Analytical solution for QCD $\otimes$ QED evolution

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.