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 of the pion in a Bethe-Salpeter approach

We calculate generalized parton distribution functions in a field theoretic formalism using a covariant Bethe-Salpeter approach for the determination of the bound-state wave function. We describe the procedure in an exact calculation in scalar Electrodynamics proving that the relevant corrections outside our scheme vanish. We extend the formalism to the Nambu-Jona-Lasinio model, a realistic theory of the pion. We go in both cases beyond all previous calculations and discover that all important features required by general physical considerations, like symmetry properties, sum rules and the polynomiality condition, are explicitly verified. We perform a numerical study of their behavior in the weak- and strong-coupling limits.Received: 12 May 2003, Revised: 25 November 2003, Published online: 8 June 2004PACS: 24.10.Jv Relativistic models - 11.10.St Bound and unstable states; Bethe-Salpeter equations - 13.40.Gp Electromagnetic form factors - 13.60.Fz Elastic and Compton scatteringL. Theußl: Present address: TRIUMF, Vancouver, B.C. V6T 2A3, Canada;     

Improved calculations of quark distributions in hadrons: the case of the pion

The earlier introduced method of calculation of quark distributions in hadrons, based on QCD sum rules, is improved. The imaginary part of the virtual photon forward scattering amplitude on some hadronic current is considered in the case, when initial and final virtualities of the current , and are different, . The operator product expansion (OPE) in , is performed. The sum rule for quark distribution is obtained using double dispersion representation of the amplitude on one side in terms of calculated in QCD OPE and on the other side in terms of physical states contributions. Double Borel transformation in , is applied to the sum rule, killing background non-diagonal transition terms which deteriorated the accuracy in previous calculations. The case of the valence quark distribution in the pion is considered, which was impossible to treat by the previous method. OPE up to dimension 6 operators is performed and leading order perturbative corrections are accounted. Valence u-quark distribution in was found at intermediate x, , and normalization point . These results may be used as input for evolution equations. Received: 14 July 1999 / Published online: 6 March 2000     

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.     

Soft-pion limits of the inclusive pion distributions in NN → π± + anything at rest

The PCAC hypothesis of extrapolating off mass shell is used to show that the invariant pion distributions in the reactions $\text{N}\text{N}\text{→ π}{}^{\mathrm{\pm }}\text{+}$ anything at rest must vanish as the pion momentum tends to zero, a behavior which is supported by recent data on $\text{p}\text{n}\text{→ π}{}^{\mathrm{\pm }}\text{+}$ anything.     

Theoretical spectral distributions and total cross sections for neutral subthreshold pion production in carbon-carbon collisions

With use of a recently developed particle-hole theory for pion production from coherent isobar formation and decay, calculations of total cross sections and kinetic-energy distributions for neutral subthreshold pions produced in carbon-carbon collisions at incident energies below 100 MeV/nucleon are made and compared with recent experimental data.     

Compton scattering on the pion and radiative pion photoproduction from the proton

The contribution of the pion polarizabilities to radiative pion photoproduction has been investigated. It has been shown how an extrapolation of future experimental data on radiative pion photoproduction to the pion pole can give information on the polarizabilities of pion.     

Probing the pion field in nuclei

It is suggested that muon absorption and doubly radiative pion absorption in nuclei can complement each other in probing the pion field in the nuclear medium. The relevant issues we consider are possible pion condensation and nuclear “renormalization” of weak and electromagnetic vertices, the adequate understanding of which is necessary to extract from nuclear experiments useful information on basic form factors in the weak Hamiltonian.     

The pion mass difference in threshold pion reactions on light nuclei

We investigate the effect of the pion mass difference in scattering and photoproduction of pions on light nuclei near threshold. The effect is found to be most easily described in a nuclear K-matrix formalism where the nuclear K-matrix elements are shown to be independent of the mass difference.     

Analysis of pion electroproduction data in terms of a pion distribution function of the proton

Experimental data on the longitudinal π⁺ electroproduction on hydrogen have been analyzed in terms of a pion distribution function of the proton. This interpretation is based on the assumption that the pion can be considered as a parton in the nucleon for low momentum transfer Q². For the kinematical conditions photon mass Q² < 1 GeV² and electron energy loss ν > 2.2 GeV the pion electroproduction in the direction of the virtual photon can be viewed upon as quasielastic eπ⁺ → e′π⁺ scattering. Using the impulse approximation the pion distribution function of the proton can be deduced. According to the distribution function evaluated from pion electroproduction in the infinite momentum frame, the physical proton has a 3% admixture of the nπ⁺ state and the π⁺ in the nπ⁺ state carries 0.6% of the proton momentum.     

Spin structure of the pion

We present the first calculation of the transverse spin structure of the pion in lattice QCD. Our simulations are based on two flavors of nonperturbatively improved Wilson fermions, with pion masses as low as 400 MeV in volumes up to (2.1 fm)(3) and lattice spacings below 0.1 fm. We find a characteristic asymmetry in the spatial distribution of transversely polarized quarks. This asymmetry is very similar in magnitude to the analogous asymmetry we previously obtained for quarks in the nucleon. Our results support the hypothesis that all Boer-Mulders functions are alike.     

Inclusive pion production by pions on nuclei and determination of root-mean-square radii of proton and neutron distributions

A brief nonanalytic survey of data available in the literature for parameters that characterize nuclear sizes is given, and the potentials of some methods for determining these sizes are compared. Attention is given primarily to determining nuclear sizes in experiments that study pion-nucleus interaction. For pointlike protons and neutrons, values of parameters that describe the radial dependences of their densities according to the Fermi distribution are presented in a tabular form. The method proposed previously at the Institute of Theoretical and Experimental Physics (ITEP, Moscow) for determining nuclear sizes on the basis of measuring differential cross sections for π^? + A → π _forw ^± + X processes in the kinematical region where the recorded interaction event occurs on a single proton (neutron) of nucleus A, the presence of the remaining nucleons manifesting itself in shadowing effects, is described, and examples of the application of this method are given.     

Inclusive pion distributions in e+e− annihilation from sum rules and duality

The threshold behaviour of pion production presented in our earlier work is successfully compared with the new SPEAR data. By using duality and sum rules we derive F_T^(π+)(x) ≈ F_L^(π+)(x) ≈ F_T^(π0)(x) ? F_L^(π0)(x) for x near 1. An accompanying results is $\text{σ}{}_{\mathrm{<mtext>\pi </mtext><mtext>A</mtext>}}{}_2\text{(s) ≈ 2σ}{}_{\mathrm{\pi \omega }}\text{(s) ≈ 4σ}{}_{\mathrm{\pi \pi }}\text{(s) ≈ 9(m}{}_{\mathrm{?}}{}^2\text{/s)}{}^3\text{σ}{}_{\mathrm{<mtext>\mu </mtext><mtext>\mu </mtext>}}$ for large s.