Optimization of factorization scale in QED Drell-Yan-like processes

TL;DR

This work analyzes how the factorization scale affects initial-state radiation corrections in a QED Drell–Yan–like process for $e^+e^-$ annihilation, employing the QED structure-function framework to organize corrections by the large logarithm $L$ and comparing LO, NLO, and known two-loop results. It systematically evaluates several scale-setting prescriptions (CSS, FAC, BLM/PMC, PMS) and identifies how scale shifts redistribute terms in the perturbative expansion, with explicit transformation relations for the coefficients when the scale is changed. The study finds that, for differential distributions, a scale near $\\sqrt{s/e}$ best concentrates corrections into higher powers of $L$, while for total cross sections a scale near $\\sqrt{sz}$ better aligns NNLO contributions with NLO; PMS suggests $\\sqrt{s}$ as a robust choice, and standard scale variation remains reasonable except in radiative-return-to-resonance regimes. These results guide improved higher-order QED corrections for current and future $e^+e^-$ colliders and may offer insights applicable to scale choices in QCD Drell–Yan computations.

Abstract

The dependence of corrections due to the initial state radiation in $e^+ e^-$-annihilation processes on the choice of the factorization scale is investigated. Different prescriptions of the factorization scale choice are analyzed within the leading and next-to-leading logarithmic approximations. Comparisons with the known complete two-loop results are used to optimize the scale choice.

Figures (7)

• Figure 1: $\mathcal{O}(\alpha^2L^1)$ contributions for different factorization scales
• Figure 2: $\mathcal{O}(\alpha^2L^0)$ contributions for different factorization scales
• Figure 3: Corrections and differences for factorization scales $\sqrt{s}$, $\sqrt{s/e}$ (left), and $\sqrt{s z}$ (right), at $\sqrt{s}=240$, $\mathcal{O} (\alpha^1)$, in $\%$
• Figure 4: NLO corrections and differences for factorization scales $\sqrt{s}$, $\sqrt{s/e}$ (left), and $\sqrt{s z}$ (right) at $\sqrt{s}=240$, $\mathcal{O}(\alpha^2)$, in $\%$
• Figure 5: NNLO corrections and difference for factorization scales $\sqrt{s}$, $\sqrt{s/e}$ (left), and $\sqrt{s z}$ (right) at $\sqrt{s}=240$, $\mathcal{O} (\alpha^2)$ , in $\%$