Using soft collinear effective field theory, we derive the factorization theorem for the quasi-transverse-momentum-dependent (quasi-TMD) operator. We check the factorization theorem at one-loop level and compute the corresponding coefficient function and anomalous dimensions. The factorized expression is built from the physical TMD distribution, and a nonperturbative lattice related factor. We demonstrate that lattice related functions cancel in appropriately constructed ratios. These ratios could be used to explore various properties of TMD distributions, for instance, the nonperturbative evolution kernel. A discussion of such ratios and the related continuum properties of TMDs is presented.
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 construct an improved implementation for combining transverse-momentum-dependent (TMD) factorization and collinear factorization. TMD factorization is suitable for low transverse momentum physics, while collinear factorization is suitable for high transverse momenta and for a cross section integrated over transverse momentum. The result is a modified version of the standard $W+Y$ prescription traditionally used in the Collins-Soper-Sterman (CSS) formalism and related approaches. We further argue that questions regarding the shape and $Q$-dependence of the cross sections at lower $Q$ are lar…
We study factorization of processes involving two fragmentation functions in the case of very small transverse momenta. We consider two-hadron production in inclusive $e^+ e^-$ annihilation and demonstrate a new simple and illustrative method of factorization for such processes including leading order $α_S$ corrections.
Based on a recent work on the quantum chromodynamic (QCD) factorization for semi-inclusive deep-inelastic scattering (DIS), we present a set of factorization formulas for the spin-dependent DIS and Drell-Yan cross sections at low transverse momentum.
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.
We demonstrate a factorization formula for semi-inclusive deep-inelastic scattering with hadrons in the current fragmentation region detected at low transverse momentum. To facilitate the factorization, we introduce the transverse-momentum dependent parton distributions and fragmentation functions with gauge links slightly off the light-cone, and with soft-gluon radiations subtracted. We verify the factorization to one-loop order in perturbative quantum chromodynamics and argue that it is valid to all orders in perturbation theory.
We calculate transverse momentum dependent quark splitting kernels $P_{gq}$ and $P_{qq}$ within $k_T$-factorization, completing earlier results which concentrated on gluon splitting functions $P_{gg}$ and $P_{qg}$. The complete set of splitting kernels is an essential requirement for the formulation of a complete set of evolution equations for transverse momentum dependent parton distribution functions and the development of corresponding parton shower algorithms.
I present some results about transverse momentum dependent distribution and fragmentation functions. Firstly I illustrate a simple model, with predictive power about the energy behavior, for T-odd, chiral odd functions. Moreover I propose a slight modification in extracting transversity from HERMES data, so as to apply correctly factorization. Lastly I suggest a method for determining the quark transverse polarization in an unpolarized or spinless hadron.
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.
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…
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…
We study the transverse single-spin asymmetry for single-hadron production in proton-proton collisions within the framework of collinear twist-3 factorization in Quantum Chromodynamics. By taking into account the contribution due to parton fragmentation we obtain a very good description of all high transverse-momentum data for neutral and charged pion production from the Relativistic Heavy Ion Collider. Our study may provide the crucial step towards a final solution to the longstanding problem of what causes transverse single-spin asymmetries in hadronic collisions within Quantum Chromodynamic…
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…
We analyze transverse thrust in the framework of Soft Collinear Effective Theory and obtain a factorized expression for the cross section that permits resummation of terms enhanced in the dijet limit to arbitrary accuracy. The factorization theorem for this hadron-collider event-shape variable involves collinear emissions at different virtualities and suffers from a collinear anomaly. We compute all its ingredients at the one-loop order, and show that the two-loop input for next-to-next-to-leading logarithmic accuracy can be extracted numerically, from existing fixed-order codes.
In this talk I am going to present some of recent phenomenological applications of the ITMD [1,2] formalism extended to account for Sudakov form factor. In particular production of dijets in the future Electron Ion Collider is discussed.
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 fac…
We study the resummation of singular distributions for electroweak-boson production at large transverse momentum in hadronic collisions. We describe the factorization properties of the cross section near partonic threshold and construct the resummed cross section at next-to-leading logarithmic accuracy in moment space in terms of soft gluon anomalous dimensions. We present full analytical results for the expansion of the resummed cross section up to next-to-next-to-leading order. Our results can be applied to W, Z, and virtual gamma production at hadron colliders.
Using partonic states we verify the transverse momentum dependent factorization for single transverse spin asymmetries in semi inclusive DIS at leading but nontrivial order of alpha_s. The factorization has been previously derived in a formal way by using diagram expansion at hadron level. We find with our partonic results that two relevant structure functions satisfy the factorization. Our results also satisfy the collinear factorization but with the perturbative coefficient different than that derived formally.
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 scattering of a single parton state with a multi-parton state to derive the complete results of perturbative coefficient functions at leading order, which appear in the collinear factorization for Single transverse-Spin Asymmetry(SSA) in Drell-Yan processes with a transversely polarized hadron in the initial state. We find that the factorization formula of SSA contains hard-pole-, soft-quark-pole- and soft-gluon-pole contributions. It is interesting to note that the leading order perturbative coefficient functions of soft-quark-pole- and soft-gluon-pole contributions are extracted…
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…
We study transverse-momentum-dependent factorization at twist-3 for Drell-Yan processes. The factorization can be derived straightforwardly at leading order of $α_s$. But at this order we find that light-cone singularities already exist and effects of soft gluons are not correctly factorized. We regularize the singularities with gauge links off the light-cone and introduce a soft factor to factorize the effects of soft gluons. Interestingly, the soft factor must be included in the definition of subtracted TMD parton distributions to correctly factorize the effects of soft gluons. We derive the…
We show that hard-scattering factorization is violated in the production of high-p_T hadrons in hadron-hadron collisions, in the case that the hadrons are back-to-back, so that k_T factorization is to be used. The explicit counterexample that we construct is for the single-spin asymmetry with one beam transversely polarized. The Sivers function needed here has particular sensitivity to the Wilson lines in the parton densities. We use a greatly simplified model theory to make the breakdown of factorization easy to check explicitly. But the counterexample implies that standard arguments for fact…
Two factorization approaches have been proposed for single transverse spin asymmetries. One is the collinear factorization, another is the transverse-momentum-dependent factorization. They have been previously derived in a formal way by using diagram expansion at hadron level. If the two factorizations hold or can be proven, they should also hold when we replace hadrons with parton states. We examine these two factorizations at parton level with massless partons. It is nontrivial to generate these asymmetries at parton level with massless partons because the asymmetries require helicity-flip a…
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…
Resummed QCD corrections to the transverse momentum ($Q_T$) distribution of the Standard Model Higgs boson, produced at the CERN Large Hadron Collider, are presented. The small $Q_T$ factorization formalism is reviewed, which is used to extend the standard hadronic factorization theorem to the low $Q_T$ region, with emphasis on the matching to the standard hadronic factorization. Comparison of the $Q_T$ predictions from the extended factorization and the parton shower method is performed.
We study the one-loop correction in Transverse-Momentum-Dependent(TMD) factorization for Drell-Yan processes at small transverse momentum of the lepton pair. We adopt the so-called subtractive approach, in which one can systematically construct contributions for subtracting long-distance effects represented by diagrams. The perturbative parts are obtained after the subtraction. We find that the perturbative coefficients of all structure functions in TMD factorization at leading twist are the same. The perturbative parts can also be studied with scattering of partons instead of hadrons. In this…
We study factorization in single transverse spin asymmetries for dijet production in proton-proton collisions, by considering soft gluon radiation at one-loop order. We show that the associated transverse momentum dependent (TMD) factorization is valid at the leading logarithmic level. At next-to-leading-logarithmic (NLL) accuracy, however, we find that soft gluon radiation generates terms in the single transverse spin dependent cross section that differ from those known for the unpolarized case. As a consequence, these terms cannot be organized in terms of a spin independent soft factor in th…
We study the spin-dependent quark distributions at large transverse momentum. We derive their transverse momentum behaviors in the collinear factorization approach in this region. We further calculate the angular distribution of the Drell-Yan lepton pair production with polarized beams and present the results in terms of the collinear twist-three quark-gluon correlation functions. In the intermediate transverse momentum region, we find that the two pproaches: the collinear factorization and the transverse momentum dependent factorization approaches are consistent in the description of the lept…
Single transverse-spin asymmetries have been studied intensively both in experiment and theory. Theoretically, two factorization approaches have been proposed. One is by using transverse-momentum-dependent factorization and the asymmetry comes from the so called Sivers function. Another is by using collinear factorization where the nonperturbative effect is parameterized by a twist-3 hadronic matrix element. However, the factorized formulas for the asymmetries in the two approaches are derived at hadron level formally by diagram expansion, where one works with various parton density matrices 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 …
In this work we consider how a parton distribution function, with an explicit transverse momentum dependence can be properly defined in a regularization-scheme independent manner. We argue that by considering a factorized form of the transverse momentum dependent spectrum for the production of a heavy lepton pair in Drell-Yan reaction, one should first split the relevant soft function into two boost invariant contributions. When those soft contributions are added to the pure collinear contributions, well-defined hadronic matrix elements emerge, i.e., the transverse momentum dependent distribut…
We investigate the gluon distribution functions and their contributions to the Higgs boson production in pp collisions in the transverse momentum dependent factorization formalism. In addition to the usual azimuthal symmetric transverse momentum dependent gluon distribution, we find that the azimuthal correlated gluon distribution also contributes to the Higgs boson production. This explains recent findings on the additional contribution in the transverse momentum resummation for the Higgs boson production as compared to that for electroweak boson production processes. We further examine the s…
We use multi-parton states to examine the leading order collinear factorization of single transverse-spin asymmetries in Drell-Yan processes. Twist-3 operators are involved in the factorization. We find that the so-called soft-gluon-pole contribution in the factorization must exist in order to make the factorization correct. This contribution comes from the corresponding cross-section at one-loop, while the hard-pole contribution in the factorization comes from the cross-section at tree-level. Although the two contributions come from results at different orders, their perturbative coefficient …
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 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…
We calculate the light cone wave functions for the QCD Fock components in a transverse virtual photon necessary for applications at next-to-leading order in the QCD coupling, including quark masses. We present a detailed calculation of both the one loop wave function for the quark-antiquark Fock component and the tree level wave function for the quark-antiquark-gluon Fock component. The quark masses are renormalized in the pole mass scheme, satisfying constraints from the requirement of Lorentz invariance. In particular the quark Pauli form factor at NLO is recovered from the on-shell limit of…
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 compare results of $k_t$-factorization approach and next-to-leading order collinear-factorization approach for dijet correlations in proton-proton collisions at RHIC energies. We discuss correlations in azimuthal angle as well as correlations in two-dimensional space of transverse momenta of two jets. Some $k_t$-factorization subprocesses are included for the first time in the literature. Different unintegrated gluon/parton distributions are used in the $k_t$-factorization approach. The results depend on UGDF/UPDF used. For collinear NLO case the situation depends significantly on whether w…
We review what is currently known about the transverse spin structure of hadrons, in particular from observables that can be analyzed within a collinear framework. These effects have been around for 40 years and represent a critical test of perturbative QCD. We look at both proton-proton and lepton-nucleon collisions for various final states. While the main focus is on transverse single-spin asymmetries, we also discuss how longitudinal-transverse double-spin asymmetries offer a complimentary, yet equally important, source of information on the quark-gluon content of hadrons. We also summarize…
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…
This talk reports on recent work where we studied the connection between the description of semi-inclusive DIS at high transverse momentum (based on collinear factorization) and low transverse momentum (based on transverse-momentum-dependent factorization). We used power counting to determine the leading behavior of the structure functions at intermediate transverse momentum in the two descriptions. When the power behaviors are different, two distinct mechanisms are present and there can be no matching between them. When the power behavior is the same, the two descriptions must match. An expli…
We compute the inclusive unpolarized dihadron production cross section in the far from back-to-back region of $e^+ e^-$ annihilation in leading order pQCD using existing fragmentation function fits and standard collinear factorization, focusing on the large transverse momentum region where transverse momentum is comparable to the hard scale (the center-of-mass energy). We compare with standard transverse-momentum-dependent (TMD) fragmentation function-based predictions intended for the small transverse momentum region with the aim of testing the expectation that the two types of calculation ro…
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 derive a factorization theorem for Drell-Yan process at low q_T using effective field theory methods. In this theorem all the obtained quantities are gauge invariant and the special role of the soft function--and its subtraction thereof--is emphasized. We define transverse-momentum dependent parton distribution functions (TMDPDFs) which are free from light-cone singularities while all the Wilson lines are defined on-the-light-cone. We show explicitly to first order in α_s that the partonic Feynman PDF can be obtained from the newly defined partonic TMDPDF by integrating over the transverse …
We discuss production of nonphotonic electrons in proton-proton scattering at RHIC. The distributions in rapidity and transverse momentum of charm and bottom quarks/antiquarks are calculated in the $k_t$-factorization approach. We use different unintegrated gluon distributions from the literature. The hadronization of heavy quarks is done by means of Peterson and Braaten et al. fragmentation functions. The semileptonic decay functions are found by fitting recent semileptonic data obtained by the CLEO and BABAR collaborations. We get good description of the data at large transverse momenta of e…
We perform calculations of Z+jet cross-section taking into account the transverse momenta of the initial partons. Transverse Momentum Dependent (TMD) parton densities obtained with the Parton Branching method are used and higher order corrections are included via TMD parton showers in the initial state. The predictions are compared to measurements of forward Z+jet production of the LHCb collaboration at $\sqrt{s}=7$ TeV. We show that the results obtained in kT-factorization are in good agreement with results obtained from a NLO calculation matched with traditional parton showers. We also demon…
We compare the transverse momentum (p_T) distribution of inclusive light-charged-particle production measured by the CDF Collaboration at the Fermilab Tevatron with the theoretical prediction evaluated at next-to-leading order in quantum chromodynamics (QCD) using fragmentation functions recently determined through a global data fit. While, in the lower p_T range, the data agree with the prediction within the theoretical error or slightly undershoot it, they significantly exceed it in the upper p_T range, by several orders of magnitude at the largest values of p_T, where perturbation theory sh…
Factorization is the central ingredient in any theoretical prediction for collider experiments. We introduce a factorization formalism that can be applied to any desired observable, like event shapes or jet observables, for any number of jets and a wide range of jet algorithms in leptonic or hadronic collisions. This is achieved by using soft-collinear effective theory to prove the formal factorization of a generic fully-differential cross section in terms of a hard coefficient, and generic jet and soft functions. In this formalism, whether a given observable factorizes in the usual sense, dep…
We discuss the single diffractive production of $c \bar c$ pairs and charmed mesons at the LHC. In addition to standard collinear approach, for a first time we propose a $k_t$-factorization approach to the diffractive processes. The transverse momentum dependent (the unintegrated diffractive parton distributions) in proton are obtained with the help of the Kimber-Martin-Ryskin prescription where collinear diffractive PDFs are used as input. In this calculation the transverse momentum of the pomeron is neglected with respect to transverse momentum of partons entering the hard process. The resul…
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 $\sin (ϕ_Λ-ϕ_S)$…
In the target fragmentation region of Semi-Inclusive Deep Inelastic Scattering, the diffractively produced hadron has small transverse momentum. If it is at order of $Λ_{QCD}$, it prevents to make predictions with the standard collinear factorization. However, in this case, differential cross-sections can be predicted by the factorization with fracture functions, diffractive parton distributions. If the transverse momentum is much larger than $Λ_{QCD}$ but much smaller than $Q$ which is the virtuality of the virtual photon, both factorizations apply. In this case, fracture functions can be fac…
We consider the one-loop factorization of the simplest twist-three process: inclusive deep-inelastic scattering of longitudinally-polarized leptons on a transversely-polarized nucleon target. By studying the Compton amplitudes for certain quark and gluon states at one loop, we find the coefficient functions for the non-singlet twist-three distributions in the factorization formula of g_2(x_B,Q^2). The result marks the first step towards a next-to-leading order (NLO) formalism for this transverse-spin-dependent structure function of the nucleon.
We study transverse-momentum-dependent (TMD) gluon distributions and factorization theorems for the gluon-initiated semi-inclusive processes at hadron colliders. Gauge-invariant TMD gluon distributions are defined, and their relations to the integrated (Feynman) parton distributions are explored when the transverse momentum is large. Through explicit calculations, soft-collinear factorization is verified at one-loop order for scalar particle production. Summation over large double logarithms is made through solving the Collins-Soper equation. We reproduce the known result in the limit that the…
