# Analytical Solution to DGLAP Integro-Differential Equation in a Simple Toy-Model with a Fixed Gauge Coupling

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## Abstract

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## 1. Introduction

## 2. Integral Transforms

#### 2.1. Mellin Transform

#### 2.2. Laplace Transform

#### 2.3. Mellin Moments

## 3. Description of Theoretical Setup

#### 3.1. Evolution Equations

#### 3.2. Parton Distributions

#### 3.3. On the Model and DIS Processes in This Model

## 4. Dglap Equation with Vanishing $\mathbf{\beta}$-function for Integrated Gluon Distribution

## 5. Dglap Equation with Vanishing $\mathbf{\beta}$-Function for Unintegrated Gluon Distribution

## 6. Contour of the Inverse Transformation from $\mathit{N}$ to $\mathit{X}$

## 7. Method to Solve the DGLAP Equation Analytically

## 8. Solution to DGLAP Equation in a Simple Toy-Model

^{2}LO for the Mellin moments of parton distribution functions with full inclusion of running coupling, the approximate solutions to the DGLAP equation corresponding to simple models still may help a lot in order to estimate physical quantities in the limits in which numerical tools and solutions show bad behavior in practical models such as QCD.

## 9. Solution to DGLAP IDE in Almost Realistic Case

## 10. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Running Coupling Case

## Appendix B. Dglap for Unintegrated PDFs: Running Coupling Case

## Appendix C. Self-Consistent Shape Function for the Running Coupling

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**MDPI and ACS Style**

Álvarez, G.; Cvetič, G.; Kniehl, B.A.; Kondrashuk, I.; Parra-Ferrada, I.
Analytical Solution to DGLAP Integro-Differential Equation in a Simple Toy-Model with a Fixed Gauge Coupling. *Quantum Rep.* **2023**, *5*, 198-223.
https://doi.org/10.3390/quantum5010013

**AMA Style**

Álvarez G, Cvetič G, Kniehl BA, Kondrashuk I, Parra-Ferrada I.
Analytical Solution to DGLAP Integro-Differential Equation in a Simple Toy-Model with a Fixed Gauge Coupling. *Quantum Reports*. 2023; 5(1):198-223.
https://doi.org/10.3390/quantum5010013

**Chicago/Turabian Style**

Álvarez, Gustavo, Gorazd Cvetič, Bernd A. Kniehl, Igor Kondrashuk, and Ivan Parra-Ferrada.
2023. "Analytical Solution to DGLAP Integro-Differential Equation in a Simple Toy-Model with a Fixed Gauge Coupling" *Quantum Reports* 5, no. 1: 198-223.
https://doi.org/10.3390/quantum5010013