DGLAP Equations in QCD

• DGLAP equations are a set of integro-differential equations that define the scale dependence of parton distribution and fragmentation functions in Quantum Chromodynamics.
• They use perturbatively derived splitting functions to model the probabilities of parton emissions, essential for extracting PDFs from experimental data in deep inelastic scattering.
• Analytic decoupling via Laplace transforms and systematic higher-order corrections in αs validate their role in accurate theoretical predictions for hadronic collider observables.

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) =