Academic literature on the topic 'Transverse momentum dependent parton distribution functions'

This talk provides a short overview of the phenomenology of transverse momentum dependent distribution and fragmentation functions, focussing on the most recent phenomenological developments in the study of their Q2 evolution and energy dependence.

Azimuthal asymmetries in high-energy processes, most pronounced showing up in combination with single or double (transverse) spin asymmetries, can be understood with the help of transverse momentum dependent (TMD) parton distribution and fragmentation functions. These appear in correlators containing expectation values of quark and gluon operators. TMDs allow access to new operators as compared to collinear (transverse momentum integrated) correlators. These operators include nontrivial process dependent Wilson lines breaking universality for TMDs. Making an angular decomposition in the azimuthal angle, we define a set of universal TMDs of definite rank, which appear with process dependent gluonic pole factors in a way similar to the sign of T-odd parton distribution functions in deep inelastic scattering or the Drell-Yan process. In particular, we show that for a spin 1/2 quark target there are three pretzelocity functions.

Recent data from semi-inclusive deep inelastic scattering collected by the HERMES collaboration allow for the first time to discuss how the transverse-momentum dependence of unpolarized distribution and fragmentation functions is affected by the flavor of the involved partons. A model built with flavor-dependent Gaussian transverse-momentum distributions fits data better than the same flavor-independent model. The current analysis is performed for totally unpolarized scattering of leptons off protons and deuterons, with detected pions in the final state. There are convincing indications of flavor dependence in the fragmentation functions, while for parton distribution functions the evidence is weaker.

We consider the cross section for semi-inclusive deep inelastic scattering in Fourier space, conjugate to the outgoing hadron's transverse momentum, where convolutions of transverse momentum dependent parton distribution functions and fragmentation functions become simple products. Individual asymmetric terms in the cross section can be projected out by means of a generalized set of weights involving Bessel functions. Advantages of employing these Bessel weights are that they suppress (divergent) contributions from high transverse momentum and that soft factors cancel in (Bessel-) weighted asymmetries. Also, the resulting compact expressions immediately connect to previous work on evolution equations for transverse momentum dependent parton distribution and fragmentation functions and to quantities accessible in lattice QCD. Bessel-weighted asymmetries are thus model independent observables that augment the description and our understanding of correlations of spin and momentum in nucleon structure.

In the description of protons, we go beyond the ordinary collinear parton distribution functions (PDFs), by including transverse momentum dependent PDFs (TMDs). As such, we become sensitive to polarization modes of the partons and protons that one cannot probe without accounting for transverse momenta of partons, in particular when looking at azimuthal asymmetries. Hadronic processes require the inclusion of gluon contributions forming the gauge links, which are path-ordered exponentials tracing the color flow. In processes with two hadrons in the initial state, such as Drell-Yan (DY), the gauge links from different parts of the process get entangled. We show that in color disentangling this gauge link structure, one becomes sensitive to this color flow. After disentanglement, particular combinations of TMDs will require a different numerical color factor than one naively might have expected. Such color factors will even play a role for azimuthal asymmetries in the simplest hadronic processes such as DY.

Using recently published HERMES data on multiplicities in semi-inclusive deep-inelastic scattering, we discuss the flavor dependence of unpolarized transverse-momentum dependent distribution and fragmentation functions. We find convincing indications that favored fragmentation into pions have smaller average transverse momentum than unfavored ones and fragmentation into kaons. We find weaker indications of flavor dependence in the transverse-momentum dependence of parton distributions.

2009/2010
The main topic of this thesis is the measurement of the transverse momentum dependent (TMD) azimuthal asymmetries in the inclusive hadron production in the deep inelastic scattering (DIS). This work has been carried on in the framework of COMPASS, a fixed target experiment at CERN's SPS, which started data taking in 2002. One of the main topic of COMPASS is the study of the nucleon spin structure using high energy (100-200 GeV) muon beam and either transversely and longitudinally polarized proton and deuteron target. Recent developments both from the experimental and theoretical side pointed out the relevance of the transverse spin effects and of new effects, related to the intrinsic transverse momentum of the quark and its correlation with the spin. The nucleon structure, in the non perturbative QCD formalism, can be described at first order, in the collinear approximation, by three parton distribution functions (PDF): the unpolarized PDF, the helicity PDF and the transversity PDF. When including the intrinsic transverse momentum of the quark other 5 PDFs are needed at first order. They are related to the correlation between the transverse momentum of the quark and the spin and are practically unmeasured. The semi inclusive DIS (SIDIS) allows to study all these functions. Indeed the the SIDIS differential cross section, taking into account the beam and target polarization, is characterized by 14 azimuthal modulations in independent linear combination of the nucleon spin angle and of the hadrons angle, as measured with respect to the virtual photon direction. Their amplitudes are related to the structure functions given by the convolution between the new PDFs and the hadron fragmentation functions. In this thesis the measurement of the azimuthal asymmetries which appears in the SIDIS cross section on the unpolarized and longitudinally polarized target. Of particular interest is the information that this measurement give on the Boer-Mulders TMD PDF, which is a very interesting topic. The analysed data have been taken using the 160 GeV positive muon beam on the deuteron target. The target configuration and the spectrometer are optimized for the spin asymmetry measurements and are described in the thesis as well as the data taking. Also, the data processing and the software used for the analysis and the SIDIS data selection are described. The measurement of the azimuthal asymmetries on the longitudinally polarized target have been done using the same data as the one used for the $\Delta$G (the contribution to the nucleon spin of the longitudinal spin of the gluons) measurement. To avoid acceptance effects in the asymmetries extraction, the data collected from the two oppositely polarized target cells (which form the target) have been opportunely combined. Other possible systematic effects have been studied in details and are described in the thesis. \\ The asymmetries have been extracted as functions of the relevant kinematical variables and separately for positive and negative hadrons. The final results, accepted for publication in European Journal C, are compared with the results produced by other experiments and with the calculations of theoretical models of the TMD functions. The measurement of the azimuthal asymmetries which appears in the spin independent part of the SIDIS cross section is more difficult since it cannot benefit from the periodically reversal of the target polarization. The measured hadron azimuthal distributions must be corrected for the possible effects introduced by the apparatus acceptance (geometrical acceptance, detectors and trigger efficiencies). The Monte Carlo simulation of the apparatus has been used to estimate those effects. Extensive studies of the description of the azimuthal acceptance of the apparatus have been done in the whole kinematical domain of interest using different SIDIS events generations. A reliable asymmetries extraction method has been developed and used. A considerable effort has been put into the study and the control of the systematics. Also in this case the asymmetries have been extracted as functions of the relevant kinematical variables and separately for positive and negative hadrons. \\ The strongest measured signals are related to the Cahn effect, which is a kinematical effect proportional to the intrinsic transverse momentum of the quarks, and to the Boer-Mulders TMD PDF, describing the correlation between the intrinsic transverse momentum of the quark and its transverse spin. \\ The results, in particular the kinematical and the charge dependencies of the asymmetries, are interpreted in terms of TMD quantities and higher effects in the non perturbative QCD. The comparisons with the results produced by the other experiments, and with the theoretical predictions are also discussed.
XXIII Ciclo
1977

The hard scattering formalism is introduced, starting from a physical picture based on the idea of equivalent quanta borrowed from QED, and the notion of characteristic times. Contact to the standard QCD treatment is made after discussing the running coupling and the Altarelli–Parisi equations for the evolution of parton distribution functions, both for QED and QCD. This allows a development of a space-time picture for hard interactions in hadron collisions, integrating hard production cross sections, initial and final state radiation, hadronization, and multiple parton scattering. The production of a W boson at leading and next-to leading order in QCD is used to exemplify characteristic features of fixed-order perturbation theory, and the results are used for some first phenomenological considerations. After that, the analytic resummation of the W boson transverse momentum is introduced, giving rise to the notion of a Sudakov form factor. The probabilistic interpretation of the Sudakov form factor is used to discuss patterns in jet production in electron-positron annihilation.