We present the first determination of transverse momentum dependent (TMD) photon densities with the Parton Branching method. The photon distribution is generated perturbatively without intrinsic photon component. The input parameters for quarks and gluons are determined from fits to precision measurements of deep inelastic scattering cross sections at HERA. The TMD densities are used to predict the mass and transverse momentum spectra of very high mass lepton pairs from both Drell-Yan production and Photon-Initiated lepton processes at the LHC.
We present the determination of Transverse Momentum Dependent (TMD) parton distributions from Monte Carlo parton showers. We investigate the effective TMD distributions obtained from the PYTHIA8 and HERWIG6 parton showers and compare them to the TMD distributions determined within the Parton Branching method.
A new calculation using off-shell matrix elements with TMD parton densities supplemented with a newly developed initial state TMD parton shower is described. The calculation is based on the KaTie package for an automated calculation of the partonic process in high-energy factorization, making use of TMD parton densities implemented in TMDlib. The partonic events are stored in an LHE file, similar to the conventional LHE files, but now containing the transverse momenta of the initial partons. The LHE files are read in by the CASCADE package for the full TMD parton shower, final state shower and…
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.)
The calculations of $Z + b{\bar b}$ tagged jet production performed in the four- and five-flavour schemes allow for detailed comparison of the heavy flavour structure of collinear and transverse momentum dependent (TMD) parton distributions as well as for detailed investigations of heavy quarks radiated during the initial state parton shower cascade. We have determined the first set of collinear and TMD parton distributions in the four-flavour scheme with NLO DGLAP splitting functions within the Parton-Branching (PB) approach. The four- and five-flavour PB-TMD distributions were used to calcul…
Calculations of Drell-Yan (DY) production at next-to-leading (NLO) order have been combined with Transverse Momentum Dependent (TMD) distributions obtained with the Parton Branching (PB). The predictions show a remarkable agreement with DY measurement from E605 experiment, consistent with previous results we obtained for R209, PHENIX, CMS and ATLAS experiments. We also present predictions for Z+jet measurements showing the importance of TMD parton shower contributions to the jet multiplicity. We show that PB-TMD parton density and the corresponding PB-TMD parton shower can be combined with lea…
Transverse-momentum-dependent distributions (TMDs) are central in high-energy physics from both theoretical and phenomenological points of view. In this manual we introduce the library, TMDlib, of fits and parameterisations for transverse-momentum-dependent parton distribution functions (TMD PDFs) and fragmentation functions (TMD FFs) together with an online plotting tool, TMDplotter. We provide a description of the program components and of the different physical frameworks the user can access via the available parameterisations.
The Parton Branching (PB) Method describes the evolution of transverse momentum dependent parton densities (TMDs). The obtained TMDs can be used in Monte Carlo generators to describe physical observables. We give an overview of recent developments, in particular the new application of PB TMDs in Drell-Yan + jets production, the four-flavor and five-flavor schemes for heavy quark contributions, and the inclusion of QED corrections in the PB evolution.
Transverse-momentum-dependent parton distributions (TMDs) provide three-dimensional images of the partonic structure of the nucleon in momentum space. We made impressive progress in understanding TMDs, both from the theoretical and experimental point of view. This brief overview on TMDs is divided in two parts: in the first, an essential list of achievements is presented. In the second, a selection of open questions is discussed.
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…
General formulations of QCD factorization for hadronic collisions extend the notion of ordinary parton distributions to transverse-momentum dependent (TMD) parton density and parton decay functions. We discuss the use of the recent high-precision deep-inelastic scattering (DIS) measurements for determination of TMD distributions. These are relevant for both low-pT and high-pT physics in hadron collisions. We comment on applications to multi-jet final states associated with electroweak gauge boson production at the LHC.
We discuss the parton branching (PB) evolution equation for transverse momentum dependent (TMD) parton distribution functions (PDFs), especially we concentrate on the angular ordering constrain and its effect on the TMD distributions. We discuss application of the PB TMDs to precise prediction of Drell-Yan transverse momentum spectrum at LHC, stressing the role of angular ordering in the low transverse momentum region. We compare the PB implementation of the angular ordering condition with the Kimber-Martin-Ryskin-Watt (KMRW), both at the analytical and numerical level.
The focus of this talk is on the transverse components of parton momenta. Like for collinear parton distribution functions (PDFs), we are also in the case of transverse momentum dependent (TMD) PDFs, talking about forward matrix elements. While the collinear PDFs describe only spin-spin correlations, the TMD PDFs (or in short TMDs) include spin-momentum correlations, including also time-reversal-odd (T-odd) correlations. The latter are important in the description of single spin asymmetries. In this way TMDs open up new ways of studying the spin structure of hadrons or they can be used as tool…
I give an account of a new definition of transverse-momentum-dependent parton densities. The new definition solves a number of difficulties and inconsistencies in earlier definitions.
Model studies play an important role for the understanding and elucidation of the nonperturbative properties of transverse momentum dependent parton distribution functions (TMDs). The parton model is often a helpful framework and starting point for first explorations of TMD properties and the description of deep-inelastic processes in which TMDs can be accessed. Based on a systematic exploration of the parton model concept, we reconcile the claims in literature that there are 2 independent structures in the quark correlator in the parton model vs the claim that there are 3, and explain the und…
Transverse momentum dependent (TMD) distributions match collinear parton density functions (PDF) in the limit of small transverse distances, which is accounted for by global extractions of TMD distributions. We study the influence of the collinear PDF value and uncertainties on the determination of unpolarized TMD distributions and the description of Drell-Yan (DY) and Z-boson production measurements at low transverse momenta. We take into account, for the first time, flavor-dependent non-perturbative TMD profiles. We carry out a Bayesian analysis to incorporate the propagation of PDF uncertai…
We study parton-branching solutions of QCD evolution equations and present a method to construct both collinear and transverse momentum dependent (TMD) parton densities from this approach. We work with next-to-leading-order (NLO) accuracy in the strong coupling. Using the unitarity picture in terms of resolvable and non-resolvable branchings, we analyze the role of the soft-gluon resolution scale in the evolution equations. For longitudinal momentum distributions, we find agreement of our numerical calculations with existing evolution programs at the level of better than 1 percent over a range…
We explore the possibility to include small-$x$ dynamics effects in the parton branching (PB) approach to transverse momentum dependent (TMD) parton distribution functions. To this end, we first revisit the PB method at leading order, presenting a new fit to inclusive-DIS precision data, and performing a numerical study of the dynamic soft-gluon resolution scale. Next we investigate the effects of modified CCFM kernels, including both Sudakov and non-Sudakov form factors.
Collinear and transverse momentum dependent (TMD) parton densities are obtained from fits to precision measurements of deep inelastic scattering (DIS) cross sections at HERA. The parton densities are evolved by DGLAP evolution with next-to-leading-order (NLO) splitting functions using the parton branching method, allowing one to determine simultaneously collinear and TMD densities for all flavors over a wide range in $x$, $μ^2$ and $k_t$, relevant for predictions at the LHC. The DIS cross section is computed from the parton densities using perturbative NLO coefficient functions. Parton densiti…
Three-body decay functions in space-like parton branches are implemented to evaluate transverse-momentum-dependent (TMD) parton distribution functions in the next-to-leading logarithmic (NLL) order of quantum chromodynamics (QCD). Interference contributions due to the next-to-leading order contribution are taken into account for the evaluation of the transverse momenta in initial state parton radiations. Some properties of the decay functions are also examined. As an example, we compare our results with an algorithm proposed in Ref.1), in which a transverse momentum distributions are evaluated…
Froissart Bound implies that the total proton-proton cross-section (or equivalently structure function) cannot rise faster than the logarithmic growth $\log^2 s \sim \log^2 (1/x)$, where \textit{s} is the square of the center of mass energy and \textit{x} is the Bjorken variable. In the present report, compatibility of such behavior will be shown in case of Transverse Momentum Dependent Parton Distributions Functions (TMDPDF) or TMDs for the model of proton based on self-similarity.
We discuss implications of the evolution of transverse momentum dependent (TMD) parton distributions on the structure of multi-jet states at high energies. In particular we analyze the theoretical systematics associated with multi-jet merging. We introduce a new merging methodology incorporating TMDs, illustrate its main features and present a comparison of our theoretical results with experimental measurements for Z-boson + jets production at the Large Hadron Collider (LHC).
QCD predictions for final states with multiple jets in hadron collisions make use of multi-jet merging methods, which allow one to combine consistently the contributions from hard scattering matrix elements with different parton multiplicities and parton showers. In this article I consider a multi-jet merging method that has recently been proposed to take into account the effects of transverse momentum dependent (TMD) evolution and parton shower, and present studies focusing on the application of this method to jets associated with Drell-Yan (DY) production in the region of high masses.
We investigate the transverse momentum dependent parton distributions (TMDs) in the quasi-parton-distribution framework. The long standing hurdle of the so-called pinch pole singularity from the space-like gauge links in the TMD definitions can be resolved by the finite length of the gauge link along the hadron moving direction. In addition, with the soft factor subtraction, the quasi-TMD is free of linear divergence. We further demonstrate that the energy evolution equation of the quasi-TMD a.k.a. the Collins-Soper evolution, only depends on the hadron momentum. This leads to a clear matching…
The parton branching formulation of TMD evolution has recently been used to make predictions for jet observables at the Large Hadron Collider (LHC), including perturbative matching at next-to-leading order (NLO). This contribution presents results for the azimuthal Δ-ϕcorrelations in events with di-jets at large transverse momentum. It focuses on the back-to-back region of large Δ-ϕand discusses prospects for detailed studies of QCD dynamics in this region at the LHC.
We review the current status of collinear and Transverse Momentum Dependent densities based on the Parton Branching method. We investigate the performance of the PB-TMD evolution as well as PB-TMD parton shower with different configurations of the hard process in the four- and five-flavour schemes.
Transverse momentum dependent parton distribution functions, also abbreviated as TMDs, offer a three-dimensional picture of hadrons by taking the intrinsic transverse momentum of the parton into consideration. Hence, they are very important for us to understand the structure of hadrons. In this article, we calculate and summarize all TMDs of quark through the spectator model, from twist-2 to twist-4. Especially, we give complete results of TMDs at twist-4. We adopt a general analytical framework to calculate TMDs, with both scalar and axial-vector spectators being considered. All TMDs are calc…
We present a complete set of transverse momentum dependent and collinear parton densities including the photon over a wide phase space. The photon density appears when evolving parton distributions with QED corrections. The photons are produced by perturbative radiation using the Parton Branching method. The QCD partons are constrained by a fit to HERA data. The lepton pair production at high masses is described within the extracted TMD densities.
If studies of electroweak gauge boson final states at the Large Hadron Collider, for Standard Model physics and beyond, are sensitive to effects of the initial state's transverse momentum distribution, appropriate generalizations of QCD shower evolution are required. We propose a method to do this based on QCD transverse momentum dependent (TMD) factorization at high energy. The method incorporates experimental information from the high-precision deep inelastic scattering (DIS) measurements, and includes experimental and theoretical uncertainties on TMD parton density functions. We illustrate …
Parton branching methods underlie the Monte Carlo (MC) generators, being therefore of key importance for obtaining high energy physics predictions. We construct a new parton branching algorithm which for the first time incorporates the off-shell, transverse-momentum dependent (TMD) splitting functions, defined from the high-energy limit of partonic decay amplitudes. Based on these TMD splitting functions we construct a new TMD Sudakov form factor. We present the first MC implementation of the algorithm for the evolution of the TMD and integrated parton distribution functions (PDFs). We use thi…
We briefly recall the main physical features of the parton distributions in the quantum statistical picture of the nucleon. Some predictions from a next-to-leading order QCD analysis are successfully compared to recent unpolarized and polarized experimental results. We will discuss the extension to the transverse momentum dependence of the parton distributions and its relevance for semiinclusive deep inelastic scattering. Finally, we will present some new positivity constraints for spin observables and their implications for parton distributions.
We present a determination of parton densities at NLO obtained with the Parton Branching method using precision measurements of deep inelastic scattering cross sections at HERA. The two sets of parton densities shown in this work are obtained with the same angular angular ordering condition for the evolution scale and they differ in the chosen scale for the strong coupling evaluation, for which we consider two scenarios: the evolution scale, and the transverse momentum qT from the angular ordering prescription. The transverse momentum dependent densities obtained with the Parton Branching meth…
We discuss prospects for Monte Carlo event generators incorporating the dynamics of transverse momentum dependent (TMD) parton distribution functions. We illustrate TMD evolution in the parton branching formalism, and present Monte Carlo applications of the method.
Parton distributions in impact parameter space, which are obtained by Fourier transforming GPDs, exhibit a significant deviation from axial symmetry when target and/or quark are transversely polarized. In combination with the final state interactions, this transverse deformation provides a natural mechanism for naive-T odd transverse single-spin asymmetries in semi-inclusive DIS. The deformation of PDFs in impact parameter space can also be related to the transverse force acting on the active quark in polarized DIS at higher twist.
We provide a concise overview on transverse momentum dependent (TMD) parton distribution functions, their application to topical issues in high-energy physics phenomenology, and their theoretical connections with QCD resummation, evolution and factorization theorems. We illustrate the use of TMDs via examples of multi-scale problems in hadronic collisions. These include transverse momentum q_T spectra of Higgs and vector bosons for low q_T, and azimuthal correlations in the production of multiple jets associated with heavy bosons at large jet masses. We discuss computational tools for TMDs, an…
We calculate in this paper the perturbative gluon transverse momentum dependent parton distribution functions (TMDPDFs) and fragmentation functions (TMDFFs) using the exponential regulator for rapidity divergences. We obtain results for both unpolarized and linearly polarized distributions through next-to-next-to leading order in strong coupling constant, and through ${\cal O}(ε^2)$ in dimensional regulator (finding discrepancy for the linearly polarized gluon TMDPDFs with a previous result in the literature). We find a nontrivial momentum conservation sum rule for the linearly polarized compo…
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…
Generalised parton distributions (GPDs) and transverse momentum dependent parton distributions (TMDs) describe complementary aspects of the three-dimensional structure of hadrons. We discuss their relation to each other and recall important theory results concerning their properties and their connection with physical observables.
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.
One of the main theoretical systematics in studies of final states with large jet multiplicities at high-energy hadron colliders is associated with the merging of QCD parton showers and hard-scattering matrix elements. We present a method to incorporate the physics of transverse momentum recoils due to initial-state shower evolution into multi-jet merging algorithms by using the concept of transverse momentum dependent (TMD) distributions and the associated parton branching. We investigate the dependence on the merging scale and illustrate the impact of the new method at the level of both excl…
This article gives an introduction to transverse momentum dependent (TMD) parton distribution functions and their use in shower Monte Carlo event generators for high-energy hadron collisions, and describes recent progress in the treatment of sea quark effects within a TMD parton-shower framework.
Transverse-momentum-dependent parton distribution and fragmentation functions describe the partonic structure of the nucleon in a three-dimensional momentum space. They are subjects of flourishing theoretical and experimental activity. They provide novel and intriguing information on hadronic structure, including evidence of the presence of partonic orbital angular momentum.
We study the transverse momentum dependent (TMD) parton distributions in the newly proposed quasi-parton distribution function framework in Euclidean space. A soft factor subtraction is found to be essential to make the TMDs calculable on lattice. We show that the quasi-TMDs with the associated soft factor subtraction can be applied in hard QCD scattering processes such as Drell-Yan lepton pair production in hadronic collisions. This allows future lattice calculations to provide information on the non-perturbative inputs and energy evolutions for the TMDs. Extension to the generalized parton d…
We present new results of our studies of soft-gluon angular ordering effects on the evolution of both collinear and transverse momentum dependent (TMD) parton distribution functions, and discuss their implications for precision predictions of Drell-Yan transverse momentum spectra at the LHC. Our method is based on the parton branching (PB) approach. We compare this with the implementation of angular ordering in the Kimber-Martin-Ryskin-Watt (KMRW) approach and with the Collins-Soper-Sterman (CSS) approach. We illustrate numerically the effects of different ordering scenarios ($p_{\bot}$, angul…
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…
We introduce a method for the separation of soft and collinear logarithms in QCD parton evolution at $\mathcal{O}(α_s^2)$ and at leading color. Using an implementation of the technique in the Dire parton shower, we analyze the numerical impact of genuine triple-collinear corrections from quark pair emission in $e^+e^-\to$ hadrons.
We present the results of T-even TMDs in a light front quark-diquark model of nucleons with the wave functions constructed from the soft-wall AdS/QCD prediction. The relations amongst TMDs are discussed. The $p_\perp$ dependence of the TMDs are compared with the $t$-dependence of the GPDs. AdS/QCD wave function provides an explanation behind the approximate $x$ and $p_\perp$ factorization observed in lattice TMD calculations.
The Parton Branching (PB) approach describes the evolution of transverse momentum dependent (TMD) parton densities. We propose to extend the PB method by including TMD splitting functions, instead of the DGLAP splitting functions which assume strong ordering in transverse momentum. We present the evolution equations and their numerical solution, which is the first Monte Carlo implementation including TMD splitting functions.
Revised formulas for the inclusive cross section of a triple parton scattering process in a hadron collision are suggested basing on the modified collinear three-parton distributions. The possible phenomenological issues are discussed.
We compute in conventional dimensional regularisation the tree-level splitting amplitudes for a quark parent in the limit where four partons become collinear to each other. This is part of the universal infrared behaviour of the QCD scattering amplitudes at next-to-next-to-next-to-leading order (${\rm N^3LO}$) in the strong coupling constant. Further, we consider the iterated limit when $m'$ massless partons become collinear to each other within a bigger set of $m$ collinear partons, as well as the limits when one gluon or a $q\bar{q}$ pair or two gluons become soft within a set of $m$ colline…
Transverse Momentum Dependent (TMD) parton distributions are a very powerful concept for the description of low and high transverse momentum effects in high energy collisions. The Parton Branching (PB) method provides TMD distributions which can be used in parton shower simulations, as already implemented in \cascade\ Monte Carlo event generator. This report gives a description of the work done during the DESY summer student program 2021, young scientists from very different time zones connecting from remote twice a day for 8 weeks, to develop a method, PS2TMD, that allows to determine effecti…
We present an extraction of unpolarized partonic transverse momentum distributions (TMDs) from a simultaneous fit of available data measured in semi-inclusive deep-inelastic scattering, Drell-Yan and Z-boson production. To connect data at different scales, we use TMD evolution at next-to-leading logarithmic accuracy. The analysis is restricted to the low-transverse-momentum region, with no matching to fixed-order calculations at high transverse momentum. We introduce specific choices to deal with TMD evolution at low scales, of the order of 1 GeV$^2$. This could be considered as a first attemp…
Transverse momentum dependent (TMD) distribution correlators can be parametrized in terms of TMD parton distribution functions (PDFs), or TMDs for short. We provide an overview of the leading-twist quark and gluon TMDs, both in momentum space and in coordinate space (also called $b_T$-space, where $b_T$ is Fourier conjugate to the partonic momentum $k_T$). We consider unpolarized, vector polarized, as well as tensor polarized hadrons, which is relevant for spin-$0$, spin-$1/2$, and spin-$1$ hadrons.
