Generalized Parton Distributions Of Pion Research Articles

Empirical Determination of the Pion Mass Distribution

Existing pion+nucleus Drell-Yan and electron+pion scattering data are used to develop ensembles of model-independent representations of the pion generalized parton distribution (GPD). Therewith, one arrives at a data-driven prediction for the pion mass distribution form factor, . Compared with the pion elastic electromagnetic form factor, is harder: the ratio of the radii derived from these two form factors is . Our data-driven predictions for the pion GPD, related form factors and distributions should serve as valuable constraints on theories of pion structure.

Pion generalized parton distributions: A path toward phenomenology

We introduce a new family of generalized parton distribution models able to fulfill by construction all the theoretical properties imposed by QCD. These models are built on standard parton distribution functions and extended to off-forward kinematics through a well-defined procedure. We apply this strategy on the pion, first handling a simple but insightful algebraic model, and then exploiting state-of-the-art computations obtained in continuum QCD. We compare these models with a more standard one relying on an xFitter extraction of the pion parton distribution functions. The results for both quark and gluon generalized parton distributions are presented and exploited for calculation of electromagnetic, gravitational, and Compton form factors. Results on the latter highlight the relevance of next-to-leading-order corrections, even in the so-called valence region.

Regularization dependence of pion generalized parton distributions * *Supported by the National Natural Science Foundation of China (11775118)

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 , when considering the effect of the contact contribution term, generalized parton distributions become not continuous at 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 are calculated. The light-front transverse-spin distributions are studied when quark polarized in the light-front-transverse direction, the transverse-spin density is no longer symmetric around , the peaks shift to , we compare the average transverse shift and in different regularization schemes. The light-cone energy radius and the light-cone charge radius 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.

Pion generalized parton distributions and light-front wave functions in the Nambu–Jona-Lasinio model

Pion generalized parton distributions and light-front wave functions are investigated in the framework of Nambu–Jona-Lasinio model. The properties of pion generalized parton distributions and light-front wave functions are examined separately. Proceeding from generalized parton distributions and light-front wave functions to form factors, parton distribution functions, parton distribution amplitudes and transverse momentum dependent parton distributions, giving a complete picture of pion structure. Moreover, comparing these distributions from the two different methods and examining their peculiarities, finding that the two methods give the same distributions. This means that generalized parton distributions and light-front wave functions give the same multi-dimensional mapping of pion in the Nambu–Jona-Lasinio model. In addition, we studied generalized form factors of pion: the quark pressure distribution θ1 and the quark mass distribution θ2, both are harder than electromagnetic form factor. The tensor anomalous magnetic moment B1,0 and the tensor generalized form factor B2,0 are harder than corresponding lattice data.

Pion generalized parton distribution from lattice QCD

We present the first lattice calculation of the valence-quark generalized parton distribution (GPD) of the pion using the large-momentum effective theory (LaMET) approach. We focus on the zero-skewness limit, where the GPD has a probability-density interpretation in the longitudinal Bjorken x and the transverse impact-parameter distributions. Our calculation is done using clover valence fermions on an ensemble of gauge configurations with 2+1+1 flavors (degenerate up/down, strange and charm) of highly improved staggered quarks (HISQ) with lattice spacing a≈0.12 fm, box size L≈3 fm and pion mass mπ≈310 MeV. The parton distribution function and the form factor are reproduced as special limits of the GPD as expected. Due to the large errors, this exploratory study does not show a clear preference among different model assumptions about the kinematic dependence of the GPD. To discriminate between these assumptions, future studies using higher-statistics data will be crucial.

Pion generalized parton distributions within a fully covariant constituent quark model

We extend the investigation of the Generalized Parton Distribution for a charged pion within a fully covariant constituent quark model, in two respects: (i) calculating the tensor distribution and (ii) adding the treatment of the evolution, needed for achieving a meaningful comparison with both the experimental parton distribution and the lattice evaluation of the so-called generalized form factors. Distinct features of our phenomenological covariant quark model are: (i) a 4D Ansatz for the pion Bethe-Salpeter amplitude, to be used in the Mandelstam formula for matrix elements of the relevant current operators, and (ii) only two parameters, namely a quark mass assumed to hold $m_q=~220$ MeV and a free parameter fixed through the value of the pion decay constant. The possibility of increasing the dynamical content of our covariant constituent quark model is briefly discussed in the context of the Nakanishi integral representation of the Bethe-Salpeter amplitude.

Modeling the Pion Generalized Parton Distribution

We compute the pion Generalized Parton Distribution (GPD) in a valence dressed quarks approach. We model the Mellin moments of the GPD using Ansätze for Green functions inspired by the numerical solutions of the Dyson-Schwinger Equations (DSE) and the Bethe-Salpeter Equation (BSE). Then, the GPD is reconstructed from its Mellin moment using the Double Distribution (DD) formalism. The agreement with available experimental data is very good.

Pion generalized parton distributions with covariant and light-front constituent quark models

We investigate the model dependence of no-helicity flip generalized parton distribution of the pion upon different approaches for the quark-hadron and quark-photon vertices, in the spacelike region. In order to obtain information on contributions from both the valence and the nonvalence regions, we compare results for spacelike momentum transfers obtained from (i) an analytic covariant model with a bare quark-photon vertex, (ii) a light-front approach with a quark-photon vertex dressed through a microscopic vector-meson model, and (iii) a light-front approach based on the relativistic Hamiltonian dynamics. Our comparisons lead us to infer the same dynamical mechanism, the one-gluon-exchange dominance at short distances, as a source of both the electromagnetic form factor at large momentum transfer and the parton distribution close to the end points. The expected collinear behavior of the generalized parton distributions at high-momentum transfer, i.e. a maximum for x{approx}1, is also illustrated, independently of the different approaches. Finally, a comparison with recent lattice calculations of the gravitational form factors is presented.

Large-Nsummation of chiral logs for generalized parton distributions

We demonstrate that in the region of Bjorken ${x}_{\mathrm{Bj}}\ensuremath{\sim}{m}_{\ensuremath{\pi}}^{2}/(4\ensuremath{\pi}{F}_{\ensuremath{\pi}}{)}^{2}$ and/or ${x}_{\mathrm{Bj}}\ensuremath{\sim}|t|/(4\ensuremath{\pi}{F}_{\ensuremath{\pi}}{)}^{2}$ the standard $\ensuremath{\chi}\mathrm{PT}$ for the pion generalized parton distributions (GPDs) fails and one must perform all order resummation of $\ensuremath{\chi}\mathrm{PT}$. We perform such resummation in the large-$N$ limit of the $O(N+1)$ extension of the chiral theory. Explicit resummation allows us to reveal novel phenomena---the form of the leading chiral correction to pion parton distribution functions (PDFs) and GPDs depends on the small $x$ asymptotic of the pion PDFs. In particular, if the pion PDF in the chiral limit has the Regge-like small $x$ behavior $q(x)\ensuremath{\sim}1/{x}^{\ensuremath{\omega}}$, the leading large impact parameter (${b}_{\ensuremath{\perp}}\ensuremath{\rightarrow}\ensuremath{\infty}$) asymptotics of the quark distribution in the transverse plane has the form (${m}_{\ensuremath{\pi}}=0$) $q(x,{b}_{\ensuremath{\perp}})\ensuremath{\sim}1/{x}^{\ensuremath{\omega}}{ln}^{\ensuremath{\omega}}({b}_{\ensuremath{\perp}}^{2})/{b}_{\ensuremath{\perp}}^{2(1+\ensuremath{\omega})}$. This result is model independent and it is controlled completely by the all order resummed $\ensuremath{\chi}\mathrm{PT}$ developed in this paper. This asymptotic interweaves with small-$x$ behavior of usual PDFs, hence it depends on the scale, at which the corresponding PDF is defined. This is a new and interesting result in which the chiral expansion meets the QCD evolution.

Deeply virtual Compton scattering on a virtual pion target

We study deeply virtual Compton scattering on a virtual pion that is emitted by a proton. Using a range of models for the generalized parton distributions of the pion, we evaluate the cross section, as well as the beam spin and beam charge asymmetries in the leading-twist approximation. Studying Compton scattering on the pion in suitable kinematics puts high demands on both beam energy and luminosity, and we find that the corresponding requirements will first be met after the energy upgrade at Jefferson Laboratory. As a by-product of our study, we construct a parameterization of pion generalized parton distributions that has a non-trivial interplay between the x and t dependence and is in good agreement with form factor data and lattice calculations.

Gravitational form factors in the axial sector from an AdS/QCD model

We calculate the stress tensor, or energy-momentum tensor, form factors of the pion and of axial-vector mesons in the chiral limit of a hard-wall AdS/conformal-field-theory model of QCD. One (of the two) pion gravitational form factors is directly related to the second moment of the pion generalized parton distribution, thus providing a sum rule for the latter. As was also the case for vector mesons, both the pion and the axial-vector mesons appear strikingly more compact measured by the gravitational form factor than by the electromagnetic form factor.

Generalized parton distributions of the pion in partially-quenched chiral perturbation theory

We consider the pion matrix elements of the isoscalar and isovector combinations of the vector and tensor twist-two operators that determine the moments of the various pion generalized parton distributions. Our analysis is performed using partially-quenched chiral perturbation theory. We work in the SU(2) and $\mathrm{SU}(4|2)$ theories and present our results at infinite volume and also at finite volume where some subtleties arise. These results are useful for extrapolations of lattice calculations of these matrix elements at small momentum transfer to the physical regime.

Color transparent generalized parton distributions

The relation between GPD's and color transparency is explored. The discovery of color transparency in pionic diffractive dissociation reactions allows us to make specific predictions for the behavior of the pion generalized parton distribution, and provide a further test of any model of the pion form factor.

Higher Fock-state contributions to the generalized parton distribution of pion

We discuss the higher Fock-state ($q\overline{q}g$) contributions to the nonzero value of the pion generalized parton distribution (GPD) at the crossover point $x=\ensuremath{\zeta}$ between the DGLAP and ERBL regions. Using the phenomenological light-front constituent quark model, we confirm that the higher Fock-state contributions indeed give a nonzero value of the GPD at the crossover point. Iterating the light-front quark model wave function of the lowest $q\overline{q}$ Fock state with the Bethe-Salpeter kernel corresponding to the one-gluon exchange, we include all possible time-ordered $q\overline{q}g$ Fock-state contributions and obtain the pion GPD satisfying necessary sum rules and continuity conditions.

Generalized parton distributions and double distributions forqq¯pions

We consider two simple covariant models for pions (one with scalar and the other with spin-1/2 constituents). Pion generalized parton distributions are derived by integration over the light-cone energy. The model distributions are consistent with all known properties of generalized parton distributions, including positivity. We also construct the corresponding double distributions by appealing to Lorentz invariance. These ostensibly constructed double distributions lead to different generalized parton distributions that need not respect the positivity constraints. This inconsistency arises from the ambiguity inherent in defining double distributions in a one-component formalism (even in the absence of the Polyakov-Weiss term). We demonstrate that the correct model double distributions can be calculated from non-diagonal matrix elements of twist-two operators.