A selection of the latest and most frequently used parton distribution functions (PDFs) is incorporated in Pythia8, including the Monte Carlo-adapted PDFs from the MSTW and CTEQ collaborations. This article examines the differences in PDFs as well as the effect they have on results of simulations and compare with data collected by the CDF experiment. Monte Carlo-adapted PDFs do a better job than leading- and next-to-leading order PDFs for many observables, but there is room for further improvements.
Associated with the use of conventional integrated parton densities are kinematic approximations on parton momenta which result in unphysical differential distributions for final-state particles. We argue that it is important to reformulate perturbative QCD results in terms of fully unintegrated parton densities, differential in all components of the parton momentum.
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…
Rapidity divergences occur when parton densities in a gauge theory are defined in the most natural way, as expectation values of partonic number operators in light-front quantization. I review these and other related divergences, and show how the definitions of parton densities can be modified to remove the divergences. A modified definition is not only essential for many phenomenological applications of QCD, but also concerns the treatment of parton densities in non-perturbative approaches. The necessity of modifications in the definition of a parton density also entails corrections in the fo…
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…
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…
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.
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.
In the paper we include Bose-Einsten correlations for identical gluons, in the evolution of the double parton densities at low $x$ (high energies). We show that these correlations result in a faster increase of the double densities, in comparison with the product of two single parton densities. In addition, we find that the typical momentum in the double parton cross section, is of the order of the hard scales in the process, including the saturation momentum. This result is in agreement with recently published experimental data.
The scale factor $σ_{eff}$, which characterizes double parton collisions in high energy hadron interactions, is a direct manifestation of the distribution of the interacting partons in transverse space, in such a way that different distributions give rise to different values of $σ_{eff}$ in different double parton collision processes. We work out the value of the scale factor in a few reactions of interest, in a correlated model of the multi-parton density of the proton recently proposed.
Partons are defined as the quanta in a Fock space description of a hadron. Gluon saturation is described in the Weizsäcker-Williams approximation for a large nucleus. The elements of DGLAP and BFKL evolution are given with the BFKL equation derived in a large-$N_c$ dipole formalism. A more general discussion of saturation is given in terms of a dipole scattering on a nucleon or nucleus. Possible evidence for saturation at HERA is discussed.
We propose a consistent treatment of divergences emerging in the computation of transverse-momentum-dependent parton densities in leading $α_s$-order of QCD perturbation theory.
We discuss how the operator product expansion (OPE) can be used to derive asymptotic expressions for certain integrals. This yields operator matrix elements (OME's) which determine the matching conditions for $\bar{\rm MS}$ parton densities across heavy flavour thresholds. Then we construct four and five-flavour densities from a three-flavour set via the AP equation using LO and NLO splitting functions.
We present recent results on transverse momentum dependent parton densities in the limit of small Bjorken x. A definition of an unintegrated seaquark density is proposed which can be used within the CCFM Monte-Carlo event generator CASCADE. We further discuss aspects of the determination of a NLO BFKL unintegrated gluon density.
We update the theoretical predictions for the t-tbar production cross-section at the Tevatron, taking into account the most recent determinations of systematic uncertainties in the extraction of the proton parton densities.
In this talk the relation between the semiclassical approach and the concept of diffractive parton densities is discussed. The proton rest frame calculation is organized in a way that exhibits the hard partonic cross section and the diffractive parton density as the two fundamental ingredients. The latter one is a non-perturbative quantity which, in the present model, is explicitly given by integrals of non-Abelian eikonal factors in the colour background field.
We draw attention to some problems in the combined use of high-Q^2 deep inelastic scattering (DIS) data and low-Q^2 hyperon β-decay data in the determination of the polarized parton densities. We explain why factorization schemes like the JET or AB schemes are the simplest in which to study the implications of the DIS parton densities for the physics of the low-Q^2 region.
Collider data can play an important role in determining the parton distribution functions of the nucleon. I outline a formalism which makes it possible to use next-to-leading order calculations in such an extraction, while minimizing the amount of numerical computation required.
The feasibility of an experimental determination of the parton distributions of virtual photons at high energy $ep$ colliders is studied in the context of the DESY-HERA collider. Recently proposed parton densities of virtual photons are utilized to evaluate the appropriate production processes and their relevant kinematical regions. It is demonstrated that high $E_T$ jet production with $E_T\,\simeq 5-7\, {\rm{GeV}}$ and $b\bar{b}$ production can probe the parton distributions of virtual photons up to virtualities of a few ${\rm{GeV}}^2$. A useful leading order parametrization of our proposed …
Collider data can play an important role in determining the parton distribution functions of the nucleon. I present a formalism which makes it possible to use next-to-leading order calculations in such a determination, while minimizing the amount of numerical computation required.
We present a solution of the DGLAP evolution equations, written in terms of Sudakov form factors to describe the branching and no-branching probabilities, using a parton branching Monte Carlo method. We demonstrate numerically that this method reproduces the semi-analytical solutions. We show how this method can be used to determine Transverse Momentum Dependent (TMD) parton distribution functions, in addition to the usual integrated parton distributions functions. We discuss numerical effects of the boundary of soft gluon resolution scale parameter on the resulting parton distribution functio…
We study the gluon parton densities [parton distribution functions (PDFs), transverse momentum distributions (TMDs), generalized parton distributions (GPDs)] and form factors in soft-wall AdS/QCD. We show that the power behavior of gluon parton distributions and form factors at large values of the light-cone variable and large values of square momentum is consistent with quark counting rules. We also show that the transverse momentum distributions derived in our approach obey the model-independent Mulders-Rodrigues inequalities without referring to specific model parameters. All gluon parton d…
The current status of the theoretical understanding of the transverse-momentum dependent parton densities (TMDs) is discussed. Special attention is payed to the difference between the operator definitions of TMDs proposed so far, the treatment of specific divergences, the geometry of the gauge links, and the role of the soft factors.
Di-jet photoproduction in polarized ep collisions at HERA is studied as a possible tool to determine the parton content of circularly polarized photons. The concept of the `effective parton density' approximation is extended to the spin-dependent case.
The Bessel-inspired behavior of parton densities at small x, obtained in the case of the flat initial conditions for DGLAP evolution equations, is used in the fixed flavor scheme to analyze precise H1+ZEUS combined data on the structure function F2.
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.
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…
I discuss the role played by large parton densities in the QCD description of high-pT observables in relativistic heavy-ion collisions. In pA collisions, high-pT particles probe large parton densities in the nucleus and provide tests of the Color Glass Condensate (CGC) picture of the small-x part of the nuclear wave function. In AA collisions, high-pT particles carry information on both the Quark-Gluon-Plasma phase, and the Glasma phase created at early times by the collision of two CGCs.
A possible application of the evolution equation for the truncated Mellin moments to determination of the parton distributions in the nucleon is presented. We find that the reconstruction of the initial parton densities at scale $Q_0^2$ from their truncated moments at a given scale $Q^2$ is exact and unique for small number of free parameters ($\leq 3$), even for the limited $x$-region of experimental data. For larger number of adjustable parameters the obtained fits are not unique and one needs an additional knowledge of the small-$x$ behaviour of the parton densities to make the reconstructi…
We have investigated the transverse charge density for longitudinal and transversely polarized $ρ$ meson in light-front quark model (LFQM). Charge densities are obtained from the elastic form factors of the $ρ$ meson calculated in LFQM including the zero-mode contributions. We have computed the different helicity matrix elements of the $ρ$ meson. In addition to this, we have also presented the results for the generalized parton distributions (GPDs) and impact-parameter dependent parton distribution functions (ipdpdfs) of the $ρ$ meson.
Motivated by the possibility that it may be misleading to assume SU(3) symmetry in the analysis of hyperon β-decays, we study the sensitivity of the polarized parton densities, derived from the world data on inclusive polarized deep inelastic lepton-nucleon scattering, to variation of the value of the nonsinglet axial charge a_8 from its SU(3) value of 3F-D.
An updated next-to-leading order (NLO) QCD analysis of all presently available longitudinally polarized deep-inelastic scattering (DIS) data is presented in the framework of the radiative parton model.
We discuss the NLO evolution of quark transversity densities and of the parton distribution function for linearly polarized gluons in a linearly polarized hadron. A supersymmetric relation between the NLO evolution kernels for transversity and for linear polarization is found. We also study the implications of NLO evolution for Soffer's inequality and the prospects of measuring transversity densities in polarized Drell-Yan at RHIC.
We use the Bessel-inspired behavior of parton densities at small Bjorken $x$ values, obtained in the case of the flat initial conditions for DGLAP evolution equations in the double scaling QCD approximation (DAS), to evaluate the transverse momentum dependent (TMD, or unintegrated) quark and gluon distribution functions in a proton. The calculations are performed analytically using the Kimber-Martin-Ryskin (KMR) prescription with different implementation of kinematical constraint, reflecting the angular and strong ordering conditions. The relations between the differential and integral formula…
A study is made of charm and bottom flavor matching conditions on parton densities. Starting from an $\bar{\rm MS}$ three-flavor density set, where the scale $μ< m_c$, and using the recently derived two-loop matching conditions, we provide a new set of four-flavor parton densities where $ m_c \le μ< m_b$ and five-flavor densities where $μ\ge m_b$. The effect of the next-to-next-to-leading order matching conditions on the evolution equations is important for scales just above the transition regions. This includes the small $x$ and small $Q^2$ domain studied by the H1 and ZEUS experiments at HER…
We evaluate both chirally even and odd generalized parton distributions(GPDs) in the leading twist in a recently proposed quark-diquark model for the proton where the light front wavefunctions are constructed from the soft-wall AdS/QCD prediction. The GPDs in transverse impact parameter space give the spin densities for different quark and proton polarizations. For longitudinally polarized proton only chiral even GPDs contribute but for transversely polarized proton both chiral even and chiral odd GPDs contribute to the spin densities. We present a detail study of the spin densities in this mo…
We investigate the Q^2 evolution of parton distributions at small x values, obtained in the case of flat initial conditions. The contributions of twist-two and (renormalon-type) higher-twist operators of the Wilson operator product expansion are taken into account. The results are in very good agreement with deep inelastic scattering experimental data from HERA.
This talk provides a short overview of the phenomenology of transverse momentum depedent distribution and fragmention functions, focussing on the most recent phenomenological developments in the study of their scale evolution and energy depedence.
We extract the pion fragmentation functions and their uncertainties from a judicious choice of e+e- and semi-inclusive DIS data. These are used to study the error propagation in the extraction of polarized parton densities from semi-inclusive DIS asymmetries. We conclude that the uncertainties on polarized PDs have been underestimated in the past.
We present a new NLO QCD analysis of the world data on inclusive polarized deep inelastic scattering. Comparing to our previous analysis: i) the values of g_A and a_8=3F-D are updated ii) the MRST'99 instead of the MRST'98 parametrization for the input unpolarized parton densities is used and iii) the recent SLAC E155 proton data on the spin asymmetry A1 are included in the analysis. A new set of polarized parton densities is extracted from the data and the sensitivity of the results to different positivity constraints is discussed.
We present results on the quark and gluon polarization in the nucleon obtained in a combined next to leading order analysis to the available inclusive and semi-inclusive polarized deep inelastic scattering data.
We consider impact parameter dependence of the polarized and unpolarized parton densities. Unitarity does not allow factorization of these structure functions over the Bjorken x and the impact parameter b variables. On the basis of the particular geometrical model approach we conclude that spin of constituent quark may have a significant orbital angular momentum component which can manifest itself through the peripherality of the spin dependent structure functions.
In this paper we found that the Bose-Einstein enhancement generates the strong correlations, which increase with energy in the BFKL evolution. This increase leads to the double parton densities ( $Φ$), that are much larger than the product of the single parton densities ($ϕ$). However, numerically, it turns out that the ratio $Φ/ϕ^2 \propto \Lb 1/x\Rb^{δ_2}$ with $δ_2 \sim \bas/\Lb N^2_c - 1\Rb^{2/3}\,\,\ll\,\,1$ and we do not expect a large correction for the accessible range of energies. However, for $N_c=3$ it tuns out that $δ_2 = 0.07 Δ_{\rm BFKL}$ where $Δ_{\rm BFKL}$ is the intercept of …
We briefly discuss recent results on the evolution of unpolarized parton densities and structure functions in massless perturbative QCD. Present partial results on the next-to-next-to-leading order (NNLO) evolution kernels prove sufficient for reliable calculations at not too small values of the Bjorken variable, x > 10^-3. One order more can be taken into account at x >= 0.2. Inclusion of these terms considerably reduces the main theoretical uncertainties of determinations of alpha_s (to about 1% at the Z-mass) and the parton densities from structure functions.
It is shown that a Bessel-like behaviour of the structure function F2 at small x, obtained for a flat initial condition in the DGLAP evolution equations, leads to good agreement with the deep inelastic scattering experimental data from HERA.
Different approaches to define transverse-momentum dependent parton distribution functions are considered from the point of view of their renormalization-group properties. The associated one-loop anomalous dimensions of these quantities are presented and compared to each other. We give arguments in favor of the "pure light-like" definition, and the use of the light-cone gauge.
We use the Bessel-inspired behavior of the structure function F2 at small x, obtained for a flat initial condition in the DGLAP evolution equations, with "frozen" and analytic modifications of the strong coupling constant to study precise combined H1 and ZEUS data for the structure function F2 published recently.
We present calculations of several pp scattering cross sections with potential applications at the LHC. Significantly large rates for momentum fraction, x, as low as 10^-7 are obtained, allowing for possible extraction of quark and gluon densities in the proton and nuclei down to these small x values provided a detector with good acceptance at maximal rapidities is used.
A new code for the scale evolution of modified-minimal-subtraction-scheme parton densities is described. Through next-to-leading order the program uses exact splitting functions. In next-to-next-to-leading order approximate splitting functions are used. For efficiency the program includes analytical results for the evaluation of the weights required for the integrations over the longitudinal momentum fractions of the partons. It also incorporates the operator matrix elements required for the matching conditions across heavy flavor thresholds in higher order perturbation theory. The more effici…
We calculate nuclear parton distribution functions (PDFs), using the constituent quark model. We find the bounded valon distributions in a nuclear to be related to free valon distributions in a nucleon. By using improved bounded valon distributions for a nuclear with atomic number $A$ and the partonic structure functions inside the valon, we can calculate the nuclear structure function in $x$ space. The results for nuclear structure-function ratio $F_2^A/F_2^D$ at some values of $A$ are in good agreement with the experimental data.
We present a detailed phenomenological study of the associated production of a prompt photon and a heavy-quark jet (charm or bottom) in proton-nucleus (p-A) and nucleus-nucleus (A-A) collisions. The dominant contribution to the cross-section comes from the gluon--heavy-quark (gQ) initiated subprocess, making this process very sensitive to the gluon and the heavy quark nuclear parton densities. We show that the future p-A data to be collected at the LHC should allow one to disentangle the various nPDF sets currently available. In heavy-ion collisions, the photon transverse momentum can be used …
We study how the components of hadronic wave functions in light-cone quantization depend on the ultraviolet cut-off by relating them in a systematic way to the matrix elements of a class of quark-gluon operators between the QCD vacuum and the hadrons. From this, we derive an infinite set of scale-evolution equations for the individual contributions to parton distributions from the Fock expansion. When summed over all the contributions, we recover the well-known DGLAP equation.
