Heavy Quark Vacuum Polarisation to Three Loops

TL;DR

The paper computes the heavy-quark contribution to the vacuum-polarisation function $Π(q^2)$ up to three loops in perturbative QCD, extending the classic Källén–Sabry result from two to three loops. It decomposes $Π^{(2)}$ by colour factors, constrains the function with high-energy, threshold, and zero-momentum data, and reconstructs the full $Π^{(2)}$ using conformal mapping and Padé approximants aided by three-loop tadpole calculations. Real and imaginary parts are obtained, yielding predictions for the ratio $R(s)$ at order ${α_s^2}$ for arbitrary $m^2/s$, with robust validation against analytically known pieces and stable Padé behaviour across approximants. This enables precise heavy-quark mass effects in $e^+e^-$ annihilation cross sections over a wide energy range, from near threshold to high energies.

Abstract

The real and imaginary part of the vacuum polarisation function $Π(q^2)$ induced by a massive quark is calculated in perturbative QCD up to order $α_s^2$. The method is described and the results are presented. This extends the calculation by Källén and Sabry from two to three loops.

Figures (3)

• Figure 1: Complete results plotted against $v=\sqrt{1-4m^2/s}$. The high energy approximation includes the $m^4/s^2$ term.
• Figure 2: High energy region. The complete results (full line) are compared to the high energy approximations including the $m^2/s$ (dash-dotted) and the $m^4/s^2$ (dashed) terms.
• Figure 3: Threshold behaviour of the remainder $\delta R$ for three different Padé approximants. (The singular and constant parts around threshold are subtracted.)