Three-Loop Polarization Function and ${\cal O}(α_s^2)$ Corrections to the Production of Heavy Quarks

TL;DR

This work computes the three-loop vacuum polarization function for a massive quark and extracts the ${\cal O}(\alpha_s^2)$ corrections to heavy-quark production in $e^+e^-$ annihilation, $R(s)$. The authors recast the problem using analyticity, dispersion relations, and a combination of high-energy and small-$q^2$ expansions, supplemented by seven Taylor coefficients and threshold information. They decompose the result into color structures and employ optimized Padé approximants with conformal mapping to reconstruct the full $\Pi(q^2)$ across energy regimes, achieving predictions with about 5% uncertainty for the ${\cal O}(\alpha_s^2)$ terms. The study provides practical formulas and numerical tools for accurate heavy-quark cross sections, and discusses scheme transitions, scale-setting, and potential for resummation in future work.

Abstract

The three-loop vacuum polarization function $Π(q^2)$ induced by a massive quark is calculated. A comprehensive description of the method is presented. From the imaginary part the ${\cal O}(α_s^2)$ result for the production of heavy quarks $R(s)=σ(e^+e^-\to \mbox{hadrons})/σ(e^+e^-\to μ^+μ^-)$ can be extracted. Explicit formulae separated into the different colour factors are given.

Figures (9)

• Figure 1: One- and two-loop diagrams contributing to $\Pi^{(0)}$ and $\Pi^{(1)}$, respectively.
• Figure 2: Three-loop diagrams contributing to $\Pi^{(2)}_l$ (inner quark massless) and $\Pi_F^{(2)}$ (both quarks have the same mass $m$).
• Figure 3: Purely gluonic contribution to ${\cal O}(\alpha_s^2)$. Diagrams with ghost loops are not depicted.
• Figure 4: Transformation between the $z$ and $\omega$ plane
• Figure 5: Complete results plotted against $v=\sqrt{1-4m^2/s}$. The high energy approximation includes the $m^4/s^2$ term.