitly show that the TMD factorization holds at one loop level. It is expected that the factorization holds…
This article presents a summary of overlapping presentations by the author to the QCD Evolution 2013 Workshop and to the Opportunities for Polarized Physics at Fermilab Workshop. It contains an introduction to the concept of Kane-Pumplin-Repko (KPR) factorization and descibes how this concept can be used in the analysis of high-precision measurements of parity-conserving transverse single-spin asymmetries. The discussion demonstrates that such measurements can not only probe directly for specific mechanisms that enhance our fundamental understanding of nonperturbative QCD but, because transver…
In this paper we study the Kotzinian-Mulders effect of a single hadron production in semi-inclusive deep inelastic scattering (SIDIS) within the framework of transverse momentum dependent (TMD) factorization. The asymmetry is contributed by the convolution of the Kotzinian-Mulders function $g_{1T}$ and the unpolarized fragmentation function $D_1$. As a TMD distribution, the Kotzinian-Mulders function in the coordinate space in the perturbative region can be represented as the convolution of the $C$-coefficients and the corresponding collinear correlation function. The Wandzura-Wilczek approxim…
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 $α_s^2$ and $α_s^3$. We also show that the "non-perturbative" functions as obtained from fits to data are equal in the t…
Transverse-momentum dependent (TMD) light-cone wave functions of a light meson are important ingredients in the TMD QCD factorization of exclusive processes. This factorization allows one conveniently resum Sudakov logarithms appearing in collinear factorization. The TMD light-cone wave functions are not simply related to the standard light-cone wave functions in collinear factorization by integrating them over the transverse momentum. We explore relations between TMD light-cone wave functions and those in the collinear factorization. Two factorized relations can be found. One is helpful for c…
The Drell-Yan hadronic tensor for electromagnetic (EM) current is calculated in the Sudakov region $s\gg Q^2\gg q_\perp^2$ with ${1\over Q^2}$ accuracy, first at the tree level and then with the double-log accuracy. It is demonstrated that in the leading order in $N_c$ the higher-twist quark-quark-gluon TMDs reduce to leading-twist TMDs due to QCD equation of motion. The resulting tensor for unpolarized hadrons is EM gauge-invariant and depends on two leading-twist TMDs: $f_1$ responsible for total DY cross section, and Boer-Mulders function $h_1^\perp$. The order-of-magnitude estimates of ang…
The Drell-Yan process is studied in the framework of TMD factorization in the Sudakov region $s\gg Q^2\gg q_\perp^2$ corresponding to recent LHC experiments with $Q^2$ of order of mass of Z-boson and transverse momentum of DY pair $\sim$ few tens GeV. The DY hadronic tensors are expressed in terms of quark and quark-gluon TMDs with ${1\over Q^2}$ and ${1\over N_c^2}$ accuracy. It is demonstrated that in the leading order in $N_c$ the higher-twist quark-quark-gluon TMDs reduce to leading-twist TMDs due to QCD equation of motion. The resulting hadronic tensors depend on two leading-twist TMDs: $…
We perform a re-analysis of Belle data for the transverse $Λ$ and $\barΛ$ polarization in $e^+ e^-$ annihilation processes within a transverse momentum dependent (TMD) factorization approach. We consider two data sets, one referring to the associated production of $Λ$'s with a light unpolarized hadron in an almost back-to-back configuration, and one for the inclusive $Λ$ production, with the reconstruction of the thrust axis. We adopt the Collins-Soper-Sterman framework and employ the recent formulation on the factorization of single-inclusive hadron production. This extends a previous phenome…
We study forward dijet production in dilute-dense hadronic collisions. By considering the appropriate limits, we show that both the transverse-momentum-dependent (TMD) and the high-energy factorization formulas can be derived from the Color Glass Condensate framework. Respectively, this happens when the transverse momentum imbalance of the dijet system, $k_t$, is of the order of either the saturation scale, or the hard jet momenta, the former being always much smaller than the latter. We propose a new formula for forward dijets that encompasses both situations and is therefore applicable regar…
We assess the current phenomenological status of transverse momentum dependent (TMD) parton distribution functions (PDFs) and fragmentation functions (FFs) and study the effect of consistently including perturbative QCD (pQCD) evolution. Our goal is to initiate the process of establishing reliable, QCD-evolved parametrizations for the TMD PDFs and TMD FFs that can be used both to test TMD-factorization and to search for evidence of the breakdown of TMD-factorization that is expected for certain processes. In this article, we focus on spin-independent processes because they provide the simplest…
In using transverse-momentum-dependent (TMD) parton densities and fragmentation functions, important non-perturbative information is at large transverse position $b_T$. This concerns both the TMD functions and their evolution. Fits to high energy data tend to predict too rapid evolution when extrapolated to low energies where larger values of $b_T$ dominate. I summarize a new analysis of the issues. It results in a proposal for much weaker $b_T$ dependence at large $b_T$ for the evolution kernel, while preserving the accuracy of the existing fits. The results are particularly important for usi…
We investigate the $\sin (ϕ_Λ-ϕ_S)$ single-spin asymmetry in the transversely polarized $Λ$ production in semi-inclusive deeply inelastic scattering process within the framework of the transverse momentum dependent~(TMD) factorization. The asymmetry is contributed by the convolution of the polarizing TMD fragmentation function $D_{1T}^\perp$ of the $Λ$ hyperon and the unpolarized TMD distribution function $f_1$ of the target nucleon. We apply two sets of $D_{1T}^{\perp,Λ^\uparrow/q}$, the spectator diquark model result and available parametrization, to numerically estimate the $\sin (ϕ_Λ-ϕ_S)$…
Transverse momentum dependent parton distributions (TMDs) appear in many scattering processes at high energy, from the semi-inclusive DIS experiments at a few GeV to the Higgs transverse momentum distribution at the LHC. Predictions for TMD observables crucially depend on TMD factorization, which in turn determines the TMD evolution of the observables with energy. In this contribution to SPIN2014 TMD factorization is outlined, including a discussion of the treatment of the nonperturbative region, followed by a summary of results on TMD evolution, mostly applied to azimuthal asymmetries.
Quarkonium production in hadron collisions at low transverse momentum $q_\perp \ll M$ with $M$ as the quarkonium mass can be used for probing transverse momentum dependent (TMD) gluon distributions. For this purpose, one needs to establish the TMD factorization for the process. We examine the factorization at the one-loop level for the production of $η_c$ or $η_b$. The perturbative coefficient in the factorization is determined at one-loop accuracy. Comparing the factorization derived at tree level and that beyond the tree level, a soft factor is, in general, needed to completely cancel soft d…
We discuss Transverse Momentum Dependent (TMD) gluon distributions within high energy factorization at next-to-leading order in the strong coupling within the framework of Lipatov's high energy effective action. We provide a detailed discussion of both rapidity divergences related to the TMD definition and its soft factor on the one hand, and rapidity divergences due to high energy factorization on the other hand, and discuss common features and differences between Collins-Soper (CS) and Balitsky-Fadin-Kuarev-Lipatov (BFKL) evolution. While we confirm earlier results which state that the unpol…
We demonstrate that partonic correlations that would traditionally be identified as subleading on the basis of a generalized TMD-factorization conjecture can become leading-power because of TMD-factorization breaking that arises in hadron-hadron collisions with large transverse momentum back-to-back hadrons produced in the final state. General forms of TMD-factorization fail for such processes because of a previously noted incompatibility between the requirements for TMD-factorization and the Ward identities of non-Abelian gauge theories. We first review the basic steps for factorizing the glu…
Inclusive production of a quarkonium $η_{c,b}$ in hadron collisions at low transverse momentum can be used to extract various Transverse-Momentum-Dependent(TMD) gluon distributions of hadrons, provided the TMD factorization for the process holds. The factorization involving unpolarized TMD gluon distributions of unpolarized hadrons has been examined with on-shell gluons at one-loop level. In this work we study the factorization at one-loop level with diagram approach in the most general case, where all TMD gluon distributions at leading twist are involved. We find that the factorization holds …
We study the evolution of the small $x$ gluon transverse momentum dependent(TMD) distribution in the dilute limit. The calculation has been carried out in the Ji-Ma-Yuan scheme using a simple quark target model. As expected, we find that the resulting small $x$ gluon TMD simultaneously satisfies both the Collins-Soper(CS) evolution equation and the Balitsky-Fadin-Kuraev-Lipatov(BFKL) evolution equation. We thus confirmed the earlier finding that the high energy factorization(HEF) and the TMD factorization should be jointly employed to resum the different type large logarithms in a process wher…
In this contribution TMD evolution of azimuthal asymmetries, in particular of the Sivers and double Collins asymmetries, is addressed. A comparison of the scale dependence is made between asymmetries described with TMD factorization at low transverse momentum and those described with collinear factorization at high transverse momentum. Finally, the advantages of Bessel weighting are discussed: convergence of transverse momentum integrals, suppression of large transverse momentum contributions, and well-defined lattice QCD evaluations of Bessel-weighted TMDs including proper gauge links.
We calculate the transverse momentum dependent gluon-to-gluon splitting function within kT-factorization [EPJC 78 (2018) 174], generalizing the framework of [NPB 175 (1980) 27; NPB 427 (1994) 475] and extending our previous works [JHEP 01 (2016) 181; PRD 94 (2016) 114013] for the quark splittings. While existing versions of kT factorized evolution equations contain already a gluon-to-gluon splitting function i.e. the leading order BFKL kernel or the CCFM kernel, the obtained splitting function has the important property that it reduces to the leading order BFKL kernel in the low z limit, to th…
The energy scale dependence of the Sivers asymmetry in semi-inclusive deep inelastic scattering is studied numerically within the framework of TMD factorization that was put forward in 2011. The comparison to previous results in the literature shows that the treatment of next-to-leading logarithmic effects is important for the fall-off of the Sivers asymmetry with energy in the measurable regime. The TMD factorization based approach indicates that the peak of the Sivers asymmetry falls off with energy scale Q to good approximation as 1/Q^{0.7}, somewhat faster than found previously based on th…
We study the production of forward di-jets in proton-lead and proton-proton collisions at the Large Hadron Collider. Such configurations, with both jets produced in the forward direction, impose a dilute-dense asymmetry which allows to probe the gluon density of the lead or proton target at small longitudinal momentum fractions. Even though the jet momenta are always much bigger than the saturation scale of the target, $Q_s$, the transverse momentum imbalance of the di-jet system may be either also much larger than $Q_s$, or of the order $Q_s$, implying that the small-$x$ QCD dynamics involved…
This article presents the review of the current understanding on the pion-nucleon Drell-Yan process from the point of view of the TMD factorization. Using the evolution formalism for the unpolarized and polarized TMD distributions developed recently, we provide the theoretical expression of the relevant physical observables, namely, the unpolarized cross section, the Sivers asymmetry, and the $\cos2ϕ$ asymmetry contributed by the double Boer-Mulders effects. The corresponding phenomenology, particularly at the kinematical configuration of the COMPASS $πN$ Drell-Yan facility, is displayed numer…
Quantifying TMD factorization breaking at tree level
Key Laboratory of Particle Physics and Particle Irradiation (MOE), Institute of Frontier and Interdisciplinary Science, Shandong University（粒子物理与粒子辐照教育部重点实验室，前沿交叉科学研究院，山东大学）；Southern Center for Nuclear-Science Theory (SCNT), Institute of Modern Physics, Chinese Academy of Sciences（南方核科学理论中心，现代物理研究所，中国科学院）
We investigate the transverse polarization of the $Λ$ hyperon in the processes $e^+e^-\toΛ^\uparrow π^\pm X$ and $e^+e^-\toΛ^\uparrow K^\pm X$ within the framework of the transverse momentum dependent~(TMD) factorization. The transverse polarization is contributed by the convolution of the transversely polarizing fragmentation function~(PFF) $D_{1T}^\perp$ of the lambda hyperon and the unpolarized fragmentation function $D_1$ of pion/kaon. We adopt the spectator diquark model result for $D_{1T}^{\perp}$ to numerically estimate the transverse polarization in $e^+e^-\toΛ^\uparrow h X$ process at…
Many apparently contradictory approaches to TMD factorization and its non-perturbative content exist. This talk evaluated the different methods and proposed tools for resolving the contradictions and experimentally adjudicating the results.
With a consistent definition of transverse-momentum-dependent(TMD) light-cone wave function of $B$-meson, we show that the amplitude of the radiative leptonic B-decay can be factorized at one-loop level as a convolution with the wave function and a perturbative coefficient function, combined with a soft factor. In this TMD factorization, the transverse momenta of partons in the $B$-meson are taken into account and all soft divergences are contained in the wave function and the soft factor. The coefficient function is infrared-safe. The factorization works on a diagram-by-diagram basis and is p…
Single inclusive particle production at next-to-leading order in proton-nucleus collisions at forward rapidities: hybrid approach meets TMD factorization
We propose a factorization formula for the cross section for forward dijet production in dilute-dense collisions. The new formula is applicable for an arbitrary value of the momentum imbalance of the two jets, $k_t$. It unifies the previously derived transverse momentum dependent (TMD) factorization for small $k_t$ (of the order of the saturation scale), and the High Energy Factorization (HEF) for large $k_t$ (of the order of the momentum of the jets). We extend the previous TMD formula, first to finite $N_c$, and then to all ranges of $k_t$ by including off-shell matrix elements. We present p…
We study the transverse momentum spectrum of dilepton produced in the unpolarized $π^- N$ Drell-Yan process, using transverse momentum dependent factorization up to next-to-logarithmic order of QCD. We extract the nonperturbative Sudakov form factor for the pion in the evolution formalism of the unpolarized TMD distribution function, by fitting the experimental data collected by the E615 Collaboration at Fermilab. With the extracted Sudakov factor, we calculate the normalized differential cross section with respect to transverse momentum of the dimuon and compare it with the recent measurement…
We study the $\sin2ϕ_h$ azimuthal asymmetry of charged and neutral pion productions off the longitudinally polarized nucleon targets in semi-inclusive deeply inelastic scattering (SIDIS) process within the transverse momentum dependent~(TMD) factorization. The asymmetry is contributed by the convolution of the TMD distribution $h_{1L}^{\perp}$ and the Collins fragmentation function $H_{1}^\perp$. We adopt the Wandzura-Wilczek-type~(WW-type) approximation for $h_{1L}^{\perp}$ and two different parametrizations on the nonperturbative part of TMD evolution for $h_{1L}^{\perp}$ and $H_{1}^\perp$. …
We calculate the transverse momentum dependent gluon-to-gluon splitting function within $k_T$-factorization, generalizing the framework employed in the calculation of the quark splitting functions in [1-3] and demonstrate at the same time the consistency of the extended formalism with previous results. While existing versions of $k_T$ factorized evolution equations contain already a gluon-to-gluon splitting function i.e. the leading order Balitsky-Fadin-Kuraev-Lipatov (BFKL) kernel or the Ciafaloni-Catani-Fiore-Marchesini (CCFM) kernel, the obtained splitting function has the important propert…
We apply the transverse momentum dependent factorization formalism to investigate the transverse single spin dependent Collins asymmetry with a $\sin(ϕ_h+ϕ_s)$ modulation in the semi-inclusive production of Kaon meson in deep inelastic scattering process. The asymmetry is contributed by the convolutions of the transversity distribution function $h_1(x)$ of the target proton and the Collins function of the Kaon in the final state. We adopt the available parametrization of $h_1(x)$ as well as the recent extracted result for the Kaon Collins function. To perform the transverse momentum dependent …
