Parton Transverse Momentum Research Articles

Investigation of the Z boson production via hadron–hadron collisions in the k_t and (z, k_t)-factorization frameworks

In this paper, we study the Z boson production via the proton–proton (p–p) collisions within the k_t and (z, k_t)-factorization frameworks, using the Martin–Ryskin–Watt (MRW) unintegrated parton distribution functions (UPDFs) and the double unintegrated parton distribution functions (DUPDFs), respectively. For calculation of the differential cross section (DCS) within the k_t-factorization (k_t is the partonic transverse momentum), the KATIE parton level event generator is used, while for the (z, k_t)-factorization, the DCS is directly computed. Up to the tree level partonic next-to-leading order (NLO) are included, beside the inclusion of branching ratios, in our calculation. It should be noted that Martin, Ryskin and Watt are originally calculated the same process, i.e., the Z boson production, within the (z, k_t)-factorization framework, while including only the lowest order tree level partonic sub-process. However, the present report extends their work from two perspectives. First, the additional sub-processes are included, second, for the processes up to the next-to-leading order (NLO), a direct calculation by considering the final state leptons is imposed. Finally, we compare our results with the 13; TeV data of the ATLAS, LHCb, CMS collaborations, the corresponding collinear factorization predictions and the Modarres, et al. reports. Our p–p DCS calculations show that the k_t and (z, k_t)-factorizations frameworks give relatively the same behavior in the central rapidity regions. While at the large rapidity regions, the (z, k_t)-factorization, predicts the p–p DCS closer to the experimental data with respect to those of k_t-factorization framework.

Toward the global fit of the TMD gluon density in the proton from the LHC data

We propose a new analytical expression for the transverse-momentum-dependent (TMD, or unintegrated) gluon density in the proton. Essential phenomenological parameters are extracted from the LHC data on inclusive hadron production in $pp$ collisions at low transverse momenta ${p}_{T}\ensuremath{\le}1\text{ }\text{ }\mathrm{GeV}$. The latter are described in the framework of a modified soft quark-gluon string model, where the gluonic state and nonzero transverse momentum of partons inside the proton are taken into account. To determine the parameters important at moderate and large $x$, we use the measurements of inclusive $b$-jet and Higgs boson production at the LHC as well as the latest HERA data on proton structure functions ${F}_{2}^{c}(x,{Q}^{2})$ and ${F}_{2}^{b}(x,{Q}^{2})$ and reduced cross sections ${\ensuremath{\sigma}}_{\mathrm{red}}^{c}(x,{Q}^{2})$ and ${\ensuremath{\sigma}}_{\mathrm{red}}^{b}(x,{Q}^{2})$. The Catani-Ciafaloni-Fiorani-Marchesini evolution equation is applied to extend the initial gluon distribution to the whole kinematical region. We achieve a simultaneous description of all considered processes with ${\ensuremath{\chi}}^{2}/\mathrm{d}.\mathrm{o}.\mathrm{f}.=2.2$, thus moving toward the global fit of the TMD gluon density from collider data. The obtained TMD gluon distribution in a proton is available for public usage and implemented in the tmdlib package and Monte Carlo event generator pegasus.

Possible solution of the puzzle for the branching ratio and CP violation in B→ππ decays with a modified perturbative QCD approach

We study $B\ensuremath{\rightarrow}\ensuremath{\pi}\ensuremath{\pi}$ decay with a modified perturbative QCD approach. The branching ratios and $CP$ violation are calculated with the transverse momenta of partons considered. Sudakov factor associated with each meson is included to suppress soft contribution in QCD. With the wave function of $B$ meson obtained in QCD-inspired relativistic potential model being used, the suppression of Sudakov factor to the soft contribution is not effective enough. Soft scale cutoff and soft form factors of $B\ensuremath{\pi}$ transition and $\ensuremath{\pi}\ensuremath{\pi}$ production have to be introduced. The main next-to-leading-order contributions of vertex correction, the quark-loop, and magnetic penguin are included. To solve the long-standing puzzle in $B\ensuremath{\rightarrow}\ensuremath{\pi}\ensuremath{\pi}$ decay, that is the theoretical prediction of the branching ratio of ${B}^{0}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{0}{\ensuremath{\pi}}^{0}$ being seriously smaller than experimental data, color-octet matrix element which is of long-distance dynamics is introduced. With parameters taken with reasonable values, all the branching ratios and $CP$ violation are well consistent with experimental data.

Transverse-momentum-dependent parton distribution functions for spin-1 hadrons

We show transverse-momentum-dependent parton distribution functions (TMDs) for spin-1 hadrons including twist-3 and 4 functions by taking the decomposition of a quark correlation function in the Lorentz-invariant way with the conditions of Hermiticity and parity invariance. We found 30 new TMDs in the tensor-polarized spin-1 hadron at twists 3 and 4 in addition to 10 TMDs at twist 2. Since time-reversal-odd terms of the collinear correlation function should vanish after integrals over the partonic transverse momentum, we obtained new sum rules for the time-reversal-odd structure functions, {\int d^2 k_T g_{LT} = \int d^2 k_T h_{LL} = \int d^2 k_T h_{3LL} =0}∫d2kTgLT=∫d2kThLL=∫d2kTh3LL=0, at twists 3 and 4. We also indicated that transverse-momentum-dependent fragmentation functions exist in tensor-polarized spin-1 hadrons. The TMDs can probe color degrees of freedom, so that they are valuable in providing unique opportunities for creating interdisciplinary physics fields such as gluon condensate, color Aharonov-Bohm effect, and color entanglement. We also found three new collinear PDFs at twists 3 and 4, and a twist-2 relation and a sum rule were derived in analogy to the Wandzura-Wilczek relation and the Burkhardt-Cottingham sum rule on the structure function {g_2}g2.

Recent Highlights from Spin-Physics Experiments

A snapshot of the current experimental status of spin-dependent nucleon structure prior to the arrival of the Electron Ion Collider will be attempted. Highlights of recent results from fixed-target experiments at JLab, Fermilab, CERN and DESY and collider experiments from RHIC are presented. Many of the recent results address the topics of transverse proton or parton spin and transverse parton momenta (TMDs), and their their (spin-orbit) correlations. Results on generalized parton distributions (GPDs), which provide an alternative pathway to mapping nucleon structure, are also discussed.

Probing Nucleon Spin Structure in Deep-inelastic Scattering, Proton–Proton Collisions, and Drell–Yan Processes

A pedagogical summary of current and past experimental results of spin-dependent nucleon structure prior to the arrival of the Electron-Ion Collider is attempted. After an introduction, results from fixed-target experiments at SLAC, Fermilab, Jefferson Lab, CERN and DESY and collider experiments from RHIC are presented, starting with the longitudinal spin structure of the nucleon, followed by generalized parton distributions (GPDs), which map the proton in transverse position space. The final part discusses transverse proton or parton spin and transverse parton momenta (TMDs), and their their (spin-orbit) correlations, which are addressed by a multitude of recent experimental results. The GPDs and TMDs provide complementary pathways to mapping multi-dimensional nucleon structure.

On transverse momentum broadening in real-time lattice simulations of the glasma and in the weak-field limit

In these proceedings, we report on our numerical lattice simulations of partons traversing the boost-invariant, non-perturbative glasma as created at the early stages of collisions at RHIC and LHC. Since these highly energetic partons are produced from hard scatterings during heavy-ion collisions, they are already affected by the first stage of the medium's time evolution, the glasma, which is the pre-equilibrium precursor state of the quark-gluon plasma. We find that partons quickly accumulate transverse momentum up to the saturation momentum during the glasma stage. Moreover, we observe an interesting anisotropy in transverse momentum broadening of partons with larger broadening in the rapidity than in the azimuthal direction. Its origin can be related to correlations among the longitudinal color-electric and color-magnetic flux tubes in the initial state of the glasma. We compare these observations to the semi-analytic results obtained by a weak-field approximation, where we also find such an anisotropy in a parton's transverse momentum broadening.

Transverse-momentum-dependent parton distribution functions up to twist 4 for spin-1 hadrons

We show possible transverse-momentum-dependent parton distribution functions (TMDs) for spin-1 hadrons including twist-3 and 4 functions in addition to the leading twist-2 ones by investigating all the possible decomposition of a quark correlation function in the Lorentz-invariant way. The Hermiticity and parity invariance are imposed in the decomposition; however, the time-reversal invariance is not used due to an active role of gauge links in the TMDs. Therefore, there exist time-reversal-odd functions in addition to the time-reversal even ones in the TMDs. We list all the functions up to twist-4 level because there were missing terms associated with the lightcone vector $n$ in previous works on the twist-2 part and there was no correlation-function study in the twist-3 and 4 parts for spin-1 hadrons. We show that 40 TMDs exist in the tensor-polarized spin-1 hadron in twists 2, 3, and 4. Some expressions of twist-2 structure functions are modified from previous derivations due to the new terms with $n$, and we find 30 new structure functions in twists 3 and 4 in this work. Since time-reversal-odd terms of the collinear correlation function should vanish after integrals over the partonic transverse momentum, we obtain new sum rules for the time-reversal-odd structure functions, $ \int d^2 k_T h_{LT}= \int d^2 k_T h_{LL} = \int d^2 k_T h_{3LL} =0$. In addition, we indicate that new transverse-momentum-dependent fragmentation functions exist in tensor-polarized spin-1 hadrons. The TMDs are rare observables to find explicit color degrees of freedom in terms of color flow, which cannot be usually measured because the color is confined in hadrons.

Gradient Tomography of Jet Quenching in Heavy-Ion Collisions.

Transverse momentum broadening and energy loss of a propagating parton are dictated by the space-time profile of the jet transport coefficient q[over ^] in a dense QCD medium. The spatial gradient of q[over ^] perpendicular to the propagation direction can lead to a drift and asymmetry in parton transverse momentum distribution. Such an asymmetry depends on both the spatial position along the transverse gradient and path length of a propagating parton as shown by numerical solutions of the Boltzmann transport in the simplified form of a drift-diffusion equation. In high-energy heavy-ion collisions, this asymmetry with respect to a plane defined by the beam and trigger particle (photon, hadron, or jet) with a given orientation relative to the event plane is shown to be closely related to the transverse position of the initial jet production in full event-by-event simulations within the linear Boltzmann transport model. Such a gradient tomography can be used to localize the initial jet production position for more detailed study of jet quenching and properties of the quark-gluon plasma along a given propagation path in heavy-ion collisions.

Collins-Soper kernel for TMD evolution from lattice QCD

The Collins-Soper kernel relates transverse momentum-dependent parton distribution functions (TMDPDFs) at different energy scales. For small parton transverse momentum $q_T\sim \Lambda_\text{QCD}$, this kernel is non-perturbative and can only be determined with controlled uncertainties through experiment or first-principles calculations. This work presents the first exploratory determination of the Collins-Soper kernel using the lattice formulation of Quantum Chromodynamics. In a quenched calculation, the $N_f=0$ kernel is determined at scales in the range 250 MeV $< q_T < 2$ GeV, and an analysis of the remaining systematic uncertainties is undertaken.

Effect of flavor-dependent partonic transverse momentum on the determination of the W boson mass in hadronic collisions

Within the framework of transverse-momentum-dependent factorization, we investigate for the first time the impact of a flavor-dependent intrinsic transverse momentum of quarks on the production of W± bosons in proton–proton collisions at s=7TeV. We estimate the shift in the extracted value of the W boson mass MW induced by different choices of flavor-dependent parameters for the intrinsic quark transverse momentum by means of a template fit to the transverse-mass and the lepton transverse-momentum distributions of the W-decay products. We obtain −6≤ΔMW+≤9MeV and −4≤ΔMW−≤3 MeV with a statistical uncertainty of ±2.5 MeV. Our findings call for more detailed investigations of flavor-dependent nonperturbative effects linked to the proton structure at hadron colliders.

Lorentz invariance and QCD equation of motion relations for generalized parton distributions and the dynamical origin of proton orbital angular momentum

We derive new Lorentz invariance and equation of motion relations between twist-three generalized parton distributions (GPDs) and moments in the parton transverse momentum, ${k}_{T}$, of twist-two generalized transverse momentum-dependent distributions (GTMDs), as a function of the parton longitudinal momentum fraction $x$. Although GTMDs in principle define the observables for partonic orbital motion, experiments that can unambiguously detect them appear remote at present. The relations presented here provide a solution to this impasse in that, e.g., the orbital angular momentum density is connected to directly measurable twist-three GPDs. Out of 16 possible equation of motion relations that can be written in the $T$-even sector, we focus on three helicity configurations that can be detected analyzing specific spin asymmetries: two correspond to longitudinal proton polarization and are associated with quark orbital angular momentum and spin-orbit correlations; the third, obtained for transverse proton polarization, is a generalization of the relation obeyed by the ${g}_{2}$ structure function. We also exhibit an additional relation connecting the off-forward extension of the Sivers function to an off-forward Qiu-Sterman term.

On the use of the KMR unintegrated parton distribution functions

We discuss the unintegrated parton distribution functions (UPDFs) introduced by Kimber, Martin and Ryskin (KMR), which are frequently used in phenomenological analyses of hard processes with transverse momenta of partons taken into account. We demonstrate numerically that the commonly used differential definition of the UPDFs leads to erroneous results for large transverse momenta. We identify the reason for that, being the use of the ordinary PDFs instead of the cutoff dependent distribution functions. We show that in phenomenological applications, the integral definition of the UPDFs with the ordinary PDFs can be used.

The 3D Entangled Structure of the Proton: Transverse Degrees of Freedom in QCD, Momenta, Spins and More

Light-front quantized quark and gluon states (partons) play a dominant role in high energy scattering processes. Initial state hadrons are mixed ensembles of partons, while produced pure partonic states appear as mixed ensembles of hadrons. The transition from collinear hard physics to the 3D structure including partonic transverse momenta is related to confinement which links color and spatial degrees of freedom. We outline ideas on emergent symmetries in the Standard Model and their connection to the 3D structure of hadrons. Wilson loops, including those with light-like Wilson lines such as used in the studies of transverse momentum dependent distribution functions may play a crucial role here, establishing a direct link between transverse spatial degrees of freedom and gluonic degrees of freedom.

Numerical analysis of the unintegrated double gluon distribution

We present detailed numerical analysis of the unintegrated double gluon distribution which includes the dependence on the transverse momenta of partons. The unintegrated double gluon distribution was obtained following the Kimber-Martin-Ryskin method as a convolution of the perturbative gluon splitting function with the collinear integrated double gluon distribution and the Sudakov form factors. We analyze the dependence on the transverse momenta, longitudinal momentum fractions and hard scales. We find that the unintegrated gluon distribution factorizes into a product of two single unintegrated gluon distributions in the region of small values of x, provided the splitting contribution is included and the momentum sum rule is satisfied.

Global extraction of partonic transverse momentum distributions from semi-inclusive deep inelastic scattering and Drell-Yan data

We discuss a first attempt at a global extraction of unpolarized partonic transverse momentum dependent (TMD) distributions and fragmentation functions from a simultaneous fit of data measured in semi-inclusive deep-inelastic scattering, Drell–Yan and Z boson production. This analysis is performed in the low transverse momentum region, at leading order in perturbative QCD. To connect data at different scales, we use TMD evolution at next-to-leading logarithmic accuracy.

Sivers effect in single spin asymmetry based on the covariant parton model

Sivers effect is describing the correlation between the transverse polarization of nucleon and the transverse momentum, [Formula: see text], of its unpolarized constituent partons. This effect is an outstanding subject and in this regard, a great deal in recent years has been considered from experimental and phenomenological points of view. It also plays an essential role to extend our understanding from nucleon structure. Semi-inclusive DIS (SIDIS) process provides us an opportunity to access to Sivers function which is dependent on transverse momentum of partons. In this paper, for the first time the covariant parton model is used to deliver us the [Formula: see text] and [Formula: see text] dependence part of Sivers function. Based on this model, this combinatory dependence is arising out from the HERAPDF parametrization group. In this paper the other required parametrized functions in Sivers function is also changed with respect to Ref. 1. The unknown parameters which exist in Sivers function can be extracted, doing a global fit over the recent available experimental data, including HERMES, COMPASS and JLAB collaborations for the single spin asymmetry (SSA) in [Formula: see text] and [Formula: see text] meson production as well as kaon production to constrain the evolved strange quark. This is done, considering advanced mathematical manipulations to overcome the difficulties which exist to compute the required multiple integrals and finally employing the CERN MINIUTE program to do a global fit. Our results for SSA are in good agreement with the available experimental data. For more confirmation a comparison between our results and the ones from Ref. 2 is also done.

Transverse parton momenta in single inclusive hadron production in e+e− annihilation processes

We study the transverse momentum distributions of single inclusive hadron production in e+e− annihilation processes. Although the only available experimental data are scarce and quite old, we find that the fundamental features of transverse momentum dependent (TMD) evolution, historically addressed in Drell–Yan processes and, more recently, in Semi-inclusive deep inelastic scattering processes, are visible in e+e− annihilations as well. Interesting effects related to its non-perturbative regime can be observed.We test two different parameterizations for the p⊥ dependence of the cross section: the usual Gaussian distribution and a power-law model. We find the latter to be more appropriate in describing this particular set of experimental data, over a relatively large range of p⊥ values. We use this model to map some of the features of the data within the framework of TMD evolution, and discuss the caveats of this and other possible interpretations, related to the one-dimensional nature of the available experimental data.

Extraction of partonic transverse momentum distributions from semi-inclusive deep-inelastic scattering, Drell-Yan and Z-boson production

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 GeV2. This could be considered as a first attempt at a global fit of TMDs.

What are the low- Q and large- x boundaries of collinear QCD factorization theorems?

Familiar factorized descriptions of classic QCD processes such as deeply-inelastic scattering (DIS) apply in the limit of very large hard scales, much larger than nonperturbative mass scales and other nonperturbative physical properties like intrinsic transverse momentum. Since many interesting DIS studies occur at kinematic regions where the hard scale, $Q \sim$ 1-2 GeV, is not very much greater than the hadron masses involved, and the Bjorken scaling variable $x_{bj}$ is large, $x_{bj} \gtrsim 0.5$, it is important to examine the boundaries of the most basic factorization assumptions and assess whether improved starting points are needed. Using an idealized field-theoretic model that contains most of the essential elements that a factorization derivation must confront, we retrace the steps of factorization approximations and compare with calculations that keep all kinematics exact. We examine the relative importance of such quantities as the target mass, light quark masses, and intrinsic parton transverse momentum, and argue that a careful accounting of parton virtuality is essential for treating power corrections to collinear factorization. We use our observations to motivate searches for new or enhanced factorization theorems specifically designed to deal with moderately low-$Q$ and large-$x_{bj}$ physics.