Generalized Parton Distributions Models Research Articles

Inside the Nucleon: Tomographic Interpretations and Universality of GPDs with DDVCS

The goal of Double Deeply Virtual Compton Scattering (DDVCS) experiments is to better understand the internal structure of the nucleon. Previous attempts to resolve the internal structure of nucleons have resulted in electromagnetic form factors and parton distribution functions for elastic scattering and deep inelastic scattering processes, respectively. Generalized Parton Distributions (GPDs) are the latest attempt to unify these models of nucleon structure. The GPDs of DDVCS give us ability to investigate off of the diagonal where x ̸= ±ξ. The main goal of our analysis is to determine the best experimental setup in order to deduce the kinematic variables on which GPDs depend from the lab observables. The effectiveness of our data collection in the laboratory is by determined the physical kinematics, Q2, Q′2,t, xi,ϕLM, ϕCMV , and θCMV . We can then run DDVCS experiments and collect data on observables to improve upon the current models for GPDs of the nucleon.

Deeply Virtual Compton Scattering at Next-to-Next-to-Leading Order.

Deeply virtual Compton scattering gives access to the generalized parton distributions that encode the information on the transverse position of quarks and gluons in the proton with dependence on their longitudinal momentum. In anticipation of the high-precision experimental data in a broad kinematic range from the Electron-Ion Collider, we have calculated the two-loop, next-to-next-to-leading order (NNLO) deeply virtual Compton scattering coefficient functions associated with the dominant Compton form factors H and E at large energies. The NNLO correction to the imaginary part of H appears to be rather large, up to factor 2 at the input scale Q^{2}=4 GeV^{2} for simple generalized parton distribution models, due to a cancellation between quark and gluon contributions.

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.

Study of exclusive processes within the GPD approach

In this report, we analyze the exclusive meson production in leptonhadron and hadron processes to study Generalized Parton Distributions (GPDs). It is shown that in both reactions, the photoproduction mechanism of vector meson (VM) production predominates. We estimate the VM production cross section and spin observables. The possibility to study the exclusive VM production at NICA was analyzed. One can investigate different observables in ρ, ω and J/ψ productions to study quark and gluon GPDs. The other important process is the exclusive Drell-Yan (D-Y) production in the proton-proton reaction determined by two GPDs contributions. This reaction gives an important test of the GPD model and hopefully can be studied at NICA energies.

Deeply virtual meson production on neutrons

In this paper we analyze in detail how the measurements of exclusive electroproduction of mesons on neutrons would complement the studies of generalized parton distributions (GPDs) of the proton, providing independent experimental observables. Some of these processes on neutrons have very distinctive features, and thus we expect that measurements on liquid deuterium would allow to clearly distinguish them from similar processes on protons, giving a very clean probe of the GPD. In the case of charged meson production, all produced hadrons are charged, and for this reason we expect that the kinematics of this process could be easily reconstructed. We estimate the cross-sections in the kinematics of the Jefferson Laboratory experiments using current phenomenological GPD models.

A Nakanishi-based model illustrating the covariant extension of the pion GPD overlap representation and its ambiguities

A systematic approach for the model building of Generalized Parton Distributions (GPDs), based on their overlap representation within the DGLAP kinematic region and a further covariant extension to the ERBL one, is applied to the valence-quark pion's case, using light-front wave functions inspired by the Nakanishi representation of the pion Bethe–Salpeter amplitudes (BSA). This simple but fruitful pion GPD model illustrates the general model building technique and, in addition, allows for the ambiguities related to the covariant extension, grounded on the Double Distribution (DD) representation, to be constrained by requiring a soft-pion theorem to be properly observed.

GPD phenomenology and DVCS fitting

We review the phenomenological framework for accessing Generalized Parton Distributions (GPDs) using measurements of Deeply Virtual Compton Scattering (DVCS) from a proton target. We describe various GPD models and fitting procedures, emphasizing specific challenges posed both by the internal structure and properties of the GPD functions and by their relation to observables. Bearing in mind forthcoming data of unprecedented accuracy, we give a set of recommendations to better define the pathway for a precise extraction of GPDs from experiment.

Generalized Parton Distributions for Nucleon in the Impact Parameter Space

We investigate the generalized parton distributions (GPDs) for the proton and neutron in both momentum and position space in the soft-wall approach of anti-de Sitter/QCD model. The soft-wall model is based upon an action which incorporates confinement through the presence of a background dilaton field and the contribution of higher Fock states are holographically incorporated via studying the dynamics of 5D fermion fields of different scaling dimension. Our results are comparable with the other phenomenological models of the nucleon GPDs.

On the universality of theJ= 0 fixed pole contribution in DVCS

S. Brodsky, F.J. Llanes-Estrada and A. Szczepaniak formulated the J = 0 fixed pole universality hypothesis for (deeply) virtual Compton scattering. We show that in the Bjorken limit this hypothesis is equivalent to the validity of the inverse moment sum rule for the D-term form factor. However, any supplementary D-term added to a generalized parton distribution (GPD) results in an additional J = 0 fixed pole contribution that violates universality. Unfortunately, one can not provide any reliable theoretical argument excluding the existence of such supplementary D-term. Moreover, the violation of J = 0 fixed pole universality was revealed in field theoretical GPD models. Therefore, J = 0 fixed pole universality hypothesis remains just an external assumption and probably will never be proven theoretically.

On the equivalence of GPD representations

Phenomenological representations of generalized parton distributions (GPDs) implementing the non-trivial field theoretical requirements are employed in the present day strategies for extracting of hadron structure information encoded in GPDs from the observables of hard exclusive reactions. Showing out the equivalence of various GPD representations can help to get more insight into GPD properties and allow to build up flexible GPD models capable of satisfactory description of the whole set of available experimental data. We review the mathematical aspects of establishing equivalence between the the double partial wave expansion of GPDs in the conformal partial waves and in the $t$-channel ${\rm SO}(3)$ partial waves and the double distribution representation of GPDs.

Measurement of the Target-Normal Single-Spin Asymmetry in Quasielastic Scattering from the ReactionHe3↑(e,e′ )

We report the first measurement of the target single-spin asymmetry, $A_y$, in quasi-elastic scattering from the inclusive reaction $^3$He$^{\uparrow}(e,e^\prime)$ on a $^3$He gas target polarized normal to the lepton scattering plane. Assuming time-reversal invariance, this asymmetry is strictly zero for one-photon exchange. A non-zero $A_y$ can arise from the interference between the one- and two-photon exchange processes which is sensitive to the details of the sub-structure of the nucleon. An experiment recently completed at Jefferson Lab yielded asymmetries with high statistical precision at $Q^{2}=$ 0.13, 0.46 and 0.97 GeV$^{2}$. These measurements demonstrate, for the first time, that the $^3$He asymmetry is clearly non-zero and negative with a statistical significance of (8-10)$\sigma$. Using measured proton-to-$^{3}$He cross-section ratios and the effective polarization approximation, neutron asymmetries of $-$(1-3)% were obtained. The neutron asymmetry at high $Q^2$ is related to moments of the Generalized Parton Distributions (GPDs). Our measured neutron asymmetry at $Q^2=0.97$ GeV$^2$ agrees well with a prediction based on two-photon exchange using a GPD model and thus provides a new, independent constraint on these distributions.

Dual parametrization of generalized parton distributions in two equivalent representations

The dual parametrization and the Mellin-Barnes integral approach represent two frameworks for handling the double partial wave expansion of generalized parton distributions (GPDs) in the conformal partial waves and in the $t$-channel ${\rm SO}(3)$ partial waves. Within the dual parametrization framework, GPDs are represented as integral convolutions of forward-like functions whose Mellin moments generate the conformal moments of GPDs. The Mellin-Barnes integral approach is based on the analytic continuation of the GPD conformal moments to the complex values of the conformal spin. GPDs are then represented as the Mellin-Barnes-type integrals in the complex conformal spin plane. In this paper we explicitly show the equivalence of these two independently developed GPD representations. Furthermore, we clarify the notions of the $J=0$ fixed pole and the $D$-form factor. We also provide some insight into GPD modeling and map the phenomenologically successful Kumeri\v{c}ki-M\"uller GPD model to the dual parametrization framework by presenting the set of the corresponding forward-like functions. We also build up the reparametrization procedure allowing to recast the double distribution representation of GPDs in the Mellin-Barnes integral framework and present the explicit formula for mapping double distributions into the space of double partial wave amplitudes with complex conformal spin.

Higher-twist contributions to neutrino production of pions

In this paper, we estimate the size of twist-3 corrections to the deeply virtual meson production in neutrino interactions due to the chiral odd transversity generalized parton distribution (GPD). We conclude that in contrast to pion electroproduction, in neutrino-induced reactions, these corrections are small. This happens due to the large contribution of unpolarized GPDs $H,E$ to the leading-twist amplitude in neutrino production. We provide a computational code, which can be used for evaluation of the cross sections accounting for these twist-3 corrections with various GPD models. Our results are particularly relevant for analyses of the pion and kaon production in the MINERvA experiment at Fermilab.

Test of two new parametrizations of the generalized parton distributionH

In 2011 Radyushkin outlined the practical implementation of the One-Component Double Distribution formalism in realistic Generalized Parton Distribution models. We compare the One-Component Double Distribution framework to the standard one and compute Deeply Virtual Compton Scattering observables for both. In particular the new implementation is more flexible, offering a greater range of variation of the real and imaginary parts of the associated Compton Form Factor while still allowing to recover results similar to the classical approach. Moreover the polynomiality property is satisfied up to the highest order. Although the comparison to experimental data may be improved, the One-Component Double Distribution modeling is thus an attractive alternative.

Topics in theory of generalized parton distributions

Several topics in the theory of generalized parton distributions (GPDs) are reviewed. First, we give a brief overview of the basics of the theory of generalized parton distributions and their relationship with simpler phenomenological functions, viz. form factors, parton densities and distribution amplitudes. Then, we discuss recent developments in building models for GPDs that are based on the formalism of double distributions (DDs). A special attention is given to a careful analysis of the singularity structure of DDs. The DD formalism is applied to construction of a model GPDs with a singular Regge behavior. Within the developed DD-based approach, we discuss the structure of GPD sum rules. It is shown that separation of DDs into the so-called “plus” part and the D-term part may be treated as a renormalization procedure for the GPD sum rules. This approach is compared with an alternative prescription based on analytic regularization.

Bethe-Heitler type radiative corrections to deeply virtual neutrino production of mesons

We study the electromagnetic Bethe-Heitler type contribution to neutrino-induced deeply virtual meson production ($\ensuremath{\nu}\mathrm{DVMP}$). Such $\mathcal{O}({\ensuremath{\alpha}}_{\mathrm{em}})$ corrections decrease with ${Q}^{2}$ in the Bjorken regime less steeply than the standard $\ensuremath{\nu}\mathrm{DVMP}$ handbag contribution. Therefore, they are relatively enhanced at high ${Q}^{2}$. The Bethe-Heitler terms give rise to an angular correlation between the lepton and hadron scattering planes with harmonics sensitive to the real and imaginary parts of the $\ensuremath{\nu}\mathrm{DVMP}$ amplitude. These corrections constitute a few percent effect in the kinematics of the forthcoming Minerva experiment at Fermilab and should be taken into account in precision tests of generalized parton distributions parametrizations. For virtualities ${Q}^{2}\ensuremath{\sim}100\text{ }\text{ }{\mathrm{GeV}}^{2}$ these corrections become on a par with $\ensuremath{\nu}\mathrm{DVMP}$ handbag contributions. A computational code, which can be used for the evaluation of these corrections employing various generalized parton distributions models, is provided in the Supplemental Material.

Flavor structure of generalized parton distributions from neutrino experiments

The analysis of deeply virtual meson production is extended to neutrino-production of the pseudo-Goldstone mesons $(\ensuremath{\pi},K,\ensuremath{\eta})$ on nucleons, with the flavor content of the recoil baryon either remaining intact, or changing to a hyperon from the $SU(3)$ octet. We rely on the $SU(3)$ relations and express all the cross sections in terms of the proton generalized parton distributions (GPDs). The corresponding amplitudes are calculated at the leading twist level and in the leading order in ${\ensuremath{\alpha}}_{s}$, using a phenomenological model of GPDs. We provide a computational code, which can be used for evaluation of the cross sections employing various GPD parametrizations. We conclude that these processes can be studied in the experiment Minerva at Fermilab, which could supplement the measurements at JLAB helping to extract the GPD flavor structure from data.

Generalized parton distributions and their singularities

A new approach to building models of generalized parton distributions (GPDs) is discussed that is based on the factorized DD (double distribution) ansatz within the single-DD formalism. The latter was not used before, because reconstructing GPDs from the forward limit one should start in this case with a very singular function $f(\ensuremath{\beta})/\ensuremath{\beta}$ rather than with the usual parton density $f(\ensuremath{\beta})$. This results in a nonintegrable singularity at $\ensuremath{\beta}=0$ exaggerated by the fact that $f(\ensuremath{\beta})$'s, on their own, have a singular ${\ensuremath{\beta}}^{\ensuremath{-}a}$ Regge behavior for small $\ensuremath{\beta}$. It is shown that the singularity is regulated within the GPD model of Szczepaniak et al., in which the Regge behavior is implanted through a subtracted dispersion relation for the hadron-parton scattering amplitude. It is demonstrated that using proper softening of the quark-hadron vertices in the regions of large parton virtualities results in model GPDs $H(x,\ensuremath{\xi})$ that are finite and continuous at the ``border point'' $x=\ensuremath{\xi}$. Using a simple input forward distribution, we illustrate implementation of the new approach for explicit construction of model GPDs. As a further development, a more general method of regulating the $\ensuremath{\beta}=0$ singularities is proposed that is based on the separation of the initial single DD $f(\ensuremath{\beta},\ensuremath{\alpha})$ into the ``plus'' part $[f(\ensuremath{\beta},\ensuremath{\alpha}){]}_{+}$ and the $D$ term. It is demonstrated that the ``DD+D'' separation method allows one to (re)derive GPD sum rules that relate the difference between the forward distribution $f(x)=H(x,0)$ and the border function $H(x,x)$ with the $D$-term function $D(\ensuremath{\alpha})$.

DEEPLY VIRTUAL COMPTON SCATTERING AT SMALL xBj

We analyze small-x deeply virtual Compton scattering data using flexible generalized parton distribution models to pin down both the skewness and t dependence. We compare our outcome at t = 0 with the full Shuvaev transformation. We point out that this integral transform is a model which is equivalent to a conformal generalized parton distribution and a minimalist "dual" parameterization. Some mathematical subtleties of conformal representations are recalled.

Novel two-to-three hard hadronic processes and possible studies of generalized parton distributions at hadron facilities

We consider a novel class of hard branching hadronic processes $a+b\ensuremath{\rightarrow}c+d+e$, where hadrons $c$ and $d$ have large and nearly opposite transverse momenta and large invariant energy, which is a finite fraction of the total invariant energy. We use color transparency logic to argue that these processes can be used to study quark generalized parton distributions (GPDs) for baryons and mesons in hadron collisions, hence complementing and adding to the studies of GPDs in the exclusive deep inelastic scattering processes. We propose that a number of GPDs can be investigated in hadron facilities such as Japan Proton Accelerator Research Complex facility and Gesellschaft f\"ur Schwerionenforschung -Facility for Antiproton and Ion Research project. In this work, the GPDs for the nucleon and for the $N\ensuremath{\rightarrow}\ensuremath{\Delta}$ transition are studied in the reaction $N+N\ensuremath{\rightarrow}N+\ensuremath{\pi}+B$, where $N$, $\ensuremath{\pi}$, and $B$ are a nucleon, a pion, and a baryon (nucleon or $\ensuremath{\Delta}$), respectively, with a large momentum transfer between $B$ (or $\ensuremath{\pi}$) and the incident nucleon. In particular, the Efremov-Radyushkin-Brodsky-Lepage region of the GPDs can be measured in such exclusive reactions. We estimate the cross section of the processes $N+N\ensuremath{\rightarrow}N+\ensuremath{\pi}+B$ by using current models for relevant GPDs and information about large angle $\ensuremath{\pi}N$ reactions. We find that it will be feasible to measure these cross sections at the high-energy hadron facilities and to get novel information about the nucleon structure, for example, contributions of quark orbital angular momenta to the nucleon spin. The studies of $N\ensuremath{\rightarrow}\ensuremath{\Delta}$ transition GPDs could be valuable also for investigating electromagnetic properties of the transition.