The generalized parton distribution of the pion in a relativistic Bethe-Salpeter quark model

We present a Poincaré covariant calculation of the generalized parton distribution of the pion. Results for different values of the kinematical parameters are shown and discussed.     

Generalized parton distributions

We discuss how generalized parton distributions (GPDs) enter in a variety of hard exclusive processes such as deeply virtual Compton scattering (DVCS) and hard meson electroproduction reactions on the nucleon. We show some key observables which are sensitive to the various hadron structure aspects of the GPDs, and discuss their experimental status.Received: 30 September 2002, Published online: 22 October 2003PACS: 13.60.Fz Elastic and Compton scattering - 13.60.Le Meson production - 12.38.Bx Perturbative calculations     

Generalized parton distributions

An introductory review of generalized parton distributions (GPDs) is given.Received: 1 November 2002, Published online: 15 July 2003PACS: 13.40.Gp Electromagnetic form factors - 13.60.Fz Elastic and Compton scattering - 13.60.Hb Total and inclusive cross-sections (including deep-inelastic processes)     

Drawing insights from pion parton distributions

A symmetry-preserving continuum approach to the two valence-body bound-state problem is used to calculate the valence,glue and sea distributions within the pion;unifying them with,inter alia,electromagnetic pion elastic and transition form factors.The analysis reveals the following momentum fractions at the scale ζ2:=2GeV:〈xvalence〉=0.48(3),〈xglue〉=0.41(2),〈xsea〉=0.11(2);and despite hardening induced by the emergent phenomenon of dynamical chiral symmetry breaking,the valence-quark distribution function,q^π(x),exhibits the x≈1 behaviour predicted by quantum chromodynamics(QCD).After evolution to ζ=5.2 GeV,the prediction for q^π(x)matches that obtained using lattice-regularised QCD.This confluence should both stimulate improved analyses of existing data and aid in planning efforts to obtain new data on the pion distribution functions.     

Regularization dependence of pion generalized parton distributions

Pion generalized parton distributions are calculated within the framework of the Nambu–Jona-Lasinio model using different regularization schemes, including the proper time regularization scheme, the three-dimensional (3D) momentum cutoff scheme, the four-dimensional momentum cutoff scheme, and the Pauli-Villars regularization scheme. Furthermore, we check the theoretical constraints of pion generalized parton distributions required by the symmetries of quantum chromodynamics in different regularization schemes. The diagrams of pion parton distribution functions are plotted, in addition, we evaluate the Mellin moments of generalized parton distributions, which are related to the electromagnetic and gravitational form factors of pion. Pion generalized parton distributions are continuous but not differential at begin{document}$ x=pm ,xi $end{document}, when considering the effect of the contact contribution term, generalized parton distributions become not continuous at begin{document}$ x=pm ,xi $end{document} in all the four regularization schemes. Generalized parton distributions in impact parameter space are considered, the width distribution of u quark in the pion and the mean-squared begin{document}$ langle {boldsymbol{b}}_{bot}^2rangle_{pi}^u $end{document} are calculated. The light-front transverse-spin distributions are studied when quark polarized in the light-front-transverse begin{document}$ +,x $end{document} direction, the transverse-spin density is no longer symmetric around begin{document}$ (b_x=0,b_y=0) $end{document}, the peaks shift to begin{document}$ (b_x=0,b_y>0) $end{document}, we compare the average transverse shift begin{document}$ langle b_{bot}^yrangle_1^u $end{document} and begin{document}$ langle b_{bot}^yrangle_2^u $end{document} in different regularization schemes. The light-cone energy radius begin{document}$ r_{E,LC} $end{document} and the light-cone charge radius begin{document}$ r_{c,LC} $end{document} are also evaluated, we found that in the proper time regularization scheme the values of these quantities were the largest, in the 3D momentum cutoff scheme they were the smallest.     

Generalized parton distributions in the deuteron

We introduce generalized quark and gluon distributions in the deuteron, which can be measured in exclusive processes like deeply virtual Compton scattering and meson electroproduction. We discuss the basic properties of these distributions and point out how they probe the interplay of nucleon and parton degrees of freedom in the deuteron wave function.     

Generalized parton distributions of the photon

We present a first calculation of the generalized parton distributions of the photon (both polarized and unpolarized) using overlaps of light-front wave functions at leading order in α and zeroth order in αs; for non-zero transverse momentum transfer and zero skewness. We present the novel parton content of the photon in transverse position space.     

Revealing pion and kaon structure via generalised parton distributions

Clear windows onto emergent hadron mass(EHM)and modulations thereof by Higgs boson interactions are provided by observable measures of pion and kaon structure,many of which are accessible via generalised parton distributions(GPDs).Beginning with algebraic GPD Ans?tze,constrained entirely by hadron-scaleπand K valence-parton distribution functions(DFs),in whose forms both EHM and Higgs boson influences are manifest,numerous illustrations are provided.They include the properties of electromagnetic form factors,impact parameter space GPDs,gravitational form factors and associated pressure profiles,and the character and consequences of allorders evolution.The analyses predict that mass-squared gravitational form factors are stiffer than electromagnetic form factors;reveal that K pressure profiles are tighter than profiles,with both mesons sustaining near-core pressures at magnitudes similar to that expected at the core of neutron stars;deliver parameter-free predictions for and K valence,glue,and sea GPDs at the resolving scale l=2GeV;and predict that at this scale the fraction of meson mass-squared carried by glue and sea combined matches that lodged with the valence degrees-of-freedom,with a similar statement holding for mass-squared radii.     

Generalized parton distributions in impact parameter space

The kinematical factor in the positivity bound (36) is incorrect. The bound correctly reads Our corrected result agrees with inequality (25) in [1], taking into account the different normalization conventions here and there.Published online: 9 October 2003Erratum published online: 10 October 2003     

Generalized parton distributions with helicity flip

We show that for both quarks and gluons there are eight generalized parton distributions in the proton: four which conserve parton helicity and four which do not. We explain why time reversal invariance does not reduce this number from eight to six, as previously assumed in the literature. Received: 7 February 2001 / Published online: 23 March 2001     

Generalized parton distributions from lattice QCD

We perform a quenched lattice calculation of the first moment of twist-two generalized parton distribution functions of the proton, and assess the total quark (spin and orbital angular momentum) contribution to the spin of the proton.     

Generalized parton distributions in impact parameter space

We study generalized parton distributions in the impact parameter representation, including the case of nonzero skewness . Using Lorentz invariance, and expressing parton distributions in terms of impact parameter dependent wave functions, we investigate in which way they simultaneously describe longitudinal and transverse structure of a fast moving hadron. We compare this information with the one in elastic form factors, in ordinary and in k_T-dependent parton distributions. Received: 29 May 2002 / Published online: 9 August 2002     

Generalized parton distributions from meson leptoproduction

Generalized parton distributions (GPDs) extracted from exclusive meson leptoproduction within the handbag approach are briefly reviewed. Only the GPD E is discussed in some detail. Applications of these GPDs to virtual Compton scattering (DVCS) and to Ji’s sum rule are also presented.     

Generalized parton distributions of the photon for nonzero skewness

We present a calculation of the generalized parton distributions of the photon when there is nonzero momentum transfer both in the transverse and longitudinal directions. We consider only the contributions when the photon helicity is not flipped and calculate those at leading order in electromagnetic coupling α and zeroth order in the strong coupling αs. We keep the leading logarithmic terms as well as the quark mass terms in the vertex. By taking Fourier transforms of the GPDs with respect to the transverse and longitudinal momentum transfer, we obtain the parton distributions of the photon in position space.     

Generalized parton distributions for the nucleon in chiral perturbation theory

We complete the analysis of twist-two generalized parton distributions of the nucleon in one-loop order of heavy-baryon chiral perturbation theory. Extending our previous study of the chiral-even isosinglet sector, we give results for chiral-even isotriplet distributions and for the chiral-odd sector. We also calculate the one-loop corrections for the chiral-odd generalized parton distributions of the pion.     

Generalized parton distributions from nucleon form factor data

We present a simple empirical parameterization of the x- and t-dependence of generalized parton distributions at zero skewness, using forward parton distributions as input. A fit to experimental data for the Dirac, Pauli and axial form factors of the nucleon allows us to discuss quantitatively the interplay between longitudinal and transverse partonic degrees of freedom in the nucleon (nucleon tomography). In particular we obtain the transverse distribution of valence quarks at given momentum fraction x. We calculate various moments of the distributions, including the form factors that appear in the handbag approximation to wide-angle Compton scattering. This allows us to estimate the minimal momentum transfer required for reliable predictions in that approach to be around . We also evaluate the valence contributions to the energy-momentum form factors entering Jis sum rule.Received: 16 September 2004, Revised: 15 October 2004, Published online: 17 December 2004