Analytical solution for QCD $\otimes$ QED evolution

TL;DR

This work delivers an analytical solution to QCD$\otimes$QED evolution at mixed order $\mathcal{O}(\alpha_S\alpha)$ for both unpolarized and polarized PDFs by applying the Abelianization technique to obtain mixed splitting kernels and solving the DGLAP equations exactly in Mellin $N$-space. It introduces two complementary methods for the singlet sector: a $U$-matrix-based approach that reuses existing QCD/QED evolution codes and a Magnus expansion method that yields a closed exponential form, with both approaches showing consistent mixed-order results. The authors also compute the mixed-order Wilson coefficients for the polarized structure function $g_1$ using Abelianization of NNLO QCD results, finding that photon- and quark-channel contributions lead to small but non-negligible corrections at high $x$. Numerically, photon PDF effects reach percent levels, while $g_1$ receives corrections of order $10^{-4}$ to $10^{-3}$, underscoring the relevance of QED effects in high-precision QCD phenomenology and polarized analyses.

Abstract

We present an analytical solution for the evolution of parton distributions incorporating mixed-order QCD $\otimes$ QED corrections, addressing both polarized and unpolarized cases. Using the Altarelli-Parisi kernels extended to mixed order, we solve the DGLAP equations exactly in Mellin $N$-space and derive the associated Wilson coefficients for the polarized structure function $g_1$. Our analytical approach not only improves computational efficiency but also enhances the precision of theoretical predictions relevant for current and future phenomenological applications.

Figures (4)

• Figure 1: Relative mixed-order corrections to the couplings, as defined in Eq.(\ref{['eq:realtivecoupl']}). We set the initial scale to $Q^2_0=M^2_z$, where $M_z$ is the mass of the $Z$ boson. For the initial values of the couplings we use $a_{\mathrm{S}}(M^2_z)\simeq 0.00946$ParticleDataGroup:2014cgoDeur:2016tte and $a(M^2_z)\simeq 0.000610$Bouchendira:2010es . We perform the evolution towards lower values of $Q^2$, evolving the couplings backwards accordingly.
• Figure 2: Integral $\int_{t_0}^t a_{\mathrm{S}}a \, dt$ with $Q_0^2 = M_z^2$. The curves show the relative difference with respect to the exact numerical solution, for the approximations given by Eq.(\ref{['eq:int11-best']}) (blue) and Eq.(\ref{['eq:int113']}) (red). The left plot corresponds to the exact solution using the LO approximation for both couplings, while the right plot includes the mixed-order evolution.
• Figure 3: Relative corrections to PDFs in the polarized case, at $Q^2=1000 \,\text{GeV}^2$, for the Magnus approach (red) and the $U$-matrix approach (blue dashed line).
• Figure 4: Structure function $g_1$ at $Q^2 = 1000~\,\text{GeV}^2$ with the factorization scale set to $\mu^2_F = Q^2$ (top plot). The red dashed line includes pure QCD and QED corrections, while the blue line incorporates mixed-order corrections. The bottom plot shows the relative corrections, as defined in Eq.(\ref{['eq:relativef']}), for the full computation (blue dashed line) and for the computation with the mixed-order Wilson coefficients subtracted (red solid line).