Theory and applications of parton pseudodistributions

We review the basic theory of the parton pseudodistributions approach and its applications to lattice extractions of parton distribution functions. The crucial idea of the approach is the realization that the correlator [Formula: see text] of the parton fields is a function [Formula: see text] of Lorentz invariants [Formula: see text], the Ioffe time, and the invariant interval [Formula: see text]. This observation allows to extract the Ioffe-time distribution [Formula: see text] from Euclidean separations [Formula: see text] accessible on the lattice. Another basic feature is the use of the ratio [Formula: see text], that allows to eliminate artificial ultraviolet divergence generated by the gauge link for spacelike intervals. The remaining [Formula: see text]-dependence of the reduced Ioffe-time distribution [Formula: see text] corresponds to perturbative evolution, and can be converted into the scale-dependence of parton distributions [Formula: see text] using matching relations. The [Formula: see text]-dependence of [Formula: see text] governs the [Formula: see text]-dependence of parton densities [Formula: see text]. The perturbative evolution was successfully observed in exploratory quenched lattice calculation. The analysis of its precise data provides a framework for extraction of parton densities using the pseudodistributions approach. It was used in the recently performed calculations of the nucleon and pion valence quark distributions. We also discuss matching conditions for the pion distribution amplitude and generalized parton distributions, the lattice studies of which are now in progress.

THEORETICAL RESEARCH OF THE SCAFFOLDER AND SOIL INTERACTION

The purpose of research is to improve the dosing of mineral fertilizers sowing unit, depending on the physico-mechanical properties of the soil. In modern technologies of precision farming with differentiated application of mineral fertilizers, data from various sensors, results of analyzes of soil samples and aerial photography are mainly used, without taking into account factor such as the depth of the humus horizon, which largely determines the soil fertility. On the basis of the Samara State Agricultural Academy, studies are underway on the effect of the depth of the humus horizon on the fertility of the soil. Several machines have been developed for differentiating mineral fertilizers with spreader and cultivator. Currently, a system of differentiated application of mineral fertilizers is being developed for sowing agricultural crops and it is planned to produce a set of equipment for a standard seeder UCS-8 Vesta. A constructive scheme of the device for determining the traction resistance is presented, which is closely related to the depth of occurrence of the humus layer. The main working element of this device is a deformer in the form of a knife, with a dihedral grinding. The article presents theoretical studies of the effect of forces acting on a deformer. Based on the results of the theoretical justification of the technological process of the proposed scheme for determining the traction resistance, the load dependencies on the strain gauge link were derived depending on the drag resistance of the deformer, which is determined by its basic design and technological parameters and the physical and mechanical properties of the soil. The obtained dependences of the traction resistance of the working organs of the tillage implement will theoretically determine the additional energy costs from the introduction of the deformer in the design stage at the design stage of the machine and outline the directions for their optimization.

Azimuthal Asymmetries of the Drell-Yan Process in pA Collisions

We discuss the azimuthal asymmetries of the Drell-Yan process in nucleon-nucleus collisions at the low transverse momentum of the lepton pair. Within the transverse-momentum-dependent (TMD) factorization formalism, the nuclear effects of these azimuthal asymmetries can be from the gauge link of the TMD quark distribution. We estimate all these nuclear effects within the assumption that all the TMD parton distributions or correlations are in Gaussian forms.

Color Entanglement in Hadronic Processes for Transverse Momentum Dependent Parton Distribution Functions

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.

The Boer-Mulders Transverse Momentum Distribution in the Pion and its Evolution in Lattice QCD

Starting from a definition of transverse momentum-dependent parton distributions (TMDs) in terms of hadronic matrix elements of a quark bilocal operator containing a staple-shaped gauge link, selected TMD observables can be evaluated within Lattice QCD. A TMD ratio describing the Boer-Mulders effect in the pion is investigated, with a particular emphasis on its evolution as a function of a Collins-Soper-type parameter which quantifies the proximity of the staple-shaped gauge links to the light cone.

UNIVERSALITY OF QUARK AND GLUON TMD CORRELATORS

Transverse Momentum Dependent (TMD) parton distribution functions (PDFs), in short referred to as TMDs, also take into account the transverse momentum (pT) of the partons. Just as the pT-integrated analogues we want to link them to quark and gluon matrix elements using Operator Product Expansion methods in QCD, involving operators of definite twist. The TMDs also involve operators of higher twist, which are not suppressed by powers of the hard scale, however. Using the expression for TMDs involving nonlocal matrix elements of quark and gluon fields there is a gauge link dependence, which also introduces an inherent process dependence. Using transverse moments, which are specific pT-weightings, we can establish the link with quark and gluon fields including the higher twist ones. We introduce (a finite number of) universal TMDs of definite rank and show how the process dependent TMDs can be written as combinations of these universal functions.

FINAL STATE INTERACTIONS AND THE TRANSVERSE STRUCTURE OF PION

In the factorized picture of semi-inclusive deep inelastic scattering the naive time reversal-odd parton distributions exist by virtue of the gauge link which is intrinsic to their definition. The link structure describes initial/final-state interactions of the active parton due to soft gluon exchanges with the target remnant. Though these interactions are non-perturbative, calculations of final-state interaction have been performed in a perturbative one-gluon approximation. We include higher-order contributions by applying non-perturbative eikonal methods to calculate the Boer-Mulders function of the pion. Using this framework we explore under what conditions the Boer Mulders function can be described in terms of factorization of final state interactions and a spatial distortion.