# Generalised Parton Distributions in Continuum Schwinger Methods: Progresses, Opportunities and Challenges

## Abstract

**:**

## 1. Introduction

**Figure 1.**Family of distributions encoding hadron structure, where x is the momentum fraction along the lightcone carried by the active parton, ${k}_{\perp}$ its transverse momentum (the Fourier conjugate of ${z}_{\perp}$). ${\Delta}_{\perp}$ is the transverse momentum transferred between the initial and final hadron state. At this stage, the integrals over ${k}_{\perp}$ should be understood as formal only, and need to be regularised, both from TMDs to PDFs [8,21,22] and from GTMDs to GPDs [23]. This picture is valid for vanishing skewness $\xi $, i.e., no momentum transfer along the lightcone is allowed.

## 2. Generalised Parton Distributions

#### 2.1. Formal Definitions and First Properties

**Figure 2.**The GPD definition domain in x and $\xi $. The so-called DGLAP (or outer) region for which $\left|x\right|\ge \left|\xi \right|$ is shaded in pink, while the ERBL (or inner) region for which $\left|x\right|\le \left|\xi \right|$ is shaded in green. The ERBL region can be extended for $\left|\xi \right|\ge 1$ (lighter green) where GPDs can be related to Generalised Distribution Amplitude [45,46,47,48] through analytic continuations thanks to the crossing symmetry.

**Figure 3.**The DGLAP or ERBL regions and their different interpretations in terms of partons. From left to right, one goes through the antiquark interpretation of the probed parton. Then, in the inner region $-\xi <x<\xi $, one recovers the ERBL region and its interpretation as the “extraction” of a quark–antiquark pair from the hadron. Finally, on the right-hand side, $\xi <x<1$ and we recover the quark interpretation of the DGLAP region.

#### 2.2. Reduction to Unidimensional Distributions

#### 2.3. Interpretation in Coordinate Space

- Collinear factorisation allows one to interpret exclusive processes in terms of GPDs for values of t much smaller than the typical hard scale of the system;
- Yet, performing the Fourier transform requires to integrate over t up to infinity, introducing model-dependent extrapolations;
- Furthermore, no experimental data is available for vanishing values of $\xi $, meaning that additional extrapolations generating more model biases are required.

**Figure 4.**3D picture of a model computation of the pion quark GPD in the impact parameter space. Figure from [60].

#### 2.4. Connection with the Energy-Momentum Tensor

#### 2.5. Double Distribution Representation

#### 2.5.1. Local Operators Analysis

#### 2.5.2. The Radon Transform and the Specific Role of the D-Term

#### 2.6. Positivity and Lightfront Wave Function Picture

#### 2.6.1. The Lightfront Wave Function Picture

#### 2.6.2. The Positivity Property

#### 2.7. Scale Dependence and Evolution

#### 2.7.1. Discussion in Momentum Space

#### 2.7.2. Properties of the Momentum-Dependent Anomalous Dimensions

#### 2.7.3. Evolution in Conformal Space

## 3. Continuum Results for Mesons

#### 3.1. Impulse Approximation and Its Limitations

- The computation (or modelling) of non-perturbative QCD correlation functions such as the Bethe–Salpeter wave function, the quark propagator and the local operator;
- The validity of the impulse approximation.

#### 3.2. From Bethe–Salpeter Wave Funtions to Lightfront Wave Functions

#### 3.3. The Covariant Extension

#### 3.4. The Sullivan Process

#### 3.4.1. Introduction to the Sullivan Process

#### 3.4.2. From the Sullivan Process to GPDs

#### 3.4.3. A Smoking Gun for Gluons at the EIC and EicC

**Figure 14.**Upper band: number of events assessed as a function of $\varphi $ for different bin in ${Q}^{2}$. The blue crosses correspond to the Bethe–Heitler signal only, the black square—to the theoretically complete model of ref. [58] with a CSM-based forward limit. The open square correspond to the same model but with gluons set to zero in the CFF. The red circles correspond to a phenomenological model with the GRS PDF [155] as an input for the forward limit. The grey band corresponds to the uncertainty of the CSM model associated with the choice of the initial evolution scale. Lower band: Associated beam spin asymmetries. The sign flip is visible with the model from ref. [58]. Figure from ref. [148].

#### 3.5. Challenges

#### 3.5.1. The Wilson Line

#### 3.5.2. Non-Perturbative Renormalisation

#### 3.5.3. The D-Term

## 4. From Mesons to Baryons

#### 4.1. Nucleon LFWFs

#### 4.2. Nucleon GPDs

#### 4.3. Transition GPDs

## 5. Conclusions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

DDs | Double Distributions |

DGLAP | Dokshitzer–Gribov–Lipatov–Altarelli–Parisi |

DVCS | Deep Virtual Compton Scattering |

EFFs | Electromagnetic Form Factors |

EIC | Electron Ion Collider |

EicC | Electron Ion collider in China |

EMT | Energy Momentum Tensor |

ERBL | Efremov–Radyushkin–Brodsky–Lepage |

GPDs | Generalised Parton Distribution |

GTMDs | Generalised Transverse Momentum dependent Distributions |

JLab | Jefferson Laboratory |

LFWFs | Lightfront Wave Functions |

NJL | Nambu–Jona-Lasinio |

PDFs | Parton Distribution Functions |

QCD | Quantum Chromodynamics |

RGE | Renormalisation Group Equation |

TMDs | Transverse Momentum dependent Distributions |

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**Figure 5.**LFWFs decomposition of GPDs.

**Left**panel, DGLAP region interpretation conserving parton number between incoming and outgoing states;

**right**panel, ERBL region interpretation not conserving parton number as the incoming state emits a pair of quark–antiquark.

**Figure 6.**Feynman diagrams used to compute the one-loop anomalous dimension in the lightcone gauge. On the first and second lines, we display non-mixing terms, while the third line display diagrams mixing quark and gluon GPDs. On top of these connected diagrams, disconnected self-energy diagrams need to be added. In covariant gauges, additional contributions come from gluon exchanges with the Wilson line.

**Figure 8.**Decomposition of contributions to the local operators when the Bethe–Salpeter amplitude is momentum-dependent. The red circles correspond the standard Bethe–Salpeter amplitudes, while the squares include modifications allowing the insertion of the local operators.

**Figure 9.**Results for the forward limit after reconstruction of the x-dependence in the case of a simple, algebraic model (see [42,113,114]). The total PDF (solid black) is symmetric under $x\to 1-x$ transformation, but the sole triangle diagram contribution (solid red) is not. The additional terms coming from the square vertices in Equation (70) provide the adequate correction (solid blue).

**Figure 10.**

**Left**: Support of the DD and example of a DGLAP line (red) and ERBL line (green).

**Right**: Support of the DD F after tessellation using a Delaunay mesh. An example of a DGLAP line is given. If numerous enough, those lines can probe every cell of the DD support. Figure from [58].

**Figure 11.**

**Left**: results of the reconstuction of the DD of Equation (79). Black, exact GPD computed from the algebraic DD (79). Blue, reconstructed GPDs with a sample of ${H}_{i}$ roughly four times the number of cells. Orange, same but with a sample of 12 times the number of cells.

**Right**: zoom on the central region to highlight the uncertainty bands.

**Figure 12.**

**Left**: Sullivan DVCS $ep\to en\gamma {\pi}^{+}$. At small t, one expects the pion pole contribution to be the leading one in such a process.

**Right**: Sullivan Bethe–Heitler, interfering with the DVCS.

**Figure 13.**Real (

**left**) and imaginary (

**right**) parts of DVCS Compton Form Factors computed with theoretically complete model developed in ref. [58] (brown curves). Dotted lines correspond to LO computation, dashed lines—to NLO without gluon contributions and the solid lines to the full NLO computation. The light blue curves are built from standard phenomenological Anzätze in the GPD framework [55], and a phenomenological forward limit [58,154] for details).

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Generalised Parton Distributions in Continuum Schwinger Methods: Progresses, Opportunities and Challenges. *Particles* **2023**, *6*, 262-296.
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Generalised Parton Distributions in Continuum Schwinger Methods: Progresses, Opportunities and Challenges. *Particles*. 2023; 6(1):262-296.
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2023. "Generalised Parton Distributions in Continuum Schwinger Methods: Progresses, Opportunities and Challenges" *Particles* 6, no. 1: 262-296.
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