---
title: DGLAP Equations in QCD
url: https://www.emergentmind.com/topics/dglap-equations
type: topic
---

# DGLAP Equations in QCD

The DGLAP equations—named after Dokshitzer, Gribov, Lipatov, Altarelli, and Parisi—formulate the renormalization group evolution of parton distribution functions (PDFs) and fragmentation functions according to perturbative Quantum Chromodynamics (QCD). These coupled integro-differential equations govern the scale dependence of PDFs and fragmentation functions as a function of the energy scale $Q^2$, providing the theoretical underpinning for scaling violations observed in deep inelastic scattering (DIS) and related processes. The DGLAP equations are central to the extraction of PDFs from experimental data, the global fitting programs, and the theoretical predictions for hadronic collider observables.

## 1. Mathematical Structure of the DGLAP Equations

At leading order (LO), the singlet DGLAP system for the quark singlet combination $F_s(x,Q^2)$ and the gluon distribution $G(x,Q^2)$ is given by a pair of coupled convolution equations:
\[
\begin{aligned}
\frac{\partial F_s(x,Q^2)}{\partial \ln Q^2} &= \frac{\alpha_s(Q^2)}{2\pi} \left[ P_{qq} \otimes F_s + P_{qg} \otimes G \right](x,Q^2) \\
\frac{\partial G(x,Q^2)}{\partial \ln Q^2} &= \frac{\alpha_s(Q^2)}{2\pi} \left[ P_{gq} \otimes F_s + P_{gg} \otimes G \right](x,Q^2)
\end{aligned}
\]
where $(P \otimes f)(x) = \int_x^1 \frac{dz}{z}\, P(z) f(x/z)$ denotes the Mellin convolution. The splitting functions $P_{ij}(z)$ encode the probability for a parton $j$ to emit a parton $i$ with longitudinal momentum fraction $z$ and are constructed perturbatively. Explicit leading-order splitting functions include
\[
\begin{aligned}
P_{qq}(z) &= \frac{4}{3} \left[ \frac{1+z^2}{1-z} \right]_+,\quad
P_{qg}(z) = 2n_f [z^2 + (1-z)^2], \\
P_{gq}(z) &= \frac{4}{3} \frac{1 + (1-z)^2}{z},\quad
P_{gg}(z) = 6 \left[ \frac{z}{1-z}_+ + \frac{1-z}{z} + z(1-z) \right] + \left(11-\frac{2n_f}{3}\right) \delta(1-z)
\end{aligned}
\]
where $n_f$ is the number of active flavors.

The DGLAP formalism generalizes to non-singlet quark distributions and to time-like evolution for fragmentation functions, and is also systematically extendable to higher orders in $\alpha_s$ (NLO/NNLO).

## 2. Laplace Transform Decoupling and Analytic Solution

To achieve analytic decoupling, the equations are transformed from $x$ space into Laplace space through the change of variable $v = \ln(1/x)$ and the Laplace transform:
\[
f(s,Q^2) =

Source: https://www.emergentmind.com/topics/dglap-equations