Non-singlet splitting functions in QED

TL;DR

The paper addresses the evolution of non-singlet electron structure functions in QED by solving the non-singlet DGLAP equation via an iterative approach, and derives explicit fourth- and fifth-order splitting functions $P^{(n)}(z)$, separating photonic and $e^+e^-$ pair contributions with careful regularization using $\Delta$- and $\Theta$-prescriptions. It uses Mellin convolution $P^{(n+1)}(z)=\int_z^1 \frac{dx}{x} P^{(1)}(x) P^{(n)}(\frac{z}{x})$ to generate higher-order terms and analyzes the soft limit through $\mathcal{D}^{NS}_{\gamma}(z,Q^2)|_{z\to1}=\frac{\beta}{2}\frac{(1-z)^{\beta/2-1}}{\Gamma(1+\beta/2)}\exp\{\frac{\beta}{2}(\frac{3}{4}-C)\}$. A key result is that the iterative expansion reproduces the known exponentiated solution, validating both methods and providing explicit kernels up to $n=5$, with numerical checks showing negligible differences between the approaches. This work offers a framework to estimate higher-order radiative corrections in QED processes and clarifies the relationship between exponentiated and order-by-order treatments of radiative effects in electron structure functions.

Abstract

Iterative solution of QED evolution equations for non-singlet electron structure functions is considered. Analytical expressions in the fourth and fifth orders are presented in terms of splitting functions. Relation to the existing exponentiated solution is discussed.